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The effects of anisotropic turbulence on fibre motion Holt, Kyle 2016

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THE EFFECTS OF ANISOTROPIC TURBULENCE ON FIBRE MOTION by  Kyle Holt  B.Sc, The University of British Columbia, 2013  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2016  © Kyle Holt, 2016 ii  Abstract  Fibre orientation and concentration distributions in a turbulent suspension are typically modeled by the Fokker-Planck equation and dispersion coefficients that relate the fluid turbulence properties to the fibre length, assuming an infinitely thin, and inertialess fibre. These predictions are used in a wide variety of engineering problems, notably the forming of paper. There is substantial literature examining the application of the Fokker-Planck equation for isotropic turbulence, however, there are few studies that examine the effect of turbulence anisotropy on fibre translational and rotational dispersion. In order to provide better estimates of fibre orientation and concentration distributions in an anisotropic turbulent suspension, the relationship between turbulence and fibres needs to be better understood.  This thesis develops a stochastic model which can be used to determine the orientational and translational dispersion coefficients. The fluid model is based on the Kraichnan turbulence fluid model [1] and the fibres are represented by rigid, inertialess infinitely thin particles assumed to be in a dilute suspension. The turbulence is made axisymmetric by utilizing the time dependent wavevector relation from Rapid Distortion Theory [2]. This enables the model to simulate the physical effects of eddy distortion by a contraction.   A range of fibre lengths and contraction ratios were modeled, and it was found that the translational dispersion coefficient increases in magnitude with contraction ratio and decreases with fibre length while the rotational dispersion coefficient decreases with both contraction ratio and fibre length. A constant was found relating the isotropic and anisotropic translational dispersion coefficients, based on contraction ratio. The simulation showed that the integral time scale for translation increased for short fibres and decreased for long fibres as contraction ratio increased for both directions. The integral time scale for rotation decreased for small fibres and iii  increased to a maximum before decreasing for long fibres as contraction ratio increased for both directions. It was shown that fibre orientation tended to the radial orientation preferentially as contraction ratio increased but spent more time in the streamwise orientation as fibre length increased. These findings will allow for improved estimates to be made for anisotropic dispersion coefficients.   iv  Preface  A version of this work will be presented at the Tokyo Paper Physics Conference 2015 under the same title.   The model described in Chapter 2 was originally developed by Dr. James Olson. This model was adapted in this work to simulate axisymmetric turbulence strained through a contraction. The model was re-written and extended by the author with advice from Dr Olson.   v  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ...........................................................................................................................v List of Tables ............................................................................................................................... vii List of Figures ............................................................................................................................. viii List of Symbols ............................................................................................................................ xii List of Abbreviations .............................................................................................................. xiiiiii Acknowledgements .................................................................................................................. xivv Chapter 1: Introduction ................................................................................................................1 Chapter 2: Literature Review .......................................................................................................6 2.1 Low Reynolds Number Fibre Dynamics ........................................................................ 6 2.2 Turbulent Reynolds Number Fibre Dynamics ................................................................ 7 2.3 Descriptions of Turbulence ............................................................................................. 8 2.4 Rapid Distortion Theory ............................................................................................... 11 2.5 Fibre Orientation in Turbulence.................................................................................... 12 Chapter 3: Numerical Model of Fibre Motion in Aniostropic Turbulence ............................16 3.1 Model of Fibre Motion in Distorted Turbulent Velocity Field ..................................... 18 3.1.1 Fibre Motion ............................................................................................................. 19 3.1.2 Turbulent Velocity Field ........................................................................................... 23 3.1.3 Anisotropic Axsiymmetric Turbulence..................................................................... 26 Chapter 4: Results........................................................................................................................31 vi  4.1 Fluid Statistics ............................................................................................................... 31 4.2 Fibre Statistics ............................................................................................................... 42 4.2.1 Translational Dispersion ........................................................................................... 42 4.2.2 Rotational Dispersion................................................................................................ 62 4.2.3 Fibre Rotation ........................................................................................................... 77 4.2.4 Timescale Relationship ............................................................................................. 81 Chapter 5: Conclusion .................................................................................................................84 5.1 Conclusions ................................................................................................................... 84 5.2 Future Work .................................................................................................................. 85 Bibliography .................................................................................................................................86 Appendices ....................................................................................................................................94 Appendix A ............................................................................................................................... 94 A.1 Proof of isotropic and anisotropic translational dispersion magnitude ..................... 94 A.2 Anisotropic estimates of long time dispersion coefficients ...................................... 95 A.3 Error on rotational dispersion coefficient ................................................................. 96 Appendix B ............................................................................................................................... 98 B.1 Fluid Model Validation ............................................................................................. 98 B.2 Fibre Model Validation ............................................................................................. 99   vii  List of Tables  Table 4.1 Time percentage spent in the streamwise orientation ................................................... 79 Table 4.2 Time percentage spent in the radial orientation ............................................................ 80 Table 4.3 Streamwise relationship of       ................................................................................. 82 Table 4.4 Radial relationship of      .......................................................................................... 83    viii  List of Figures  Figure 1.1 An axisymmetric contraction ratio, 1 and 2 are orthogonal principal directions .......... 4 Figure 3.1 A straight rigid fibre of length L. v is the fibre velocity and u is the fluid velocity. The fibre is oriented to point in the direction of the unit vector p ....................................................... 20 Figure 3.2 Sample unit wavenumber spheres generated by the wavevector distortion for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4. ........................................................................................................ 29 Figure 4.1 Streamwise Eulerian spatial velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4 ........................................................................................................................................ 32 Figure 4.2 Radial Eulerian spatial velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4....................................................................................................................................................... 33 Figure 4.3 Streamwise numerically determined longitudinal velocity correlation equation fit for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 .......................................................................................... 35 Figure 4.4 Radial  numerically determined longitudinal velocity correlation equation fit for  (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 .......................................................................................... 36 Figure 4.5 Comparison of the simulated and expected length scales for all contraction ratios .... 38 Figure 4.6 Eulerian and Lagrangian time scales for all contraction ratios ................................... 39 Figure 4.7 Eulerian and Lagrangian temporal velocity correlations for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ..................................................................................................................................... 40 Figure 4.8 Radial Eulerian and Lagrangian temporal velocity correlations  for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ......................................................................................................................... 41 Figure 4.9 Streamwise fibre translation for all fibre lengths for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4 ........................................................................................................................................ 44 ix  Figure 4.10 Radial fibre translation for all fibre lengths for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4....................................................................................................................................................... 45 Figure 4.11 Comparison of the streamwise dispersion to the radial dispersion ........................... 47 Figure 4.12 Streamwise component of the translational dispersion coefficient for all contraction ratios for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 .......................................................................... 49 Figure 4.13 Radial component of the translational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ......................................................................................................................... 50 Figure 4.14 Streamwise translational velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4 ........................................................................................................................................ 52 Figure 4.15 Radial translational velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 54 Figure 4.16 The difference in relative fibre length depending on how many eddies are averaged....................................................................................................................................................... 55 Figure 4.17 The streamwise fibre Lagrangian integral time scale for all contraction ratios ........ 57 Figure 4.18 The radial fibre Lagrangian integral time scale for all contraction ratios ................. 57 Figure 4.19 Small fibres behave like fluid particles vs averaging of many eddies over the length of long fibres ................................................................................................................................. 58 Figure 4.20 Long term streamwise translational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 .............................................................................................................................. 60 Figure 4.21 Long term radial translational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ........................................................................................................................................ 61 Figure 4.22 Illustration of how angular velocity changes in a contraction ................................... 63 Figure 4.23 The increase in eddy size does not impact the small fibres significantly .................. 63 x  Figure 4.24 Orientation correlation in the streamwise direction for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ........................................................................................................................................ 64 Figure 4.25 Radial orientation correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ................ 65 Figure 4.26 Streamwise rotational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4 ........................................................................................................................................ 67 Figure 4.27 Radial rotational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 . 68 Figure 4.28 Streamwise angular velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 70 Figure 4.29 Radial angular velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ........ 71 Figure 4.30 The streamwise fibre Lagrangian angular velocity integral time scale for all contraction ratios ........................................................................................................................... 73 Figure 4.31 The radial fibre Lagrangian angular velocity integral time scale for all contraction ratios .............................................................................................................................................. 73 Figure 4.32 Long time rotational dispersion coefficients for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4 ........................................................................................................................................ 75 Figure 4.33 Long time radial rotational dispersion coefficients for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ........................................................................................................................................ 76 Figure 4.34 Streamwise heat maps for fibres of length     = 0.25 for (a) C = 1 (b) C = 2  (c) C = 3 (d) C = 4 ......................................................................................................................... 78 Figure 4.35 Streamwise heat maps for fibres of length     = 3 for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 ........................................................................................................................................ 79 Figure 4.36 Streamwise heat maps for fibres of length     = 10 for (a) C = 1 (b) C = 2  (c) C = 3 (d) C = 4 ......................................................................................................................... 80 xi  Figure B.1 The rms velocity for different times and the velocity mean for all contraction ratios tested ............................................................................................................................................. 98  Figure B.2 The rms velocities of the simulation compared with the Townsend equations and with Prandtl's formulae ................................................................................................................. 99      xii  List of Symbols    = Contraction Ratio    = Rotational dispersion coefficient  (            )     = Long-time rotational dispersion coefficient (            )          = Anisotropic long-time rotational dispersion coefficient (            )    = Translational dispersion coefficient (            )     = Long-time translational dispersion coefficient (            )          = Anisotropic long-time translational dispersion coefficient (            )      = Longitudinal velocity correlation      = Transverse velocity correlation    = Energy spectrum max (        )       = Energy spectrum max at specific contraction ratio (        )   = Fibre length, dimensional (      )    = Rotational time scale (    )    = Translational time scale (    )   = Instantaneous fluid velocity (           )        = Turbulent intensity    = Root mean square (           )   = Fibre velocity (           )   = Taylor Microscale (    ) xiii  List of Abbreviations  DNS = Direct Numerical Simulation RDT = Rapid Distortion Theory   xiv  Acknowledgements  I would like to thank Dr. Olson for his supervision over my thesis. His questions helped me to understand and appreciate turbulence, and without his help, this thesis would never have been finished.  I would also like to thank my parents for their love and support.  Lastly, I would like to thank my partner, Tiffany, whose support was invaluable to me while I worked on this thesis.   1  Chapter 1: Introduction  The behaviour of fibres in turbulent flows greatly impacts the properties of suspensions such as rheology, orientation and concentration. Turbulent fibre suspensions are of particular interest in the pulp and paper industry, where the orientation distribution of the suspension can drastically affect the properties of the final product. In papermaking, a floc fibre suspension is fluidized in turbulence. The fibre suspension is then accelerated through a headbox and is evenly spread out along a permeable moving wire where the water is drained and the paper formed. The industrial qualities of the paper depends on the final orientation of the fibres. For instance, in papermaking, the strength properties of the manufactured paper can be anisotropic if the fibre orientation distribution is anisotropic. Similarly, the paper could be much stronger in one direction if the fibres are aligned in that direction. One of the primary parameters influencing orientation distribution is the shape and rapidness of the headbox contraction, ahead of being dewatered. The larger the contraction ratios the more the fibres will be aligned. The effects of headbox contraction on fibre orientation and the properties of paper has been investigated by many but is not yet fully understood.  The investigation of fibre motion is split into two methods. There is the Lagrangian method, which is the most common, where the equations of motion are integrated through a flow field to determine the fibre trajectories. This method also allows for the investigation of fibre-fibre collisions. The other method is the Eulerian method where a control volume is used to calculate both the fibre orientation and concentration distribution through the flow field simultaneously. The Eulerian approach is more computationally efficient and accounts for turbulent dispersion and fibre-fluid interactions. 2   The flow of a fibre suspension is described by a convection-dispersion equation. The effect of turbulence on the suspension is modeled using dispersion coefficients. These coefficients are influenced by the fluid and fibre properties. Fibre length is one of the key parameters in determining the properties of a suspension for papermaking. In pulping, the fibre length depends on the raw materials and the pulping process. Short, stiff fibres result from mechanical pulping and longer, more flexible fibres from chemical pulping. Olson and Kerekes [3] were the first to investigate how fibre length affects rotational and translational turbulent dispersion coefficients. They developed an Eulerian model of fibre motion for an ideal fibre in a dilute suspension and used Cox’s [4] expression for the drag force acting on a fibre. To apply this expression to a turbulent fibre suspension, they implemented the free-draining approximation, which assumes the fibre does not affect the flow field. They related the rotational and translational fluctuations of the fibre to the length and time scales of the turbulence and estimated the dispersion coefficients for an Eulerian model of fibre motion.  To validate the analytic findings of Olson and Kerekes [3], a numerical model of a fibre in isotropic turbulence was developed by Olson [5]. Kraichnan’s [1] model was used to generate a stochastic turbulent velocity field. The equations of fibre motion were integrated through the flow and the dispersion coefficients were estimated. One of the main hypothesis was confirmed, as it was shown that as fibre length increases, the integral time scale of fibre motion moves from the fluid’s Lagrangian time scale to the fluid’s Eulerian time scale and the rotational and translational dispersion coefficients decrease. There was qualitative agreement between the approximate expressions of Olson and Kerekes [3] and the simulation of Olson [5]; however, it was found that the rotational dispersion coefficient was over-estimated by Olson and Kerekes. 3   Shin and Koch [6] refined these estimates using direct numerical simulation to simulate an isotropic turbulent field and slender-body theory [7] to determine the fibre motion. Estimates of the translational dispersion coefficient qualitatively agreed with Olson [5] but differed quantitatively. Shin and Koch [6] concluded that while the approximate expression for the translational dispersion coefficient was qualitatively correct, the expression for the rotational dispersion coefficient was not as Olson and Kerekes [3] assumed the fibre’s orientation was uncorrelated with the local axes of strain and rotation. Shin and Koch [6] presented their own expression for the rotational dispersion coefficient which was on the order of the inverse of the fibre Eulerian rotational time scale,   .   The literature contains several investigations of the rotational and translational dispersion coefficients for fibre suspensions in isotropic turbulent flows. However, many fibre processes utilize a contraction, which can alter the properties of turbulent motion by imposing distortions upon it, rendering it anisotropic [2]. There have been numerous studies regarding the effects of contractions on turbulence but there has been little investigation into the effect on fibre orientation in suspensions in anisotropic flow. To better understand this, a suitable method of modeling the effects of a contraction is required in order to examine how a contraction affects the properties of turbulence. Physically, eddies are stretched along the contraction and contracted radially. Batchelor and Proudman’s [2] Rapid Distortion Theory, or RDT, predicts that many of the turbulent properties are modified by this same kind of stretching and contracting. As a result of this, RDT was used to model the effects of a contraction. We compared RDT predictions to experimental measurements in contractions with contraction ratios that have been previously validated against experiment, where a contraction ratio is defined by the following, 4           (1)  where    is the radius at the inlet and   is the local radius. This method can be applied to both planar and axisymmetric contractions, as several studies have illustrated that contraction type does not qualitatively change the turbulence characteristics [8], [9].  Axisymmetric turbulence is the simplest form of anisotropic turbulence [10], which for low contraction ratios can be predicted by Prandtl’s formulae [11], given by the following,                              (2)  where   is the exit velocity,   is the inlet velocity.  ,  ,   are the standard representations of the velocity in the 1, 2, and 3 direction, respectively. This is illustrated in Fig. 1.1. Prandtl’s formula and RDT are reasonably predictive for axisymmetric turbulence of contraction ratios less than 4 [12]. While this is less than the contraction ratio in a typical headbox, we are interested in the statistics as anisotropy initially develops with contraction ratio, in essence, before the return to isotropy. This makes RDT a reasonable model to investigate fibre statistics in anisotropic turbulence.         Figure 1.1 An axisymmetric contraction ratio, 1 and 2 are orthogonal principal directions. 5   This thesis focuses on studying the statistics of axisymmetric flow as well as the statistics of fibre trajectories in axisymmetric turbulence. The literature has investigated the effects of geometry on suspension behaviour, but there are currently no studies that relate the fibre statistics to the fluid statistics in axisymmetric turbulence, without the complex near-wall region. To achieve this, the wavenumbers from isotropic turbulence will be distorted such that they agree with Prandtl’s formulae for a given contraction ratio. From these distorted wavenumbers, we develop a stochastic flow field which represents anisotropic turbulence. We follow the work of Olson [5] instead of Shin and Koch [6] as we want to model a spatially uniform flow field, free of any geometric effects. Following Shin and Koch [6], we would have to model the geometry rendering the turbulence anisotropic. To the author’s knowledge, this method would be extremely difficult to isolate the near-wall effects and then obtain a spatially uniform flow field. We utilize this spatially uniform flow field to isolate the effect anisotropy has on fibre dispersion when free of geometric effects. This flow field will allow us to control the anisotropy by a single parameter, the contraction ratio. This contraction ratio acts as an index in anisotropy and is not the same as a papermaking headbox contraction ratio. We will use this model anisotropic flow to study how the rotational and translational dispersion coefficients relate to the scales of axisymmetric turbulence and how they evolve as the turbulence grows in anisotropy. Only dilute suspensions, which ignore fibre-fibre interactions, will be considered for a range of fibre lengths, ranging from those smaller than the Kolmogorov length and those larger than the integral length scale. The statistics for the fibres will be related to the statistics from the flow field to examine how relationships established in Olson and Kerekes [3], Olson [5], and Shin and Koch [6] are impacted by the degree of anisotropy.   6  Chapter 2: Literature Review 2.1 Low Reynolds Number Fibre Dynamics  The equations of fibre motion were first derived by considering the net forces acting on a fibre in Stokes flow. Jeffrey [13] was the first to investigate the motion of ellipsoidal particles in Stokes flow. He applied a no-slip boundary condition on the fibre surface and used a matched asymptotic analysis to solve for the flow field around the fibre. From this study, it was found that fibres in simple shear flow rotate in a family of closed orbits. This has been verified by Aidun et al. [14] and Ding and Aidun [15], though a negligible time delay was found to develop over several orbits. Jeffrey’s equations have been verified by several investigators [16], [17], [18], for fibres, discs, cylindrical particles, spheroids and later arbitrary shapes; it has also been used in predicting the fibre orientation of a semi-dilute suspension in a homogeneous Stokes flow [19].   Several investigators have studied various shapes in creeping flows [18], [20]. Cox [4] extended this work to include long, thin and not necessarily straight bodies. He expressed the Stokes flow equations in terms of the inner part of the body and then separately in terms of the outer part. Using matched asymptotics, he solved the resulting Stokes equation for a thin body and was able to generalize the result to an arbitrary shape in an arbitrary non-homogeneous flow. He developed a drag tensor that was independent of fibre position and also derived relations for the force on two interacting bodies as well as with a body interacting with a wall. Cox’s equations were developed for a low Reynolds number flow where the Reynolds number was a function of the fibre diameter. Strictly, this would only be applicable in turbulent flows if the fibre length was small enough such that viscosity dominates over the inertial forces. Several investigators have applied Cox’s expression to fibres exceeding these length scales by using the free-draining approximation [3], [5], [21], [22], [23], [24], [25]. 7   Batchelor [7] obtained the equations of motion by approximating the body as a distribution of Stokeslets over a line for the field surrounding the fibre. This modeled the field around the fibre as a distribution of singular point forces, which when taken together would yield the total force acting on the fibre. Collectively, the work of Cox and Batchelor is known as slender-body theory and has been used by several investigators to determine the properties of fibre suspensions in turbulence. These include the translational and rotational dispersion coefficients [6] and the orientation distribution in an axisymmetric contraction [25]. The method has also been modified to model the rheology of semi-dilute suspensions [26]. 2.2 Turbulent Reynolds Number Fibre Dynamics  One of the fundamental challenges in understanding turbulent fibre motion is relating the rotational and translational motion to the properties of turbulence. The fluctuating component of turbulence causes particles to have a continuous quasi-random motion that behaves much like molecular diffusion. Taylor [27] investigated this phenomenon by relating the mean square displacement from a point source to the Lagrangian fluid velocity correlation. By assuming that the Lagrangian particle velocity correlation was the same as the fluid, which is known as the frozen turbulence hypothesis, Taylor was able to relate the dispersion of the particle to the turbulence.  In contrast to what is predicted by Taylor’s frozen turbulence hypothesis, the Lagrangian particle velocity correlation is not the same as the Lagrangian fluid velocity correlation. This is because real particles do not follow the fluid exactly, as shown by Snyder & Lumley [28], and Reeks [29]. To obtain the Lagrangian particle velocity correlation, the fibre motion can be integrated through the fluid velocity field. Many studies have looked at integrating fibre motion through a variety of model flow fields. Kraichnan [1] investigated fluid particles to relate the 8  Lagrangian and Eulerian statistics. This method has since been used to model inflow in LES and DNS, as well as for noise generation and the calculation of turbulent fluctuations in other models [30], [31], [32], [33]. Olson [5] integrated fibre motion through a series of stochastic Fourier nodes super-imposed on one another.  Olson et al. [34] studied fibre motion through a model one-dimensional headbox.  yens   [35] used this model in conjunction with computational fluid dynamics and experiment to model a dilute fibre suspension and the effect of vane placement on orientation. Shin and Koch [6] derived equations of motion using slender-body theory and integrated these through a model isotropic flow field. Lin, Liang and Zhang [25] used a combination of the Reynolds Averaged Navier-Stokes and stochastic Fourier nodes to model the turbulence and they modeled the fibre motion with slender-body theory. Homann and Bec [36] considered point-particles and used a pseudo-spectral method to investigate their dynamics. They found that particle dynamics could be approximated by the advection of a synthetic flow. Macinnes and Bracco [37] investigated different flow models and found that they produced a range of Lagrangian statistics, making a favorable method elusive.  Recently, there have been several investigations on how a fibre suspension affects suspension flow. Nikbakht et al. [38] studied fibre suspensions during the transition from laminar to turbulent flow. They were able to determine that plug size and yield stress are a function of Reynolds number and found that plug size reduces initially to overcome an increase in frictional pressure drop. Yang, Shen and Ku [39] investigated a turbulent fibre suspension in pipe flow. They found that the concentration and aspect ratio of the suspension can have a significant effect on flow characteristics. 9  2.3 Descriptions of Turbulence  Following Taylor’s frozen turbulence hypothesis, investigations [40] using hot-wire techniques were used to quantitatively measure velocity fluctuation which made it possible to evaluate the correlations of turbulence in space and time. Taylor [41] assumed that the turbulent properties were equal in all directions, thereby introducing the concept of isotropic turbulence, which was then generalized mathematically by Robertson [42]. vón Karman [43] and vón Karman and Howarth [44] described the two-point velocity correlations for general isotropic turbulence and were able to demonstrate that isotropic turbulence could be fully described by this pair of two-point correlation functions.   This description of isotropic turbulence has not been extended to the case of a general anisotropic flow. In lieu of this, a branch of anisotropic flow that has been extensively studied is axisymmetric flow. Batchelor [45] and Chandrasekhar [46] were able to demonstrate that it was necessary to have four different two-point velocity correlations to fully describe axisymmetric turbulence, with related length and time scales. These have been shown to be analogous to the two-point longitudinal and transverse correlations for isotropic turbulence by Ertúnc [47]. A full description of anisotropy is elusive; however, many investigators have been able to characterize axisymmetric turbulence by studying how it behaves in a contraction, as the simplest model of anisotropy. Prandtl [11] showed that Kelvin’s circulation theorem was able to predict the intensity of vortices in an axisymmetric wind tunnel contraction and that the streamwise component of turbulence decays through the contraction. These are occasionally referred to as Prandtl’s formulae for axisymmetric contractions.  Uberoi [12], [48] comprehensively studied the effect of isotropic grid-generated turbulence, strained through a contraction. He examined the spectrum before and after the 10  contraction to study the distortion of the turbulence structure. For any contraction ratios described by Eq. (1), Prandtl’s formulae were found to diverge for contraction ratios greater than 4. Uberoi found that under axisymmetric strain, the mean square streamwise fluctuations decreased while the transverse fluctuations increased. After the contraction nozzle, it was noted that the turbulence tends to isotropy as it decays. Mills and Corrsin [49] also noticed this phenomena, which is now known as return to isotropy and has been investigated downstream of the outlet by several investigators [50], [51].   Building on the work of Uberoi, many investigators have looked into the effect of contraction ratio on turbulence. Ramjee and Hussain [52] compared Prandtl’s formulae with their own experiment and found good agreement for contraction ratios less than 4. For contraction ratios greater than 4, their experiment showed good agreement with Uberoi. There was a deviation in fluctuating velocity where instead of a monotonic increase, as predicted by Prandtl, the anisotropy began to decrease starting at      . This deviation was investigated by Ertúnc [47] and was called the high contraction ratio anomaly. Ramjee and Hussain [8] found that the contraction shape and Reynolds number did not have a significant effect on turbulence characteristics, rather the inlet condition had the most impact. The contraction types they tested were axisymmetric so their results do not describe a planar contraction. Brown et al. [9] later investigated a planar contraction and found that the variation of turbulence characteristics with contraction angle was nearly independent of the contraction angle, showing qualitative similarities between axisymmetric and planar contractions.   The phenomena of anisotropy peaking has been examined by several investigators. Tsuge [53] found that small eddies decay through the contraction but larger eddies are amplified from vortex stretching. This stretching eventually causes the eddies to be too large for the contraction, 11  which leads to the decrease in anisotropy. Brown et al. [9] found that anisotropy peaked for               and that the rate of strain was important when      . However, when     , the orienting effect of the convective flow dominated the randomizing effect of turbulence. It has been found that turbulence entering a contraction does not become anisotropic immediately. It decays for some time before anisotropy becomes significant [9], [54], [55], [56].  2.4 Rapid Distortion Theory  The effect of a contraction on turbulence was first investigated because of its application to wind tunnels in jet flows. Prandtl [11] and Taylor [57] were the first to study the distortion of free stream turbulence through contractions. Prandtl found that the transverse fluctuations were due to the vortex filaments that were aligned in the flow direction whereas the streamwise fluctuations were due to the vortex filaments aligned perpendicular to the flow direction. Taylor found that Prandtl’s model did not account for mutual interactions of continuum vortices. He proposed a new model using Cauchy’s equations and the conservation of circulation [57]. Building on the vortex analysis and using the Cauchy form of the vorticity equation, Ribner and Tucker [58] and Batchelor and Proudman [2] independently introduced a formal analysis on the distortion of vortices that is now known as Rapid Distortion Theory, or RDT. They recognized that the strain could be treated by decomposing it into the fluctuations of the local mean flow and treating them as separate distortions. By assuming the time scale of the distortion is much less than the time scale of the eddies, the distortion occurs in a time scale such that viscous dissipation will not have an appreciable effect, allowing both viscosity and eddy interactions to be ignored. These assumptions allowed them to neglect non-linear processes and calculate the changes of vorticity by a linear equation, which allows the calculation of the velocity and vorticity fluctuations. This makes RDT a strictly linear theory dependent on its initial conditions. 12  For an in-depth review of both the theory and history of RDT, the reader is directed to Sreenivasen and Narasimha [59], Savill [60], Hunt and Carruthers [61] and Durbin and Pettersson-Reif [62].  Several investigators have since investigated the validity and applicability of RDT to experiments. The works of Ramjee and Hussain [8], Brown et al. [9], Uberoi [12], Reynolds and Tucker [63], [64], Warhaft [65], and Lee [66] have all demonstrated phenomena regarding turbulence in contractions that is well predicted by RDT; Ramjee and Hussain [8] found that although no contraction was rapid enough to adequately cause the distortion as predicted by RDT, the trends predicted by RDT were observed as long as      . For      , RDT fails to describe the phenomena, likely due to the eddy-geometry interactions described by Tsuge [53]. The balance between the rapid-pressure strain term and the production of anisotropy in the transport equation for anisotropy determines the state of isotropy. Brown et al. [9] found that at      , the balance shifts from production to rapid-pressure strain, which RDT fails to account for; in fact, the rapid pressure-strain term and production are the only active terms in RDT. Ertúnc [47] tested RDT with a correction made for the high contraction ratio anomaly against different contraction rapidness and found that RDT was not very accurate for slow contractions but became quite accurate as the rapidness became very large. This correction allowed RDT to handle much higher contraction ratios than those predicted by Uberoi [12] and Ramjee and Hussain [52]. 2.4.1 Fibre Orientation in Turbulence  The properties of products produced from a fibre suspension are highly dependent on the orientation of the suspension. Krushkal and Galliley [67] and Berstein and Shappiro [68] studied the orientation distribution of glass fibres and non-spherical aerosol particles in turbulence, 13  respectively. It was concluded that as the turbulent intensity increased, particles became more randomly oriented until they reached an almost isotropic distribution. However, if there was a strong enough mean flow this would result in an anisotropic distribution. Ullmar and Norman [69], Ullmar [70] and Zhang [71] concluded that orientation was a function of the contraction ratio and that increasing it significantly increases fibre alignment, but they were unable to relate the orientation anisotropy to the turbulent characteristics.   Exploring this relationship between the contraction ratio   and the fibre orientation, Olson et al. [34] showed that increasing   significantly increased the alignment of fibres, whereas the inlet fluid velocity only had a small influence. They showed that turbulent dispersion greatly influenced fibre orientation and that fibre orientation could be described by a dimensionless headbox Péclet number. Parsheh et al. [55] used a model based on high speed imaging data. They saw that turbulent fluid variations in the contraction were independent of the contraction shape but that the orientation anisotropy was dependent on the contraction shape due to the variation of the rotational Péclet number in the contraction. The inlet turbulent kinetic energy of the fluid in contractions was found to have a negligible effect on the orientation. At large  , orientation distribution develops close to Stokes flow, which suggests that the effect of fibre inertia is negligible. From this, they concluded that fibre diameter is the appropriate length scale for the fibre Reynolds number. Wu and Aidun [72] investigated flexible fibre suspensions and found that the fibre orientation distribution became more anisotropic as fibres became more rigid. As fibres became more flexible, an asymmetry appeared in the orientation distribution due to fibre deformation or fibre-fibre contact.  Most models assume a dilute suspension for simplicity. Recently, many investigators have investigated semi-dilute models. Krochak et al. [73] developed a model of fibre orientation 14  that accounts for fibre-fibre interactions by modeling them with a rotational diffusion term. They measured the evolution of the orientation distribution of a semi-dilute suspension in a linear contracting channel. They noted that the diffusion coefficient increased up to a certain point before decreasing, likely due to fibre clumping or flocculation. It was shown that as the concentration of the suspension increased, the coupled and uncoupled solutions converged, though the greatest difference was only 2%.  Several investigators have described the evolution of fibre orientation in turbulence with a Fokker-Planck equation [3], [21], [34], [54], [55], [67], [73] given by                                                                        (3)  where   and   are the fluid velocity field components in the 1 and 2 direction, respectively.   and   are the azimuth and zenith orientation angles, used in the orientation vector  .     and   are the rotational velocities for the 3D orientation vector.   is the probability density function describing the evolution of fibre orientation. Finally, the rotational diffusion coefficient    can be viewed as the effect of turbulence on orientation anisotropy. Olson et al. [34] found that the rotational diffusion coefficient was constant throughout the contraction and gave an estimate that agreed with the experimental studies by Ullmar [70] and Zhang [71]. The model from Olson et al. [34] has been used in conjunction with CFD simulations to understand how adjusting the rotational dispersion coefficient could account for the variation of orientation anisotropy with varying turbulence levels [74]. Parsheh et al. [54], [55] used grid-generated isotropic turbulence to investigate the influence of turbulence on the orientation of stiff fibres. They found that the rotational dispersion coefficient is dependent only on the inlet flow conditions and is 15  independent of turbulent intensity. It was also found that the rotational dispersion coefficient decayed exponentially with the contraction ratio.   Many investigators have related the dispersion coefficients to the turbulence properties [3], [5], [54], [55], [67]. Olson and Kerekes [3] derived analytical expressions for both coefficients in both the limit of large fibres and very small fibres. These expressions were then tested in a stochastic simulation by Olson [5] using Kraichnan’s [1] energy spectrum and it was found that they predicted qualitative trends about turbulent fibre motion. Both coefficients were found to decrease with increasing fibre length; small fibres were found to more closely follow the Lagrangian fluid velocity correlation whereas larger fibres followed the Eulerian fluid velocity correlation.    Shin and Koch [6] simulated isotropic turbulence with DNS to study the translational and rotational motions of fibres using slender-body theory [7]. They found that the rotational dispersion coefficient is influenced by the scales between the Kolmogorov scale and the integral time scale. This allowed them to express the rotational diffusion coefficient in terms of the fibre orientation integral time,   . They confirmed Olson & Kerekes [3] and Olson’s [5] findings that both the rotational and translational dispersion coefficients decrease as fibre length increases. Olson and Kerekes [3] expression for the translational dispersion coefficient was found to quantitatively agree while their expression for the rotational dispersion coefficient was found to over-estimate the rotary dispersion, similar to the findings of Olson [5]. The rotational dispersion coefficient was also found to depend on the Reynolds number. These results have been used to investigate the modulating effect of long rods as well as rotation of rods in turbulence as well as axisymmetric particles in turbulence [75], [76], [77], [78].  16  Chapter 3: Numerical Model of Fibre Motion in Anisotropic Turbulence  The quantities of turbulence are described using the Reynolds decomposition, a statistical tool to analyze the fluid flow as the sum of mean and fluctuating component. Fibres in turbulence are subjected to this mean and fluctuating motion and the resulting fibre motion is composed of translational and rotational components, given in the follow equation,           . (4)   These components of motion are used in an Eulerian description of fibre motion, where the fibre position   and orientation   at any time   is governed by the probability distribution function         .  is given by the evolution of the following convection-dispersion equation, which is also equivalent to Eq. (3),                                 ), (5)  where    is the rotational dispersion,    is the translational dispersion and   is the mean fibre translation [21].    is the rotational operator, expressed as              (6)  The fibre’s angular velocity   is expressed in relation to the fibre’s orientation vector as        . (7)   In order to model the effect of turbulence on the fibres, we relate    and    to the properties of the turbulence flow field. Olson and Kerekes [3] investigated these dispersion coefficients and by assuming a free-draining, inertialess, rigid fibre of length   in a homogeneous, isotropic fluid were able to derive the following expressions, 17                                               (8)  and                                                        (9)  where        is the Lagrangian fibre velocity correlation through time.   is difficult to express analytically, so Olson and Kerekes approximated   as the product of the Eulerian fluid spatial velocity correlation and the Lagrangian particle temporal velocity correlation using the argument that they are statistically independent. They were able to show that the velocity correlation of the fibre was equal to the Lagrangian fluid velocity correlation for short fibres and they hypothesized it was equal to the Eulerian fluid velocity correlation for long fibres. Shin and Koch [6] found that the expression for    was quantitatively correct but that the expression for    was not. Olson and Kerekes assumed that fibre orientation was uncorrelated with the local axes of strain and rotation, however, Shin and Koch found that the orientation tends to correlate with the axes of strain and this results in a significant slowing of the rotational motion. Shin and Koch suggested another expression from their DNS simulation,              (10)  where    is the fibre Eulerian rotational time scale.   In axisymmetric turbulence, the mean and fluctuating parts of the turbulence change in quantity but are still qualitatively the same, that is, the relations above would still be valid for fibres suspended in an axisymmetric flow. However, there are no estimates of    and    for anisotropic turbulence analogous to those made by Olson and Kerekes. In anisotropic turbulence, 18     and    would be tensors, however, the orthogonal components of these tensors could be treated separately as scalars.  In this thesis, a numerical simulation of individual fibres moving in a random velocity field is used to develop relations between the Lagrangian fibre velocity correlations and a specified Eulerian fluid velocity correlation as required by the equations found by Olson and Kerekes [3] and Shin and Koch [6]. The Eulerian fluid correlations correspond to the Kraichnan [1] spectrum, but other spectrums could be used. Using RDT, the random velocity field will then be distorted to simulate an ideal contracted flow based on Prandtl’s formula [11]. The distorted velocity field will be used to examine the relationships between the isotropic and anisotropic rotational and translational dispersion coefficients. 3.1 Model of Fibre Motion in Distorted Turbulent Velocity Field  In this section, the equations of motion for an infinitely thin, rigid fibre are given and the model for the turbulent velocity field is described and extended to the case of axisymmetric turbulence. The fluid and fibre system is assumed dilute and combined in a one-way coupling such that fibres do not affect the turbulence [5], [6], [34], [79]. Although a one-way coupling is used, it is noted that two-way coupling has recently been modeled for fibre orientation by Krochak et al. [73] but was found to have only a very small difference in a contracting channel. Despite this difference being small, real fibres would affect the turbulence, especially at high concentrations and non-dilute concentrations as they would raise the effective viscosity [39]. While there is accuracy gained by using a two-way coupling method, the gain is small and the computational cost is large. Thus, in order to isolate a single fibre and relate it to the turbulent properties, the dilute suspension will be used. 19   Fibre trajectories are calculated using a 5th order Runge-Kutta method for numerical integration where the step size is based on error tolerance, which is     , as well as global adaptive quadrature with the same error tolerance. The simulation calculates 1000 individual fibre trajectories and 1000 flow fields, from which the Lagrangian statistics are gathered.  3.1.1 Fibre Motion  To model fibre motion, a model of the force imposed on the fibre by the fluid is required. Shin and Koch [6] used the matched asymptotic expansions of slender-body theory [7] to derive the equations of motion. Olson [5] and Olson et al. [34] modeled the fibre by assuming it was rigid and inertialess and obtained the equations of motion by using a constant drag tensor [4]. By considering that the region around a fibre is characterized by the no-slip condition, there would be a region enveloping the fibre that is governed by Stokes flow. Following Olson [5], this region will let us assume the form of the force on the fibre is equivalent to Cox’s [11] expression for creeping flow. This method has been validated by the two-way coupled DNS simulation of Shin and Koch [6]. Cox’s expression is given as                     (11)  where for infinitely thin fibres, the drag   is independent of the position from the fibre centre,  , and is given by                    (12)  where   is the aspect ratio of fibres and assumed to be large,   is the identity matrix, and   is the unit vector of the fibre. Fibre length must also be considered as Eq. (11) was derived for low Reynolds numbers where the Reynolds number was based on fibre length,  . Large fibres will then have a Reynolds number that is too large for Cox’s expression. This means that Eq. (11) is 20  only valid for infinitely thin fibres that are also smaller than the Kolmogorov length scale,  . To resolve this, we impose the free-draining approximation which has been used to model long rigid fibres [5], [34], flexible fibres [23], and polymers [21], [22]. This approximation models the fibre as a series of elements, each smaller than   and hydrodynamically independent. Therefore, each of these individual elements can have the force on them described by Cox’s expression. Their hydrodynamic independence allows them all to be considered separately which means each fibre element is described by Cox’s expression and therefore the whole fibre is. Of course, these simplifying assumptions are neglecting some properties of long fibres, limiting the quantitative predictive qualities of this model.        Figure 3.1 A straight rigid fibre of length L. v is the fibre velocity and u is the fluid velocity. The fibre is oriented to point in the direction of the unit vector p.   The fibres considered in this work are straight and rigid. A model fibre is illustrated in Fig 3.1. Flexible fibres have been investigated recently but will not be considered here. Rigid fibre motion is separated into translational and rotational components, so the velocity at any point on the fibre will be the sum of the fibre’s translational velocity,  , and rotational velocity, 21     , where    is the rate of change of orientation, defined as the time derivative of a unit vector in the direction of the fibre, and   is the distance along the fibre from the center. Integrating Eq. (11) along the length of the fibre will give the net external force,  . Noting that      is the sum of the translational and rotational velocity, such that                 , we have                                  (13)  The net external moment acting on the fibre is then obtained by considering                                          (14)  Assuming that the inertial forces are negligible for an infinitely thin particle, this will result in                 , which yields       and     . With this, we have the velocity and angular velocity of the fibre as                            (15)  and                                  (16)  respectively. If the mean fluid velocity is substituted for the instantaneous velocity   in   and  , then this reduces to the mean velocity equations that are required in the convection-dispersion equation. The rate of change of orientation,   , on a rotating body with a fixed origin point is          (17)  The trajectory of the fibre can be obtained by numerically integrating  ,  , and   . This method is computationally intensive and an alternative was used to model fibres by Olson [5]. 22   The orientation can be expressed in terms of the fibre orientation angles   and  .   can then be expressed using the Cartesian coordinate transformation to spherical coordinates,                            (18)    and   have corresponding unit vectors, which are given as the standard spherical unit vectors in relation to the Cartesian unit vectors                               (19)  and                    (20)  The orientational velocity is given by the differential equation                  (21)  The angular velocity is given by a pair of differential equations                                       (22)  and                                   (23)  By using an iterative method, the fibre trajectory is determined numerically by integrating  ,   , and   . This is the method that will be employed in this work. 23  3.1.2 Turbulent Velocity Field  The details of this stochastic turbulent model are given by Kraichnan [1], who used the model to test his direct interaction approximation of turbulent self-diffusion for an initial Reynolds number,      . It neglects non-linear terms which can be addressed separately. The velocity field is given as a sum of stochastic Fourier nodes superimposed by the following expression,                                                          (24)  To ensure that the fluid model is incompressible, we enforce             (25)  by defining coefficients      and     , which represent random velocity components, as                         (26)  and                         (27)  The components of the random coefficients      and      are chosen from a Gaussian random distribution with zero mean and standard deviation            , where    is the root mean square of the fluctuating velocity and   is the number of Fourier nodes.    From a given energy spectrum     , a wavenumber      is randomly chosen from the surface of a sphere of radius     . That is, a point on the surface is selected at random, and the 24  corresponding co-ordinate becomes the wavevector      corresponding to the wavenumber     . The total energy over all wavenumbers in isotropic turbulence is given by                        (28)  where                                        (29)  This energy spectrum represents low Reynolds number turbulence behind a grid [10] and is given by Kraichnan [1]. Kraichnan also investigated other models for 2-dimensional turbulence. The energy spectrum is minimized at       and is maximized at       . The corresponding longitudinal and transverse velocity correlations, defined from     , are given by                      (30)  and                             , (31)   where      is defined in terms of      by the following relation [80], by continuity and homogeneity,                         (32)  where for Eq. (30)-(32),       , meaning it has lost its directional information. Other spectra have been used to model higher Reynolds number turbulence, such as in the study by Wang and Stock [81], where both the Kraichnan spectrum and the Karman-Pao spectrum were used. 25   The frequencies      are chosen from a Gaussian distribution with zero mean and zero standard deviation, given by the following               (33)  The resulting energy spectrum is given by                                               (34)  which shows good agreement with the DNS of Hunt et al. [83] for        . This model has been used for fibre suspensions [5], which also showed good agreement with the DNS of Shin and Koch [6], who utilized a Reynolds number,             .  Many investigators have used this stochastic method of modeling isotropic turbulence [1], [5], [81], [83], [84], [85], [86],[87]. Some shortcomings of this model were discussed by Maxey [83]. The foremost of these is that there were elements of the energy cascade that were not accounted for, such as the representation of energy transfer between large and small eddies and the advection of small eddies by larger eddies. Shin and Koch [6] found that stochastic models that consider the fluid strain and rotation as independent would over-estimate the rotary velocity of fibres. They also noted that the Kraichnan spectrum lacks an inertial subrange and does not mimic the spectrum of isotropic turbulence exactly. Despite this, the model is still capable of reliably producing the two-point statistics and making qualitative predictions. In response to these shortcomings, many investigators have altered the model. Wang and Stock [81] modified the model to account for the effect of turbulent decay in the wake of a grid by incorporating time dependent    and   . This model was found to agree with the experimental findings of Snyder and Lumley [28] on particle dispersion in grid-generated turbulence. Fung et al. [86] modified the model to account for the advection of small eddies by larger eddies and 26  used it to study relations between Eulerian and Lagrangian structures of turbulence. They suggested that this approach could account for advection by coupling it with techniques like large eddy simulation. 3.1.3 Anisotropic Axisymmetric Turbulence  This section details the model and methods for the distortion of the turbulent velocity field. Prandtl’s formulae [11] describe the root mean square velocities in the streamwise and transverse direction. Eq. (2) can be given in terms of the root mean square velocities as,                  (35)  These relations have been experimentally verified to be valid for        [12], [52], where   is the contraction ratio given by Eq. (1). The length is implicitly assumed such that our model contraction will be sufficiently rapid. The major reason for the       limitation is that the eddies begin to become too large for the contraction [53], which would lead to a transfer in the production and rapid-pressure strain terms [9]. To model this, we assume that eddy distortion is the source of all anisotropic statistics. The problem of modeling then requires a method of distorting the wavevectors such that the root mean square velocities agree with Eq. (35). To resolve this and relate the root mean square velocities to the strain induced by the contraction, we turn to Batchelor and Proudman’s [2] Rapid Distortion Theory.   Batchelor and Proudman considered the Cauchy form of the vorticity equation, given as                             (36)  where    is the vorticity of a fluid element, and    and   are the pre- and post-contraction position vectors of that fluid element. The velocity components in the axes of strain are assumed to be          so that particle trajectories satisfy 27                (37)  with the corresponding solutions,                   (38)  where      is the strain given in the     direction. The deformation matrix given in Eq. (36) is                                   (39)  with                                Thus the solutions of the vorticity equation are predicted by the time dependent dilation,  . Taking the Fourier transform of the vorticity field yields                   (40)  Substituting Eq. (39) and Eq. (40) into the Cauchy form gives                          (41)  Knowing the solution for the particle trajectory from Eq. (38), the Fourier transform term can be re-written as                    (42)  where expanding the exponent     then results in                    (43)  Batchelor and Proudman then simplified this expression, yielding the time-dependent wavenumber evolution expressed by the following relation                 (44)  28  where the exponent is the dilation in the   direction and the exponential is a time dependent value. If the dilation is assumed to be instantaneously maximized, then the time dependency is dropped, resulting in               (45)  Tucker [88] described a contraction as containing two negative rates of strain which would both be in the transverse direction. This would correspond to    being positive and    and    being both negative and by the principles of axisymmetry,        . The constants can then be numerically determined by comparing the simulated root mean square velocities with Eq. (35). Sample wavenumber spheres are given for all contraction ratios in Fig 3.2.               (a) 29        Figure 3.2 Sample unit wavenumber spheres generated by the wavevector distortion for (a) C = 1 (b) C = 2  (c) C = 3 (d) C = 4    (b) (c) (d) -3-2-10123-0.500.5-0.500.5C = 2kxkykz-3-2-10123-0.500.5-0.500.5C = 1kxkykz-3-2-10123-0.500.5-0.500.5C = 3kxkykz-3-2-10123-0.500.5-0.500.5C = 4kxkykz(a) (c) 30   Batchelor [45] and Chandrasekhar [46] were able to show that      and      do not adequately describe axisymmetric turbulence. They derived four two-point correlations, one pair describing the streamwise direction and the other pair describing the transverse direction. These have been related to      and      by Ertúnc [47] and are given as                                                                  (46)                                                                   (47)                                                                   (48)                                                                  (49)  These are analogous to      and      in isotropic turbulence and will be used to determine the velocity correlations in this work. This method still uses Kraichnan’s [1] spectrum and the same turbulence model as the isotropic model. It has the same shortcomings as in the isotropic case, however, Kraichnan’s model produces the two-point velocity correlations reliably and it would be expected that it will do so for the axisymmetric case as well.   31  Chapter 4: Results  The isotropic turbulent flow field is defined by the wavevector    that maximizes the energy spectrum      and the root mean square velocity   , both parameters are equal to 1.0 in all flow realizations. The axisymmetric flow field is defined by an additional parameter, the contraction ratio  , which will equal 1, 2, 3, and 4, where       refers to the isotropic flow field. In both the isotropic and axisymmetric flow fields, the energy spectrum is assumed to be the Kraichnan energy spectrum. The range of length scales for isotropic turbulence is expected to be                 , with the Kolmogorov length scale   estimated as approximately        .   For each fibre length, 1000 trajectories were calculated, each through a new realization of the flow field. The fibre lengths calculated were     equals 0.25, 1, 3, 5, 7, 10, and 20. Fibre lengths were kept constant throughout all contraction ratios tested, regardless of the change in      . For all realizations, the flow field had the number of Fourier nodes,  , equal to 100. From the ensemble of realizations, the fibre and fluid correlations were calculated. The time step in both simulations is given as 5 time steps per         .    4.1  Fluid Statistics  The isotropic Eulerian longitudinal and transverse fluid velocity correlations calculated from the numerical simulation are compared with the exact correlations given by Eq. (30), (31) and are given in Fig. 4.1a. The results agree closely with the analytic solution, demonstrating the ability of this model to converge to the exact Eulerian correlations for an undistorted flow. The distorted flow conforms to both Prandtl’s equations and Townsend’s equations, shown in the appendix. The axisymmetric streamwise Eulerian longitudinal and transverse fluid velocity 32  correlations are given in Fig 4.1b-d. Currently, there are no analytic solutions for the axisymmetric fluid velocity correlations for any form of anisotropic turbulence. Since the model converges to the exact isotropic solutions, the numerical solutions can be used to make an estimate for      and      in axisymmetric turbulence.    Figure 4.1 Streamwise Eulerian spatial velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 (a) (b) (c) (d) 0 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationStreamwise Direction Spatial for C = 1  f(r)R22x1g(r)R11x10 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationStreamwise Direction Spatial for C = 2  R22x1R11x10 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationStreamwise Direction Spatial for C = 3  R22x1R11x10 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationStreamwise Direction Spatial for C = 4  R22x1R11x133    Figure 4.2 Radial Eulerian spatial velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   The transverse correlation is defined analytically as a function of      and is given by Eq. (31). The axisymmetric form of the Eulerian transverse fluid velocity correlation can then be estimated by knowing the Eulerian longitudinal fluid velocity correlation. The numerical (a) (b) (c) (d) 0 1 2 3 4 5 6 7 8-0.4-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationRadial Direction Spatial for C = 1  f(r)R11x2g(r)R22x20 1 2 3 4 5 6 7 8-0.4-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationRadial Direction Spatial for C = 2  R11x2R22x20 1 2 3 4 5 6 7 8-0.4-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationRadial Direction Spatial for C = 3  R11x2R22x20 1 2 3 4 5 6 7 8-0.4-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationRadial Direction Spatial for C = 4  R11x2R22x234  simulation was then used to fit the following function for all contraction ratios, illustrated in Fig 4.3,                              (50)   In the radial direction, given in Fig. 4.2a-d, the numerical simulation was used to estimate the following functions, given in Fig 4.4                                 (51)                                  (52)                                  (53)  The radial evolution of      as   changes can be approximated for all contraction ratios by the following relation and is shown in Fig 4.4,                                (54)   35                          Figure 4.3 Streamwise numerically determined longitudinal velocity correlation equation fit for  (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   (a) (b) (c) (d) 0 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Streamwise Eulerian for C = 1  R22x1Eq. (49)0 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Streamwise Eulerian for C = 2  R22x1Eq. (49)0 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Streamwise Eulerian for C = 3  R22x1Eq. (49)0 2 4 6 8 10 12 14 16-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Streamwise Eulerian for C = 4  R22x1Eq. (49)36       Figure 4.4 Radial numerically determined longitudinal velocity correlation equation fit for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4  (a) (b) (c) (d) 0 1 2 3 4 5 6 7 8-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Radial Eulerian for C = 1  R11x2Eq. (50)Eq. (51)Eq. (52)0 1 2 3 4 5 6 7 8-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Radial Eulerian for C = 2  R11x2Eq. (50)Eq. (51)Eq. (52)0 1 2 3 4 5 6 7 8-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Radial Eulerian for C = 3  R11x2Eq. (50)Eq. (51)Eq. (52)0 1 2 3 4 5 6 7 8-0.200.20.40.60.81rk0Eulerian Spatial Velocity CorrelationAnalytic Radial Eulerian for C = 4  R11x2Eq. (50)Eq. (51)Eq. (52)37   The length scale is calculated by integrating the longitudinal and transverse fluid velocity correlations, shown in the following relations                       (55)                      (56)                       (57)                       (58)  where the subscript R refers to the radial length scales. As the contraction ratio increases, both the streamwise longitudinal and transverse length scales increase. This is because the eddies elongate in the streamwise direction, as shown in Fig 3.2, resulting in lower velocity fluctuations. Lower velocity fluctuations results in the velocities being spatially correlated for longer times. The amount of elongation, normalized by the isotropic length scale, can be approximated by   , shown in Fig. 4.5. This follows the experimental relationship established by Ertúnc [47]. The radial-longitudinal and radial-transverse length scales decreased, with the radial-longitudinal decreasing to close to half of the isotropic value while the radial-transverse length scale quickly approaches 0 as   increases. The length scales in the radial directions had no pattern similar to that observed in the streamwise direction.     38          Figure 4.5 Comparison of the simulated and expected length scales for all contraction ratios   The Eulerian and Lagrangian temporal fluid correlations were calculated from the simulation. The correlations are given in Fig. 4.7 for the streamwise direction and in Fig. 4.8 for the radial direction, for all contraction ratios. The estimated and analytic form of the Eulerian temporal correlation and the resulting time scale is given by the following relations, respectively,                                                (59)    and                    (60)                (61)  It is noted that the dependency of Eq. (60) on the evolving wavevector is from   , defined in Eq. (33). However, this does not change significantly as   increases. As the anisotropy increases, the 1 1.5 2 2.5 3 3.5 41.522.533.544.555.566.5CIntegral Length ScalesLongitudinal integral length scales  0 R11x10 R11x2Expected 0 R11x1Expected 0 R11x239  spread of possible frequencies increases and the mean value does not differ significantly from the isotropic mean, only differing by about 10% for      . This means that using Eq. (60), a significant change would not be expected in the Eulerian temporal scales as contraction ratio increases beyond the isotropic condition, which can be seen in Fig. 4.6. The Lagrangian fluid velocity correlation is determined by integrating the velocity field given by Eq. (24) and is presented along with the Eulerian temporal correlation. As the contraction ratio increased, there was a significant relative change in the streamwise Lagrangian temporal correlation but it was much less than the Eulerian time scale. These results adhere to the expected isotropic relationship of           [10], [28], [80]. As the contraction ratio increased, there was a small change in the Eulerian temporal correlation beyond the initial spike for    , suggesting that there is only a small dependence on contraction ratio for the Eulerian temporal velocity correlations in contrast to the Eulerian spatial correlation, where there is a very large dependence on contraction ratio.           Figure 4.6 Eulerian and Lagrangian time scales for all contraction ratios  1 1.5 2 2.5 3 3.5 40123456CTime ScalesEulerian and Lagrangian Time Scales  Eulerian RadialEulerian StreamwiseEq. (60)Lagrangian StreamwiseLagrangian Radial40     Figure 4.7 Eulerian and Lagrangian temporal velocity correlations for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   (b) (a) (c) (d) 0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationTemporal Correlations for C = 1  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationTemporal Correlations for C = 2  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationTemporal Correlations for C = 3  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationTemporal Correlations for C = 4  EulerianLagrangianEulerian Analytic41             Figure 4.8 Radial Eulerian and Lagrangian temporal velocity correlations for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4  (d) (b) (a) (c) 0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationRadial Temporal Correlations for C = 1  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationRadial Temporal Correlations for C = 2  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationRadial Temporal Correlations for C = 3  EulerianLagrangianEulerian Analytic0 1 2 3 4 5 6 7 8-0.200.20.40.60.81tu0k0Temporal Velocity CorrelationRadial Temporal Correlations for C = 4  EulerianLagrangianEulerian Analytic42  4.2 Fibre Statistics 4.2.1  Translational Dispersion  All fibre trajectories were tested with two separate sets of initial conditions; one orients the fibre in the streamwise orientation and the other in the radial orientation. The streamwise condition is represented by      ,       at       and the radial condition by      ,       at      . It was found that the fibre statistics are independent of initial conditions. The fibre translation is calculated by integrating the fibre velocity, given by the following equation,                   (62)  The mean square translation      of all fibre lengths as a function of time is given in Fig. 4.9a and 4.10a for isotropic turbulence. In isotropic turbulence, the fibres disperse slowly initially because the velocity is strongly correlated with time. After several eddy interactions they disperse approximately linearly with time. As a consequence of isotropic turbulence, the fibres disperse equally in all directions. Fibres that are shorter than the medium scales of turbulence, which for this isotropic simulation are those with      , disperse like fluid particles, which is expected because all fibres are assumed to be inertialess. As the fibre length increases beyond the small scales of turbulence, the rate of translational dispersion decreases because the turbulent fluctuations are spatially averaged over the length of the fibre.  The axisymmetric results of the mean square fibre translation is given in Fig 4.9b-d, 4.10b-d. In axisymmetric turbulence, longer fibres disperse more slowly than shorter fibres, similar to isotropic turbulence,    . However, the fibres do not disperse equally in all directions. It is only fluctuations along the principal axis that cause translation in a given direction, so the magnitude of fluctuations has a significant impact on translation.  43   The mean square translation       in the streamwise direction has all fibres dispersing much more slowly than in isotropic turbulence because these fluctuations are much less in axisymmetric turbulence than isotropic turbulence. The streamwise mean square translations get closer together for all fibre lengths because the length scales get larger, so the relative length of the fibres decrease, causing more of them to act like fluid  particles or close to it.  In the radial direction, the mean square translation       of all fibres is much larger than in isotropic turbulence. This is amplified as   increases; for instance, for      , fibres of length          disperse at approximately the same rate radially as fibres of length         in isotropic turbulence. This amplification of translation occurs because the turbulent fluctuations are so large in magnitude in the radial direction when compared to the streamwise direction that even the longest fibres are averaging very large fluctuations, so they translate a significant amount. The radial mean square translation of the fibre lengths get closer together as   increases because the radial turbulent fluctuations get so large that all fibre lengths disperse linearly almost immediately. This occurs because when fluctuations are weak, fibre velocities stay correlated over more eddy interactions. Since the radial fluctuations are larger, it takes fewer eddy interactions for the fibres to decorrelate. 44   Figure 4.9 Streamwise fibre translation of all fibre lengths for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 1 2 3 4 5 6 7 802.557.51012.515(a)k02<y12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(b)k02<y12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(c)k02<y12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(d)k02<y12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2045   Figure 4.10 Radial fibre translation of all fibre lengths for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 1 2 3 4 5 6 7 802.557.51012.515(a)k02<y22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(b)k02<y22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(c)k02<y22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 802.557.51012.515(d)k02<y22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2046   The translational dispersion coefficient,   , is calculated by relating it to the mean translation of fibres from a point source [10] by the following relation                 (63)   Eq. (63) is a scalar in isotropic turbulence, however, in anisotropic turbulence, the translational dispersion coefficient would be a tensor. By invoking orthogonality, we may consider the component in the streamwise and radial directions separately using Eq. (63), as long as we use the appropriate translational component in place of         .  The translational dispersion coefficients were calculated for all fibre lengths and plotted in Fig 4.12a for isotropic turbulence. In isotropic turbulence, the dispersion coefficient is initially zero before asymptotically increasing to a long time constant value,    . The time to reach     corresponds to how long it takes for the translations to be uncorrelated, which for this simulation is about            for shorter fibres. The smallest fibres have a large error in calculation due to the large amount of dispersion they undergo. However, the dispersion coefficient they give is still approximately constant, allowing for a good estimate of     to be made. For the longest fibres, the dispersion coefficient increases slowly and does not reach a constant until much later times.  The axisymmetric translational dispersion coefficient was calculated and given in Fig 4.12 and 4.13b-d. In the streamwise direction, it takes longer to reach          as   increases. For increasing  , the dispersion coefficient increases more slowly asymptotically and takes longer to reach the long time constant value. This occurs because the turbulent fluctuations in the streamwise direction are weaker as   increases, resulting in less dispersion and therefore a longer time for the fibres to randomize. In the radial direction, the time to reach          also increases 47  with anisotropy, but unlike in the streamwise direction, this time does not increase much beyond      . The time to become uncorrelated is much greater in the streamwise direction than the radial direction. This is because the fluctuations in the radial direction dwarf those in the streamwise, so all fibres will disperse much faster radially than streamwise. The streamwise dispersion coefficients are extremely small when compared to the radial dispersion coefficient because the fibres do not translate very much in the streamwise direction but translate significantly in the radial direction. There is more ambiguity in           for shorter fibres, as the increased fluctuations greatly increase the dispersion of small fibres. The translational dispersion coefficient for smaller fibre lengths, relative to the fluid length scales, can be approximated by the following relation,                  (64)  The relations are illustrated in the following plot,          Figure 4.11 Comparison of the streamwise dispersion to the radial dispersion 0 2 4 6 8 10 12 14 16 18 201.522.533.544.555.566.5h(C)k0LComparison of streamwise dispersion to radial dispersion  C = 2C = 3C = 448  From Fig 4.11, it is seen that for fibres smaller than        , the difference between the isotropic and anisotropic translational dispersion coefficients is approximately constant. For the largest fibres, the difference between the two coefficients is equal to approximately the contraction ratio  . Fibres that are in between         and           are also seen to have similar valued constants, suggesting that fibres are grouped together discretely, depending on their relative length.    It is shown in the appendix that for both the isotropic and axisymmetric case, that                     (65)  By using Eq. (65) in conjunction with Eq. (64), it is possible to estimate the magnitude of the translational dispersion as contraction ratio increases. This is made possible as the streamwise translational dispersion coefficient becomes increasingly small as contraction ratio increases, as shown in Fig 4.12, while the radial translational dispersion coefficient becomes extremely large.  This behavior would be expected to continue as contraction ratios increase, such that Eq. (65) would only be inaccurate for a contraction ratio of 2.   49    Figure 4.12 Streamwise component of the translational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4  0 1 2 3 4 5 601234567(a)(k0/u0)Dt1k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(b)(k0/u0)Dt1k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(c)(k0/u0)Dt1k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(d)(k0/u0)Dt1k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2050    Figure 4.13 Radial component of the translational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3  (d) C = 4  0 1 2 3 4 5 601234567(a)(k0/u0)Dt2k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(b)(k0/u0)Dt2k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(c)(k0/u0)Dt2k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 601234567(d)(k0/u0)Dt2k0L  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2051    To examine how the integral time scale is affected by increasing fibre length, the fibre’s temporal velocity correlation is estimated from the simulation and plotted in Fig. 4.14, 4.15 from the following relation,                   (66)  In isotropic turbulence, shown in Fig 4.14a, 4.15a, the fibre velocity correlation is bounded by the fluid Lagrangian and Eulerian temporal velocity correlation [5]. It was found that for the shortest fibres, the Lagrangian fibre velocity correlation tends towards the Lagrangian fluid velocity correlation for short fibres and would then approach the Eulerian fluid velocity correlation as the fibre length increased. In agreement with Olson [5], it was seen that as     approaches 0 or  , the Lagrangian fibre velocity correlation gets infinitely close to the fluid’s Lagrangian or Eulerian velocity correlation, respectively.  For streamwise axisymmetric turbulence, shown in Fig. 4.14b-d, the Lagrangian fibre velocity correlation for fibres shorter than          collapse towards one another and all fibre lengths move away from the Lagrangian fluid velocity correlation and approach the Eulerian fluid velocity correlation. This happens for almost all fibre lengths because the non-dimensional length scale     gets so large that more fibres are decreasing in relative size such that        . This effect is amplified with increasing  , until there is little difference between the short and long fibres. For all  , the longest fibres remain correlated for about the same amount of time whereas for fibres of length         , the correlations become more compact until      , when there is very little difference in correlation coefficient between fibre lengths. This is because the streamwise turbulent fluctuations are getting significantly smaller with increasing   52  so the fibre velocity of the small fibres changes very slowly for each time-step, resulting in the velocity staying correlated for longer times.   Figure 4.14 Streamwise translational velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 0 1 2 3 4 5 6 7 8-0.200.20.40.60.81(a)<v 1(0)v1(t)>/v12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(b)<v 1(0)v1(t)>/v12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(c)<v 1(0)v1(t)>/v12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(d)<v 1(0)v1(t)>/v12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2053     For the largest fibres, the axisymmetric correlation coefficient is always approximately equal to the isotropic correlation coefficient. In the radial direction, shown in Fig. 4.15b-d, the smallest fibres move away from the Lagrangian fluid velocity correlation and become tightly packed, similar to the streamwise direction. For all fibres, the correlation coefficient for the radial correlation is less than or equal to the correlation coefficient for the streamwise correlation, for all  . This is because the fluctuations in the radial direction are much larger than the fluctuations in the streamwise direction and will cause fibre velocity to fluctuate more dramatically, leading to velocities being less correlated. The different fibre lengths still behave similarly in terms of grouping to the streamwise direction because the length scales have decreased, leading to more fibres being much larger than     and thereby experiencing spatial averaging over their increased relative lengths. This leads to a smaller net velocity, much as the smaller fluctuations in the streamwise direction do. For all  , the correlation coefficient is greater than in isotropic turbulence, except for         , where it is approximately equal. The Lagrangian velocity correlation of the largest fibres decreases only slightly as contraction ratio increases. This is because the magnitude of the fluctuations in the radial direction are very large compared to the isotropic fluctuations, so the largest fibres will deviate slightly despite the spatial averaging of the velocity fluctuations along their length.        54    Figure 4.15 Radial translational velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 1 2 3 4 5 6 7 8-0.200.20.40.60.81(a)<v 2(0)v2(t)>/v22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(b)<v 2(0)v2(t)>/v22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(c)<v 2(0)v2(t)>/v22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(d)<v 2(0)v2(t)>/v22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2055   To calculate the dispersion as given by Eq. (8), we require the translational time scale. The time scale is determined by integrating Eq. (66) for each fibre length and is given in Fig. 4.17, 4.18 for isotropic turbulence. Olson [5] found that initially the increase in integral time scale with increasing fibre length is approximately linear and then approaches the Eulerian integral time scale as fibre length becomes large, which the simulation has agreed with. In this simulation, the Eulerian integral time scale was found to be        , which is taken from Fig 4.5 as the value of      .   In order to understand how the anisotropic integral time scale changes, it is necessary to consider the notion of relative fibre length. For reference, consider a unit fibre length under the influence of a unit eddy. If this eddy is less than unity, the unit fibre will experience more of these eddies, in essence, it will experience some spatial averaging, making it act like a larger fibre than the original unit fibre. Likewise, if the eddy is larger than unity, the unit fibre will act like a smaller fibre and experience less spatial averaging. This is illustrated in Fig 4.16.  Figure 4.16 The difference in relative fibre length depending on how many eddies are averaged  56   The streamwise integral time scale for all fibre lengths in axisymmetric turbulence is plotted in Fig 4.17. The streamwise integral time scale of a small fibre is greater than the corresponding isotropic time scale and increases slightly as   increases. The streamwise integral time scale of the larger fibres increases from the isotropic time scale when       but then quickly decreases as   increases, corresponding to the decrease in Eulerian time scale as   increases. This spike occurs because at      , relative fibre length has decreased by about 30% while the velocity fluctuations have decreased by about 50%, so large fibres stay correlated for longer times. As   increases, the relative fibre length goes down by approximately 55% while the velocity fluctuations decrease by about 75%. The difference in how much the fibre length and velocity fluctuations decrease is always 20%, so as the relative fibre length gets smaller, the effect of this 20% difference will increase, resulting in a slight decrease in correlation time. The opposite relationship is true in the radial direction.  The radial integral time scale, shown in Fig 4.18, changes similarly with  , and is always approximately          less than the streamwise time scale. The radial integral time scale of the largest fibres decreases steadily as   increases. The largest fibre length in isotropic turbulence had a time scale equal to the Eulerian time scale, as suggested by Olson [5]. In the radial direction,   decreases as   increases which results in relative fibre length increasing such that the large fibres experience more spatial averaging. However, the time scale of these large fibres does not increase as they get larger because velocity fluctuations go up drastically in the radial direction, resulting in a larger velocity change which affects the time scale of the fibres more than the increased relative length.    57            Figure 4.17 The streamwise fibre Lagrangian integral time scale for all contraction ratios   Figure 4.18 The radial fibre Lagrangian integral time scale for all contraction ratios  0 2 4 6 8 10 12 14 16 18 2011.522.533.5Streamwise Lagrangian Integral Time ScaleT t1u0k0k0L  C = 1C = 2C = 3C = 40 2 4 6 8 10 12 14 16 18 2011.522.533.5Radial Lagrangian Integral Time ScaleTt2u0k0k0L  C = 1C = 2C = 3C = 458   For both directions, the axisymmetric time scale of the smallest fibres is greater than or equal to the isotropic time scale because the velocity correlations all approach the Eulerian correlation as   increases. Olson [5] noted that the tendency of the fibre Lagrangian time scale was to increase towards the fluid Eulerian time scale as fibre length increased. Conceptually, this occurs because small fibres travel like fluid particles, due to the inertialess assumption. They therefore experience the Lagrangian time scale. Larger fibres average many eddies over their length at one time, so they act like a control volume. Because of this, they experience the Eulerian time scale. This is shown in Fig 4.19. Olson noted this was similar to the crossing-trajectory effect [28], [89], where as particle inertia increases the particle experiences the Eulerian statistics. This relationship is shown through Fig. 4.17, 4.18 to be maintained as anisotropy increases, however, due to an increase in the Eulerian time scales in both directions and the relative decrease of fibre length in the streamwise direction and the increase in velocity fluctuations in the radial direction, the Lagrangian fibre time scale relative to the Eulerian fluid time scale has actually decreased. As fibre length approaches the largest length, the time scales tend to approximately the Eulerian time scale and this behavior does not change as   increases.         Figure 4.19 Small fibres behave like fluid particles vs. averaging of many eddies over the length of long fibres 59   Now that we have the translational time scale of the fibres, we are able to compare the dispersion given by the simulation with the analytic estimate of the isotropic long time dispersion coefficient     that was derived by Olson and Kerekes [3], given by the following,                              (67)  Fig 4.20a, Fig 4.21a  plots the long time isotropic dispersion coefficient,    , which is estimated as the long time value of    for all fibre lengths, against Eq. (67).  This estimate was tested for the axisymmetric case, shown in Fig 4.20 for the streamwise direction and Fig. 4.21 for the radial direction. The simulation showed good agreement in the streamwise direction but was inaccurate in the radial direction.     in isotropic turbulence decreases as fibre length increases in both directions.  This behavior is seen in streamwise axisymmetric turbulence as well, shown in Fig. 4.20b-d, but the magnitude of the axisymmetric coefficients is significantly smaller than the isotropic translational dispersion coefficient. The estimate for the long time dispersion in the radial direction is shown in Fig. 4.21. It can be seen that     for a given fibre length increases significantly radially as   increases, shown in Fig 4.21b-d. Just as in the isotropic case,    , decreases as fibre length increases.  60    Figure 4.20 Long time streamwise translation dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4  (a) (c) 0 2 4 6 8 10 12 14 16 18 2000.3750.751.1251.5(k0/u0)Dtk0L(a)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2000.3750.751.1251.5(k0/u0)Dtk0L(b)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2000.3750.751.1251.5(k0/u0)Dtk0L(c)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2000.3750.751.1251.5(k0/u0)Dtk0L(d)  DtEq. (67)61    Figure 4.21 Long term radial translation dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   (c) (a) 0 2 4 6 8 10 12 14 16 18 2001.534.567.5k0L(k0/u0)Dt(a)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2001.534.567.5(k0/u0)Dtk0L(b)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2001.534.567.5(k0/u0)Dtk0L(c)  DtEq. (67)0 2 4 6 8 10 12 14 16 18 2001.534.567.5(k0/u0)Dtk0L(d)  DtEq. (67)62  4.2.2 Rotational Dispersion  The orientation correlation equation is given as                    (68)  Using Eq. (68) for the ensemble of fibre trajectories, the orientation correlation is calculated and given for all fibre lengths in Fig 4.24a for isotropic turbulence. For isotropic turbulence, Olson [5] found that small fibres are randomized at long times while larger fibres are still strongly correlated. This simulation found that short fibres are completely randomized after          whereas long fibres are strongly correlated beyond         . Because of the spatial averaging of the angular velocity fluctuations over the fibre length, larger fibres experienced a smaller net angular velocity fluctuation and thus remain correlated longer than shorter fibres. This increased orientation correlation corresponds to a decrease in rotational diffusion with fibre length.  For the streamwise axisymmetric case, shown in Fig 4.24b-d, small fibres exhibited a similar time to randomize as the isotropic case, approximately          for all  . However, as   increased, the correlation coefficient decreased at a much faster rate for smaller fibres while still taking the same time to randomize. This increased rate of decrease occurs because fibres are rotated by the angular fluctuations which are much greater in the streamwise direction than the radial direction, pictured in Fig 4.25, thus small fibres are experiencing much greater angular velocity fluctuations in the streamwise direction as   increases while experiencing insufficient spatial averaging to dampen the fluctuations, resulting in the streamwise correlation coefficient decreasing faster. Larger fibres were still strongly correlated beyond         . This occurs because as   increases, the increase in fluctuations result in a decrease in eddy size. The decreased radial eddy size results in large fibres experiencing a more significant effect from 63  spatial averaging than from increased angular fluctuations. Smaller fibres do not benefit from the decreased eddy size because there is not enough spatial averaging to offset the increased angular velocity fluctuations. This behavior shift is shown to occur around      , from Fig 4.24.   In the radial direction, shown in Fig 4.25b-d, the smallest fibres are more strongly correlated for increasing contraction ratio while the correlation of the largest fibres is slightly less correlated with increasing contraction ratio. The increased correlation coefficient of the smaller fibres is due to the decreased angular velocity fluctuations. The eddy size also increases, however, this does not impact the small fibres as they did not experience spatial averaging in isotropic turbulence. This is visualized in Fig 4.22, 4.23. The larger fibres, on the other hand, experience less spatial averaging because of the increase in eddy size but simultaneously the angular velocity fluctuations orienting them also decrease, resulting in the orientation correlation decreasing only slightly as anisotropy increases.      Figure 4.22 Illustration of how angular velocity changes in a contraction     Figure 4.23 The increase in eddy size does not impact the small fibres significantly 64    Figure 4.24 Orientation correlation in the streamwise direction for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 1 2 3 4 5 6 7 8-0.200.20.40.60.81(a)<p1(0)p1(t)>/p12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(b)<p1(0)p1(t)>/p12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(c)<p1(0)p1(t)>/p12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(a)<p1(0)p1(t)>/p12tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2065    Figure 4.25 Radial orientation correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 1 2 3 4 5 6 7 8-0.200.20.40.60.81(a)<p2(0)p2(t)>/p22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(b)<p2(0)p2(t)>/p22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81(c)<p2(0)p2(t)>/p22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5 6 7 8-0.200.20.40.60.81C = 4<p2(0)p2(t)>/p22tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2066   The rotational dispersion coefficient characterizes turbulent motion in the convection-dispersion equation and is calculated using the following relation, derived by Olson and Kerekes [3],                                    (69)  In isotropic turbulence, the rotational dispersion is initially zero, similar to the translational dispersion, and then increases asymptotically over time until it reaches a constant value. This constant is the longtime constant value of    and is denoted as    .    depends significantly on and decreases with fibre length, seen in Fig 4.26. There is some difficulty in determining     for shorter fibres as the orientation correlation is very close to zero, meaning that     is determined from noisy statistics. Beyond         , standard error analysis shows that    for the shortest fibres is unreliable, the calculation of which is in the appendix.   The evolution of         does not change qualitatively as   changes.         is initially zero and increases asymptotically until it reaches a max,         . In the streamwise direction,         increases slightly but maximizes much faster as   increases, while in the radial direction,         grows faster but decreases in magnitude as   increases. For large fibres,          does not change much in the streamwise direction while it decreases significantly in the radial direction. In the streamwise direction, the initial decrease in the orientation correlation of the small fibres is so drastic that Eq. (69) tends to  immediately before decreasing to a finite value and then going back to  after several eddy interactions.   As   increases, the time at which the error in Eq. (69) gets large occurs sooner. The shortest fibres in the streamwise direction are compared to the estimation for the rotational 67  dispersion coefficient given by Eq. (10). It is seen in Fig 4.26 that the estimation given by Shin and Koch, from Eq. (10), is close to that predicted by Eq. (69), before it tends to .  Figure 4.26 Streamwise rotational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 0 1 2 300.10.20.30.40.50.6(a)Dr1/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 20Eq. (10), k0L = 1Eq. (10), k0L = 30 1 2 300.10.20.30.40.50.6(b)Dr1/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 20Eq. (10), k0L = 1Eq. (10), k0L = 30 1 2 300.10.20.30.40.50.6(c)Dr1/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 20Eq. (10), k0L = 1Eq. (10), k0L = 30 1 2 300.10.20.30.40.50.6(d)Dr1/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 20Eq. (10), k0L = 1Eq. (10), k0L = 368   In the radial direction, shown in Fig 4.27, Eq. (69) does not tend to  immediately for higher contraction ratios as it does in the streamwise direction. In the streamwise direction there is not much deviation in the long time dispersion coefficient of small fibres for isotropic and axisymmetric turbulence while in the radial direction there is a significant difference for small fibres.   Figure 4.27 Radial rotational dispersion coefficient for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 0 1 2 3 4 500.10.20.30.40.5(a)Dr2/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 500.10.20.30.40.5(b)Dr2/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 500.10.20.30.40.5(c)Dr2/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 500.10.20.30.40.5(d)Dr2/(u0k0)tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2069   To calculate the dispersion given by Olson and Kerekes [3] in Eq. (9), we require the rotational time scale. The rotational time scale is the integral time scale of the angular velocity and can be calculated by integrating the angular velocity,                                    (70)   Assuming    is small such that   does not change significantly over the integral time scale, we may employ the relation proposed by Olson [5],                                     (71)  The isotropic angular velocity correlation has the smallest fibres being completely randomized by           , while the largest fibres are completely randomized only when           It is seen from Fig. 4.28a that as fibre length increases, the angular velocity correlation decreases.   As   increases, the streamwise angular velocity correlation, given in Fig. 4.28b-d, randomizes faster. Fibres randomize slightly faster as   increases due to the increase in eddy size, but it is still only after            that they are completely randomized for all  . The correlation coefficients of all fibres of length less than          bunch up more tightly in the streamwise direction but spread out in the radial direction as   increases. This bunching up occurs because the eddies in this direction get larger as   increases so more fibres are acting like fluid particles or very small fibres, meaning they have very similar angular velocity correlations. The spreading out occurs radially, shown in Fig. 4.29b-d, because the length scales gets smaller, so fibres are relatively larger, with more discrete sizing, resulting in more spread out correlations. Smaller fibres experience this as well, but their correlation coefficient decreases because they are experiencing an increase in angular velocity due to the conservation of angular 70  momentum [91]. In the streamwise direction, the largest fibres decrease in relative length which results in them being less correlated as   increases, whereas in the radial direction the correlation coefficient of the largest fibres actually increases because the relative fibre length increases. The radial direction sees the smallest fibres randomize by          for       while the longest fibres randomize more slowly as   increases, still being weakly correlated by         .   Figure 4.28 Streamwise angular velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4 0 1 2 3 4 5-0.200.20.40.60.81(a)<1(0)1()>/<12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(b)<1(0)1()>/<12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(c)<1(0)1()>/<12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(d)<1(0)1()>/<12>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2071    Figure 4.29 Radial angular velocity correlation for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4    0 1 2 3 4 5-0.200.20.40.60.81(a)<2(0)2()>/<22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(b)<2(0)2()>/<22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(c)<2(0)2()>/<22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 200 1 2 3 4 5-0.200.20.40.60.81(d)<2(0)2()>/<22>tu0k0  k0L = 0k0L = 1k0L = 3k0L = 7k0L = 2072   Olson [5] derived the following approximate expression for the shape of the isotropic Eulerian rotational velocity correlation, which the simulation has agreed with,                                      (72)  This equation was derived for the rotation of short fibres, using relationships found in Hinze [10]. It assumes that the Eulerian statistics are valid for long fibres that do not rotate significantly during the duration of    and can be used as an upper bound for the Lagrangian integral time scale.   The isotropic rotational integral time scale was found to be, for short fibres,             and for longer fibres,            , shown in Fig 4.30a. In the streamwise direction, shown in Fig 4.30a-d, the time scale of the shortest fibres was approximately             for all  . The time scale of the largest fibres decreased steadily as   increased. In the radial direction, shown in Fig 4.31a-d, the time scale for the smallest fibres decreased significantly as   increased, with the time scale at       being approximately half that of the isotropic time scale. For the longest fibres in the radial direction, the time scale increased initially with anisotropy but then decreased slightly as   increased.           73             Figure 4.30 The streamwise fibre Lagrangian angular velocity integral time scale for all contraction ratios            Figure 4.31 The radial fibre Lagrangian angular velocity integral time scale for all contraction ratios  0 2 4 6 8 10 12 14 16 18 2000.511.522.533.5Streamwise Lagrangian Angular Velocity Integral Time ScaleTrxu0k0k0L  C = 1C = 2C = 3C = 40 2 4 6 8 10 12 14 16 18 2000.511.522.533.5Radial Lagrangian Angular Velocity Integral Time ScaleTryu0k0k0L  C = 1C = 2C = 3C = 474   Now that we have the rotational time scales, we are able to compare the dispersion approximated by the simulation against the approximate expression for    that was derived by Olson and Kerekes [3], given by the following,                                        (73)  Eq. (73) was derived as a scalar quantity for use in isotropic turbulence. However, in anisotropic turbulence, the rotational dispersion would be a tensor. Similar to how we treated the translational dispersion, we will invoke orthogonality to treat the rotational dispersion in the principal directions separately. Eq. (73) has been shown to be a reasonable estimate for the isotropic long time dispersion coefficients by Olson [5], which this simulation agrees with, shown in Fig 4.32a, 4.33a. This estimate offers a close approximation in the streamwise direction, shown in Fig. 4.32b-d, but a much less accurate description in the radial direction, shown in Fig. 4.33b-d. It is seen that in the streamwise direction for    , the values of          from the simulation do not change much with anisotropy, whereas Eq. (73) is affected by the increase in anisotropy. In the radial direction, the simulation shows         decreases for short fibre lengths and approaches a constant value for large fibre lengths. At    ,         is very close in value for all fibre lengths.        75     Figure 4.32 Long time rotational dispersion coefficients for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4    0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Drk0L(a)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Drk0L(b)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Drk0L(c)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Drk0L(d)  DrEq.(73)76     Figure 4.33 Long time radial rotational dispersion coefficients for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Dr yk0L(a)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Dr yk0L(b)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Dr yk0L(c)  DrEq.(73)0 2 4 6 8 10 12 14 16 18 2000.050.10.150.20.250.30.35(k0/u0)Dr yk0L(d)  DrEq.(73)77  4.2.3 Fibre Rotation  Eq. (18) is expressed in terms of the spherical to Cartesian transformation, allowing us to plot the orientation of the fibre in Cartesian 3-space. The fibres are plotted at each time-step such that by the law of large numbers we can expect the plots to represent the probability distribution function for fibre orientation. The resulting pdfs are given in Fig 4.34, 4.35, 4.36. The time spent in a region is calculated by considering which component of   is greatest. For example, if    is greatest, the fibre is said to be oriented mostly in the streamwise direction. There are some limitations to this simple counting method but it will still be a useful numerical interpretation of the distribution functions. This is calculated from the ensemble of fibres and given in Table 4.1, 4.2.  In isotropic turbulence, all fibres lengths are approximately uniformly distributed. The center is the only area of greater intensity because all fibres are initially inserted in that orientation. In axisymmetric turbulence, the smallest fibres behave like fluid particles and spend almost all of their time in the radial orientation. Regardless of starting orientation, the magnitude of fluctuations in the radial direction as opposed to the streamwise direction will rotate fibres to the radial orientation. As fibre length increases, there is a small amount of time spent in the streamwise orientation that increases with fibre length. The largest fibres spend at least 25% of their time in the streamwise orientation as opposed to the 33% expected of a uniform distribution, resulting in a distribution that is close to uniform. 78    Figure 4.34 Streamwise heat maps for fibres of length          for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4   (d) (c) (b) (a)                   (b) (d) (c) (a) 79   Figure 4.35 Streamwise heat maps for fibres of length       for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4  C     = 0.25     = 1     = 3     = 5     = 7     = 10     = 20 1 32.6 32.7 33 32.3 33.7 34.7 35.2 2 5 4.8 5.8 7.8 11.6 16 33.9 3 1.4 1.7 2.2 3.9 6.1 9.6 28.9 4 0.7 0.8 1.2 2.7 4 7.2 25.9  Table 4.1 Time percentage spent in the streamwise orientation             (d) (c) (b) (a) (d) (b) (c) (a) 80  C     = 0.25     = 1     = 3     = 5     = 7     = 10     = 20 1 67.4 67.3 67 67.7 66.3 65.3 64.8 2 95 95.2 94.2 92.2 88.4 84 66.1 3 98.6 98.3 97.3 96.1 93.9 90.4 71.1 4 99.3 99.2 98.8 97.3 96 92.8 74.1  Table 4.2 Time percentage spent in the radial orientation  Figure 4.36 Streamwise heat maps for fibres of length        for (a) C = 1 (b) C = 2 (c) C = 3 (d) C = 4                   (d) (c) (b) (a) (b) (c) (d) (a) 81   These results suggest that fibres preferentially align in a contraction because of the presence of a mean flow and the steric effects of the contraction. The presence of a mean flow will act to rotate the fibre towards the streamwise orientation or translate it if it is already in that orientation. Krochak et al. [91] found that when fibres are infinitely close to the wall they flip and move away from the wall. Considering this, fibres in a contraction would rotate radially and flip when they get close to the wall, which would be every radial position in an axisymmetric contraction, until they flip or rotate, whether by mean flow or fluctuations, to the streamwise orientation, where the steric effect of the contraction would not permit them to rotate radially anymore, thus preferentially aligning them. This would be similar for long fibres, except they would more often rotate to the streamwise orientation, whereupon the steric effect would hold them there. 4.2.4 Timescale Relationship  The streamwise Lagrangian integral time scale for short fibres was found to be                                          (74)  and                                           (75)  for           and  , respectively. The radial Lagrangian integral time scale was found to be                                          (76)  and                                          (77)  82   for           and  , respectively. In the simulation by Olson [5], the relationship between    and    was found to be, for short fibres,            (78)  while in the limit of infinitely long fibres,            (79)  This simulation found the isotropic relationship for both short and long fibres to agree with the established estimates. In anisotropic turbulence, these relationships changed, with    becoming larger when compared to    in the streamwise direction. This is because increasing the contraction ratio   causes the eddies to distort which alters the fluctuations of translational velocity. However, this distortion does not have as much of an impact on rotation. In the radial direction,     and    are vastly different in size for small fibres before becoming similar in magnitude in the limit of long fibres. As the contraction ratio increased, the streamwise relationship of       was found to approach 0.3 for the smallest fibres and 0.6 for the longest. The radial direction was found to approach 0.18 for the smallest fibres and was approximately 1.0 for the largest fibres.  C     = 0.25          = 1     = 3     = 3     = 5     = 10     = 20 1 0.5529 0.5910 0.6398 0.6686 0.7078 0.7522 0.7862 2 0.3925 0.4388 0.5006 0.5132 0.5176 0.5376 0.6388 3 0.3382 0.3873 0.4997 0.5056 0.5072 0.5190 0.6205 4 0.3168 0.3557 0.4791 0.4895 0.4895 0.5447 0.6221  Table 4.3 Streamwise relationship of       83   C     = 0.25           = 1     = 3     = 3     = 5     = 10     = 20 1 0.5464 0.5791 0.6154 0.6375 0.6578 0.7316 0.8010 2 0.4151 0.4439 0.5735 0.6296 0.7797 0.8522 0.9979 3 0.2758 0.2919 0.4324 0.5506 0.6895 0.8378 1.0599 4 0.1829 0.2058 0.3332 0.4530 0.5882 0.7447 1.0924  Table 4.4 Radial relationship of      84  Chapter 5: Conclusion 5.1  Conclusions  A numerical simulation of an ideal anisotropic axisymmetric turbulent fibre suspension flow was developed and used to investigate the effect of fibre length and contraction ratio on rotational and translational dispersion coefficients. The translational dispersion coefficient was found to decrease in the streamwise direction and increase in the radial direction with contraction ratio   but decreased with fibre length for both directions. The rotational dispersion coefficient was found to decrease with both   and fibre length in both directions. An estimate, based on contraction ratio, was found that related the isotropic and anisotropic translational dispersion coefficients. Estimates were made for the Eulerian longitudinal and transverse correlations for all contraction ratios tested based on the simulation. Fibres were shown to spend more time aligned in the radial direction as contraction ratio increased but less time as fibre length increased.  The Lagrangian integral time scales for translation of short fibres were found to increase in both the streamwise and radial direction as   increased while the rotational time scales were found to decrease in both directions. The integral time scales for long fibres were found to increase at      , but then decreased as   increased in the streamwise direction for translation and both directions for rotation. The radial translational time scale for long fibres decreased steadily from the isotropic value.    was found to be greater in magnitude to    for all cases except for the longest fibres in the radial direction, where the time scales were found to be approximately equal.   This work has a number of simplifying assumptions inherited from the models used. The fluid model lacks an inertial sub-range as well as a method of modeling the energy cascade, so neither of these elements are well represented. The fibres are assumed infinitely thin, so this 85  model is unable to examine the interactions of fibre height with decreasing eddy size. This becomes increasingly significant as anisotropy increases because at larger contractions, eddy size is getting thin enough that some averaging over the fibre height may occur.  5.2 Future Work  Future work will consider more robust fluid and fibre models as well as anisotropic turbulence in the near wall region along with the fibre-fibre interactions that occur there. It is important to develop an experimentally validated model of near wall phenomena as the boundary conditions of fibre suspension flow are formed there. Currently, the literature does not isolate the effect of the wall on fibre dispersion. This will be investigated with a similar stochastic model, where the orientation and concentration distributions of both dilute and semi-dilute fibre suspensions will be modeled for a range of fibre and flow conditions near boundaries with a range of roughness types and magnitudes. It is expected that a relation between the anisotropy found in the turbulence near a solid boundary can be related to the expected orientation anisotropy of the suspension. The fibre model assumes an infinitely thin fibre, which has no radius. A fibre model of finitely thin fibres of varying radii will be explored to examine the effect of radius on dispersion coefficients. This work utilizes the work of Cox [4], which has been used extensively for inertialess fibres. Future work will explore the effects of inertia on the rotational coefficients, perhaps using the model of Khayat and Cox [92]. The interaction of boundaries with varying roughness will be investigated using an anisotropic flow model to relate the expected anisotropy in the concentration distribution to the anisotropy found in the turbulence near the boundary. Together, these works should give a clearer picture of the effect of anisotropy on turbulent fibre suspensions.  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For the isotropic case, we have                 (80)  Now, substituting Eq. (80) into the right hand side of Eq. (65) yields                                                  (81)  which simplifies to                             (82)  In isotropic turbulence, we have             (83)  So, Eq. (65) becomes               (84)  Considering the magnitude of translation and the relation in Eq. (65) yields,                       (85)  Using this expression in Eq. (80) and squaring gives, 95                     (86)                                             (87)  Then, employing the equivalency given in Eq. (83) yields                     (88)  which proves Eq. (65) for isotropic turbulence.  For axisymmetric turbulence, Eq. (83) is not true and only the   and   components are equal. Eq. (84) then becomes,                     (89)  Similarly, Eq. (85) would become                (90)  and noting this change for Eq. (86) as well, we would have                           (91)  thereby demonstrating that Eq. (65) is also valid for axisymmetric turbulence. A.2 Anisotropic estimates of long time dispersion coefficients  Eq. (67), (73) from Olson and Kerekes [3] was implemented for each principal axis separately by using Onsager’s reciprocity relation as described by Knauth and Vona [93]. A 96  minor change was made to reflect the appropriate root mean square velocity and the time scale for both equations, given by the following,                                                   (92)  and                                                             (93)  Neither of these forms were able to accurately predict      and     with the same degree of precision as seen in isotropic turbulence. The results are given in Fig 4.20, 4.21, 4.32, 4.33. A.3 Error of rotational dispersion coefficient  By considering the chain formula for   , the error can be calculated as                                                                              (94)  which simplifies to                                             (95)  the given time step is           , which yields an error of                                             (96)  which will very quickly go to infinity as             , which occurs at approximately          for isotropic turbulence and occurs sooner as   increases. This will also quickly go 97  to infinity as                   , which occurs for small fibres in contraction ratios larger than 2 almost immediately.    98  Appendix B   B.1 Fluid Model Validation  The model is comprised of several components that require validation. These are separated into the fluid and fibre models. The isotropic fluid model is verified by ensuring that it’s properties are equal in all directions; these will be verified by ensuring that the root mean square velocities in all directions is equal to the input    as   increases for   realizations of the flow. The Eulerian longitudinal and transverse correlations are well known for the Kraichnan spectrum, and this model is shown to converge to them in Fig. 4.1. Eq. (24) depends on a zero-mean flow, so it will be necessary to ensure that the flow has a zero-mean in both isotropic and axisymmetric models. This is demonstrated in the following figures,  Figure B.1 The rms velocity for different times and the velocity mean for all contraction ratios tested 1 1.5 2 2.5 3 3.5 4-1-0.8-0.6-0.4-0.200.20.40.60.81Velocity mean for all CVelocity meanC0 100 200 300 400 500 600 700 800 900 10000.90.920.940.960.9811.021.041.061.081.1u0 as t increases, 100 realizationsu0t(a) (b) 99  The anisotropy of the model needs to be verified against predictions for axisymmetric flows. Prandtl’s equations predict the rms velocity through a contraction, so the axisymmetric root mean square velocity will be verified against Eq. (35), for each   tested.  Figure B.2 The rms velocities of the simulation compared with the Townsend equations and with Prandtl’s formulae  The isotropic flow model was made anisotropic by distorting the wavenumbers as described by RDT in Eq. (44). This is verified by considering the analytic root mean square velocities for specific strain rates, given by Townsend [94], in Fig. B.2a B.2 Fibre Model Validation  The fibre model is validated for isotropic turbulence against the results of Olson [5]. Different integration schemes were used, so qualitative similarities are what will be considered. Foremost is the relation of the translational and rotational timescale, which Table 4.1, 4.2 has 1 1.5 2 2.5 3 3.5 4 4.5 500.511.522.5Prandtl rms velocitiesu1, u2C1 1.5 2 2.5 3 3.5 400.511.522.53RDT rms velocitiesu1, u2e-t(a) (b) 100  shown good agreement with. The analytic estimates given by Eq. (69), (73) from Olson and Kerekes [3] was shown to have good agreement with the simulation of Olson [5]. Fig 4.20a, 4.32a demonstrate that this simulation also provides good agreement with the estimates for fibres in isotropic turbulence. 

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