Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Petersen, Kasper Juel
Doctor of Philosophy - PhD
University of British Columbia
Boiling and cavitation—the phenomena which describe thermally- and pressure-driven, first-order phase transitions of liquids to their gaseous states—are central to many industrial, natural, and medical flows. By example, “rapid phase transitions” can cause vapour explosions in paper and metal processing, nuclear reactors, and cryogenics technologies when superheated volatile liquids become perturbed out of their equilibrium condensed state. In particular, phase-transitions are (i.) metastable, instigated by (ii.) thermal fluctuations, and are (iii.) multiscale, (iv.) heterogeneous processes; vapour bubbles O(< 10−⁶ m) form preferentially at microscopic nucleation sites, and are fully grown on the macroscale O(10−² m). From two literature reviews of phase-change modelling and simulation, it was found that (i.–iv.) are rarely considered in engineering numerical simulations except for (i.,ii.) in fluctuating hydrodynamics theories solved with the finite-volume method. Rather, the physics are frequently modelled by empirical models, leaving little opportunity for peering into realistic phase change dynamics. The fluctuating lattice-Boltzmann method (FLBM) has become a popular choice for studying thermal fluctuations. However, similar to fluctuating hydrodynamics, the FLBM relies on sampling techniques of probability distributions for imposing thermal noise in to the simulation variables. In an alternative effort, this research revolves around modelling thermal fluctuations by the Fokker-Planck equation (FPE) where LBM-techniques are repurposed to solve the FPE on the lattice. To introduce metastability and accommodate a high level of thermo-hydrodynamic nonideality into the model, its kinetics are recast with Particles-on-Demand—a recently proposed semi-Lagrangian realization of the LBM. The primary contribution is the mathematical derivation of the resulting latticeFokker-Planck-Boltzmann model. Asymptotic analysis shows that the limit of the model is the stochastic Navier-Stokes-Fourier equations with a stochastic stress tensor analogous to the deterministic equivalent derived on the basis of the Boltzmann equation. An initial fluctuation-dissipation theorem was formulated indicating equivalent behaviour to the FLBM. Numerical tests of the model are reported and show promise in computing thermal fluctuations evolved from the initial conditions in simulations. The research opens the door to intriguing numerical, theoretical, and computational research with the FPE, which has not yet been employed to investigate phase-transitioning liquids.
A lattice-Fokker-Planck model of thermal noise inliquidsbyKasper Juel PetersenB.Sc. Energy Engineering, Aalborg University, 2016M.Sc. Thermal Energy and Process Engineering, Aalborg University, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)August 2024© Kasper Juel Petersen, 2024The following individuals certify that they have read, and recommend to the Collegeof Graduate Studies for acceptance, a thesis/dissertation entitled:A lattice-Fokker-Planckmodel of thermal noise in liq-uidssubmitted by Kasper Juel Petersen in partial fulfillment of the requirements of thedegree of Doctor of PhilosophyProf. Joshua Brinkerhoff, School of Engineering, UBCOSupervisorProf. Sina Kheirkhah, School of Engineering, UBCOSupervisory Committee MemberProf. Sunny Li, School of Engineering, UBCOSupervisory Committee MemberProf. Anthony Wachs, Department of Mathematics, UBC VancouverUniversity ExaminerProf. Alexander Wagner, Department of Physics, North Dakota State UniversityExternal ExamineriiAbstractBoiling and cavitation—the phenomena which describe thermally- and pressure-driven,first-order phase transitions of liquids to their gaseous states—are central to many indus-trial, natural, and medical flows. By example, “rapid phase transitions” can cause vapourexplosions in paper and metal processing, nuclear reactors, and cryogenics technologieswhen superheated volatile liquids become perturbed out of their equilibrium condensedstate. In particular, phase-transitions are (i.) metastable, instigated by (ii.) thermalfluctuations, and are (iii.) multiscale, (iv.) heterogeneous processes; vapour bubblesO(< 10−6 m) form preferentially at microscopic nucleation sites, and are fully grown onthe macroscale O(10−2 m).From two literature reviews of phase-change modelling and simulation, it was foundthat (i.–iv.) are rarely considered in engineering numerical simulations except for (i.,ii.)in fluctuating hydrodynamics theories solved with the finite-volume method. Rather,the physics are frequently modelled by empirical models, leaving little opportunity forpeering into realistic phase change dynamics. The fluctuating lattice-Boltzmann method(FLBM) has become a popular choice for studying thermal fluctuations. However, similarto fluctuating hydrodynamics, the FLBM relies on sampling techniques of probabilitydistributions for imposing thermal noise in to the simulation variables. In an alternativeeffort, this research revolves around modelling thermal fluctuations by the Fokker-Planckequation (FPE) where LBM-techniques are repurposed to solve the FPE on the lattice.To introduce metastability and accommodate a high level of thermo-hydrodynamic non-ideality into the model, its kinetics are recast with Particles-on-Demand—a recentlyproposed semi-Lagrangian realization of the LBM.The primary contribution is the mathematical derivation of the resulting lattice-Fokker-Planck-Boltzmann model. Asymptotic analysis shows that the limit of the modelis the stochastic Navier-Stokes-Fourier equations with a stochastic stress tensor analogousto the deterministic equivalent derived on the basis of the Boltzmann equation. Aninitial fluctuation-dissipation theorem was formulated indicating equivalent behaviour tothe FLBM. Numerical tests of the model are reported and show promise in computingthermal fluctuations evolved from the initial conditions in simulations. The researchiiiopens the door to intriguing numerical, theoretical, and computational research with theFPE, which has not yet been employed to investigate phase-transitioning liquids.ivLay SummaryBoiling and cavitation are phase-change processes responsible for transforming liquidsinto vapour bubbles. A “metastable” phase-changing liquid can remain in its liquidstate despite being heated beyond its boiling point. Central to phase change, is thatthermal fluctuations (noise) in energy provoke a liquid out of metastability, and thusit is important to account for noise in simulations of phase changing flows. In thisdissertation, a mathematical model of thermal noise in liquids is developed on the basisof the Fokker-Planck equation. Implementation of the theory in computer code, witha customized numerical solution methodology enables the use of the Fokker-Planck forevolving thermal noise in space and time.Phase change is central to numerous industrial, natural, and engineered processes;from energy conversion in electrical power plants to cavitation leveraged in medical ther-apies to destroy cancer tumours. To further improve phase change technologies it is ben-eficial to investigate the behaviour of the microscale and non-equilibrium phenomenon ofthermal noise. As an example, insight into metastability and transition mechanisms in-duced by noise could in the future inform the control of metastability for efficient energytransfer in for example heat exchangers, among other applications.vPrefaceThis dissertation is written solely by K.J. Petersen, and J.R. Brinkerhoff has providedreview and corrections of drafts. The initial conceptualization of the research was madeby J.R.B., including suggesting early on to adopt lattice-Boltzmann methods in the re-search. K.J.P. conducted the literature review including material in chapter 2 identifyingthe Fokker-Planck equation as a direction for the research. All the mathematical deriva-tion and analysis found in chs. 3,4 was carried out by K.J.P. The software used in ch. 5alongside simulations, data analysis, and visualization was created by K.J.P. Moreover,the dissertation includes the following published and peer-reviewed material:[J4] K.J. Petersen, J.R. Brinkerhoff. Towards a stochastic lattice-Fokker-Planck-Boltzmann method for modeling thermal fluctuations in non-ideal fluids. Phys.Rev. E. Undergoing review. Preprint available on the arXiv. K.J.P.: conceptual-ization, analysis, writing. J.R.B.: corrections.[J3] K.J. Petersen, S. Rahbarimanesh, J.R. Brinkerhoff. Phase change in cryogenicfluids: a review of numerical methods and mathematical models. Appl. Math.Mod. 116 327-349 (2023). K.J.P.: conceptualization, literature review, writing.S.R.: literature review. J.R.B.: conceptualization, literature review, corrections.[J1] K.J. Petersen, J.R. Brinkerhoff. On the lattice Boltzmann method and its ap-plication to turbulent, multiphase flows of various fluids including cryogens: Areview. Phys. Fluids 33, 041302 (2021). Editor’s Pick. K.J.P.: conceptualization,literature review, writing. J.R.B.: conceptualization, corrections.The publications are referred to throughout the dissertation. The research was alsopresented at the following two venues:[C4] K.J. Petersen, J.R. Brinkerhoff. Thermal fluctuations in lattice-Fokker-Plancksimulations. Proc. In: 33rd Disc. Sim. Fluid Dyn. 2024. Zurich, Switzerland.[C3] K.J. Petersen, J.R. Brinkerhoff. A Fokker-Planck-Boltzmann kinetic model ofnucleation in cavitating and boiling flows. Proc. In: Int. Conf. Multi. Flow 2023.Kobe, Japan.viPermissions for reprinting of [J1] and [J3] were obtained from AIP Publishing and Else-vier, respectively. Both publications are copyrighted material. [J4] is not yet copyrightedas it is under review. [J1] and [J3] are included in §2. The literature review in [J4] is alsofound in §2, and the remainder of the publication in §3. Readers wishing to cite excerptsfrom this dissertation are asked to cite the individual publications. Generative artificialintelligence was not used for any parts of this work.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thermally and inertially-driven phase transitions . . . . . . . . . . . . . . 11.1.1 Nucleate pool boiling . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Metastability and thermal noise . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Entropy extrema and thermal fluctuations . . . . . . . . . . . . . . 91.2.2 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Research questions and objectives . . . . . . . . . . . . . . . . . . . . . . . 131.4 Methodology overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Dissertation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 2: Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Phase-change models in CFD . . . . . . . . . . . . . . . . . . . . . . . . . 20viiiTABLE OF CONTENTS2.1.1 Inertially-driven, liquid-vapour phase-change . . . . . . . . . . . . 202.1.2 Thermally-driven, liquid-vapour phase-change . . . . . . . . . . . . 262.2 Deterministic LBMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Streaming and collision solution algorithm . . . . . . . . . . . . . . 302.2.2 Colour-gradient LBMs . . . . . . . . . . . . . . . . . . . . . . . . . 352.2.3 Pseudo-potential LBMs . . . . . . . . . . . . . . . . . . . . . . . . 382.2.4 Free-energy LBMs . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2.5 Phase-field LBMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.2.6 LBM cryogenic applications . . . . . . . . . . . . . . . . . . . . . . 482.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4 Fluctuating LBMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.5 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . 61Chapter 3: Lattice-Fokker-Planck method . . . . . . . . . . . . . . . . . . 653.1 Semi-Lagrangian advection . . . . . . . . . . . . . . . . . . . . . . . . . . 663.1.1 Gauge transformations from moment-matching conditions . . . . . 713.2 Semi-Lagrangian diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.2.1 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3 Lattice discretization of the FPE . . . . . . . . . . . . . . . . . . . . . . . 773.3.1 Gauss-Hermite quadrature . . . . . . . . . . . . . . . . . . . . . . . 773.3.2 The lattice Fokker-Planck operator . . . . . . . . . . . . . . . . . . 813.3.3 The lattice acceleration operator . . . . . . . . . . . . . . . . . . . 833.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Chapter 4: Theoretical consistency . . . . . . . . . . . . . . . . . . . . . . 884.1 Chapman-Enskog multiscale analysis . . . . . . . . . . . . . . . . . . . . . 884.1.1 Perturbation series . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.1.2 Expansions: Taylor series, perturbations, separation, moments,and recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . . . . . . . . 103Chapter 5: Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1.1 Computational domain and boundary conditions . . . . . . . . . . 1085.1.2 Initial-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.1.3 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118ixTABLE OF CONTENTS5.2.1 Fokker-Planck-perturbations in a Gaussian impulse . . . . . . . . . 1185.2.2 Fokker-Planck-perturbations in a noisy initial condition . . . . . . 1195.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Chapter 6: Conclusion and future work . . . . . . . . . . . . . . . . . . . 1266.1 Research summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2 Impact of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.4 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Appendix A: Thermodynamic identities . . . . . . . . . . . . . . . . . . . . . . 192Appendix B: Eigenfunctions of the Fokker-Planck and acceleration operators . . 200Appendix C: Hermite coefficients for the operators . . . . . . . . . . . . . . . . 204Appendix D: Diagrams, tensors, and misc. . . . . . . . . . . . . . . . . . . . . . 208xList of TablesTable 2.1 Selected computational fluid dynamics studies of cavitating flows since 2013. (†):All Zwart cavitation models are correlated to evaporation and condensation constantsFvap, Fcond that need to calibrated to the pertinent fluid, in lieu of key thermo-physicalproperties. In most of the studies information about temperature and pressure de-pendence in pertinent thermo-physical properties is lacking and as such it is assumedthat these are modelled as constant quantities. . . . . . . . . . . . . . . . . . 25Table 2.2 Selected CFD studies on pool boiling with various cryogens, refrig-erants, and water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Table 2.3 Selected numerical experiments with the LBM of various turbu-lent flows since 2018. Collision models employed: Entropic [1, 2],MRT [3–7], BGK [8]. Nomenclature: cylinder/sphere diameter D,time-averaged root-mean-square velocity U rms, Taylor micro-scaleλ, continuous phase kinematic viscosity νc, friction velocity Uτ ,channel-height h, plate/filament/channel length l, minimum andmaximum stroke ϕ, flapping frequency f , span-wise length of awing lw, mean wing-chord c, and lid-velocity Ulid. . . . . . . . . . . 35Table 2.4 Overview of published works on cryogenic fluids using colour-gradient(CG), pseudo-potential (PP), and finite-difference (FD) multi-speedLBMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 3.1 Hermite coefficients for the FP and acceleration lattice operatorsevaluated with (C.10, C.17) as well as the Hermite coefficients forthe populations F(l)α (fi) (3.37) and G(l)α (gi) (3.38). The operatorcoefficients are ultimately used to complete the series expansionyielding Ω(FP)i and Ω(η)i . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 5.1 Model solution configuration. . . . . . . . . . . . . . . . . . . . . . 108Table 5.2 2D computational domain—lattice dimensions in lattice-units (LU). 109xiLIST OF TABLESTable 5.3 Universal physical constants and thermo-physical properties andinitial conditions of the VDW fluid. . . . . . . . . . . . . . . . . . . 111Table 5.4 Unit-conversion factors derived from the basic conversion factors(5.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Table 6.1 Strategies to be applied for furthering the LFPM towards realisticsimulations of noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 138Table F.2 The Maxwell relations [9] used for the current derivations. . . . . . 192Table F.3 Summary of thermodynamic identities [9]. . . . . . . . . . . . . . . 193Table F.4 The cyclic relations for three thermodynamic variables x, y, z [9]. . 197xiiList of FiguresFigure 1.1 Semantic overview of the topics relevant to liquid-vapour phasetransitions with the current research stream highlighted in orange.The blue background connections indicate coupling to thermalnoise. Nomenclature: Rayleigh-Taylor instability (RTI), Taylor-Helmholtz instability (THI), Navier-Stokes-Fourier (NSF), equa-tion of state (EOS), Cahn-Hillard (CH), Allen-Cahn (AC), fluid-structure interaction (FSI), conjugate heat transfer (CHT). . . . . 3Figure 1.2 Pool-boiling curve with the (free convection), nucleate, transition,and film-boiling regimes. Nomenclature: onset of boiling (ONB),critical heat flux (CHF). . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.3 Photographs of water pool boiling (a,c,d) on a laser-etched 2.7×2.7cm copper surface by Allred et al. [10]© American Physical Soci-ety 2022. Film boiling of ethanol (b) with bubbles intermittentlyshed at the “most dangerous wavelength” λc © Wildeman et al.[11] 2022. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.4 Reduced temperature-density diagram with the coexistence curve(binodal) for a van der Waals fluid. Boiling occurs along isobarsand the greyed area between the bi and spinodals is the metastableregion. The spinodal region is the envelope of the spinodal locus.The reduced temperature is T = T/Tcrt. Adapted from Schmelzeret al. [12]. Isobars are not to scale. . . . . . . . . . . . . . . . . . . 8Figure 1.5 Reduced pressure-volume diagram for a van der Waals fluid. Cav-itation occurs along isotherms and the grayed area between thebi and spinodals is the metastable region. θ = p/pcrt, v = v/vcrt.Adapted from Schmelzer et al. [12]. . . . . . . . . . . . . . . . . . 8Figure 1.6 Image of the Wiener process in space with an example stochasticsample path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13xiiiLIST OF FIGURESFigure 1.7 Overview of the employed methods for obtaining a LFPM forevolving thermal fluctuations in VDW fluids. Nomenclature: Fokker-Planck equation (FPE), semi-Lagrangian advection (SLA) anddiffusion (SLD), equation of state (EOS), Navier-Stokes-Fourier(NSF), fluctuation-dissipation theorem (FDT). Dashed lines indi-cate auxiliary components used to enable the analysis. . . . . . . . 17Figure 2.1 Evolution of a cavity collapsing on a wall in water with an ini-tial radius of 3.8 mm, as simulated by Yang et al. [13]. Colourcontours show the volume-fraction field. The simulation atteststo the intricacy and complexity of the collapse process of a sin-gle cavity. Reproduced with permission from Phys. of Fluids 33,073311 (2021). American Institute of Physics © 2021. . . . . . . . 24Figure 2.2 With their DI-FH solver Magaletti et al. [14] found that subcriti-cal embryos in water pool boiling coalesce with ones neighbouringmaking them survive for longer, which makes it possible for themto self-sustain nucleation for longer on ultra-smooth surfaces. Thiscorroborates low onset superheat levels found in various experi-mental studies [15, 16]. Reproduced with permission from Int. J.Multiph. Fl. 130, 103356 (2020). Elsevier © 2020. . . . . . . . . . 28Figure 2.3 Illustration of the five streaming paths (including no streaming inc0) in the D2Q5 lattice model. . . . . . . . . . . . . . . . . . . . . 31Figure 2.4 General solution procedure for an LBM. . . . . . . . . . . . . . . . 32Figure 2.5 Simulation snapshot at 2× 104δt of nucleate boiling in water in a6002×300 computational domain with a cascaded pseudo-potentialLBM [17]© American Institute of Physics 2020. . . . . . . . . . . 42Figure 2.6 Instantaneous velocity contour of LO2 propellant undergoing lat-eral sloshing in a tank. Solution from an isothermal simulationwith the pseudo-potential method. Reproduced from Orr et al. [18]. 51Figure 2.7 Collapse of H2O, LN2, and LH2 cavities near a solid wall by Ez-zatneshan and Vaseghnia [19]. Reproduced with permission fromPhysical Review E 102, 053309 (2020)© 2020 American PhysicalSociety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54xivLIST OF FIGURESFigure 2.8 Freezing droplet morphology as reported in the paper by Li et al.[20]. The ice phase is purple and the liquid phase blue. Repro-duced with permission from International Journal of Heat andMass Transfer 114, 628-639 (2017). © 2017 Elsevier. . . . . . . . 56Figure 3.1 Upscaling of the FPE mesoscale effects to the continuum governedby the NSFEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 3.2 Vector fields of the discrete velocities vi (3.1) and ci. The examplearbitrarily includes a unity velocity set ci, u = [0.1, 0.3]†, and√θ = 1.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 3.3 Solution algorithm with Particles-on-Demand considering an ar-bitrary collision functional Ω(f) with explicit equations in theform of (3.4). The nomenclature follows: the seed gauge λn=0comprises all the solution variables, including pre-advected pop-ulations prior to the first predictor-corrector iteration is carriedout. v(n)i are the adapted discrete velocities at the n-th iterationand xd the associated departure points. F is the subset containingall pre-advected, collocation data at collocation points xp definingF(xp, t− δt) ⊆ {f, g, p, ρ,u} at n = 0. The “ ” accent representsreconstructed quantities and Λ(xd,xp) an arbitrary interpolationkernel. It is implied that λ′ is the adapted gauge at the n-thiteration at which the moments of the reconstructed populationspredict the advected macroscopic observables at the (n+1)-th iter-ation, from which any EOS can be employed to determine the localthermodynamics and λn+1 thereby closing the predictor-correctorloop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Figure 3.4 Illustration of the collocation stencil in relation to the entire lattice. 70xvLIST OF FIGURESFigure 3.5 Overview of the collocation stencil with mp×np collocation pointsxp for a current monitoring point x (destination of all discretevelocity vectors) with discrete velocities vi advecting from depa-ture poitns xd = x − viδt (solid red bullets). F(xp) is the setthat contains all the pre-advection data coinciding with xp. Semi-Lagrangian advection and diffusion over a finite period δt is facil-itated by reconstructing Q populations fλ′(in which the accentand superscript are omitted for brevity) to be advected to x. Theblue shaded areas are reconstruction stencils, specifically a cubic,fourth-degree kernel and bilinear, second-degree kernel for fλ′5 andfλ′6 , respectively, by example. . . . . . . . . . . . . . . . . . . . . . 72Figure 3.6 Solution algorithm for the lattice-Fokker-Planck equation withsemi-Lagrangian diffusion: the implicit FPE (3.18) is solved byaccounting for the seed and advected states in the predictor. First,the LHS is mediated on the seed populations at t− δt, after whichthe populations are gauge transformed, reconstructed. Then theRHS is realized before the corrector step causing the populationsto be “diffused”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 4.1 Example perturbation-order contributions to a macropscopic den-sity ρ = ρ(0) + ϵρ(1) + ϵ2ρ(2) as it evolves in time at an arbitrarypoint in space. By the perturbation-series expansion the signalis decomposed into diminishing time scales whereto the smallnessparameter ϵ is associated with some characteristic timescale t(1).One can then interpret the first-order perturbations to oscillatewith periods O(δt) ∼ t(1) and similarly for the second-order witht(2). In addition to the temporal dimension, the multi-scale natureof the perturbations can also be imagined to extend out of the pagein the spatial dimension (x), forming a hyperplane which analo-gously to the density response in time is increasingly non-uniformfor ϵ→ 0. Figure is not to scale. . . . . . . . . . . . . . . . . . . . 90Figure 5.1 Computational domain. . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 5.2 The stitching of periodic boundaries are illustrated of the foldingof a 2D plane into a toroidal shape, connecting the north-southand west-east faces in pairs. . . . . . . . . . . . . . . . . . . . . . . 110xviLIST OF FIGURESFigure 5.3 FP operator parameter space for fi (top row) and gi (bottom) atu = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.4 Moments ∆ρ =∑Q−1i=0 Ω(FP)i (f),∆E =∑Q−1i=0 Ω(FP)i (g) of the op-erators at u = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 5.5 FP operator parameter space for fi at u = [0.5, 0]T . . . . . . . . . 113Figure 5.6 FP operator parameter space for gi at u = [0.5, 0]T . . . . . . . . . 114Figure 5.7 FP operator parameter space for fi and gi at u = [0, 0.5]T . . . . . 115Figure 5.8 Collocation (bounded by the dashed blue line) and Lagrange re-construction (shaded blue area for xd(i = 5)) stencils relative todeparture points (red bullets). . . . . . . . . . . . . . . . . . . . . 116Figure 5.9 Fourth-degree Lagrange kernel used to reconstruct a populationwith an arbitrarily populated reconstruction stencil. Bullets arecollocation point values. . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 5.10 Convergence rate in the semi-Lagrangian diffusion scheme in pres-sure, density, and temperature. Percent-wise change is computedas ∆%φ(i) =∣∣φ(i)− φ(i− 1)∣∣/φ(i− 1)× 100%. . . . . . . . . . . 119Figure 5.11 Isolines of total energy, density, temperature, and pressure fieldsafter 50δt for γ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 5.12 Isolines of velocity field u(x) components, and velocity magnitude|u| = (u2x + u2y)1/2 after 50δt for γ = 0. . . . . . . . . . . . . . . . 120Figure 5.13 y-centerline cross-section of Fig. 5.11 compared with the initialcondition and PonD with (γ = 0.5) and without (γ = 0) a FPoperator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.14 Isolines of total energy, density, temperature, and pressure fieldsafter 75δt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 5.15 Isolines of velocity field components, and velocity magnitude after75δt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure F.1 Each of the 4× 4 Lagrange polynomials in Λ. . . . . . . . . . . . . 211xviiNomenclatureLatin model variablesVariable Interpretation Dimensionsa van der Waals interaction constant m5/s2–kgA Helmholtz free energy J/kgAα Variance tensor (for FDTs) -b van der Waals excluded volume m3ci Peculiar velocity: lattice-discretized molecular velocities LUcp Isobaric specific heat capacity J/kg–Kςs Speed of sound m/scv Isochoric Specific heat capacity J/kg–KC Unit-conversion factor -D No. of spatial dimensions -D Diffusion coefficient -δx Spatial discretization size mδt Time-step size se Internal energy J/kgE Total energy J/kgE = 2ρE Total energy moment J/m3f Continuous population f(v;x, t) not yet mapped by GH quadrature kg/m3fi Population discretized by GH quadrature fi(x, t) kg/m3f˜i Remapped population kg/m3fi Reconstructed population kg/m3f ′i Advected population kg/m3F Force vector NF Phase indicator (for VOF) -Fα Hermite coefficient for f population -F Collocation subset -g Energy population J/kg–KG Gibbs free energy J/kgG Transformation matrix -Gα Hermite coefficient for g population -h Enthalpy J/kg–KH Total enthalpy J/kg–KHα Hermite polynomial -i Quadrature node index -j = ρu Momentum vector kg/m3× m/sxviiik Thermal conductivity W/m–skB Boltzmann’s constant (kB ≡ 1.3806505× 10−23) J/Kk Wave vector -K Truncation order in GH quadrature -Kn ≡ λ/Lc Knudsen number -l Order size in GH quadrature -m Molecular mass kgm˙ Mass sink/source kg/smp Number of collocation points in spatial x-dimension -M Molar mass kg/molM Moment matrix -M Mapping matrix -np Number of collocation points in spatial y-dimension -NA Avogadro’s constant (NA ≡ 6.0221415× 1023) 1/molO() On the order of () -vi Discrete velocity: lattice-discretized, recast version of ci LUp Pressure PaPαβ Pressure tensor PaPr Prandtl number Pr = cpµ/k -q Heat flux J/m2–sQ Number of quadrature nodes -Qαβγ Third-order Hermite coefficient kg/s3R Specific heat constant J/kg–KRαβ Higher-order energy flux tensor J/m–s2R Space of real numbers -Re Reynolds number Re = ρLu/µ -S Entropy J/kgS Source termt Time sT Temperature °KTL Lattice temperature LUU Internal energy J/kgvT =√kBT/m Mean thermal speed m/sv = [vx, vt, vz]T Velocity m/sV Volume m3w Quadrature weight -W Wiener process m/sx = [xx, xy, xz]T Spatial coordinate. xd (departure point), xp (collocation point) mxixGreek model variablesVariable Interpretation DimensionsαT Coefficient of thermal expansion, mass fractionβT Isothermal compressibilityγ Friction factor (in FPE) 1/sδ Dirac’s delta -δαβ Kronecker’s delta -ϵ Smallness parameter (in perturbation series) -ζ Volume viscosity Pa–sη Acceleration m/s2θ = p/(ρRTL) Reduced pressure -κ Korteweg surface tension coefficient m6s3/Paλ Galilean reference frame -Λ Reconstruction kernel -µ Dynamic viscosity Pa–sν Kinematic viscosity m2/sρ Density kg/m3σ Surface tension Paτ Deterministic stress tensor, relaxation time in BGK kernels Pa, sτ˜ Stochastic stress tensor PaΨ Free-energy functional Pa–m3ω Relaxation frequency 1/sΩ Operator kernelxxAcronymsAcronym DescriptionBGK Bhatnager-Gross-KrookCE Chapman-EnskogCEMA Chapman-Enskog multiscale analysisCFD Computational fluid dynamicsCG Colour-gradientCNT Classical nucleation theoryCPU Central-processing unitCSF Continuum-surface forceEOS Equation of stateFD Finite differenceFDT Fluctuation-dissipation theoremFHT Fluctuating-hydrodynamics theoryFLBE Fluctuating lattice-Boltzmann equationFLBM Fluctuating lattice-Boltzmann methodFP Fokker-PlanckFPB Fokker-Planck-BoltzmannFPE Fokker-Planck equationGH Gauss-HermiteGPU Graphics-processing unitLBE Lattice-Boltzmann equationLBM Lattice-Boltzmann method/model (used interchangeably)LFPBE Lattice-Fokker-Planck-Boltzmann equationLFPBM Lattice-Fokker-Planck-Boltzmann method/model (used interchangeably)LFPM Lattice-Fokker-Planck method/model (used interchangeably)LU Lattice unitsMB Maxwell-BoltzmannMD Molecular dynamicsMPI Message-passing interfaceMRT Multi-relaxation timeNA Not applicableNSF Navier-Stokes-FourierPonD Particles on DemandPP Pseudo-potentialPU Physical unitsSC Shan-ChenSLA Semi-Lagrangian advectionSLD Semi-Lagrangian diffusionVDW van der WaalsxxiAcknowledgementsThroughout my undergraduate and graduate educations I aspired to become a re-searcher. During my M.Sc. in 2017, I applied to come to UBC Okanagan from Denmarkas a Visiting International Student for coursework. While I needed to follow four courses,only one of my requested eleven courses ended up being scheduled for the semester. Byluck, Professor Joshua Brinkerhoff and I found each other before I cancelled my plans,and he took me under his wings as a Visiting International Research Student. That onesemester of research led to me staying for another, and he offered me to return for myPh.D. Was it not for Brinkerhoff, it is unlikely I would have had a chance to realize myaspirations. I am grateful for all the advice, encouragement, support, and teaching andresearch opportunities he has given me over the years.Integral to my research was the support from several colleagues in Brinkerhoff’s re-search group; CS Wang, Alejandra Garcia, Marjan Sharifi, Shahab Zeerati, Andrew Mills,Aaron Stroda, and many others at UBCO. Over the course of conducting research to-gether I developed lasting friendships with them. I also thank Professors Sunny Li andSina Kheirkhah for serving on my committee.In 2022 I visited Professor Tim Colonius’ group, and in 2024 Professor Brian Wet-ton. Visiting these inspiring researchers, their students, and new academic environmentswas a hugely defining experience for my growth as a person and scientist. I gratefullyacknowledge these individuals for their support and advice.Studying outside of Denmark for six years during my B./M.Sc. and this Ph.D. re-quired a significant amount of resources, resources I was awarded by many funders,enabling me to dedicate my efforts to research. Without a doubt and without theirsupport, I would not have completed this dissertation. I acknowledge the following foun-dations; Hedorf’s Fond, Viet Jacobsen Fonden, Augustinus Fonden, Knud HøjgaardsFond, Reinholdt W. Jorck og Hustrus Fond, Henry og Mary Skovs Fond, Familien HedeNielsens Fond, Otto Mønsteds Fond, Schurs Støttefond, William og Hugo Evers Fond,Oticon Fonden, Inge og Jørgen Larsens Mindelegat, Dansk Tennis Fond, Julie DammsStudiefond, and Fonden Julius Skrikes Stiftelse.I extend my gratitude to all my friends who supported me through this research. Axxiisurprising side effect of doing a Ph.D. is that a majority of my friends are academics,and most academics enjoy scientific, philosophical, and political conversation which havecontributed to varying extents to my research. I thoroughly enjoyed the innumerable,engaging conversations during ski-touring adventures to Roger’s Pass, climbing in theRockies, mountain-biking, and camping trips around British Columbia among other ac-tivities with Michael Kudla, Jonathan Laumer, Sebastiano Stipa, and Piotr Ahmad, andin the USA; José Chreim and Micheal Sleeman, and many other fantastic individuals.Despite the distance and inconvenient nine-hour time-zone difference, I am fortunate tostill have very close friendships back in Denmark; I thank especially Paul Yogasundram,Helle Hjort Petersen, Marius Nørgaard, and Adrian Expethit for our friendships. Youhave been very supportive. To Sara’s parents and brother; Dobrila, Ivica, and MarkKolomejac, I am so grateful for how you have welcomed me in your family.To see your son and brother leave Denmark so often, and for such long periods I canimagine is a simultaneously difficult and joyous experience. To my mom, dad, and sister,I am deeply grateful that you have always been so supportive of my endeavours abroad.Truly, this research is the culmination of your careful dedication to my own as well asmy sister’s education.Not the least, to my dearest Sara, thank you. I am not sure what is more testing; to bea Ph.D. student or to be the partner of a Ph.D. student. Your sacrifices and unwaveringsupport during my Ph.D. and a global pandemic is a testament to your compassion,selflessness, endurance, and resolve. You are an inspiration to me, our friends, and yourclients. I would not have completed this Ph.D. without you. If you ever decide to pursuea Ph.D. I would not hesitate to support you with my best efforts, even though I probablywill not be able to handle it as gracefully as you did.xxiiiDedicated to Sara, my mom, dad, and sister.xxivChapter 1Introduction1.1 Thermally and inertially-driven phase transitionsPhase transitions, or phase change, are common phenomena occurring in a myriad ofscenarios in nature, industrial applications, and many engineered devices and processes.First-order, liquid-vapour phase transitions are generally thought to be either pressure-driven in highly non-uniform pressure-gradient fields e.g. in flows with high advection,or thermally-driven in diabatic flows with large temperature-gradient fields. Specifically,by “first-order” is meant that extensive volume V and entropy S evolve by the firstorder gradient terms in pressure p and temperature T , V =(∂G/∂p)T,Nand S =−(∂G/∂T )p,N[21] of Gibbs free energyG. The equalities apply along isobars and -thermsfor an isolated system of N moles, denoted by subscripts p, T,N . The two isothermaland isobaric phase change processes are more commonly known as cavitation and boiling.In addition, boiling and cavitation occur either homogeneously in pure fluids [22–24], orheterogeneously in the presence of nucleation sites in crevices of the surface topology ofa boiling surface or the surface of a solid impurity in cavitation sheltering microscopicparcels of gas, or undissolved gasses in the liquid bulk. Such undissolved gasses arevirtually impossible to eliminate even in the most controlled laboratory settings [25]. Thepresence of these is generally accepted to markedly alter the probability of a nucleationevent [22–24].Cavitation is predominantly an adverse phenomenon that is sought to be avoided inapplications not limited to; turbomachinery including marine-propeller erosion and effi-ciency loss [26], cavitation in the xylem of drought-stricken trees [27], cavitation arounddolphins’ fins that limits their maximum speed [28, 29], skeletal joint cavitation [30],phase roll-over in storage of liquefied natural gas (LNG) [31], among others. Cavitationcan also be used as a tool that is beneficial in medical drug delivery, medical histotripsyand lithotripsy for treating cancer tumors and kidney stones, and as a predatory tool bymantis and pistol shrimp in their appendages [32, 33], by exemplification. In addition tothe common example of a kitchen kettle, boiling is a process with broad applicability inRankine cycles in combined-heat-and-power plants, mechanical coolers and liquid/solid11.1. THERMALLY AND INERTIALLY-DRIVEN PHASE TRANSITIONScryostats in deep-space radiators [34], pool boiling in spills of liquefied fossil fuels suchas LNG from marine vessels [35], cooling of electronics in high-performance computing[36], etc.Phase transitions, especially heterogeneous ones, are relatively complicated and gov-erned by several physical phenomena and intricate dynamics. From a thermodynamicsperspective the driving forces in phase transitions are mechanical, thermal, and osmoticand are thus determined by local pressure, temperature, and concentration gradients. Inthe current author’s recent reviews [37, 38], it was found that very few simulation studiesexist (especially on cryogenic fluids) which attempt to simultaneously account for all theaspects that paint the full picture (Fig. 1.1) of pool boiling and cavitation, includingimperfect solid surfaces, large temperature and pressure ranges, heterogeneous boiling,or non-ideal thermodynamic effects not limited to thermal noise. In fact, even thoughit is widely accepted [14, 39, 40] that thermal fluctuations are responsible for perturbingliquids out of their metastable states to yield nucleation of embryonic vapour bubbles,thermal fluctuations are rarely studied in computational fluid dynamics literature. Thisis presumably owing to the fact that thermal fluctuations occur on length scales thatare difficult to access in experiments, and computationally prohibitive for computationalsimulations of flows on the macroscales. Reference to micro, meso, and macroscales isdrawn throughout the dissertation. “Microscales” are defined broadly with characteris-tic lengths at and below some magnitude O(10−6 m) where molecular, non-equilibriumeffects are prevalent in the mean fields of density and energy. In contrast, “macroscales”refer to characteristic lengths > O(10−3 m) where non-equilibrium effects are infinitesi-mal relative to other hydrodynamic structures such as turbulent eddies, solid bodies (e.g.hydrofoils), etc. “Mesoscales” are more elusive to define, but are generally interpreted tobe those that overlap both micro and macroscales at which non-equilibrium effects andcontinuum phenomena are interconnected.One aspect of boiling as well as cavitation rarely studied in CFD is the physics ofmetastability, i.e. the phenomenon that enables liquids to remain in their condensedstate despite their local temperature exceeding or pressure falling below the pertinentsaturation state, for extended periods in time. Ultimately it is the aim, beyond theresearch presented in this dissertation, to unify all of the aspects in a single theory andmodelling framework which will make it possible to answer outstanding and importantquestions regarding boiling and cavitation. The physics of thermal fluctuations andmetastability in first-order liquid-vapour phase transitions, as well as its mathematicalmodelling, is the main focus of this dissertation.21.1. THERMALLY AND INERTIALLY-DRIVEN PHASE TRANSITIONSCavitationand boilingI.MetastabilityEntropyextremaThermalnoiseIV.Hetero-geneityMixturesMiscibilityFSI incl.CHTInterfacetrackingCH, ACequationsIII.Thermo-dynamicsEOSOrder pa-rameter(s)Structurefactor(s)NSFequationsII.Hydro-dynamicsRTI,THICapillarityTurbulenceCompres-sibilityNSequationsFigure 1.1: Semantic overview of the topics relevant to liquid-vapour phase transitions with thecurrent research stream highlighted in orange. The blue background connections indicate coupling tothermal noise. Nomenclature: Rayleigh-Taylor instability (RTI), Taylor-Helmholtz instability (THI),Navier-Stokes-Fourier (NSF), equation of state (EOS), Cahn-Hillard (CH), Allen-Cahn (AC),fluid-structure interaction (FSI), conjugate heat transfer (CHT).1.1.1 Nucleate pool boilingAs opposed to flow boiling, pool boiling liquids are characterized by their mass andthermal transport being defined by thermal convection without external forcing, resultingin unique boiling dynamics. Regardless, pool boiling is a relevant scenario occurring ine.g. spills of cryogenic liquids, thermal management in high-power electronics, and steamgeneration in nuclear power plants [21, 34, 36]. For liquids in contact with a hotter solidsubstrate, or another liquid, a number of boiling regimes are encountered with increasingtemperature differences [21]. Given a small temperature difference, e.g. between a coldliquid at or above its saturation temperature and a slightly warmer substrate, the liquidis metastable with heat transfer exclusively facilitated by convection. At the onset of31.1. THERMALLY AND INERTIALLY-DRIVEN PHASE TRANSITIONSSuperheating ∆TCHFONBLeidenfrost pointTransition FilmNucleatePartially FullyHeatfluxdevelopedFreeconvectionFigure 1.2: Pool-boiling curve with the (free convection), nucleate, transition, and film-boilingregimes. Nomenclature: onset of boiling (ONB), critical heat flux (CHF).boiling, at some temperature characteristic to the liquid-substrate pair, the nucleateboiling regime (Fig. 1.2) starts with the appearance of small bubbles. With furtherincrease in temperature difference, the bubble density becomes critically high, causinga sudden reduction in heat flux within an unstable transition regime to the Leidenfrostpoint, beyond which the film boiling regime commences. Therein, a contiguous vapourfilm covers the entire substrate and the heat flux become positively proportional to thetemperature difference yet again.Curiously, the nucleate boiling regime further hosts partially and fully developedboiling (sub-)regimes. In the former, spherical bubbles are intermittently shed fromnucleation sites which in the fully developed regime start coalescing into longer vapourcolumns. During the fully developed regime, increasing temperature differences changethe dried area at the base of the columns, rapidly restricting the heat transfer rate in thedepleting wetted area. Ultimately, a critical heat flux characterizes the onset of a lateralTaylor-Helmholtz instability (THI) in the bubble-column interfaces [41, 42]. Anotherhydrodynamic instability defines the film boiling regime; a Rayleigh-Taylor instability(RTI) in the interface which dictates the shedding rate of large bubbles. Whereas the RTIand the film boiling regime is relatively frequently simulated in literature, the nucleateand transition regimes are much less studied. Presumably, this is owing to the filmboiling regime being less complicated than the nucleate regime. Film boiling is more soan interfacial problem where the phase change occurs in the oscillating interface with largevapour parcels shed by the RTI, as opposed to nucleate boiling where it is unavoidableto model the heterogeneous solid-liquid interface as well as the more chaotic bubbledynamics. Fig. 1.3 depicts experiments of the pool-boiling phenomena. In nucleate41.1. THERMALLY AND INERTIALLY-DRIVEN PHASE TRANSITIONS(a) Growing and detaching vapour nucleus.�cThin film(b) Rayleigh-Taylor instability in a Hele-Shaw cell.(c) Nucleate boiling. (d) Film boiling.Figure 1.3: Photographs of water pool boiling (a,c,d) on a laser-etched 2.7× 2.7 cm copper surface byAllred et al. [10] © American Physical Society 2022. Film boiling of ethanol (b) with bubblesintermittently shed at the “most dangerous wavelength” λc © Wildeman et al. [11] 2022.boiling surface wettability, metastability, and thermal noise are central to understandingthe boiling dynamics.From a thermodynamic perspective, the surface wettability as a macroscopic propertymakes for either a hydrophilic, neutral, or hydrophobic interaction with the liquid whichcan be correlated with the free energy incident at the wetted area and the solid-liquid-vapour (i.e. triple or slv) contact line (TCL). On the microscale, boiling substrates arerepresented by a roughness profile made up of infinitesimally small crevices of varyingsizes. Such crevices can host parcels of undissolved gasses [43] which can readily act asnucleation sites when the local thermodynamic state is favourable for a liquid-vapourphase transition. A gas nucleus occupying a crevice or (on a larger continuum scale)a grown, surface-attached vapour bubble are associated with capillary stresses in theliquid-vapour interface which can be captured in a stress tensor such as the Kortewegmatrix Kαβ .In the TCL, K is dependent on the free energy made up by the mutual interaction of51.1. THERMALLY AND INERTIALLY-DRIVEN PHASE TRANSITIONSliquid, vapour and solid molecules, and in a macroscopic sense a contact angle θ, whereasthe stresses in the slv interface correlate with a coefficient of curvature κ. The suscepti-bility of an embryonic nucleus to growing to the larger macroscopic scales is related toθ which can retain a constant equilibrium value during the growth, or in contact-anglehysteresis an intermediate, varying value delimited by the receding and advancing con-tact angles θR ≤ θ ≤ θA, respectively, beyond which the TCL becomes non-stationary.This hysteresis “window” can be temperature dependent as indicated by a characteristicwetting temperature Twet at which a wetting transition may occur between partial andfull-wetting states [44] which has been observed in cryogens [45–50]. Moreover, Twet isexceptionally prone to thermal hysteresis [50–54] rendering analysis of the heat transfertaking place at the TCL more complicated. To that end, research has shown that boilingheat transfer predominantly takes place around the bubble meniscus [55], i.e. the micro-scopic, diffuse-interface region of the TCL. It has further been remarked that the effectivenucleation rate (i.e. the number density of nucleation events ×m−2s−1) is more sensitiveto surface treatments in cryogens compared to water [56]. What is more, the strong heattransfer incident at the TCL results in strong temperature fluctuations in a solid sub-strate. For boiling in the transition regime, these fluctuations may reach magnitudes of70°K [57]. The non-uniformity is hypothesized to be greater towards lower superheatinglevels (i.e. in the nucleate boiling regime) in which a substrate surface experiences morewetted area, contrasting higher levels (in the film-boiling regime) where vapour filmsoccupy the surface and have an insulating effect. This concurrent communication in thetemperature profiles in the solid and fluid which effectively dictates the heat flux is stud-ied in the context of solid-fluid conjugate heat transfer (CHT). CHT is rarely accountedfor in computational studies of boiling despite the non-trivial temperature fluctuations[58, 59].Focusing on the liquid within the thermal boundary layer of a boiling substrate, andin the bulk a host of other topics of interest are identified. As previously mentioned,phase transitions can be either homogeneously or heterogeneously driven where the for-mer case occurs in pure, single-component fluids in the absence of solid impurities, solidsubstrates (i.e. occurring in the fluid bulk, away from boundary layers), and suspended,undissolved gasses. The homogeneous mode is relatively simpler than its counterpart,and is often and accurately modelled and simulated by classical nucleation theory (CNT)[60–62], which predicts nucleation rates based on an Arrhenius-equation-inspired nucle-ation barrier proportional to exp(−∆G⋆/(kBT )), where the numerator is the Gibbs freeactivation energy and kBT the average molecular kinetic energy [63], occasionally usedin population-balance models [64, 65].61.2. METASTABILITY AND THERMAL NOISEIn CNT the incipience of a new phase comprises microscale clusters of moleculeswhich can grow and shrink due to the initial thermodynamic state in which the growthrate is owed predominantly to local fluctuations in density. CNT can be extended to ac-count for heterogeneous nucleation, though mostly for phase transitions in glass [66–69].Multi-species (ternary), miscible mixtures such as liquefied-natural gas (LNG), or similarhydrocarbons, are useful but complicated examples: upon spill on e.g. concrete the het-erogeneity comprised of air pockets within the roughness topology of a concrete substratecan act as nucleation sites. Furthermore, the constitutive components (methane CH4,ethane C2H6, propane C3H8, butane C4H10, and trace amounts of CO2) of LNG phase-transition at dissimilar rates dependent on their separate thermodynamic behaviour thatmay obey different empirical laws for cryogenic fluids [70]. According to a pertinent cubicEOS1, a phase transition can be associated with an entropy contribution (and theretoa Gibbs ∆G or Helmholtz free energy ∆A) which fundamentally defines the relaxationtime of the transition. Having contributions to the free energy from each species of amixture only add to the complexity, with additional degrees of freedom which need tobe modelled. Core to nucleation and phase transition rates is metastability and thermalfluctuations that cause a liquid to depart from a stable, interim equilibrium state.1.2 Metastability and thermal noiseMetastability is prefaced with an example of vapour explosions in spilled LNG onwater, which will only serve as an illustrative example motivating the study of metasta-bility in this dissertation. Vapour explosions, also commonly referred to as rapid phasetransitions, are of concern in several industrial settings where there is a risk of a hot(relatively) non-volatile liquid coming into contact with a (relatively more) volatile coldliquid [21]. Such setting can be found in paper production [21, 71], aluminum and othermetals processing [21, 71], nuclear reactors [72], and more broadly in cryogenics indus-tries [21, 71–73]. Superheating is initially defined as the local temperature exceeding,or the pressure falling below the saturation state, without causing a phase transition inthe now metastable liquid. In the above cases, the routes to superheating of the coldliquid are sudden depressurization or isobaric heating, where the extent of superheatingis sensitive to the presence or absence of nucleation sites which promote heterogeneousphase transitions. The magnitude with which the binodal curve is penetrated into themetastable region during a superheating is inversely proportional to the number density1The formulation has to be cubic in specific volume v for it to be non-monotonic, and being able ofhosting an energy minimum and entropy maximum.71.2. METASTABILITY AND THERMAL NOISEReduced density ρ/ρcrtpcrtSpinodal000.20.40.60.811.231 2 2.51.50.5ReducedtemperatureTBinodalFigure 1.4: Reduced temperature-density diagram with the coexistence curve (binodal) for a van derWaals fluid. Boiling occurs along isobars and the greyed area between the bi and spinodals is themetastable region. The spinodal region is the envelope of the spinodal locus. The reduced temperatureis T = T/Tcrt. Adapted from Schmelzer et al. [12]. Isobars are not to scale.Reduced volume v110.80.60.40.2010Tcrt0.9TcrtReducedpressureθSpinodalBinodalFigure 1.5: Reduced pressure-volume diagram for a van der Waals fluid. Cavitation occurs alongisotherms and the grayed area between the bi and spinodals is the metastable region.θ = p/pcrt, v = v/vcrt. Adapted from Schmelzer et al. [12].of solid impurities and the contact area with irregularities in solid substrates. Given suf-ficient superheating and time, the liquid will spontaneously undergo a phase-transition,activated by homogeneous nucleation. In vapour explosions, the homogeneous nucleationtemperature for most liquids at atmospheric pressure has been reported around 90% oftheir respective critical temperatures Tcrt, requiring that the temperature of the hot liq-uid is Th ≳ 0.9Tcrt [21, 71]. As such, the liquid resides in its metastable state betweenthe liquid binodal and spinodal (indicated in Figs. 1.4 for boiling and 1.5 for cavitation)for an extended period of time before equilibrating to a globally stable vapour state [39].The study of metastability as it pertains to thermodynamics and phase transitionscan be treated either phenomenologically in continuum interpretations or statistically81.2. METASTABILITY AND THERMAL NOISEin microscopic interpretations [21]. The former rely on stability criteria represented bya limit locus (i.e. the spinodal) of values outside which the metastable state cannotpersist. The derivation of phenomenological stability criteria rely on EOS and can beused to investigate the stability of pure as well as fluid mixtures. Given the foundationbeing empirical via EOS, the practical importance of this avenue cannot be understated.Nevertheless, for more rigorous analyses one needs to look towards statistical me-chanics methodologies and kinetic theory which address some of the inconsistencies ofphenomenological approaches, particularly considering that the transition across the spin-odal locus in reality is not sharp but smooth and relaxed in time [21], and responsefunctions such as the isothermal compressibility βT (susceptibility), the isobaric heat ca-pacity cp, the coefficient of thermal expansion αT , and speed of sound ςs diverge [74–76]. The study of metastability is traditionally practised by physicists and chemists withmolecular-dynamics simulations approaching the Ångström scale. Owing to their pro-hibitive computational cost, very few studies on the mesoscales exist, although recentefforts made by Magaletti et al. [14], Gallo [40], Lulli et al. [77] will be discussed later.1.2.1 Entropy extrema and thermal fluctuationsCore to the phenomenological approach is the entropy-extremum principle that estab-lishes the necessary and sufficient criteria for thermodynamic equilibrium and metasta-bility [21, 78–80]. Considering an isolated system (i.e. an infinitesimal control volume)that does not exchange mass, heat, nor work with its surroundings and therefore containsconstant energy, volume V , and mass, then it can be stated that the following for changesin entropy δS and internal energy δU apply [21]:δS|U,V ≤ 0, (1.1)δU |S,V ≥ 0. (1.2)Conversely, if measuring entropy or internal energy in computational and physical exper-iments is inconvenient the constraints can be Legendre transformed [78] into free-energyrepresentations [21],δA|T,V ≥ 0, (1.3)δG|T,p ≥ 0. (1.4)The system is in equilibrium if the Helmholtz free energy A = U−TS over the manifoldsof states in volume and temperature (V, T ) is minimized, and analogously for minimiza-91.2. METASTABILITY AND THERMAL NOISEtion of Gibbs free energy G = U − TS + pV = A + pV over (p, T ). The inequalities(1.1–1.4) are equivalent and indicate that a spontaneous thermodynamic process is asso-ciated with a production of entropy. A further entropy subsidy is provided by thermalfluctuations, which on the longer time scales appear as thermal noise in large collectionsof molecules [81] and in ensembles fluctuate on relatively large scales [74].Considering the isobars in the T–ρ (Fig. 1.4) and isotherms in the p–v (Fig. 1.5)diagrams, thermal fluctuations will result in oscillations in the density of the liquid(and concentrations in case of multiple species) around its Maxwell-Boltzmann equi-librium ρ(eq) value in molecular velocity space represented by the sketched translucentbands along the isotherms and isobars. Thus, approaching the liquid spinodal, pertur-bations make it more likely for the thermodynamic state to be disequilibrated acrossthe spinodal region until settling somewhere beyond the vapour spinodal. Within thefluctuating density field, momentum conservation implies that the continuum velocityfield will also fluctuate and it is well established that the statistics of the fluctuations inρ(x, t),u(x, t), T (x, t) correlate with an entropy deviation away from its maximum valueassociated with a local thermodynamic equilibrium [82]. The entropy deviation, andby extension the system free energy(A(x, t) or G(x, t)), are generally dictated by thefluctuating state variables i.e. ρ, p which define the instantaneous conditions of a givensystem and alter its energy [82]. Consequently, the fundamental thermodynamic deriva-tives obtained via Maxwell’s relations, not limited to the speed of sound ς2s ≡(∂p/∂ρ)s[9], are also affected by thermal noise.This dissertation focuses on the metastability of first-order, liquid-vapour phase tran-sitions. The principal challenge in the study of metastability is establishing the finitelifetimes that elapse prior to a transition to a new stable phase. These lifetimes, thecorresponding thermophysical properties, and the stability of metastable states are fun-damentally connected to the nature of the fluctuation processes responsible for the decayof a metastable state [74]. In analyzing the relaxation of metastable states, especiallyapproaching the spinodal, a number of challenges arise [74]: firstly, appropriately rep-resenting the pertinent thermodynamic system by a representative thermal noise is im-portant, as the colouring of fluid-systems of interest may not simply be white (i.e. flatfrequency spectrum). Fluctuations in the “order” parameter non-trivially affect the ther-modynamic derivatives2 which must be accurately modelled, simultaneous accounting forthe non-linear nature of the creation and growth of large scale “heterophase”3 fluctua-2βT (p), cp(T ), αT , ςs(ρ)3in the sense that a control volume is diluted with minuscule vapour phases that all fluctuateanomalously—denoted “fluctuation clusters”101.2. METASTABILITY AND THERMAL NOISEtions comprised in nuclei of an embryonic phase. The formation and growth of fluctuationclusters near the spinodal closely follow fractal structures [83] which must be accuratelycaptured. Lastly, the use of one or more accurate EOS (throughout the binodal region)is of paramount importance.1.2.2 Thermal noiseResearch into metastable states in liquid-vapour phase transitions is traditionally notwithin the interest of the research community in computational fluid dynamics (CFD),but more so practised by computational physicists. The study of metastability is initial-ized with an effort of modelling by asymptotic methods thermal fluctuations which onthe macroscopic spatio-temporal scales appear as thermal noise with some characteristicfrequency spectrum. Given the complexity of thermal fluctuations as well as their non-linear coupling with the thermo-hydrodynamics, this dissertation primarily contributesnew mathematical and methodological insights for future use in mesoscale simulations ofphase-transitioning liquids.The commotion cause by thermal noise in fluids is commonly described by Ornstein-Uhlenbeck processes of Brownian motion in which trajectories of particles are consid-ered memoryless (Markovian) and Gaussian defined by “white noise” [84]. In Langevinequations white noise imposes independence of frequency in the noise intensity of thefluctuation spectrum. The noise intensity, i.e. the power-spectral density is effectivelyconstant (flat) with frequency. Alternatively, the spectrum can be “coloured” when thetrajectory of a particle does depend on its past motion which effectively implies thatthe thermal forces in the fluid becomes frequency dependent and spectrally non-uniform.In this case, the motion is described as a non-Markovian process with memory and thefluctuation process is often referred to as coloured noise [85].It is only recently that the actual colour of thermal noise in liquids has been able tobe probed experimentally; Berg-Sørensen and Flyvbjerg [86] in 2005 discussed how thepower spectrum of coloured noise of Brownian motion may be measured in experimentsusing “optical tweezers” where the resonance frequencies of solid micro-spheres are usedas proxies for investigating the thermal fluctuations of molecules. Subsequently, Franoschet al. [87] measured the colour of thermal noise in water and acetone, and Jannasch et al.[88] in water, both studies indicating colorations in the thermal noise causing an inertial-memory effect. Whether interfacial phenomena, compressibility effects, and turbulencecan affect the coloration of thermal noise is not clear.In a relatively large (compared to the Ångström scale) domain with Knudsen numbers111.2. METASTABILITY AND THERMAL NOISEfar below unity Kn≪ 1, noise is perceived as a chaotic muddle of molecules that assem-bled in ensembles of various number sizes4 compete for influence in a densely populatedspace. Isolating individual molecules to consider their individual, continuous trajectoriesin space and time establishes that the trajectory inherently has stochastic property inits evolution, and to some extent mimics Brownian motion. Fundamentally, fluctuationsarise from the discreteness of matter, and a Langevin equation seeks to reintroduce noiseinto a system represented in the continuum.As increasingly larger spatial domains are observed, i.e. include larger length scalesuntil some upper threshold at which the immediate distances covered by the strongestthermal fluctuations are maximized, the full spectrum of thermal noise is realized inwhich innumerable ensembles intermingle at all length and time scales. This intermin-gling may consist of the coalescence of ensembles where larger fluctuation clusters ab-sorb smaller ones, clusters of similar sizes mutually resonate constructively or destruc-tively. In such a constructed perspective, one can identify individual molecules havinghad their own stochastic paths in space and time, as well as ensembles of moleculesthat owed to their individual trajectories through which they have interacted consti-tute some probability-density function (PDF) evolving in space and time generally ac-cording to the local thermo-hydrodynamics. The sample paths of single particles obeyan idealized Langevin equation which imposes a so-called Wiener process W (t) to thestochastic molecular velocity (illustrated in Fig. 1.6). “Idealized” is here meant as op-posed to more sophisticated models which effectively impose non-Markovian, frequency-dependent, coloured-noise spectra into the stochastic variables. Realizing a large numberof Langevin “particles” with separate, intertwining stochastic paths effectively imposes afluctuation spectrum on the macroscopic density and temperature fields.The noise in the macroscopic fluctuating fields is captured in the each of the variables’PDFs, e.g. f(v;x, t) being the integrand of the fluctuating density[ρ(eq)+ρ˜](x, t), wherethe argument of the functional is the continuous velocity vector v(x, t) ∈ RD. The evolu-tion of the functional f is dictated by the local thermo-hydrodynamics. Thermal noise isa non-equilibrium effect which affects fluids in equilibrium such that if the local state ofthe fluid obeys a Maxwell-Boltzmann (MB) distribution f (eq)(v;x, t), then the thermalnoise continuously perturbs the state away from the MB equilibrium with decaying in-tensity with reducing temperature. The Langevin dynamics reflected in the evolution off in velocity space and time is equivalent to a particular stochastic differential equation(SDE), namely the Fokker-Planck equation (FPE) [89, 90].4These are expected to span scales from Å to mm121.3. RESEARCH QUESTIONS AND OBJECTIVESSpace xx0SpaceyW(t)|x(t)|Stochastic sample pathFigure 1.6: Image of the Wiener process in space with an example stochastic sample path.1.3 Research questions and objectivesIn the previous text it was established that in CFD simulations in literature, there islimited focus on the interplay between thermal noise and non-equilibrium fluid dynamics.Currently, there only exists molecular methods within the domain of molecular dynam-ics that are able to study the Langevin dynamics of single molecules as they interactin a computational domain of limited size, and continuum fluctuating-hydrodynamicsmodels based on the theory originally proposed by Landau and Lifshitz [91, 92]. TheFHTs rely on imposing thermal noise into selected stochastic macroscopic variables thatobey fluctuation-dissipation theorems. Alternatively, one can utilize kinetic theory inwhich the primitive variable is a molecular velocity closer to the Langevin interpretation.As this concept is still relatively immature within computational kinetic theories, thework presented herein is rather preliminary and anticipates that a significant amount ofdevelopment work will be needed before the concept can be applied to large-scale com-putational simulations of engineering relevance. Consequently, the reader will find thatthe scope of this dissertation is predominantly on establishing the mathematical detailsand solution strategy for the integration of the Fokker-Planck equation (FPE) within acomputational kinetic theory.The current work employs the FPE first and foremost as a means of studying ther-mal fluctuations. Alternatively, thermal noise could also be computationally simulatedby molecular dynamics (MD) models, fluctuating-hydrodynamics theory (FHT), the fluc-tuating lattice-Boltzmann method (FLBM), which will be discussed throughout the dis-sertation. MD is prohibitive in that it is limited to computational molecular domainsizes as MD relies on resolving the stochastic sample paths of individual molecules, butMD datasets are instrumental to validating course grained models. FHTs and FLBMs131.3. RESEARCH QUESTIONS AND OBJECTIVESare realized on the mesoscales and have their own strengths and weaknesses in e.g. com-putational performance, theoretical consistency, computational size limits, fluctuationintensity limits, and other aspects. The FPE is a relatively computationally unexploredfoundation which eventually will be exploited in extensions of the present work for mod-elling and simulation of phase-transitioning, metastable liquids.The overarching physical research questions, which the present work seeks to enableinsight into with the developed methodology, revolve around the effects of fluctuationson thermo-hydrodynamic behaviour:Q1. How are thermal fluctuations currently modelled in CFD?Q2. Is the Fokker-Planck equation a suitable foundation for simulating thermal fluctu-ations in liquids?Q3. (How) can the FPE be solved with techniques from the LBM?Q4. How can the physical consistency of a prospective lattice-based solution be verified?Q5. How does the FP-operator behave in its parameter space, i.e. the zero, first, andsecond moments and other relevant thermo-hydrodynamic observables?In the presents author’s literature reviews on phase-change modelling in CFD [37, 38],numerical evidence from simulations pertaining to the relaxation rate and finite lifetimesof metastable states was not found. As these lifetimes are hypothesized to be dependenton fluid-solid interaction (FSI), wettability, fluctuations, miscibility in fluid mixtures inpool boiling as well as cavitation, it is worthwhile to procure a general framework forstudying their mutual effects. The implications of enabling this are non-trivial. An ex-ample for the case of boiling, in addition to the aforementioned case of rapid phase tran-sitions in LNG, is in contemporary power-plants relying on the Rankine cycle. Therein,boilers work most efficiently in the nucleate-boiling regime [93], but given the compli-cated dynamics of the nucleate and transition boiling regimes, establishing stable heattransfer can be elusive. Insight into how the chaos caused by thermal fluctuations can becontrolled and exploited with the aim of maximizing boiler efficiency can have significantimpact. Because thermal fluctuations are so determinative for the stability of a liquidphase, creating a tool for their modeling within a CFD framework is a strong propo-sition. It is hypothesized that the strongest and slowest fluctuations on the mesoscaleare more probable to be responsible for disequilibration in metastable fluids, and thatfluctuations on diminishing length scales and increasingly faster time scales are less im-portant. Thus, given scale-independence in a prospective computational solution of the141.4. METHODOLOGY OVERVIEWFPE, resolution in simulations is only required down to the scales associated with thestrongest fluctuations.In establishing the FPE model the below objectives are followed and the associatedresults are presented throughout the dissertation.O1. Derive lattice Fokker-Planck equations for two distributions evolving mass andmomentum, and total energy, for single-component van der Waals fluids.O2. Establish a solution method for solving the lattice-Fokker-Planck equations withParticles-on-Demand (PonD) for kinetic theory.O3. Perform a Chapman-Enskog multiscale analysis proving the hypothetical, fluctuat-ing thermo-hydrodynamic limit: the stochastic Navier-Stokes-Fourier (NSF) equa-tions.O4. Develop a C++ code for solving the system of kinetic equations.O5. Conduct preliminary numerical tests of the developed model applied to van derWaals fluids perturbed by thermal fluctuations.Consequently, the thesis culminating from the work presented herein is that thermalfluctuations instilled in initial conditions can be sustained by the dynamics in the Fokker-Planck equation in computer simulation whilst accommodating metastability and phasetransitions by a suitable equation of state. Two distribution functions are adopted toevolve momentum and energy in the NSF equations, and to accommodate a non-idealEOS.1.4 Methodology overviewThe analysis in the current research springs out of the FPE, and seeks to discretizethe equation onto a (computational) lattice, assess the theoretical consistency of the dis-cretized equation, and devise the necessary numerical solution strategy to eventually usethe theory in computational trials. Firstly, continuous FPEs for evolving a density pop-ulation f(v;x, t) and a total energy population g(v;x, t) are projected onto the latticeby Gauss-Hermite quadrature, which primarily revolves around fitting a series expansionof molecular-velocity polynomials of varying order to the populations and their velocityspace. The velocity polynomials are further formulated on the basis of an adaptive setof discrete velocities vi borrowed from “Particles on Demand” (PonD) for kinetic theory151.5. SCOPE[94]—a solution method promising unconditional Galilean invariance and numerical sta-bility in computational simulations of non-equilibrium fluids. The analysis results in twolattice-FPEs (LFPEs) which require a tailored solution method to be evolved computa-tionally. Initially, semi-Lagrangian advection (SLA), as originally proposed to be usedwith PonD, is adapted to solve the LFPEs.On the basis of SLA, a Chapman-Enskog (CE) multiscale analysis is carried outto elucidate the thermo-hydrodynamic limit reflected by the microscale LFPEs, whichproved to be the stochastic Navier-Stokes-Fourier equations. Central to the analysis is theuse of perturbation theory which can be used to assess non-equilibrium effects in selectedmodel variables. In particular, the stochastic FPE effects are assessed as they perturbthe viscous stresses resulting in a closed form of a stochastic stress tensor τ˜ , which isfurther analyzed for theoretical consistency via a fluctuation-dissipation theorem (FDT).As will be shown, realizing SLA necessitates that the LFPEs are remapped in thepopulations to ensure that the equations can be solved explicitly, but this turns out to beinconvenient. Thus, an alternative solution method to SLA is devised under the guise of a“semi-Lagrangian diffusion” (SLD) scheme, in which streaming and molecular collision aretreated concurrently, rather than sequentially as in the SLA. Preliminary computationaltests are conducted with SLD and presented herein. The main methodological ingredientsare synthesized in Fig. 1.7.1.5 ScopeThe research effort started with this dissertation purports to create a theory of fluctu-ations in metastable liquids undergoing phase change, and a numerical solution method-ology for computing the evolution of the FPE in computational simulations. Neverthe-less, due to the uniqueness, combined physical and mathematical complexity of thermalnoise, metastability, and phase stability, the objectives are limited to developing thenoisy component of the theory, despite the initial aim to apply the method to simula-tions of phase-changing liquids. I.e. the work is not at a stage yet where phase changecan be trivially evolved in simulations. As a consequence, the reader will find that theprimary contribution of this dissertation is the derivation and theoretical analysis of amathematical model, rather than application in the form of simulation of physical flowscenarios. At this stage, the model has only been applied to preliminary numerical tests.These tests are not yet validated by empirical data and thus only serve to showcase thenumerics of the scheme.Even though experiments suggest that even pure fluids have slightly coloured noise,161.5. SCOPEa white-noise variant of the FPE is initially considered. Furthermore, whereas the FPEis equivalent to a Wiener process, the projection of the FPE onto the lattice is mediatedwith asymptotic methods which truncate the modelled noise at an arbitrary amplitudeand frequency threshold, and as such result in a “coarse-grained” numerical solution. Itwill still capture the most important, strong thermal fluctuations that are more likely totrigger a phase transition, but rare events may not be captured in the solution.The thermal fluctuations are coupled to the thermodynamics manifested by a van derWaals EOS which is the simplest non-monotonic, cubic EOS supporting phase-changeavailable. While the EOS can be modified to represent multiple fluid components in amixture, the formulation adopted herein is representative for a pure, single-componentfluid. While the EOS served as a convenient model in the mathematical derivations yetto be presented, it has proven to be non-physical in correlating ρ, T, p of substances ofinterest, e.g. water, methane, ethane, and nitrogen. This also makes the traditionalContinuous FPEs forf(v;x, t), g(v;x, t)Gauss-Hermite quadrature (§3)SLA SLDChapman-Enskoganalysis (§4)NSF equationsFDT for stochasticstress tensor τ˜ (§4)EOSGauge transformationsParticles on DemandReconstruction kernelsNumericaltests (§5)Lattice FPEs forfi(vi;x, t), gi(vi;x, t)Figure 1.7: Overview of the employed methods for obtaining a LFPM for evolving thermalfluctuations in VDW fluids. Nomenclature: Fokker-Planck equation (FPE), semi-Lagrangian advection(SLA) and diffusion (SLD), equation of state (EOS), Navier-Stokes-Fourier (NSF),fluctuation-dissipation theorem (FDT). Dashed lines indicate auxiliary components used to enable theanalysis.171.6. DISSERTATION STRUCTUREprocess of unit conversion from the LBM extraneous and thus all solutions given hereinare in lattice-units (LU). LU are dimensionless units of pertinent simulation variablescommonly used to balance numerical accuracy and stability. LU are facilitated by theBuckingham-π theorem, and will be discussed briefly in chapters 5 and 6.The theoretical consistency assessed with the CE analysis only reports the stochas-tic effects imposed in the stress tensor via the second-order momentum flux moment ofthe density populations f(v;x, t). For a full elucidation of the non-equilibrium, stochas-tic effects in the model, the CE analysis should be expanded to density ρ˜, energy E˜,momentum ȷ˜, heat flux q˜, and the second-order, energy-equivalent R˜ to the stress ten-sor τ˜ . The consistency of the theory in the form of the derived FDT for τ˜ does notyield any information about the correlations in the rest of the stochastic variables. Thus,full thermo-hydrodynamic consistency is first validated when separate FDTs for the sameaforementioned variables are derived, in addition to the covariance matrix for the Wienerprocess imposed on the primitive stochastic variables (molecular velocities v).Given that the theory and framework is relatively unique, and that PonD is signifi-cantly more complicated to implement in code than conventional LBMs, it is expectedthat there is still a significant amount of numerical solution-methodology and code de-velopment to be done to optimize the code performance. At the stage of writing, theserial C++ implementation of the LFPM is rather inefficient and once parallelized shouldreveal more detailed insight into the kinetic behaviour of the theory. Moreover, the re-construction kernel used in the SLA and SLD schemes is formulated with cubic Lagrangepolynomials, and it is expected that the accuracy of the code is dependent on the choice ofinterpolation scheme given the non-uniformity of the fluctuating fields. Throughout thedissertation the reader will find mentions of the lattice-Fokker-Planck-Boltzmann method(LFPBM) and lattice-Fokker-Planck method (LFPM). A distinction is here noted thatwhen the LFPBM is referred to, it is implied that the considered kinetic equations includethe combined Fokker-Planck and Boltzmann (BGK) operators, as opposed to exclusivelythe FP operator in the LFPM.1.6 Dissertation structureThis dissertation revolves around the derivation of lattice solution of the FP oper-ator, and is organized as follows. In chapter 2 a literature review is presented. Thereview considers phase-change modelling in more traditional computational fluid dynam-ics (CFD), kinetic theory as it pertains to multiphase lattice-Boltzmann methods andespecially fluctuating lattice-Boltzmann methods, and lastly the Fokker-Planck equation.181.6. DISSERTATION STRUCTURESubsequently, chapter 3 presents the mathematical groundwork enabling computationalsolution of the LFPEs, including SLA and SLD of the LFPEs, the Gauss-Hermite quadra-ture carried out on the FPEs, and the thermodynamics which are inferred in the theory.In chapter 4 the CE analysis is carried out and the fluctuation-dissipation theorem de-rived. As such, chapter 3 encompasses the work pertaining to objectives O1 and O2, andchapter 4 pertains to O3.In chapter 5 more details on the solution strategy and case setup are presented.Numerical “tests” comprising diffusion in a Gaussian impulse and in a noisy initial fieldare conducted to assess the numerical stability and performance of the SLD scheme.A discussion of the results is presented. The chapter addresses O4 and O5. Finallyin chapter 6, the outcomes of the research and suggested future work are described.Appendices A–D contain mathematical detail supporting the derivations documentedthroughout the dissertation.19Chapter 2Literature reviewThis literature review aims to establish the state of phase change modelling in com-putational fluid dynamics (CFD) as it pertains to the modelling and simulation of phasechange, metastability, thermal noise, and thermodynamics. The contents have beenpeer-reviewed and published, and revolve around the following. §2.1 is based on the pub-lication [J3] and outlines the use of Navier-Stokes solvers for modelling and simulation ofboiling and cavitating flows in especially cryogens, i.e. liquids below 120 K [95]. The con-tents of [J1] are revisited in §2.2 where as an alternative to Navier-Stokes-equation-basedtheory, the use of kinetic theory and the Boltzmann equation is explored as it is usedin the lattice-Boltzmann method (LBM), for modelling phase transitions. Thirdly, theliterature review from [J4] is presented in 2.4 elaborating on fluctuating LBMs in whichthermal noise is added to the Boltzmann equation. Finally, the chapter is concluded withan introduction to the Fokker-Planck equation in §2.5, which will be the foundation ofthe present research methodology.2.1 Phase-change models in computational fluid dynamics2.1.1 Inertially-driven, liquid-vapour phase-changeIn cavitation, if the static pressure at any location in a turbomachine drops be-low a fluid’s saturation pressure, localized evaporation events may occur, followed byrapid collapse of the vapour cavities [96]. Cavitation in turbomachinery creates sheetsof vapour films or dispersed bubbles, which induce high-amplitude, high-frequency reso-nance within components that can cause accelerated erosion and losses in efficiency [26].Although water and some cryogens exhibit comparable temperature variations in experi-ments, 5 the cavitation evolves differently in cryogenic liquids because cryogens generallyhave markedly lower liquid-vapour density ratios, thermal conductivities, as well as moretemperature-sensitive saturation pressures [100], which in turn makes it fallacious to as-5Dular and Coutier-Delgosha [97] reported a temperature decrease in the water surrounding a cavityliquid-vapour interface of ∼ 3 K during growth and ∼ 4 K increase during collapse of a cavity producedin their tube-arrest experiments, and Hord [98, 99] 2–4 K in hydrofoil and ogive flows.202.1. PHASE-CHANGE MODELS IN CFDsume an isothermal condition in pertinent equations of state (EOS) and phase-changemodels. Moreover, cryogenic cavitation is characterized by significant variations in com-pressibility, lower latent heats [101] and a sensitivity to thermo-physical properties nearthe critical temperatures of cryogenic fluids, and to temperature fluctuations [102, 103].The phase morphology is significantly different with cavities being smaller in cryogensthan in water (by about 4 times in tip vortices in LN2 [104]) producing opague cavitationfields in cryogens with dispersed vapour bubbles. Visualizations of cavitating cryogensand water can be found in the works of Brennen [25], Ito [103], Ito et al. [104], Higuchiet al. [105], Frost and Sturtevant [106].Numerical studies of cavitation in isothermal flows have been conducted for decades[107], mostly with the use of two-fluid models and single-fluid (mixture) models. Two-fluid models comprise a set of conservative, compressible continuity equations, for thephase-changing liquid and vapour phases, and additional non-phase-changing components(denoted by subscript i) that are solved with the volume-of-fluid (VOF) method, in agiven multiphase flow system [108]:∂t(ρlαl)+ ∂α(ρlαluα)= m˙ = m˙+ − m˙−, (2.1a)∂t(ρvαv)+ ∂α(ρvαvuα)= −m˙ = m˙− − m˙+, (2.1b)∂t(ρiαi)+ ∂α(ρiαiuα)= 0, (2.1c)where the density and mass-fraction α fields are conserved with the condition αl + αv +αi = 1, and the mixture velocity u = αlul + αvuv + αiui is applied exclusively acrossinterfaces. The mass transfer rate m˙ is computed as a balance between the condensationand evaporation rates m˙+ and m˙−, and are popularly correlated cf. the Kunz et al. [109],Schnerr and Sauer [110], and Zwart et al. [111] mass-transfer models [108],m˙−Kunz =Cvρvαlmin(p− pv, 0)t∞1/2ρlu2∞, (2.2a)m˙+Kunz =Ccρv(αl − αi)2max(1− αl − αi, 0)max(p− pv, 0)t∞max(p− pv, 0.001pv) , (2.2b)m˙−S—S =3ρlρvραlmax(1− αl − αi, 0)R−1b×√23ρl(|p− pv|+ 0.001pv) min(p− pv, 0), (2.2c)m˙+S—S =3ρlρvραlmax(1− αl − αi, 0)R−1b212.1. PHASE-CHANGE MODELS IN CFD×√23ρl(|p− pv|+ 0.001pv) max(p− pv, 0), (2.2d)m˙−Zwart = Fvap3αnuc(1− αv)R−1b√23pv − pρlmin(p− pv, 0), (2.2e)m˙+Zwart = Fcond3αvρvR−1b√23p− pvρlmax(p− pv, 0). (2.2f)Therein, Cv, Cc, Fvap, Fcond are empirical constants, t∞ a mean flow-time scale, R−1b isan inverse cavitation nuclei radius, and pv the vapour pressure. αnuc is a nucleation-sitevolume fraction that must be tuned a-priori. Effectively, the Kunz model immediatelyintercepts cavitation when the static pressure drops below the vapour pressure, and con-densation only occurs when vapour is present in a computational cell [108]. The Schnerrand Sauer model is based on the Rayleigh-Plesset equation. In addition, interfaces aretracked with an advection equation for a VOF indicator F(x, t) which assumes unityvalue in the liquid, 0 in the gas, and 0 < F < 1 at an interface [112],∂tF + u · ∇F = 0. (2.3)Mixture models omit slip velocities between phases and assume the flow to consists ofa “single-fluid” mixture with a local mixture density ρ = ρvαv + ρl (1− αv) obeyingcontinuity and D-dimensional momentum conservation [113, 114],∂tρ+ ∂α(ρuα)= 0, (2.4a)∂t(ρuα)+ ∂β (ρuαuβ) = −∂αp+ ∂βταβ, (2.4b)ταβ =(µ+ µt)(∂αuβ + ∂βuα − 2D(∂γuγ)δαβ), (2.4c)where µt is the turbulent eddy viscosity c.f. the Boussinesq approximation [115]. Thermaleffects are modelled via a variety of energy equations, closed with an equation of state,often a stiffened-gas type [114]. Most of the reviewed material applies a total-energy(E = e + uαuα/2)based transport equation [114, 116–118] comprising the internal eand kinetic uαuα/2 energies (2.5a). To also include the pressure work, enthalpy-basedvariants [119, 120] with (2.5b) and without kinetic energy and transient effects [121] wereapplied. Few authors focused on internal energy-based transport [122, 123], which is lesssuitable for flows with significant compressibility effects in its assumed form (2.5c).Dt(ρE)= −∂α(puα)+ ∂α(ταβuβ)+ ∂αqα + Slv, (2.5a)222.1. PHASE-CHANGE MODELS IN CFDDt(ρ(h+ k))− ∂tp = ∂α(ταβuβ)+ ∂αqα + Slv, (2.5b)Dt(ρe)= −∂α(puα)+ ∂α(ταβuβ)+ ∂αqα + Slv. (2.5c)In the equations, Dt(·) = ∂t(·) + ∂α(uα·)is the material derivative, qα = keff∂αT isthe heat flux due to the effective thermal conductivity keff which can be written in amixture form keff = αl(klam + kturb)+ αv(kv + kturb)[124] with e.g. the eddy diffusivitykturb = cpµturb/Prturb. The source term Slv = hlvm˙ [125], in which the mass-transfer ratem˙ is approximated by cavitation models such as (2.2), is omitted in some studies [118,122]. For complete discussions of the mathematical formulations the reader is referredto the reviews of Kharangate and Mudawar [126] and Wörner [127] in which numericalschemes for interface tracking, surface-tension modelling, interfacial mass, momentum,and energy-transfer modelling, in boiling and condensation simulations, are discussed.Cavitation models are distinguished in three families: Interfacial-dynamics models(IDMs), density-based cavitation methods (DCMs), and transport equation-based models(TEMs) in which the cavitation model is formulated in terms of the Rayleigh-Plessetequation [113, 128]. A subvariant of the TEMs are full-cavitation models (FCM) thatseek to instil barotropic phase-change through Young-Laplace capillary-laws and Gibbs-Duhem theory [129, 130]. Surface tension forces are accounted for automatically inthe IDM, while the DCM correlates the density field with a homogeneous equilibriummixture (HEM) model [131] through closure relations—commonly barotropic equationsof state where ρ = φ(p) [100]—and have been extensively used to simulate cryogenic flows[132]. Cavitation inception can also be inferred via population-balance equations basedon classical nucleation theory [64]. The reviewed works that utilized these frameworksare found in Tab. 2.1.232.1.PHASE-CHANGEMODELSINCFDDespite the strong development of numerical methods for modelling cavitation in thermo-sensitive fluids they still re-quire further work [133] due to the intricate coupling between heat transfer mechanisms, interfacial interactions, instabilities,turbulence, phase change and variable thermo-physical properties that render detailed simulation of macro-scale, cavitationdynamics difficult to achieve. As was presented, only few studies resolved the spatial domain down to the Kolmogorov lengthscale. Clearly, the multitude of energy exchanging mechanisms make cavitation research constrained as the computationaldemand can only be met with massively-parallel computing resources. As such, numerical cavitation research is very compu-tationally prohibitive and at the current state of computing is especially saturated with studies addressing turbulence withReynolds-averaged Navier-Stokes models predominantly implemented in commercial codes. The topic benefits from higher-fidelity studies, of which there are few, to address the interaction between turbulent structures and cavity fields. DNS iscurrently only computationally feasible for the study of single cavity dynamics and has been done for water [13] with resultsshown in Fig. 2.1.Figure 2.1: Evolution of a cavity collapsing on a wall in water with an initial radius of 3.8 mm, as simulated by Yang et al. [13]. Colour contoursshow the volume-fraction field. The simulation attests to the intricacy and complexity of the collapse process of a single cavity. Reproduced withpermission from Phys. of Fluids 33, 073311 (2021). American Institute of Physics© 2021.242.1. PHASE-CHANGE MODELS IN CFDTable 2.1: Selected computational fluid dynamics studies of cavitating flows since 2013. (†): All Zwartcavitation models are correlated to evaporation and condensation constants Fvap, Fcond that need to calibratedto the pertinent fluid, in lieu of key thermo-physical properties. In most of the studies information abouttemperature and pressure dependence in pertinent thermo-physical properties is lacking and as such it isassumed that these are modelled as constant quantities.Ref. Case(s) Fluid(s) Methods, softwareOperating and geometrical conditions[113] 2D quartercaliber hydrofoil.LN2. Calibration of modified [111] cavitation model(†), RNG k-ε turbulence model. pmax ∼ 500 kPa • ANSYS CFX.[132] Hydrofoil. LN2. Simplified non-equilibrium finite rate cavitation model mlv ∝ Cu, τlv, τvl, where Cu is the inertial cavitation number and τ time constants.k-ε turbulence model • CRUNCH.[128] (i.) hydrofoil,(ii.) ogive.LN2, LH2. TEM, modified RNG k-ε turbulence model, modified Zwart et al. [111] cav. model with temp. sensitivity.Constant latent heat • ANSYS CFX.[129] (i.) hydrofoil,(ii.) ogive.LN2, LH2.FCM, Young-Laplace equation, k-ε turbulence model, investigation of phase-change thermal effects pmax ∼ 400 kPa.ρ, pv , µ, Cp = φ(T ) • ANSYS Fluent 6.3.[134] diesel injectornozzle.diesel fuel. Schnerr and Sauer [110] cavitation model, k-ε turbulence model • ANSYS Fluent.[135]Barber-Nichols HEP35-000 pump.LHe. k-ω–SST turbulence model, varying superheating, no energy equation. Cavitation assessment by evaluation of NPSH •Software not disclosed.[136] Nozzle orifice. n-heptane. Orifice geometry modifications. Eulerian-Lagrangian, KH-ACT model • CONVERGE.[137] Turbine expander. Liquefied air. k-ε turbulence model. HMM. Zwart et al. [111] cavitation model. Geometric optimization algorithm for fairing cone andimpeller geometries. Cp, k, µ = φ(T, p), ρ = φ(T ), pmax ∼ 400 kPa. Gov. eqs not described • ANSYS CFX.[138] Orifice nozzles. Water. k-ε turbulence model, Schnerr and Sauer [110] cavitation model, geometric optimization for number of orifices andtheir sizes • ANSYS Fluent.[139] Turbine expander,hydrofoil.LNG. k-ε turbulence model, Zwart et al. [111] cavitation model, optimization of impeller and exducer blade geometries •ANSYS CFX.[140] Flow controlventuri tube.Water. HMM, k-ε turbulence model, Schnerr and Sauer [110] cavitation model, optimization of diffuser angle • ANSYS Fluent18.2.[141] Bi-directionalcone flowmeter,hydrofoil.LN2. HMM, RNG k-ε turbulence model, Zwart et al. [111] cavitation model, pv = φ(T ). Optimization of operating conditions and structuralparameters • Software not disclosed.[142] Turbine expander,draft tube.LNG, LH2. HMM, k-ω–SST turbulence model, Zwart et al. [111] cavitation model. pmax ∼ 400 kPa, No info on modelling of µ, k, Cp, hlv .Optimization of blade loading distributions and blade lean angles • ANSYS CFX.[143] Cryogenic pump,hydrofoil.LN2. RNG k-ε turbulence model, modified Zwart et al. [111] cavitation model considering thermal effects, pv = φ(T ). Parametric studyof inlet pressure, surface roughness, and flow rate • ANSYS CFX.Shearing instabilities in cavitating flows[144] Conv.-divergingnozzle,submerged jet.Water. HMM, RNG k-ε turbulence model, Zwart et al. [111] cavitation model with shear-stress extension, complemented byproprietary experiments • Software not disclosed.[145] 3D Channel. Fluid not given. Orifice geometry modifications, direct and LESs, modified Chen cavitation model, isothermal, Reh = 104 • Software notdisclosed.[120]Mixing behind flatplate splitter.LNG. Direct numerical simulation, homogeneous equilibrium mixture model, barotropic EOS, non-isothermal, Reω ∈ [44, 215],Eu ∈ [0.1, 1.1] • OpenFOAM.Surface-tension forces[108] diesel injectornozzles.diesel, air. LES, VOF, Schnerr and Sauer [110] and Kunz et al. [109] cavitation models, Rew ∈ [1817, 5452, 215] • OpenFOAM.[133] Hydrofoil. LH2, LN2 VOF, k-ω–SST turbulence model, Schnerr and Sauer [110] non-isothermal cavitation model, convective and (const.) latent-heatsource terms. Nucleation model dependent on crit. nucleation diameter and site density. Rew = {1.76, 1.08} × 107 • OpenFOAM.[146] NACA66hydrofoil.Fluid not given. VOF, simplified Reyleigh-Plesset model, incompressible, Reh = {48, 600} • CFDShip-Iowa 6.2.Turbulent, cavitating flows[147] Orifice. Fluid w.ρcr = ln 2.DNS, pseudo-potential LBM, cavitation inception by the Joseph’s minimum tension criteria [148], non-ideal EOS,Rec = 2× 106 • Software not disclosed.[149] (i.) hemisphericalheadform,(ii.) hydrofoil,(iii.) 2D venturi,(iv.) propeller.(i.) water,(ii.) LH2,(iii.) Freon R-114,(iv.) water.k-ε, RNG k-ε, k-ω turbulence models, Launder-Reece-Rodi Reynolds stress model [150], k-ω–SST turbulence model,compressible barotropic EOS φ(the speed of sound), Kunz et al. [109] cavitation model. Gov. eqs not described • OpenFOAM.[151] 3D NACA0015hydrofoil.LH2. HMM, LES, Schnerr and Sauer [110] cavitation model, non- and isothermal simulations, const. nucleation-site density nb = 108,const. hlv, µ • ANSYS Fluent.[152] 3D NACA0015hydrofoil.Fluoroketone. PANS, k-ε turbulence model, [111] cavitation model, Rec = 1.05× 106 • Software not disclosed.[119](i.) circular orifice,(ii.) Laval nozzle.LN2, LNG DNS, homogeneous equilibrium model, barotropic EOS, isothermal cavitation model based on theory ofKärrholm et al. [153] • OpenFOAM.[124] NACA0015hydrofoil.Fluoroketone. PANS, HMM, extended Schnerr and Sauer [110] cavitation model with thermal effects, k-ω–SST turbulence model,force-element analysis • Software not disclosed.Perturbed cavitation[154] (i.) linear waveconvection,(ii.) bubbly flow,(iii.) 2D cylinder.Water. DNS, density-based homogeneous equilibrium model, cavitation pressure field compared with acoustic analogy ofFitzpatrick and Strasberg [155], Red = 200 • Software not disclosed.[156] Spraynozzle.LN2. HMM, realizable k-ε turbulence model, Zwart et al. [111] cavitation model, fluctuating inlet pressure and varying inlettemperature • ANSYS Fluent.INSTABILITIES AND INTERACTION MECHANISMS IN CAVITATING FLOWS[157] Ogive. LH2. LES, HMM, Schnerr and Sauer [110] cavitation model, const. latent heat • ANSYS Fluent.[158] Venturi tube. LN2. TEM, Schnerr and Sauer [110] cavitation model, realizable k-ε turbulence model, const. Pr, hlv , Cp, µ,nucleation-site density = 108, proprietary experimental validation • Software not disclosed.[159] (i.) ogive,(ii.) NACA0015hydrofoil.LN2. HMM, Zwart et al. [111] cavitation model, const. Pr, hlv , Cp, µ, thermal effects inferred via crit.time constant tc = φ(pv , T∞), RNG k-ε turbulence model • Software not disclosed.[125] (i.) ogive,(ii.) NACA0015hydrofoil.LH2, LN2. HMM, compressible Zwart et al. [111] cavitation model, gas phase assumed ideal, pv = φ(T ), RNG k-ε turbulence model,validated with experimental results [98, 99] • Software not disclosed.[64] 2D quartercaliber hydrofoil.LH2, LN2. HMM with population balance equations coupled with classical nucleation theory, Schnerr and Sauer [110] cavitation model,k-ω–SST turbulence model, Rec = 1.08× 107 • OpenFOAM 3.0.1.[160] Inducer. LN2. HMM, bubble-size distribution model, Rayleigh-Plesset equation, no turbulence model • OpenFOAM.[161] Ogive. LH2. RNG k-ε turbulence model, HMM, Schnerr and Sauer [110] cavitation model • ANSYS Fluent 14.5.[162] NACA0015hydrofoil.Fluoroketone. k-ε turbulence model, HMM, liquid-vapour two-phase EOS, study of cavitation vortices effect on hydrofoil forcecoefficients • ANSYS CFX.[163] NACA0015hydrofoil.Fluoroketone. HMM, FBDC-k-ε turbulence model, Zwart et al. [111], study on thermosensitivity and shedding dynamics • ANSYS CFX.[164] NACA0066hydrofoil.Fluid not given. HMM, modified k-ω–SST turbulence model, Schnerr and Sauer [110] cav. model, study on cavity evolution • OpenFOAM.[165] NA. NA. Review on fluid-solid interaction in cavitating flows.[166] CD nozzle. LN2. Density-corrected turbulence model, Hertz-Knudsen [167, 168] cav. model • Software not disclosed.252.1. PHASE-CHANGE MODELS IN CFD2.1.2 Thermally-driven, liquid-vapour phase-changeIn literature, VOF solvers modelling separate phases in a flow are superior in pre-diction accuracy and interpretability in comparison to the cheaper and popular mixturemodels, similar to those presented in §2.1.1. A prominent disadvantage of mixture mod-els is that they do not resolve separate phases and their interfacial relative velocities,which otherwise can induce viscous dissipation and interfacial instabilities. Neverthe-less, their computational cost and practicality for industrial flows favour their usage.In mixture models, the multiple components and/or phases are represented in terms ofmixture densities and velocities. Key to these are closure models that infer the effectsof relative velocities between individual phases. However, for fundamental studies ofmulti-physics boiling and cavitation systems, such simplifications sacrifice interpretabil-ity. Tab. 2.2 reviews CFD studies of varying fidelity of pool boiling with cryogens, water,and refrigerants. The weakness of both the mixture, and two-fluid frameworks is thatthe source-term models are often dependent on empirical formulations that limit theapplicability to specific physical cases. Murallidharan et al. [169] reviewed a number ofuseful wall-boiling models, and Yeoh and Zhang [170] dozens of models for nucleation-site density, nucleation frequency, and departure diameter for cryogenic boiling—whichare crucial for simulating more advanced boiling systems. However, these are difficult togeneralize to larger parameter spaces.From a fundamental perspective, Navier-Stokes solvers are deterministic, and in theirmost basic form, cannot spontaneously nucleate vapour without phase-change models andother extensions. However, they can still be used to investigate the bubble developmentand morphology in various boiling systems. Nabil and Rattner [171] developed a toolboxwith the capability of simulating thermal phase change. Through a VOF formulationwith interface capturing capabilities, their model allows simulation of nucleate phasegrowth with an initialized volume of vapour in a square cavity on a planar surface.In this system, four source terms were used to model the multi-phase effects, includingvarious phase change formulations that were used for computing the evaporation rateacross interfaces. Capillary forces from surface tension were modelled following Brackbillet al. [172], Lafaurie et al. [173] and Raeini et al. [174]. An energy equation was employedbased on static enthalpy in which pressure transport and viscous effects were neglected, asconductive and convective fluxes were assumed to be more significant, which aligns withearlier studies [175–177]. The most rigorous method to evaluate the heat transfer rateswas deemed to be direct evaluation of cross-interface temperature gradients, but thisapproach is computationally prohibitive and requires geometric reconstruction, which is262.1. PHASE-CHANGE MODELS IN CFDdifficult to implement on unstructured meshes [171, 178, 179]. Sato and Niceno [58, 59]extended the concept of Nabil and Rattner [171] by randomly initializing nucleation sitesin the simulation on the heated surface. An activation temperature was then selectedbased on available experimental data for water, and single-cell sized vapour embryos wereexplicitly triggered in a CFD cell when it exceeded the activation temperature.The VOF approach has been used in the investigation of nucleate and film boilingin various fluids [175, 176, 180], including cryogens by Liu et al. [181], Bouazaoui et al.[182], Zhou et al. [183], Xiaobin et al. [184] and Ahammad et al. [185, 186]. However, tosimulate boiling processes with an inherent ability to nucleate a liquid, other theories arerequired to augment the Navier-Stokes equations. The diffuse interface (DI) approachof Liu et al. [187] represents a liquid-vapour interface at the nanoscale, and naturallyembeds phase change and interfacial tension [14]. The interfacial tension automaticallyemerges in the approach because DI models are able to resolve steep, continuous densityfields approaching the molecular scale. It couples the varying density and temperaturefields through the Helmholtz free-energy operator Ψ over the entire fluid domain:Ψ(ρ, T ) =∫ [ψ (ρ, T ) +κ2|∇ρ|2]dx. (2.6)The capillary coefficient κ curtails variations in the density field, and subsequently theformation of interfaces between regions with high density gradients [188–191]. Galloet al. [192] showed that with an equation of state for the free energy ψ, a constant κcan reproduce an accurate representation of the dependence between temperature andcapillarity. The total energy density in the energy equation (2.8c) was then formulatedasE = ψ + Ts+12ρ |u|2 + κ2|∇ρ|2, (2.7)where the entropy density s = −∂ψ/∂T is computed from a fitting EOS for the bulkfree energy ψ(p, T ) [14, 192]. The DI method works well in continuum mechanics andat the mesoscale. However, when approaching the length scale of an interface in nucle-ation, thermal fluctuations become predominant, necessitating a stochastic descriptionrather than the determinism of the DI method [14, 193], or alternative phase-field formu-lations [194]. Fluctuating hydrodynamics theory incorporates such stochastic behaviourinto the Navier-Stokes equations. The founders of the theory—Landau and Lifshitz[91]—proposed that the stress tensor and energy flux should be augmented by stochasticfunctionals. Thus, the evolution of two-phase nucleating fluid systems was computed by272.1. PHASE-CHANGE MODELS IN CFDcombining these two theories in terms of the conservation equations,∂tρ+ ∂α(ρuα)= 0, (2.8a)∂tuα + uα∂α (ρuβ) = ∂βταβ + ∂β τ˜αβ, (2.8b)∂tE + uα∂α (e) = uα∂β (ταβ + τ˜αβ)− ∂α (qα + q˜α) , (2.8c)ταβ = −pδαβ + µ(∂αuβ + ∂βuα − 23∂γuγδαβ)+ κ(12|∇ρ|2 + ρ∇2ρ)δαβ − κ∂αρ∂βρ, (2.8d)where the last two terms in ταβ make up the diffused capillary stresses. The fluctuatinghydrodynamics theory was invoked via delta-distributed, white-noise symmetric tensorsη(τ˜)αβ , η(q˜)α , in the stress tensor and heat flux τ˜αβ, q˜α,τ˜αβ =√2kBTµ(η(τ˜)αβ − δαβ13η(τ˜)ζζ), (2.8e)q˜α =√2kBT 2k · η(q˜)α . (2.8f)Coupling the fluctuating hydrodynamics theory and diffuse interface modelling provedto proficiently model spontaneous bubble nucleation as well as cavitation [14, 192, 195].Two successive time instants of the simulations of Magaletti et al. [14] are depicted inFig. 2.2.Even with off-the-shelf commercial software, nucleation was simulated in pool boil-ing with a sub-grid model for surface roughness topology [196], where the superheatingrequired for inception is modelled as dependent on the sub-grid roughness. In additionFigure 2.2: With their DI-FH solver Magaletti et al. [14] found that subcritical embryos in water poolboiling coalesce with ones neighbouring making them survive for longer, which makes it possible forthem to self-sustain nucleation for longer on ultra-smooth surfaces. This corroborates low onsetsuperheat levels found in various experimental studies [15, 16]. Reproduced with permission from Int.J. Multiph. Fl. 130, 103356 (2020). Elsevier © 2020.282.1. PHASE-CHANGE MODELS IN CFDto cryogenic and water pool boiling, several works have been published recently on flowboiling [169, 197, 198] including cryogenic fluids [199], as well as pool boiling in nanofluids[200–204] with recent works synthesized by Dadhich and Prajapati [205]. The numeri-cal sophistication of the pool-boiling simulations are non-trivial. However, few of themconsider fluid-solid interaction effects that are central to pool boiling.Table 2.2: Selected CFD studies on pool boiling with various cryogens, refrigerants, and water.Ref.SubstrateFluid(s)Methods•Software[170]NA.Cryogens.Review.Two-fluid.Populationbalancemodelling.Synthesisof(empirical)closurerelationsfornucleationsitedensity,bubbledeparturediameter,nucleationfrequency.Forcebalancemodels.[14]Ideallysmoothsurface.Water.Nucleateboiling,diffuseinterface/fluctuatinghydrodynamicsmodel,vanderWaalsEOS,bubble-detectionalgorithm,delta-correlatedwhitenoisestressandheatfluxtensors•Softwarenotdisclosed.[181]Spillonground.LN2.Nucleate,transition,andfilmboilingregimes,VOFmodel,source-termmodelsforvapour-liquidphasechangecharacterizedbysinglerelaxationparameter,surfaceroughnessneglected,novalidation•ANSYSFluent13.[175]Horiz.wall.Water.Multimodefilmboiling,DNS,finite-differencemethod.Numericalmodeldetailedin[176]•In-housecode.[206]Singlenucl.site.Water,FC72,R22,R113,R134a.Dettachmentinthenucleateboilingregime,sensitivitytowettability,superheat,andgravitation.VOFw.sensibleandlatentheattransfer•OpenFOAM2.2.1.[207]Horiz.wall.Subcooledwater.Multi-scalepoolboiling.k-ω–SSTturbulencemodel.“Generalizedtwophaseflow”model.Smallerbubblesaremodelledasadispersedphasewhereaslarge-scalestructuresarestatisticallyresolved.Forcebalancemodel.Phasechangemodel•ANSYSCFX.[208]Vert.wall.Water.Nucleateboiling.Heattransferpartitioningmodel.Microlayerevaporationmodel.Bubbleslidingmodel.Statisticaltreatmentofmutualinteractionbetweenbubblesandmechanisticpredictionofdrysurfacearea•STAR-CCM+.[209]Vert.tube.Water.Realizablek-εturb.model.Hertz-Knudsenbasedphase-changemodel.VOFw.piecewiselinearinterfacereconstruction.Continuumsurfaceforcemodel•ANSYSFluent15.0.[182]Brassribbon.LN2.Singlenucleatesite,atmosphericpressure.Phase-changemodelnotdisclosed.Two-fluidmodel•COMSOLmulti-physics.[184]Notgiven.LN2.Nucleateboiling.Heattransferpartitioningmodel.RensselaerPolytechnicInstitutephasechangemodel•ANSYSFluent.[59]Copperblock.Water.Nucleate,transition,andfilmboiling.Immersedboundarymethodwithconjugatesolid-fluidheattransfer.Micro-layermodel.Colour-functioninterfacesharpeningandtrackingmethod.LES.Contact-angletreatment•MPIparallelizedin-houseCFDcode“PSI-BOIL”.[58]Horiz.plate.Water.Nucleateboiling.Conjugateheattransfer.Micro-layermodel.Colour-functioninterfacetrackingmethod.LES•In-housecode.[171]Structuredsurface.Water.Wavyfallingfilmcondensation,Taylorflowevaporation,nucleateboiling,dropwisecondensation.VOFwithphase-changesource-term.Surfacetensionforcemodel.RANS/LES.MPIparallelism•OpenFOAM.[185]Horiz.wall.LN2,L-methane.FilmboilingandRTinstability.VOFwithgeometricreconstruction.Continuumsurfaceforcemodel•ANSYSFluent14.0.[186]Horiz.wall.LN2,L-methane.FilmboilingandRTinstability.VOFwithpiecewiselinearinterfacereconstruction.Continuumsurfaceforcemodel•ANSYSFluent14.0.[196]Vert./horiz.flatsurfaces,3D,porousenhancedsurface.R134a,water.Simulationsofnucleate-through-filmboiling.VOFwithtime-explicitgeo-reconstruction.Adhesiveandspreadingforcesincludedinmomentumeq.Brackbilletal.[172]continuumsurface-forcemodel.Subgridmodelroughness-scalefeaturesusedtopredictthesuperheatrequiredfornucleation•ANSYSFluent.292.2. DETERMINISTIC LBMS2.2 Deterministic lattice-Boltzmann models ofmulticomponent, multiphase fluids2.2.1 Streaming and collision solution algorithmKinetic theory revolves around describing the dynamics of non-equilibrium processesand their relaxation towards a thermodynamic equilibrium. Central to the theory is themathematical cornerstone of the Boltzmann equation conceived by Ludwig Boltzmann fordescribing dilute gasses [210]. Pivotal to the equation is the evolution of the probabilitydensity f(c;x, t) for finding a molecule at location x at time t, with a velocity c. TheBoltzmann equation is a seemingly innocent equation describing the competition betweenadvection and collisions of particles [210–214],∂f∂t+ c · ∇f = Ω. (2.9)Here, c is the particle velocity and f is the probability density function representing theproportion of particles with a particular c. The probability density function f is depen-dent on the time evolution of the particle following its trajectory, which is inferred throughthe local particle collision kernel Ω. Ω encompasses all physical phenomena that definesits behaviour in relation to the thermodynamic equilibrium state. For example, immedi-ately after an m-th particle collides with a l-th particle that has a probability density f (l)of colliding with the m-th particle, Ω changes f (m) by an amount δΩ(m) = G −L [211].G describes the gain in particle-m momentum expressed by the probability of a particle-j colliding with the m-th within the ranges[x(m),x(m) + δx(m)],[c(m), c(m) + δc(m)],[x(l),x(l) + δx(l)],[c(l), c(l) + δc(l)], and [t, t+ δt] as a function of the particle surfacearea, and x(m), x(l), c(m), c(l), and t. In a similar fashion, the loss term L describes theloss in momentum of the m-th particle.The discretized lattice-Boltzmann equation (LBE) (2.10) can be written into a stream-ing process (left-hand side) and a collision process (right-hand side):fi (x+ δx, t+ δt)− fi (x, t) = Ωi. (2.10)The collision process is correlated with an offset between the non-equilibrium and equi-librium distributions and a tensor containing characteristic relaxation coefficients, whichrepresent the (streaming) time between individual collisions. The solution of the equa-tion is performed along i streaming directions—including the zeroth representing nostreaming—as exemplified in Fig. 2.3 for a D2Q5 lattice model, and computed in a dis-302.2. DETERMINISTIC LBMS-1 0 1-101c1c0c2c3c4Space xSpaceyFigure 2.3: Illustration of the five streaming paths(including no streaming in c0) in the D2Q5 lattice model.crete manner according to the algorithm in Fig. 2.4. The populations in the Boltzmannequation are discretized by Gauss-Hermite quadrature resulting in the discrete stream-ing directions, and the Chapman-Enskog multiscale expansion bridges the mesoscale tothe continuum limit of the Navier-Stokes equations. These concepts will be describedin depth in §3. The macroscopic thermo-hydrodynamic variables density ρ, velocity u,and temperature T , are correlated to the sums of the statistical particle distributions ffor mass and momentum, and a second g for energy, in the Q total streaming directions(NB; the formulation varies between LBMs) asρ =Q−1∑i=0fi, (2.11)ρu =Q−1∑i=0fici, (2.12)T =Q−1∑i=0gi. (2.13)Body forces F can also be imposed in such a model by recasting the continuum velocityu← u+F δt/2ρ, where a-priori u = 1ρ∑ni fici followed by the computation of a forcingsource term applied to the populations in the collision step. The physics describing theparticle interactions in multiphase flows is imposed through models using equations-of-state, free energy, interaction energies, and others.Approaches for modelling phase transitions in multiphase flows can be categorizedinto explicit and implicit methods [18]. In explicit methods, phase interfaces are tracked312.2. DETERMINISTIC LBMSInitialization fi, ρ,u, etc.Collisions, compute ΩiSteaming stepBoundary conditionsCompute moments and ρ,u, T , etc.Next time step, t← t+ δtFigure 2.4: General solution procedure for an LBM.in time and space, and in implicit methods—such as the model of Shan and Chen [215]—interfaces are produced by calculating intermolecular action between adjacent popula-tions of molecules. In implicit methods, the interaction forces are spatially continuousacross several lattice nodes. This is advantageous for modelling the behaviour of cryo-gens in environments where the temperature is not strongly dissimilar to the criticaltemperature requiring a high accuracy in predicting interfacial phase-change [18].Core to the lattice-Boltzmann method is the way with which the thermodynamicstate of a medium is altered towards one in equilibrium with a Maxwellian distributionf(eq)i approximating the continuous equilibrium function,f (eq) =ρ(2πv2T)2/2 exp [−(v − u)22v2T]. (2.14)The inequilibrated distribution fi, which in the continuum limit corresponds to a density,changes with a rate dictated by the intermolecular collisions. The effects of these collisionsare reflected by a collision operator Ωi which relaxes the distribution function towardsan equilibrium state. In the simplest form the evolution of fi is represented by a singlerelaxation time τ in the lattice-Boltzmann equation [216]:Ωi = −1τ(fi − f (eq)i). (2.15)However, with the collisions represented by a single relaxation time, some multi-speedlattice models with this Bhatnagar-Gross-Krook (BGK) model [217] suffers from somelimitations, e.g. the Prandtl number is fixed at unity [216]. Consequently, numerous322.2. DETERMINISTIC LBMSformulations of the collision operator have been proposed since the contribution of Bhat-nagar et al. [217], including multi relaxation time (MRT) schemes and models based onentropy that also provide numerical stability at high Reynolds numbers.To ensure stability in the simulation of turbulent cryogenic flows, more sophisticatedcollision models with multiple relaxation times beyond the single relaxation-time BGKvariant have been used [218–220]. Pareschi et al. [220] presented a two-way coupled con-jugate heat transfer model applied at the fluid-solid boundary, with a two-populationentropic LBM to simulate the bulk fluid kinetics. The model was validated for caseswith turbulent Reynolds numbers and complex boundaries. Since the original formu-lation of the entropic-class LBM by Karlin et al. [221], the method was considered animportant milestone in the development of the LBM. It enjoyed the attention due to itsunconditional stability at high Reynolds number flows in immersed complex geometriesas the sub-grid physics were modelled. However, this comes at higher computationalcost owing to the solution of non-linear equations for entropy. With inspiration from theH-theorem by Boltzmann, Karlin et al. [221] exploited that the equilibrium populationsf (eq) coincide with the minima in a convex entropy function,H =n∑i=1[fi ln(fiwi)], (2.16)with wi is the i-th direction-dependent weight from the particular DmQn m-dimensionallattice model with n streaming directions. In order to accommodate turbulent flows,Pareschi et al. [220] employed the multi-relaxation Karlin-Boesch-Chikatamarla (KBC)entropic LBM [222, 223] in which the double distribution functions are chosen so as tominimize H:fi(x+ ciδt, t+ δt)− fi(x, t) = ωf(f(eq)i − fi)+ (ωf − ω1,f )(f⋆i − f (eq)i), (2.17a)gi(x+ ciδt, t+ δt)− gi(x, t) = ωf(g(eq)i − gi)+ (ωf − ω1,f )(g⋆i − g(eq)i). (2.17b)In these kinetic equations pseudo-equilibrium distributions denoted by “⋆” were adoptedto allow for variable Prandtl numbers that otherwise are fixed to 1 in collision modelswith single relaxation times. The relaxation parameters ωf and ω1,f , where subscript“f ” denotes fluid, allow for variable kinematic viscosity ν and thermal diffusivity α viaPr =να=(2− ωf )ω1,f(2− ω1,f )ωf . (2.18)332.2. DETERMINISTIC LBMSThe collision operators in eqn. (2.17) replaced the conventional BGK collision model,which is deficient at relatively high Reynolds numbers due to numerical instabilities in-ducing deleterious divergence. Nevertheless, because of the simplicity and computationalefficiency, the BGK collision model remains the workhorse of most LBM-implementations[223].Owing to the BGK popularity, the LBM was conjectured to be unsuitable for sim-ulating high Reynolds number flows. A number of recent studies have shown this tobe incorrect, as in addition to the use of the entropic LBM [1, 2], recent studies [3–7]have simulated turbulent flows with MRT collision operators—all outlined in Tab. 2.3.Whereas the BGK collision model adopts a single fixed rate 1/τ for f , in the MRTcollision model all moments are relaxed by individual rates [3] from relaxation (S) andtransformation (M) matrices [224]:Ωi = −M−1SM(fi − f (eq)i)δt. (2.19)From the basic notion that the moments in an LBM—i.e. the lower-order density (ze-roth), momentum (first), momentum flux (second), and those of higher order—relaxtowards their equilibrium states at dissimilar rates [3] brings to reason how representingthe turbulence energy-cascade with a single rate can be deleterious and produce non-trivial errors. Whereas the low-order moments are inferrable from the fluid properties[7], the higher-order moments are non-interpretable and are normally fixed at a con-stant value [3]. Being able to fine tune each of the rates adaptively during runtime isparamount to eliminating numerical dispersion.Other methods for relaxing the higher-order moments than with entropy as in theKBC model have been formulated. Li et al. [3] recently introduced a “measurementfunctional” that adjusts the higher-order moment relaxation rates on-the-fly. Otherturbulence-studies using MRTs such as those of Wu et al. [5], Zecevic et al. [7] showedgood agreement with the Navier-Stokes equations from other direct numerical simula-tions (DNS) [225, 226] in terms of turbulence-energy budgets. With more sophisticatedformulations of the relaxation rates in MRT schemes and entropic extensions, stable sim-ulations of turbulent flows are becoming a part of the repertoire of the LBM. In additionto generically relaxing the solution of the LBE, the collision kernel can be modified toaccount for phase-separation and multiple fluid components. Popular operator modelsfor this purpose are the colour-gradient, pseudo-potential, free-energy, and phase-fieldmodels which will be elaborated in the following.342.2. DETERMINISTIC LBMSTable 2.3: Selected numerical experiments with the LBM of various turbulent flows since 2018.Collision models employed: Entropic [1, 2], MRT [3–7], BGK [8]. Nomenclature: cylinder/spherediameter D, time-averaged root-mean-square velocity U rms, Taylor micro-scale λ, continuous phasekinematic viscosity νc, friction velocity Uτ , channel-height h, plate/filament/channel length l, minimumand maximum stroke ϕ, flapping frequency f , span-wise length of a wing lw, mean wing-chord c, andlid-velocity Ulid.Ref. Case(s) Reynolds no.(s)Morrisonand Leder [1](a)wall-mountedcylinder(b)sediment transportaround cylinder(a)ReD =U∞Dν= 2× 102(b)ReD= 1.7× 104Feuchter et al. [2] wall-mountedcylinderReD = 3.2× 104Li et al. [3] (a)tornado(b)falling box(c)plank drop(d)turbulent jet(a)8× 104(b)105(c)1.4× 105(d)2× 104Zhong andKomrakova [4]liq. droplet breakup Reλ = Urmsλνc ={56.7, 57.1, 57.1}Wu, Zhouand Lin [5]Channel flows Reτ = Uτνh2={180, 395, 640}Xu et al. [6] (a)vertical plate(b)stationary/oscillatingcylinders(c)flapping foils(d)flexible filaments(e)wavy wall(f)stationary sphere(g)hovering dragonfly(a)Rel =U∞lν= 126(b)ReD = {100, 185}(c)Rec =U∞cν{1, 1.1, 10}×103(d)Rel,f = {200, 103,7.8× 103, 9× 103}(e)Reh = 2.4× 103(f)ReD ={1, 2.5, 3, 5} × 102(g)Re = 2(ϕmax−ϕmin)flwcν= 2.495× 103Zecevic et al. [7] Channel flow Rel = 180Hegele et al. [8] Cavity flow Re = Ulid lν= 5× 1042.2.2 Colour-gradient LBMsThe colour-gradient LBM was introduced by Gunstensen et al. [227] based on thetwo-component lattice-gas model of Rothman and Keller [228], which applies to immis-cible fluids. The colour-gradient LBM applies red and blue particle distributions fRiand fBi , representing the velocity distributions of two fluids. The inherent disadvantagesof the method are susceptibility to Galilean variance and spurious currents, and that352.2. DETERMINISTIC LBMSphase separation requires source term models. Moreover, the accuracy significantly de-cays at high Reynolds numbers and density ratios, which have to be compensated forwith closure models. Lishchuk et al. [229] modelled surface tension at a two-fluid in-terface through a continuum surface-force equation, which significantly reduced but didnot completely eliminate the occurrence of spurious currents. Latva-Kokko and Roth-man [230] proposed a diffusion method for creating symmetric distributions of particlesaround the interface to mitigate problems with lattice-pinning, which occurs when a lat-tice close to an interface experiences inadequate transport of either blue or red particles,so that the interface is artificially “pinned” to the lattice. Newer variants of recolouringschemes have since eliminated these problems. For example, Ba et al. [231] successfullyremoved spurious currents and improved accuracy at high Reynolds numbers and den-sity ratios by decomposing the k-coloured collision parameter Ωki that constitutes themomentum loss or gain in the i-th direction. The decomposition yielded three compo-nents: a multiple-relaxation-time operator(Ωki)(1) which improves stability, a perturba-tion operator(Ωki)(2) that computes high-curvature interfacial forces while eliminatingspurious currents, and a recolouring operator(Ωki)(3) for inferring a discrete immisciblefluid boundary:Ωki =(Ωki)(3) · [(Ωki )(1) + (Ωki )(2)] . (2.20)The multi relaxation time operator(Ωki)(1) (2.21) projects the colour-specific distributionfunction fki from the discrete space to the moment space, i.e. the continuum so thatmki =∑jMijfkj is a vector including stress-tensor components, densities, and otherthermo-physical properties:(Ωki)(1)=∑j(M−1S)ij(mkj −mk,(eq)j)+∑j(M−1)ijCkj . (2.21)S is a diagonal matrix related to fluid viscosity, Ckj are source terms which recover theNavier-Stokes equations through the Chapman-Enskog analysis, and Fs is an interfacialtension force introduced in (2.23). fk′i is the post-perturbation distribution function andfk′′i is the post-segregation distribution function. The perturbation operator(Ωki)(2) in(2.22) generates interfacial tension [232] using a continuum surface-force (CSF) model(2.23), such as those of Halliday et al. [233] and Lishchuk et al. [229], through thecurvature K = − [I − n · n] · ∇ · n, interfacial tension σ, and the phase-field function362.2. DETERMINISTIC LBMSρN =(ρR − ρB) / (ρR + ρB) [227, 234]:(Ωki)(2)= Ak wi(1− ωk2)[3 (ci − u) + 9 (ci · u) ci] · Fs, (2.22)Fs = −12σ K ∇ρN. (2.23)Employing this operator and the CSF effectively reduces the occurrence of spuriouscurrents and the anisotropy of the interface. The recolouring operators formulated byLatva-Kokko and Rothman [230], defined as(ΩRi)(3) (f ′i) ≡ (f ′′i )R = ρRρ f ′i + βwi ρRρBρ cos(φi)|ci|, (2.24)and(ΩBi)(3) (f ′i) ≡ (f ′′i )B = ρBρ f ′i + βwi ρBρRρ cos(φi)|ci|, (2.25)ensure efficient phase segregation and a reasonably sharp interface. ϕi is the angle be-tween the colour gradient ∇ρN and ci, and β is an interface-thickness parameter ∈ [0, 1].It is noted, that even though the combined use of (2.22) and the CSF scheme does notaccurately recover the kinematic condition, it was successfully used to predict multi-species flows in bubble coalescence [235], droplet formation [236], and deformation undershearing [237].Recently, Wen et al. [238] proposed a Galilean invariant 3D colour-gradient LBM withimproved stability for immiscible two-phase flows. The first 3D colour-gradient LBM wasconceived by Tölke [239] with a D3Q19 lattice. Chapman-Enskog analysis of the LBMindicated that the following error terms in k-fluid velocity ∆ukα were produced in thehydrodynamic momentum equations [240]:∆ukα = ∂β{(τ − 12)[13e2 −(cks)2] · [ukβ∂αρk + uα∂βρk + ∂γ (ρkuγ) δαβ]}, (2.26)where e = δx/δt is the lattice speed. In fluids with equal densities (2.26) will disappear,but for dissimilar densities, Galilean invariance is lost and may impair the numericalaccuracy. Wen et al. [238] used an improved equilibrium distribution function fk,(eq)i witha MRT collision operator to eliminate the error and ensure Galilean invariance. Whena circular droplet was simulated in a uniform flow field with a LBM without Galileaninvariance, the droplet morphed into an elliptic shape, while it retained its circular shape372.2. DETERMINISTIC LBMSwith Galilean invariance [238, 241, 242]. Another contribution is from Lee and Ahn [243]who used a combined colour-gradient-LBM and smoothed-profile-method framework tosimulate spinodal decompositions and wetting of solid particles in two-fluid systems.2.2.3 Pseudo-potential LBMsThe pseudo-potential LBM, also known as the interparticle-potential model, Shan-Chen (SC), or Shan-Doolen model, is the simplest and most popular of the multiphasemodels [215, 244]. The primary advantage in using the pseudo-potential LBM is that itautomatically resolves, tracks, and develops phase separation, without interface-trackingclosure algorithms. This is due to the particle-particle interaction (for single speciessystems) being described by a simple force equation [244]. However, Chin [245], Kanget al. [246, 247], and Zhang et al. [248] have reported issues with the scheme not beingexclusively immiscible and not creating sharp interfaces. This is in the nature of theSC model which is a diffuse-interface model—the density is non-discrete across tens oflattices. Given that the interfaces are not tracked, the grid resolution must be uniform,which increases the requirement for the total number of lattices to adequately resolveinterfaces [18]. Moreover, the particle-particle interaction must be relatively strong tomaintain an interface which may incur stability issues [249]. For a multispecies, mul-tiphase fluid system consisting of m components, the SC lattice-Boltzmann equation(LBE) is written routinely with a relaxation scheme, for the k’th species, as [216, 244]fki (x+ δx, t+ δt)− fki (x, t) = −1τk(fki − fk,(eq)i), (2.27)where k ∈ {1, · · · ,m}. In order to determine the streaming process, the equilibriumpopulation for the k’th species was formulated as an arbitrary function of the com-ponent density ρk =∑i fki , equilibrium velocity ck,(eq)i , and the lattice configuration;fk,(eq)i(ρk, ck,(eq)i). The equilibrium velocity distribution was used for determining fk,(eq)i .The velocity distribution (2.28) is dependent on the mixture velocity u defined byρu =∑k ρkuk =∑k∑i cifki , where ρ =∑k ρk is the mixture density:ck,(eq)i (x) = u(x) + τkδtF kρk. (2.28)For m phases, G makes up the m ×m symmetric interaction matrix Gk−k˜ (2.30). Theintensity and direction of the interaction force is inferred through Gk−k˜ so that thenegative and positive signs create attractive and repulsive forces, respectively.382.2. DETERMINISTIC LBMSThe non-monotonic force F k (2.29)—responsible for phase separation—is inferredfrom the interaction between the phase k and its adjacent lattice phase k˜ from theeffective density ρeff and Gk−k˜ asF k(x) = −ρkeff(x)∑k˜Gk−k˜∑iρk˜eff(x+ ciδt) · ci, (2.29)where Green’s function Gk−k˜ is defined by Shan and Chen [215, 244] and Kwok [250] asGk−k˜ =0 |x− x′| > δx,Gk−k˜ |x− x′| = δx. (2.30)The Boltzmann equation is coupled with this interaction force to adjust the particlevelocities. The added benefit of this particular method is that it is stable at high densityratios ρl/ρv ∼ 103 and relatively high Reynolds numbers, as compared to other LBMs.It has been applied to many fluid dynamics problems. Green’s function can be usedto represent fluid-fluid interactions as well as fluid-solid interactions such as wettability[251]. Recently, Reyhanian et al. [252] formulated the force in terms of the Kortewegstresses to represent phase-transitions and surface tension in miscible and immisciblespecies.Yang et al. [253] recently examined droplet morphology by invoking contact anglehysteresis—that is, the difference between the minimum (receding) and maximum (ad-vancing) contact angles of a droplet on a surface—through a modified MRT pseudo-potential LBM with tunable surface tension. With f i = fi−δtF i/2 andΛij =(M−1SM)ij,the relaxed LBE wasf i(x+ δx, t+ δt) = f i(x, t)−Λij[f j − f (eq)j]∣∣∣∣∣(x,t)+(I − Λij2)F i(x, t)δt, (2.31)where the forcing terms were computed with Guo’s force model [216, 254] as below, withthe total force F = FΨ+Fg including the pseudo-potential interaction force FΨ and bodyforces Fg. The relaxation rates in Λ are normally calculated through the thermophysicalproperties of the working fluid. Writing the collision process from (2.31) to the momentspace through the linear transformation matrix M yielded:m⋆ =m− S(m−m(eq))+(I − S2)δtF . (2.32)392.2. DETERMINISTIC LBMSHere, the superscript “⋆” denotes the state at (x, t), and m =Mfi,m(eq) =Mf(eq)i andF =M F. Following the collision process, the streaming process prescribes f i(x+ ciδt,t + δt)= f⋆i , where f⋆i = M−1m⋆. The flow properties in the macro perspective werethen ρ =∑i f i and ρu =∑i f ici +δtF2 . In order to simulate advancing and recedingcontact angles, the collision process was adjusted to:(2.32)+ δt ·C. (2.33)Yang et al. [253] used the correction term δt ·C for which the D3Q19 lattice model wasmodelled with inspiration from Li and Luo [255] asC =[0,765sν(Qxx +Qyy +Qzz), 0, 0, 0, 0, 0, 0, 0,−sν(2Qxx −Qyy −Qzz), 0,−sν(Qyy −Qzz), 0,sνQxy,−sνQyz,−sνQxz, 0, 0, 0]−1,where the tensor Q with the surface tension adjustment parameter κ isQ = κG2ψ(x)∑iw(|ci|2)[ψ(x+ ciδt)− ψ(x)]ci · ci. (2.34)This reformulation of the pseudo-potential framework gave rise to simulation of advancingand receding contact angles in hysteresis windows.The pseudo-potential LBM has become the popular choice for simulating pool boilingprocesses due to the natural interface treatment without the need for interface trackingalgorithms. Recently Zhou et al. [256] used a two-population (often referred to as double-distribution function) pseudo-potential LBM similar to the one described above with thePeng-Robinson EOS to simulate the nucleate heat transfer on square micro-pillar surfaces[256], and square, circular, trapezoidal and inverted trapezoidal cavities [257]. Zhou et al.[257] found that arranging a grid of micro-pillars created nucleation sites between pairsof four pillars, whereas a smooth surface at the same thermophysical state would notexperience nucleation. The nucleate heat transfer reported for the cavity boiling cases isall attributed to microlayer evaporation. Other studies on pool boiling also exist [258–260], but to the authors’ knowledge, not concerning cryogenic fluids. Other authors haveused the pseudo-potential model for simulating droplet dynamics [261], and cavitation[19, 124, 147, 262].The accuracy of the two-population pseudo-potential models in predicting heat trans-402.2. DETERMINISTIC LBMSfer have rarely been documented. Hu et al. [263] provided a comprehensive evaluationof the accuracy in the most widely used thermal models for correlating the macroscopicvariables, including T , to the equilibrium populations f (eq)i and g(eq)i . These are theinternal-energy-based (ITLB), temperature-based (TTLB), and simplified temperature-based (STTLB) thermal lattice-Boltzmann models, respectively, with equilibria definedasg(eq)i = cpTf(eq)i (ITLB), (2.35)g(eq)i =Tf(eq)iρ(TTLB), (2.36)g(eq)i = wiT (x, t)[1 +ci · c(eq)ς2s](STTLB). (2.37)Hu et al. [263] compared the thermal LB models to a finite-difference simulation of natu-ral convection in a cavity and static phase change in a 1D channel. They found that thethermal LB models contained non-trivial errors. Unfortunately, no experimental valida-tion was performed; the numerical simulations were merely compared to the numericalheat transfer model coupled with the k-ε turbulence model of Barakos et al. [264]. Thus,no clear benchmark was provided to establish the true physical errors in the simulations.Likely, the computationally most demanding pseudo-potential-LBM simulation ofnucleate pool boiling in water was conducted by Fei et al. [17] in which a single populationmodel was coupled to a temperature equation with finite-differences in a cascaded LBM.The authors correlated the interaction forces with a Peng-Robinson equation of staterepresentative of homogeneous nucleation. While the simulations (illustrated in Fig.2.5) pushed the computational bounds of the LBM solid-fluid interaction forces incidentat the boiling substrate were not accounted for.2.2.4 Free-energy LBMsProposed by Orlandini et al. [265], and Swift et al. [266, 267], the free-energy LBMis purely rooted in thermodynamics, so that the particle collision parameter is based onthe free energy functional Ψ. For one component non-ideal fluids, the Helmholtz freeenergy functional Ψ (2.38), is written as a function of the bulk free energy ψ, and thefree energy correlated to surface tension ψκ = κ/2(∇ρ (x))2 [216, 266–268],Ψ(x) =∫ [ψ (T (x) , ρ (x)) +κ2(∇ρ (x))2]dx. (2.38)412.2. DETERMINISTIC LBMSFigure 2.5: Simulation snapshot at 2× 104δt of nucleate boiling in water in a 6002 × 300computational domain with a cascaded pseudo-potential LBM [17]© American Institute of Physics2020.The capillary coefficient κ gives rise to a changing energy balance and interfacial tensionin areas of high density gradients near triple junctions. From the van der Waals EOS,the non-local pressure is [266]p (x) = ρδΨδρ−Ψ = p0 − κρ∇2ρ− κ2|∇ρ|2 , (2.39)wherep0 = ρ · ∇Ψ(ρ)−Ψ(x) . (2.40)The thermodynamic pressure tensor P˜ (x) is then correlated with the mixed partialderivative of the density and the surface tension as [266]P˜αβ(x) = p(x)δαβ + κ∂αρ∂ρ∂xβ. (2.41)It is noted that P˜ excludes the shear stress tensor, so that the tensor representing thetotal pressure is P = P˜ + τ . Then the LBE for the BGK collision model is [266]fi (x+ ciδt, t+ δt)− fi (x, t) = −1τ[fi − f (eq)i (ρ, c,∇ρ)], (2.42)where f (eq)i is formulated so as to be thermodynamically consistent and upon performinga Chapman-Enskog analysis, will recover the macroscopic equations in the hydrodynamic422.2. DETERMINISTIC LBMSlimit with insignificant error [266] given the equilibrium moments,∑if(eq)i = ρ,∑if(eq)i ci = ρu,∑ici,αci,βf(eq)i = P˜αβ + ρuαuβ. (2.43)For binary fluid systems, Swift et al. [267] introduced a free energy form based ontwo lattice populations fi and gi,Ψ =∑[ψ (ρ, T, φ) +κ2|∇ρ|2 + κ2|∇φ|2]dx, (2.44)where the bulk free energy density ψ is defined as a functional of the density and thesum of the higher order moment φ =∑i g(eq)i including the binary interaction strengthG [267]:ψ =RT2(ρ− φ) ln(ρ− φ2)+G4ρ(1− φ2ρ2)− ρRT + RT2(ρ+ φ) ln(ρ+ φ2).(2.45)In order to reproduce the Navier-Stokes equations the following equilibrium constraintsfor gi must be obeyed [267]: ∑ig(eq)i = φ,∑icig(eq)i = φu,∑ci,αci,βg(eq)i =Mi∆µδαβ + φuαuβ. (2.46)Herein, Mi is the fluid “mobility” yet to be introduced in (2.55). From the free energydensity defined in (2.45), the chemical potential ∆µ between the pertinent componentsand the pressure tensor P˜ are derived as [267]∆µ = −Gφ2ρ+RT2ln(1 + φ/ρ1− φ/ρ)− κ∇2φ (2.47)andP˜ (x) = p(x)δαβ + κ∂αρ∂βρ+ κ∂αφ∂βφ. (2.48)432.2. DETERMINISTIC LBMSThe non-local pressure is then determined as [267]:p(x) = ρRT − κ (ρ∇2ρ+ φ∇2φ)− κ2(|∇ρ|2 + |∇φ|2). (2.49)As with the original colour-gradient method, the original free-energy LBM lackedGalilean invariance. However, a handful of studies have for certain DmQn lattice modelsmitigated the adverse effects from the variance. Notably, Inamuro et al. [241], Kalarakiset al. [242], Swift et al. [267], Zheng et al. [269, 270] improved the triangular lattice Frischet al. [271] (FHP)-model and the D2Q9, D2Q5 lattice models. Similarly to Swift et al.[266], Wen et al. [272] recently modified a chemical-potential based free-energy LBMthrough manipulation of the pressure tensor P . Instead of only exerting (2.41) on themomentum through (2.43), a non-ideal force was constructed and augmented by an idealgas pressure tensor P (0) = c2sρI coupled with the Redlich-Kwong and the Peng-RobinsonEOS, yieldingF (x) = −∇ · P +∇ · P (0), ∇ · P = ρ∇µ. (2.50)This formulation is thermodynamically consistent and Galilean invariant [273], with theforce computed through chemical potentials. Because the pressure tensor is not computeddirectly via (2.41), as was done in previous studies [241, 242, 267, 269, 270], computationalefficiency improved and lower spurious currents were present. Zheng et al. [270], Inamuroet al. [274] modified the model to extend the applicability to high-density ratios, whileWen et al. [272] also developed a chemical potential BC to simulate the wettability ofsolid surfaces.Recently, Wöhrwag et al. [275] proposed a thermodynamically consistent hybrid free-energy, entropic LBM for ternary, immiscible fluids with large density ratios of the orderO(103), which subsequently was extended by Bala et al. [276] to wetting boundariesincident at solid walls with varying geometric and forcing schemes. Therein, the colli-sions of particles were correlated with the production of entropy when a particular fluidtends towards a thermodynamic equilibrium state. Approaches prior to Wöhrwag et al.[275] were thermodynamically inconsistent [215, 277], and could not handle significantdensity ratios [278, 279]. Moreover, the entropic LBM proposed by Wöhrwag et al. [275]accommodated a broad spectrum of interfacial and surface-tension ratios enabling thesimulation of partial wetting states in Neumann triangles in liquid lenses, through fullwetting with complete fluid encapsulation. The interfacial tension in the model was flexi-bly tuned by simulating a liquid lens and varying the Neumann angles until the existence442.2. DETERMINISTIC LBMScurves predicted by the model matched theoretical solutions. Wöhrwag et al. [275] in-cluded double-well potentials to model two or more fluids, whether gaseous or liquid. Thepotentials were analogous to those introduced in the original free-energy model proposedby Swift et al. [267].A study of the optimal EOS for the entropic LBM was also carried out by Wöhrwaget al. [275]. A Helmholtz free-energy volumetric functional Ψ was formulated that de-composed the free energy into bulk and interfacial contributions:Ψ =∫ψB + ψI dx. (2.51)The bulk free energy ψB was defined as the sum of potentials for one gas and two liquids,expressed asψB =λ12[ψEOS (ρ)− ψ0] (gas) (2.52)+λ22· C2l,1 (1− Cl,1)2︸ ︷︷ ︸liquid 1+λ32· C2l,2 (1− Cl,2)2︸ ︷︷ ︸liquid 2,where the potentials for the liquids were of the double-well form. A thorough study onthe bulk free energy with various EOS was performed by Boyer and Lapuerta [280] withthe Cahn-Hillard model. Their results were for three liquid potentials. Similarly to theoriginal free-energy formulation of Swift et al. [266, 267], interfacial forces were modelledasψI =κ12(∇ρ)2 + κ22(∇Cl,1)2 + κ32(∇Cl,2)2 , (2.53)where λ and κ were relaxation parameters, Cl the liquid concentration, and ψ0 thereference state free energy. The free energy ψEOS was predicted by the Carnahan-Starling(CS) EOS [249],ψEOS = ρ[C − aρ− 8RT (−6 + bρ)(−4 + bρ)2 +RT log (ρ)]. (2.54)in which C, a, and b are fluid-critical constants. Wöhrwag et al. [275] noted that thePeng-Robinson and van der Walls EOS also can be implemented relatively effortlessly.Yuan and Schaefer [249] evaluated existence curves for various EOS including the CS,Peng-Robinson, van der Walls, Shan-Chen, Redlich-Kwong, Redlich-Kwong-Soave andcompared them to the Maxwell construction. Yuan and Schaefer [249] noted that the CS452.2. DETERMINISTIC LBMSEOS is more generic as it has no free parameters, which may lead to inferior predictive ac-curacy for some fluids in comparison to other EOS, despite being defined from coexistencedata. For cryogenic liquid oxygen, Orr et al. [18] reported that the CS EOS over-predictedpressure. So, the accuracy of the Shan-Chen LBM is subject to the ability of the EOSin reproducing empirical data—it is important to validate an EOS for the specific fluidto be simulated to ensure sufficient solution accuracy. The fourth-order polynomial bulkfree energy from Boyer and Lapuerta [280] was used to determine the chemical potentialsµρ = δΨ/δρ∣∣T,ϕ, µϕ = δΨ/δϕ∣∣T,ρand the pressure tensor ∇·P = ρ∇µρ+ϕ∇µϕ. In orderto solve the equations of motion at a solid-fluid boundary, the Cahn-Hillard wall-collisionmodel (2.55) was employed:∂ϕk∂t+∇ · (ϕku) =Mk∇2µϕk . (2.55)The authors assumed a value for the fluid mobility Mk constant when it is in realitytightly coupled with the velocity field [281]. The interparticle collisions were solvedroutinely via the popular BGK collision operator.2.2.5 Phase-field LBMsRepresenting interfaces in ternary fluid systems is an elusive task requiring computa-tionally intensive interface-tracking models. The most common interface theories that areused in LBM are the Cahn-Hillard (2.55) and Allen-Cahn models [282, 283]. The Allen-Cahn model includes second-order derivatives which makes it more efficient, althoughthe use of a fourth-order approximation of the diffusion term makes it susceptible todivergence [284, 285]. The Cahn-Hillard models advantage is its conservative featuresin comparison to the original Allen-Cahn model, although recent efforts have producedconservative versions [286].One of the latest phase-field LBMs was formulated by Abadi et al. [281]. The authorsextended the tracking algorithm of Geier et al. [287] for a two-fluid interface to ternaryfluid systems by deriving the equilibrium state volume fraction |ϕk|(eq) in (2.56) from theEuler-Lagrange (EL) differential in (2.57). The EL differential equation correlates thebulk free energy Φ, the capillary force term ψ, and the chemical potential in the k-thphase fraction ϕk, which were introduced by Boyer and Lapuerta [280]:|ϕk|(eq) = 4ξ· |ϕk (1− ϕk)| , (2.56)∂Φ∂ϕk−∇ · ∂ψ∂ϕk− µϕk = 0. (2.57)462.2. DETERMINISTIC LBMSThe advection of phase k was subsequently modelled through the following conservationequation:∂ϕk∂t+∇ · (uϕk) = ∇Mk(∇ϕk − nk|∇ϕk|(eq) + 133∑k=1nk|∇ϕk|(eq)). (2.58)The mobility of a phase Mk—a measure of the efficacy with which one phase diffusesinto a neighboring phase, as a function of boundary curvature and interfacial tension—istightly coupled to the phase-field distribution of mass, as well as momentum transfer.Significant effort was invested in formulating mobility relations [288–291]. The mobility isespecially dependent on density and the phase-field, augmenting non-linearities. Closureis achieved through a ternary formulation of the interface-tracking LBE formulated byGeier et al. [287], Fakhari et al. [292], with a D2Q9 lattice configuration. Moreover,hydrodynamic stability is enforced through an LBE which accommodates high densityand viscosity ratios, which reduced spurious currents to below 10−7 lattice units. Sadeghiand Shadloo [293] used this phase-field LBM with the Cahn-Hillard interface-trackingmodel to simulate single and multimode film boiling in agreement with the Berenson[41] and Klimenko [294] correlations. In addition to this, the method has been usedto simulate a wide array of problems. Ma et al. [295] proposed a spectral-differencephase-field LBM for incompressible two-fluid systems and used it to simulate rigid-bodyrotation of Zalesak’s disk, layered Poiseuille flows, bubble deformation caused by shear,Rayleigh-Taylor instabilities, and bubble coalescence. Kalantarpour et al. [296] employeda ternary phase-field LBM to simulate static and dynamic bubble interaction at a liquid-liquid interface, Rayleigh-Taylor instabilities, as well as coalescence of a gas bubble risingin a column of water with an oil phase. Fields of bubbles in turbulent water-air pipeflows were simulated in the direct numerical simulations of Sitompul et al. [297]. Thephase-field LBM has also been accelerated to GPUs by Alpak et al. [298] to investigatetransient binary flows in capillary tubes as well as flows in various geological formations.Moreover, the LBM is often used to simulate the convection and growth of dendriticnetworks of solidifying alloys [299–301].As an alternative to the previous multiphase LBMs with phase-change Reyhanianet al. [252, 302] recently formulated a kinetic theory which uses adaptive velocities (2.59),comprising both the thermodynamic velocity and the velocity of a translating referenceframe, in the streaming and collision processes:vi =√pρRTLci + u. (2.59)472.2. DETERMINISTIC LBMSTherein, p/(ρRTL) is the reduced pressure relative to the ideal gas pressure manifested bythe underlying lattice temperature TL. For thermal models in which not only hydrody-namic behavior is important, but other thermophysical phenomena such as phase-changeare relevant, thermodynamic consistency implies that the model possesses a well-definedtemperature and specific entropy across the simulated physical regimes [303, 304]. Nat-urally, this relies on the accurate coupling and concurrent evolution of hydrodynamicand thermodynamic behavior. As a consequence, thermodynamic consistency is oftenassociated with a correct implementation of equations of state (EOS). The kinetic theoryof Reyhanian et al. [252] is thermodynamically consistent and eliminated issues relatedto non-satisfaction of Galilean invariance and allowed for very stable simulations of com-pressible flows as well as phase-change processes.2.2.6 Lattice-Boltzmann model applications in cryogenic fluidssystemsThis section highlights applications of the LBM to cryogenic fluid systems. The stud-ies that have used LBMs to simulate cryogenic phase-change, summarized in Tab. 2.4, arefew despite the success with which the LBM has been applied to other fluids. In the ear-lier days of LBM maturity, Tatsumi and Nishijima [305] used the colour-gradient LBMto investigate the stability of superconductivity in a wire submerged in liquid Helium(LHe2). In more recent years, Orr et al. [18] reported their development of a pseudo-potential LBM for spacecraft propellant slosh simulation in micro-gravity environments.In space, low Bond numbers Bo =(ρgL2c/σ)are useful for characterizing the hy-drodynamic behaviour of multiphase cryogenic flows—such as the oxidizer liquid oxygen(LO2) simulated by Orr et al. [18]—due to the low gravitational acceleration relative tocapillary forces. It was crucial to be able to alter gravitational effects, so they incorpo-rated body forces into the LBE following the exact-difference method of Kupershtokh[312]. This method applies body forces simultaneously to the advection step and theequilibrium population where the subsequent shift in velocity due to the forcing is com-puted as ∆u = F/ρ, where F is the momentum density, which equates to g if thegravitational field is uniform. The lattice velocity distributions are shifted to a modifiedvelocity u⋆ which is subsequently used to compute the collision process. The authors re-ported that the equilibrium populations must be updated at least twice during collision.482.2. DETERMINISTIC LBMSTable 2.4: Overview of published works on cryogenic fluids using colour-gradient (CG),pseudo-potential (PP), and finite-difference (FD) multi-speed LBMs.Ref.LBMCasesConditions[305]CGNaturalconvectionaroundsuperconductingwireSubcooled,LHe 2,thermalresponseandconvectivestabilitytovaryingtrans-portcurrents,dynamicviscosityµ=5×10−6Pa·s[306]PPHeterogeneousnucleationContinuationof“(a)”withphasetransition[307]PPSpinodalestimationLennard-JonesArgonat128K,liquidfilminvapour,gravitation-free,accompaniedbymoleculardynamicssimulations[18]PPVaryingwetting(i.)Lateralsloshingintank,frequencyanalysis(ii.)(i.):LO2,cornergeometry,negative(hydrophobic),neutral,andpositive(hydrophilic)wallinteractionpotentials,contactangleθ<π/2(ii.):LO2,surfacetensionσ=0.0122N/m,Bond/EötvösnumberBo=∆ρgL2/σ∼20,acc.g=0.001·g0[308]FDLid-drivencavity(i.)Riemannshocktube(ii.)Saturatedfilmboiling(iii.)(i.):Gaseousnitrogenat114Kand0.306MPa,laminar,incompressibleflow,Re<5000,ideal-gasmodel(ii.):Supercritical/gaseousoxygen,ρt=0∈[43.6,170]( kg/m3) ,T t=0∈[139,170](K),Ma∈[1.118,1.139],ideal-gasmodel(iii.):LO2,Tt=0=Tsat=139K,ρ=723kg/m3,1.2MPa,∆T∈[5,62][19]PPValidationstudy(i.)Heterogeneouscavitation-bubblenucleation(ii.)Cav.-bubblecollapsenearsolidsurface(iii.)(i.):H2O,LN2,LH2dropletssuspendedinvapour,verificationwithLaplacelawandtheMaxwellconstruction,validationfromNIST[309](ii.):H2O,LN2,LH2,ρl/ρv∈[13,112],phase-transitioninceptionwiththeenergy-barrierequation[310],verificationfromtheRayleigh-Plessetequation[311](iii.):H2O,LN2,LH2,collapsetimesensitivitytovaryingbubble-to-walldistanceandcontactangleθ={30◦ ,90◦ ,150◦ }492.2. DETERMINISTIC LBMSThe (n+1)’th time-step was equated to the below, where (pc) is the post-collision state:f [n+ 1] =(I−M−1SM) (f (pc)[n]− f (eq) (x,u, ρ))+ f (eq) (x,u⋆, ρ) . (2.60)With the discretized LBE fi (x+ δx, t+ δt) = fi (x, t)−Λ[fi (x, t)−f (eq)i (x,u, ρ)], thecollision matrix Λ expands toΛαβ = Λ0 + ωαβ∆Λαβ =M−1 (S0 +∆Sαβ)M, (2.61)where S0 is the hydrodynamic part of the relaxation matrix. The matrix ∆Λαβ onlyrelaxed the diagonal and off-diagonal entries of the pressure tensor Pαα and Pαβ , re-spectively. With ∆f [n] = f (pc)[n] − f (eq) (x,u⋆, ρ), the collision equation for the MRTscheme with body forcing isf [n+ 1] = (I−Λ0)∆f [n] + f (eq) (x,u⋆, ρ)− ωαβ∆Λαβ∆f [n]. (2.62)Crucial to the simulation of spacecraft dynamics is the total force acting on the entirefluid domain, which was computed as the difference between body forces and internalforces (in terms of momentum density):Fd[n] =∑αβFαβ −∑αβραβUαβ[n]−∑αβραβUαβ[n− 1] . (2.63)Orr et al. [18] used the above approach to compute the lateral sloshing of LO2 in anasymmetrical tank with a gravitational field of g = 0.001g0 at 15◦ in relation to theaxial direction. Their 2D solution at t = 9.8 s is depicted in Fig. 2.6 which shows thevelocity field in lattice units. For spacecraft dynamics in micro-gravity environments,convection caused by gradients in interfacial tension is characterised through the Bondnumber, Bo = ρgL2c/σ. The primary mode oscillated with a frequency of O(10−2)Hz with a modest lateral force of O(10−4)N. The simulation showed strong advectivepatterns in the vapour phase in the form of rotational structures emanating from thetriple junctions at the tank walls. The interface behaviour at the triple junctions suggestthat the wettability of the wall produces an apparent attractive effect at the pertaininglow Bond number of the problem.Traudt and Schlechtriem [308] used a van der Waals-based finite-difference LBM tosimulate transient two-phase flow of cryogenic propellants in rocket engine feed systems,including test cases on single LO2 bubble evaporation and the Riemann shock problem.502.2. DETERMINISTIC LBMSLO2InterfaceFigure 2.6: Instantaneous velocity contour of LO2 propellant undergoing lateral sloshing in a tank.Solution from an isothermal simulation with the pseudo-potential method. Reproduced from Orr et al.[18].Their model showed the capability to resolve shocks and supersonic flow. In the designof rocket engines and their propellant feeding systems, the complexity of valve config-urations and timing upon start-up incur high costs in prototyping, as the integrity ofthe engine is highly prone to damage from the timing of said valves. The fuel and ox-idizer are most often cryogenic and cavitate when the valves open, and also experienceheterogeneous nucleation when contacting the solid engine parts. Moreover, after cool-ing the rocket nozzle and being pressurized in turbopumps, the cryogens may becomesupercritical. Thus, numerical methods for simulating the transient start-up processesare required to be stable at high Reynolds numbers, compressible, sub- and supercriticalmultiphase flows of non-ideal fluids, in complex geometries.Traudt and Schlechtriem [308] used the multi-speed model of Watari and Tsutahara[313] which has proven to be accurate for supersonic flow simulations [314], with anextension for simulating non-ideal van der Waals fluids [315]. The real-gas model ofGonnella et al. [315] implemented a force term similar to the pseudo-potential LBMthat emulates the interaction forces between molecules to simulate phase transitions andcontinuous density gradients across interfaces. Since the lattice speeds are lattice-speedindependent, a finite difference discretization was used with a van-Leer limiter in the flux512.2. DETERMINISTIC LBMSF¨ at the half-way distance j ± 1/2 to the neighboring node j:fi(n+ 1) = fi(n)− ckδtwiδx[F¨j+1/2i − F¨ j−1/2i](n)− δtτ[fi(n)− f (eq)i](n)− δtFi, (2.64)whereF¨j+1/2i = fi +12(1− ckδtwiδx)[f j+1i − fi](n)Ψ˜ (Θ) (2.65)and Θji is a smoothness function and Ψ˜ is a flux delimiter:Θji =f ji − f j−1if j+1i − f ji, (2.66)Ψ˜ =0, Θji ≤ 0,2Θji1+Θji, Θji > 0. (2.67)The multi-speed model produced no error through the Chapman-Enskog expansion andimproved stability in multiphase, compressible flows. The van der Waals EOS containsderivatives that Traudt and Schlechtriem [308] computed with a fourth-order accuracythrough a nine-point discretization. The authors’ simulations comprised a Riemann shockproblem with LO2. In comparison to the analytical solution of Wagner [316], predictionsfor the velocities, pressures, temperatures, and densities for four different simulationsincurred < 1% error. Traudt and Schlechtriem [308] also simulated the film boiling ofLO2 with good agreement with experiments.The comparative study of Peng et al. [317] found that of the velocity shifting method(VSM), exact difference method (EDM) and Guo’s force schemes, the VSM and EDMallowed for the highest density ratios. Thus, Ezzatneshan and Vaseghnia [19] employedthe VSM and EDM in a pseudo-potential LBM for studying the nucleation and collapseof cavitation bubbles in water, LN2, and LH2. In the VSM the hydrodynamic velocity isshifted by the sum of the pertinent body forces (e.g. interaction, gravitational, fluid-solid)F so that the velocity used in the equilibrium distribution isu⋆ = u+τFρ, (2.68)and the real velocity field is redefined asu˜ = u+F δt2ρ. (2.69)522.2. DETERMINISTIC LBMSIn the EDM, the body forces are directly added to the RHS of the LBE:fi(x+ ciδt, t+ δt)− fi(x, t) = −δtτ[fi − f (eq)i (ρ,u)]+ f(eq)i (ρ,u,∆u)− f (eq)i (ρ,u),(2.70)where ∆u = F δt/ρ. Ezzatneshan and Vaseghnia [19] conducted a sensitivity analy-sis of the bubble nucleation/collapse dynamics with varying vapour-to-liquid pressuredifferences ∆p = pl − pv via an energy-barrier equation,∆E = 4πR2σ +43πR3∆p, (2.71)relating the total system energy to the cavity radius R and the surface tension σ. Theinequality of this equation leads to phase-transition, but the equality preserves any liquid-vapour pair in an equilibrium state where a critical pressure difference ∆pc = −σ/R (theLaplace law) is unique for a fluid—i.e. ∆pc = {−0.0090285,−0.0063359,−0.0090285}(lattice-units) for H2O, LN2, and LH2, respectively [19]. The authors were able to verifythe growth dynamics from the Rayleigh-Plesset equation [311] and investigated the nu-cleation of a cavity in a liquid filled with micro-bubbles, as well as the collapse of a cavitynear a solid wall with varying wettability and wall-to-cavity distance. The authors foundthat the collapse of the water cavity, in comparison to the cryogens, was more aggressive(as indicated in Fig. 2.7), in agreement with the experiments of Dular and Petkovšek[318]. The distinct collapsing behaviour was owed to the implementation of the acentricfactor in the authors LBM which is a measure of the molecular structure and polarity ina fluid, where a low acentricity is associated with spherical molecules with low polarity[319].A stable liquid is bounded by its equilibrium curve between its freezing and boil-ing points, but outside of this in the metastable region the liquid will become unsta-ble and prefer a new thermodynamic state. However, beyond the equilibrium with ei-ther superheating or subcooling, substances can exist in their liquid state under certainconditions—i.e. if no impurities exist and the system is not disturbed by external forces[320]. However, the metastable region is similarly bounded by stability limits in either ofthe directions of superheating and subcooling, beyond which vapours and liquids cannotexist [307]. Such a limit is termed a “spinodal” and these are impossible to determineexperimentally [307]—a full elimination of impurities and disturbances in a given sub-stance is impractical to achieve. Accordingly, Imre et al. [307] simulated a liquid filmsurrounded by vapour in a zero-gravity environment, with a pseudo-potential LBM (and532.2. DETERMINISTIC LBMSFigure 2.7: Collapse of H2O, LN2, and LH2 cavities near a solid wall by Ezzatneshan and Vaseghnia[19]. Reproduced with permission from Physical Review E 102, 053309 (2020)© 2020 AmericanPhysical Society.a molecular-dynamics simulation). They found that the employed equation of state [215]in the LBM—extended deep into the metastable extremes—was able to reproduce theliquid spinodal for argon, derived from experimental results for interfacial thickness andsurface tension. The authors’ simulations are useful for further development of EOSapplicable to metastable systems.Modelling accidental spills of cryogenic fluids on water is relevant to shipping of LNGand its use as a marine fuel. Accurate models of cryogenic spills require elucidatingthe heat transfer characteristics of the ice formed in water following contact with thecryogenic LNG, as well as a better understanding of how water turbulence affects thefreezing rate and overall heat transfer between the two fluids. Recently, Li et al. [20]proposed a pseudo-potential LBM with a concurrent evaporation, condensation, andfreezing phase-change model based on enthalpy. The authors employed two populationsfor momentum (2.72) and energy (2.75), to resolve the velocity, pressure, and temperaturefields:fi(n+ 1)− fi(n) = −1−Bτf[fi − f (eq)i]+BΩsi + (1−B)∆fi. (2.72)542.2. DETERMINISTIC LBMSΩsi = fi− − fi + f (eq)i (ρ,us) ,−f (eq)i (ρ,u) is a bounce-back collision term for the non-equilibrium part of f with us being the velocity of the solid. The f -population wasweighted by B which relaxes the collision process as a function of the relaxation time τfand the solid volume fraction fs:B =fs (τf − 1/2)1/2− fs + τf . (2.73)The solid volume fraction is unity for an iced phase and zero for liquid, and 0 < fs < 1for multiple phases including vapour, and was determined based on enthalpy [251]:fs =0, h > hl,h−hshl−hs , hs < h < hl,1, h < hs.(2.74)The population corresponding to energy is a function of two source terms ϕvl and ϕls:gi(n+ 1)− gi(n) = − 1τT[gi − g(eq)i]+ δtwi (ϕvl + ϕls) . (2.75)The algorithm computed the freezing phase change by computing ϕvl and ϕls prior to thecollision step, and enthalpy and fs after computing the macroscopic variables. In termsof Fourier number, Fo = αst/l20, Li et al. [20] simulated a cryogenic spot with watervapour undergoing condensation and freezing phase change. The phase morphology pre-dicted by Li et al. for a droplet solidifying on a vertical substrate is depicted in Fig. 2.8.In agreement with Yang et al. [253], Li et al. noted that surface wettability affects themorphology of liquid droplets but not the growth of ice under concurrent evaporation,condensation, and freezing phase-change. The wettability was inferred through the bodyforces F = FΨ + Fs + Fg with Fs representing fluid-solid interaction. Preceding thiswork, Sun et al. [251] developed a LBM for modelling the freezing phase-change heattransfer within a droplet as well as surface frost formation on a horizontal surface fromair humidity. To resolve the frost layer, they adopted porous media permeability theoryso that the fluid streams according to f ′i (x+ δx, t+ δt) = fi (x, t) when fs = 0, the pop-ulations remained stationary when fs = 1, and if 0 < fs < 1 the populations contained asolid-fluid mixture and only a fraction of fi streams with a modified streaming equation:f ′i (x+ δx, t+ δt) = (1− fs) fi (x, t) + fsfi (x+ δx) . (2.76)552.3. SUMMARYFigure 2.8: Freezing droplet morphology as reported in the paper by Li et al. [20]. The ice phase ispurple and the liquid phase blue. Reproduced with permission from International Journal of Heat andMass Transfer 114, 628-639 (2017). © 2017 Elsevier.In addition, Zhao et al. [321] simulated the impact dynamics of a droplet on freezingsurfaces. The importance of concurrent evaporation, condensation, and freezing phase-change in cryogenic spills is exemplified in the fact that the cryogen comes in contactwith water in ocean spills in which an ice film will interact with both the ambient humidair and water phases. Furthermore, the subsequent dispersion of the cold vapour triggersthe condensation of the unsaturated vapour in the air.2.3 SummaryOn basis of the reviewed approaches to modelling phase change it is concluded thatthe majority of studies are more pertinent to engineering science than the fundamentalsof phase change; predominantly, phase-change is interpreted to comprise a mass-transferprocess in multi-component fluids, and to a lesser extent one driven by non-equilibriumthermodynamics and infinitesimally micro-length-scale and rapid time-scale fluid dynam-ics. Especially this span of scales in CFD makes rigorous phase-change modelling elusive.562.4. FLUCTUATING LBMSIn “conventional” CFD approaches within the finite-volume method, very few studiesconsider heterogeneity and free energy in simulated flows, and fewer still investigate theeffects of thermal fluctuations and metastability on phase-transition rates. These areall concepts central to the dynamics of phase transitions. Studies conducted using theLBM were found to be more rigorous owing to their statistical mechanics roots whichallow for more versatility when considering non-equilibrium effects. Particularly, thepseudo-potential and free-energy-based LBMs have shown promise for predicting phasetransitions in mesoscale simulations. However, even in the LBM which is more commonlyused in the computational-physics community, more effort is needed for concurrently im-posing all of the above phenomena into a centralized modelling framework. At thecurrent state of computing and mathematical modelling, this is a quite complicated task.The focus can now be turned from thermodynamics towards how to manifest numericalthermal noise. In the following, a review is presented of fluctuating LBMs which arethe kinetic-theory equivalent to fluctuating-hydrodynamics theory used in Navier-Stokessolvers.2.4 Fluctuating lattice-Boltzmann models of non-idealfluidsThe concept of including thermal-noise in macroscopic simulations is the fundamentof fluctuating-hydrodynamics theory (FHT) conceived by Landau and Lifshitz [322]. Theauthors introduced classical fluctuations (i.e. with non-quantum oscillation frequenciesω ≪ kBT/ℏ) into macroscopic observables of the hydrodynamic equations by a zero-mean Gaussian random “outside” stress tensor τ˜αβ in the Navier-Stokes equation, and anoutside heat flow vector q˜ in the conduction equation q = −k∇T+q˜, both being mutuallyuncorrelated. The cornerstone of the FHT are the spatio-temporally ensemble-averaged,continuum quantities,〈τ˜(x, t)τ˜(x′, t′)〉= Aαβγδδ(x− x′)δ(t− t′), (2.77)〈q˜(x, t)q˜(x′, t′)〉= Aαβδ(x− x′)δ(t− t′), (2.78)which are tightly correlated with the rate of change of total entropy [92] through thevariance tensors Aαβγδ, Aαβ . The averages are seminal and still derived today with thefluctuation-dissipation theorem (FDT) in stochastic models as they inherently validatethe consistency of modelled fluctuations. In recent works by Magaletti et al. [14], Gallo[40], Gallo et al. [192, 195] the authors combined the theory with Navier-Stokes simula-572.4. FLUCTUATING LBMStions of boiling and cavitation using the finite-volume method. Alternatively to imposingthe fluctuations on the macroscale their introduction into kinetic theories was exploredstarting in the 1990’s, specifically in the fluctuating lattice-Boltzmann method (FLBM)[210] which by virtue of its molecular roots can be extended to include the interactionforces in liquids that ultimately are responsible for thermal fluctuations.FLBM research originated with the work by Ladd [323] who proposed a fluctuat-ing lattice-Boltzmann model for colloidal suspensions. The Brownian motion of thesuspension was manifested by separately tracking the Newtonian dynamics of solid par-ticles and incorporating the thermal fluctuations directly into the governing equationsfor an isothermal immersion liquid. Using the same FHT of Landau and Lifshitz [91],Ladd added coarse-grained fluctuations exclusively to the fluid-stress tensor for a single-population lattice-Boltzmann equation where fluctuating populations were designed soas to obey the FDT (2.77). The computation of the fluctuating stresses were mediatedby solving a discrete Langevin-type equation (the Langevin equation will be detailedmomentarily in §2.5).Adhikari et al. [324] extended the work of Ladd [323] with noise satisfying a FDTdirectly at the lattice level such that the fluctuations in mass, momentum, and stresseswere consistently mechanically equilibrated for all wave vectors k ≡ k[cos(θ), sin(θ)]T .The authors in addition to the hydrodynamic observables (referred to as modes) notedthe existence of local, non-hydrodynamic “ghost modes” [325, 326] which must be ac-counted for in the FDT and models with single as well as multi-relaxation time kernelsto maintain hydrodynamic isotropy and Galilean invariance. If unaccounted for, thesemodes may drain thermal fluctuations from the hydrodynamic modes. In simulationsthe computational overhead associated with the ghost modes was reported to be ∼ 10%in runtime. The noise in the mentioned degrees of freedom was imposed by a set ofcoupled Langevin equations evolving discretely in time to predict the fluctuating den-sities ξi in their single-population FLB model. The model was numerically realized bycomputing the covariance matrix of n − D − 1 noises of n macroscopic variables in Dspatial dimensions after which the noise terms were projected back to lattice space us-ing eigenvectors connecting the continuum moments with the lattice-Boltzmann velocityspace. As such, the noise is incorporated in a top-down manner. With their algorithmthey recommended setting kBT ≤ 1/3000 to achieve accurate incompressibility, which isinformed by the low-Mach-number stability threshold kBT ≪ ς2s = 1/3 common for allclassical LB models.In an alternative effort, Dünweg et al. [327] returned to the lattice-gas roots of theLBM and derived a generalized lattice-gas model in which ensembles of population densi-582.4. FLUCTUATING LBMSties locally realize probability distributions rather than mass densities. Then, the collisionrules for the model were described as Monte-Carlo processes satisfying a detailed balance[89] for the transition probability between pre and post-collision states. Particularly, alinear relaxation model exactly solving a Langevin equation for each mode was adopted.Thus, achieving correct “thermalization”—i.e. FDT consistency—for each degree of free-dom requires modelling of the same number of random variables.Gross et al. [328] furthered the FLBM by extending the theory to represent non-ideality with a square-gradient free-energy functional∫dx(ϕ(ρ) + κ/2(∇ρ)2) and adouble-well Landau potential ϕ. The authors thoroughly assessed the thermalizationof all degrees of freedom (density, momentum, stresses, internal energy, heat flux, andghost modes) by first establishing the FDTs for each, including the structure factor fordensity fluctuations and the covariance matrix being the variance in the FDT for theLangevin fluctuating sources. The FDTs were then central in determining the “equilibra-tion ratios”〈|M˜α(k)|2〉/Gαα(k) in Fourier space for the fluctuating component of eachmode in the vector M˜α from the diagonal components of the correlation matrix Gαβ .Moreover, the parameter space of the covariance matrix was shown to correlate withinterface width and predict the stability of simulations of multi-phase flows. Lastly, cap-illary fluctuations in a liquid-vapour interface were simulated to showcase the applicationof the FLBM to multi-phase flows.With the previous works the FLBM saw concerted development from the initial intro-duction of noise in the momentum-flux of a single component, “simple” ideal gas [323],through the equilibration of all mechanical degrees of freedom including ghost modes[324], further study of mechanical equilibration by assessing detailed balances of thestochastic “collisions” in FLBMs [327], and then extension of the FLBM to non-ideal,square-gradient fluids with focus on the full thermalization of mechanical as well as ther-mal modes [328]. Belardinelli et al. [329] generalized the work of Gross et al. [328] to ad-dress multi-component fluids. A single FLBE was employed and the fluctuating strengthof the various thermo-hydrodynamic modes were controlled by a multi-relaxation timemodel. The long-wavelength consistency in terms of FDTs for the degrees of freedomwas analyzed with the linearized non-equilibrium thermodynamics theory by Onsager andMachlup [330, 331]. Furthermore, the authors investigated the thermalization in homo-geneous as well as heterogeneous cases and showcased proper relaxation via equilibrationratios computed from simulations.Recently, Lulli et al. [77] utilized a FLBM to study metastability in phase transitions.Phase separation was brought about with a pseudo-potential (Shan-Chen) interaction-force model perturbed directly by the stochasticity imposed in a single FLBE. In sim-592.4. FLUCTUATING LBMSulations with open-source code implemented on CPU/GPU architectures, Lulli [332]confirmed equilibrium behavior in metastable states perturbed by small fluctuations inthe long-wavelength limit. The density structure factor was also numerically verified tohigh precision.As suggested by Belardinelli et al. [329] and Moroni et al. [333], it may be worthwhileemploying the stochastic Fokker-Planck equation (FPE) as an alternative basis to theFLBM for studying thermal fluctuations in non-ideal fluids. Accordingly, a brief intro-duction to the equation is presented. The FPE by Fokker [334] and Planck [335] has beensuggested to be a suitable model of liquids [336] compared to the popularly employedBoltzmann equation in lattice-Boltzmann methods, which arguably is a powerful toolin studying phase-changing flows [37]. This is owed to the FPE being directly derivedfrom the Langevin equation, which models the velocity space of particles including theirchaotic fluctuations that arise due to the continuous interaction between the electro-static potentials of neighbouring molecules and other external forces. The FPE therebymimics the stochastic behaviour of the Langevin equation with an evolving probabilitydistribution, from the underlying molecular dynamics that manifests as thermal noise inthe FPE. It is of interest to investigate the suitability of the FPE as a versatile modelof liquids; as the continuum density, momentum and total energy can be recovered fromthe moments of the primitive variables of the FPE, the inherited Langevin noise natu-rally projects into the macroscopic variables as additional noisy components u˜, T˜ , p˜, etc.Moreover, entropy generated by thermal noise can be directly derived from the FPE [90]and as entropy maxima dictate phase stability [78], the FPE framework may present anavenue for rigorously studying the non-equilibrium residence times of fluids in metastableregimes. The infinitesimal oscillations in pressure and temperature fields are presumedto enable the perturbation of (respectively) stretched and superheated metastable fluids,and thereby spontaneous nucleation.The FPE has already enjoyed success in investigating the turbulence-energy cas-cade and its associated entropy generation [337], modelling rarefied-gas dynamics [338–341], ion transport in clays [342], homogeneous and heterogeneous condensation [65],whole-brain modelling [343], wealth modelling [344], among other topics. The work thuspurports to determine the potential numerical and theoretical capabilities of the FPEtailored to nucleation and first-order phase transitions. Initially, a lattice-based solutionframework of the FPE is sought after for a simple, pure van der Waals fluid based ona cubic equation of state (EOS). It is acknowledged that real noise in fluids may notbe white, but attributed with a coloured spectrum, pending further investigation in adedicated literature review. Nevertheless, prior to exploring coloured noise, the lattice602.5. THE FOKKER-PLANCK EQUATIONsolution with white noise should be established. Eventually the work can be extended toaccommodate more unique spectra for various fluids, more sophisticated EOS, as well asmulti-component kinetic models. To understand how nucleation is mediated by thermalnoise, section §2.5 lays out the Langevin foundation of the FPE and explores its benefitsas a kinetic model of liquids [210].2.5 The Fokker-Planck equationFirst-order phase transitions occur when a critically large population of moleculesexperience sudden large fluctuations in the momenta that manifest a perturbation inentropy and hence a transition between stable states associated with the concavity ofentropy [78]. Since the 1970’s the Fokker-Planck equation has proven a powerful toolfor studying problems that involve stochastic fluctuations (noise), and how fluctuationsaffect systems around critical transition points [345]. In describing the FPE, the 1Dlinear Langevin equation for Brownian acceleration of a particle with mass m and speedξ is considered,η = ∂tξ = −γξ + Γ˜(t), (2.79)in which γ .= α/m = 1/τ is a friction factor that correlates with a Stokes-like damping-force coefficient α, and a relaxation time τ [345]. For simplicity only the Langevinequation for a single velocity component is included, but it generalizes to a molecular-velocity vector v =[ξx, ξy, ξz]† in three dimensions. Bold-font indicates vector quantitiesbut regular font is adopted for higher-rank tensors including the viscous stresses ταβ . Thestochasticity in the equation is facilitated by the per-unit mass Langevin fluctuation forceterm Γ˜ = F˜ /m. Assuming that Γ˜(t) is Gaussian and δ-correlated, a set of key propertiesarise from the fluctuation-dissipation theorem (FDT); namely that the average over anensemble is zero〈Γ˜(t)〉= 0, and that the ensemble-average of a product of two Langevinforce-terms at different times is zero if t − t′ is greater than the correlation time τc, i.e.〈Γ˜(t)Γ˜(t′)〉= 0, |t − t′| ≥ τc. Conversely, when timescales smaller than the correlationtime are of interest, the ensemble average is Dirac δ-correlated,〈Γ˜(t)Γ˜(t′)〉= Aδ(t− t′), (2.80)where the noise-strength (i.e. variance) can be shown to be A = 2γkBT/m, where kBis Boltzmann’s constant and T is the absolute temperature. In a three-dimensional caseΓ˜(t) would be a vector quantity with mutually independent components for each of the612.5. THE FOKKER-PLANCK EQUATIONspatial dimensions. Conversely, if the noise components were mutually dependent thestrength A would be a matrix quantity. “White” noise is characterized by the spectraldensity of the Langevin equation (2.79) being independent of frequency. On the otherhand, for non-δ-correlated terms, the spectral density is frequency-dependent and hencethe noise is attributed as being “coloured”.Directly solving the Langevin equation for a finite number of particles is not desired,but instead wish to solve an equivalent equation of motion for a distribution f(v;x, t)of Langevin particles. With this notation of the arguments generally adopted in thedissertation, it is implied that f is inherently dependent on the molecular velocitiesbeing the quadrature nodes of the population, whereas the spatio-temporal dependenceis indirectly owed to the time-integration of the FPE, as will be detailed later. The bridgebetween the FP and Langevin equations is the Kramers-Moyal (KM) expansion [346, 347]around a single or larger set of fluctuating variables, considering the molecular velocityvector v in one and two spatial dimensions. If the Langevin equation with Gaussianδ-correlated noise governs v it can be shown [345] that the KM expansion,∂tf(v;x, t)=∞∑n=1(−∂v)nD(n)f(v;x, t), (2.81)reduces to the FPE, with vanishing coefficients D(n) for the higher-order terms with n ≥3. In this case the expansion comprises the deterministic drift coefficient D(1) (n = 1),and the diffusion coefficients D(2) (n = 2) accounting for the fluctuations in the vari-able. Truncation to any finite order n > 2 results in introducing vanishingly small noisecontributions from the Langevin equation, guaranteeing that the process is statisticallycontinuous except for some non-vanishing cases [348] where discrete, anomalous eventsplay a significant role [337]. The drift and diffusion terms read,D(1) = −γv − η, (2.82)D(2) = 1/2q = γkBT/m, (2.83)which in concert with (2.81) for f(v;x, t) in position and velocity-space produces the FPequation [345],(∂t + vα∂α + ηα∂vα)f = Ω̂(FP) ◦ f .= γ∂vα(vα + v2T∂vα)f, (2.84)where vT =√kBTm−1 is the thermal velocity. The LHS body-forcing term with acceler-ation FB/m = η is included, which is not to be confused with the Langevin acceleration.622.5. THE FOKKER-PLANCK EQUATIONThe notation for the FP operator Ω̂(FP) ◦ f imposed on the populations is employed,and apply the Einstein notation for which repeated subscripts denote summation overpertinent indices.At this point, it bears emphasizing that the FP operator effectively models liquidstates [210, 336] and so hard-sphere binary collisions must also be considered to modelgaseous states. To that end, the Boltzmann operator Ω(BGK) .= τ−1c(f (eq) − f) with theMaxwell-Boltzmann (MB) equilibrium distribution,f (eq) =ρ(2πRT)D/2 exp(−(v − u)22RT), (2.85)introduced into the RHS of (2.84) yields a Fokker-Planck-Boltzmann (FPB) type equa-tion,(∂t + vα∂α)f = −ηα∂vαf + γ∂vα(vα + v2T∂vα)f +1τc(f (eq) − f), (2.86)where the FP and Boltzmann operators share the same MB equilibrium [336], and theacceleration term has been moved from the advection side. In the equilibrium (2.85)ρ is the mass density, R the gas constant, D the number of spatial dimensions, v themolecular velocity, and u the continuum velocity. A unique, transient solution of theFPB equation is elicited in 1 + 2D discrete degrees of freedom, i.e. (v,x, t), and itsmoment equations enabling simulation of large-scale fluid-dynamics problems with phasetransitions. The FP operator is discretized on the lattice following Moroni et al. [333]to derive kinetic equations for phase-changing flows of pure van der Waals fluids. TheBoltzmann operator is modelled following the conventional lattice-Boltzmann methodtogether with the Bhatnagar-Gross-Krook (BGK) single-relaxation time collision model[349].It is here emphasized that in contrast to the aforementioned FLBM literature inwhich energy conservation was predicted by a square-gradient functional of f , only twopopulations f and g and two kinetic lattice FPBEs are employed to separately evolvethe hydrodynamics and thermodynamics. The FPE constitutes a deterministic evolutionof a distribution function, in the same manner as a Boltzmann equation, although withdifferent innate dynamics. The FPE effectively models “glancing” interactions apt forrepresenting liquids whereas the BGK operator purports to model collisions in gases[349]. It follows, that across a liquid-vapour interface, relaxation towards equilibriumis going to be mediated by a combination of glancing and hard-sphere collisions. A632.5. THE FOKKER-PLANCK EQUATIONsimple numerical realization of the two in combination is a superposition in which theoperators’ relative relaxation towards equilibrium is tuned by two independent relaxationfrequencies. The approach is a simplified analogue of the Skinner and Wolynes [350]collision kernel where the proportionality of the FP and BGK operators is captured in asingle relaxation frequency. However, projecting the kernel onto the lattice is arguablymore complex than treating the operators separately as in the current work.64Chapter 3Lattice-Fokker-Planck methodIn this chapter a lattice-solution strategy for (2.86) is presented. The main limitationof FLBMs is that the numerical stability is limited by the fluctuation strength propor-tional to the perturbing thermal speed v2T = kBT/m ≪ ς2s ≪ 1/3 (as is similarly thecase for the hydrodynamic velocity in deterministic LBMs), and that it is complicatedcompared to deterministic LBMs. To not limit the method to weakly fluctuating fluidsand to allow for highly compressible and diabatic flows, the solution strategy incorpo-rates the “Particles on Demand” (PonD) approach by Dorschner et al. [94, 252] in a semi-Lagrangian diffusion (SLD) solution scheme for double-distribution functions, rather thana semi-Lagrangian advection scheme as is customary with PonD. Furthermore, thermo-dynamic non-ideality is mediated by correlating thermo-hydrodynamic variables T, e, ρ, pby a van der Waals equation of state. After establishing the solution procedure for lattice-FP equations, the FP operator is projected onto the lattice by Gauss-Hermite quadrature(GHQ) which effectively integrates the FP populations over the velocity space of the FPE,yielding a discrete set of equations for the FP operator. This effort is inspired by theworks of Moroni et al. [333, 336]. Theoretical consistency of the FPE is investigated inthe next chapter 4 via a Chapman-Enskog multiscale analysis (CEMA). Moreover, for thethermal fluctuations manifested in the FPE, a preliminary fluctuation-dissipation theo-rem is formulated for stochastic viscous stresses. Both chapters are partially reproducedfrom [J4], with concepts related in Fig. 3.1.CEMAFPEs(2.84)LFPEs(3.18)NSFEs(4.42)CHQEOSSLD(3.48)§4.1§3.3.1§3.2Figure 3.1: Upscaling of the FPE mesoscale effects to the continuum governed by the NSFEs.653.1. SEMI-LAGRANGIAN ADVECTION3.1 Semi-Lagrangian advection:Particles on Demand for kinetic theoryAs the stochastic sample paths of ensembles of many Langevin particles are inher-ently realized by evolution of the velocity space v(x, t) and f(v;x, t) in the FPE, it isnecessary to account for the associated non-equilibrium effects that arise in the popula-tions themselves. To that end, it is hypothesized that a conventional discrete velocityset of the LBM will incur error as the set will not numerically account for the ther-mal fluctuations. Therefore, the lattice-discrete velocity space is adapted to the localthermo-hydrodynamics, in which the fluctuations are inherently captured, by means ofthe recently proposed Particles-on-Demand for kinetic theory with the discrete velocities[94] (shown relative to ci in Fig. 3.2 with arbitrary values),vi =√θci + u, (3.1a)such that the inertial reference frame is shifted by the continuum velocity u(x, t) ren-dering ci the constant relative velocities as given in the conventional LBM. The discretevelocities are further scaled by the local thermodynamics through the reduced pressureθ(x, t), which are predicted by the FPB populations. As such the non-linearity from thethermal noise in the discrete velocity space is intrinsically captured via the thermody-namics which is coupled to the diffusion driven by velocity gradients in the FPE. Thesubscript i corresponds to each of the n quadrature abscissae of the pertinent DmQnmodel. The real-gas formulation,θ =pρRTL, (3.1b)is adopted where TL is the lattice temperature of a corresponding isothermal, ideal-gasrealization of the LFPBE, represented by a known constant for everyDmQn lattice model[351, 352]. The recast velocities (3.1) enable simulation of fluids far from equilibrium,whether that be at high Mach numbers or strongly thermal flows owing to the factthat the Hermite-expansion of the equilibrium populations shared by both the FP andBoltzmann equations become exact and velocity-error free [94, 302, 352],f(eq)i = ρwi, (3.2)g(eq)i = ρwi(2e−Dpρ+ v2i). (3.3)663.1. SEMI-LAGRANGIAN ADVECTION-1 0 1-101|c1||c2 |√θc1,x√θc2,yc1uc8c2c3c4c6c7Space xSpaceyv 5Figure 3.2: Vector fields of the discrete velocities vi (3.1) and ci. The example arbitrarily includes aunity velocity set ci, u = [0.1, 0.3]†, and√θ = 1.17.Therein, the equilibria are correlated with the density field ρ(x, t), D, the quadratureweights wi, internal energy e(x, t), and real-gas pressure p(x, t). v2i = vi ·vi is the velocitydot product. Mitigating velocity errors in a prospective simulation with a lattice-FPBEis especially attractive as previous lattice-solutions [336] of the FPE have suffered fromsignificantly reduced stability limits with increasing Reynolds numbers in comparison toconventional lattice-Boltzmann models.The benefits of PonD are contrasted by the complexity of the required solution meth-ods for the considered lattice equations; in the conventional LBM the discrete velocity setsensure exact streaming between lattice nodes, but this is not the case with (3.1) whichis more likely to incur populations streaming between off-lattice locations and latticenodes. Thus, establishing the populations at off-lattice locations demands a predictor-corrector scheme and a reconstruction procedure on the basis of the adjacent knownon-lattice populations. Moreover, the off-lattice nature of the discrete velocities requiresthat spatio-temporal integration of (2.86) is carried out backwards in time along thecharacteristic lines. Adopting the second-order accurate trapezoidal integration resultsin the (interim) kinetic lattice equations,fi(x, t)− fi(x− viδt, t− δt) = ω(FP)Ω(FP,η)i (fi) + ω(B)Ω(B)i (fi), (3.4a)673.1. SEMI-LAGRANGIAN ADVECTIONgi(x, t)− gi(x− viδt, t− δt)︸ ︷︷ ︸Advection= ω(FP)Ω(FP,η)i (gi)︸ ︷︷ ︸Perturbation+ω(B)Ω(B)i (gi)︸ ︷︷ ︸Collision, (3.4b)where the populations f and g recovering density and total energy, respectively, arerelaxed by the frequencies ω(FP) and ω(B) which are imposed on the lattice-discretizedFP-interaction Ω(FP)i and BGK-collision Ω(B)i operators. Going forward, for the sake ofbrevity, where relevant subscript i is omitted in f and it is implied that one of the Qpopulations are being treated.These equations get their explicit formulation in LBMs by recasting f to f ←f − δt/2Ω(f). However, as will be shown later, time integration of the FPE with thesame mapping function used for integration of the BE, does not render the kinetic equa-tions explicit but rather implicit, necessitating an alternative numerical methodology yetto be presented in §3.2. The work on the consistency of the LFPBEs presented in theChapman-Enskog multiscale analysis (CEMA) of the work [J4] is underpinned by (3.4)and assumes—for the sake of simplicity—that the common mapping methods used forobtaining explicit lattice-Boltzmann equations also exist for the LFPE. The derivationcarried out in the CEMA in §4.1 anticipates the explicit kinetic equations (3.4). Never-theless, to make the LFPBM more versatile it is necessary to extend the CEMA assumingnon-existence of mapping functions. This should be treated in future work.The predictor-corrector algorithm in PonD needs to be modified to reflect the im-plicitness in the time integration of the LFPE. This algorithm will be detailed in §3.2.Regardless, the reconstruction procedure in a “traditional” PonD solution with its gaugetransformations and interpolation schemes remain unchanged in an implicit formulation.The solution algorithm for the explicit kinetic equations (3.4), including the aforemen-tioned predictor-corrector scheme necessitated by the backwards-in-time integration gen-erally follows the steps in Fig. 3.3 in order to predict fi(x, t) and gi(x, t) during advection[302]. For completeness the following elaborates on it.In PonD the solution algorithm strides over an entire computational domain on aper-lattice node basis where a current node is labelled as a “monitoring” point at x. Thenode is interpreted in PonD to be monitoring in the sense that it perceives populationsstreaming from its surrounding space originating at t − δt and forwards in time untilthe populations coincide at x after a period δt. At each locale x the prediction offi(x, t) is sought after by a fully converged gauge λn = {ρ(n), p(n),u(n)} after n predictor-corrector iterations. Given the initial fields including ρ(x, t0), p(x, t0), T (x, t0),u(x, t0),etc. for a given problem being studied, the semi-Lagrangian advection is carried outuntil convergence at the single monitoring point. Firstly in the predictor, the seed gauge683.1. SEMI-LAGRANGIAN ADVECTIONλ0 is determined as either the initial fields at t0, or the advected fields from a previoustime instance for t ≥ t0 + δt. From the defined thermo-hydrodynamic observables, theQ discrete velocities vi can be determined, as well as the associated Q departure pointsPopulate F(f(xp))at n = 0Converged?AdvectionInteraction and collision01 CorrectorPredictorλn=0 = {ρ0, p0,u0}Reconstruct populations at dep. pointsfλ′i (xd, t− δt) =∑pΛ(xd,xp)fλ′i (xp)n++Update gaugeen+1 = En+1 − u2n+1/2Tn+1 = EOS(en+1, pn+1)pn+1 = EOS(ρn+1, Tn+1)λn+1 = {pn+1, ρn+1, Tn+1}Initialization at t0ρ(x), p(x), T (x),u(x)Define seed gaugev(n)i =√θci + uDiscrete velocitiesxd = x− v(n)i δtDeparture pointsfλ′i (xp) = Gλ′λp(xp)Gauge transformationAdvect populationsfλ′i (x, t) = fλ′i (xd, t− δt)Compute momentsρn+1 =∑i fλ′iρn+1un+1 =∑i fλ′i vi2ρn+1En+1 =∑i gλ′iFigure 3.3: Solution algorithm with Particles-on-Demand considering an arbitrary collision functionalΩ(f) with explicit equations in the form of (3.4). The nomenclature follows: the seed gauge λn=0comprises all the solution variables, including pre-advected populations prior to the firstpredictor-corrector iteration is carried out. v(n)i are the adapted discrete velocities at the n-th iterationand xd the associated departure points. F is the subset containing all pre-advected, collocation data atcollocation points xp defining F(xp, t− δt) ⊆ {f, g, p, ρ,u} at n = 0. The “ ” accent representsreconstructed quantities and Λ(xd,xp) an arbitrary interpolation kernel. It is implied that λ′ is theadapted gauge at the n-th iteration at which the moments of the reconstructed populations predict theadvected macroscopic observables at the (n+ 1)-th iteration, from which any EOS can be employed todetermine the local thermodynamics and λn+1 thereby closing the predictor-corrector loop.693.1. SEMI-LAGRANGIAN ADVECTIONxd at each monitoring point,xd = x− viδt. (3.5)The main goal is to predict the populations at the departure points. This can be facili-tated in two approaches by either establishing a single fixed reconstruction stencil sharedby all xd for reconstruction of their respective populations, or alternatively Q adaptivereconstruction stencils each used for a single departure point. To organize the data usedfor Q reconstructions, two stencils are distinguished between; a collocation stencil withnode coordinates coinciding with the underlying lattice, and Q reconstruction stencilswith sizes set by the degree of the interpolation kernel within the collocation stencil.In the current effort, adaptive reconstruction stencils are used. Given xd, the dimen-sions of the reconstruction stencil including an adaptive number of collocation pointsxp can be computed for a given interpolation kernel Λ(xd,xp). Let F(f(xp))(depictedin Fig. 3.4) be the subset of a lattice containing finite fields of populations f(xp) ina collocation stencil with an arbitrary number of collocation points with individual co-ordinates xp. F(xp) is once populated with pre-advected populations and macroscopicobservables (ρ0, p0, T0,u0 required for computing a transformation matrix Gλ′λp) in theseed gauge λ0. With Gλ′λp, the collocation populations in F are carried from their localinertial frames to that of the monitoring point x after which the transformed popula-tions are used to reconstruct the Q departure-point populations. These populations areinterim and Galilean variant until their moments converge in the gauge λn. The gaugeis defined by a fitting an EOS which computes Tn+1, pn+1 from said moments. DerivingGλ′λpfrom the moments is described in the following section.SpaceySpace xxF(xp)x dxpFigure 3.4: Illustration of the collocation stencil in relation to the entire lattice.703.1. SEMI-LAGRANGIAN ADVECTION3.1.1 Gauge transformations from moment-matching conditionsOwing to the off-lattice origin of the advecting populations as well as the individualvectors vi, their naïve computation is susceptible to modelling error, particularly infields of large continuum velocity and temperature gradients. To eliminate such errorsin simulations requires that Galilean invariance is upheld, i.e. first-principles laws ofmotion must be consistent in all inertial frames of reference and points in space, whichcan be attributed to a set of macroscopic quantities in λ′. As such, when advection iscarried out in the reference frame of the monitoring point at x on the basis of adjacentcollocation points, each at their local inertial frames λp, mass ρ, momentum ρu, and totalenergy E must be conserved irrespective from which location the f and g populations aremonitored. This is facilitated by Galilean transformations in which the transformationmatrix Gλ′λp is constructed by matching the moments of two dissimilar frames, i.e. λ′ andλp, by the condition [94],Mλ′fλ′ =Mλpfλp , (3.6)where the Q×Q linear mapping matrix M maps the populations to Q moments Mλ′ =Mλ′fλ′ . In 2D, these linearly independent moments are computed explicitly as the sumMλ′mn =Q−1∑i=0fλ′i vmi,xvni,y (3.7)along the integer dimensions m and n representing the order of the moments. By find-ing the inverse linear transformation M−1λ , e.g. by LU decomposition, the populationsat collocation points can be explicitly transformed to the same gauge as the currentmonitoring point,fλ′=M−1λ′ Mλpfλp , (3.8)in which Gλ′λp =M−1λ Mλp is the Q×Q transformation matrix. A transformation matrixfor the g population can be similarly realized. Due to the complexity of Gλ′λp , realizingthe transformations of fλp in code can alternatively be done in two relatively simplersteps by the equivalent outer productfλ′=M−1λ′ Mλp(fλp). (3.9)713.2. SEMI-LAGRANGIAN DIFFUSIONThe expanded moments and mapping matrices are presented in App. D for the D2Q9lattice model. If mp, np are the dimensions of a collocation stencil (Fig. 3.5) of arbitrarysize for a single monitoring point to which it is purported to advect Q populations fromthe departure points, Q ×mp × np gauge transformations are performed per predictor-corrector iteration. Following the gauge transformation of all populations in F(xp), theycan be used with an interpolation kernel of choice to reconstruct populations incident atthe departure points. The same gauge transformations are used in both SLA and SLD.In the following, it is described for SLD.3.2 Semi-Lagrangian diffusion: concurrent advection andinteraction3.2.1 Time integrationIn the lattice solution of the continuous Fokker-Planck equation, the aim is to dis-cretize the primitive variables along three dimensions. As covered in §3.3, the velocityspace v(x, t) = [vx, vy, vz]T of the equation can be discretized by means of Gauss-Hermitequadrature which requires populations to be advected to single points at x in space on thelattice. The spatio-temporal integration in space x = [x, y, z]T and time t is in conven--3 -2 -1 0 1 2 3-3-2-10123SpaceySpace xf8 f4f6f3f7xpmpnpf5f2f 0f1Figure 3.5: Overview of the collocation stencil with mp × np collocation points xp for a currentmonitoring point x (destination of all discrete velocity vectors) with discrete velocities vi advectingfrom depature poitns xd = x− viδt (solid red bullets). F(xp) is the set that contains all thepre-advection data coinciding with xp. Semi-Lagrangian advection and diffusion over a finite period δtis facilitated by reconstructing Q populations fλ′(in which the accent and superscript are omitted forbrevity) to be advected to x. The blue shaded areas are reconstruction stencils, specifically a cubic,fourth-degree kernel and bilinear, second-degree kernel for fλ′5 and fλ′6 , respectively, by example.723.2. SEMI-LAGRANGIAN DIFFUSIONtional lattice-Boltzmann models most often mediated by the trapezoidal rule constitutingthe “streaming” step with peculiar velocities ci in LBMs. Given a kinetic partial differ-ential equation with an arbitrary collision model Ωi(fi), its formulation usually reads,(∂t + vi,α∂α)fi(x, t) = Ωi(fi(x, t)). (3.10)Discretizing this equation in space and time can be done by trapezoidal integrationfrom t to t + δt, where one can denote the continuously time-shifted populations asf(s) = fi(x + cis, t + s), where the subscript i is omitted and it is implied that theQ populations reside at each lattice node. Then, the trapezoidal rule approximates theintegration with second-order accuracy byf ′ − f =∫ δt0ds Ω(f(s)) ≃ δt2[Ω(f ′) + Ω(f)]+O(δt3), (3.11)where f ′ = f(δt) denotes streamed states fi(x + ciδt, t + δt) and conversely f(0) non-streamed fi(x, t). A similar approach can be followed for the g population. However,this approximation is difficult to realize in conventional LBMs owing to the presenceof the streamed populations f ′ occurring on both sides of the equation, rendering itimplicit. A popular workaround is to introduce a remapping function which eliminatesthe streamed-state dependence by recasting f to [353–357]f˜ = f − δt2Ω(f), (3.12)as well as the integral equation to a more solvable and explicit form with one occurrenceof f ′,f˜ ′ − f˜ = δtΩ(f) +O(δt3), (3.13)and similarly for g. Applied to the BGK collision model, the familiar Boltzmann evolutionequation is obtained,f˜ ′ − f˜ = − τ−1δt1 + τ−1δt/2Ω(BGK)(f˜), (3.14)in which the origin of the relaxation frequency ω(B) = τ−1δt/(1+τ−1δt/2)often is takenfor granted. Conveniently, with this particular mapping function, the zeroth and firstmoments of f and f˜ coincide in the sense that the equalities∑Q−1i=0 f˜ = ρ˜ ≡∑Q−1i=0 f = ρand ȷ˜α ≡ jα hold. Only the second moment does not hold, i.e. P˜αβ(f˜) ̸= Pαβ(f). As733.2. SEMI-LAGRANGIAN DIFFUSIONthe BGK model only requires evaluation of the Maxwell-Boltzmann equilibrium f (eq),which is itself a function of jα, sampling can be done exclusively on the basis of f˜and information pertaining to f and its “bare” moments is not required, which couldotherwise be achieved by inverting (3.12). However, with the peculiar velocity set ci,Rotenberg and Moroni [357] showed that in the integration of the FP operator with thesame mapping function, only the zeroth moments coincide ρ(f) = ρ˜(f˜) and the first andsecond moments do not, i.e. jα(f) ̸= ȷ˜α(f˜), Pαβ(f) ̸= P˜αβ(f˜). Thus, the evolution off˜ can be used directly to compute only the density field. Yet, they were still able toformulate a recursive procedure where the coinciding densities could be used to furthercompute the bare and remapped first and second moments required for evaluating theequilibrium state and the FP operator Ω˜(FP)(f˜).Inconveniently, the same recursive procedure may be unsuitable for the FP operatorwith PonD discrete velocities vi =√θci + u, where the trapezoidal integration is eval-uated on the interval t− δt to t. In this case, the (backwards) time-shifted populationsdenoted by f(s) = fi(x+ vis, t+ s) yields the integral equation,f ′ − f︸ ︷︷ ︸Advection=∫ 0−δtds Ω(FP)(f(s)), (3.15)with f ′ = f(0) being the advected-state populations, fi(x, t), and f(−δt) those yet to beadvected from their respective departure points, fi(x−viδt, t−δt). Thus, by trapezoidalintegration one would alternatively solve the integral,∫ 0−δtΩ(FP)(f(s))ds ≃ −δt2(Ω(FP)(f ′) + Ω(FP)(f))︸ ︷︷ ︸Perturbation+O(δt3), (3.16)that similarly contains both pre- and post-advection states. With the mapping function,(3.15) becomesf˜ ′ − f˜ = −δtΩ(FP)(f ′) +O(δt3), (3.17)which would make a solution algorithm including collisions realizable if not for the fol-lowing. Following some algebra considering the isotropy conditions (App. D), it can beshown that with the FP operator in PonD-velocity space, none of the aforementionedpopulations nor moments (for both mass and energy) coincide, i.e. f˜ ̸= f, ρ˜ ̸= ρ, ȷ˜α ̸=jα, P˜αβ ̸= Pαβ, g˜ ̸= g, E˜ ̸= E, q˜α ̸= qα, R˜αβ ̸= Rαβ . Furthermore, it is found that theremapped moments are not only linear combinations of their bare moments, they are743.2. SEMI-LAGRANGIAN DIFFUSIONalso dependent on vT and u which themselves are dependent variables of the bare mo-ments, e.g. ρ = ρ(ρ˜, γ, δt,u(j/ρ), vT(T (E, ρ))). Consequently, the technique presentedby Rotenberg and Moroni [357] likely cannot be exploited in the current framework withthe lattice FP-operator (3.42a) (yet to be derived in §3.3), unless another mapping func-tion which inherently coincides ρ˜ = ρ and E˜ = E is constructed. Summarily, the analysisresults in a non-linear, implicit system of equations for obtaining Ω˜(FP)i(f˜i)and Ω˜(FP)i(g˜i)that can be computed, although possibly requiring a non-trivial computational cost.Alternatively, one can digress and question the trapezoidally evolving integral relation(3.15) in the scope of the recast discrete velocities vi. PonD provides a substantial amountof leeway for allowing virtually any non-equilibrium thermo-hydrodynamic effect beingimposed into the computation of the departure points xd = x−viδt, and henceforth thereconstructed populations f(xd, t − δt) at those locations. The fixed peculiar velocitiesci presume that the streamed trajectories account for distances of δx(i) = ciδt which inmost implementations in literature are unity for non-diagonal components. At the currenttime of writing, the LBM with PonD implements advection and collision sequentially inits solution algorithm [94], as is traditional in LBMs. What if both processes are insteadallowed to occur concurrently as a single diffusion process where the semi-Lagrangiannature of the particles and their departure points are exploited to carry out the numericalintegration implicitly? In this case, a convergence criterion is already provided from(3.16) that can be rewritten to a balanced form with isolated pre- and post-advectionstates,f ′ +δt2Ω(FP)(f ′) ≃ f − δt2Ω(FP)(f) +O(δt3). (3.18)Then, thermodynamic consistency would require that this balance is enforced in additionto the thermodynamic observables p, ρ,u converging in the predictor-corrector algorithmof PonD.The semi-Lagrangian diffusion is realized as follows. Let F(f(xp))be the subset ofa lattice containing the finite fields of populations f(xp) in a collocation stencil withan arbitrary number of collocation points with individual coordinates xp. Then, theheuristics of the diffusion comprise the computation of the RHS of (3.18), which iscarried out across all collocation points in their respective frames of reference λp(xp, t−δt) = {p, ρ,u}(xp, t − δt) and storing the resulting populations in F(f⋆λp(xp)). Thesepopulations,f⋆λp(xp).= fλp(xp)− δt2Ω(FP)(fλp), (3.19)753.2. SEMI-LAGRANGIAN DIFFUSIONrepresent the pre-advected states at time t − δt where the FP operator has been mani-fested, and thus “⋆” denotes “perturbed” states.Subsequently, the aim is to carry the perturbed populations f⋆λp(xp) from their localreference frames λp(xp, t−δt) to the same inertial reference frame as that of a monitoringpoint λ′(x, t) = {p′, ρ′,u′}(x, t) by the same gauge-transformation presented earlier,f⋆λ′(xp).= Gλ′λpf⋆λp , (3.20)warranting their use in the reconstruction procedure. Transforming the collocation pop-ulations to the monitoring gauge is tantamount to representing them at time t after thedeparture-point populations have travelled from xd to x. During this travel, the parti-cles are expected to experience the remainder of the trapezoidally-integrated (effectivelyat time t) FP-perturbations amounting to +δt/2 · Ω(FP)(fλ′) from the LHS of (3.16),such that the perturbed populations analogous to (3.19), but representative at t, can becomputed. Prior to this, populations are reconstructed at departure points by a genericinterpolation kernel Λ,fλ′(xd, t− δt) =∑pΛ(xd,xp)f⋆λ′(xp). (3.21)Lastly, to balance the trapezoidal integration, the predictor step and diffusion of particlesare concluded by perturbing the departure-point populations and transferring them tothe advected-population array f ′(x, t):f ′(x, t) = fλ′ +δt2Ω(FP)(fλ′). (3.22)Eventually, the corrector step of the semi-Lagrangian diffusion can be performed inline with the conventional advection scheme of PonD by first computing the moments,ρn+1 =Q−1∑i=0f ′i , (3.23)ρn+1un+1 =Q−1∑i=0f ′iv(n)i , (3.24)2ρn+1En+1 =Q−1∑i=0g′i, (3.25)required for establishing the full thermo-hydrodynamic state from an EOS of choice. The763.3. LATTICE DISCRETIZATION OF THE FPEinternal energy can be established from the zeroth total-energy moment of g′, and themomentum moment of f ′ as e = E − u2/2. Given the zeroth density moment of f ′the temperature can be established as T = EOS(e, ρ), which further closes the correctedframe of reference λ′(x, t) = {p′, ρ′,u′} via the pressure p = EOS(ρ, T ). The additionalconvergence criterium (3.18), effectively evolving f by f ′ − f = ∆Ω(FP) would thenbe equivalent to monitoring the equality fλ′(x, t) = fλ′(xd, t − δt) + O(∆f) assuminga user-defined convergence-error threshold O(∆f), excluding the temporal integrationerror O(δt3). If f ′ (or one or more of the thermodynamic variables) have not converged,the corrected monitoring gauge is used as a seed gauge for a new predictor step. Thediffusion algorithm would then follow the flow and logic in Fig. 3.6, amended from Fig.3.3.3.3 Lattice discretization of the velocity space in theFokker-Planck equationHaving devised a solution-algorithm for the lattice-FPE, this section presents thediscretization of the FP operators Ω(FP)(f ; g). To enable the evolution of (3.18) and (3.4),it is necessary to establish a lattice model for the FP operator, which will be facilitatedwith Hermite quadrature. Some of the work documented in the article by Moroni et al.[333] is repeated for the sake of completeness. However, as will be clear later, the explicitformulation for the operator is markedly different from that reported by Moroni et al.[333] owing to a characteristic eigenvalue property of the FP operator modified by thediscrete velocity set (3.1) of PonD. A notation is adopted for the operators in whichcircumflex-accented ( ̂ ) operators denote that the operators are to be considered onlyas the operational expressions excluding the populations. This notation is important inthe CE series expansion.3.3.1 Gauss-Hermite quadratureThe discretization of the operators as well as the populations f, g is facilitated bya Hermite-polynomial tensor H(l)α (v) basis in D dimensions. Just as these tensors canbe exploited to transform gradient terms in the continuous FPE (2.84), the continuouspopulation f(v;x, t) in phase-space can be expanded in terms of its Hermite coefficients773.3. LATTICE DISCRETIZATION OF THE FPEF(l)α (x, t) as [333]f(v;x, t) = w(v)∞∑l=01v2lT l!F(l)α H(l)α (v), (3.26)Compute arguments for Ω(FP)iρ(x, t) =∑i fλ0i , jα, PαβE(x, t) =∑i gλ0i , qα, Rαβv2T = kBT (x, t)/mConverged?Initialization at t = 0ρ, T, p,u(x)Define seed gaugeλ0 = {p0, ρ0,u0}Perturb seed populationsf⋆λ0 (x, t) = fλ0 − δt2Ω(FP)i (fλ0 )Discrete velocities, departure pointsvni =√θci + uxd = x− viδtGauge transform populationsf⋆λ′ (xp) = Gλ′λpf⋆λp (xp)Reconstruct populationsfλ′ (xd, t− δt) =∑p Λ(xd,xp)f⋆λ′Diffuse populationsf ′(x, t) = fλ′ +δt2Ω(FP)i(fλ′)t = t+ δtUpdate gauge λn+1en+1 = En+1 − u2n+1/2Tn+1 = EOS(en+1, ρn+1)pn+1 = EOS(ρn+1, Tn+1)Compute momentsρn+1 =∑i f′iρn+1un+1 =∑i f′ivi2ρn+1En+1 =∑i g′i01n++PredictorCorrectorFigure 3.6: Solution algorithm for the lattice-Fokker-Planck equation with semi-Lagrangian diffusion:the implicit FPE (3.18) is solved by accounting for the seed and advected states in the predictor. First,the LHS is mediated on the seed populations at t− δt, after which the populations are gaugetransformed, reconstructed. Then the RHS is realized before the corrector step causing the populationsto be “diffused”.783.3. LATTICE DISCRETIZATION OF THE FPEwhere the populations are proportional to the Gaussian weight function [333],w(v) =1(2πv2T)D/2 exp [−(v − u)22v2T]. (3.27)The subscript α = α1 · · ·αl in the Hermite coefficients and polynomials contracts the lindices of the l-rank Hermite tensor H(l)α . To that end, the zeroth- through third-ranktensors read [358, 359],H(0)(v) = 1, (3.28a)H(1)α (v) = vα, (3.28b)H(2)αβ (v) = vαvβ − v2T δαβ, (3.28c)H(3)αβγ(v) = vαvβvγ − v2T(vαδβγ + vβδαγ + vγδαβ), (3.28d)and are further detailed by Moroni et al. [333]. The Hermite coefficients for the popula-tions are the velocity integrals which occupy RD,F(l)α (x, t) =∫dv f(v;x, t)H(l)α (v). (3.29)To obtain a tenable series expansion, (3.26) is truncated at an order K over l ranks andthose exceeding K are discarded, such that f(v;x, t) is approximated as [333],f(v;x, t) = w(v)K∑l=01v2lT l!F(l)α H(l)α (v). (3.30)By Gauss-Hermite quadrature, the velocity integrals are evaluated over a finite set ofvelocity quadrature-nodes, or abscissae, such that the continuous integrals become Qeasily-digestible discrete sums over the index i = 0, ..., Q − 1. Crucially, given anypolynomial p(v) with a lower order than 2K, the continuous integral of its product witha Gaussian function becomes the quadrature [333, 360],∫dv w(v)p(v) =Q−1∑i=0wip(vi), (3.31)accompanied by vi ∈ RD and wi identified as the quadrature nodes (3.1) and weights,respectively. The above result is the main feature of the Gauss-Hermite polynomials,which allows numerical integration of any 1D function at discrete points. In (3.30),793.3. LATTICE DISCRETIZATION OF THE FPEf/w can be characterized as a maximum K-order polynomial, which conveniently can beexploited to define the familiar discrete populations fi, gi. Starting with the conventionalm-order moments of f ,M (m)α.=∫dv f(v;x, t)vα1 · · · vαm , (3.32)M(m)α can be rewritten as [333],∫dvw(v)w(v)f(v;x, t)vα1 · · · vαm =Q−1∑i=0wiw(vi)f(vi;x, t)vi,α1 · · · vi,αm.=Q−1∑i=0fivi,α1 · · · vi,αm , (3.33)arriving at the familiar discrete population fi with the ratio,fi(x, t)f(vi;x, t)=wiw(vi). (3.34)K ≥ 2 is required to reproduce the correct thermo-hydrodynamic behaviour governed bythe Navier-Stokes-Fourier (NSF) equations. The series up to these K-orders require theconventional quadratures of the fi, gi populations which represent mass and total energy,respectively [302, 361],ρ.=∫dv f(v;x, t) =Q−1∑i=0fi, (3.35a)jα.=∫dv f(v;x, t)vα =Q−1∑i=0fivi,α, (3.35b)Pαβ.=∫dv f(v;x, t)vαvβ =Q−1∑i=0fivi,αvi,β, (3.35c)Qαβγ.=∫dv f(v;x, t)vαvβvγ =Q−1∑i=0fivi,αvi,βvi,γ , (3.35d)andE.=∫dv g(v;x, t) =Q−1∑i=0gi, (3.36a)803.3. LATTICE DISCRETIZATION OF THE FPEqα.=∫dv g(v;x, t)vα =Q−1∑i=0givi,α, (3.36b)Rαβ.=∫dv g(v;x, t)vαvβ =Q−1∑i=0givi,αvi,β. (3.36c)Here, it is emphasized that E = 2ρE, and ρ, uα = jα/ρ, and E = e + uαuα/2 arethe macroscopic observables density, continuum velocity, and total energy, which are ofinterest in simulations, and recoverable as in any other conventional lattice-Boltzmannmodel. They also participate in the thermo-hydrodynamic limit yet to be derived in §4.1.Furthermore, it can be observed that the Hermite coefficients F(l)α ,G(l)α are merely linearcombinations of these,F(0) = ρ, (3.37a)F(1)α = jα, (3.37b)F(2)αβ = Pαβ − v2Tρδαβ, (3.37c)F(3)αβγ = Qαβγ − v2T(δαβjγ + δαγjβ + δβγjα), (3.37d)for fi and,G(0) = E = 2ρE, (3.38a)G(1)α = qα, (3.38b)G(2)αβ = Rαβ − v2T(2ρE)δαβ, (3.38c)for gi. In the following subsection, the Hermite series for Ω(FP)i and Ω(η)i will be estab-lished. As these reside in the same subspace as (3.30), the quadrature of F(l)α and G(l)αcan be utilized in the series.3.3.2 The lattice Fokker-Planck operatorThe above methodology can now be applied to the operators Ω̂(FP) = γ∂vα(vα +v2T∂vα)and Ω̂(η) = −ηα∂vα in order to derive explicit functionals for Ω(FP)i(fi; gi)andΩ(η)i(fi; gi). First, the FP operator is treated. Extending the function Ω̂(FP) ◦ f bywi/w(vi) enables us to write the action of the Hermite expansion on that function [333],813.3. LATTICE DISCRETIZATION OF THE FPEΩ(FP)i = wiK∑l=01v2lT l!Ω(FP)α H(l)α (vi), (3.39a)where the Hermite coefficient for the operator is computed as,Ω(FP)α =∫dv Ω̂(FP) ◦ fH(l)α (v). (3.39b)The goal is finding an operational expression which can be used to rewrite Ω̂(FP) in a moreconvenient, explicit form, to which end it is needed to exploit the eigenvalue property ofthe characteristic Gaussian w(v). The product Ω̂(FP)[w(v)H(l)α (v)], which is identifiablein (3.39) by insertion of (3.30), can be constructed and will comprise a set of gradientand divergence terms up to the second order. Expanding the product necessitates theintroduction of an important eigenvalue property of the characteristic Gaussian w(v),which makes the continuous-velocity partial derivatives directly solvable. In App. B,this property is used to expand the product, where the property is found to be,∂vβw(v) = −√θcβv2Tw(v), (3.40)which indeed is an eigenfunction with the eigenvector −√θcβ/v2T . This result accountsfor the recast velocities (3.1), whereas the corresponding eigenvalue −cβ/v2T found inthe analysis of Moroni et al. [333] considers the conventional peculiar velocity set ci ∈RQ used by all LB models with exact streaming. As is demonstrated in App. C, theoperational expression derived from Ω̂(FP)[w(v)H(l)α (v)]can be used to compute theHermite coefficients Ω(FP)α , melded with the previously found Hermite coefficients forf(v;x, t) and F(l)α . With the eigenvalue property, it is found that the product expandstoΩ̂(FP)[w(v)H(l)α (v)]= γ(1 + uβvi,β + uβv2T− l)[w(v)H(l)α (v)]− γ 2uβv2T[w(v)H(l+1)βα (v)],(3.41)whereas Moroni et al. [333] reported the result as Ω̂(FP)[w(v)H(l)α (v)]= −γl[w(v)H(l)α (v)]with the peculiar velocity set. In (3.41) vi,β are the Q recast velocity vectors (3.1), andH(l+1)βα is the (l + 1)–rank tensor relative to the current rank being considered for H(l)α .Thus, the local thermo-hydrodynamics imposed by vi directly affect the FP dynamics.823.3. LATTICE DISCRETIZATION OF THE FPETable 3.1: Hermite coefficients for the FP and acceleration lattice operators evaluated with (C.10,C.17) as well as the Hermite coefficients for the populations F(l)α (fi) (3.37) and G(l)α (gi) (3.38). Theoperator coefficients are ultimately used to complete the series expansion yielding Ω(FP)i and Ω(η)i .l Ω(FP)α (l) ◦ {fi; gi} Ω(η)α (l) ◦ {fi; gi}0 γ[1 + uγvi,γ+uγv2T]{F(0);G(0)}−[ηγuγv2T]{F(0);G(0)}1 γ[uγvi,γ+uγv2T− 2]{F(1)α ;G(1)α} [ηα{F(0);G(0)}− ηγuγv2T]{F(1)α ;G(1)α}2 γ[(uγvi,γ+uγv2T− 1){F(2)αβ ;G(2)αβ} [ηα{F(1)β ;G(1)β}+ ηβ{F(1)α ;G(1)α}−2(uα{F(1)β ;G(1)β}+ uβ{F(1)α ;G(1)α})]−ηγuγv2T{F(2)αβ ;G(2)αβ}]As (3.41) is fully explicit and F(l)α (3.37) are merely linear combinations of the populationmoments (3.32), all the ingredients are gathered for computing the Hermite coefficientsΩ(FP)α . The first three are computed in Tab. 3.1 using the explicit form of the coefficients(C.10). To compute the integral in the coefficients Ω̂(FP) the Hermite orthonormalityrelation was utilized as is also showcased in App. C. These coefficients, combined with(3.39a), yield the series expanded up to l ≤ K = 2, for both fi, gi:Ω(FP)i (fi) = wiγ[ρ(1 + uγvi,γ + uγv2T)+vi,αv2T(uγvi,γ + uγv2T− 2)jα+vi,αvi,β − v2T δαβ2v4T({uγvi,γ + uγv2T− 1}{Pαβ − v2Tρδαβ}− 2 {uαjβ + uβjα})], (3.42a)Ω(FP)i (gi) = wiγ[E(1 + uγvi,γ + uγv2T)+vi,αv2T(uγvi,γ + uγv2T− 2)qα+vi,αvi,β − v2T δαβ2v4T({uγvi,γ + uγv2T− 1}{Rαβ − v2T (2ρE)δαβ}− 2 {uαqβ + uβqα})]. (3.42b)3.3.3 The lattice acceleration operatorA similar analysis is carried out for Ω̂(η) via its corresponding Ω̂(η)[w(v)H(l)α (v)]function. Moroni et al. [333] found the operational expression Ω̂(η)[w(v)H(l)α (v)]=833.3. LATTICE DISCRETIZATION OF THE FPEηβv2T[w(v)H(l+1)βα (v)]with the peculiar velocity set. With the recast velocities, the resultΩ̂(η)[w(v)H(l)α (v)]=ηβv2T[w(v)(H(l+1)βα (v)− uβH(l)α (v))], (3.43)is reported instead. This result can again be expanded in the same Hermite series (3.39a)as for the FP operator, recognizing that the aim is to obtain the combined lattice operatorΩ(FP,η)i = Ω(FP)i +Ω(η)i whereΩ(η)i = wiK∑l=01v2lT l!Ω(η)α H(l)α (vi), (3.44a)and the Hermite coefficient for the operator is computed asΩ(η)α =∫dv Ω̂(η)[f ]H(l)α (v). (3.44b)Using (3.43), one arrives at the relation for the Hermite coefficients (C.17) derived inApp. C, which was used to obtain the first three Hermite coefficients in Tab. 3.1. Thesecoefficients, in concert with (3.44a) yield the full (l ≤ K = 2)–kinetic operators,Ω(η)i (fi) = wiγ[−v(η)γ uγv2Tρ+vi,αv2T(v(η)α ρ−v(η)γ uγv2Tjα)+vi,αvi,β − v2T δαβ2v4T({v(η)α jβ + v(η)β jα}− v(η)γ uγv2T{Pαβ − v2Tρδαβ})], (3.45a)Ω(η)i (gi) = wiγ[−v(η)γ uγv2TE +vi,αv2T(v(η)α E −v(η)γ uγv2Tqα)+vi,αvi,β − v2T δαβ2v4T({v(η)α qβ + v(η)β qα}− v(η)γ uγv2T{Rαβ − v2TEδαβ})], (3.45b)where v(η)α = ηαγ−1 was adopted as a pseudo velocity designed by the dimensionality of[γ] = s−1. Having rewritten the acceleration term this way, terms with correspondingranks of F(l)α ,G(l)α may be factorized into the same groups, as all coefficients now areprefixed by γ. Having discretized both operators on the lattice one can combine them inthe more compact form,Ω(FP,η)i (fi) = Ω(η)i +Ω(FP)i843.3. LATTICE DISCRETIZATION OF THE FPE= wiγ[ρ+vi,αv2Tȷα +vi,αvi,β − v2T δαβ2v4TPαβ], (3.46a)Ω(FP,η)i (gi) = Ω(η)i +Ω(FP)i= wiγ[E +vi,αv2Tqα +vi,αvi,β − v2T δαβ2v4TRαβ], (3.46b)where the f -moments were recast yielding density, momentum flux, and pressure andhigher-order tensors into new functionals,ρ =(1 + uγvi,γ + uγ − v(η)γv2T)ρ, (3.46c)ȷα = j(η)α +(uγvi,γ + uγ − v(η)γv2T− 2)jα, (3.46d)Pαβ =(v(η)α − 2uα)jβ +(v(η)β − 2uβ)jα+(uγvi,γ + uγ − v(η)γv2T− 1)(Pαβ − v2Tρδαβ), (3.46e)where j(η)α = ρv(η)α is a pseudo momentum. The quadrature-node index notation inthe recast moments is omitted and it is noted that their dependence on vi,γ is impliedgoing forward. Evidently, v(η)γ originates from the redefinition of ηγ . Similarly the recastfunctionals for the gi-population moments read,E =(1 + uγvi,γ + uγ − v(η)γv2T)E, (3.46f)qα = uγvi,γ + uγ − v(η)γv2Tqα +(v(η)α − 2uα)E, (3.46g)Rαβ =(v(η)α − 2uα)qβ +(v(η)β − 2uβ)qα+(uγvi,γ + uγ − v(η)γv2T− 1)(Rαβ − v2TEδαβ). (3.46h)The lattice operators (3.46a–3.46h) are the final results for the kinetic scheme. Theydefine the “perturbation” process and can be implemented in semi-Lagrangian advectionand diffusion to successively simulate the spatio-temporal evolution of ρ,u, E and othermacroscopic quantities of interest. To correlate the moments to the thermodynamics,the employed van der Waals equation of state is presented in the following.853.4. THERMODYNAMICS3.4 ThermodynamicsPhase transformations include non-equilibrium thermodynamics and large variationsin viscosity and density which evolve on a multitude of spatio-temporal scales. Such scaleshave historically been difficult to treat in the continuum without resorting to limitingassumptions that appear in many empirical mass-transfer models for multi-componentmixtures, as was described in §2 and [J3].Whereas mixture models account for the latent heat transfer between species m˙hlv,the present single-component implementation evolves T, p and phase transitions underthe saturation curve in tandem with the empirical van der Waals (VDW) equation ofstate (EOS). Even though the EOS is empirical its simple non-monotonic topology witha distinct energy minimum enables nucleation in kinetic theories, as was shown in [252] inwhich the same EOS was employed. Moreover, the cubic VDW-EOS is the simplest EOSthat retains the ability to predict spinodals [362] and is inherently mechanically unstablein the spinodal region and metastable in the binodal region. When a metastable fluidis perturbed the fluid state will transition towards the binodal curve where the localfree energy is minimized [363]. In order to carry out a multi-scale analysis (§4.1) it isnecessary to establish explicit relations for the the internal energy and pressure. It isduly noted that the kinetic equations conserve the density (3.35a), momentum (3.35b),and total energy (3.36a), where the latter predicts the internal energy e via,E = e+uαuα2. (3.47)Thus, given e, the temperature can be computed by reorganizing the EOS,e = cvT − aρ− pρ, (3.48a)in which the term p0/ρ is included to retain positive temperatures in the subcritical regionT/Tcr < 0.84375 [302]. Then, the analysis proceeds with defining the thermodynamicpressure as,p =ρRT1− bρ − aρ2 + p, (3.48b)which contributes the non-ideal energy contribution T(∂p/∂T)v∂αuα in the Fourier equa-tion, as will be documented later in the Chapman-Enskog multiscale analysis in §4.1.2.This analysis further necessitates deriving the isobaric specific heat and speed-of-sound,to arrive at the thermo-hydrodynamic limit. According to the fundamental thermody-863.4. THERMODYNAMICSnamic derivatives [9] the former reads,cp.=(∂h∂T)p, (3.49)and the real-gas speed-of-sound,ς2s.=(∂p∂ρ)s, (3.50)and can be rewritten using the cyclic and Maxwell relations [9] as,cp =(∂e∂T)p+ p(∂v∂T)p(3.51)= cv +R2TRT − 2aρ(1− bρ)2 , (3.52)ς2s =(∂p∂ρ)T+Tρ2cv(∂p∂T)2v(3.53)=RT(1− bρ)2 (1 + δ)− 2aρ, (3.54)where cv = R/δ [252, 364]. The derivations for these correlations can be found in App.A. The empirical, thermodynamic coefficients of the VDW-EOS can be computed fromthe critical state of the pertinent fluid, as follows [9],a =2764R2T 2crtpcrt, (3.55a)b =RTcrt8pcrt. (3.55b)Lastly, explicit correlations for chemical potential can be derived for the VDW EOS[12, 365, 366] although these have not been pursued.87Chapter 4Theoretical consistency4.1 Chapman-Enskog multiscale analysisIn order to utilize the kinetic models (3.4) and (3.18) in mesoscale simulations wherethe macroscopic quantities (3.35, 3.36) are recovered from microscopic populations, it iscrucial to investigate how the LFPBEs behave across scales. In this chapter, the con-sistency of (3.4) is reported. Specifically, a Chapman-Enskog multiscale analysis [361] isconducted. Initially, the post-advection state is approximated by a Taylor series, afterwhich the modified kinetic equations are expanded in terms of perturbation series of allthe degrees of freedom of the LFPBEs. Subsequently, all terms in the equations can beseparated by their representative orders forming a hierarchy of perturbation relations forthe advection and collision sides of the equations. Then, the zeroth- through second-ordermoments of these relations can be taken to form moment equations, i.e. scale-dependentforms of the continuum conservation laws. This step is crucial as it comprises the mo-ments stemming from non-equilibrium, higher-order contributions of the perturbationseries which can be chosen to either be included or excluded with the aim of assessingtheir mathematical formalism and physical implications in the continuum, thus creatinga direct avenue between the microscopic and macroscopic scales. Finally, the momentequations can be recombined by inverting the perturbation series, resulting in the Navier-Stokes-Fourier (NSF) equations. By including selected non-equilibrium populations inthe moment equations, it is possible to elucidate the asymptotic behaviour of differentparts of the NSF hierarchy, focusing first on the stochastic shear-stress tensor.4.1.1 Perturbation seriesThe mesoscale behaviour, especially that stemming from the fluctuations of the FPE,evolves at different characteristic length and time scales. Analysis of the scale effectsas they manifest in the continuum can be treated with multiscale methods seeking toseparate derivative terms, populations, and macroscopic observables into contributionsthat scale with a “smallness” parameter ϵ, where ϵ ≪ 1 is associated with the thermo-884.1. CHAPMAN-ENSKOG MULTISCALE ANALYSIShydrodynamic limit [210]. By exemplification, the viscous effects of FP may reside onlonger time scales whereas shocks propagate on shorter time scales. Specifically, thedistributions fi, gi, spatial and temporal gradients ∂α, ∂t, and macroscopic moments aredecomposed into a family of perturbation series around ϵ raised to different exponentvalues. Firstly, the populations expand to,fi = f(0)i + ϵf(1)i + ϵ2f(2)i +O(ϵ3), (4.1a)gi = g(0)i + ϵg(1)i + ϵ2g(2)i +O(ϵ3), (4.1b)where the zeroth-order contributions are the known equilibrium distributions, such thatthe higher orders act as unknown, stochastic components. Propagation and diffusionphenomena are captured by decomposing the time derivative into two time variables,∂t = ϵ∂(1)t + ϵ2∂(2)t +O(ϵ3). (4.1c)As for the spatial derivatives, these are evolved on the hydrodynamic scale appropriatelyresolved as,∂α = ϵ∂(1)α +O(ϵ2). (4.1d)Moreover, the interest is in the zeroth- through to the second-order perturbations in allof the macroscopic observables in order to recover the correct continuum-conservationlaws,ρ = ρ(0) + ϵρ(1) + ϵ2ρ(2), (4.1e)E = E(0) + ϵE(1) + ϵ2E(2), (4.1f)jα = j(0)α + ϵj(1)α + ϵ2j(2)α , (4.1g)qα = q(0)α + ϵq(1)α + ϵ2q(2)α , (4.1h)Pαβ = P(0)αβ + ϵP(1)αβ + ϵ2P(2)αβ , (4.1i)Rαβ = R(0)αβ + ϵR(1)αβ + ϵ2R(2)αβ , (4.1j)and similarly for the linear recombinations of the above, ρ = ρ(0) + ϵρ(1) + ϵ2ρ(2) andso forth. The zeroth-order terms are interpreted as “equilibrium” contributions whichdictate the phenomena that evolve on the macroscales, whereas increasing orders rep-resent perturbations on diminishing scales. Consequently, one can imagine that micro-scopic fluctuations due to thermal noise may be observable on the ϵ1, ϵ2 scales, where the894.1. CHAPMAN-ENSKOG MULTISCALE ANALYSIScoarser fluctuations associated with ϵ1 are likely more relevant to analyze in the contextof mesoscopic nucleation. Fig. 4.1 depicts the perturbation series components of density.4.1.2 Expansions: Taylor series, perturbations, separation, moments,and recombinationPerturbation-series expansionThe previously derived lattice-FPBEs (3.4) are expanded to prove their consistencyin the thermo-hydrodynamic limit. In the following the notation for the pre-advectionpopulations fi = fi(x − viδt, t − δt) and advected populations f ′i = fi(x, t) is adopted.First, the advection side of the equations is treated. The purpose is to approximate theadvected state f ′i by its Taylor expansion around the datum fi, exploiting the commonansatz that third and higher order terms n > 2 are very small and do not significantlyaffect the macroscopic behaviour [361]. As such, to find the Navier-Stokes-Fourier equa-tions, only the two lowest orders in the Taylor series expansion are retained,f ′i = fi + δt(∂t + vi,δ∂δ)fi +δt22(∂t + vi,δ∂δ)(∂t + vi,ε∂ε)fi +O(δt3), (4.2)and similarly for gi. It follows that the lattice equations (3.4) can be rewritten to,δt(∂t + vi,δ∂δ)fi +δt22(∂t + vi,δ∂δ)(∂t + vi,ε∂ε)fi = ω(FP)Ω(FP)i + ω(B)Ω(B)i , (4.3)δt ∼ t(2) δt ∼ t(1)ϵ2ρ(2)ϵρ(1)ρ(0)Density(x,t)Time tFigure 4.1: Example perturbation-order contributions to a macropscopic densityρ = ρ(0) + ϵρ(1) + ϵ2ρ(2) as it evolves in time at an arbitrary point in space. By the perturbation-seriesexpansion the signal is decomposed into diminishing time scales whereto the smallness parameter ϵ isassociated with some characteristic timescale t(1). One can then interpret the first-order perturbationsto oscillate with periods O(δt) ∼ t(1) and similarly for the second-order with t(2). In addition to thetemporal dimension, the multi-scale nature of the perturbations can also be imagined to extend out ofthe page in the spatial dimension (x), forming a hyperplane which analogously to the density responsein time is increasingly non-uniform for ϵ→ 0. Figure is not to scale.904.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISwhere the higher-order O(δt3)terms were neglected. Before treating the collision side ofthe equation, the initial perturbation analysis is made on the advection terms, retainingthem for further analysis. The analysis starts with another common ansatz : only thetwo lowest orders in the Knudsen number Kn ≡ λ/Lc (where λ is the molecular mean-free path and Lc the pertinent characteristic length scale) are required to obtain thecontinuum NSEs where only the coarse non-equilibrium effects in f (1)i are considered inaddition to the equilibrium f (0)i [361]. Substituting in the pertinent variables from (4.1)yields the expanded advection side,{δt[(ϵ∂(1)t + ϵ2∂(2)t)+ ϵvi,α∂(1)α]+δt22[(ϵ∂(1)t + ϵ2∂(2)t)+ ϵvi,α∂(1)α]×[(ϵ∂(1)t + ϵ2∂(2)t)+ ϵvi,β∂(1)β]}(f(0)i + ϵf(1)i + ϵ2f(2)i), (4.4)where contributions separated by orders in ϵ are,ϵ0 : 0, (4.5a)ϵ1 : δt(∂(1)t + vi,α∂(1)α)f(0)i , (4.5b)ϵ2 :[δt∂(2)t +δt22(∂(1)t + vi,α∂(1)α)(∂(1)t + vi,β∂(1)β)]f(0)i+ δt(∂(1)t + vi,α∂(1)α)f(1)i . (4.5c)These contributions will eventually equate with the corresponding-order contributionsfrom the analogously expanded collisions operators. To that end, the BGK operator isexpanded with the perturbation series yielding the separated orders,ϵ0 : ω(B)(f(eq)i − f (0)i), (4.6a)ϵ1 : −ω(B)f (1)i , (4.6b)ϵ2 : −ω(B)f (2)i . (4.6c)Expansion of the FP operator is similarly performed by parts in Ω(FP)i = Ω(FP,0)i +ϵΩ(FP,1)i + ϵ2Ω(FP,2)i (wherein the perturbation order nϵ should not be confused withthose of the Hermite series), in which each of the orders are further decomposed bynϵ = {0, 1, 2},Ω(FP,nϵ)i = wi[ρ(nϵ) +vi,αv2Tȷ(nϵ)α +vi,αvi,β − v2T δαβ2v4TP(nϵ)αβ]. (4.7)914.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISThus, separation of orders simply yields,ϵ0 : ω(FP)Ω(FP,0)i , (4.8a)ϵ1 : ω(FP)Ω(FP,1)i , (4.8b)ϵ2 : ω(FP)Ω(FP,2)i , (4.8c)where the various functionals (3.46c)–(3.46h) are linear combinations of the perturbationseries of the macroscopic variables (4.1e)–(4.1j), such that the corresponding ϵ-orders arealso linearly retained.The next step is to recombine the different orders of the perturbation series of theadvection terms, and the BGK and FP operators, resulting in,ϵ0 : 0 = ω(B)(f(eq)i − f (0)i)+ ω(FP)wi[ρ(0) +vi,αv2Tȷ(0)α +vi,αvi,β − v2T δαβ2v4TP(0)αβ], (4.9a)ϵ1 :(∂(1)t + vi,δ∂(1)δ)f(0)i = −ω(B)δtf(1)i+ω(FP)δtwi[ρ(1) +vi,αv2Tȷ(1)α +vi,αvi,β − v2T δαβ2v4TP(1)αβ], (4.9b)ϵ2 :[∂(2)t +δt2(∂(1)t + vi,δ∂(1)δ)(∂(1)t + vi,ε∂(1)ε)]f(0)i+(∂(1)t + vi,δ∂(1)δ)f(1)i = −ω(B)δtf(2)i+ω(FP)δtwi[ρ(2) +vi,αv2Tȷ(2)α +vi,αvi,β − v2T δαβ2v4TP(2)αβ], (4.9c)to which end a third ansatz is made: there exists multiple roots in the zeroth-orderequality (4.9a) where f (0)i = f(eq)i is a candidate, which is conventionally the case inmulti-scale analyses of the Boltzmann equation. It is further inferred that for the equalityto hold, it must be required that,ρ(0) = 0, (4.10)ȷ(0)α = 0α, (4.11)P(0)αβ = 0αβ. (4.12)The zero equalities are the only information extracted from the zeroth-order relation fornow. Prior to taking the moments of the perturbation relations, the first- and second-924.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISorder relations are simplified. If f (1)i is factorized out of Ω(FP,1)i in (4.9b) and the notationΩ̂(FP,1)i ◦ f (1)i is adopted as the FP expression operating on the first-order populations,the material derivative of the equilibrium populations can be written as,ϵ1 : D(v,1)t f(0)i = f(1)i(ω(FP)δtΩ̂(FP)i −ω(B)δt), (4.13)employing the notation D(v,1)t = ∂(1)t +vi,δ∂(1)δ for conciseness, and recognizing that Ω̂(FP)iis in fact independent of the perturbation order. This result is useful for eliminating thefirst-order derivatives of f (0)i in (4.9c). Combining all prefactors of f(1)i by substitutingin this reformulated ϵ1 contribution yieldsϵ2 : ∂(2)t f(0)i +D(v,1)t(1 +ω(FP)Ω̂(FP)i2− ω(B)2)f(1)i= −ω(B)δtf(2)i +ω(FP)δtΩ(FP,2)i . (4.14)This allows us to again in a reverse manner recast the relation with f (1)i isolated in (4.13),such that the second-order contributions now obeyϵ2 : ∂(2)t f(0)i +(∂(1)t + vi,δ∂(1)δ)(∂(1)t + vi,ε∂(1)ε)δt×(1 +2− ω(B)2ω(FP)Ω̂(FP)i− 2 + ω(FP)Ω̂(FP)i2ω(B))f(0)i= −ω(B)δtf(2)i +ω(FP)δtΩ(FP,2)i , (4.15)where an equilibrium differential equation was obtained with f (0)i instead of f(1)i , wherethe non-equilibrium effects are administered through the ω(B) and ω(FP) coefficients off(0)i . The perturbation equations (4.9a), (4.13), (4.15) are now at a stage where theanalysis can be furthered by forming the moment equations.Moment equationsInitially, one can take the zero, second, and third moments of (4.9b), from whichthe continuity and momentum equation are obtained. The zeroth-moment equation isobtained by summing both sides of the relation over i, the first-moment equation bysumming over i multiplied by vi,ε, the second-moment equation by vi,εvi,ζ , and so forth.Thereto, the effects of perturbations on the different macroscopic observables can be934.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISinvestigated by including or excluding various moments. In the simplest analysis, theansatz is made that the non-equilibrium moments are nulled [367],Q−1∑i=0{f(neq)i , g(neq)i}={ρ(neq),E(neq)}= 0, (4.16a)Q−1∑i=0{f(neq)i , g(neq)i}vi,δ ={j(neq)δ , q(neq)δ}= 0δ, (4.16b)Q−1∑i=0{f(neq)i , g(neq)i}vi,δvi,ε ={P(neq)δε , R(neq)δε}= 0δε. (4.16c)for the non-equilibrium states in nϵ ≥ 1. This conventionally renders the continuityand Euler equations, and does not give any insight into non-equilibrium effects. As it isinteresting to investigate the non-equilibrium behaviour of the FP equation on smallerscales, it may be worthwhile to contest the conventional nullassumption of (4.16a–4.16c)and include the coarse first-order perturbations in the analysis. As it is known thatthe entropy-extremum principle corresponds to a minimization of energy [78], one couldinclude the contributions from E(1) =∑i g(1)i . Additionally, for thermally-driven phase-change processes such as pool boiling, it is apt to consider the fluctuations in the heatflux q(1)δ =∑i g(1)i vi,δ, and for cavitation which is a stress-induced phenomenon, themomentum flux P (1)δε =∑i f(1)i vi,δvi,ε. As the inclusion of P(1)δε in the second-momentequation of the ϵ-relation based on the Boltzmann equation results in the viscous-stresstensor τδε with the dynamic viscosity µ =(1/ω(B)−1/2)pδt [252], one could suspect thatω(FP) could be attributed to a pseudo-viscosity from the γ-scaling of thermal fluctuations.Thus, the analysis revolves around the same non-equilibrium moments and how they areaffected by the FP dynamics, assuming that the values and correlations of all non-equilibrium moments (apart from the unknown and non-zero P (1)δε moment) are zero,ρ(2).= 0, (4.17a)j(1)α = j(2)α = ȷ(1)α = ȷ(2)α.= 0α, (4.17b)P(2)αβ = P(2)αβ.= 0αβ, (4.17c)E(1) = E(2) = E(1)= E(2) .= 0, (4.17d)q(2)α = q(2)α.= 0α, (4.17e)R(1)αβ = R(2)αβ = R(1)αβ = R(2)αβ.= 0αβ, (4.17f)944.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISin addition to the non-zero counterpart,P(1)αβ ̸= 0. (4.18a)This ultimately allows us to take the zero-through-second moments of (4.9b, 4.14), whichusually are necessary to recover the NS equations. Moreover, for the FP contributions, theprimary interest is the non-equilibrium effects on the momentum and energy equations,and as such the density-fluctuations in the zero-moment continuity equation are omitted.As for the Boltzmann collision operator, the analysis relies on the usual methodology ofomitting non-equilibrium contributions from the zero- and first-order moments of theperturbation equations [361]. Moments of the first-order perturbation expansion yieldsthe (first-order) continuity, Euler, and a higher-order equation,∂(1)t ρ(eq) + ∂(1)δ j(eq)δ = 0, (4.19a)∂(1)t j(eq)δ + ∂(1)ε P(eq)δε = 0, (4.19b)∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ = −ω(B)δtP(1)δε +ω(FP)δtΩ̂(FP)P(1)δε , (4.19c)in which the zero-non-equilibrium moments of f (1)i were exploited, and∑Q−1i=0 fi factorizedout from Ω(FP), leaving the moment of the operational expression itself Ω̂(FP). It willlater be shown that the last moment equation, as in the LBM, recovers the viscous-stress tensor, albeit with additional stresses manifested by Ω̂(FP)P (1)δε . To that end, theexplicit formalism of P (1)δε is not recovered directly from (4.19c) but rather appears in thefirst-moment of the second-order perturbation expansion (4.15). Lastly, the equilibriummoments are defined from the equilibrium populations (3.2) as,ρ(eq) =Q−1∑i=0f(eq)i = ρ, (4.19d)j(eq)δ =Q−1∑i=0f(eq)i vi,δ = ρuδ, (4.19e)P(eq)δε =Q−1∑i=0f(eq)i vi,δvi,ε = ρuδuε + pδδε, (4.19f)Q(eq)δεζ =Q−1∑i=0f(eq)i vi,δvi,εvi,ζ = ρuδuεuζ + p[uδ]δεζ, (4.19g)954.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISwhere[uδ]δεζ= uδδεζ + uεδδζ + uζδδε. Applying the product rule to ∂(1)δ (ρuδ) in thecontinuity equation (4.19a) enables rewriting the material derivative as,D(1)t ρ = −ρ∂(1)δ uδ. (4.20)This material derivative appears in the product-rule expanded momentum equation con-sidering the equilibrium moments j(eq)δ and P(eq)δε , which after some algebra allows us torewrite the momentum equation as,D(1)t uδ = −1ρ∂(1)ε pδδε. (4.21)Noting that the same perturbation series apply to gi, its corresponding moment equationhierarchy is,∂(1)t E(eq) + ∂(1)δ q(eq)δ = 0, (4.22a)∂(1)t q(eq)δ + ∂(1)ε R(eq)δε = −ω(B)δtq(1)δ +ω(FP)δtΩ̂(FP)q(1)δ , (4.22b)where the moments of the equilibrium populations (3.3) read,E(eq) =Q−1∑i=0g(eq)i = 2ρE, (4.22c)q(eq)δ =Q−1∑i=0g(eq)i vi,δ = 2ρuδH, (4.22d)R(eq)δε =Q−1∑i=0g(eq)i vi,δvi,ε = 2ρuδuε(H + p/ρ)+ 2pHδδε, (4.22e)and H = E+ p/ρ = e+u2/2+ p/ρ is the total enthalpy. All of the equilibrium momentsexcept Q(eq)δεζ are symmetric tensors. To arrive at the NSF equations, the second gi-moment equation is not required, and is thus excluded. After expansion with the productrule, and substituting the van der Waals internal energy (3.48a), the derivative relationT(∂p/∂T)v= aρ2 + p on a VDW-EOS basis, and the rewritten momentum equation(4.20), it can be shown that the first-order contribution to the energy equation is,D(1)t T = −TρCv(∂p∂T)v∂(1)δ uδ. (4.23)964.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISFurthermore, an analogous pressure equation can be established by considering thatp = p(ρ, T ) where applying the corollary chain rule,D(1)t p =(∂p∂ρ)TD(1)t ρ+(∂p∂T)ρD(1)t T, (4.24)can be rewritten with the first-order continuity and energy equation contributions (4.20,4.23),as well as the speed-of-sound (3.53), yielding,D(1)t p = −ρς2s∂(1)δ uδ. (4.25)This relation will be exploited later to derive the viscous stresses in the moment equations.Now that the necessary zero and first-moment equations of fi, gi have been obtained,the next step is deriving an operational expression for the moment P (1)δε in the second-moment equation (4.19c). This last part of the analysis revolves around the second-orderperturbations (4.14) from which the zero moment becomes,∂(2)t ρ = 0, (4.26)and consequently one can conclude that the density predominantly evolves on the ϵscales via (4.20). Formulating the first-moment equation of the ϵ2 scales necessitatesmore algebraic work. First, it is noted that in the expanded first-moment equationstemming from (4.15) successive instances of the momentum equation (4.21) appear andsignificantly simplify the result to,∂(2)t ρuδ = ∂(1)ε(∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ)δt[(1ω(B)− 1)+(ω(FP)2ω(B)Ω̂(FP) − 1− ω(B)/2ω(FP)1Ω̂(FP))], (4.27)where the moment Ω̂(FP) of the operator is∑Q−1i=0 Ω̂(FP)i . Nevertheless, given that Ω̂(FP) =Ω̂(FP)(x,u, t) the relaxation-frequency-dependent terms cannot immediately be excluded,except for(1/ω(B) − 1), in the derivative terms, and thus it is needed to dissect theequation further. After some algebra it follows that the second-order contributions tothe momentum equation take the form,∂(2)t uδ =δtρ{(1ω(B)− 1)∂(1)ε(∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ)974.1. CHAPMAN-ENSKOG MULTISCALE ANALYSIS+ω(FP)2ω(B)∂(1)ε[Ω̂(FP)(∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ)+(P(eq)δε ∂(1)t +Q(eq)δεζ ∂(1)ζ)Ω̂(FP)]− 1− ω(B)/2ω(FP)∂(1)ε[(Ω̂(FP))−1 (∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ)+(P(eq)δε ∂(1)t +Q(eq)δεζ ∂(1)ζ)(Ω̂(FP))−1]}, (4.28)in which the superposition of the three prefactors of ∂(1)ε ( · · · ) recover the dynamicviscosity in the thermo-hydrodynamic limit which together with ∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζconstitute the viscous-stress tensor, as will be shown in the following. In the result threemajor components to the stresses can be observed: the ordinary Boltzmann-equationcontributions by(1/ω(B) − 1) as well as contributions from the FP operator with boththe material derivative of itself (either in the continuous (2.84) or derived Hermite-spaceformalism) linearly dependent on the moments, as well as more FPE-analogous materialderivatives of the moments operated on by the operator.In an analogous analysis of the perturbation expansion of gi, only the zero-momentis required for recovering the Fourier equation. Considering the perturbation expansionof not fi but gi in (4.15), the resulting moment equation simplifies to∂(2)t T =12ρCv{∂(1)δ(∂(1)t q(eq)δ + ∂(1)ε R(eq)δε)δt[(1ω(B)− 1)+(ω(FP)2ω(B)Ω̂(FP) − 1− ω(B)/2ω(FP)1Ω̂(FP))]− 2ρuδ∂(2)t uδ}, (4.29)which requires exploiting the thermodynamic pressure (3.48b), the internal energy (3.48a),the continuity equation (4.20), and the aforementioned pressure equation (4.25). Finally,Ω̂(FP) = Ω̂(FP)(x,u, t) necessitates further expansion yielding,∂(2)t T =12ρCv{δt(1ω(B)− 1)∂(1)δ(∂(1)t q(eq)δ + ∂(1)ε R(eq)δε)− 2ρuδ∂(2)t uδ+ δt(ω(FP)2ω(B))∂(1)δ[Ω̂(FP)(∂(1)t q(eq)δ + ∂(1)ε R(eq)δε)+(q(eq)δ ∂(1)t +R(eq)δε ∂(1)ε)Ω̂(FP)]− δt(1− ω(B)/2ω(FP))∂(1)δ[(Ω̂(FP))−1 (∂(1)t q(eq)δ + ∂(1)ε R(eq)δε)+(q(eq)δ ∂(1)t +R(eq)δε ∂(1)ε)(Ω̂(FP))−1]}. (4.30)This hierarchy of equations—specifically the continuity (4.20,4.26), the momentum (4.21,984.1. CHAPMAN-ENSKOG MULTISCALE ANALYSIS4.28), and the energy (4.23,4.30) equations—form the foundation for recovering the NSFequations by recombination of the perturbation components into the native series formu-lation (4.1).Thermo-hydrodynamic limit: the stochastic Navier-Stokes-Fourier equationsAt this stage, the stochastic Navier-Stokes-Fourier equations can be recovered fromthe hierarchy of perturbed moment equations by recombining each of the orders in ϵ.Merging (4.20,4.26), and recognizing that ϵ∂(1)t + ϵ2∂(2)t = ∂t and ϵ∂(1)δ = ∂δ yields thecontinuity equation,∂tρ+ uδ∂δρ = −ρ∂δuδ. (4.31)Similarly, the momentum equation becomes,ρDtuδ = −∂εpδδε − ∂ε(τδε + τ˜δε), (4.32a)where the viscous-stress tensor stemming exclusively from Boltzmann-type collisions is,τδε = −µ(B)(∂δuε + ∂εuδ − 2D(∂ζuζ)δδε)− ζ(B)(∂ζuζ)δδε. (4.32b)In addition to this “traditional” stress tensor, the analysis also yields the stochastic stress-tensor components (4.32g)—displayed on the next page—perturbed directly by the FPoperator and its inverse, separated by complementing perturbative components (denotedby “+” and “−” subscripts) that concurrently amplify and attenuate stresses (assumingΩ̂(FP) > 1).In (4.32b,4.32g) the viscosities manifest due to both the Boltzmann and FP relaxationfrequencies for the deterministic viscous stresses,µ(B) =(1ω(B)− 1)pδt, (4.32c)ζ(B) =(1ω(B)− 1)(D + 2D− ρς2sp)pδt, (4.32d)and the stochastic stresses,µ(FPB)+ =(ω(FP)2ω(B))pδt, µ(FPB)− =(−1− ω(B)/2ω(FP))pδt, (4.32e)994.1.CHAPMAN-ENSKOGMULTISCALEANALYSISζ(FPB)+ =(ω(FP)2ω(B))(D + 2D− ρς2sp)pδt, ζ(FPB)− =(−1− ω(B)/2ω(FP))(D + 2D− ρς2sp)pδt, (4.32f)respectively. Thus, four pair-wise complementing, coarse-grained “stochastic” stress tensors in (4.32g) are obtained: twotraditional stress tensor components (I,II) pre-factored by the FP-operator Ω̂(FP) and its inverse counterpart 1/Ω̂(FP). Theremaining two viscous-stress components (III,IV) are due to the spatial and temporal derivatives of Ω̂(FP) in (4.32g), andaccount for the diffusion contributed from the FP operator itself as it varies due to external forces (via v(η) in the ∂v(η)ζterms), thermal fluctuations (via v2T in the ∂v2T terms), and continuum velocity (u in the ∂uζ terms), respectively. It is dulynoted that by stochastic it is implied that the dynamics of the Langevin equation is projected onto the velocity space of theFPE rather than computed directly as discrete, stochastic events sampled at random from a characteristic Gaussian whitenoise distribution, and that this noise permeates the stresses. The term stochastic is adopted rather than “fluctuating” to notconfuse the current methodology with the aforementioned fluctuating-hydrodynamics theory originating from Landau andLifshitz [322].τ˜δε =(I)︷ ︸︸ ︷Ω̂(FP)[−µ(FPB)+(∂δuε + ∂εuδ − 2D(∂ζuζ)δδε)− ζ(FPB)+(∂ζuζ)δδε]+1Ω̂(FP)[−µ(FPB)−(∂δuε + ∂εuδ − 2D(∂ζuζ)δδε)− ζ(FPB)−(∂ζuζ)δδε] }(II)+ µ(FPB)+P(eq)δεp{[(∂v(η)ζΩ̂(FP))Aζρς2sγm+(∂v2TΩ̂(FP)) kBTmρcv(∂p∂T)v] (∂ζuζ)+(∂uζ Ω̂(FP)) 1ρ(∂δp)δδζ} }(III)+ µ(FPB)−P(eq)δεp{[(∂v(η)ζ1Ω̂(FP))Aζρς2sγm+(∂v2T1Ω̂(FP))kBTmρcv(∂p∂T)v] (∂ζuζ)+(∂uζ1Ω̂(FP))1ρ(∂δp)δδζ}︸ ︷︷ ︸(IV). (4.32g)1004.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISIn obtaining the velocity gradient and divergence terms in the stresses, i.e.(∂δuε +∂εuδ − 2/D(∂ζuζ)δδε)and(∂ζuζ)δδε, the expanded formulation,∂(1)t P(eq)δε + ∂(1)ζ Q(eq)δεζ = p(∂(1)δ uε + ∂(1)ε uδ)+(p− ρς2s)∂(1)ζ uζδδε, (4.33)previously identified in (4.28) was exploited. The stochastic stresses arise analogouslyto the stresses by the Boltzmann operator. The recombination resulted in the auxiliarycomponents with the derivatives(P(eq)δε ∂(1)t +Q(eq)δεζ ∂(1)ζ){Ω̂(FP); 1/Ω̂(FP)}imposed onthe FP operational expressions and its inverse counterpart, respectively. This form canbe recast to a convenient material-derivative basis as Q(eq)δεζ = P(eq)δε uζ and thus,(P(eq)δε ∂(1)t +Q(eq)δεζ ∂(1)ζ)= P(eq)δε D(1)t . (4.34)This trick enables reuse of the chain rule to effectively rewrite P (eq)δε D(1)t{Ω̂(FP), 1/Ω̂(FP)}where it is noted that Ω̂(FP) = Ω̂(FP)(un,v(η), vnT (T ))is a function of velocity polynomialsof various even orders n,D(1)t Ω̂(FP) =(∂Ω̂(FP)∂uζ)v(η)ζ ,v2TD(1)t uζ+∂Ω̂(FP)∂v(η)ζv2T ,uζD(1)t v(η)ζ+(∂Ω̂(FP)∂v2T)uζ ,v(η)ζD(1)t v2T . (4.35)Therein, one can in addition to (4.21) identify the following material derivatives derivedpreviously by reformulating v2T , v(η)ζ . Firstly, the previously derived material deriva-tive (4.23) can be exploited which combined with the direction-independent, root-mean-square formulation of the average thermal molecular speed,v2T.=kBTm, (4.36)for molecules of mass m, yields,D(1)t v2T =kBmD(1)t T = −kBTmρcv(∂p∂T)v∂(1)ζ uζ . (4.37)1014.1. CHAPMAN-ENSKOG MULTISCALE ANALYSISAs a force model has not been specified in this study Newton’s second law of motionηζ = Fζ/m and (4.25) can be used to formulate the material derivative of v(η)ζ ,D(1)t v(η)ζ = −Aζρς2sγm∂(1)ζ uζ , (4.38)where the unit area Aζ was further introduced as a consequence of converting the forceinto a pressure p = Fζ/Aζ , such that Aζ in a simulation scenario would be the grid-cellcross-sectional area. The non-ideal speed of sound ς2s was previously found in (3.53).Consequently, the explicit result is obtained for,P(eq)δε D(1)t Ω̂(FP) = −P (eq)δε{[(∂v(η)ζΩ̂(FP))Aζρς2sγm+(∂v2TΩ̂(FP)) kBTmρcv(∂p∂T)v](∂ζuζ)+(∂uζ Ω̂(FP)) 1ρ(∂δp)δδζ}, (4.39)and similarly for the 1/Ω̂(FP) destructive components with µ(FPB)− , where in both cases∂(1)ζ was recombined according to the previously defined perturbation series.Considering (4.30) wherein (4.28) is substituted results in the Fourier equation. Theshear velocities arise from the derivatives of the moments,∂(1)t q(eq)δ + ∂(1)ε R(eq)δε = 2(p− ρς2s)∂(1)ζ uζuδ + 2puε(∂(1)ε uδ + ∂(1)δ uε)+ 2p∂(1)δ h, (4.40)where h = e+ p/ρ is the enthalpy. As noted by Reyhanian et al. [252, 302] the resultingheat flux would be q = −µ∇h and consequently the gi-kinetic equation needs to beaugmented with an energy correction term M0 = 2∂α(−µ∂αh + k∂αT ) to recover thecorrect Fourier law. For real gasses ∂αh = cp∂αT + v(1 − β)∂αp with β = ρ(∂v/∂T )pbeing the thermal-expansion coefficient. After applying the product rule to the shear-velocity terms, the substituted result can be recombined with the first-order perturbationcontribution (4.23), confirming the conservation of the Fourier equation,ρcvDtT = −(τδε + τ˜δε)∂δuε − T(∂p∂T)v∂δuδ − ∂δqδ, (4.41)where the normal and stochastic shear stresses organize as expected in the same formu-lation of the tensors (4.32b,4.32g).1024.2. FLUCTUATION-DISSIPATION THEOREMIn summary, the compressible, stochastic Navier-Stokes-Fourier hierarchy of equationsare recovered:Dtρ = −ρ∇ · u, (4.42a)ρDtu = −∇p−∇ · (τ + τ˜), (4.42b)ρcvDtT = −(τ + τ˜) : ∇u− T(∂p∂T)v∇ · u−∇ · q, (4.42c)in which the Fourier heat flux is defined as q = −k∇T . To enable simulations, thepresented viscosities are superimposed into effective shear and volume viscosities,µ = µ(B) + µ(FPB)+ + µ(FPB)−=[(1ω(B)− 1)+(ω(FP)2ω(B))−(1− ω(B)/2ω(FP))]pδt, (4.43)ζ = ζ(B) + ζ(FPB)+ + ζ(FPB)−=[(1ω(B)− 1)+(ω(FP)2ω(B))−(1− ω(B)/2ω(FP))](D + 2D− ρς2sp)pδt, (4.44)where known initial values of µ, ζ can be defined and subsequently used to tune thetime-step size δt and the kinetic relaxation frequencies ω(B), ω(FP)(γ), which in turn setsthe thermal noise intensity governed by the Langevin equation. This result, along withthe kinetic equations (3.4), constitute the foundation of the numerical method developedwithin this dissertation.4.2 Fluctuation-dissipation theoremAs a preliminary theoretical verification of the derived model, the fluctuation-dissipationtheorems (FDTs) for the stochastic-stress tensor τ˜ (4.32g) are presented and compared tothe analogous FDT of the stress tensor used in Landau-Liftshitz fluctuating-hydrodynamicstheory in space and time [328, 368] namely,〈τ˜αβ(x, t)〉= 0αβ, (4.45)〈τ˜αβ(x, t)τ˜δε(x′, t′)〉= 2kBT[µ(δαγδβδ + δαδδβγ − 2Dδαβδγδ)+ ζδαβδγδ]δ(x− x′)δ(t− t′), (4.46)1034.2. FLUCTUATION-DISSIPATION THEOREMwhere the shear and volume viscosities correspond to µ(B), ζ(B) in (4.32b) originatingfrom the BGK model. To do this the behaviour of a perturbed quiescent homogeneousequilibrium state is studied at ρ(eq) and a zero macroscopic velocity u = 0α. If onlylong-wavelength deviations from the equilibrium are considered one can simplify the non-linear NSFEs into their corresponding linear forms where the macroscopic observables arelinearly approximated as ρ(x, t) = ρ(eq) + δρ(x, t), p(eq) + δp(x, t) [369]. By substitutioninto (4.42a,4.42b) the linearized stochastic hydrodynamic equations are achieved,∂tδρ = −ρ(eq)∇ · u, (4.47)ρ(eq)∂tu = −ς2s∇δρ−∇ · (τ + τ˜), (4.48)in which the bulk pressure difference was expressed in terms of the speed of sound δp ≃ς2s δρ from its fundamental definition ς2s ≡ (∂p/∂ρ)s [9]. At this stage it is necessary toisolate and decouple the longitudinal and transverse components of the velocity u andstochastic stress tensor τ˜ , which is facilitated by Fourier transformations of δρ,u, τ˜ . Thedensity Fourier transform is carried out in space asδρ(k, t) =(12π)3 ∫dx δρ(x, t) exp(ik · x), (4.49)and likewise for u and τ˜ . Thus, Fourier-transforming the continuity equation results in[369, 370],∂tδρ− iρ(eq)k · u. (4.50)Therein, u = u(k, t) can be further separated into its constituent longitudinal and trans-verse (perpendicular to k) modes,u = u∥k̂ + u⊥, (4.51)in which the normalized wave-vector reads k̂ ≡ k/|k|. Given the identity k ·u(k, t) ≡ ku∥[369], where by the definition k · u⊥ ≡ 0, the continuity equation finally becomes,∂tδρ− iρ(eq)ku∥ = 0. (4.52)1044.2. FLUCTUATION-DISSIPATION THEOREMAs Gross et al. [328] showed for (4.48), the analogous wave equations for the longitudinalu∥ and transverse u⊥ ≡ u ·(I − k̂k̂) velocities, after some amendment, simplify to,∂2t u∥ = −k2ς2su∥ − ν∥k2∂tu∥ +ikρ(eq)∂tτ˜∥, (4.53)∂tu⊥ = −ν⊥k2u⊥ + ikρ(eq)τ˜⊥, (4.54)in which the longitudinal and transverse stochastic stress components are τ˜∥ ≡ k̂ · τ˜ · k̂and τ˜⊥ ≡ k̂ · τ˜ ·(I − k̂k̂), respectively, and the associated kinematic viscosities are,ν∥ =[ζ(B) + µ(B)(2− 2D)]1ρ(eq), ν⊥ =µ(B)ρ(eq). (4.55)It is further noted that the above results hold true for homogeneous fluids without surfacetension forces, as well as square-gradient fluids since the square-gradient tension termρ(eq)κk2 would absorb into the speed of sound ς2s (k) = ς2s+ρ(eq)κk2 [328, 371]. By Fouriertransforming the longitudinal wave equation in time, it is observed that with the abovefoundation one arrives at the same linear-response function in frequency space ω (not tobe confused with the relaxation frequencies ω(B), ω(FP)) as in fluctuating hydrodynamicstheories,u∥(k, ω) =ωkρ(eq)(ω2 − k2ς2s − iων∥k2) τ˜∥(k, ω), (4.56)in which τ˜∥(k, ω) acts as a perturbation on the longitudinal susceptibility χ∥ such thatu∥(k, ω) ≡ χ∥(k, ω)τ˜∥(k, ω). In Fourier space the purpose is to determine the variancematrix Aαβ(k) which correlates the white-noise FDT〈|τ˜αβ(k, ω)|2〉 = Aαβ(k). To obtainthis, the analysis further follows Gross et al. who defined the equal-time correlationfunction from the inverse Fourier transform F−1t of the modulus-square-mean of u∥,〈|u∥(k, t = 0)|2〉 = A∥2π∫dω |χ∥(k, ω)|2, (4.57)in which A∥ is the longitudinal component of Aαβ . Then, following contour integrationof∫dω |χ∥(ω)|2 = π/(ρ2(eq)ν∥)and assuming thermal equilibration, and equipartition ofkinetic energy〈|u∥(k)|2〉 = kBT/ρ(eq), it is found that the FDT in Fourier space for thelongitudinal stochastic stresses reads,〈|τ˜∥(k, ω)|2〉 = 2kBTρ(eq)ν∥, (4.58)1054.2. FLUCTUATION-DISSIPATION THEOREMand similarly for the transverse components:〈|τ˜⊥(k, ω)|2〉 = 2kBTρ(eq)ν⊥. (4.59)Finally, inverse-Fourier transforming these components back to real space and time resultsin the classical, uncorrelated FDT (4.46) with µ = µ(B), ζ = ζ(B), which is valid forsimple, homogeneous fluids with white thermal noise and viscosities independent of k[91, 328, 369].106Chapter 5Numerical testsThe LFPBM described in chapter 3 is not yet fully validated and needs more investi-gation into its theoretical consistency, especially when it comes to correct equilibration ofthe thermal noise. Moreover, due to PonD being an integral part of the kinetic model, thenumerical implementation in code is significantly more complicated than a conventionalLBM with exact streaming between lattice nodes. It is, in particular, challenging to im-plement the gauge-transformation and reconstruction kernel into the predictor-correctorin a manner that yields fast and unconditional convergence in the numerical solution ofthe predictor-corrector scheme. Furthermore, the implementation of boundary conditionsis trivial in conventional LB models with exact streaming, but the off-lattice nature ofthe recast discrete particles v makes imposing boundary conditions much more complex.Nevertheless, a preliminary numerical test of the method at its current state is carriedout in this chapter, which is intended to demonstrate the method as well as to inform itsfuture development.In fluctuating LBMs, as presented in the literature review of [J4] and in §2.4, simula-tion relies on frequent sampling of probability distributions governing the thermal noisein order to impose the effect of thermal noise on the macroscopic variables. In contrast,for the LFPBM developed in this dissertation, the thermal noise is induced by the veloc-ity space of the model, i.e. indirectly via ∂v in the FPE-drift and diffusion terms, whichvary spatially and temporally in the lattice. As the discrete velocities are coupled to thethermodynamics by semi-Lagrangian diffusion, any thermodynamic non-uniformity willinduce thermal fluctuations.5.1 Numerical methodDuring the Ph.D. project, the author wrote a C++ software to solve the LFPE withSLD. The code is configured using CMake and supports prospective development includ-ing multi-core parallelization with MPI and thread-optimization with OpenMP, as well asgraphics-processing unit (GPU) acceleration by OpenACC and CUDA. Furthermore, unittests can be performed with gtest (Google Test). Computation output is organized in1075.1. NUMERICAL METHODthe HDF5 format for data visualization. The code further can compile documentationin html, latex (publishable on that platform ReadTheDocs), as well as PDF. The codeis fully transient, and allows for D2Q9 lattices of arbitrary size. All memory struc-tures are vectorized and formatted to double precision (64-bit) and can relatively easilybe reformatted to single precision. Population arrays are contiguous in fi, gi and thenthe first spatial dimension, and macroscopic variables are contiguous in the first spa-tial dimension. Although the code is configured to support multi-core parallelization,thread-optimization, and acceleration via GPUs, its present naïve implementation is notyet parallelized.In the following numerical test, the previously presented semi-Lagrangian diffusionscheme (§3.2) is employed to iteratively advect and collide populations with the LFPE(3.18). For now, BGK collisions are excluded. To eliminate Galilean variance, the trans-formation procedure laid out in §3.1.1 is adopted. To interpolate off-lattice populations,a simple Lagrange kernel is used. The thermodynamics are recovered by the van derWaals EOS described in §3.4. The configuration of the computations are summarized inTab. 5.1.5.1.1 Computational domain and boundary conditionsFor the numerical tests, a (m+2mgn)×(n+2ngn) computational domain is configured,where m and n are the number of lattice nodes in the first and second spatial dimensionsand mgn = 2 and ngn = 2 are the number of ghost nodes in each of the two separatelayers per dimension. An aspect ratio of 1 (i.e. m/n = 1) is used, allocating a lattice-node spacing with δx = L/m = H/n where L,H are the total length and height of thedomain. The grid is uniform and Cartesian. Periodic boundary conditions tailored tothe discrete velocities vi are applied along the lattice boundaries. The lattice-setup isTable 5.1: Model solution configuration.Condition ValueLattice model D2Q9Lattice temperature TL 1/3Quadrature weights wi [4/9, 1/9, 1/9, 1/9, 1/9, 1/36, 1/36, 1/36, 1/36]TPeculiar velocity components cx [0, 1, 0,−1, 0, 1,−1,−1, 1]TPeculiar velocity components cy [0, 0, 1, 0,−1, 1, 1,−1,−1]TImplicit solution Semi-Lagrangian diffusionGauge transformations By moment matchingReconstruction kernel Lagrange polynomials (D.6)EOS VDW (3.48)1085.1. NUMERICAL METHODTable 5.2: 2D computational domain—lattice dimensions in lattice-units (LU).Variable ValueL 128H 128m,n,mgn, ngn 128, 128, 2, 2nnode 16384δx = δy 1δt 0.2No. of populations 2× 147456illustrated in Fig. 5.1 and summarized in Tab. 5.2. The temporal dimension is alsouniformly discretized with δt = T/nt being the time step size over a total period T withnt number of time steps.mLL nFigure 5.1: Computational domain.The periodic boundaries follow the conventional LBM-interpretation of joining op-posed boundaries in a torus as illustrated in Fig. 5.2, but require more sophisticatednumerics, as the semi-Lagrangian diffusion must also be carried out in the boundarynodes with populations streaming across the periodic boundaries. In the torus, dataflows pairwise across the joined north-south (→) and west–east (↑) boundaries. How-ever, as the origin of all Q populations at each boundary node is predicted by the localthermo-hydrodynamics, the populations must be more carefully tracked, especially atcorner nodes.5.1.2 Initial-value problemTo establish the initial condition for the tests, fitting of the VDW EOS to the thermo-physical properties of water, nitrogen, methane, carbon dioxide, and hydrogen was at-1095.1. NUMERICAL METHODFigure 5.2: The stitching of periodic boundaries are illustrated of the folding of a 2D plane into atoroidal shape, connecting the north-south and west-east faces in pairs.tempted across a broad range of T, ρ in the liquid and supercritical regimes. The outputof the EOS was compared with data available from the National Institute of Standardsand Technology (NIST) [309]. Despite the VDW EOS being highly favoured in recentreviews [372] for liquids and liquid mixtures with appropriate mixing rules relevant topetro-chemical industries, it is the experience of the author that it is difficult to findthermodynamic conditions in the EOS that are consistent with the NIST database. Assuch, establishing physical conditions based on the EOS for the selected liquids is diffi-cult. On the basis of the critical states of the substances, their characteristic interactionstrength and excluded volume parameters a, b (3.55) were found. However, for p = 0,the EOS unconditionally returned negative pressures. To counteract this, one can offsetthe thermodynamic pressure (3.48b) by a positive base pressure p, but this in turn yieldsnegative internal energies (3.48a), unless either a is arbitrarily reduced or cv increasedto a value not representative of the pertinent liquid. Thus, a VDW fluid is opted forwith parameters configured to obtain a numerically reliable solution, acknowledging thatthe parameters may not correspond to a realistic thermo-physical condition for a specificliquid. This selection is justified on the basis that the present chapter aims at conduct-ing a numerical test of the developed method as a first step towards a fully-consistentphysical model. Together with the thermo-physical properties given, the initial conditionof the test is configured as summarized in Tab. 5.3. The computational domain is occu-pied by a quiescent fluid with an overlaid Gaussian “impulse” in the centre defined by astandard deviation σρ and a mean of 1.05 × ρ0. The LFPB solution can be sensitive toinconsistent implementations of the initial fields or strong spatial discontinuities withinthe initial fields. To that end, the initial conditions were constructed via the followingprocedure:1. Define the initial uniform state with independent T0(x, t0), ρ0(x, t0);2. Overlay the Gaussian impulse into the density field ρ0;3. Compute the equilibrium populations for density f (eq)i (ρ0) (3.2);1105.1. NUMERICAL METHODTable 5.3: Universal physical constants and thermo-physicalproperties and initial conditions of the VDW fluid.Parameter ValueFP γ 0.5kB/m 10EOScv, R [1, 1]Ta (3.55a) 0.1b (3.55b) 0.5p 0Initialfields Density ρ0(x, t0) (free variable) 1Std. dev. σρ 0.1Temperature T0(x, t0) (free variable) 1Pressure p0(x, t0) (3.48b) 1.9Velocity u0(x, t0) 04. Compute the pressure field p0(T0, ρ0) (3.48b);5. Compute the internal and total energy fields e0(T0, ρ0) (3.48a);6. Compute vi =√p0/(ρ0RTL)ci + u0 (3.1), where u0(x, t).= 0;7. Compute the equilibrium populations for energy g(eq)i (ρ0, e0, p0,vi) (3.3);8. Compute total energy from the moment E0 = (2ρ0)−1∑Q−1i=0 g(eq)i (4.22c). Asu0 = 0 it follows that E0 ≡ e0;9. Compute T0(e0, ρ0) = (e0 + aρ0)/cv by rearrangement of (3.48a);10. Set fi(x, t0) = f(eq)i and gi(x, t0) = g(eq)i .Performing the calculations once does not ensure that the Gaussian shape is fully carriedover in T, p, e, E, fi, gi. Therefore, steps 4–10. should be repeated until the Gaussianshape is correctly imposed in all the variables. The LFP operators without externalforces (3.42) are employed, i.e. representative of (2.84), and it is noted that in the(initially) quiescent fluid with u(x, t0) = 0 and consequently jα = 0α, the operatorsassume the values in Figs. 5.3 and 5.4. The values for non-zero velocity components areexemplified in Figs. 5.6 and 5.7.1115.1. NUMERICAL METHODFigure 5.3: FP operator parameter space for fi (top row) and gi (bottom) at u = 0.Figure 5.4: Moments ∆ρ =∑Q−1i=0 Ω(FP)i (f),∆E =∑Q−1i=0 Ω(FP)i (g) of the operators at u = 0.5.1.3 Solution methodGauge transformationIn code, two separate “seed” and “advected” state arrays for each of p, ρ,u, f, g arerequired. These are denoted by p and p′ for pressure as an example, where the primedenotes the converging state λ′(x) at t, whereas the seed gauge is found at t − δt. Inthe solution algorithm the seed gauge is defined once at the start of a time step, and theλ′ gauge adapts to the conditions imposed by the seed gauge. The seed gauge remainsintact until it is overwritten by the converged gauge at the completion of a timestep.Until a swapping algorithm equivalent to conventional LBMs is formulated, it is veryimportant to carefully organize these two seed and advected states in separate arrays forthe variables used in the gauge transformation.The gauge transformation is carried out as follows. At the start of a timestep, priorto initiating the predictor-corrector scheme, the stencil used to reconstruct populations1125.1. NUMERICAL METHODFigure 5.5: FP operator parameter space for fi at u = [0.5, 0]T .at departure points is populated. Currently, an adaptive reconstruction stencil tailoredto each specific departure point is employed. As such, two distinct stencils are used;a collocation stencil (with collocation points coinciding with the lattice nodes) and thereconstruction stencil, where the latter is a subset of the collocation stencil. The colloca-tion stencil is described as the set F(xp) containing mp×np× [ρ, T, p,u, Q× fi, Q× gi]T .As a first step, for a current monitoring point, the purpose is to compute mp, np toestablish temporary data structures separate from the seed and advected arrays. Thisis done by considering the degree of the used reconstruction stencil. A fourth-degreeLagrange kernel requires 4×4 surrounding lattice nodes to interpolate population valuesat a departure point. In order to have each of the Q departure points centred withinthis reconstruction stencil, the extents of the collocation stencil can be established byrounding the departure point coordinates. E.g. the south-west corner can be found fromthe departure point of the fifth population (as this one will always be the populationwith the south-west most origin) by,x(min)p = ⌊xd(i = 5)⌋ − 1,1135.1. NUMERICAL METHODFigure 5.6: FP operator parameter space for gi at u = [0.5, 0]T .and the north-east corner by,x(max)p = ⌈xd(i = 7)⌉+ 1 = ⌊xd(i = 7)⌋+ 3.Thus, if the departure points fall within a distance of |x−xd| < δx from the monitoringpoint and Lagrange interpolation is used, the collocation stencil size will bemp = np.= 5.Alternatively, a fixed stencil could be used where all departure points use the samecollocation stencil for population reconstruction [373].Next, the established collocation stencil is populated with seed-gauge variables nec-essary for transforming the seed populations fλpi , gλpi to the adaptive gauge λ′ of thecurrent monitoring point by the transformation (3.9). It is here emphasized that thepopulations are assumed to have been recast by the FP operator evaluated on the RHSof the trapezoidal balance in (3.18). The transformation is likely the most costly opera-tion among the numerical computations and is carried out once per predictor-correctorstep. It is possibly worthwhile to employ projection operators such as Grad’s projec-tion [358, 359]. When all the populations in the temporary collocation stencil have beengauge transformed, they can be used to consistently reconstruct populations at departure1145.1. NUMERICAL METHODFigure 5.7: FP operator parameter space for fi and gi at u = [0, 0.5]T .1155.1. NUMERICAL METHOD-3 -2 -1 0 1 2 3-3-2-10123SpaceySpace xf5F(xp)ΛFigure 5.8: Collocation (bounded by the dashed blue line) and Lagrange reconstruction(shaded blue area for xd(i = 5)) stencils relative to departure points (red bullets).points fλ′i (xd, t− δt).ReconstructionThe reconstruction scheme (3.21) with the Lagrange kernel Λ in (D.6), is used for2 × Q instances, per monitoring point. Given that the polynomials in the kernel arecubic, the numerical realization in code should be treated carefully as extrapolationoutside of the kernel is likely to cause the solution to diverge. The converged solution infiλ′(xd, t− δt) and gλ′i (xd, t− δt) is overwritten into the aforementioned advected arraysfor populations f ′i(x, t) and g′i(x, t), and used to update the moments and remainingthermo-hydrodynamic variables prior to treating the LHS of the trapezoidal balance in(3.18). The collocation and reconstruction stencils are illustrated in Fig. 5.8. An exampleof the Lagrange kernel applied to reconstruct a population is depicted in Fig. 5.9. Eachof the 4× 4 Lagrange polynomials are illustrated in Fig. F.1.Boundary treatmentBoundary treatment with PonD is more complicated than in traditional LBMs withexact streaming. Even the “simplest” periodic boundary condition becomes prone tobugs in code as the reconstruction has to be carried out across boundaries. To that end,two strategies have been attempted. In the first more complicated variety, the departurepoints associated with each boundary node are used to identify which populations origi-nate from outside and inside the computational domain. The departure points that falloutside of the domain are projected into the fluid domain originating from an opposingperiodic boundary and used to identify the reconstruction stencil inside the fluid domain,1165.1. NUMERICAL METHODFigure 5.9: Fourth-degree Lagrange kernel used to reconstruct a population with an arbitrarilypopulated reconstruction stencil. Bullets are collocation point values.which can then be copied to the ghost nodes surrounding the boundary node currentlybeing treated.An alternative approach is adopted, more aligned with periodic boundary conditionsin the LBM [361], the boundary treatment comprises populating the ghost-node layerswith all seed-gauge populations and macroscopic variables from opposing fluid node lay-ers. I.e. given two ghost-node layers (mgn = ngn = 2) encompassing a domain of sizem × n, data for the southern-most ghost nodes at integer locations with y-componentlattice index iytrgt would be fetched from lattice nodes adjacent to the northern boundarywith y-component lattice index iysrc = iytrgt + ngn + n. This approach requires no logicstatements, is faster than the previous, and is much less susceptible to bugs.Summary: execution order1. Configure the initial condition at t0: u(x) = 0, Gaussian discontinuity in ρ(x),establish remaining macroscopic fields with EOS;2. Enter time loop:(a) Prior to starting the predictor-corrector, compute Ω(FP)i (fi) and Ω(FP)i (fi);(b) Recast fi ← fi − δt2 Ω(FP)i (fi) and gi ← gi − δt2 Ω(FP)i (gi) across entire lattice,including ghost nodes;(c) Recompute moments ρ,E and the EOS-derived macroscopic observables u, e,T, p prior to entering the predictor-corrector loop;(d) Initiate SLD on a per-lattice node basis (excl. ghost-nodes); for the currentmonitoring point, compute the number of collocation points required to re-construct all 2 × Q populations at their respective departure points. After1175.2. RESULTSthis, populate the stencil with the moments ρ, jα, Pαβ,E, qα, Rαβ as well asT, p,u, f, g;(e) Execute the predictor-corrector loop with a specified number of iterations:i. Perform the predictor step:A. For each collocation point, transform f and g to the gauge of thecurrent monitoring point (λp → λ′);B. For each of the Q departure points, reconstruct fi and gi with thepertinent interpolation kernel Λ. These are the advected populationsf ′i and g′i.ii. Perform the corrector step:A. Compute Ω(FP)i (f′i) and Ω(FP)i (g′i);B. Recast f ′i ← f ′i + δt2 Ω(FP)i (f′i) and g′i ← g′i + δt2 Ω(FP)i (g′i);C. Recompute moments ρ′,E′ and the derived macroscopic observablesu′, E′, e′, T ′, p′;D. Monitor convergence (optional).iii. Exit the loop when the solution has converged;(f) Advance time t← t+ δt;(g) Across all fluid nodes, reset the seed-gauge populations fλ0i (t − δt) = f ′i andgλ0i (t− δt) = g′i, and similarly the macroscopic variables pλ0 , ρλ0 ,uλ0 ;(h) Treat boundary conditions.(i) Save the macroscopic solution fields to disk for data visualization and post-processing.5.2 Results5.2.1 Fokker-Planck-perturbations in a Gaussian impulseFirst, the convergence in the predictor-corrector scheme is reported. Here, it wasfound that the scheme in both SL advection and diffusion, with and without the FPoperator, converges to single-precision within two-three iterations, and machine doubleprecision within three-five iterations. This has also been the experience in fallacious im-plementations of the FP operator. The predictor-corrector scheme is very reliable albeitcomputationally costly. The percentage convergence rate for SLD with γ = 0.5, δt = 0.1,at the centre lattice node, is reported in Fig. 5.10. After six-seven iterations the per-1185.2. RESULTScentage change falls below machine precision. Even though the scheme has been quitedependable throughout this research, a more formal analysis of convergence rates fordifferent reconstruction and transformation strategies, variable δt, and anomalous, non-uniform and highly disequilibrated fields within the stencils, should be carried out.Figs. 5.11–5.13 show the total energy, density, temperature, pressure, velocity fields,as well as the centerline cross-section of the scalar fields compared with the initial con-dition and bare PonD advection with γ = 0 (no FP action). The preliminary results areapparently affected by a numerical anomaly wherein the total energy and density evolveasymmetrically despite all variables being initialized as Gaussian distributed with radialuniformity. The origin of this anomaly remains to be identified.Furthermore, Fig. 5.13 illustrates that mass and energy are not conserved (to the tuneof ∼ O(0.1%/100δt) in ρ and E) in the current implementation of the semi-Lagrangiandiffusion with FP action. A suspected source for the artifact is that the adaptive La-grange stencil reconstructs populations preferentially from one spatial dimension overthe other. Possibly, the effect is further compounded by the magnitudes of lateral op-erator components(Ω(FP)1−4)being greater than the diagonal components(Ω(FP)5−8), owedpartially to the values of the quadrature weights wi. As noted by Zakirov et al. [374]adaptive stencils are prone to non-conservativity, and fixed stencils are generally moreconservative. But otherwise, the solution strategy is reliable so long as the macroscopicfields are relatively continuous.5.2.2 Fokker-Planck-perturbations in a noisy initial conditionThe discontinuity in the previous case was finely resolved. To establish the sensitivityin the thermal fluctuations to spatial resolution a highly non-uniform case is now pre-Figure 5.10: Convergence rate in the semi-Lagrangian diffusion scheme in pressure, density, andtemperature. Percent-wise change is computed as ∆%φ(i) =∣∣φ(i)− φ(i− 1)∣∣/φ(i− 1)× 100%.1195.2. RESULTSFigure 5.11: Isolines of total energy, density, temperature, and pressure fields after 50δt for γ = 0.Figure 5.12: Isolines of velocity field u(x) components, and velocitymagnitude |u| = (u2x + u2y)1/2 after 50δt for γ = 0.sented. In this, the same configuration as previously outlined in Tab. 5.3 is used exceptthat ρ0 is set to 0.04 which has been gauged to be the stability limit for the fluctuationswhere they are strongest according to the non-linear region around the value in Ω(FP)i(Fig. 5.4). The initial field comprises uniform fields for x > δx/2 × m, y > δx/2 × nand otherwise the density field is sampled from a random Gaussian distribution with amean of 0 and and a standard deviation of 0.01. The initial condition is evolved withsemi-Lagrangian diffusion for 75δt, after which the solution results in the fluctuatingthermodynamic fields and hydrodynamic fields in Figs. 5.14 and 5.15, respectively. In1205.3. DISCUSSIONFigure 5.13: y-centerline cross-section of Fig. 5.11 compared with the initial conditionand PonD with (γ = 0.5) and without (γ = 0) a FP operator.the datasets ghost-nodes are included. From the figure, it is evident that diffusion wavesemanate from the initialized fluctuating fields. Due to the non-ideality in the EOS, thepressure field is the most sensitive to anomalous fluctuations, which do occur throughoutthe simulation. It could be expected that there would be more diagonal diffusion in allthe fields, rather than the clear advective patterns in the current results. The anisotropicdiffusion could be a numerical artifact from inadequate DmQn resolution and/or fromthe previously mentioned adaptive stencil causing false advection. Switching to a fixedstencil might mediate the issue.5.3 DiscussionThese preliminary results purport to illustrate that the FP operator causes an effec-tive diffusion which changes the evolution of the macroscopic fields. Further work needsto be conducted to understand the parametric behaviour of the amplitude of fluctuatingdiscontinuities as a function of the macroscopic variables as well as γ and vT (T ). Thepresent model solves the continuous FPE (2.84) by proxy, which in its native form isa linear PDE. However, because the semi-Lagrangian diffusion introduces the effects ofthe thermal fluctuations directly into the velocity space vi =√θci + u of the model,it is expected that the solution would be more quasi-linear as vi(p, ρ,u), v2T (T ), andthe populations are non-linearly coupled in the solution algorithm. Prospective realistic1215.3. DISCUSSIONFigure 5.14: Isolines of total energy, density, temperature, and pressure fields after 75δt.Figure 5.15: Isolines of velocity field components, and velocity magnitude after 75δt.1225.3. DISCUSSIONsimulations including thermal noise should evolve highly non-uniform fields containingnumerous discontinuities of varying amplitude and standard deviation. With such su-perposed fluctuating sources the accuracy of the reconstruction stencil in all directionsbecomes paramount for accurately capturing and diffusing the thermal fluctuations inspace. An assessment of various interpolation kernels should be made to establish themost suitable for the task. The currently serial (slow) realization in code makes it diffi-cult to conduct any dedicated analysis of larger domains, with more non-uniformity. So,after the asymmetry in the solution identified in section 5.2.1 is fixed, MPI and GPUacceleration should be prioritized to make analyses more streamlined.Another outstanding issue is the use of unit conversion between physical units (PU)and lattice units (LU). In simulations, the conventional LBM relies heavily on unit-conversion of the macroscopic observables to obtain numerically stable and accuratecomputations. In the current implementation in code and the SLD, the conversion im-pacts stability to a lesser extent. The code uses double precision allowing for a broadspan in the magnitude of the simulation variables. The current realization of the code isequally reliable in single precision (32 bit format). If greater variability in the simulationvariables allows for the use of single precision variables is pending further investigation,even though the traditional LBM has been reported to be sufficiently accurate with singleprecision [375]. Incorrect unit conversion especially impacts hydrodynamic behaviour—ifthe characteristic dimensionless groups, i.e. Re, Pr, etc. are not equivalent across unitsystems, then the thermo-hydrodynamic behaviour anticipated on a PU basis will notbe the same in a LU basis. Furthermore, unit conversion is important to correctly scalevT =√kBT/m, as its magnitude is very important in (3.42) which contains a mix ofquadratic, cubic, and quartic polynomials of vT ,vi,u. The average thermal speed isdefined from the universal Boltzmann constant kB = 1.3806505 × 10−23 [J/K] [9], andAvogadro’s constant NA = 6.0221415 × 1023 [1/mol] [9], which together with the per-tinent fluids molar mass M define the molecular mass m = M/NA. The speed is thedeterminant dictating the fluctuation intensity in a fluid via its presence in the diffusioncoefficient of the FPE (2.84). As such, it is inherently coupled to fluid temperature.In the above tests, the solutions were reported in LU, revealing limited informationabout the true nature of the fluid being represented in the tests. As previously mentioned,attempts were made to compute the correct lattice units on the basis of the VDW EOS,but due to its inconsistent representation of fluids of interest, it was deemed futile toconvert units. Nevertheless, for future efforts, the methodology for establishing a unitconversion is as follows. The physical units of the macroscopic variables are convertedinto the LU basis by establishing a set of basic conversion factors from which all degrees1235.3. DISCUSSIONof freedom of the model can be converted PU→ LU. The consistency of the conversion isconfirmed by computing the relevant dimensionless groups (i.e. Reynolds, Peclet, Prandtlnumbers, etc.) on an PU and LU basis. Given equality across these bases, the conversionis consistent. One can choose the basic conversion factors for mass, time, length, andtemperature as,Cl.= δx, Ct.=Xtδt, Cρ.= ρ0, CT.= T0. (5.1)In this case, it is prudent to scale δt with a prefactor Xt ∈ [0, 1] to be able to controlthe advection distance |xd − x| carefully, which non-trivially influences the numericalrealization of the predictor-corrector scheme. It follows that all the model variables canbe converted by considering their physical dimensions. By exemplification, the internalenergy [e] = J/kg = m2s−2 would be converted by,e(LU) = e(LU)× C2tC2l, etc.,where the derived conversion factor for internal energy would be, Ce.= C2l C−2t = δx2δt−2.It then follows that the dimensionless group can also be converted. E.g.:Re(LU) = Re(PU)× CνClCuetc.Selected, derived dimensionless factors are summarized in Tab. 5.4. The time-stepsize (now recast to δt ← Xtδt) ought to be established on the basis of the viscositycoefficients derived from the Chapman-Enskog analysis. However, only the coefficientsfor semi-Lagrangian advection (4.43) were established in the present dissertation, andthe same analysis should be carried out for semi-Lagrangian diffusion. As such, it is alsounclear what the physical value for viscosity is in the tests. In prospective simulations,it is important that the specific heats be defined from their real-gas correlations (3.52)(or with other realizations than VDW) rather than with constant values from e.g. NIST,and evaluated during runtime.For accuracy of prospective simulations, it matters which basic conversion parametersare used. Choosing the set C (5.1), the order of accuracy in the lattice-unit based vari-ables tends to machine precision as δx(LU) = δx(PU)/Cl ≡ 1, δt(LU) = δt(PU)/Ct =Xt,etc. The predictor-corrector loop consists of firstly establishing the departure points ateach monitoring point where xd = xd(x,vi, δt) and vi = vi(p, ρ,R, TL,u). Secondly, theaccuracy of the gauge-transformation procedure is bounded in the accuracy in fi, gi,vi,1245.3. DISCUSSIONTable 5.4: Unit-conversion factors derived from the basic conversion factors (5.1).Correlation UnitsCm CρC3l [kg]Cp CρC2l C−2t [Pa = kg·s−2]Cu ClC−1t [m·s−1]Cµ CpCt [Pa·s]Cν C2l C−1t [m2s−1]Ce C2l C−2t [J·kg−1 = m2s−1]Ca C2l C−2t C−1ρ [m5s−2kg−1]Cb C−1ρ [m3kg−1]CR CeC−1T [J·kg−1K−1 = m2s−2K−1]CCv CR [J·kg−1K−1]CCp CR [J·kg−1K−1]CF CmClC−2t [N = kg·m·s−2]Ck CFC−1t C−1T [W·m−1K−1 = N·s−1K−1]as Gf = Gf (fi,vi) (3.8). It is exact given the moment-matching condition (3.6). Consid-ering vi as the “primitive” variable of the model, and that it inherently establishes theaccuracy bounds of the predictor-corrector scheme, it would be prudent to choose thebasic conversion factors such that O(vi(LU)) ≃ 1/2. Then, if δx = δt = 1, the departurepoints fall predictably within the bounds of the reconstruction stencils.125Chapter 6Conclusion and future work6.1 Research summaryThe presented literature review showed that continuum-based models formulated inCFD rely on reduced-order models to predict instantaneous, predominantly homogeneousphase change, especially for cases concerning cavitation. In most cases, phase change isrepresented by reduced-order mass-transfer models tuned by empirical coefficients leavinglittle leeway for investigating phase-change physics that are too complicated to be treatedempirically. The CFD community working on boiling research is more detail oriented andhas selectively included metastability, thermal noise, conjugate heat transfer, and free-energy in some research efforts. However, to paint a full picture of heterogeneous phasechange, all of those aspects need to be combined into a single study, which so far hasnot been done owing to the mathematical complexity in the pertinent models. Phasechange is most rigorously represented in fluctuating-hydrodynamics theory (FHT) inrecent efforts. More details of the review are found in [J3].After considering these more common engineering models, a review [J1] was con-ducted of the lattice-Boltzmann method, which is a theoretical framework for castingvariants of the Boltzmann equation into a discrete form to be solved computationally ona lattice. It was found that LBMs are very powerful for modelling and simulating complexmultiphase flows and have to date considered conjugate heat transfer, solid-fluid inter-action via wettability conditions, free energy, and thermal noise in phase-transitioningfluids. However, as in conventional CFD, most research revolves around homogeneousphase change, and rarely tries to combine multiple of the physical phenomena that re-alitiscally represent phase change. Thermal fluctuations can be imposed in fluctuatingLBMs, but these are sophisticated and require a significant specialization in physics,computing, and applied mathematics. FLBMs were reviewed in [J4]. Answering Q1,FLBMs and FHTs impose thermal fluctuations by explicitly sampling random distribu-tions of density, heat flux, and viscous stresses during runtime and they were the onlymethodologies found enabling simulation of thermal noise. FLBMs have not yet beenapplied to phase-change processes, as most of the efforts invested into their research have1266.1. RESEARCH SUMMARYrevolved around theoretical validation of the intermittent fluctuations in the models.Crucially, they are also limited to relatively weak noise intensities and lower values ofthermal speeds vT =√kBT/m. A particular theory called Particles-on-Demand (PonD)that realizes a type of semi-Lagrangian advection (SLA) in LBMs was identified as usefulfor modelling fluids undergoing strong disequilibration, including thermal and compress-ibility effects, and as such was chosen as a foundation for modelling the discrete velocityspace in a prospective lattice model. To model thermal fluctuations, the Fokker-Planckequation was deemed suitable for modelling noise in a significantly different numericalrealization compared to FHTs and FLBMs. With regards to Q2, the FPE is a suitablemodel of liquids in that it can represent both white (fluctuation-frequency independent)and coloured noise (fluctuation-frequency dependent), and can account for molecularforces either directly in forcing terms, or by correlation to an external equation of state.However, realizing these features in a lattice model is non-trivial and, as a result, makeproviding direct numerical evidence for answering Q2 elusive.The primary contribution of the current research was to create a lattice model ofthe FPE along with a complementary numerical solution methodology and computa-tional implementation in C++ code to solve the resulting kinetic lattice-Fokker-Planckequations (LFPEs) in simulations. Initially, PonD was considered to be used for solvingthe LFPEs using its original SLA realization, but it was deemed unsuitable for solvingthe LFPEs. To overcome this limitation, a “semi-Lagrangian diffusion” solution methodwas created which purports to treat streaming and collisional effects concurrently duringtime integration rather than sequentially as is done in SLA. In addition to this significantcontribution, the Fokker-Planck equation, including a forcing term and the adaptive dis-crete velocities of PonD, was discretized with Gauss-Hermite quadrature with inspirationfrom Moroni et al. [333], resulting in fully discretized lattice equations projected ontothe lattice. In a line of first efforts in assessing the theoretical consistency of the derivedmodel, asymptotic expansion by Chapman-Enskog multiscale analysis was performedto obtain the thermo-hydrodynamic limit. This was done on the basis of lattice-Fokker-Planck-Boltzmann equations with SLA simultaneously representing collisional effects as acombination of FP operator perturbations and Bhatnagar-Gross-Krook (single-relaxationBoltzmann) collisions. The reasoning was to enable representation of dense liquids aswell as dilute gasses in a singular lattice-Fokker-Planck-Boltzmann model. The thermo-hydrodynamic limit was shown to be the stochastic Navier-Stokes-Fourier equations witha stochastic stress tensor. Further to the stress tensor, a fluctuation-dissipation theoremwas derived, which together with the asymptotics indicated preliminary theoretical con-sistency. The derivations associated with the analyses are presented in [J4] in addition to1276.1. RESEARCH SUMMARYthis dissertation. Conclusively, the answer to Q3 is indeed that common LBM techniquescan be used to solve the FPE, albeit modification to their solution strategy is necessary.With the established theory and methodology, the constructed C++ solver was used togain insight into the evolved behaviour of the thermal fluctuations captured in the LFPEs.In the first tests, a 2D computational domain was configured exclusively with periodicboundary conditions and a VDW fluid. To gauge the performance of the semi-Lagrangiandiffusion scheme, a simple Gaussian “impulse” in the centre of the domain was relaxedwith γ = 0 and γ = 0.5 which would manifest an Euler-type advection equation in the firstcase, and in the second case, the full NSF equations with FP perturbations and resultingviscous effects. It was found that while the semi-Lagrangian diffusion was applied withhigh spatial resolution to a relatively smooth problem with weak gradients in its thermo-hydrodynamic observables, the convergence of the scheme is comparably fast to what wasreported for SLA with PonD and BGK collisions [94]. That is, the SLD scheme convergesto machine single-precision in 2-3 predictor-corrector iterations, and machine double-precision in 5-7 iterations. From this test it was found that a spatial asymmetry arises intime in the macroscopic variables, which causes especially the hydrodynamic velocitiesto be advected faster in the lateral directions as compared to the diagonal directions. Itis suspected that it is associated with the adaptive nature of the reconstruction stencilbeing centred at each departure point as opposed to being fixed about each monitoringpoint. Moreover, mass and energy is not conserved in the current implementation, butthat is likely due to the adaptivity of the reconstruction stencil [373].The uniqueness of the develop LFPM as opposed to FHTs and FLBMs lies in itsdesign that does not necessitate advanced ad-hoc sampling methods for imposing therandomness into the solution variables. Rather, the randomness is only imposed in theinitial condition to perturb the thermal noise into numerical existence. Thus, stochas-ticity is manifested through the high-order Hermite polynomials in the Gauss-Hermitequadrature of the FP operator. Using a random number generator to initialize a simula-tion, different generator seeds should yield distinct spatio-temporally fluctuating fields.A simulation was carried out with an initial field containing Gaussian-randomly samplednoise in the x-y-lower quadrant immersed in uniformly initialized values. Here it wasfound that the LFPEs cause diffusion waves emanating from the noisy quadrant, wherethe variables in the wake of these waves appear to be noisy as opposed to smeared aswould be expected from e.g. a BGK collision operator with SLA to which the associatedviscous effects can only be resistive as opposed to intermittently resistive and perturba-tive in the FP operator. Given the high degree of non-uniformity in the fluctuating fieldsthat are reliably evolved by the semi-Lagrangian diffusion, it is clear that the scheme is1286.1. RESEARCH SUMMARYvery robust. Thus far, the solution has been evolved up to 103δt and yet shows no signof divergence, ignoring the slowly accumulating loss in mass and energy stemming fromthe lack of conservativity.In relation to Q4, the research has shown that consistency of the LFPM should beassessed both theoretically and by verification/validation of simulation datasets. Aspreviously mentioned, a CE analysis was carried out to assess the thermo-hydrodynamiclimit of the model, and a FDT was presented for the associated stochastic stress tensor.These concepts are useful for theoretical validation but were only carried out for selectedvariables. For a fuller picture, the CE analysis and the perturbation series expansiontherein should trace not only perturbations in the stresses but also density, total energy,momentum, heat flux, and eventually entropy. Moreover, FDTs should be establishedfor the structure factor (density fluctuations), fluctuations in energy, momentum, heatflux, and the so-called “ghost-modes” mentioned in §2.4. On the basis of such FDTs,equilibration ratios can be derived which can be exploited in numerical simulations togauge physical consistency as they all should tend to unity. Furthermore, given a datasetincluding numerous timesteps, a discrete Fourier-transform analysis could be carried outto establish the spectra of fluctuation intensities in the LFPE solution which should yieldfrequency-independent statistics considering the current white-noise realization of theFPE. Summarily, while initial indicators of physical consistency were found, completelyproving the physical consistency of the LFPM entails further mathematical and numericalanalysis.Although the majority of the present work revolved around mathematical analysis andcode development, preliminary numerical tests were conducted and presented in §5. In-line with Q5, it is noted that the derived LFPEs are asymptotic expansions approximatingthe native continuous FP partial-differential equation, where the FP operator is predictedby (non-)linear combinations of the discrete velocity set vi, and the thermo-hydrodynamicmoments. In §5 explicit computations of Ω(FP)i across ρ, T ∈ [0, 1] were presented. Whileit is not immediately clear how the parameter space can be interpreted physically, it isnoted that the parameter space of Ω(FP)i is dependent on ρ, T, p in vi(p, ρ) and vT (T ) andthus the cubic, non-monotonic tendencies of the isobars, isochors, and isotherms shouldto some extent be reflected in the Ω(FP)i (ρ, T ) surfaces. However, distinguishing thesefrom the meddle of quadratic, cubic, and quartic polynomials in the operators makesfor a strongly non-linear numerical representation of the operators. Add to that, Figs.5.3–5.7 are evaluated with equilibrium populations f (eq)i , g(eq)i , limited values of u, andmost importantly do not account for the SLD algorithm. SLD evaluates Q computationsof Ω(FP)i (xd) at departure points indirectly from Q×mp×np significantly different non-1296.2. IMPACT OF CONTRIBUTIONSequilibrium values of Ω(FP)i (xp) in the collocation stencil, at two time instances duringtime integration making the prediction of Ω(FP)i more sophisticated. The inclusion ofthe EOS thermodynamics directly into the FP operator is expected to pave the way forinvestigating the influence of fluctuations on the metastable effects and phase-changedynamics that are directly involved in many EOS.6.2 Impact of contributionsProving that the Chapman-Enskog multiscale analysis (CEMA) can be applied to theFPE while consistently reproducing the NSFEs is a significant contribution. It is likelypossible to incorporate models tailored to the LBM, e.g. forcing schemes, interactionforces directly in the FPE, arbitrary EOS, and multiphase models previously reviewed(§2), into the LFPM making it more realistic. Furthermore, as shown in the expansionof the stochastic stress tensor τ˜ , it is possible to identify the mathematical formalismunderlying the thermal noise effect on viscous stresses and the correlations for the viscouscoefficients µ, ζ (4.43) and (4.44). The versatility of CEMA is evidently significant andis likely to give more insight into the thermo-hydrodynamic behaviour of fluctuationsacross scales.Choosing the FPE as the groundwork for a fluctuating kinetic theory is a strongproposition in its own right; the equation has not been previously used to study complexphysical problems in fluid dynamics with even the initial number of degrees of freedompresented in the current work. The importance is pronounced considering that thermalfluctuations are rarely studied computationally in the context of phase change, and havenever been investigated with the FPE (at least to the extent of the information uncoveredin this dissertation). Having inaugurated the LFPM tailored for thermal fluctuations6opens up an avenue of lattice research with many years worth of future investigations.Considering the diversity in continuous FPE formulations that exist in literature (seecited examples in §2), as well as their areas of applications, there are a lot of opportunitiesfor testing the LFPM on more complex and non-linear FPEs including higher-orderdiffusion tensors (representative for coloured noise) than the current scalar correlationfor white noise resulting in the Langevin dynamics in (2.80).The SLD algorithm devised specifically for the LFPM has turned out to be morerobust than expected and does not appear to contribute any numerical error. While thescheme is not yet mature, it may prove very powerful, in that complicated kinetic modelscan be reliably solved in the predictor-corrector, as proven by the incorporation of the6While Moroni et al. [333] formulated the first LFPM, it was not applied to thermal fluctuations.1306.2. IMPACT OF CONTRIBUTIONSFP operator. Furthermore, it is possible that higher-order numerical integration schemesin addition to the trapezoidal rule can be used by splitting each timestep and δt intomultiple instances with separate FP-perturbations in e.g. a Runge-Kutta integration.This is a powerful alternative considering that virtually all LBMs are implemented witheither second or third-order rectangular and trapezoidal integration. The original SLArealization of PonD in [94, 252] evidently can be used in highly compressible (tested upto Ma ≃ 102) and thermal problems due to the exact velocity-error free prediction ofthermo-hydrodynamics. As SLD is similarly exact in its velocity space, it is likely thatit can be applied in highly compressible and thermal scenarios now including thermalfluctuations. Here it is noted that it is “exact” to machine precision, not considering un-certainties in the various asymptotic expansions, numerical methods, and specific DmQnlattice models used in the research. The versatility of the reconstruction procedure alongwith the accuracy of SLD also is likely to enable weak as well as arbitrarily strong fluc-tuation intensities (2γkBT/m), which considering the literature review §2 is not possiblein current FLBMs. Thus, in the future, the LFPM could be used to simulate thermalfluctuations in cryogens and liquid metals, conditional on the availability of a suitableEOS.In addition to the unconditional exactness of SLD, the LFPM is very unique com-pared to MD, FHTs and FLBMs in that it evolves the LFPEs as “initial-value problems”where the predicted simulation variables in space and time are unique to the initial con-dition and will produce essentially random predictions for small changes in the initialconditions. With no need for advanced sampling methods of random distributions tocause fluctuations in the simulation variables, it is possible that the LFPM is less com-putationally demanding compared to FHTs and FLBMs in identical simulations. Thisis important in the context of phase change, as it is currently only possible to simulatedomains with volumes limited to nm3. What is more, lattice-site updates are local, andcan be evolved independently from neighbouring sites during a time step. This allows forarbitrary spatial update-patterns in lattice variables, and freedom to solve the LFPEscontiguously in memory arrays in code. The contiguous dimension can be chosen arbi-trarily and does not have to be fi and gi as in the current vectorized implementation (asopposed to x, y, or z for future 3D implementations). Numerical tests with the SLD haveshown that the numerical solution of the LFPEs is not sensitive to the update pattern.These features of the LFPM favour efficient computations in both CPUs and GPUs andcan be exploited to conduct spatially large-scale simulations of non-equilibrium phasechange.SLA-PonD has already proven to be stable in transcritical-flow simulations [252, 302]1316.2. IMPACT OF CONTRIBUTIONSin addition to the density dimension associated with compressibility. Thus, it shouldbe possible to study metastability anywhere under the saturation curve, given a consis-tently implemented EOS. The scientific implications of this are that metastability can becarefully investigated across manifolds of ρ, T and therefore noise intensities, where themetastability of phase transitions are well-established, at least theoretically [21, 74, 89].Of specific interest is the Taylor-Helmholtz instability (THI) which has been rarely in-vestigated [42, 376] despite its importance in delimiting the critical heat flux (CHF) inthe pool boiling curve (previously encountered in Fig. 1.2). As it is hypothesized thatthe wetted area of a boiling substrate across the nucleate-boiling regime determines thesusceptibility to THI, it is worthwhile to determine the importance of thermal noise along∆T increasing from the onset of boiling (ONB), as it may impact the nucleation rateincident at the solid-liquid interfaces and the wetted area. Knowledge about the THinstability and how to control the ONB and CHF should have direct industrial impact asthe CHF is the most efficient state for heat transfer at moderate temperature differences,central to many boiling technologies as exemplified in §1 and 2.The impacts of thermal noise in cavitation is not clear in the reviewed literature. Inrecent experiments on hydrofoil cavitation [377], it was shown that nano-microscale airnuclei introduced upstream of a hydrofoil caused the large-scale cavitation-cloud dynam-ics to behave markedly different compared to the same experimental conditions but withno nucleation sites (i.e. pure water). In light of such a coupling spanning the micro andmacroscales, it is questioned to which extent thermal fluctuations promote/delay the on-set of cavitation in space and time. Foreseeably, with appropriate theoretical and numer-ical extensions, the current LFPM will enable investigation of the nucleation dynamicsowed to its semi-Lagrangian nature. Furthermore, coloured noise becomes relevant in thepresence of solid impurities suspended in a liquid [88]. If thermal (coloured) noise doesinfluence cavitation rates, the impact would be non-trivial in scenarios where the work-ing fluid is strongly impure (e.g. seawater), contributing to research in marine-propellererosion [378], and bio-fluid dynamics in e.g. mantis-shrimp predation [33]. Evidentlyfrom the initial simulation data in Fig. 5.14 as well as the VDW formalism (3.48b), thepressure p is non-linearly sensitive to thermal fluctuations and thus could be importantin stretched liquids approaching the saturation state.While the current state of the LFPM can simulate thermal noise, there is still alot of development required before heterogeneous nucleation can be investigated. Untilheterogeneity is introduced into the theory and numerics, it is unlikely that engineerswill adopt the LFPM. Compared to CFD methods already used in industry such as theFVM, it takes significant specialization to use LBMs and the LFPM, even though the1326.3. LIMITATIONSLBM is continuously gaining interest in non-specialized communities.6.3 LimitationsIn the following, the limitations of the LFPM at its current state are outlined, thoughkeeping in mind that they can be eliminated or mitigated in the method with suitable the-oretical and numerical extensions. Such extensions will be mentioned where applicable.Firstly, theoretical limitations are considered.Chiefly, it must be emphasized that the FPE is a deterministic diffusion equationwhich does not have the ability to inherently create noise in a uniform macroscopicfield. It is not ruled out that via its dynamics, including fluctuating information retainedfrom the Langevin description and non-ideality, it might be able to initiate a transition tochaos in cases where the FPE has a large number of degrees-of-freedom in its phase-space,thereby sustaining coarse-grained, thermal fluctuations instilled as an initial condition.However, generally the dynamics of the FPE is not well-understood for more than twospatial degrees-of-freedom [369], and the properties of such mimicked fluctuations mightnot agree with reality. Reintroducing fundamental fluctuations into the equations by asampling step is possible albeit not trivial [379–381]. Such discrete fluctuations wouldaccommodate spontaneously arising thermal noise from uniform initial conditions.To predict the thermodynamics in the numerical tests, a VDW EOS was employed,correlating p, ρ, T . However, it was found that the applicability of this EOS to real fluidsis limited, and thus different EOS should be employed to accurately represent fluids ofinterest. A prospective EOS should be carefully chosen considering the problem underinvestigation including its applicable thermodynamic states in ρ, T as well as accuracy inpredictions of relevant thermo-physical properties (e.g. cp (3.52), ςs (3.54) including theirsensitivity to the ρ, T manifold) compared to available experimental data. The LFPMcurrently assumes a constant Pr, but in cryogenic liquids Pr is variable—this can al-ways be mitigated by adopting two relaxation frequencies in each LFPE as was done in[382] for the LBM. Consider also that the pertinent EOS is directly used within the CEanalysis to arrive at the NSF equations, and changing EOS in the LFPM necessitatesa partial re-derivation of the perturbation-series expansion. Notably, the current VDWEOS applies to single-component fluids, and multiple components will require more so-phisticated formulations. However, the LFPM is EOS agnostic and can accommodatemany different EOS.For realistic phase-change modelling, the current CEMA and LFPEs are lacking twopotentially important phenomena: firstly, there is no explicit source or sink term for mass1336.3. LIMITATIONStransfer in the thermo-hydrodynamic limit, as is the case in phase-change NSE solvers(see ch. 2). However, in single-component liquids, such a term may prove insignificantas it generally is driven my osmotic pressure defined by gradients in chemical potentialand consequently concentration. Contrarily, in miscible fluid mixtures where concentra-tion gradients are non-zero mass transfer cannot be neglected. The mass transfer ratescould be facilitated numerically by correlating cross-species diffusion with free-energyin prospective multi-component formulations of the LFPM. Secondly, the cost of nucle-ation is associated with capillary forces, which are currently not accounted for in theLFPM. These can prospectively be captured phenomenologically in a Korteweg stresstensor or in a square-gradient free energy functional formulation. These features shouldbe incorporated into the CE analysis. Furthermore, the presented CEMA 4.1 is for SLA-PonD and it should be rederived for SLD-PonD to achieve the viscous coefficients for thenew algorithm, and particular attention should be applied to the moment conservationthroughout the steps of the predictor-corrector scheme, to theoretically prove that theLFPM is consistent.It is thought that the thermal fluctuations mimicked by the LFPEs are scale inde-pendent. This requires that the thermal noise intensity be calibrated for the fluid underconsideration. Therefore, the lattice may need to be implemented in such a way as toadequately resolve the fluctuation spectrum, as the LFPE itself does not inherit anyscale-dependence. Finally, whereas the accuracy of the various DmQn lattice modelsused in the LBM is rarely questioned, their accuracy should be assessed for the LFPM.The limitations in the numerics of the LFPM should also be addressed. The conser-vativity in PonD, SLA and SLD is known to be sensitive to the reconstruction stencil,specifically the degree of accuracy and which type (cubic, WENO, linear, etc.) of recon-struction kernel is used, besides the fixed or adaptive realizations of the stencil. Adap-tive stencils do not appear to be conservative, especially in mass, and induce artificial,direction-dependent numerical diffusion in the lateral quadrature directions i ∈ [1, 4].Each candidate interpolation kernel should be assessed in highly non-uniform fields, andif predictions fall within the values of the surrounding collocation points. This bringsinto question the accuracy of the current Lagrange polynomials which do not necessarilyfall within the values of the collocation point populations, as is evident from Fig. 5.9.Compared to “conventional” LBMs, the LFPM with SLD is significantly more com-plicated to implement in code and is susceptible to bugs, especially in the programmingof gauge transformation. The current LFPM in code accommodates uniform Carte-sian grids which is less convenient than e.g. the FVM where meshes can be smoothlybody-fitted, unstructured, non-uniform and curvilinear. Nevertheless, highly complex1346.4. RECOMMENDATIONS FOR FUTURE WORKgeometries (walls) can be represented in the uniform lattice with the bounce-back wallboundary conditions of the LBM. Such schemes should be modified in the LFPM due toSLD, which can be mediated by Grad’s projection. Furthermore, owing to the LBM andLFPM tracking Q populations per lattice node, the use of adaptive-mesh refinement isdifficult to implement theoretically and in code. The current implementation in code isserial and MPI parallelization should be prioritized. This will make searching the phase-space of the LFPEs for phase transitions faster. This is also worthwhile in the currentVDW-EOS realization which should accommodate metastability and phase transitions,which were not analyzed in §5.Lastly, validation was not yet reported in this dissertation. Validation of the LFPMcould be facilitated by experimental data for the structure factor from e.g. opticaltraps [86, 88], light scattering and shadowgraphy experiments for single-component flu-ids and binary mixtures [383], and from molecular-dynamics-lattice-gas simulations [380,381]. Comparing simulation datasets from the LFPM would require numerous time-stepdatasets. In addition to the structure factor, equilibration ratios for the model param-eters would suggest correct calibration of the fluctuations. But this requires that therelevant FDTs in addition to the one for τ˜ (4.46) are derived. In prospective valida-tions, it is important to also verify that the null-condition for ⟨τ˜⟩ holds, which is notimmediately clear from the equal-time correlation (4.45). In addition, a prospective unitconversion on the basis of a new, more accurate EOS may incur that the simulationvalues vary anomalously. Thus, the basic conversion factor in a unit conversion shouldbe chosen with care to scale these quantities to unity, whilst producing departure pointsthat generally do not fall further than δx away from each monitoring point.6.4 Recommendations for future workThis section will present extensions to the LFPM necessary for rendering it able tosimulate phase-transitioning, metastable liquids with homogeneous and heterogeneousnucleation triggered by thermal noise. The state-space of the FPE (2.84) from whichthe LFPM is derived is (only) delta-correlated in time, whereas FHTs and FLBMs aredelta-correlated in space δ(x − x′) and time δ(t − t′), or wave space δ(k + k′). In thefuture, it may be worthwhile to establish the FPE in Fourier space, or pending furtherliterature review “generalized” FPEs [384–386]. In any case, for planned simulations ofparticular liquids and cases, it should be verified that the initialized noise is physicallyconsistent for the setup. Cataloguing the noise in liquids with supporting experimentaldatasets would be valuable for future simulations.1356.4. RECOMMENDATIONS FOR FUTURE WORKCentral to the triggering of a phase transition is the metastability of a liquid phaseoutside the spinodal locus. The thermodynamics modelled in the LFPM should in-herently capture metastability for accurate predictions of nucleation. Given that themetastability is tightly correlated with the entropy-extremum principle and entropy gen-erated by thermal fluctuations themselves [82, 90], it is relevant to extend the state-spaceof the LFPM with entropy and (Helmholtz or Gibbs) free energy. Thereto, square-gradient theory with the aforementioned generalized functional (2.38) can be employedto predict free energy including capillary forces. Therein, the bulk free-energy term ispredicted by empirical EOS or well-potentials such as the Lennard-Jones variant. Specif-ically for cryogenic fluids, a literature review should be conducted to identify the mostsuitable EOS where [70, 387] may offer a suitable vantage point.Furthermore, it is clear that metastability is succinctly predicted by the “order” pa-rameter ϕ from phase-field methods (§2.2.5). Accounting for fluctuations in the orderparameter is essential. ϕ is correlated with fluctuations at large distances and timescalesin the neighbourhood of the critical point as well as the spinodal locus [74, 388–392]. Forexample, the mean-square density fluctuations in an ensemble are known to be corre-lated with the isothermal compressibility [21, 74] relating kBTβT /V =〈(δρ)2〉/⟨ρ⟩2 to amicroscopic fluctuation. This density ensemble further separates regions under the sat-uration curve of no fluctuational influence(kBTβTρ2[vcrt(ρ− ρ˜bi)2]−1 ≪ 1) and strongdependence(kBTβTρ2[vcrt(ρ− ρ˜bi)2]−1 ≫ 1), where ρ˜bi is understood to be the densitydelineating the inequality close to the binodal locus and vcrt the volume of the criticalnucleus. In vicinity to this inequality in the p-ρ diagram a transition regime betweenclassical nucleation and spinodal nucleation exists, dictated by fluctuations in ϕ [74]. Fur-thermore, the magnitude of the (free) energy barrier associated with formation of criticalvapour nuclei, and consequently the possible penetration depth into the metastable regionincluding the transition regime, is correlated with non-local and non-linear fluctuationsin ϕ [74, 92, 393, 394]. A full review of stability criteria should be studied with the pur-pose of establishing a methodology for assessing prospective metastability encounteredin the LFPM.In future research, the treatment of heterogeneous fluids should be considered. Inthe context of nucleation, heterogeneity refers to the presence of infinitesimally smallundissolved, suspended vapour nuclei or solid particles, which can act as nucleation sitesin a liquid by decreasing the energy barrier. If such nuclei inhabit the same lengthscales as thermal fluctuations, then they should be resolved directly in simulations viamulti-component formulations of the FPE and/or immersed-boundary methods for solidparticles. If the characteristics length scales of the nuclei fall below the correlation lengths1366.4. RECOMMENDATIONS FOR FUTURE WORKof the fluctuations, then it may be worthwhile to pursue sub-grid modelling of dispersednuclei in a pseudo-single component liquid. Thereto, the Fokker-Planck equation maybe a candidate foundation, as it has been used to evolve nuclei-size densities in kinetictheory of nucleation [63, 65, 395–397].The current and future theoretical formalism of the LFPM needs to be supportedby sophisticated numerical methods, given the non-uniformity in the fluctuating ther-modynamic variables. Predominantly, parallelizing the current serial implementation ofthe LFPM is important as it will accelerate the theoretical and numerical-method de-velopment. Furthermore, to accelerate very large scale simulations, the code should beported to GPU architectures by a suitable platform, e.g. CUDA, OpenACC, or Kokkos.Central to the accuracy of the LFPM is the applied reconstruction method. Given thespatial variability in the fluctuating fields, it is possible that Lagrange interpolation isinaccurate, especially in capturing the spatially and temporally correlated fluctuatingfields. Alternative reconstruction kernels should be reviewed for reconstructing fi andgi. In addition to “deterministic” interpolation methods used often in CFD, it may provefruitful to use statistical regression techniques, e.g. Kriging [398–400]. In addition to thereconstruction process of SLD, gauge transformations are computationally intensive inthe method. These, as well as boundary conditions and source terms, can be approxi-mated by Grad’s projection [302, 358, 401, 402].In prospective simulations, it would be interesting to enable the configuration of noisespectra in the initial fields for representing real thermal noise in the FPE—i.e. initiallyfrequency-independent (white) noise and eventually frequency-dependent (coloured) noise.The initial derivations were done based on a combined LFP-Boltzmann equation antic-ipating that a gaseous state would be more accurately described by a BGK operator.However, the standard BGK-form of the deterministic stress tensor was captured in-herently in τ˜ (4.32g) and perhaps it is already approximated in the FP operator inwhich the leading expansion order contains the equilibrium populations ρwi relaxed byγ. Having undertaken significant theoretical and methodological work, the LFPM is ata stage where its development can be increasingly informed by more numerical testing.Ch. 5 gave initial insight into the behaviour of the model, but not beyond showingthat the SLD scheme converges whilst accounting for non-ideality in the VDW EOSand FP perturbations. However, there does not exist a tried methodology for validatingthermal noise in CFD, nor does there exist benchmark test cases for comparing LFPEsolutions. Devising a testing and validation strategy should be done with attention toseveral aspects in a systematic manner. This strategy is synthesized in Tab. 6.1. Giventhat a meaningful unit-conversion with the VDW EOS is elusive, there are no physical1376.4. RECOMMENDATIONS FOR FUTURE WORKconstraints to the parameter space, i.e. the primitive variables ρ, T,u and the derivedvariables p, e, E, nor the model constants γ, kB,m, a, b. With the several degrees of free-dom, and the unconventional SLD scheme it is difficult to identify in simulations whatis physical noise and what are numerical instabilities and artifacts arising from e.g. thereconstruction procedure. Thus, individual tests that can isolate the phenomena needto be created. In conducting the tests in ch. 5 numerical anomalies arose that should beexamined in future work. Lattice-pinning issues were apparent in the fields of the totalenergy moment and especially the density moment. This numerical phenomenon occursin a simulation when there is insufficient transport and/or excessive non-uniformity inthe lattice variables. This effect should be eliminated by designing the unit conversionwith the aim of achieving a significant magnitude in u O(100)responsible for the trans-port, or initializing a smoother initial condition (than the Gaussian fields in Figs. 5.14and 5.15) via δx. Furthermore, the model has a propensity to cause stronger advectionin the directions of c1–4 with weakened diagonal components coinciding with i ∈ [5, 8]resulting in anisotropic diffusion patterns. It especially causes the velocity fields to besmeared over time. It is suspected that the feature arises in the computation of theFP operator and/or the reconstruction procedure. Possibly, the D2Q9 lattice model isnot adequate for capturing the non-equilibrium effects from the FP perturbations in fiand gi and higher-order variants should be considered, e.g. D2Q13, D2Q17, etc. In theLFPM in (3.42) γ linearly scales the FP perturbations and relates to viscosity. Pre-dominantly, the magnitude∣∣Ω(FP)i ∣∣ is set by the relative magnitudes of vi,u, and vT ,where the latter directly fixes the fluctuation strength in (2.80). As such, the velocitiesshould be set consistently when varying other parameters, which could be facilitated byfixed Courant-Friedrichs-Lewy conditions for discrete velocities |vi|δt/δx, thermal speedTable 6.1: Strategies to be applied for furthering the LFPM towards realistic simulations of noise.Modification Tasks and anticipated effectsDmQn models Mitigate directionality and velocity smearing.Interpolation schemes Kernels tailored to noisy signals. Thorough literature review.Mitigate numerical dispersion and velocity smearing, increase accuracy.Initialization strategies Parameter variation in kB , γ, a, b and ρ0, T0, p0,u0(x, t0).Sensitivity analysis of |vi|/vT .Procedure for establishing non-zero velocity field u0 consistent with ρ0 and p0.Introduction of Courant-Friedrichs-Lewy conditions for[vT , |vi|, |u|]T × δt/δx.Spatio-temporal resolution Varying δx and δt to resolve faster fluctuations. Eliminate lattice pinning.Solution algorithm Impose seed FP perturbation before/after gauge transformation.Unit conversion factors Tune order of magnitude of relevant parametersVerification Derivation of FDTs for primitive variables and associated equilibration ratios.Discrete Fourier transform of fluctuating variables to verify noise spectrum.1386.4. RECOMMENDATIONS FOR FUTURE WORKvT δt/δx and hydrodynamic velocity |u|δt/δx where their relative magnitudes determinethe advection distances |xd−x| and thereby the numerical stability and accuracy of theSLD scheme. The SLD scheme also needs to be systematically assessed for conservativity.The FP dynamics and the non-ideality of the EOS is coupled in the predictor-correctorand the moments in the operator (3.42), thus the configuration of a and b also definesthe fluctuation dynamics. To assess if there are other sources of instability in the modeland/or code the coefficients should be zeroed a = b .= 0 resulting in ideal gas behaviour.After establishing a detailed and comprehensive validation strategy and completingthorough tests of the numerics of the LFPM, the model has the opportunity to open formore exciting investigations into thermal fluctuations and future simulations of liquid-vapour phase change caused by noise.139Bibliography[1] Helen E. Morrison and Alfred Leder. Sediment transport in turbulent flows withthe lattice boltzmann method. Computers & Fluids, 172:340–351, 8 2018. ISSN00457930. doi: 10.1016/j.compfluid.2018.04.015. URL https://linkinghub.elsevier.com/retrieve/pii/S0045793018302111. → pages xi, 34, 35[2] C Feuchter, O Wagner, A Stief, and T Beisswenger. Turbulent flow simulationsaround a surface-mounted finite cylinder using an entropic multi-relaxation latticeboltzmann method. Fluid Dynamics Research, 51:055509, 9 2019. 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URL https://linkinghub.elsevier.com/retrieve/pii/S0021999115002636. → pages 137191AppendicesAppendix A: Thermodynamic identitiesDerivation of van der Waals internal energyTo infer the correct thermodynamics in the model the internal energy of a real gas[9] is considered, which is not only a function of temperature as for the ideal case,de = cvdT +(∂e∂v)Tdv. (A.1)If the Maxwell relation, (∂e)/(∂v)T = T (∂p/∂T )v − p in Tab. F.2 is used, then onearrives at,de = cvdT +[T(∂p∂T)v− p]dv. (A.2)This can be modified as follows,dedv− cv dTdv︸ ︷︷ ︸(LHS)=[T(∂p∂T)v− p]︸ ︷︷ ︸(RHS). (A.3)Table F.2: The Maxwell relations [9] used for the current derivations.−(∂p∂s)v=(∂T∂v)s,(∂v∂s)p=(∂T∂p)s,(∂s∂v)T=(∂p∂T)v,(∂s∂p)T= − ( ∂v∂T )p .192If de/dv is interpreted as an isentropic state change such that de/dv → (∂e/∂v)s = −pand similarly for (dT/dv) → (∂T/∂v)s = −(∂p/∂s)v, then—with the definition of cv ≡T (∂s/∂T )v from Tab. F.3—the left-hand side can be rewritten as,dedv− cv dTdv=(∂e∂v)s− cv(∂T∂v)s(A.4)= −p− T(∂s∂T)v[−(∂p∂s)v]= −p+ T(∂p∂T)v︸ ︷︷ ︸= (LHS). (A.5)This result is rewritten in terms of the van der Waals (VDW) equation of state (EOS),which reads [9],pVDW =RTv − b −av2=RT1/ρ− b − aρ2 (A.6)pVDW =ρRT1− bρ − aρ2, (A.7)such that,(LHS) = −(ρRT1− bρ − aρ2)+ T(∂∂T[ρRT1− bρ − aρ2])v(A.8)= − ρRT1− bρ + aρ2 + T(ρR1− bρ)(A.9)Table F.3: Summary of thermodynamic identities [9].Isochoriv specific heat cv =(∂e∂T)v= T(∂s∂T)v,Isobaric specific heat cp =(∂h∂T)p= T(∂s∂T)p=(∂e∂T)p+ p(∂v∂T)p,Speed of sound ς2s =(∂p∂ρ)s,Isothermal compressibility βT = − 1v(∂v∂p)T,Isentropic compressibility βs ≡ −v(∂p∂v)s,Temperature T =(∂e∂s)v=(∂h∂s)p,Pressure p = −(∂e∂v)s,Specific volume v =(∂h∂p)s,Enthalpy h = e+ pv = e+ pρ,Int. energy relation de = Tds− pdv,Enthalpy relation dh = Tds+ vdp,Helmholtz free energy da = de− Tds− sdT = −sdT − pdv,Gibbs free energy dg = dh− Tds− sdT = −sdT + vdp.193= aρ2. (A.10)Thus,aρ2 =(RHS)︷ ︸︸ ︷T(∂p∂T)v− p . (A.11)Let us reconsider (A.2). With the above result this can be rewritten to,deVDW = cvdT + aρ2dv. (A.12)Hence one arrives at the integral equation,∫deVDW =∫cvdT +∫aρ2d(1ρ), (A.13)which after a change of variables and integration simplifies to,eVDW = cvT − aρ. (A.14)Recasting with thermodynamic pressure off-set by base pressureWith a base pressure p, so that the thermodynamic pressure is rewritten to a totalpressure p = pVDW + p, (A.8) is redefined to,(LHS) = aρ2 − p, (A.15)when the derivation starting with the thermodynamic pressure in (A.6) is recast topVDW + p = ρRT/(1− bρ)− aρ2 + p. Similarly the right-hand side becomes,(RHS) = T(∂p∂T)v− (pVDW + p), (A.16)and as such,aρ2−p = T(∂p∂T)v− (pVDW + p) (A.17)↕T(∂p∂T)v= aρ2 + pVDW. (A.18)194In this case, (A.2) becomes,∫deVDW =∫cvdT +∫ [aρ2 − p] d(1ρ), (A.19)which upon integration yields,eVDW = cvT − aρ− pρ. (A.20)Considering the enthalpy it remains unchanged by the recasting, as the base pressurecancels out:H = e+uαuα2+pρ=(cvT − aρ−pρ)+uαuα2+pVDW + pρ(A.21)= cvT − aρ+ uαuα2+pVDWρ. (A.22)Derivation of the specific heatIn the framework, the specific heat of a real gas is temperature dependent shown withthe following derivation. By definition the isobaric specific heat is (Tab. F.3),cp ≡(∂h∂T)p=(∂e∂T)p+ p(∂v∂T)p, (A.23)which means that the rate of change to internal energy with temperature is,(∂e∂T)p= cp − p(∂v∂T)p. (A.24)If the differential equation for internal energy (A.2) is considered and rewritten with asmall perturbation in temperature the following result is obtained:(∂e∂T)p= cv +[T(∂p∂T)v− p](∂v∂T)p. (A.25)The two results are combined in the equality,(∂e∂T)p= cp −p(∂v∂T)p= cv + T(∂p∂T)v(∂v∂T)p−p(∂v∂T)p, (A.26)195such that,cp = cv + T(∂p∂T)v(∂v∂T)p. (A.27)This generic result can be further manipulated to arrive at an expression specifically fora VDW fluid by considering the specific VDW EOS (A.6) and evaluating the partialderivatives,c(VDW)p = cv + T[∂∂T(ρRT1− bρ − aρ2)]v(∂v∂T)p(A.28)c(VDW)p = cv +ρRT1− bρ(∂v∂T)p(A.29)From this, the final result isc(VDW)p = cv +R2TRT − 2aρ (1− bρ)2 . (A.30)Derivation of the speed of soundTo support the derivation of the hydrodynamic limit, it is needed to establish thespeed of sound of a real gas starting with the fundamental definition of the speed ofsound:c2s ≡(∂p∂ρ)s= −v2(∂p∂v)s. (A.31)The cyclic relations outlined in Tab. F.4 can be conveniently used to rewrite derivative,(∂p∂v)s(∂v∂s)p= −(∂p∂s)v(A.32)↕(∂p∂v)s= −(∂p∂s)v÷(∂v∂s)p. (A.33)On the right-hand side of the result the Maxwell relations from Tab. F.2 can be used196to rewrite the two derivatives to−(∂p∂s)v=(∂T∂v)s, (A.34)(∂v∂s)p=(∂T∂p)s. (A.35)The new right-hand sides can be further expressed in terms of the cyclic relations.(∂T/∂v)s becomes, (∂T∂v)s= −(∂T∂s)v(∂s∂v)T, (A.36)in which the isothermal contribution can be rewritten from the Maxwell relations in Tab.F.2 and the isochoric by the identity in Tab. F.3 to,−(∂T∂s)v(∂s∂v)T= − Tcv(∂p∂T)v. (A.37)Similarly (∂T/∂p)s can be expanded with the cyclic relations to,(∂T∂p)s= −(∂T∂s)p(∂s∂p)T=Tcp(∂v∂T)p, (A.38)where the isobar and isotherm was manipulated with Tabs. F.2, F.3. Subsequently, onecan combine these and rewrite (A.33) to,(∂p∂v)s= −(∂p∂s)v÷(∂v∂s)p= −cpcv(∂p∂T)v÷(∂v∂T)p, (A.39)where the isochor and isobar can be rewritten using the Maxwell relations to,(∂p∂v)s=cpcv(∂s∂v)T÷(∂s∂p)T=cpcv(∂p∂v)T(A.40)=cpcv(∂p∂(1/ρ))T= −ρ2cpcv(∂p∂ρ)T. (A.41)Table F.4: The cyclic relations for three thermodynamic variables x, y, z [9].(∂z∂x)y(∂x∂y)z= −(∂z∂y)x,(∂x∂y)z(∂y∂z)x(∂z∂x)y= −1.197Thus, the speed of sound can be written in term thereof,c2s = −v2(∂p∂v)s= −(1ρ2)[−ρ2cpcv(∂p∂ρ)T], (A.42)∴ c2s =cpcv(∂p∂ρ)T. (A.43)With this result for specific heat (A.27), the previous result can be manipulated to:c2s =1cv[cv + T(∂p∂T)v(∂v∂T)p](∂p∂T)v. (A.44)Therein, the isobar can be rewritten to (∂v/∂T )p = −(∂s/∂p)T via the Maxwell relations(F.2) such that,c2s =1cv[cv − T(∂s∂ρ)T](∂p∂T)v, (A.45)where (∂s/∂ρ)T can be written as −1/ρ2(∂s/∂v)T , and −1/ρ2(∂p/∂T )v with the Maxwellrelations applied. Thus, the speed of sound in a real gas is,c2s =(∂p∂ρ)T+Tρ2cv(∂p∂T)2v. (A.46)This generic formulation can be further applied to obtain explicit relations for van derWaals fluids by substituting pVDW = ρRT/(1− bρ)−aρ2 (A.6), resulting in the isothermand isochor expanding to,(∂p∂ρ)T=(∂∂ρ[ρRT1− bρ − aρ2])T=RT (1− bρ) + (bRTρ)(1− bρ)2 − 2aρ=RT[1−bρ+bρ](1− bρ)2 − 2aρ, (A.47)(∂p∂T)v=(∂∂T[ρRT1− bρ − aρ2])v=Rρ1− bρ. (A.48)Therefore, with cv = R/δ [252, 364] the VDW-speed-of-sound can be expanded to,(c(VDW)s)2=(∂p∂ρ)T+Tρ2cv(∂p∂T)2v(A.49)198=[RT(1− bρ)2 − 2aρ]+Tδρ2R[Rρ1− bρ]2(A.50)=RT + TδR2ρ2/(ρ2R)(1− bρ)2 − 2aρ. (A.51)Thus, the real-gas speed-of-sound for a VDW fluid simplifies to,(c(VDW)s)2=RT(1− bρ)2 (1 + δ)− 2aρ. (A.52)199Appendix B: Eigenfunctions of the Fokker-Planck andacceleration operatorsIn this section, the aim is to derive the explicit eigenfunctions of the continuousFP and acceleration operators. A crucial step in discretizing the operators is to recastthe entire continuous operators in convenient, functional forms that do not include anygradient terms. Reiterating, two Hermite series should be elucidated that approximateeach of the continuous operators,{Ω(η)i ,Ω(FP)i}= wiK∑l=01v2lT l!{Ω(η)α ,Ω(FP)α}H(l)α (vi), (B.1a)where the Hermite coefficients integrate the operators acting on the population f(v;x, t),in phase space, {Ω(η)α ,Ω(FP)α}(l) =∫dv{Ω̂(η), Ω̂(FP)}◦ fH(l)α (v), (B.1b)In the following derivations the hat-operators are treated as actions on f , but omittingthe population itself. f in phase-space also needs to be discretized with its Hermitecoefficients F(l)α ,f(v;x, t) = w(v)K∑l=01v2lT l!F(l)α (x, t)H(l)α (v). (B.2)As it turns out, one can exploit the eigenvalue property of the operators on the productsΩ̂(FP)[w(v)H(l)α (v)]identified in the Hermite series of both operators, to eliminate thederivative terms and arrive at an explicit formulation. This eigenvalue property is derivedfrom the Gaussian,w(v) =1(2πv2T )D/2exp(−(v − u)22v2T), (B.3)which is the common, null-space of the acceleration, FP, and BGK operators [336]. Thevelocity-derivative of this results in the eigenvector −√θcα/v2T as,∂vαw(v) = −√θcαv2Tw(v), (B.4)200where θ = p/(ρRTL) is the reduced pressure [252], cα the conventional peculiar velocityfor any DmQn lattice model, and v2T =√RT2the squared mean thermal speed. Con-sidering Ω(FP) excluding f , and omitting the implied notation w(v) the product can beexpanded,Ω̂(FP)[w(v)H(l)α (v)]= γ∂vβ(vβ + v2T∂vβ)[wH(l)α]= γ[(wH(l)α + vβH(l)α ∂vβw + vβw∂vβH(l)α)+(v2T∂vβ∂vβwH(l)α)]. (B.5)Using the eigenproperty (B.4) in the first parenthesis and rewriting the second yields,Ω̂(FP)[w(v)H(l)α (v)]= γ[(wH(l)α − vβH(l)α√θcβv2Tw+ vβw∂vβH(l)α)+ v2T∂vβ(w∂vβH(l)α +H(l)α ∂vβw)]. (B.6)For now the first parenthesis’ terms cannot be manipulated further and the attention isturned to the second where one can use the eigenproperty as well as the chain-rule,(B.6) = γ[(wH(l)α − vβH(l)α√θcβv2Tw + vβw∂vβH(l)α)+ v2T(∂vβ{w∂vβH(l)α}− 1v2T∂vβ{H(l)α√θcβw})]= γ[(wH(l)α − vβH(l)α√θcβv2Tw + vβw∂vβH(l)α)+ v2T({w∂vβ∂vβH(l)α+(∂vβw)︸ ︷︷ ︸−√θcβv2Tw(∂vβH(l)α)}− 1v2T∂vβ{H(l)α√θcβw})], (B.7)which yields the expanded result,(B.7) = γ[(wH(l)α − vβH(l)α√θcβv2Tw + vβw∂vβH(l)α)+(v2Tw∂vβ∂vβH(l)α −√θcβw∂vβH(l)α)− (∂vβH(l)α √θcβw)]. (B.8)201Now the task is to expand the term in the third parenthesis that cf. the product ruleyielding,(B.8) = γ[wH(l)α − vβH(l)α√θcβv2Tw + vβw∂vβH(l)α+ v2Tw∂vβ∂vβH(l)α −√θcβw∂vβH(l)α− (√θcβw∂vβH(l)α +H(l)α √θcβ* (B.4)∂vβw)], (B.9)wherein√θ, cβ are independent of vβ and thus drop out with the partial derivative. Thisleaves the result(B.9) = γ[wH(l)α − vβH(l)α√θcβv2Tw + vβw∂vβH(l)α + v2Tw∂vβ∂vβH(l)α−√θcβw∂vβH(l)α −(√θcβw∂vβH(l)α −H(l)αθcβcβv2Tw)]. (B.10)In there vβ =√θcβ+uβ was substituted, such that a number of terms cancel out resultingin the fully expanded result,Ω̂(FP)[w(v)H(l)α (v)]= γw[1 + uβ(∂vβ −√θcβv2T)+ v2T∂vβ∂vβ −√θcβ∂vβ]H(l)α . (B.11)To transform the partial-derivative terms, one can exploit the Hermite relation [333],v2T∂vβ∂vβH(l)α =(vβ∂vβ − l)H(l)α , (B.12)so that the previous result can be rewritten to,(B.12) = γw[1− uβ√θcβv2T− l + 2uβ∂vβ]H(l)α , (B.13)where vβ −√θcβ = uβ was exploited. Finally, the remaining partial derivative can betransformed using the recurrence relation [333, 359],∂vβH(l)α =1v2T[vβH(l)α −H(l+1)βα], (B.14)202that enables us to write,(B.14) = γw[1− uβ√θcβv2T− l]H(l)α + γw2uβv2T[vβH(l)α −H(l+1)βα]. (B.15)Thereto, vβ =√θcβ + uβ can be utilized and the vβH(l)α term factorized into the firstparenthesis yielding,(B.15) = γw[1 + uβvβ + uβv2T− l]H(l)α − γw2uβv2T[H(l+1)βα], (B.16)where it was further exploited that 2vβ −√θcβ = vβ + uβ . Thus, the explicit relationcan be used in the (l ≤ K)–Hermite series expansion for Ω(FP)i in (B.1a):Ω̂(FP)[w(v)H(l)α (v)]= γw[(1 + uβvβ + uβv2T− l)H(l)α −2uβv2TH(l+1)βα]. (B.17)A similar and conveniently easier analysis can be carried out for the acceleration-operator product Ω̂(η)[w(v)H(l)α (v)]. Firstly, an expansion of this exploiting the linearityof the composite partial derivative renders,Ω̂(η)[w(v)H(l)α (v)]= −ηβ∂vβ[wH(l)α]= −ηβ[w∂vβH(l)α +H(l)α ∂vβw]. (B.18)Using the eigenproperty (B.4) and the Hermite recurrence relation (B.14) this can berewritten to,(B.16) = −ηβw(vβ −√θcβ)H(l)αv2T−H(l+1)βαv2T . (B.19)Thus, the explicit eigenfunction for the acceleration operator reads,Ω̂(η)[w(v)H(l)α (v)]=ηβwv2T[H(l+1)βα − uβH(l)α], (B.20)that analogously can be used in the expansion of Ω(η)i in (B.1a). In App. C these areused to compute the Hermite coefficients for the series expansions.203Appendix C: Hermite coefficients for the operatorsNow that the operator eigenfunctions{Ω̂(FP); Ω̂(η)}[w(v)H(l)α (v)]have been computed the Hermite coefficients Ω(FP)α ,Ω(η)α can be establishedby evaluating the integrals (B.1b). For the FP operator coefficient as a composite functionof f(v;x, t), it is necessary to utilize the Hermite series (3.30) for that population, suchthat the integral can be rewritten to account for F(l)α ,Ω(FP)γ =∫dv H(m)γ[Ω̂(FP) ◦ f]=K∑l=0F(l)αv2lT l!∫dv H(m)γ × Ω̂(FP)[w(v)H(l)α (v)], (C.1)where wH(l)α was relocated such that the previous result in App. B conveniently can beidentified and exploited therein. Substituting in the eigenfunction one can expand theintegral to,∫dv H(m)γ[Ω̂(FP) ◦ f]=K∑l=0F(l)αv2lT l!∫dv H(m)γ×γw[(1 + uβvβ + uβv2T− l)H(l)α −2uβv2TH(l+1)βα]. (C.2)To compute the integral, the Hermite orthonormality relation [333, 359] can be usedwhich reads, ∫dv w(v)1vl+mTH(l)α (v)H(m)β (v) = δlmδ(l)αβ, (C.3)where δ(l)αβ is the sum of products of l Kronecker δ’s where each of them associates withone index in α and one in β such that there are l! terms in δ(l)αβ [359]. δlm simplydenotes that only tensors of identical rank l = m contribute to the sum, and all othercombinations l ̸= m evaluate to zero for the integral. As it is necessary to apply (C.3)twice in (B2), once for each of the H(l)α ,H(l+1)βα terms, they are treated separately asfollows, and combined after. The l-rank term becomes,K∑m=l=0F(l)αl!× γ[1 + uβvβ + uβv2T− l] ∫dv w1v2lTH(m)γ H(l)α204=F(l)α γl![1 + uβvβ + uβv2T− l]δ(l)γα, (C.4)where it can be observed that 1/vl+mT = 1/v2lT , as m = l, and that only F(l)α associateswith the indices in δ(l)γα. This results in the discrete sum,(C.4) = γ[1 + uβvβ + uβv2T− l]F(l)γ1,...,γl , (C.5)where the l! sums of F(l)α cancelled out with the 1/l! prefix in (C.4). Now the analysisis repeated for the second part of (C.2) with −2uβH(l+1)βα /v2T . In this case, it is notedthat only the l = m − 1 ranks of H(l+1)βα yield a non-zero contribution to the integral.Consequently, the goal is to compute the integral,K∑m=0F(l)αv2lT l!∫dv H(m)γ × γw[−2uβv2TH(l+1)βα]= −K∑m=02γuβF(l)αv2l+2T l!∫dv wH(m)γ H(l+1)βα (C.6)= −K∑l=m−1=02γuβF(m−1)α(m− 1)!∫dv w1v2(m−1)+2TH(m)γ H(m)βα . (C.7)Therein, it is noted thatH(m)γ ,H(m)βα are of the same ranks, and that indeed v−[2(m−1)+2]T =1/vl+mT , such that one again can apply the orthonormality relation (C.3). Substitutionyields the result,(C.7) = − 2γ(m− 1)!uβF(m−1)α δ(m−1)γ,βα , (C.8)where both uβ,F(m−1)α associate with the indices in δ(m−1)γ,βα . Ultimately, the discrete sumis computed as,(C.8) = −2γ[uγ1F(m−1)γ2,...,γm + · · ·+ uγmF(m−1)γ1,...,γm−1]. (C.9)205Combining the two results yields the coefficients,Ω(FP)γ (l) = γ[1 + uβvβ + uβv2T− l]F(l)γ − 2γ[uγ1F(m−1)γ2,...,γm + · · ·+ uγmF(m−1)γ1,...,γm−1].(C.10)The procedure is repeated for the acceleration operator, to which end the aim is tocompute the coefficients,Ω(η)γ =∫dv H(m)γ[Ω̂(η) ◦ f]=K∑l=0F(l)αv2lT l!∫dv H(m)γ × Ω̂(η)[w(v)H(l)α (v)](C.11)=K∑l=0F(l)αv2lT l!∫dv H(m)γ ×ηβwv2T[H(l+1)βα − uβH(l)α](C.12)=K∑l=0F(l)α ηβl!∫dv w1v2l+2TH(m)γ[H(l+1)βα − uβH(l)α](C.13)=K∑l=0F(l)α ηβl![∫dv w1v2l+2TH(m)γ H(l+1)βα− uβv2T∫dv w1v2lTH(m)γ H(l)α]. (C.14)In the last line the terms are grouped in a convenient form aligned with the orthonor-mality relations. Again, letting l ← m − 1 and then applying (C.3), the first integralbecomes:K∑l=0F(l)α ηβl![∫dv w1v2l+2TH(m)γ H(l+1)βα]=K∑l=m−1=0F(m−1)α ηβ(m− 1)![∫dv w1v2(m−1)+2TH(m)γ H(m)βα]=1(m− 1)!F(m−1)α ηβδ(m−1)γ,βα=[ηγ1F(m−1)γ2,...,γm + · · ·+ ηγmF(m−1)γ1,...,γm−1]. (C.15)206The second integral in (C.15) can be directly manipulated with (C.3) such that oneobtains the following:K∑l=0F(l)α ηβl![−uβv2T∫dv w1v2lTH(m)γ H(l)α]= −K∑m=l=0ηβuβF(l)αv2T l![∫dv w1v2lTH(m)γ H(l)α]= −ηβuβv2T l!F(l)α δ(l)γ,α= −ηβuβv2T[F(l)γ1,...,γl]. (C.16)Combining the two integrals yields the sought after coefficients,Ω(η)γ (l) =[ηγ1F(m−1)γ2,...,γm + · · ·+ ηγmF(m−1)γ1,...,γm−1]− ηβuβv2T[F(l)γ1,...,γl], (C.17)that together with (C.10) can be used to compute the full Hermite series. Furthermore,instead of using the f distribution one can also use the g distributions given G(l)α insteadof F(l)α in the coefficients.207Appendix D: DiagramsLattice isotropy conditionsLattice-isotropy conditions [361]:Q−1∑i=0wi = 1, (D.1a)Q−1∑i=0wici,α = 0, (D.1b)Q−1∑i=0wici,αci,β = v2T δαβ, (D.1c)Q−1∑i=0wici,αci,βci,γ = 0, (D.1d)Q−1∑i=0wici,αci,βci,γci,δ = c4T(δαβδγδ + δαγδβδ + δαδδβγ), (D.1e)Q−1∑i=0wici,αci,βci,γci,δci,ε = 0. (D.1f)In the Hermite series, it is only required including up to the third moment condition,where it is found that projecting the above conditions for the peculiar-velocity set ontothe recast velocities vi =√θci + u yields,Q−1∑i=0wi = 1, (D.2a)Q−1∑i=0wivi,α = uα, (D.2b)Q−1∑i=0wivi,αvi,β = θv2T δαβ + uαuβ, (D.2c)Q−1∑i=0wivi,αvi,βvi,γ = θv2T[uδ]αβγ+ uαuβuγ , (D.2d)Q−1∑i=0wivi,αvi,βvi,γvi,δ = θ2v2T [δδ]αβγδ + θv2T [uuδ]αβγδ + uαuβuγuδ, (D.2e)208where [uδ]αβγ = uγδαβ + uβδαγ + uαδβγ , [δδ]αβγδ = δαβδγδ + δαγδβδ + δαδδβγ , and[uuδ]αβγδ = uαuβδγδ + uαuγδβδ + uαuδδβγ + uβuγδαδ + uβuδδαγ + uγuδδαβ are short-hand notations for conciseness. The condition (D.2b) follows from,Q−1∑i=0wivi,α =Q−1∑i=0wi(√θci,α + uα)=Q−1∑i=0wiuα = uα, (D.3)wherein, (D.1b) was used. The second-order moment relation (D.2c) arises from expand-ing the discrete velocity product,Q−1∑i=0wivi,αvi,β =Q−1∑i=0wi(√θci,α + uα)(√θci,β + uβ)=Q−1∑i=0wi(θci,αci,β + uαuβ + uβ√θci,α + uα√θci,β)= θv2T δαβ + uαuβ, (D.4)wherein (D.1b, D.1c) were exploited. Lastly, (D.2d) is computed by considering thethird-order velocity product,Q−1∑i=0wivi,αvi,βvi,γ =Q−1∑i=0wi(√θci,α + uα)(√θci,β + uβ)(√θci,γ + uγ)=Q−1∑i=0wi(θci,αci,β + uαuβ + uβ√θci,α + uα√θci,β)(√θci,γ + uγ)= θv2Tuγδαβ + uαuβuγ + θv2Tuβδαγ + θv2Tuαδβγ , (D.5)where the first (D.1b), second (D.1c), and third-order (D.1d) conditions were utilized.Expanded tensorsThis appendix expands relevant, higher-order tensors referenced throughout the text,as they are implemented in code.Lagrange polynomialsThe reconstruction procedure utilizes 16 unique Lagrange polynomials Λmn(xd) toreconstruct populations at Q departure points xd(i), from the surrounding 16 colocation209points. These polynomials read [94],Λmn =Λ00 Λ01 Λ02 Λ03Λ10 Λ11 Λ12 Λ13Λ20 Λ21 Λ22 Λ23Λ30 Λ31 Λ32 Λ33 =∏0≤k≤3k ̸=n(xd − x0)− kn− k∏0≤l≤3l ̸=m(yd − y0)− lm− l , (D.6a)where one can denote{(xd − x0), (yd − y0)}={∆xd,∆yd}. As such the tensor compo-nents become,Λ00 =(∆xd − 10− 1)(∆xd − 20− 2)(∆xd − 30− 3)×(∆yd − 10− 1)(∆yd − 20− 2)(∆yd − 30− 3),(D.6b)Λ10 =(∆xd − 10− 1)(∆xd − 20− 2)(∆xd − 30− 3)×(∆yd1− 0)(∆yd − 21− 2)(∆yd − 31− 3),(D.6c)Λ20 =(∆xd − 10− 1)(∆xd − 20− 2)(∆xd − 30− 3)×(∆yd − 02− 0)(∆yd − 12− 1)(∆yd − 32− 3),(D.6d)Λ30 =(∆xd − 10− 1)(∆xd − 20− 2)(∆xd − 30− 3)×(∆yd − 03− 0)(∆yd − 13− 1)(∆yd − 23− 2),(D.6e)which concludes the first column of the matrix. The remaining are similarly defined,Λ01 =(∆xd)(∆xd − 2−1)(∆xd − 3−2)×(∆yd − 1−1)(∆yd − 2−2)(∆yd − 3−3), (D.6f)Λ11 =(∆xd)(∆xd − 2−1)(∆xd − 3−2)× (∆yd)(∆yd − 2−1)(∆yd − 3−2), (D.6g)Λ21 =(∆xd)(∆xd − 2−1)(∆xd − 3−2)×(∆yd2)(∆yd − 1)(∆yd − 3−1), (D.6h)Λ31 =(∆xd)(∆xd − 2−1)(∆xd − 3−2)×(∆yd3)(∆yd − 12)(∆yd − 2), (D.6i)Λ02 =(∆xd2)(∆xd − 1)(∆xd − 3−1)×(∆yd − 1−1)(∆yd − 2−2)(∆yd − 3−3), (D.6j)Λ12 =(∆xd2)(∆xd − 1)(∆xd − 3−1)× (∆yd)(∆yd − 2−1)(∆yd − 3−2), (D.6k)Λ22 =(∆xd2)(∆xd − 1)(∆xd − 3−1)×(∆yd2)(∆yd − 1)(∆yd − 3−1), (D.6l)210030.53$ 002 211 10 03030.5$ 102 211 10 03030.5$ 202 211 10 0030.53$ 302"yp211 10 03030.52$01211 10 03030.52121 10 03030.52121 10 03030.52 211 10 03030.52$02211 10 03030.52121 10 03030.52121 10 03030.52 211 10 0030.532$03211 10 03030.52 211 10 03030.52 211 10 0030.532 2"xp11 10 0Figure F.1: Each of the 4× 4 Lagrange polynomials in Λ.Λ32 =(∆xd2)(∆xd − 1)(∆xd − 3−1)×(∆yd3)(∆yd − 12)(∆yd − 2), (D.6m)Λ03 =(∆xd3)(∆xd − 12)(∆xd − 2)× (∆yd − 1−1)(∆yd − 2−2)(∆yd − 3−3), (D.6n)Λ13 =(∆xd3)(∆xd − 12)(∆xd − 2)× (∆yd)(∆yd − 2−1)(∆yd − 3−2), (D.6o)Λ23 =(∆xd3)(∆xd − 12)(∆xd − 2)× (∆yd2)(∆yd − 1)(∆yd − 3−1), (D.6p)Λ33 =(∆xd3)(∆xd − 12)(∆xd − 2)× (∆yd3)(∆yd − 12)(∆yd − 2). (D.6q)211Gauge-transformationThe moment-matching criterion exploits the invariance property of the moments,Mλαβ =Q−1∑i=0fλi(√θc(i)x + ux)α(√θc(i)y + uy)β, (D.7)where {α, β} = {0, 1, 2} such that the moments for a D2Q9 model organize in the columnvector (“†” denotes the transpose),Mλαβ =[Mλ00,Mλ10,Mλ20,Mλ01,Mλ11,Mλ21,Mλ02,Mλ12,Mλ22]†. (D.8)Therein, the individual moments extend to,Mλ00 = fλ0 + fλ1 + · · ·+ fλ8 , (D.9)Mλ10 = fλ0(√θc(0)x + ux)+ · · ·+ fλ8(√θc(8)x + ux), (D.10)Mλ20 = fλ0(√θc(0)x + ux)2+ · · ·+ fλ8(√θc(8)x + ux)2, (D.11)Mλ01 = fλ0(√θc(0)y + uy)+ · · ·+ fλ8(√θc(8)y + uy), (D.12)Mλ11 = fλ0(√θc(0)x + ux)(√θc(0)y + uy)+ · · ·+ fλ8(√θc(8)x + ux)(√θc(8)y + uy), (D.13)Mλ21 = fλ0(√θc(0)x + ux)2(√θc(0)y + uy)+ · · ·+ fλ8(√θc(8)x + ux)2(√θc(8)y + uy), (D.14)Mλ02 = fλ0(√θc(0)y + uy)2+ · · ·+ fλ8(√θc(8)y + uy)2, (D.15)Mλ12 = fλ0(√θc(0)x + ux)(√θc(0)y + uy)2+ · · ·+ fλ8(√θc(8)x + ux)(√θc(8)y + uy)2, (D.16)Mλ22 = fλ0(√θc(0)x + ux)2(√θc(0)y + uy)2+ · · ·+ fλ8(√θc(8)x + ux)2(√θc(8)y + uy)2,(D.17)where the superscript in cx, cy denote their association to the Q populationsfλi =[fλ0 , fλ1 , fλ2 , fλ3 , fλ4 , fλ5 , fλ6 , fλ7 , fλ8]†, (D.18)such that the gauge transformation assumes the same order in the peculiar velocities,cx =[0, 1, 0,−1, 0, 1,−1,−1, 1]†, (D.19)cy =[0, 0, 1, 0,−1, 1, 1,−1,−1]†. (D.20)The solver assumes this order of the individual moments and discrete velocity compo-nents. The linear transformation of the populations involve the Q × Q linear map Mλ212such that the moments follow the matrix-vector multiplication Mλ =Mλfλ. Thus, thelinear system takes the full form,M00 M01 M02 M03 M04 M05 M06 M07 M08M10 M11 M12 M13 M14 M15 M16 M17 M18M20 M21 M22 M23 M24 M25 M26 M27 M28M30 M31 M32 M33 M34 M35 M36 M37 M38M40 M41 M42 M43 M44 M45 M46 M47 M48M50 M51 M52 M53 M54 M55 M56 M57 M58M60 M61 M62 M63 M64 M65 M66 M67 M68M70 M71 M72 M73 M74 M75 M76 M77 M78M80 M81 M82 M83 M84 M85 M86 M87 M88λfλ0fλ1fλ2fλ3fλ4fλ5fλ6fλ7fλ8=Mλ00Mλ10Mλ20Mλ01Mλ11Mλ21Mλ02Mλ12Mλ22. (D.21)Irrespective of gauge the components (by row) of Mλ(and Mλ′)become,Mλ0β = {1}, (D.22)Mλ1β = {√θc(β)x + ux}, (D.23)Mλ2β ={(√θc(β)x + ux)2}, (D.24)Mλ3β = {√θc(β)y + uy}, (D.25)Mλ4β ={(√θc(β)x + ux)(√θc(β)y + uy)}, (D.26)Mλ5β ={(√θc(β)x + ux)2(√θc(β)y + uy)}, (D.27)Mλ6β ={(√θc(β)y + uy)2}, (D.28)Mλ7β ={(√θc(β)x + ux)(√θc(β)y + uy)2}, (D.29)Mλ8β ={(√θc(β)x + ux)2(√θc(β)y + uy)2}. (D.30)The moment matching condition implies that Mλ′ = Mλ, and thus Mλ′fλ′ = Mλfλ.Thus, one can transform f from gauge λ to λ′ via the transformation matrix Gλ′λ ,fλ′= Gλ′λ fλ =M−1λ′ Mλfλ. (D.31)A MATLAB code (Algo. 1) has been developed with inspiration from Kallikounis et al.[351] for finding the unique D2Q9 transformation including the inverse M−1λ′ found withLU decomposition. The full forms of Gλ′λ ,M−1λ′ and Mλ are found on the following page.Unmodified, Gλ′λ =M−1λ′ Mλ is also used to transform Sλi during the collision step.213Algorithm 1: MATLAB code for obtaining the transformation matrix Gλ′λ .% Define symbolic variables:syms ux uy sqrt_th ux_p uy_p sqrt_th_p M M_p;Q = 9;% Discrete velocities:cx = [0 1 0 -1 0 1 -1 -1 1];cy = [0 0 1 0 -1 1 1 -1 -1];% Moment exponents:m = [0 1 2 0 1 2 0 1 2];n = [0 0 0 1 1 1 2 2 2];for a = 1:1:Q dofor b = 1:1:Q doM(i,j) = (sqrt_th*cx(j)+ux)ˆ (m(i))*(sqrt_th*cy(j)+uy)ˆ (n(i));M_p(i,j) = (sqrt_th_p*cx(j)+ux_p)ˆ (m(i))*(sqrt_th_p*cy(j)+uy_p)ˆ (n(i));endendG = inv(M_p)*M;214Mλ =1 1 · · · · · · 1{√θc(0)x + ux} {√θc(1)x + ux}· · · · · ·{√θc(8)x + ux}{√θc(0)x + ux}2 {√θc(1)x + ux}2· · · · · ·{√θc(8)x + ux}2{√θc(0)y + uy} {√θc(1)y + uy}· · · · · ·{√θc(8)y + uy}{√θc(0)x + ux}{√θc(0)y + uy} {√θc(1)x + ux}{√θc(1)y + uy}· · · · · ·{√θc(8)x + ux}{√θc(8)y + uy}{√θc(0)x + ux}2{√θc(0)y + uy} {√θc(1)x + ux}2{√θc(1)y + uy}· · · · · ·{√θc(8)x + ux}2{√θc(8)y + uy}{√θc(0)y + uy}2 {√θc(1)y + uy}2· · · · · ·{√θc(8)y + uy}2{√θc(0)x + ux}{√θc(0)y + uy}2 {√θc(1)x + ux}{√θc(1)y + uy}2· · · · · ·{√θc(8)x + ux}{√θc(8)y + uy}2{√θc(0)x + ux}2{√θc(0)y + uy}2 {√θc(1)x + ux}2{√θc(1)y + uy}2· · · · · ·{√θc(8)x + ux}2{√θc(8)y + uy}2=1 1 1 1 1 1 1 1 1ux√θ + ux ux ux −√θ ux√θ + ux ux −√θ ux −√θ√θ + uxu2x{√θ + ux}2u2x{√θ − ux}2u2x{√θ + ux}2 {√θ − ux}2 {√θ − ux}2 {√θ + ux}2uy uy√θ + uy uy uy −√θ√θ + uy√θ + uy uy −√θ uy −√θux · uy uy ·{√θ + ux}ux ·{√θ + uy}−uy ·{√θ − ux}−ux ·{√θ − uy} {√θ + ux}·{√θ + uy}−{√θ + uy}·{√θ − ux} {√θ − ux}·{√θ − uy}−{√θ + ux}·{√θ − uy}u2x · uy uy ·{√θ + ux}2u2x ·{√θ + uy}uy ·{√θ − ux}2 −u2x · {√θ − uy} {√θ + ux}2·{√θ + uy} {√θ + uy}·{√θ − ux}2 −{√θ − ux}2·{√θ − uy} −{√θ + ux}2·{√θ − uy}u2y u2y{√θ + uy}2u2y{√θ − uy}2 {√θ + uy}2 {√θ + uy}2 {√θ − uy}2 {√θ − uy}2ux · u2y u2y ·{√θ + ux}ux ·{√θ + uy}2 −u2y · {√θ − ux} ux · {√θ − uy}2 {√θ + ux}·{√θ + uy}2 −{√θ + uy}2·{√θ − ux} −{√θ − ux}·{√θ − uy}2 {√θ + ux}·{√θ − uy}2u2x · u2y u2y ·{√θ + ux}2u2x ·{√θ + uy}2u2y ·{√θ − ux}2u2x ·{√θ − uy}2 {√θ + ux}2·{√θ + uy}2 {√θ + uy}2·{√θ − ux}2 {√θ − ux}2·{√θ − uy}2 {√θ + ux}2·{√θ − uy}2215Columns 0 through 3:Gλ′λ =((ux − u′x)2(uy − u′y)2 − θ′ ((ux − u′x)2 + (uy − u′y)2)) /θ′2 + 1 1− (θ′((uy − u′y)2 + (√θ + ux − u′x)2)− (uy − u′y)2 (√θ + ux − u′x)2) /θ′2 1− (θ′((ux − u′x)2 + (√θ + uy − u′y)2)− (ux − u′x)2 (√θ + uy − u′y)2) /θ′2 1− (θ′((uy − u′y)2 + (√θ − ux + u′x)2)− (uy − u′y)2 (√θ − ux + u′x)2) /θ′2((ux − u′x)(√θ′ + ux − u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y))/(2θ′2) ((√θ + ux − u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y)(√θ +√θ′ + ux − u′x))/(2θ′2) −((ux − u′x)(√θ′ + ux − u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + uy − u′y)) / (2θ′2) ((√θ − ux + u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x)) / (2θ′2)((uy − u′y) (√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ′ + uy − u′y))/(2θ′2) −((uy − u′y) (√θ′ + uy − u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ + ux − u′x)) / (2θ′2) ((√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ + uy − u′y)(√θ +√θ′ + uy − u′y)) / (2θ′2) −((uy − u′y) (√θ′ + uy − u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ − ux + u′x)) / (2θ′2)−((ux − u′x)(√θ′ − ux + u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y))/(2θ′2) ((√θ + ux − u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x))/(2θ′2) ((ux − u′x)(√θ′ − ux + u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + uy − u′y))/(2θ′2) ((√θ − ux + u′x)(√θ′ + uy − u′y)(√θ′ − uy + u′y)(√θ +√θ′ − ux + u′x))/(2θ′2)−((uy − u′y) (√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ′ − uy + u′y))/(2θ′2) ((uy − u′y) (√θ′ − uy + u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ + ux − u′x))/(2θ′2) ((√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ + uy − u′y)(√θ −√θ′ + uy − u′y))/(2θ′2) ((uy − u′y) (√θ′ − uy + u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ − ux + u′x))/(2θ′2)((ux − u′x)(uy − u′y) (√θ′ + ux − u′x)(√θ′ + uy − u′y))/(4θ′2) ((uy − u′y) (√θ + ux − u′x)(√θ′ + uy − u′y)(√θ +√θ′ + ux − u′x))/(4θ′2) ((ux − u′x)(√θ′ + ux − u′x)(√θ + uy − u′y)(√θ +√θ′ + uy − u′y))/(4θ′2) ((uy − u′y) (√θ − ux + u′x)(√θ′ + uy − u′y)(√θ −√θ′ − ux + u′x))/(4θ′2)−((ux − u′x)(uy − u′y) (√θ′ − ux + u′x)(√θ′ + uy − u′y))/(4θ′2) ((uy − u′y) (√θ + ux − u′x)(√θ′ + uy − u′y)(√θ −√θ′ + ux − u′x))/(4θ′2) −((ux − u′x)(√θ′ − ux + u′x)(√θ + uy − u′y)(√θ +√θ′ + uy − u′y)) / (4θ′2) ((uy − u′y) (√θ − ux + u′x)(√θ′ + uy − u′y)(√θ +√θ′ − ux + u′x)) / (4θ′2)((ux − u′x)(uy − u′y) (√θ′ − ux + u′x)(√θ′ − uy + u′y))/(4θ′2) −((uy − u′y) (√θ + ux − u′x)(√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x)) / (4θ′2) −((ux − u′x)(√θ′ − ux + u′x)(√θ + uy − u′y)(√θ −√θ′ + uy − u′y)) / (4θ′2) −((uy − u′y) (√θ − ux + u′x)(√θ′ − uy + u′y)(√θ +√θ′ − ux + u′x)) / (4θ′2)−((ux − u′x)(uy − u′y) (√θ′ + ux − u′x)(√θ′ − uy + u′y))/(4θ′2) −((uy − u′y) (√θ + ux − u′x)(√θ′ − uy + u′y)(√θ +√θ′ + ux − u′x)) / (4θ′2) ((ux − u′x)(√θ′ + ux − u′x)(√θ + uy − u′y)(√θ −√θ′ + uy − u′y)) / (4θ′2) −((uy − u′y) (√θ − ux + u′x)(√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x)) / (4θ′2).Columns 4 through 8:1−(θ′((ux − u′x)2 +(√θ − uy + u′y)2)− (ux − u′x)2 (√θ − uy + u′y)2) /θ′2 ((√θ +√θ′ + ux − u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ + uy − u′y)) /θ′2 ((√θ +√θ′ − ux + u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ + uy − u′y)) /θ′2 ((√θ +√θ′ − ux + u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ − uy + u′y)) /θ′2 ((√θ +√θ′ + ux − u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ − uy + u′y)) /θ′2−((ux − u′x)(√θ′ + ux − u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − uy + u′y))/(2θ′2) −((√θ + ux − u′x)(√θ +√θ′ + ux − u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ − ux + u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ − ux + u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ − uy + u′y)) / (2θ′2) −((√θ + ux − u′x)(√θ +√θ′ + ux − u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − uy + u′y)) / (2θ′2)((√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ − uy + u′y)(√θ −√θ′ − uy + u′y))/(2θ′2) −((√θ + uy − u′y)(√θ +√θ′ + ux − u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + ux − u′x)) / (2θ′2) −((√θ + uy − u′y)(√θ +√θ′ − ux + u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ − ux + u′x)) / (2θ′2) −((√θ − uy + u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ − ux + u′x)(√θ −√θ′ − uy + u′y)) / (2θ′2) −((√θ − uy + u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ + ux − u′x)(√θ −√θ′ − uy + u′y)) / (2θ′2)((ux − u′x)(√θ′ − ux + u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − uy + u′y))/(2θ′2) −((√θ + ux − u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ − ux + u′x)(√θ +√θ′ − ux + u′x)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ − ux + u′x)(√θ +√θ′ − ux + u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − uy + u′y)) / (2θ′2) −((√θ + ux − u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ − uy + u′y)) / (2θ′2)((√θ′ + ux − u′x)(√θ′ − ux + u′x)(√θ − uy + u′y)(√θ +√θ′ − uy + u′y))/(2θ′2) −((√θ + uy − u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ + ux − u′x)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ + uy − u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ − ux + u′x)(√θ −√θ′ + uy − u′y)) / (2θ′2) −((√θ − uy + u′y)(√θ +√θ′ − ux + u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x)) / (2θ′2) −((√θ − uy + u′y)(√θ +√θ′ + ux − u′x)(√θ +√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x)) / (2θ′2)((ux − u′x)(√θ′ + ux − u′x)(√θ − uy + u′y)(√θ −√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ + uy − u′y)(√θ +√θ′ + ux − u′x)(√θ +√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ + uy − u′y)(√θ +√θ′ + uy − u′y)(√θ −√θ′ − ux + u′x))/(4θ′2) ((√θ − ux + u′x)(√θ − uy + u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ − uy + u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ − uy + u′y))/(4θ′2)−((ux − u′x)(√θ′ − ux + u′x)(√θ − uy + u′y)(√θ −√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ + uy − u′y)(√θ +√θ′ + uy − u′y)(√θ −√θ′ + ux − u′x))/(4θ′2) ((√θ − ux + u′x)(√θ + uy − u′y)(√θ +√θ′ − ux + u′x)(√θ +√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ − uy + u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ − uy + u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ − uy + u′y))/(4θ′2)−((ux − u′x)(√θ′ − ux + u′x)(√θ − uy + u′y)(√θ +√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ + uy − u′y)(√θ −√θ′ + ux − u′x)(√θ −√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ + uy − u′y)(√θ +√θ′ − ux + u′x)(√θ −√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ − uy + u′y)(√θ +√θ′ − ux + u′x)(√θ +√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ − uy + u′y)(√θ +√θ′ − uy + u′y)(√θ −√θ′ + ux − u′x))/(4θ′2)((ux − u′x)(√θ′ + ux − u′x)(√θ − uy + u′y)(√θ +√θ′ − uy + u′y))/(4θ′2) ((√θ + ux − u′x)(√θ + uy − u′y)(√θ +√θ′ + ux − u′x)(√θ −√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ + uy − u′y)(√θ −√θ′ − ux + u′x)(√θ −√θ′ + uy − u′y))/(4θ′2) ((√θ − ux + u′x)(√θ − uy + u′y)(√θ +√θ′ − uy + u′y)(√θ −√θ′ − ux + u′x))/(4θ′2) ((√θ + ux − u′x)(√θ − uy + u′y)(√θ +√θ′ + ux − u′x)(√θ +√θ′ − uy + u′y))/(4θ′2).216
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Brinkerhoff, Joshua
A lattice-Fokker-Planck model of thermal noise in liquids
Text