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The interaction of opposing jets Quick, Jeffrey William 1994

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THE INTERACTION OF OPPOSING JETS By Jeffrey William Quick B. A. Sc. (Eng. Phys.) University of British Columbia , 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY m THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1994 © Jeffrey William Quick, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my v^ Titten permission. (Signature Department of ^U(J^A^//CAL / ^ A / ^ M / > : > ^ / g / A / ^ The University of British Columbia Vancouver, Canada Date W C^X. / 9 ? 4 / DE-e (2/88) Abstract Opposing jets under conditions typical of industrial furnace air systems are studied using isothermal numerical and physical modelling. The prime motivation for the study comes from an interest in the development of modelling capabilities for the recovery furnaces used in the wood pulping process. The instability of the jet interaction and nature of the resulting flow states are the foci of the work. Cases for opposing jets in confined ducts with either a cross flow or a closed floor are investigated, which exhibit sonae of the main features of recovery furnaces and a number of other applications. The numerical modelling is based on the UBC-MGFD finite volume, multi grid code, under development in the Department of Mechanical Engineering at U.B.C., which is modified for transient computations in this work. Two and three dimensional computa-tions are performed for a variety of parametric configurations; resulting in bifurcations to stable asymmetric flows in the 2D cases and unsteady, oscillatory flows in the 3D cases. Extensive testing of the numerical method is carried out for one case of opposing jets in a closed floor cavity to determine convergence level requirements and time step and grid dependence; which shows that reliable results can be obtained. Physical modelling is performed for one case of opposing jets in a closed floor cav-ity. Laser doppler velocimetry measurements of velocity time series are compared with corresponding time series data from the numerical modelling and excellent agreement is obtained for the total root mean square velocity at a strategic point where the jets are colliding. Flow visualization studies using a laser light sheet with a seeded flow, and a particle image velocimetry analysis reveal flow patterns that are in good, qualitative agreement with the numerical results. 11 Table of Contents Abstract ii Table of Contents iii List of Tables v i List of Figures vii Nomenc la ture x iv Acknowledgement xvii 1 Introduct ion 1 1.1 Motivation 1 1.2 Background 2 1.3 Posing the Model 4 1.4 Dimensionless Form 5 1.4.1 Mass Flow Ratio 6 1.5 Objective and Scope of the Work 6 2 Literature R e v i e w 12 2.1 Furnace Flow Modelling 12 2.2 Fluid Amplifiers 14 2.3 Jets in Cross Flow 15 2.4 Opposing Jets without Cross Flow 17 iii 3 C F D M e t h o d 19 3.1 Mathematical Model 19 3.2 Numerical Method 20 3.3 Multi-grid Procedure 22 3.3.1 Multi-grid convergence rates 26 4 Opposing Je ts in Cross Flow 29 4.1 2D Reference Case B/b = 20, Rej = 5.6 x lO^M = 25 29 4.1.1 Variations in Momentum Flux Ratio 31 4.1.2 Transient Perturbations and Stability Analysis 32 4.1.3 Jet Tracer Analysis 32 4.2 3D Reference Case AR = 4, S/b = 5, B/b = 10, Rej = 10% M = 20 . . . . 34 4.2.1 Steady State Computations 34 4.2.2 Transient Computations 36 4.2.3 Dimensional Analysis for the Time Scale 37 4.2.4 Analysis of Transient Results and Time Averaging 38 4.2.5 Computer animated flow visualization 39 4.2.6 Jet Tracer Analysis 39 4.3 Grid Refinement 40 4.4 Asymmetric Firing 42 4.5 Variations of parameters 42 5 Opposing Jets in a Closed Floor Cavity 67 5.1 Closed Floor Case AR = 4, S/b = 4, B/b = 10, Rej = 10% Di = 15.5 . . . 67 5.1.1 Wall Shear Stress Effects 68 5.2 Asymmetric Firing Geometry 68 5.3 Time Step Dependence 69 iv 5.4 Convergence Level Dependence 70 5.5 Grid Dependence 72 5.5.1 Wall Shear Stress Effects 75 5.6 Combined time step and grid refinement 75 6 Physical Exper iments 91 6.1 Apparatus 91 6.2 Flow Measurement and Calibration 94 6.3 Drift Error 95 6.4 Laser Doppler Velocimetry 96 6.5 LDV and CFD time series comparison 96 6.6 Flow Visualization 99 6.7 Particle Image Velocimetry 101 7 Conclusions and Recommendat ions 117 7.1 Conclusions 117 7.2 Recommendations 119 Bibliography 121 Appendices 125 A Wall Shear Stress Effects 125 A.l Standard Level Two Grid 125 A.2 Refined Level Two Grid 127 B Additional Closed Floor Resul ts 135 List of Tables 3.1 Constants for the k ~ e model 20 3.2 Multi-grid parameters in the UBC-mgfd code 27 4.1 Summary of calculated Strouhal Numbers and U2 Amplitudes 44 5.1 Summary of calculated Strouhal Numbers 73 5.2 Summary of calculated U2 Amplitudes 77 6.1 Experimental Geometry 92 VI List of Figures 1.1 Schematic of the Kamloops Recovery Furnace 9 1.2 Schematic of Opposing Jets in Cross Flow; Vertical Plane 10 1.3 Schematic of Opposing Jets in Cross Flow; Horizontal Plane 10 1.4 Schematic of Opposing Jets in a Closed Floor Cavity; Vertical Plane. . . 11 1.5 Schematic of Opposing Jets in a Closed Floor Cavity; Horizontal Plane. . 11 3.1 Multi-grid Triangle Diagram 28 3.2 Multi-grid V-Diagram 28 4.1 Enforced symmetry steady state flow. 2D reference case 45 4.2 Asymmetric steady state flow. 2D reference case 45 4.3 Bifurcation diagram showing steady state reattachment lengths versus mo-mentum flux ratio. 2D flow, long domain with zero gradient exit 46 4.4 Transient perturbation of the 2D reference case enforced symmetry result. A 0.5 s ± 5 % Vj pulse is applied at t = 0 46 4.5 Transient perturbation of the 2D reference case asymmetric result. A 0.5 s ±20% Vj pulse is applied at t = 0. The asymmetry is not reversed. . . 47 4.6 Transient perturbation of the 2D reference case asymmetric result. A 2.0 s ±20% VJ pulse is applied at t = 0. The asymmetry is reversed 47 4.7 Enforced symmetry steady state jet tracer concentration (in %) . 2D ref-erence case. ^Bidk — 52.79% 48 4.8 Asymmetric steady state jet tracer concentration (in %). 2D reference case. ^Buik = 52.79% 48 vii 4.9 3D Reference case enforced symmetry, steady state flow. Vertical plane through jets 49 4.10 3D Reference case enforced symmetry, steady state flow. Horizontal plane through jets 50 4.11 3D Reference case enforced symmetry, steady state flow. Convergence performance for level 2 (32 x 14 x 48) multi-grid and single-grid 50 4.12 3D Reference case enforced symmetry, steady state flow. Convergence performance for level 2 (32 x 14 x 48) and level 3 (64 X 28 x 96) multi-grid. 51 4.13 3D Reference case enforced symmetry, steady state flow. Profiles for W velocity and pressure coefficient at xjh = 10, on duct centre-line, for three grid levels 51 4.14 3D Reference case enforced symmetry, steady state fiow. Profiles for tur-bulent intensity and viscosity at xjh = 10, on duct centre-line, for three grid levels 52 4.15 3D Reference case time series from start-up 52 4.16 3D Reference case time series showing periodic result 53 4.17 3D Wide spacing, S/h = 10, time series from start-up 54 4.18 3D Reference case transient flow when U2 ~ ^ {t* = 2980). Vertical plane through jets 55 4.19 3D Reference case transient flow when U2 = maximum {i* = 3080). Ver-tical plane through jets 56 4.20 3D Reference case time average flow over one cycle. Vertical plane through jets 57 4.21 3D Wide spacing, Sjh = 10, nearly steady result at t* = 960. Vertical plane through jets 58 vui 4.22 3D Reference case transient flow when U2 = 0 (t* = 2980). Horizontal plane through jets 59 4.23 3D Reference case transient flow when U2 = maximum {t* = 3080). Hor-izontal plane through jets 59 4.24 3D Reference case time average flow over one cycle. Horizontal plane through jets 60 4.25 3D Wide spacing, S/b = 10, nearly steady result at t* = 960. Horizontal plane through jets 60 4.26 3D Reference case enforced symmetry, steady state concentration fleld. Vertical plane through jets 61 4.27 3D Reference case transient concentration field when U2 = 0 (t* = 2980). Vertical plane through jets 62 4.28 3D Reference case transient flow concentration fleld when U2 = maximum, {t* = 3080). Vertical plane through jets 63 4.29 3D Reference case time average concentration fleld over one cycle. Vertical plane through jets 64 4.30 3D Reference case enforced symmetry, steady state concentration fleld. ^Buik = -39. Horizontal plane through jets 65 4.31 3D Reference case transient concentration fleld when U2 = 0 {t* = 2980). ^Buik = -39. Horizontal plane through jets 65 4.32 3D Reference case transient concentration fleld when U2 = maximum {t* = 3080). ^Buik = -39. Horizontal plane through jets 66 4.33 3D Reference case time average concentration field over one cycle. ^Buik = .39. Horizontal plane through jets 66 5.1 Closed floor case time series showing periodic result 78 IX 5.2 Closed floor case time series for asymmetric firing geometry, showing nearly periodic result 78 5.3 Closed floor case time series for U2 showing effect of time step size. . . . 79 5.4 Convergence performance for one time step, restart from t* = 1920, for transient run on standard 2 level grid 80 5.5 Closed floor case time series for U21 showing effect of convergence level. . 80 5.6 Closed floor case time series for U2, showing restart from standard grid level 2 to continue on level 3 81 5.7 Closed floor case time series for U21 showing grid dependence for standard levels 2 and 3 plus level 2 refined grids 81 5.8 Closed floor case Strouhal numbers versus number of grid nodes, showing grid dependence for standard levels 2 and 3 plus level 2 refined grids. . . 82 5.9 Closed floor case Strouhal numbers versus cube root number of grid nodes, showing grid dependence for standard levels 2 and 3 plus level 2 refined grids 82 5.10 Closed floor case profiles of U* and Cp along the axis of opposing jets when U2 = maximum, showing grid dependence for levels 2, 2R and 3 grids. . 83 5.11 Closed floor case profiles of TI and u^ along the axis of opposing jets when U2 = m-axmum, showing grid dependence for levels 2, 2R and 3 grids. . 83 5.12 Closed floor case time series for f/2, showing a restart from standard grid level 2 to continue on level 3, with a further continuation for a refined time step 84 5.13 Closed floor case time series for U2 on the refined level two grid (2R), with a continuation for a refined time step 84 5.14 Closed floor case transient flow when C/2 = 0 {t* = 2080). Vertical plane through jets 85 x 5.15 Closed floor case transient flow when U2 = maximum {t* = 2190). Verti-cal plane through jets 86 5.16 Closed floor case with asymmetric firing geometry, transient flow when t* = 5000. Vertical plane through jets 87 5.17 Closed floor case transient flow when U2 = m^aximum {t* = 4535). Level 3 grid, refined time step, Ai* = 5. Vertical plane through jets 88 5.18 Closed floor case transient flow when U2 = m^aximum {t* = 4535). Level 3 grid, refined time step, Ai* = 5. Every second vector shown for com-parison with level 2 result. Vertical plane through jets 89 5.19 Closed floor case transient flow when U2 — maximum, {t* = 3675). Refined level 2 grid (2R), refined time step, At* = 5. Vertical plane through jets. 90 6.1 Schematic of Experimental Apparatus; Vertical Plane 103 6.2 Schematic of Experimental Apparatus; Horizontal Plane 104 6.3 Time series for C/2: a) LDV result from experiment, b) computed closed floor result 105 6.4 Fourier spectrum of LDV result from experiment: a) log-log scale over measurable frequency range, b) linear-log scale at low frequency 106 6.5 Transient streakline photograph from experiment, when jet impingement is maximally off-centre to the left. Vertical plane through jets 107 6.6 Transient streakline photograph from experiment, when jet impingement is maximally off-centre to the right. Vertical plane through jets 108 6.7 Closed floor case transient flow when U2 = maxim^um, {t* — 2190). Verti-cal plane through jets 109 6.8 PIV flow fields, 2.5 s averages between 0 and 10 s 110 6.9 PIV flow fields, 2.5 s averages between 10 and 20 s I l l XI 6.10 PIV flow fields, 2.5 s averages between 20 and 30 s 112 6.11 PIV flow fields, 10 s averages between 0 and 30 s, and a 0 to 30 s average. 113 6.12 PIV flow field: 30 second average, axis box of closed floor CFD case. Positions of recirculation centres agree with time average CFD result. . . 114 6.13 Computed closed floor time average of oscillating flow. Vertical plane through jets 115 6.14 Computed enforced symmetry steady flow. Vertical plane through jets. . 116 A.l Closed floor case time series with free slip walls, showing periodic result. 128 A.2 Closed floor case time series with wall functions, showing nearly periodic result 128 A.3 Closed floor case time series for f/a on the refined level two grid (2R), showing a restart with wall functions 129 A.4 Closed floor case transient flow when U2 = maximum (t* = 2190), with free slip walls. Standard level 2 grid. Vertical plane through jets 130 A.5 Closed floor case transient flow when U2 = m,axim,um, (t* = 3930), with wall functions. Standard level 2 grid. Vertical plane through jets 131 A.6 Closed floor case transient flow when U2 = m,axim,um (t* = 2950), with free slip walls. Refined level 2 grid (2R). Vertical plane through jets. . . . 132 A.7 Closed floor case transient flow when U2 is nearly w,aximum [t* = 3670), with wall functions. Refined level 2 grid (2R). Vertical plane through jets. 133 A.8 Closed floor case transient flow when U2 = m^aximum {t* = 3680), with wall functions. Refined level 2 grid (2R). Vertical plane through jets. . . 134 B.l Closed floor case transient flow when U2 = m,axim,um, (t* = 4535). Level 3 grid, refined time step. At* = 5, every second vector shown. Vertical plane through jets 136 xii B.2 Closed floor case Cp field when U2 = maximum (t* = 4535). Level 3 grid, refined t ime step, At* = 5. Vertical plane through jets 137 B.3 Closed floor case TI field when U2 = maxim,um. {t* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane through jets 138 B.4 Closed floor case transient u* field when U2 = w,aximum (t* — 4535). Level 3 grid, refined time step. At* = 5. Vertical plane through jets. . . . 139 B.5 Closed floor case transient flow when U2 = m,axim,um, (t* = 4535). Level 3 grid, refined time step, At* = 5, every second vector shown. Vertical plane between jets 140 B.6 Closed floor case Cp field when U2 = m^aximuw, {t* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane between jets 141 B.7 Closed floor case TI field when U2 = m,aximum, {t* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane between jets 142 B.8 Closed floor case transient i^^ field when U2 = m,axim,uw, (i* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane between jets. . . . 143 B.9 Closed floor case transient flow when U2 = m,axim,um, (t* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets 144 B.IO Closed floor case Cp field when U2 = m,axim,um, (t* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets 144 B . l l Closed floor case TI field when U2 = m,axim,um, (t* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets 145 B.12 Closed floor case transient u* field when U2 = m,aximum, (t* = 4535). Level 3 grid, refined time step, At* = 5. Horizontal plane through jets. . 145 xiu Nomenclature AR Jet aspect ratio ( = h/w ). b Jet characteristic size ( = yhw ). B Cavity half-depth between opposite jets. Cfi, Ci, C2 Turbulence model constants. Cp Pressure Coefficient ( = Il/~pVj ). Di, D2 Cavity heights below and above the jets respectively. DiFi DiB Cavity heights below jets on front and back walls. d Defect or residual in the numerical method. / Frequency. h Jet height. J Jet momentum flux ( = PUj ). k Specific turbulent kinetic energy. L Computational grid level. Li,L2 Jet reattachment lengths. I Turbulent length scale. M Momentum flux ratio (jet to cross flow). m Mass flow ratio (jet to cross flow). TV Number of computational grid nodes. i^ , i^  Nonlinear operators representing the governing and correction equations. p Pressure. Rej Jet Reynolds number ( = Ujb/v ). XIV S Jet spacing in the lateral, or spanwise direction of the array. St Strouhal number (dimensionless frequency). t Time. A i Time step in CFD calculations. A i , Still camera shutter speed. T Oscillation period ( = 1 / / ). TI Turbulent intensity. Ui,u,v,w Velocities in the Xi, x, y, and z directions. Ui, U, V, W Reynolds mean velocities in the Xi, x, y, and z directions. Ur Friction Velocity ( = JT^JIp ). u[,u',v',w' Fluctuating velocities in the Xi, x, y, and z directions. u Exact solution to the governing equations. Uh Exact solution to the discretized equations on grid h. Uh Approximate solution to the discretized equations on grid h. Vh Exact correction to the approximate solution {uh = Uh-\-Vh)• W Cavity width in the spanwise direction. w Jet width. Xi, Xj, X, y , z Cartesian coordinates (i = 1,2,3, j = 1, 2, 3). 2/+ Wall coordinate ( = yUr/ty ). e Dissipation rate of specific turbulent kinetic energy. $ Reynolds mean scalar: mass fraction of jet fluid, jet fluid tracer concentration, or temperature parameter ( = (T — Tc)/{Tj — Tc) )• H Dynamic viscosity. p Kinematic viscosity ( = pLJp ). n Reynolds mean pressure. XV p Fluid density. Turbulent Prandtl numbers in the turbulence model. Wall shear stress. Subscripts b B C h,H 1,2,3 J s t Made dimensionless using jet size b. Made dimensionless using duct half-depth B. Cross Flow. Finer and coarser computational grids. Cartesian tensor indices. Points on the vertical duct center-line for time series data. Jet. Still camera shutter. Effective turbulent value. Superscripts 0' 6 0 0 Dimensionless quantity. Quantity at the previous time step. Fluctuating quantity, with respect to the Reynolds mean value. Time rate of change quantity. Approximate quantity. Time average value. XVI Acknowledgement The author gratefully acknowledges the support of the United States Department of Energy, The Institute of Paper Science and Technology at Atlanta Georgia, Industry, Science and Technology Canada, Energy, Mines and Resources Canada, the Natural Sciences and Engineering Research Council of Canada, the British Columbia Science Council, and H.A. Simons of Vancouver B.C. The author would also like to thank the University of British Columbia's University Computing Services for assisting with the animated flow visualization. The author would like to express his appreciation to his supervisors. Professor I.S. Gartshore and Professor M. Salcudean, for their tremendous support, their inspiration and their confidence in the author throughout the work. The support of Mr. Douglas M. Bruce of H.A. Simons, who introduced the author to industrial furnace practices, has been greatly appreciated. Thanks are due to Dr. Zia AbduUa for his guidance in many aspects of the UBC recovery furnace modelling program. Mr. Michael Savage has been invaluable in very many ways and his efforts and his friendship are greatly appreciated. Thanks are due to Mr. Edward Abel, who with great expertise and ingenuity provided ready and effective solutions for modifying the experimental apparatus. The author found great pleasure in working with Mr. Abel. Throughout the author's course of graduate studies, Mr. Jian-Ming Zhou has been a most stimulating colleague and a friend. His ability to translate between the worlds of mathematics, physics and engineering is brilliant. A most sincere thank you is owed to Mr. Zhou. There are many others at U.B.C. who have been tremendously supportive and the author hopes that they understand his appreciation. xvn Chapter 1 Introduction 1.1 Motivat ion The motivation to study the interaction of opposing jets comes from an interest in the modelling of industrial furnaces. This work is part of a project in the Department of Mechanical Engineering at the University of British Columbia to develop numerical and physical modelling capabilities specifically for recovery boiler applications. The recovery boiler poses unique and particularly difficult operational problems to the industry, owing to its dual function of combustion and chemical reduction in the wood pulping cycle. The economic implications of these operational problems are critical to competitiveness in the industry. The chemical processes in the boiler make for an extremely harsh envi-ronment, in which experimental measurements or even visualization are very difficult to perform. The development of improved modelling capabilities is therefore of great inter-est to researchers, manufacturers and operators. The modelling effort is a focal point for fundamental research into the physical and chemical processes, development of improved operational and control strategies, improved retrofit configurations and primary design concepts. The approach to development of both physical and numerical modelling capabilities is to isolate important components of the overall process. Firstly, considering the isothermal flow due to the operation of the air system alone isolates problems associated with gross flow patterns. In fact the major motivation for this work are recent indications that Chapter 1. Introduction 2 the gross flow patterns can be inherently unstable for certain air system configurations. Geometric simplification is a necessary step to isolate problems associated with certain furnace cavity shapes and, in particular, portions of the air system, such as studying the flow induced by one level of a possibly three level configuration. Combustion can be studied in isolation from the full geometric and chemical complications and so on. In each of these sub problems there is valuable physical insight and understanding to be gained, which in many cases will be of more fundamental and wider interest to related fields of work in fluid mechanics, combustion and chemical sciences and engineering. The ultimate goal is to put all of these aspects together and improve the modelling capability for the entire process. By considering isothermal jet interaction for simplified geometry, this work is intended to address a fundamental problem in fluid mechanics and make a contribution to the development of recovery boiler modelling capability. 1.2 Background The recovery boiler is the largest single piece of capital investment in a pulp mill. Typ-ically, the technology and hence capacity of other parts of the pulping process improve, well within the service life of the recovery boiler, to the point where it becomes economi-cally very attractive to elevate the recovery boiler capacity. There are examples (Verloop et al. 1990 [45]) of improved boiler capacity of 5-15% through attention to detail in al-most every aspect of the machine and it's firing practice, including air delivery system upgrades which tend to involve higher speed jets that strongly interact with those from opposing walls. Figure 1.1 shows a schematic diagram of a B&W recovery furnace at the Weyerhaeuser pulp mill in Kamloops B.C. There are textbooks such as Adams and Frederick 1988 [1] Chapter 1. Introduction 3 which concern the components and processes of a complete recovery boiler plant. The complex geometry of a typical three level air system is evident in Figure 1.1. Primary air enters near the floor of the furnace through ports of the smallest dimensions, and greatest numbers, located on four walls. The penetration of the primary jets into the furnace is usually the weakest and they often control the perimeter of the molten smelt bed which accumulates on the floor. The secondary jets are larger and generally penetrate to the the furnace core. Together with the primary air, they generally provide the necessary air for char combustion in the bed and volatiles combustion in the lower furnace. The tertiary level provides additional air for completing combustion and can be important for flow pattern control higher in the furnace. The individual tertiary ports are the largest and are often fired from only two walls. There are often special designs used which may fire the tertiary jets in directly opposed or interlaced arrangements, and port sizes may be unequal for deliberate variations in jet penetration. Some designs fire the top level of air tangentially to produce a swirl in the upper furnace. The division of air between the various levels also varies between manufacturer and operators, but it is typical to find about 30 % primary, 40-50 % secondary and 20-30 % tertiary air splits. Air system configuration is currently a subject of great activity in the field and new design and retrofit concepts are appearing all the time. The many problems which can limit the throughput of recovery boilers are described by Grace 1984 [22]. Grace 1990 [23] discussed the major limits to throughput: airborn emissions and fireside fouling, both of which are primarily governed by the firing practice i. e. the introduction of air and fuel. Airborne emissions include oderous total reduced sulpher (TRS) and SO2 gases and dust loading, or opacity. These are subject to re-strictions based on what is acceptable to the local industry, community and government regulations. They result from insufficient combustion oxygen. In fact, furnaces could be run oxygen starved and raise throughput because the capability to process inorganic Chapter 1. Introduction 4 chemicals generally exceeds combustion capability. To maintain airborn emissions below acceptable limits requires 15-20% excess air, depending on the particular unit and the local restrictions. Under such conditions the combustibles lost in the flue gas are very low so there is little to gain on combustion and thermal efficiencies. Fireside fouling, or plugging, is caused by impaction of molten unburnt liquor or smelt particles entrained in the gas flow or deposition of microscopic inorganic fume particles on heat transfer sur-faces. The two critical parameters controlling fouling are carry-over rate and the furnace exit temperature proflle. In a region of the convective heat transfer section where the temperature is too high, carry-over and fume particles are more likely to stick to the surfaces. Airborn emissions and fouling can be reduced, or maintained within manage-able limits with increased throughput, via improved flow and mixing in the furnace. The improved flow and mixing must minimize particle entrainment, maximize residence time, improve combustion in the lower furnace, and give more uniform flow and temperature profiles at the exit. The average potential capacity gain is estimated at 20-25% beyond initial design specifications. 1.3 Posing the Mode l In order to study the role of opposing jet interaction in furnace flows, two simplified problems are considered. Firstly, two symmetrically opposed jets are injected normally into a uniform cross-fiowing stream of like fluid,as shown in Figures 1.2 and 1.3. The second problem concerns opposing jets issuing into a closed floor cavity, as shown in Figures 1.4 and 1.5. There is a two dimensional limiting case of two conceptually infinite opposing slot jets of height h (to —> oo), and the three dimensional case of two opposing discrete jets as illustrated in the figures. In the latter case, the cross-flow is bounded on either side of the Chapter 1. Introduction 5 jets by free slip impenetrable walls which may be interpreted as planes of symmetry in a conceptually infinite array of opposing jets. The spacing S between these boundaries is equivalent to the spanwise spacing between jets in the corresponding array. This target problem is a relevant step toward the study of jet interaction in furnaces at a secondary or tertiary air level, where there may be finite arrays of opposed jets on two or four walls. 1.4 Dimensionless Form Specific cases for the target problems are described by a set of parameters. The geometry is described in terms of a reference length scale equal to the square root of the flow area of one jet b=Vh^. (1.1) The geometric parameters are then the jet aspect ratio (AR=h/w) , spanwise spacing ratio (S/b) , cross-flow duct half-width ratio (B/b) , and duct length ratios (Di/b) (z = 1,2). A jet Reynolds number Rej = ^ (1.2) relates the jet reference length scale, jet velocity and fluid molecular viscosity. A final specifying parameter is the ratio of momentum flux of one jet to that of half the cross-flow; (hw)Uj SBUl Vs) [B] \%. (1.4) which can be simplified for a 2D slot jet where h becomes b, u; -^ oo[AR -^ 0) and S —^  oo; Chapter 1. Introduction 1.4.1 Mass Flow Rat io Dilution or mixing studies require an additional parameter to specify the ratio of mass flows in the jet and cross flow streams, (hwpj m M /M fUj\ SBUc \S) \B) \UcJ which can be simplified for a 2D slot jet where h becomes h, w —^ oo[AR —> 0) and (1.6) (1.7) S —> oo; - 5 ( I I- (-) The mass flow ratio, m, can be expressed in terms of the momentum flux ratio and the geometric paramaters as follows which can again be simplified for the 2D case as / . \ 1/2 m = ^ j M^/^ (1.10) 1.5 Object ive and Scope of the Work It is pointed out in the next chapter that results have been reported in the literature on furnace flows which strongly indicate some form of gross unsteadiness. The frequencies are rather low and quite uncharacteristic of the turbulence expected to be present in such a shear flow environment, and the amplitudes appear to be significant. In the works concerned, and in furnace operation in general, the boundary conditions for air jet velocity are expected to be essentially steady. It therefore appears that the unsteadiness must result from some kind of instability inherent to the internal flow regime of the Chapter 1. Introduction 7 furnace, or interaction thereof with the jet air delivery system. The obvious candidate is the collision, in the centre core of the furnace, of the many jets issuing normally and at typically high velocity from the surrounding walls. Indeed, studies of certain single jet trajectories in a furnace environment, reviewed in the next chapter, show that they may be influenced only slightly by the typical local cross flow strength due to the surrounding flow pattern. This would not be true of typical primary jets which are generally set up for shallow penetration to control the perimeter of the smelt bed, but is very realistic for some secondary jets, which generally pass over the top of the smelt bed, and for tertiary jets used higher in the furnace to add additional combustion air and break up channelling or core flow in the centre of the furnace. Such jets can easily penetrate the furnace and impinge on an opposite wall quite abruptly, in the absence of neighbors or opposites. When fired directly at one another, or even interlaced between opposing jets, there is likely then to be an abrupt collision and deflection of the flows. The role of the primary and secondary jets in creating a central core flow pattern, whether or not the tertiary jets break up the core and make the flow more uniform and better mixed, and whether or not these are even desirable flow characteristics are currently points of some contention in the field. However, it would appear that the jet collision, or impingement flow is a major determinant of furnace flow patterns and gives rise to a physical instability that seriously complicates the nature of the flow. The objective of this work is to study the interaction of opposing jets; to indentify any instability that can arise, and examine the possible flow states by considering t ime-dependant numerical modell ing for the equations of motion with a two equation turbulence model. The opposing jet problem is isolated from the greater complexity of the furnace geometry by considering only the simplified models of opposing jets in weak cross fiows and closed floor cavities. Furthermore, only isothermal modelling is being considered in the present scope of the work. It is shown in the literature review section Chapter 1. Introduction 8 that there exists a very sound foundation for using isothermal modelling in studying furnaces and related combusting flow phenomena. This is true despite the fact that there are density variations of a factor of about four between incoming air jets and hot furnace gases in the combustion zone. A good deal of success is reported using jet nozzle scaling, which attempts to correct for these effects in an isothermal model. Of course isothermal modelling is not the ultimate answer but it is an important stage for understanding the fluid mechanics. In addition to numerical modelling, this work includes a limited program of physical experiments designed to provide some degree of validation and additional insight into the physical phenomena. The physical, turbulent flows will contain a great range of finer scales of motion which can only be averaged and lumped together in the numerical modelling. While it is true that direct simulations of turbulent flows are now a reality, they are so expensive as to be beyond the scope of most practical analyses. The physical experiments are therefore considered to be an integral part of this work, however, they need only consider one characteristic case for comparison with a corresponding numerical case. The numerical modelling lends itself much more readily to the study of parametric variations in geometry and so on. Chapter 1. Introduction LIQUOR GUNvl OBSERVATION PORTS BULLNOSE TERTIARY STARTER BURNER SECONDARY PRIMARY RIGHT Figure 1.1: Schematic of the Kamloops Recovery Furnace. Chapter 1. Introduction 10 a h a V J m 'I Outflow I % Point 3 Point 2 Point 1 ^Y Figure 1.2: Schematic of Opposing Jets in Cross Flow; Vertical Plane. s x^— '^//////////////////////////////////////^ Point 2 • 11/ ' ^ i f Y V J 2B Figure 1.3: Schematic of Opposing Jets in Cross Flow; Horizontal Plane. Chapter 1. Introduction 11 a a V, •^ ^Outflow i Point 3 Point 2 • Point 1 • ; :^??^??^?^^?^^^^?^^?^ 2a 1/, z i x-^ ^ ey Figure 1.4: Schematic of Opposing Jets in a Closed Floor Cavity; Vertical Plane. X - ^ ^z S y////////////////////////////////////^^^^^ Y ^ ^ Point 2 IV r J 2B Figure 1.5: Schematic of Opposing Jets in a Closed Floor Cavity; Horizontal Plane. Chapter 2 Literature Rev iew 2.1 Furnace Flow Model l ing The most significant problem in isothermal scale modelling of furnace flows is the fact that in the real furnace combustion air is introduced via the jets at much lower temperature, and therefore higher density, than the hot furnace gases. Methods for overcoming this problem are discussed by Davison 1968 [13], including that of Thring and Newby 1952 [42] which is relatively easy to apply. The correction of Thring and Newby enlarges the dimensions of the air ports in an isothermal scale model by the factor {PolPfY •, the ratio of air jet density to furnace gas density, which enables the preservation of dynamic and kinematic similarity at the correct mass flow rate. It is geometric similarity near the jet port which is sacrificed, however, the typical distortion required is less than 2 and the region affected is small compared with furnace dimensions. Isothermal scale model results of Perchanok et al. 1989 [37], Blackwell et al. 1989 [6], Llinares 1989 [31] and Dykshoorn 1986 [18] demonstrate the general nature of furnace flows. The model of Perchanok et al 1989 [37] is of a wood waste fired spreader stoker furnace consisting of an under grate plenum air supply and a single level, two side-wall opposed jet array over-fire air supply, which is not very closely 2D because the jet arrays terminate some distance from the front wall to permit fuel delivery. The nozzle scaling of Thring and Newby is applied. Two distinct flow regimes are identified depending on the parameter M, the ratio of the momentum flux of the jet array on one wall to that of 12 Chapter 2. Literature Review 13 one half the cross flow duct width. At low M, the interaction of opposed arrays is weak. At high M values (from 10 to 32) the interaction of opposed jets is strong with regions of recirculation both downstream and upstream, and high turbulence intensity near the jet nozzle plane. The authors suggest that high M operation is best for good gas mixing and furnace performance. Comparison of jet trajectories to that of a single jet in unconfined cross flow is poor and emphasizes the importance of the confined, opposed jet interaction in furnace flows. The very high turbulent velocities (root mean square values) measured at the nozzle plane, and in particular toward the furnace exit which are approximately five times the mean flow velocity for the cross-section, are not indicative of truly steady mean flow with shear generated turbulence, but rather of a grossly unsteady, turbulent flow. The model of Blackwell et al. [6] is of an older Combustion Engineering Kraft Recovery furnace with a two level air system, the second of which is tangentially flred. Typical recirculatory flow is induced by the primary air level and persists to roughly the secondary air level. Velocities measured by hot wire anemometer (HWA) vary "from second to second" and maximum, minimum and typical values are determined over an observation interval of approximately 60 s. These measurements are reported for two horizontal planes, one at the liquor gun level between primary and secondary air, and the second at the furnace exit. The typical values reflect the recirculatory flow at the liquor gun level, a central upflowing core with downflow near the walls, and more uniform upflow at the exit. The ratio of (max-min)/typical velocities is 29-133% at the liquor gun level and 46-250% at the exit. Corresponding measurements in a cold flow test on the actual furnace are higher (24-280% and 63-650% respectively) than in the model. These numbers indicate grossly unsteady flow. There are numerical model results for the Kraft Recovery furnace by Jones 1989 [24] and of similar pulverized fuel furnaces by Robinson 1985 [38] and Fiveland and Wessel Chapter 2. Literature Review 14 1988 [19]. These models include 3D control volume formulation, first order accurate hy-brid and power law differencing and combustion models. Predicted gas temperatures and wall heat flux capture overall trends in available experimental data, but are inaccurate in predicting details. Results of additional isothermal flow simulations by Jones 1989 [24] of an existing configuration and one with more numerous, and higher velocity tertiary or over-fire air jets are compared to the corresponding isothermal scale model data of Dyk-shoorn 1986 [18] with qualitative agreement. The Thring Newby criterion is not applied by Jones 1989 [24], so that kinematic and dynamic similarity are not preserved, but it is used by Dykshoorn 1986. The numerical models discussed assume steady flow and that of Jones 1989 [24] assumes symmetry where there is geometric symmetry by modelling one half of the furnace. 2.2 Fluid Amplifiers The technology of fluid amplifiers or fiuidic devices began with a device described by Yeaple (1960) [48] that was developed at Harry Diamond Laboratories ( then Diamond Ordinance Fuse Laboratories ) in Washington D.C. It can switch a main stream or jet of fluid between different output chambers by means of a weak control stream. The concept had the advantages, over mechanical switching, of no moving parts to wear and ease of construction. A number of control applications were suggested: com-puters, automatic washing machines, machine tools, missiles and aircraft. Rapid development of the technology followed for ten years or so. The ASME held a symposium on fluidic devices in 1967 (Brown 1967) [11]. An analysis of the literature and bibliography was published by Brock (1968) [10] of the British Hydrodynamics Research Association. It appears now that in many applications, however, digital electronic am-plifiers and switches have become more favorable. The technology is still active, though, Chapter 2. Literature Review 15 and works are reported in the Journal of Fluid Control, published by Delbridge, which encompasses fluid control. Hydraulics and Pneumatic Instrumentation, and fluidics. The bistable jet inherent in the original devices is exhibiting a bifurcation phenomenon with two stable states. The term wall at tachment is used in the literature to describe how the main jet is attracted to either of the two walls which are continuous between the input main stream and either output chamber. Presumably there is a theoretical, symmetrical flow solution, in which fluid issues equally into both outputs simultaneously, which is unstable and never realized in practice. 2.3 Jets in Cross Flow The study of jets in cross flow includes a single jet or an array of jets discharging at some angle into a stream of like fluid (which may have different properties such as temper-tature, density etc. ) which may be effectively unconfined or conflned by a cylindrical surface of arbitrary cross-section. Examples of these phenomenae are found in chimney plumes, gas turbine combustors and blade cooling, dispursion of effluents in rivers and estuaries, v/stol aircraft and recovery and similar furnaces. A comprehensive review of experimental, analytical and numerical work in the field was compiled by Demuren 1986 [15] . Of particular interest with respect to furnace flows are confined and opposed arrays of jets. The author contributed to a report for Energy, Mines and Resources Canada, undertaken by Blackwell et al. 1990 [5] of the consulting firm Sandwell Swan Wooster in Vancouver B.C., which reviews literature relevant to air jets in recovery boilers. Daniel Tse (1994) [43] also reviews much of the literature relevant to single side jet injection into a cross flow and reports results for wide slot jets and arrays of rectangular jets, at various spacings, in confined cross flows and furnace-like cavities. There are experimental data on the confined single jet in cross flow [41, 26, 20, 12], Chapter 2. Literature Review 16 the confined array [26, 28], the confined 2D or slot jet [33] as well as the directly opposed single jet and jet arrays [26, 3, 29]. Corresponding computational model results are also available for the confined single jet [25, 12, 4], the confined array [28, 14], the nearly 2D jet in open channel flow [32, 16] and a model of an opposed array of laminar jets in asymmetric geometry [39]. Atkinson et al. 1982 [3], in a study of opposed jets in cross flow, found that apparently symmetric geometry does not necessarily give a symmetric flow field and noted that this might be a serious problem in practical applications such as combustors. A flow visu-alization experiment by Sivasgaram and Whitelaw 1986 [40] of opposed arrays without cross flow shows that small asymmetries in jet speed or geometric alignment, e. g. a 7% diff^erence in jet speed, causes large asymmetries in the flow field. All the computational models mentioned employ a control volume formulation, similar to a finite difference method, with a one or two equation turbulence model. Differencing schemes include the hybrid central/upwind scheme [25, 28, 32, 4], the power law scheme [12] and both the hybrid and quadratic upstream weighted (QUICK) schemes [14, 17, 16, 4]. The QUICK scheme is said to eliminate the second order truncation error terms which cause a numerical or false diffusion in the upwind scheme, and thus creates a second order accurate method. The QUICK method predicts significantly narrower jet velocity and jet fluid concentration profiles within the first one or two duct widths downstream which agree better with experimental observations. An interesting experimental technique in the recent literature is that of Vranos and Liscinsky 1988 [46], called planar nephelometry, used in a study of essentially unconfined jets. The method uses digitized vidicon camera images of laser illuminated planes in the flow field containing light scattering aerosol particles injected with the jet fluid. This high speed imaging system can provide both the mean and fiuctuating parts of the jet fluid concentration field. Chapter 2. Literature Review 17 2.4 Opposing Je ts without Cross Flow A study of reaction injection moulding (RIM) of polymer parts by Wood et al. (1991) [47] demonstrates the nature of opposing jet instabilities. Two cylindrical, laminar jets are impinged head-on near the closed end of a cylindrical mixing chamber. Jet Reynolds numbers (Rej) are between 50 and 300. The jet diameter is one tenth of the mixing chamber diameter and the mixing chamber has a long exit length of 9.5 times its diameter. The closed end is realized by a moveable piston located at between one half and one chamber diameter away from the jets. While the laminar regime of this flow is very different from the high Reynolds number, turbulent flow in a recovery furnace, the relative geometry of the jets and chamber are similar to that of a recovery furnace. Flow visualizations are reported in which the flow is seeded with polystyrene spheres and observed by illuminating the model with a laser sheet passing through the jet and chamber axes. Axial and lateral velocity measurements are made by laser doppler anemometry. Above a threshold value of Rej — 75 the flow becomes unsteady. This is evident first in the downstream region of what the authors describe as the radial jet or "pancake" of fluid flowing away from the jet impingement region, which begins to "meander" back and forth slightly, with a long wavelength. Above Rej = 90 there is a more pronounced oscillation with significant lateral motion. At Rej = 135 the jets no longer remain impinged on each other but move back and forth to either side or above and below one another, impinging on a far wall, and occasionally reimpinging on each other. Above Rsj = 150, the jets remain off centre and never reimpinge on one another. In the range of regular oscillations 90 < Rej < 135 there is a fairly well defined dominant frequency. A Strouhal number {St) based on this frequency, jet velocity and size varies with Rej and is of order 0.01. Chapter 2. Literature Review 18 Numerical calculations of this flow show very similar results, although at higher val-ues of Rej up to 300 the jets never miss one another completely, as observed in the physical experiments, and the more regular oscillation remains. At Rej = 125 the com-puted Strouhal number is very close to the experimental value. The computed St rises with increased Rej but appears to level off at a sufficiently high value, suggesting Rej independence at higher values. This configuration of opposing jets in a chamber closed at one end is similar to opposing jets in cross flow where the cross flow is weak or zero compared to the jets. The case of zero cross flow does not necessarily imply that the chamber is closed, however, as it may be open at both ends. Fluid could then flow to either exit. The effect of even a very weak cross flow is to limit the penetration of jet fluid in the upstream direction, and in this sense it is perhaps more similar to the case of zero cross flow in a chamber closed at one end. It is reasonable then to expect that the results of Wood et al. (1991) [47] might be similar to cases with a weak cross flow. The fact that this type of instability appears in a laminar flow regime suggests that fluctuations observed in a turbulent furnace flow might not be a turbulence phenomenon but could be associated with a mean component of the flow distinct from the turbulence. This mean flow can be distiguished from the turbulence if it has a characteristic time scale longer than the longest scales in the turbulence. In this regime a turbulence field affects the mean flow in the form of an effective viscosity which is in general non-uniforn and anisotropic. The flow may then be similar to a laminar flow where the viscosity is of the same magnitude as the effective viscosity level on average, or in key regions of the turbulent flow. The turbulent viscosity fields can be non-dimensionalized using the form of a Reynolds number based on the characteristic velocity and length scales, the jet velocity and size for example. It can then be interesting to compare the flow behaviour to that in laminar, or constant viscosity flows, at similar, true Reynolds numbers. Chapter 3 C F D M e t h o d This chapter describes the computational fluid dynamics (CFD) modelling. The under-lying governing equations for the Reynolds mean flow with & k — e turbulence model are presented first. The UBC-MGFD finite volume numerical code is then outlined. This is a multi-grid code under development at the University of British Columbia, a steady state version of which was modified by the author for time-dependent flow. The multi-grid procedure for convergence acceleration is described in more detail, from the point of view of the user, as the convergence performance is critical for the large, transient problems considered here. The user must set up the multi-grid procedure to be effective for a given problem and assess the effectiveness of the method and the degree of convergence required. Some theoretical limits on multi-grid performance are also discussed. 3.1 Mathemat ica l Mode l The mathematical model consists of the Reynolds equations for the ensemble mean flow coupled, via the effective viscosity hypothesis, to the k — e turbulence equations. This has become a rather standard set of equations for engineering computational fluid dynamics (CFD). For this work, the flow is assumed to be incompressible and of constant density. The model equations are breifly described here. The Reynolds equations consist of the mass conservation, or continuity equation and the momentum conservation equations for the ensemble mean flow. Subscripts {i,j) refer to any of the three cartesian coordinates (1,2,3) or (x,y,z) and repeated indices imply a 19 Chapters. CFD Method 20 summation; dUj dxi dU, , , dUi 1 dU d 0 \dxj dxi 0 (3.1) (3.2) dt dxj p dxi dxj where the effective viscosity is incorporated as the sum of laminar and turbulent viscosi-ties (Ueff = {v + Ut)). The turbulence model consists of two Reynolds type equations for the specific turbu-lent kinetic energy, k, and its rate of dissipation, e, as proposed by Launder and Spalding (1974); d ^ U — - (^ dUAdUi dt ^ dxi \dxj dx{ } dxi dx {v + Vt) dk CTk dxj + e = 0 de ^^ de ot dxj ^ e (dUi dUj\ dUi d I (J _ g k \ dxj ' dxi J dxj dxj a^ dxj k {v + Vt) de The turbulent viscosity is calculated from k and e; Vt = C^k'^le. The constants in this equation are summarized in Table 3.1. Table 3.1: Constants for the k — e model. (3.3) (3.4) (3.5) c. 0.09 Ci 1.44 C2 1.92 o-* 1.0 0"e 1.3 For three dimensional flow these equations result in a coupled system of six nonlinear, partial differential equations. An iterative numerical solver is therefore required. 3.2 Numerical M e t h o d The numerical method is based on the the UBC-MGFD code written by Paul Nowak (1984,1991) [34, 35] for steady state flow. The code is modified for time-dependent flow Chapter 3. CFD Method 21 in this work. The equations are discretized by a finite volume method on a staggered grid, using a power law weighting of first order upwind and second order central differences for the nonlinear convection terms, and second order central differences for the diffusion terms as discussed by Patankar (1980) [36]. A fully implicit first order time step is used which guarantees stability in the time integration. A line relaxation version of Vanka's method (1986) [44] is used, which is simultaneous in velocity and pressure. In Vanka's method, a matrix equation is defined in terms of one unknown pressure and four unknown neighbor velocities, together. This is quite distinct from the pressure correction algorithms, such as the SIMPLE method of Patankar and Spalding (1980) [36] which decouples the calculation of velocity and pressure fields at each iteration, and uses point-wise equations. The iterative multi-grid method is the full approximation scheme (FAS), as proposed by Brandt (1977, 1980) [8, 9]. This is outlined in a following section. The duct walls are specified with a free slip condition in much of this work. This is done to keep the model simple, as the boundary layers will be thin, giving little displacement effect from the walls, and the wall shear stress may not affect the internal pressure field appreciably over the short lengths of duct in which the jets interact. The effects of wall shear stress are investigated for certain cases, however, using the wall function method of Launder and Spalding (1974) [30]. The free slip condition is also applied at the boundaries, or planes of symmetry, on either side of the jets and, in the case of an enforced symmetry calculation, on the geometric symmetry plane normal to the jets. Inlet velocity profiles are uniform, and two alternative treatments are applied at the exit. The first is a zero gradient condition which amounts to a fully developed flow assumption, and requires a suitable length of duct. The other treatment prescribes a uniform exit velocity profile. This outlet condition is the simplest way to allow use of duct lengths comparable to real furnaces, which are too short for a fully developed flow Chapter 3. CFD Method 22 assumption at the exit. In a real furnace, the pressure drop across the convective heat transfer section tends to make the flow more uniform, so the uniform flow assumption is believed to be a good one for the present purpose. Inlet turbulence quantities, k and e, are defined in terms of turbulence intensities {Tlin) a^ nd length scales (4n); hn = liTUnfUl (3.6) 1.3/2 e.n = -f-. (3.7) Hn Values used are Tlin equal to 10 % and 4n equal to the square root of the area of one jet for the jet flow, and one half the cross-flow duct width for the cross-flow. Results are not very sensitive to the turbulence boundary values, since turbulence generated by shear within the calculation domain is dominant. 3.3 Multi-grid Procedure The multi-grid procedure is applied to a steady state problem, or at each t ime step of a transient problem which uses an implicit time integration scheme. Following the notation of Ascher (1991) [2], the general, nonlinear set of governing equations, described in a preceding section, are denoted by K(u,ii°) = 0 (3.8) where ^^  is a nonlinear operator, u is the exact solution, and u° is the solution at a previous time step. When discretized on a grid of characteristic size /i, we have )Xh{uHXH) = ^ (3-9) where Uh is now the exact solution to the discretized equations. Chapter 3. CFD Method 23 Relaxation sweeps, or smoothings, are performed using the line method of Vanka (1986) [44], already discussed. Relaxation of this type is an inherently local process and therefore reduces errors of high spatial frequency effectively, but is less effective in reducing errors having relatively low spatial frequency, or long wavelength in comparison to the grid spacing h. As relaxation proceeds the effectiveness of each sweep diminishes. The approximate solution, Uh, is associated with a defect, d^, defined by MUh,ul) = -df, (3.10) where the ' —' sign is for convenience. One seeks a correction, 14, such that Uh = UH + VH (3.11) which will satisfy a correction equation obtained by substituting Equation (3.11) into (3.9) and subtracting (3.10), MVh,K) = MUh + VH,UI) - MUh,ul) = 4 . (3.12) The idea is to then find an approximation to this correction problem MVh,ul) = MUH + Vh,ul) - MUH,UI) = k- (3.13) which is easier to solve and, as part of an iterative process with relaxation of the governing equations on the finer grid, will give convergence toward the exact solution Uh. As the defect, dh, is associated with errors of low spatial frequency, the correction problem can be more efficiently solved on a coarser grid, H, so that UH^UH VH^-VH Chapter 3. CFD Method 24 ^H'—^h dn <— dh where coarse grid representations of the variables, such as Ujj, are obtained by a suitable method, termed a restriction, which usually takes a geometric weighting of neighboring fine grid values. The coarse grid correction problem is written then as ^HiVH,u°H) = MUH + VH,U°H) - '^H{UH,U°H) = dn. (3.14) Relcixation of the coarse grid correction problem then proceeds, with a sweep cost of about 1/4 for a 2D problem, and 1/8, for a 3D problem, of the cost for a finer grid relaxation sweep. As the correction problem is only an approximation, it is only sensible to solve it approximately numerically before interpolating, or prolongating the correction, Vjji back to the finer grid and updating the solution UH^UH + Vh. (3.15) The prolongation of V will introduce some new high frequency errors to Uh, so further relaxation of the governing equations is required after adding the correction, to complete one coarse grid correction cycle. The coarse grid correction problem has associated with it another defect, which should be made at least an order of magnitude smaller than the current defect, dh, in the governing equations on the finer grid. If the coarser grid, H, is still rather fine, relaxation of the correction problem may again be unacceptably slow, and the correction procedure can be applied again. In this way, the procedure becomes nested with an arbitrary number of grids. The finest grid problem solves the actual governing equations and is denoted by the correction level of 0. The first correction problem is denoted by correction level 1, and the second correction Chapter 3. CFD Method 25 problem, providing a correction to the first correction problem, is denoted by correction level 2 etc. The multi-grid procedure is illustrated by the triangle diagram of Figure 3.1, where the vertical side represents grid levels from coarse to fine, and the horizontal side repre-sents correction levels. One multi-grid cycle, or iteration, starts on a fine grid with an initial number of relaxation sweeps, or smoothings . This is followed by restriction to a coarser grid level and first correction level. In the case of only two grids, there is only one correction level and the resulting corrections are subsequently prolongated back to the fine grid, where the corrected solution is again smoothed a number of times. In the case of three grids, there are two levels of correction etc. The multi-grid cycle is repeated until the desired level of residual, or defect, is obtained for the solution to the governing equations on the finest grid. Also shown on the triangle diagram of Figure 3.1 is a start-up procedure, often used for steady state problems, in which the governing equations are solved first on a coarse grid. The solution (not a correction) is then prolongated to a finer grid as an initial guess for the solution there and so on. The complete multi-grid procedure is also illustrated by the V-diagram of Figure 3.2, where the progression is from left to right. The circled nodes represent the ends of loops which may be repeated. The multi-grid paramaters, as denoted in the UBC-MGFD code by Nowak (1991) [35], are indicated on both Figures 3.1 and 3.2. Typical values of these parameters are summarized in Table 3.2 for the steady and transient 3D calculations done in this work. Chapter 3. CFD Method 26 3.3.1 Multi-grid convergence rates Convergence for the numerical method is measured in terms of the residuals remaining in the discretized governing equations The computer work done in reducing the residuals is measured in terms of the cost for one relaxation sweep on the finest grid used, defined as one work unit (WU). The actual computing cost for a work unit depends on the number of operations required which is proportional to the number of nodes, N, on the finest grid. The cost of one work unit is therefore said to be of order N, or 0(N). Work done on a coarser grid is expressed in terms of the same work units, so that one relaxation sweep will cost about 1/8 WU for one level lower, as there are 1/8 the number of nodes there. The work required to achieve a given level of residuals, or convergence level, is proportional to the number of multi-grid iterations times the number of operations per multi-grid iteration. There is theory in the literature for linear problems which states that as the number of grid nodes on the finest grid, N, increases, the number of multi-grid iterations required to reach a given convergence level approaches a constant. For a rather coarse finest grid, this number will generally be smaller and increase for finer and finer finest grids to approach a constant number. Furthermore, the number of operations required per multi-grid iteration approaches a constant also, i.e. the number of relaxation sweeps at each grid level becomes independent of N. Therefore the computational load approaches a constant number of iterations, at a cost of 0(N) operations each, giving a total cost of order N. This result for linear problems is known in the literature as the order N algorithm, or 0[N) algorithm. For the same linear problem, relaxation on the finest grid only will require 0{N) iterations, at a cost of 0{N) operations each, giving a total cost of O(N^) operations, e.g. Gauss Seidel relaxation. The order N algorithm is Chapter 3. CFD Method 27 Table 3.2: Multi-grid parameters in the UBC-mgfd code. Description No. smoothings for solution on coarsest grid No. smoothings for corrections on coarsest grid, of less than highest order No. of smoothings for corrections of highest order No. of initial smoothings for solutions on fine grids, except finest No. of initial smoothings for solution on finest grid No. of initial smoothings for corrections on fine grids of correction levels No. of smoothings after adding a correction No. of multi-grid steps for solutions on fine grids, except finest No. of multi-grid steps for solutions on finest grids No. of multi-grid steps for corrections Name nslO nslc nsh nsO nsfO nsfc nsac nmO nmfO nmc Steady 400 50 50 10 10 10 10 8 8 1 Transient N /A N / A 50 N / A 10 10 10 N/A 1 1 a useful concept even when working with nonlinear problems, as in this work, because it represents the best possible convergence performance. Chapter 3. CFD Method 28 nsfO CD > CD -J CD nsO ns10 0 1 2 Correction Level Figure 3.1: Multi-grid Triangle Diagram. nsfO nsac nsO ns10 nmO V nmc(1 = V,2=W) y nmfO Figure 3.2: Multi-grid V-Diagram. Chapter 4 Opposing Jets in Cross Flow This chapter presents results for two and three dimensional cases of opposing jets in cross flowing streams. The two dimensional cases are based on conditions in a waste wood funace, the Number 4 Power Boiler at the Woodfibre pulp mill near Squamish B.C. This boiler was the subject of cold flow physical modelling work at U.B.C. by Perchanok et al. (1989) [37], a collaborative effort with the consulting firm of H.A. Simons in Vancouver B.C. who continued to support this work also. This furnace is a spreader stoker type with opposed jets on two walls and under grate air from below. The 2D cases only approximate this configuration. The three dimensional cases are based on conditions in an operating B&W recovery furnace at the Weyerhaeuser pulp mill in Kamloops B.C. Recovery furnaces are the main focus of attention for the work at UBC, due to their economic siginificance in a pulping operation and the concurrent interest and support from industry and governments for research. A scale water model of the Kamloops furnace, originally built by Weyerhaeuser Co. in Tacoma Washington, is at U.B.C. and is used for the physical experiments done in this work. 4.1 2D Reference Case 5 / 6 = 20, Rej = 5.6 x 10^ M = 25 The two dimensional case represents symmetrically opposing slot jets, of height b, pene-trating a cross flow in a duct of width 2B/b = 40. The duct extends upstream a distance of Di/h = 23.0, where the cross flow is specified with a uniform inlet profile. This up-stream distance is made long enough to not disturb the resulting interaction with the 29 Chapter 4. Opposing Jets in Cross Flow 30 jets significantly. The exit plane is located at a distance downstream from the jets of D2/b — 77.22, or D2/2B = 1.931, which corresponds to a point mid way between the mud and steam drums of the Woodfibre No. 4 power boiler, and a uniform exit velocity profile is specified there. The no slip condition is applied to the duct walls using the wall function method of Launder and Spalding (1974) [30]. The numerical grid has 48 nodes along the duct axis and 40 along the jet axis. The two dimensional TEACH code of Gosman [21] with the SIMPLE algorithm [36], hybrid differencing and a single grid is used. At the momentum flux ratio of M = 25, a symmetrical flow field exists, but it can only be computed by enforcing symmetry mathematically at the geometric symmetry plane. The result is shown in Figure 4.1, where the results for the half domain computation are reflected to the other half for plotting. This halving of the computation domain to exploit symmetry is often used in CFD practice to reduce costs. However, in the case of full domain computations there is a bifurcation, and there exist two asymmetric steady flow fields, mirror images of one another with respect to the duct walls. Figure 4.2 shows one of these. The most important features of the 2D flow fields in Figures 4.1 and 4.2 are the two spanwise recirculation zones adjacent to the walls, downstream of each jet, which are associated with reattachment of each jet to its respective wall. Also, the jets do not collide head-on, as they are rapidly deflected by the cross-flow fluid which must pass between them in a 2D flow. It is found that the bifurcation is quite sensitive to the computational grid size; if the grid is too coarse, the symmetric steady flow will be stable and the bifurcation will not appear. For a case of similar geometry at M = 10, using the UBC-MGFD code and two-level multi-grid, the initial solution on the coarse grid of 20 x 20 nodes is symmetrical and stable. Chapter 4. Opposing Jets in Cross Flow 31 4.1.1 Variations in M o m e n t u m Flux Ratio The importance of momentum flux ratio is demonstrated in Figure 4.3, which shows the resulting reattachment lengths of the two jets for a series of runs over a wide range of momentum flux ratio. These results are from work using the TEACH single-grid code. There is a zero gradient condition applied at the exit which must consequently be placed at a large distance from the jets; far enough away so as not to influence greatly the resulting flow and reattachment lengths. The length used here is D2/2B = 6.921. The computational grid has 48 nodes along the duct axis and 40 along the jet axis direction. At low momentum flux ratios, of less than M = 2, the jets interact very weakly, or not at all, and each jet is deflected by the cross-flow in a symmetrical manner, i. e. the flow field obeys the symmetry of the firing geometry. Above a critical momentum flux ratio, M « 2, the jets interact more strongly, resulting in a bifurcation. The plot of Figure 4.3 includes also the enforced symmetry reattachment lengths for reference. The enforced symmetry reattachment lengths reach a maximum value at about M = 15, and are apparently limited in size by the duct. The longer of the asymmetric reattachment lengths, however, is still growing at the maximum computed momentum flux ratio, M = 25, because the duct is twice as wide in the full domain calculations, and it appears to be approaching roughly twice the limiting enforced symmetry reattachment length. These results indicate that the confined flow pattern shape can become independent of the relative strengths of the fluid stream inputs, a result reported for single side jet injection into a cross flow in experiments by Mikhail et al. (1975) [33]. This can not happen for a single jet issuing into an infinite, unbounded cross fiow, where the reattachment length will grow indefinitely with jet momentum flux. Notice that the longer reattachment length at M = 25 of L2IB = 7.7, in this very long duct, is about twice as long as the shorter downstream duct length, D2IB — 3.862, used for the results of Figures 4.1 and 4.2 to simulate the Chapter 4. Opposing Jets in Cross Flow 32 realistic furnace dimensions of the Woodfibre No. 4 power boiler. The shorter duct length therefore limits the long reattachment significantly. 4.1.2 Transient Perturbat ions and Stabil ity Analysis If the enforced symmetry result of Figure 4.1 is used as an initial guess for a full domain transient calculation, the symmetry remains undisturbed by the solver. Perturbing the jet velocities slightly, however, will then yield a transient response leading to one of the asymmetric steady states. This is shown in Figure 4.4 for a 0.5 s duration, ± 5 % Vj pulse disturbance in the jet velocities applied at t = 0. The calculation uses a real t ime step of 0.1 s, where the actual Woodfibre No. 4 power boiler depth of 2B = 5.18m is used, and 80 time steps for an 8.0 s transient simulation. At the 81st t ime step indicated, the time step is actually reset to infinity to confirm convergence to a steady state. An asymmetric steady state is reached relatively quickly, in about 5.5 s, with no oscillation or overshoot in the reattachment lengths, suggesting that the asymmetric states are very stable. The stability of the asymmetric states is then tested directly by applying pulse per-turbations, as shown in Figures 4.5 and 4.6. Figure 4.5 shows that a 0.5 s ±20% Vj pulse only disturbs the flow weakly, and the same asymmetric state is recovered at about 3.0 s. The longer duration pulse of Figure 4.6, a 2.0 s ±20% Vj pulse, is sufficient to completely reverse the asymmetry with the opposite state being reached at about 5.5 s. 4.1.3 Je t Tracer Analysis A passive scalar transport equation is solved by numerical computation to observe the dilution of the jet fluid with that of the cross flow, inside the duct. The equation is of identical form to the governing equations for the flow variables, with only the convection and difl"usion terms present, and a turbulent Prandtl number, (7$, equal to 1.0, where the Chapter 4. Opposing Jets in Cross Flow 33 subscript $ denotes mass fraction of the jet fluid in the flow or je t tracer concentra-tion. Boundary values are 1.0 at the jet inlet and 0.0 at the cross flow inlet. The solid boundaries require only a zero gradient treatment. The scalar transport equation is analagous to the energy equation for small tem-perature variations (hence negligible density variations or buoyancy forces) where solid boundaries are considered to be adiabatic (a zero gradient condition on heat flux there). One can think of the mass fraction or tracer concentration level at a point where the two fluid streams have partially mixed, then, as being analagous to a local, dimensionless mean temperature parameter deflned by ^ = 1 ^ (4-1) where T is then the local, Reynolds mean temperature of the flow as a result of jet dilution. If the jet and cross flow streams can be mixed completely and uniformly at some point downstream in the duct, then the jet fluid mass fraction or tracer concentration will equal the bulk value, ^Buik = ^ ^ ^ (4.2) ^ (4.3) 1 + ^^ mj (4.4) " 1 + i m where m is the mass flow ratio (jet to cross flow). For the present 2D case of B/b = 20 and M = 25, equation 1.10 gives m = 1.118 so that ^Buik = 0.5279. The jet tracer dilution results are plotted as iso-contours for the enforced symmetry and asymmetric full domain computations in Figures 4.7 and 4.8 respectively. The geo-metric flow patterns are clearly reflected by the concentration contours. The degree of Chapter 4. Opposing Jets in Cross Flow 34 mixedness in the vicinity of the jets is similar for both the flow states, with significant gradients observed there. $ approaches the uniform bulk value towards the exit, showing that the flows become well mixed within the residence time inside the duct. 4.2 3 D Reference Case AR = 4, S/b = 5, B/b = 10, Rej = 10^ M = 20 This case approximates conditions at the tertiary level in a B&W recovery furnace at the Weyerhaeuser pulp mill in Kamloops B.C. There is a scale water model of this furnace at UBC, which is used in the physical experiments of this work. The computational domain (Figures 1.2 and 1.3) extends upstream Di/b = 18 units where the cross flow is prescribed as a uniform stream. This length is made just sufflcient to not disturb the resulting interaction with the jets, and is only 0.56 longer than the maximum depth to the floor of the Kamloops furnace. The exit plane is located at -D2/6 = 55, or D2/2B = 2.75, which corresponds to a point mid way between the mud and steam drums of the Kamloops furnace. The standard numerical grid has 32 nodes along the jet axis, 14 along the spanwise axis and 48 along the duct axis, defined as the x, y, and z directions respectively. Two level multi-grid is used, so a coarser grid of one half the number of nodes in each coordinate direction is also used in the calculations. 4.2,1 Steady State Computat ions As found in the 2D work, for this case at M=20 a symmetric steady state fiow can only be obtained by enforced symmetry calculations. Figures 4.9 and 4.10 show this result in two views, where the results calculated for one half of the domain have been refiected about the geometric symmetry plane for the plot. This flow is signiflcantly diff'erent from the 2D results, in that there is a strong, head-on impingement of the jets. This can occur because the jets are deflected less by the cross-flow, which is able to flow around either Chapter 4. Opposing Jets in Cross Flow 35 side of the jets in a 3D flow. From the impingement region there is radial flow outwards in all directions, forming core flows directed both upwards and downwards, with the downward core penetrating quite far upstream against the cross flow. This produces some spanwise recirculation upstream of the jets, however, owing to the three dimensionality of the flow there is no reattachment of the jets to the walls as in the 2D flows. This flow pattern is qualitatively very similar to the laminar reaction injection moulding (RIM) flow fields described by Wood et al. 1991 [47], for cylindrical geometry, and Reynolds numbers in the vicinity of 100. The convergence performance of the UBC-MGFD code for the enforced symmetry flow is shown in Figure 4.11 which compares two-level multi-grid and single-grid computations of the solution on the 32 x 14 x 48 level two grid. The flgure plots root mean square (RMS) residuals for each of the equations, non-dimensionalized with respect to the jet velocity, Vj, and size, b, versus work units (WU), a measure of the computer work required. One work unit is the computer work required to make one relaxation sweep on the finest grid, the level two grid in this case. In the case of multi-grid computation, the work done in each relajcation sweep on a coarser grid is expressed in terms of the same units, with respect to the finest grid, for consistency. One relaxation on a grid that is one level below the finest then requires approximately 1/8 WU, as there are 1/8 the number of nodes there. The multi-grid convergence is clearly better than that for the single grid computation. On the semi-log scale used the multi-grid convergence lines are decreasing nearly linearly with WU, for a nearly constant logarithmic convergence rate, but the logarithmic convergence rate of the single-grid computation worsens with continued smoothing. In full calculations through both jets, there is no convergence at all when starting from an arbitrary initial guess, which suggests that the symmetrical flow is unstable. Unlike the 2D work, stable steady state flow fields can not be found. For repeated iteration Chapter 4. Opposing Jets in Cross Flow 36 of the solver, the root mean square residual error in the equations will not decrease below a level of about 10~^, made dimensionless with respect to Vj and b. At successive iterations the current flow field will be asymmetrical and varying from one side of the duct to another. 4.2.2 Transient Computat ions A time dependent run initiated from one of these unconverged steady state runs does converge rapidly to lower residuals. A dimensionless time step At* = At~ (4.5) 0 of 10.0 is used and two multi-grid iterations are performed at each time step. The root mean square (RMS) dimensionless residual error at the end of each time step ranges from 10~® to lO"'^, or three to four orders of magnitude lower than for the unconvergent steady state solver runs. It should be noted that a normalized residual error magnitude of 10"^ is quite typically used as a convergence threshold in CFD practice. In this case it is not sufficient. Furthermore, it is essential to confirm that the problem is truly convergent, i. e. that the residual error does not stall at a given level but will continue to decrease with continued iteration. The transient run from start-up is shown as a time series in Figure 4.15. This figure plots velocities on the duct geometric centre-line, at the geometric jet impingement point (point 2), 26 upstream (point 1), and 46 downstream (point 3). The eventual result is a periodic solution which is shown as time series plots in Figure 4.16. Figure 4.16a) plots U velocity components, parallel to the jet axis, and figure 4.16b) plots W velocity components, parallel to the duct axis. The oscillation period for the U velocities is T: = T^ (4.6) Chapter 4. Opposing Jets in Cross Flow 37 equal to 422.2 (dimensionless). The W velocities actually oscillate at twice this frequency, or half the period. The inverse of this quantity is a dimensionless frequency or Strouhal number S4 = i = /A (4.T) equal to 0.237 x 10"^ (dimensionless). This is very low compared to typical turbulent motion. In fact, even the Strouhal numbers for the dominant frequency of oscillation in the laminar RIM flows reported by Wood et al. (1991) [47] are an order of magnitude higher. In the Kamloops recovery furnace, for example, with a tertiary jet velocity of 69.2 m / s and b=0.567 m, after Thring Newby enlargement of the air ports for isothermal modelling (Thring and Newby 1952 [42]), one has a real t ime step of 0.082s and oscillation period of 3.5s. 4.2.3 Dimensional Analysis for the Time Scale A time scale more representative of the oscillation period can be defined from the jet momentum flux J = b^Uj, and cavity dimension B, giving B^/J^'^ (s). It is related to the original jet t ime scale by ^' " ' ^ ' ^ (4.8) Jl /2 \hJ Vj where {B/hy is equal to 100.0 in the present case. The oscillation period is then expressed as J l / 2 n = T^^ (4.9) equal to 4.22 (dimensionless). The corresponding Strouhal number can also be redefined as StB = f ^ , (4.10) equal to 0.237. This time scale may be more generally applicable as it does not involve the jet velocity and size specifically, and therefore can be defined for limiting cases of Chapter 4. Opposing Jets in Cross Flow 38 very small jets approaching point sources of momentum flux, J, issuing into a very large furnace. This is probably not entirely accurate, but it is an interesting limiting case, and the relative jet size b/B can always be added again as an additional parameter to complete an analysis. 4.2.4 Analysis of Transient Resul ts and T ime Averaging Figures 4.18 and 4.19 show transient flow fields at times when U2, the U velocity at the geometric impingement point, is at a zero crossing and a peak respectively. The impingement point where the jets collide is moving back and forth across the duct and the core flows moving away from the impingement point fluctuate from side to side also. The core flows moving downward below the jets do not penetrate as far against the cross flow as in the enforced symmetry flow, and generate alternating larger and smaller spanwise recirculations there. The core flows above the jets now produce similar spanwise recirculations also, which grow alternately on either side of the core and are convected away downstream. Notice from the time series of Figure 4.16 that centre-line U velocities at the three points are out of phase with one another; Ui is leading and t/3 is lagging with respect to 1/2- They are never zero at the same time and so the transient flow never obeys the symmetry of the firing geometry. Observed individually, each centre-line U velocity does oscillate symmetrically about a zero value. This suggests that a time average of the periodic flow field will obey the symmetry of the firing geometry. The centre-line W velocities oscillate asymmetrically about nonzero values. Figure 4.16b) includes the corresponding values obtained from the enforced symmetry steady state result as a reference. Clearly, the time average of the periodic solution can not equal these enforced symmetry steady state values, and this proves that the time average flow is distinct from the enforced symmetry steady flow. The time average flow computed over one period of oscillation is shown in Figure 4.20. Chapter 4. Opposing Jets in Cross Flow 39 It is clearly distinct from the enforced symmetry steady state in that gradients in velocity are generally smoother. The jet impingement is less abrupt, and the upstream penetration of the core flow from the impingement point, and associated spanwise recirculations, are weaker. 4.2.5 Computer animated flow visualization The instantaneous behaviour of the unsteady, periodic flow is best observed using com-puter animation. A view of the flow field such as the vertical plane through the jets of Figures 4.18 and 4.19 can be taken. One period of oscillation is animated by displaying the flow field at successive time steps. This process can be repeated in a loop to observe continuous oscillations. The vizualization techniques used include a standard vector field, where velocity magnitude is represented by vector length, and a field of vectors having fixed size and velocity magnitude mapped to a color spectrum. The latter method is a useful addition in that it shows more clearly the flow in areas where velocities are much weaker than in the near vicinity of the jets. 4.2.6 Jet Tracer Analysis The results for the reference case computations are plotted as iso-contours of the jet tracer concentration for both the enforced symmetry steady state and the periodic flow in Figures 4.26 - 4.33. In this case for B/b = 10, S/b = 5, and M = 20, the mass flow ratio from equation 1.9 is TTT. = 0.6325, and the bulk jet tracer concentration from equation 4.4 is therefore ^Buik = 0.3874. From the plots in the vertical plane through the jets, it is observed that in the enforced symmetry steady state flow the jet fluid penetrates much farther upstream against the cross flow, as suggested by the corresponding velocity fields already discussed. Chapter 4. Opposing Jets in Cross Flow 40 The transient plots for the periodic, full domain computation show there is signifi-cant variation in the concentration field with time. The time average concentration field, evaluated over one oscillation cycle, or period of the motion, exhibits the same weaker penetration of the jets observed in the corresponding velocity field. The time average concentration field is also more uniform in the vicinity, and downstream of the jet in-teraction region. However, the instantaneous mixing in the transient concentration field is of a similar degree of non-uniformity to that of the enforced symmetry, steady state result. Of course the mixing is more uniform towards the exit, as seen in the 2D flows, and $ approaches the bulk value, ^Buik- The instantaneous dilution behaviour is more physically significant to operation of a furnace than the time average behaviour. It is the time average behaviour that is much more readily measured if one does experiments, though, and it could be misleading in the case of an oscillatory flow. 4.3 Grid Refinement Grid refinement effects are tested for the 3D reference case of opposing jets in cross flow at M = 20. A third multi-grid level is used which doubles the number of grid nodes in each coordinate direction to 64 x 28 x 96, and increases the total number of grid nodes by eight times. Due to limited computer memory at the time of this work, only the enforced symmetry case is investigated because it requires half the number of grid nodes, and half the number of unknowns for each node, giving a much smaller memory demand compared to the full domain, transient problem. Grid refinement for a full transient problem is discussed in the next chapter concerning the similar case of opposing jets in a closed fioor cavity. Runs are carried out using the typical multigrid parameters outlined in the description of the multi-grid method in an earlier chapter. The convergence performance of the Chapter 4. Opposing Jets in Cross Flow 41 UBC-MGFD code using the multi-grid method is shown in Figure 4.12 for the level two and level three grids. The work units for these two graphs now refer to the level 2 and 3 grids respectively. The convergence rate for level three multi-grid is about one half of that for level two multi-grid. Theoretically, the convergence rate for a finer grid solution approaches that of the next coarser grid solution for sufficiently large N in l inear problems, according to the 0(N) algorithm concept discussed in the section on multi-grid convergence performance. Prior to reaching such a limit, however, the convergence rate is always slower on the finer grids. The level three multi-grid convergence rate, however, is still significantly better than the level two single-grid convergence rate shown in Figure 4.11, and discussed in a previous section on the preliminary enforced symmetry steady state calculations. The results are compared for each of the three grid levels as profiles along the vertical duct centre-line, or symmetry axis, of vertical velocity, pressure coefficient, turbulent intensity and turbulent viscosity in the following dimensionless form: W W* = ~ (4.11) ^p = T ^ (4-12) TI = ^ (4.13) "' = -ij]r- (*•"' Figure 4.13 shows that both the vertical velocity and pressure coefficient profiles change substantially with each grid level. Figure 4.14 shows that the same is true for the turbu-lent intensity, but that the turbulent viscosity profiles agree very well on all grid levels. The turbulent viscosities agree exactly near the jet impingement point, and differ the most upstream, near the the cross flow inlet, consistent with the fact that the jets pene-trate farther upstream, against the cross flow, on the finer grids, as can be seen in the W Chapter 4. Opposing Jets in Gross Flow 42 velocity profiles of Figure 4.13. Clearly the level two grid is insufficiently refined, but the question remains as to how close the level three grid is to grid independence. A fourth grid level represents an unreasonably large calculation. What is done in the next chapter for the closed floor case is to make a new, unrelated grid, which at level two has a node density between that of the original (standard) level two and three grids. 4.4 A s y m m e t r i c Firing To investigate the effect of differences in plenum pressures for opposed jets, arising due to small variations from control set points, for example, a case is run for a pressure differential of 5.0%, or a 2.5% difference in jet velocities. The result is only a very slight shift of the flow to one side, with the amplitude of U2 shifting by at most 3.3 %. The main features and frequency of the motion are essentially unchanged. There is no drastic change in the flow state for a small, asymmetric perturbation in jet velocities. Real furnaces may also have slight asymmetries in geometric alignment, but it is reasonable to expect that these too will have a small effect on the flow and bifurcation behaviour. Geometric misalignment is investigated for the closed floor case in a subsequent chapter. 4.5 Variations of parameters For sufficiently weak jets, relative to the cross flow, the bifurcation disappears and there is a stable, symmetric steady flow field. This transition or bifurcation point occurs for the reference case geometry at a value of M between 1.25 and 5.0. Increasing the jet spacing by a factor of two to S/b=:10, which changes M to 10.0, gives a different behaviour entirely. One jet passes to the side of the other in the spanwise direction and the flow is symmetrical in the vertical centre plane. The flow is practically steady, as can be seen in the time series for the U velocities in Figure 4.17, and the Chapter 4. Opposing Jets in Cross Flow 43 jets do not oscillate from one side to another in the lateral or spanwise direction either. The steady state solver does not converge, however, indicating a very weak instability and transient fluctuation. Figures 4.21 and 4.25 show the flow field in the vertical and horizontal planes through the jets respectively. A reduction in jet spacing to S/b=4, which changes M from 20 to 25, increases the oscillation frequency by 25% as well as increasing the amplitude of the motion by 9 % for 1/2- A case with S/b=4 and velocities adjusted to preserve M = 20 has a further 3.0 % increase in frequency, and a 6 % decrease in amplitude for f/2. For a given jet spacing, S/b, decreasing M raises the oscillation frequency or Strouhal number, St, and lowers the amplitude. For sufiiciently low M, the oscillation vanishes and there is a steady symmetrical flow. For a given M, decreasing the jet spacing, S/b, raises the oscillation frequency or Strouhal number, St, and raises the amplitude also. For sufficiently wide jet spacing, the oscillation essentially vanishes, but there is a bifurcation whereby the jets pass to either side of one another in the horizontal plane, giving a spanwise asymmetry. The results show that both the spacing and momentum fiux ratio parameters are important. Calculated Strouhal numbers and oscillation amplitudes for U2 are summarized in Table 4.1, including the result from the next chapter for the closed floor case on the similar, standard level two grid used there. Chapter 4. Opposing Jets in Cross Flow 44 Table 4.1: Summary of calculated Strouhal Numbers and U2 Amplitudes S/b 5 5 5 4 4 10 4 ^ - sJv^ 20 5.0 1.25 25 20 10 Closed Floor StB — fjlj2 .237 .315 (Very Weak) Steady Symmetrical .296 .305 Nearly Steady Spanwise Asymmetry .229 U2 Amplitude ±.32 - ^ 0 0 ±.35 ±.33 ^ 0 ±.25 Chapter 4. Opposing Jets in Cross Flow 45 0.0 2.5 5.0 Cavity Dimensions (m) Figure 4.1: Enforced symmetry steady state flow. 2D reference case. 5.0 -2.5 -0.0 I - I - ^^^--'^k:^H^r"^--^-^--|~'^-H H - ^ ^ P ^ ^ I ^ J- H ^ I ^ I ^ I --2.5 0.0 2.5 5.0 Cavity Dimensions (m) 7.5 10.0 Figure 4.2: Asymmetric steady state flow. 2D reference case. Chapter 4. Opposing Jets in Cross Flow 46 8.0 CD g' 5.0 C 4.0 O E O 3.0 S to O 2.0 0.0 10 ' - B Lg Enforced Symmetry - A L, Full Domain -A L, Full Domain 10° 10' Momentum Flux Ratio (M) Figure 4.3: Bifurcation diagram showing steady state reattachment lengths versus mo-mentum flux ratio. 2D flow, long domain with zero gradient exit. i o . ( i I I I r I I I I I I I I I I I 'i i I I [ I I I r I I I I I I I I I I I I a £ 7.5 5.0 2.5 0 . 0 ' I I I ' ' I ' ' ' ' • ' ' • ' • • I r I t • ri 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Tiine(s) Figure 4.4: Transient perturbation of the 2D reference case enforced symmetry result. A 0.5 s ±5% Vj pulse is applied at t = 0. Chapter 4. Opposing Jets in Cross Flow 47 lOQ I I I I '^ 7.5 t J I (S 5.0 2.5 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1^  0 . 0 r t I • I I I I I I I I . I I I . I I I I I I I , I I , , I I , , I I I I 1 I I r 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Tiine(s) Figure 4.5: Transient perturbation of the 2D reference case asymmetric result. A 0.5 s d=20% Vj pulse is applied at t = 0. The asymmetry is not reversed. io.q-r ? 7.5 S 5.0 £ Z5 -I I r I I I [ I I I I I I [ I I I I I I I I I I I I I I [ I I I I I I I I r [ 0 0 1 " ' I 1 I I I I I I I I I I I I I I I I I I t I t I • 1 I • I I • • I I I I • I r 0.0 1.0 . 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Timed) Figure 4.6: Transient perturbation of the 2D reference case asymmetric result. A 2.0 s ±20% VJ pulse is applied at t = 0. The asymmetry is reversed. Chapter 4. Opposing Jets in Cross Flow 48 «ve L K J 1 H G F E 02 100.0 95.00 90.00 85.00 80.00 75.00 70.00 65.00 2.5 5.0 Cavity Dimensions (m) 10.0 Figure 4.7: Enforced symmetry steady state jet tracer concentration (in %) . 2D reference ^Buik = 52.79%. case. 5.0 -2.5 -0.0 -2.5 Leve L K J 1 H G F E 0 c B A 9 8 7 6 5 4 3 2 1 02 100.0 95.00 90.00 85.00 80.00 75.00 70.00 65.00 60.00 55.00 50.00 45.00 40.00 35.00 30.00 25.00 20.00 15.00 10.00 5.00 0.00 - I 1 1 i 1—u 2.5 5.0 Cavity Dimensions (m) 7.5 10.0 Figure 4.8: Asymmetric steady state jet tracer concentration (in %) . 2D reference case. ^Buik = 52.79%. Chapter 4. Opposing Jets in Cross Flow 49 20 15 T — I — r I ( t I -\—I—r , t t t < I t t , I I t / . t t . , t t . . / » ^ . , I > - ' / TTT 1 1 1 1 1 1 t f T — r t t t t ^ - - - - . - . ^ - ^ y 10 0 -5 \ V -\ \ I \ t \ V V \ \ \ \ \ \ \ V \ \ V \ \ V N \ -.— ..-*- — — . . - ' -^ \ <• < — « — t f — * -<—<— — M" t ! ' / ' ' ' ' ' t i ' M ( T T 1 W \ \ \ V - , ^ ^ ' ' ^ ^ . t I \ \ \ \ \ \ V v v , ^ ^ ^ / / / f t t ^ ^ ^ / / / / t I f t f / I f / I f / / f / ' H t I t » \ \ \ \ \ \ \ \ V I I > \ > ^ \ . \ \ \ -J I I I I I I I I — I L X / / / / / / / I I I I I l_ 0 10 15 20 V Jet Figure 4.9: 3D Reference case enforced symmetry, steady state flow. Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 50 5.0 2.5 0.0 I I . I T^^T^I^'^^T^'^'^^T^"^^'^'^ L-LJ U L J I - I r \ [ \ ] I - I I - I I - I ' • I 10 15 20 Jet 0 5 Figure 4.10: 3D Reference case enforced symmetry, steady state flow. Horizontal plane through jets 10" 10" 10' 10"" 10" 10"' 10" 10"' r Level 2 One Grid n 10"" in" ' 10"'° 10"" 10"'^  10"" r - -^-ur vr wr mr kr er _L J. U. 50 100 150 200 250 0 50 100 150 200 Work Units Work Units L.bJl 250 Figure 4.11: 3D Reference case enforced symmetry, steady state flow. Convergence performance for level 2 (32 x 14 x 48) multi-grid and single-grid. Chapter 4. Opposing Jets in Cross Flow 51 50 100 150 200 Work Units 250 50 100 150 200 Work Units 250 Figure 4.12: 3D Reference case enforced symmetry, steady state flow. Convergence performance for level 2 (32 x 14 x 48) and level 3 (64 X 28 x 96) multi-grid. o o > 0.50 0.25 ->< 0.00 -0.25 -0.50 I i 1 1 1 1 1 1 1 1 1 1 1 < 1 1 1 1 1 o f } J'« f ^^^^^^^^^^&M J] I . I .1 J . I . I . i 1 1 1 . , i 1 1 1 1 r 1 1 1 1 1 1 1 1 B A e , , , , ! , , 1 1 1 1 1 1 1 1 1 1 1 1 W-, Grid Level 1 W-i Grid Level 2 W-, „ Grid Level 3 -^ P . . 1 . . . . 1 . . . . 0.20 0.10 -0.00 -0.10 -0.20 CM Z) Q. OJ Q. 11 a o c 0) H— o O Z3 w 0) OL -10 -5 5 10 15 Elevation (z / b) 20 25 30 Figure 4.13: 3D Reference case enforced symmetry, steady state flow. Profiles for W velocity and pressure coefficient at x/b = 10, on duct centre-line, for three grid levels. Chapter 4. Opposing Jets in Cross Flow 52 -^ 0.20 cvi, 0.10 II P ^ -0.00 '</) c g -0.10 c , 3 -0.20 -10 I . . . . I T I - , Grid Level 1 T i n Grid Level 2 V , - • Tl- i Grid Level 3 V , - I 0.100 0.050 CO =1 o II 0.000 -0.050 CO o o w c -0.100 0 5 10 15 Elevation (z / b) 20 25 30 Figure 4.14: 3D Reference case enforced symmetry, steady state flow. Profiles for turbu-lent intensity and viscosity at x/b = 10, on duct centre-line, for three grid levels. 0.75 -0.75 400 800 1200 1600 2000 2400 2800 3200 Time [ t / ( b / U , ) ] Figure 4.15: 3D Reference case time series from start-up. Chapter 4. Opposing Jets in Cross Flow 53 0.75 -0.75 2400 2600 2800 3000 3200 Time [ t / ( b / U j ) ] - 3 Z> "^ ^ w " • 1 — ' o o > 5 u./o 0.50 0.25 ( 0.00 -0.25 -0.50 . n 7CL 1 1 1 1 1 1 B - - - W , a W^ • • • « W 3 _ W, VV2 _ .. ..".."".; .VYa ^ ^ ^ ' ' * % « , B . ^ 0 * ^ * ' > . . e 3 , . ^ ^ ^ P"^  Ki^^fifi^^^^ IS T^^f ip *^^ n ^ ^ i ^ d p * ^ . J — — 1 , 1 — 1 1 1 — Transient -Enforced Symmetry ', -^ ' ^ ^ ^ ^ W * ^ ^ ^ ^ ^ ^ B ^ ^ ^ ^ ; 13^^^^^^ --1 2400 2600 2800 3000 3200 Time [ t / (b /Uj ) ] Figure 4.16: 3D Reference case time series showing periodic result. Chapter 4. Opposing Jets in Cross Flow 54 0.75 -0.75 200 400 600 800 Time [ t / ( b / U j ) ] Figure 4.17: 3D Wide spacing, S/b = 10, time series from start-up. Chapter 4. Opposing Jets in Cross Flow 55 T — I — I — I — I — I — I — I — I — r 1 I I I r , ^ V V V X \ 1 — I — I — I — I — I — r \ \ \ \ \ 20 15 . , , , 1 t \ M \ \ \ t t t t t , , , , . , , . / / / / ( f t t t f f / / / , . , . . , / / f t t t t f t t / , , . . ^ ^ / / / / f f f f f / . . . . .^///fjjf, ^^^^//J/J f , / ^ .^^_*-*_-._^_*._-.-*-^ / / ' . : : : , - * - ^ > ^ ^ - ^ • = ^ = ^ ^ ^ 3 ^ ^ ^ - ^ . < < « . . . -, 10 • > > : > • j» > j ^ ^ ^ ^ ^ ^ ^ ' k j - / 7 f t t - r T \ \ t t t 1 ' / X 4 ^ ^ -- V \ > A \ t \ \ \ \ \ \ \ \ \ \ \ V . •^  ^,f/f^fTl - X V t t f / / 1 M \ \ \ \ \ \ \ \ N . , -^  l;i//, , , . , > I t t t f i \ \ \ \ \ \ w \ \ \ . , ^1. U. W ^ , . . / / f t t A \ \ \ \ \ \ \ \ \ \ \ ' ^ ' . ' , \ \ \ \ \ ^ ^ ^ . y / f f t T t \ \ \ \ \ \ \ N \ \ v ^ , , , , , ^ . , ^ ^ / f t \ \ \ \ N \ \ \ V I V \ \ -^  0 ^ \ \ \ \ t I \ 1 t I I \ \ \ \ \ \ \ \ v \ \ \ \ \ \ \ \ \ N N \ \ \ \ \ \ K \ \ \ \ \ » \ \ \ \ \ \ \ \ \ t I I t t t t I t \ S \ t t \ I I t ( 1 t i ( If f t / t / f r I I 0 5 V 10 15 20 Jet Figure 4.18: 3D Reference case transient flow when f/2 = 0 (t* = 2980). Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 56 T — I — r T — r 20 15 T ^ T I I I I I I I I , ( ( t t t ' t t t f t f / / ' f / / / , , . r I / / y ^ y -I I / / J- y . ^ f / / t t t t t t t t t t t f > I , , , y y /•//// f f f f f f f f f / , , , y y ^ ^ / / / / / / / / / / ' / / / ' , , , . 10 1 \ \ \ \ \ v ^ . t \ \ \ \ V ^ ^ \ \ \ V ^ -. 0 -5 -X ^ / / / / / / / / ^ ^ , ^ I / / ^ y ^ y y ^ / / / / / / / / / , , - 7 / t / ^. / / / / / / / / / , , . . , , , , , . t / / ^ . '^ ^ JA'J f f ' ' - - , , , . - . • I , . ^V/fU^ '^ •  '^ / / / I / / I t I • / I \ \ ^ \ \ \ \-| . V ^ > t t t / / / / / I t" , ( / / / / / / / / / M t , I t I I t t t i \ \ \ \ \_ , / / / f t f t t i \ \ \ \ y y / / / / f I t t \ \ \ ^ J — J L I . I ^ ^ />//// t t y y ^ / / / I t ^ / f / I I t t I I I I t i l l t t t I I t , I I / / I n I t t \ I I t J I I I I—-I I I L 0 5 V 10 15 20 Jet Figure 4.19: 3D Reference case transient flow when U2 = maximum {t* = 3080). Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 57 T — I — I — r t t \ 1—I—I—r - I — I — I — I — I — I — I — r 0 J t \ \ \ \ \ \ \ "t \ \ \ \ \ \ \ \ I I _ — - ' - ' / / / / / / f f ^ ^ ^ y / / f t t t t , , , y / / I t i t I , , , , / / I I I t t / / I I I I I I I t i l l . J I l__l I L 1 , 1 t t < I I I I I J L 0 5 V 10 15 20 Jet Figure 4.20: 3D Reference case time average flow over one cycle. Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 58 20 15 10 0 I I I t t t t t t t f t t I , t 1 i /J\\^ \ \ V t I t I I I I t t t I I I I I ( I t t I t I t » I t ( t i l l I t I I - I I I L I ' ' ' J_J L 1 I I I L I < I I J I L. 0 5 V 10 15 20 Jet Figure 4.21: 3D Wide spacing, S/h = 10, nearly steady result at t* through jets. 960. Vertical plane Chapter 4. Opposing Jets in Cross Flow 59 5.0 2.5 0.0 . . . . . . | . . .^ . -^  I . . . . . ^ I . . . . . . . " . ' I • . . r . ." I '. , " . ' I . ' , ' 0 10 15 20 ^ V Jet Figure 4.22: 3D Reference case transient flow when U2 = 0 (t* = 2980). Horizontal plane through jets. 5.0 2.5 0.0 — r - r r _ _* — • < * t - 1 1 . 1 , *-— * • ~^ —*" 5 J "». ' 1 - 1 JJ-T-, — • -.*-—*-—^  — • * 1- 1 L , f ^ — — V 1-i "N •V. • » ^ • ^ .^ ^ ^ f 1 -' .^  .*_ .«_ -* <a • • — *-1 -1 1 . 1 ^ -«-. .^ _ -« ^ • * — *-I - 1 ^ ^ " *_ .*^  ^ ., -« .^  *— — 1 -1 ^ • * -.«~_ •-* • ^ — • -1 • r -s i _ 1 . 1 ^ , .«_ >- — -' 0 5 10 15 20 Figure 4.23: 3D Reference case transient flow when U2 = maximum (i* = 3080). Hori-zontal plane through jets. Chapter 4. Opposing Jets in Cross Flow 60 5 .0 I I . I I . I I . I I. I - C 3 - I .1 -I—VT r I I - I 2.5 - ^ g Q _ 0 I I ' I I ' I I • I L I [ I L. 0 5 10 j _ ^ j I -1 I ' I 15 20 Figure 4.24: 3D Reference case time average flow over one cycle. Horizontal plane through jets. ^ V Jet Figure 4.25: 3D Wide spacing, S/b = 10, nearly steady result at t* = 960. Horizontal plane through jets. Chapter 4. Opposing Jets in Cross Flow 61 Concentration B A 9 8 7 6 5 4 3 2 1 1 .OOOEO 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO 0 10 15 20 Figure 4.26: 3D Reference case enforced symmetry, steady state concentration field. Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 62 -5 -I I I Concentration B A 9 8 7 6 5 4 3 2 1 1.000E0 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO 0 10 15 20 Figure 4.27: 3D Reference case transient concentration field when U2 = 0 {t* — 2980). Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 63 -5 Concentration ' ' ' I i ' • • I • I i I I I I i B A 9 8 7 6 5 4 3 2 1 1.000E0 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO 0 10 15 20 Figure 4.28: 3D Reference case transient flow concentration field when U2 = maximum {t* = 3080). Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 64 -5 I I I I ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ j Concentration B A 9 8 7 6 5 4 3 2 1 1 .OOOEO 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO 0 10 15 20 Figure 4.29: 3D Reference case time average concentration field over one cycle. Vertical plane through jets. Chapter 4. Opposing Jets in Cross Flow 65 Concentration B A 9 8 7 6 5 4 3 2 1 1 .OOOEO 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO Figure 4.30: 3D Reference case enforced symmetry, steady state concentration field. ^Buik = -39. Horizontal plane through jets. Concentration B A 9 8 7 6 5 4 3 2 1 1 .OOOEO 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1 .OOOE-1 O.OOOEO Figure 4.31: 3D Reference case transient concentration field when U2 = 0 {t* = 2980). ^Buik = -39. Horizontal plane through jets. Chapter 4. Opposing Jets in Cross Flow 66 I I I I 7-10 15 Concentration 20 B A 9 8 7 6 5 4 3 2 1 1.000E0 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO Figure 4.32: 3D Reference case transient concentration field when U2 = maximum (t* = 3080). ^Buik = -39. Horizontal plane through jets. 20 Concentration B A 9 8 7 6 D 5 4 3 2 1 1.000E0 9.000E-1 8.000E-1 7.000E-1 6.000E-1 5.000E-1 4.000E-1 3.000E-1 2.000E-1 1.000E-1 O.OOOEO Figure 4.33: 3D Reference case time average concentration field over one cycle. ^Buik = -39. Horizontal plane through jets. Chapter 5 Opposing Jets in a Closed Floor Cavity This chapter focuses on the closed floor case CFD computations which are representative of the experimental case described in the next chapter. The preliminary closed floor calculations using the standard level two grid of moderate density are described. The effects of asymmetric firing geometry are investigated. Issues of computational accuracy and convergence are then investigated using this case. These include time step and grid dependence, and convergence level requirements. 5.1 Closed Floor Case AR = 4, S/h = 4, B/h = 10, Rej = 10^ A = 15.5 This case is representative of the geometry in the physical experiments described in the next chapter. There is a closed floor in the cavity at Di/h = 15.5 below the jets. In the absence of a cross flow there is no momentum flux parameter, and so the only flow parameter is the Reynolds number, upon which the CFD results are essentially independent at these high values. A reference standard numerical grid is defined with 32 nodes along the jet axis, 14 along the spanwise axis, and 48 along the jet axis, defined as the X, y and z axes respectively. Two level multi-grid is used, so a coarser grid of one half the number of nodes in each coordinate direction is also used in the calculations. The results are shown as a time series plot in Figure 5.1, and instantaneous flow fields in Figures 5.14 and 5.15. The main feature of the transient flow field is that the recirculations below the jets can grow to a size limited by the floor. The calculated Strouhal number of Sts = 0.229 is lower than for the cases presented having a cross flow 67 Chapter 5. Opposing Jets in a Closed Floor Cavity 68 (e.g. S/b=4, M=20) which is probably due to the larger recirculations below the jets. 5.1.1 Wall Shear Stress Effects The effects of wall shear stress are investigated in Appendix A, where the wall function method of Launder and Spalding (1974) [30] is applied to the two vertical walls and the bottom floor of the cavity. The results are not very different from those with free slip walls: the U2 oscillation amplitude decreases by about 11 % and the Strouhal number increases by only 1 %. There is also a slight difference in phase between the impingement point and surrounding core flow motions, observable in the time series traces for Ui, U2, and U3, and in the instantaneous velocity fields. It is shown later in the chapter that the results on this standard level two grid are somewhat grid dependent. On a more suitably refined grid, it is also shown in appendix A that the wall function treatment has a lesser effect on the results. 5.2 A s y m m e t r i c Firing Geometry The effect of raising the jet on the right hand wall upwards by one characteristic jet size unit, b, is investigated here. This displacement is equivalent to one half the jet height of 26. The computational grid used is nearly identical to that of the symmetrically fired case described in the preceding section, except that the grid expansion, in the vertical direction away from the jets, is delayed by two cells to facilitate placement of the elevated jet on the right hand side. The time series trace of Figure 5.2 shows that a periodic oscillation is nearly reached, but this is after about twice the length of time required to reach periodicity in the symmetrically fired case. The oscillation is asymmetrical, with respect to U = 0, and appears to consist of two unequal sub-oscillations, resulting in four zero crossings rather Chapter 5. Opposing Jets in a Closed Floor Cavity 69 than the usual two. The period measured from this trace is Tg = 9.47. Half of this period, 4.735 time units, represents an average for the two sub-oscillations. The corresponding Strouhal numbers , Sts, are 0.106 and 0.211 respectively. The sub-oscillation frequency, then, is 92 % of that for the symmetrically fired case, or 8 % slower, on average. This double oscillation period may also explain the longer time required to reach a periodic state in the computations. The peak to peak amplitude for U2 is practically unchanged from the symmetrical case. The flow field at the end of the time series trace is shown in Figure 5.16. It is apparent that the jets do not miss one another, but still impinge near the center of the cavity. This is due to the fact that by the time a jet reaches the center of the cavity, it has spread more than the lb displacement of the right hand jet. In real furnaces, some geometric misalignment is likely to be present, due both to port location, and plenum conditions which may direct the jet slightly anti-normal to the wall. From these results, however, it appears that such minor misalignments from a directly opposing design will not remove the inherent instability causing an oscillatory flow. 5.3 T ime Step Dependence For the standard level 2 grid, a series of CFD runs are performed using different time steps. This is done by restarting the calculation from a converged, transient result using the reference time step Vj At* = A i - / b of 10.0, and continuing with the adjusted time step. Results are shown in Figure 5.3 where the restart has been made from t* = 1920, or after 192 steps at Ai* = 10, and continued for time steps of At* = 40, 20, 10, and 5. Clearly there is severe degradation for the longer time steps of 20 and 40. In fact, at Ai* = 40 the oscillation is no longer Chapter 5. Opposing Jets in a Closed Floor Cavity 70 symmetrical about the geometric symmetry plane. The asymmetry is associated with a generally poor level of convergence that is obtained with a longer time step. This might improve with more multigrid iterations per time step. On the other hand, one observes that the time step (A^*) of 5 gives quite a similar result to a time step of 10, particularly for the oscillation frequency which is 99.1 % of the result for a time step of 10. The amplitude increases slightly also. Of course, the time step may affect results to a varying degree according to refinement of the grid size and proximity to a grid independent result. Grid dependence is examined further in a subsequent section. 5.4 Convergence Level Dependence For a given amount of computer work at each time step, it is pointed out in the previous section that the level of convergence becomes severely degraded for longer time steps, and indeed the results become unphysically asymmetrical. This raises the question of what level of convergence is actually required at each time step, for a reasonable choice of grid size and time step, and hence what the minimum computing cost will be without affecting the result obtained. Firstly, the convergence rate for a single time step is examined in Figure 5.4, for the standard grid and time step of 10, by restarting from t* = 1920. The root mean square residual for pressure, or the continuity equation, is plotted versus work units (one smoothing on the finest grid is one work unit) for three multi grid iterations and for smoothing on the fine grid only. The multi grid procedure is indicated by a third curve which tracks the switching from one grid to another, in this case from grid level 1 to 2 and vice versa. One observation is that the lower the convergence level that will be required, the greater the advantage of the multi grid procedure will be in reducing the necessary Chapter 5. Opposing Jets in a Closed Floor Cavity 71 computer work, due to the relatively higher curvature of the single grid residual curve. To test the effect of convergence level, it is necessary to run a computation through some period of time, for example one full oscillation cycle, with varying degrees of com-puter work per time step. This is summarized in Figure 5.5, where the runs are restarted from the standard transient case at t* = 1920. The legend in this figure warrants some explanation. The label MG20,(50,10) x 2 , for example, indicates a multi grid method with 20 initial smoothings on the fine grid, then two multi grid iterations consisting of 50 smoothings of the correction problem on the coarser grid and 10 smoothings on the fine grid after adding the correction. The label 1G25 refers to smoothing on the fine grid only 25 times. Note that the number of work units for one smoothing on a grid that is coarser by one level is approximately 1/8, so the correction problem may be solved to a generous degree of convergence without significantly affecting the overall cost. The three multi grid cases shown on the figure all agree exactly after one cycle. The 1G25 case is off by 0.481 %, and the IGIO case is off by 15.4 %. The 1G25 case requires the same computer work as the one multi grid iteration case MGIO, (50,10) x 1, but it appears to be just above the threshold for the convergence level required. Referring to Figure 5.4, which shows the convergence performance for the first t ime step following the restart, it is clear that the MGIO, (50,10) x 1 convergence level is significantly lower than the 1G25 convergence level. It is concluded that one multi grid iteration, MGIO, (50,10) x 1, is a reasonably safe procedure for this standard level two grid, and that the original calculations done with two multi grid iterations were more than adequately converged. Chapter 5. Opposing Jets in a Closed Floor Cavity 72 5.5 Grid D e p e n d e n c e Once t ime step and convergence level effects are studied on a reference grid, it is sensible to study grid refinement effets. This is done initially by use of the multi grid formulation, where the problem is raised to a third grid level from the standard level two grid, which doubles the number of grid nodes in each of the three coordinate directions, or increases the total number of nodes by a factor of eight. While this is quite a dramatic increase in the number of nodes, it is quite an attractive approach in the context of multi grid, as one may restart a three level run from a periodic two level transient result as shown in Figure 5.6. The effect of going up one grid level is quite a sigificant change in the oscillation frequency, with a lesser change in amplitude. From these two results, however, it is not possible to say how near the level three result is to grid independence. Going to a fourth level grid would give rise to a prohibitively large number of nodes for the computers available here, and so it is necessary to define a grid somewhat in between the levels 2 and 3 of the standard grid. A so called refined level two grid is therefore defined, which contains 40 nodes along the jet axis, 20 along the spanwise ajcis, and 72 along the duct axis. With this new grid, the computation must be started from scratch with an initial, unconvergent steady state run, followed by a transient until a periodic result is obtained. The results for the standard level two (L = 2), refined level two (L = 2R), and standard level three (L = 3) grids are shown as time series in Figure 5.7. The figure plots only the final periodic results for U2 over one oscillation period for the standard level two grid, which has the longest period, with the starting points for each curve adjusted to put them all in phase initially. It is quite clear that the oscillation frequency of the level two refined grid is much closer to the level three result than to the level two Chapter 5. Opposing Jets in a Closed Floor Cavity 73 Table 5.1: Summary of calculated Strouhal Numbers. Grid L = 2 (32 X 14 X 48) L = 2R (40 X 20 X 72) L = 3 (64 X 28 X 96) Sts — fjr/2 .229 .279 .293 ^ ( % ) .782 .952 1.00 standard grid result. This suggests that grid independence is being rapidly approached for the two finer grids of the three used. This grid refinement eff"ect can be seen in another way by plotting the dimensionless oscillation frequencies, or Strouhal numbers, against the number of grid nodes ( N ) as in Figure 5.8. Figure 5.9 shows the same data, but plotted in terms of the cube root of N, which is like the average number of nodes in each coordinate direction. From either Figures 5.8 or 5.9, it is apparent that grid independence of the Strouhal number is rapidly being approached for the refined level 2 and standard level 3 grids. Even the standard level 2 Strouhal number is 78.2 % of the level three result, which is quite reasonable. Calculated Strouhal numbers are summarized in Table 5.1. The transient flow fields for the three different grids can be compared at points of equivalent phase in the cycle, a zero crossing or maximum being the most obvious choices. Profiles along the axis passing through the two jet centres are plotted for U velocity, pres-sure coefficient, turbulent intensity and turbulent viscosity in the following dimensionless form: U* = ^ a TI U_ Uj n Ik Uj (5.1) (5.2) (5.3) Chapter 5. Opposing Jets in a Closed Floor Cavity 74 Figures 5.10 and 5.11 show these profiles when the U velocity at point 2, U2, is a maximum on each of the grids. From these plots one can observe the rather better agreement between level 2R and 3 results than those for levels 2 and 2R, over most of the depth of the cavity. In fact this agreement is more convincing than just looking at the peak values, or amplitudes, for U2 in the time series plots (summarized in Table 5.2), which show sizeable discrepancy, particularly between grids 2R and 3. The reason for this is the strong gradient in U velocity near the impingement point, evident in the profile plots. A very slight translation of the impingement point will cause a m.uch larger change in the U velocity observed at point two on the cavity center line. It is really the impingement point motion that is physically significant, and is the main focus of attention when observing the oscillating flow field in computer animations, for example. The impingement point motion is captured only indirectly by observing the velocity at a fixed point on the cavity center line, which is certainly the conventional way to extract data, either from calculations or experimental measurements. It is interesting to look at the magnitudes of the turbulent viscosity. The dimensionless turbulent viscosity defined in Equation 5.4 can be viewed as the inverse of a Reynolds number like quantity, Ret^\ = ^ (5.5) which we can refer to as the turbulent Reynolds number. This quantity varies throughout the field, of course, but examine the magnitudes along the profiles between the two jet centers shown in Figure 5.11. The turbulent viscosity, u^, varies between .01 near the jets and .05 near the impingement point, so that Rct varies between 100 and 20 respectively. In Appendix B there are iso-contour plots of turbulent viscosity in various planes of the flow field for the refined time step results described in a later section. They Chapter 5. Opposing Jets in a Closed Floor Cavity 75 show that the turbulent viscosity varies within this same range throughout much of the region where the significant jet interaction and flow pattern development occurs. These results are particularly interesting because the regular oscillations reported by Wood et al. (1991) [47] for laminar RIM flows occur at true Reynolds numbers of about the same order of magnitude. The qualitative similarities between these flows is quite sensible because the role of the turbulent viscosity (plus molecular viscosity) in the Reynolds mean flow equations for a turbulent flow is analagous to that of the molecular viscosity in the Navier Stokes equations for a laminar flow. 5.5.1 Wall Shear Stress Effects The effects of wall shear stress are revisited for the refined level two grid (2R) in Appendix A. On this refined grid, the application of wall functions has a lesser efi"ect than on the coarser, standard level two grid discussed in a foregoing section. The U2 amplitude is less than 1 % lower, and the Strouhal number is about 1.4 % lower at Stg — .275. The comparison between the flow fields obtained with free slip and wall function treatments is also better than on the coarser level two grid. It is shown that the computational grid is not strictly fine enough near the wall to satisfy the logarithmic law criterion of y+ < 200. However, the results strongly suggest that the underlying instability and oscillation behaviour are mainly determined by the jet flows and duct geometry, and are only weakly affected by wall shear stress. 5.6 Combined t ime s tep and grid refinement For the standard level two grid, it is shown in a foregoing section that a refinement of the time step (Ai*) from the standard value of 10, to a value of 5, has only a small effect on that grid. However, given that the results on that grid are not grid independent, as Chapter 5. Opposing Jets in a Closed Floor Cavity 76 demonstrated in the preceding section, the question remains as to the effect of time step on the finer grids, 2R and 3. Both the grid 2R and 3 calculations are continued for the refined time step, as shown in Figures 5.12 and 5.13, with less than 1 % changes in the respective oscillation frequencies or Strouhal numbers. There is a greater change in the U2 amplitudes, however, which are about 10 % higher in each case. The U2 amplitudes are summarized in Table 5.2. It is concluded that the time step (Ai*) of 10 is quite reasonable, although a time step of 5 is perhaps optimal. For the level 3 grid, and a time step (Ai*) of 10, it is found that two multi grid iterations per time step are preferable, or rather that one iteration is a little marginal in terms of convergence to a truly periodic state. Only one multi grid iteration is required for a time step of 5. The computer work for one multi grid iteration is actually .625 (25WU/4:0WU) of that for two iterations, so the total work is still 25 % higher for the refined time step of 5. The flow fields for the level 3 and level 2R grid results, with the refined time step (Ai*) of 5, are shown as vertical planes through the jets, when U2 = m,axim,um,, in Figures 5.17 and 5.19 respectively. Figure 5.18 shows the same results as Figure 5.17, for level 3, but plots every second vector only, for comparison with the level 2 results described in a preceding section. The results on grids 3 and 2R agree very well, as one observes the similarity between the flow patterns. The recirculation centers are practically coincident. In the level 2 result, however, one can see the difference in the flow pattern and the recirculation centers are slightly displaced from those of the finer grids. A more complete set of results is presented in appendix B for one case: the finest, level three (3) grid used and the refined time step of At* = 5, when the U velocity at point 2 is a m,aximum. (t* ~ 4535). Velocity fields are shown in vertical planes through the jets, in the planes of symmetry between the jets, and in horizontal planes through the jets. The scalar fields for pressure coefficient (Cp), turbulence intensity {TI\ and Chapter 5. Opposing Jets in a Closed Floor Cavity 77 Table 5.2: Summary of calculated U2 Amplitudes. Grid L = 2 (32 X 14 X 48) L = 2R (40 X 20 X 72) L = 3 (64 X 28 X 96) U2 Amplitude At* = 10 ± .248 ± .252 ± .226 A r = 5 ± .266 ± .295 ± .271 turbulent viscosity (z/^ *) are also shown as iso-contours in the same three planes. The pressure coefficient has a peak in the jet impingement region which moves back and forth as the flow oscillates. The magnitude of the pressure peak reaches a maximum of 13 % of the jet dynamic head when U2 is approximately a majcimum, and the impingement point is maximally off-center. The pressure reaches negative minima down to - 9 % of the jet dynamic head in the recirculation zones. The turbulence intensities illustrate the generation of turbulence in the shear layers of the jets, and show a maximum of about 20 % in the jet impingement region. Recall that the TI boundary values in the jets are set moderately, at 10 %, and have little effect on the results. The turbulent viscosity fields are discussed in a previous section where it is pointed out that the inverse of the dimensionless turbulent viscosity (Equation 5.4), the turbulent Reynolds number (Equation 5.5), is of a similar range of magnitudes to the true Reynolds numbers for which regular oscillations occur in the laminar RIM flows of Wood et al. (1991) [47]. Chapter 5. Opposing Jets in a Closed Floor Cavity 78 0.50 0.25 0 0.00 o _o > 3 -0.25 -0.50 2000 2200 2400 2600 Time [ t / ( b / U j ) ] 2800 Figure 5.1: Closed floor case time series showing periodic result. 0.50 I — > — I — : — t — : — I — > — r 0.25 3 Q 0.00 O O (U > 3 " 1 — ' — I — ' — I — ' — r -0.25 -0.50 J I L J I I I I I • I 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 Time [ t / ( b / U , ) ] Figure 5.2: Closed floor case time series for asymmetric firing geometry, showing nearly periodic result. Chapter 5. Opposing Jets in a Closed Floor Cavity 79 3 >. 'o _o > 0.50 0.25 0.00 -0.25 -0.50 2000 2200 2400 2600 2800 Time [ t / ( b / U j ) ] 3000 3200 Figure 5.3: Closed floor case time series for U2 showing effect of time step size. Chapter 5. Opposing Jets in a Closed Floor Cavity 80 10" • ! " " l " " ' 1 " ^ ' 10 MG Grid Level P Residual MG - a P Residual 1G 20 30 40 Work Units (WU ) 2.0 -Qj > 0) _J ;g 1.0 50 Figure 5.4: Convergence performance for one time step, restart from t* = 1920, for transient run on standard 2 level grid. 0.50 -0.50 2000 2200 Time [ t / ( b / U j ) ] 2400 Figure 5.5: Closed floor case time series for U2, showing effect of convergence level. Chapter 5. Opposing Jets in a Closed Floor Cavity 81 CM 3 o o > -0.25 -0.50 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 Time [ t / ( b / U j ) ] Figure 5.6: Closed floor case time series for U2, showing restart from standard grid level 2 to continue on level 3. 3 o _o (1) > 0.50 0.25 0.00 -0.25 -0.50 100 200 300 400 Time [ t / ( b / U , ) ] Figure 5.7: Closed floor case time series for U2, showing grid dependence for standard levels 2 and 3 plus level 2 refined grids. Chapter 5. Opposing Jets in a Closed Floor Cavity 82 - 3 OJ CD 0.300 0.275 0.250 g 0.225 5.00E4 1.00E5 1.50E5 Number of Grid Nodes ( N ) - 1.000 0.700 2.00E5 Figure 5.8: Closed floor case Strouhal numbers versus number of grid nodes, showing grid dependence for standard levels 2 and 3 plus level 2 refined grids. 0.300 0.200 20 30 40 50 Cube Root Number of Grid Nodes ( NN ) 1.000 - 0.700 60 Figure 5.9: Closed floor case Strouhal numbers versus cube root number of grid nodes, showing grid dependence for standard levels 2 and 3 plus level 2 refined grids. Chapter 5. Opposing Jets in a Closed Floor Cavity 83 o o > 5 10 15 Axial Position (x / b) Figure 5.10: Closed floor case profiles of U* and Cp along the axis of opposing jets when 1/2 = maximum, showing grid dependence for levels 2, 2R and 3 grids. —3 II w o o w c XI 5 10 15 Axial Position (x /b ) Figure 5.11: Closed floor case profiles of TI and v* along the axis of opposing jets when U2 — m,axim,um,, showing grid dependence for levels 2, 2R and 3 grids. Chapter 5. Opposing Jets in a Closed Floor Cavity 84 3 OJ 3 o o 0) > ID 0.50 | — I — I — I — I — \ — I — I — 1 — I — 1 — I — 1 — r 0.25 0.00 -0.25 -0.50 I ' I ' ~ r I ' I L = 2,At =10 L = 3,At' = 10 -0 L = 3, At' = 5 -e-J L — L J 1 I I I I L _L 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 Time [ t / ( b / U j ) ] Figure 5.12: Closed floor case time series for U2, showing a restart from standard grid level 2 to continue on level 3, with a further continuation for a refined time step. 3 CM 3 o o (U > 3 0.50 0.25 0.00 -0.25 -0.50 '—^ -^ L = 2R,At =10 -e L = 2R, At' = 5 _L J_ J_ _L J_ J_ 2000 2200 2400 2600 2800 3000 3200 3400 3600 Time [ t / ( b / U j ) ] Figure 5.13: Closed fioor case time series for U2 on the refined level two grid (2R), with a continuation for a refined time step. Chapter 5. Opposing Jets in a Closed Floor Cavity 85 20 15 \T, -I I J * I I 1 I J t — I — T — r I I t I ( T I I I I I - | — I — I — I — I — | — T — I — I — r T — r 10 Js. ~^' 0 T ] r-, / / / / / / / /////'^^^^ ^ ^ ^ , / / / t t f f f / / / / ' ^ . - v v v v v ; ; _ . v N \ \ \ \ \ \ \ \ V N '' ( * \ V \ \ - • ^ ^ c s ^ ^ ^ ^ ^ ^ S ^ ^ " • • • ' ' r 1 t / / y ^ ^ •_ 4 t t t 1 t ^ ^  • ^ ^  M W \ \ \ \ v ^ - / / / / / / / I t ' * -t t 1 \ \ \ \ N . , 1 \ \ \ > . ^ _ _ ^ ^ ---^z / / / / / I « ^ / / / / / I I V -/ / / ( I \ \ V •- -t I t \ \ V ^ ".• -* -* , ' ' / . y f / ^ ^ A / t t-l ( / i A /• / r I t I I -1-.-I I—_l L_l L J I L 0 5 ^ V 10 15 20 Jet Figure 5.14: Closed floor case transient flow when 1/2 = 0 {t* = 2080). Vertical plane through jets. Chapter 5. Opposing Jets in a Closed Floor Cavity 86 -r T — I — I — r . . 1 I t ( I'/ / / / / / / JTZTTl I I I—I—r—r - -v \ V V V \ V I t ^ '' V \ \ » \ ( I 4. ' ' I I I I t I t r ' ' I I I I i \ i '• •' f / I I I I i n 1 F 20 - ' 1 < \ \ \ \ M t t t f f / / / / . , . . N W W W W \ \ \ V V ^ * ^ . v ^ N v W W W w \ V . ' V - V . V W W N \ \ N V ^ - _ 1 l!i:;r.-^ntn\N:v:::::::::::::: 0 -5 ' ^ ^ * ^ ^ * * * « ^ ^ « « S J S //ft / n t y/t t ^ / f t - ~ ' - - ' ^ ^ y /' / / / / I I I \ \ \ V V «v ' ^ • 1 \ \ \ \ V — I I — I I — I I I I I I I I I I I I ' ' ' ' 0 5 10 15 20 V, Jet Figure 5.15: Closed floor case transient flow when U2 = maximum (t* = 2190). Vertical plane through jets. Chapter 5. Opposing Jets in a Closed Floor Cavity 87 20 15 10 0 ^ V N W W W W \ \ \ X _ . v \ \ \ \ \ \ \ \ \ \ \ ^ , _ v N \ \ \ \ \ \ \ \ \ \ , \ \ \ W \ U t t t / M t t t t f, t t t / . / / / / / / t t t / / ^ / / / f flLlt f ^y^///fLU 11 , '-zMjii. •T~~r T — I — I — r 1 1 1 ) ) I I ( I ( ) ) I I t J-I I ) 1 ( )• f I ( I ) t" t t t t t t t t M \ \ \ \ . ^ . . V \ \ \ < t \ \ \ \\\\\^^^ I n \\\ w^^^" \ X . V V - , , t \ \ \ \ \ V V ~ _ * t t t t t f t , / / / f t f t f. / / / / f f t. ^ _ ^ / / / / / / t . X / / / / / 4-. . - / - / / / / ! -y / I I I i , y / I I I • \ \ \ X -K ' ' ' ' -y / / I I \ \ \ J I I L ' ' ' ' ' ' ' I 0 5 ^ V 10 15 20 Jet Figure 5.16: Closed floor case with asymmetric firing geometry, transient flow when t* = 5000. Vertical plane through jets. Chapter 5. Opposing Jets in a Closed Floor Cavity 88 i\ I'l I i\t I Hi ft 1 i\t t r/ / /'///^//'^>^^^, I I 11 I n t I ( H / M /• ttttfff/////'^^, Ti I u 11 M 1111 M f / / / / / / / / / / / / / ^ ^ ^ ^ ^ ^ - ^ H M M l U H t t t t t t f / / / / / / / / 20 0 W W \ \ \ v \ w \ \ V vT ^^WWVWWVUV U 11 I v v x v v > k N \ \ \ \ V ^ \ \ \ \ ' W , 1 1 , 1 1 , , , , , , , , , , , , / / / / / / 1 1 n • / • / / / / / / i / / / f f 11 • •>• / / / -5 -tut ((it ^ ^ ^•^•^''''''''-'^^-'^-^^^1^^^$^ 111111 H t t ) 11 ( / / / / / . , — vC'C->^ >CV> 11111111 i H ( m 1 1 1 1 1 . , , . . :^?'>>>v> H H \ \ n \ \ \ \ \ \ \ \ \ \ \ \ \ w . - . , , ""•^'^^yy///// / / / / ) ) I 1 1 > > > > • , V V N \ \ S WWNVNNVC^^VNN-^ '-^^^^^^y// / / n i l III • • • - N W W W V V V N . V . W . - - J V > . - ^ > . . . . • . . ^ ^ ^ ^ ^ ^ ^ ^ ^ , , , , f I I , , , S VNX W _1_U L J I L J I I L ' ' ' 0 5 V 10 15 20 Jet Figure 5.17: Closed floor case transient flow when U2 = maximum {t* = 4535). Level 3 grid, refined time step, Ai* = 5. Vertical plane through jets. Chapter 5. Opposing Jets in a Closed Floor Cavity 89 20 15 10 .1 'I I' I ( ft /\ / '/ / / I' ^ i - ' J ' ^ U - L - . L J - . 1 I I I I I t t f / / t t / / /• ^ . ^ ^ \ I r ^ N. S \ \ \ V t , v , V > s \ \ \ \ \ \ \ \ \ N V . , , , V V \ \ \ \ V V \ X ' ' ) I I I ) t I 1 ' ' I I I t \ \ 1 \ i I 1 I I i \ V ^ 0 -»—^ / / y /////•^^^-^-^^ t f / / / / y ^ . / / / / / / / / - ^ t ( ( / / / / / ^ - , ,t t t t t t t 1 t t t I , . I t t \ \ \ \ \ \ \ \ \ \ \ * - ^ / / / / / • ^ ^ ^ / / /-f \ \ \ \ V N V - v , ' » , - . , . ^ » . _ ^ / / I I \ \ \ \ ^ • / / / / ' _1 I L J I I I U _ l I I I ' ' ' ' I I I I ' 0 5 V 10 15 20 Jet Figure 5.18: Closed floor case transient flow when JJ^ = maximum {t* = 4535). Level 3 grid, refined time step. At* = 5. Every second vector shown for comparison with level 2 result. Vertical plane through jets. Chapter 5. Opposing Jets in a Closed Floor Cavity 90 t I I f / / / / / / y^^^ . .^^^ 20 15 10 l i l t I I I I t i l l V \ \ \ \ n t t 11 / / f //^^. \ \ \ \ \ \ \ \ \ \ \ 1 1 , , , \ \ \ \ \ \ \ \ u \ 1 I I I , , . _ . ^ V V N V W W W N W W N V .. .^-vWsNWWW \ W W ^^  V V V \ \ \ \ t I I t. ' ' < 1 I I 1 t 1 I ( 1 ' ' ' / / / ( ( I ( I r 0 \ \ \ \ \ V V \ \ \ \ \ V V \ \ \ \ \ \ N N V . \ \ \ \ N S W - N , ^ ^ \ \ \ S N V V ^ - ^ - ^ - ^ I t \ WW v \ \ \ \ v ^ ' ' I I \ \ \ \ \ \ \ \ ^ - ' - . ' / / / / M M * • — ' • y / / / / I I I I - " " ^ f / - / / / t i l / / I I i I I I I I \ I I \ I I l __ l I l - _ i I I I L. 0 5 10 15 20 Figure 5.19: Closed floor case transient flow when C/g = maximum [t* = 3675). Reflned level 2 grid (2R), refined time step, A^* = 5. Vertical plane through jets. Chapter 6 Physical Exper iments The interaction of opposing jets in a closed floor cavity is investigated experimentally in a modified isothermal, scale model of a recovery furnace. This apparatus and the par-ticular experimental case are described. The flow measurement and calibration methods are outlined and experimental errors are discussed. Laser doppler velocimetry (LDV) measurements are made for velocity time series data at a strategic point where the jets are colliding. Flow visualization is performed with a laser light sheet and flow patterns are recorded with streak photographs and S-VHS video. The S-VHS video is also used in a particle image velocimetry (PIV) analysis. 6.1 Apparatus The physical apparatus is a 1:28 scale, isothermal model of a B&W recovery furnace using water as the working fluid. The model is one of two in a facility donated to U.B.C. by Weyerhaeuser Co. of Tacoma Washington, and represents an operating furnace at the Weyerhaeuser pulp mill in Kamloops B.C. Only the tertiary level of a three level air system is used, which has been modified from an interlaced design to fire 5 jets from each of the front and back walls, symmetrically, and directly opposed. The geometry is shown in Figures 6.1 and 6.2. Each jet has an aspect ratio of 4.3 and characteristic size b = 19.8 mm. Each bank of five jets is located on a wall of width W / b = 21, and the cavity depth between opposite banks of jets is 2B/b = 20.74. The spacing ratio between centre and first neighbor jets is Sijh = 4.2, that between first and second neighbors is 91 Chapter 6. Physical Experiments 92 Table 6.1: Experimental Geometry. Parameter AR b W/b 2B/b Si/b S2/b 53/26 Dip/b DiB/b Value 4.3 1.98 cm 21 20.74 4.2 4.6 1.7 15.5 17.5 52/6 = 4.6 and the side walls are S^/lb = 1.7 from these. The floor slopes from the front wall at Dip/b = 15.5, downward, toward the back wall at Dis/b = 17.5. The geometric parameters are summarized in Table 6.1. Each jet is a rectangular, sharp edged orifice, of 0.95 cm width and 4.11 cm height, cut in a 0.50 cm thick plate. The flow is supplied to each jet from a common header via a 1.9 cm nominal flexible pipe system with a gate valve and orifice plate for flow control and measurement, which are discussed in a following section. Prior to the jet, the flow is expanded suddenly to a 3.8 cm pipe system consisting of a vertical, 10 diameter long straight section, gradual 90° bend of 5 diameter radius, and a second horizontal, 10 diameter long straight section feeding a small rectangular plenum. Data from Blevins (1984) [7] indicates that such an expansion would reattach to the walls within less than three diameters of the larger 3.8 cm pipe, so that fully developed flow would be expected within the 10 diameters before the bend. The plenum measures 3.8 cm deep in the streamwise direction, and 6.0 cm high by the jet spacing in width. The original design used a 1.3 cm diameter feed pipe, requiring a one to three expansion in the flow cross sectional area. The small plenum could not facilitate such an expansion and the jet Chapter 6. Physical Experiments 93 velocity profile would certainly be concentrated near the centre of the orifice. By changing to a 3.8 diameter feed pipe, the author achieves a three to one contraction ratio from the circular feed pipe to the rectangular jet. The three central jets of each bank have their centres concentric with their respective feed pipes, but the two jets near each side wall are slightly closer to each wall than the axis of their feed pipes. In scale modelling, the jet velocity is also scaled down from full scale values. If the jet velocity is scaled down in exact proportion to the geometric scaling, then the time scales for the flow will match in both model and full scale. A typical full scale jet velocity of 69.2 m / s gives a model velocity of 2.41 m/s at 1:28.7 scale. The flow rate to each jet would be 15.0 US gal/min, which exceeds the capacity of the flow control and measurement apparatus, so the model flow rates are slowed further, by a factor of 2.5, to 6.0 US gal/min. The time scales for fluid motions in the model are therefore 2.5 times longer than in a typical full scale prototype. The flow rates of 6.0 US gal/min, per jet, give a jet velocity of 0.964 m/s and Reynolds number of 2 x 10^, which is reasonably high, and typical of scale, isothermal furnace models. The t ime scale based on the jet size, b, is — = 2.04 X 10-^5. The ratio of furnace depth, B, to jet size, b, gives (y) = '"'-'^ not quite 100.0 as in the corresponding CFD calculations. The time scale based on furnace depth, B, is then B^ ~T = 2.19s. For the purpose of comparison with the CFD calculations, where the Strouhal number is StB = f-r = -229, J 2 Chapter 6. Physical Experiments 94 or the dimensionless period, Tg = ^ ^ , is 4.37, the real oscillation frequency under the experimental conditons would be 0.105 Hz, or the real period would be 9.56 s. 6.2 Flow Measurement and Calibration The water flow is supplied to the model through a closed loop system using a large reservoir. Water is pumped from the reservoir by a Bingham centrifugal pump, powered by a U.S. Electrical RC-1, 30 kW AC motor. There is a system of header pipes with primary control valves feeding the various arrays of jets on two water models in the facility. From a common header, flow is directed individually to each of the tertiary jets used in this work, through a 1.9 cm nominal flexible pipe system incorporating a gate valve and Gerand D7 orifice plate for flow control and measurement. The out flow from the model is collected in a zero pressure return header and directed back to the reservoir. A Gerand M-125 differential pressure meter, calibrated by the manufacturer for the D7 orifice plate over a range from 1.0 to 7.5 US gal/min, is used to set the flows. Independent calibration tests were carried out by Ketler (1993) [27], with assistance from the author, by removing a section of the model out flow pipe and directing the flow to a cylindrical column, calibrated to 12.0 US gallons, and measuring the filling time. A total of 22 of the Gerand D7 orifice plates on one model were tested. The following results are quoted for flows between 4.0 and 7.5 US gal/min only, as the set value in these experiments is 6.0 US gal/min. It was found that the orifice plates and meter underpredicted the flow by about 10.0 %, on average. This appears to be a systematic error. In addition, there was scatter in the data about the mean which amounted to approximately ± 6.0 %, for ± three standard deviations, and stastical modelling by Ketler (1993) [27] gives a standard error (square root of the variance) of ± 2.2 %. As the objective of these experiments is to observe the behaviour of balanced, opposing Chapter 6. Physical Experiments 95 jets, it is the scatter, or random error, which is of more concern to the quality of the results, and the numbers quoted are actually quite reasonable in comparison to other errors likely to be present, such as human error in reading the meter etc. The systematic error, underpredicting the flows, affects the actual velocity from each jet, which will alter the t ime scale for the motion proportionally. The corresponding change in Reynolds number, however, would be comparatively negligible. 6.3 Drift Error Another error in flow control is an observerd tendency for the measured flows to drift with time, probably due to the valve screws and gates moving. While at tempts are made to minimize backlash effects when initially setting the valves, they do drift. Ketler (1993) [27] ran some drift tests which show a tendency for flows to decrease from their set point, by 12 to 14 % in the first hour or two, and then to a lesser degree in the next several hours. The set point for these tests, 2.5 US gal/min, is rather low. As a low flow setting requires a valve to be closed down to a small opening, perhaps it will be more sensitive to any given movement of the gate, and, furthermore, the hydrodynamic loading on the gate will also be greater and may affect the gate motion to a greater degree. In running the present experiments the flows were checked at intervals of about one hour or less, over a total run time of from three to four hours, with small adjustments of less than ± 0.25 US gal/min, or ± 4 %, typically required. This degree of monitoring would be quite impractical in studies involving many more than the ten orifice plates and valves used here. There is certainly room for improvement in the flow control, but with regular monitoring the errors in flow balance are tolerable. Chapter 6. Physical Experiments 96 6.4 Laser Doppler Velocimetry Measurements are made using a TSI Inc. back scatter, two component laser doppler ve-locimetry (LDV) system, powered by a Coherent 7 W argon ion laser. The measurement volume obtained where the two 514.5nm (green) beams cross is an ellipsoid, approxi-mately 150fim in diameter and 2.0mm in length. The velocity is measured in a direction in the plane of the two beams and orthogonal to the long axis of the ellipsoid. The long axis of the measurement volume is roughly one tenth of the characteristic jet size, b, and one fifth of the jet width of 9.5mm, and is therefore considered to be sufficiently small for making measurements near the jet collision point. A time trace for the u velocity at point 2 (see Fig. 1.1 for location) between the centre pair of jets is plotted in Fig. 3. The mean and rms velocities are given by u/Uj = .026 and Vu^/Uj = .277 respec-tively. Oscillations at relatively low frequency can be distinguished from the turbulence, although the amplitudes of the turbulence are of comparable magnitude. The oscillations are somewhat irregular. They do not exhibit the perfect periodicity of the CFD results. 6.5 L D V and C F D t ime series comparison The closed fioor CFD case described in Chapter 5 is similar to the present experimental case, but more simplified. The CFD case has a level floor at an elevation equivalent to the higher limit of the sloping floor in the experiment. The cavity plan area below the jets is expected to be more important than an accurate representation of the slightly sloping floor geometry, as this area limits the growth of the recirculation zones there. In retrospect, it would probably be better to locate the CFD case floor at the mid point betwen the upper and lower limits of the sloping floor, giving the same plan area below the jets, however, the error in plan area is only 6.2%. The CFD case only includes one pair of jets bounded on either side by planes of Chapter 6. Physical Experiments 97 symmetry, rather than the five pairs of opposing jets, and should therefore be most representative of the centre pair in the experiments. The CFD calculated time series for U-velocity at point 2 on the centre plane between the jets, using the standard level two grid, is re-plotted in Figure 6.4, using the same time scale as the LDV result for comparison. The experimental oscillations are somewhat irregular, but the oscillation cycle times are similar and appear to range from roughly a factor of two higher and lower than the fixed period of the CFD result. The LDV Fourier spectrum of Figure 6.4 supports this. There is a peak at /* = 0.1 and two peaks near /* = 0.4 observable in the linear-log plot at low frequency, which can be compared to a Strouhal number of 0.229 for the CFD period, based on the same time scale of •Q21 ji/2 Notice in the log-log plot over the entire measurable frequency range that there appear to be three regions; a low frequency peak and decay, flatter mid range and a high frequency -5/3 decay region, showing a distinction between the frequencies of the mean flow unsteadiness and the higher frequencies of the turbulence. The oscillation amplitudes can be compared by considering both the unsteady mean flow and the turbulence in the CFD results, together. The U-component of velocity at point 2 in the CFD model can be expressed as the sum of ensemble mean and turbulent parts, U2{t)^U2{t)^u'^{t). (6.1) The root-mean-sqare (RMS) velocity evaluated over an oscillation period in t ime would be expressed as lul = ^U^-Vu'i + 2U2u'^, (6.2) where the correlation term 2f/2W2 = 0 because the turbulence and mean flow can not be correlated in the k — e model. The Chapter 6. Physical Experiments 98 turbulent velocity, or turbulent intensity (TI) term is expressed in terms of the computed specific turbulent kinetic energy, k, as & = ^=^1^ Uj ' ' Uj So we have 2 ^ + Th'. (6.3) The root-mean-square of the ensemble mean velocity is computed from one cycle of the CFD time series data of Figure 6.3 as ^ =0.1725, UJ and the root-mean-square turbulence intensity is calculated from the time average CFD result for k as ^== J2/3k^ = 0.155, v^=Vp UJ so that Equation (6.3) gives ^ ( .1725)2+ (.155)2 = 0.232, UJ which is 16.2% lower than the experimental value of 0.277. Expt For the refined level two grid (2R) CFD result from Chapter 5, Equation (6.3) gives = Y / ( .1688)2-K (.180)2 = 0.247, UJ which is only 11 % lower than the experimental result. As discussed in an earlier section, the experimental flow rates, and hence the velocities in the model were systematically about 10 % higher than their set points, with a standard error of ± 2.2 %, and drift Chapter 6. Physical Experiments 99 adjustments of roughly ± 4 % being required. The level two grid (2R) CFD results therefore agree with the experimental result for the total RMS velocity at point 2, within the range of experimental errors. It is also interesting that the mean flow and turbulent RMS velocities are of comparable magnitude in the CFD results. 6.6 Flow Visualization Flow visualization is done with a laser light sheet from a plano-concave cylindrical lens. The flow is seeded with polystyrene seed particles of about 300 fim diameter and specific gravity of 1.05. The lens has a virtual focal length of -6.4 mm, aperture of 6.35 mm, and a clear aperture of 80 % of this value, giving an expansion half angle of 21". The thickness of the resulting light sheet is about 5 mm, which is narrower than the 0.95 cm wide tertiary jets. A vertical cross section of the model can be entirely illuminated from below by sending the sheet upward, through the clear plexi-glass floor. Horizontal cross sections can be illuminated from the sides of the model. The particle motion is recorded in real t ime with a Super VHS video camera at 30 frames per second (fps) and a shutter speed of 1/60 s or faster. A still camera with a motor winder is used to record sequences of streak photographs at 2.25 fps and a slow shutter speed of 1/4 s. The reference time scale based on furnace depth of B'^/J^'^ = 2.19 s gives a dimensionless shutter speed of j l / 2 Atls = A i , - ^ = 0.114 or, in terms of the time scale based on jet size of b/Vj = 2.04 x 10"^ s At% = A t , ^ = 12.25, 0 which is of the order of one time step in the CFD calculations. Each frame captures a streak flow pattern over the shutter opening time, as in Figures 6.5 and 6.6 for a vertical plane through the centre pair of jets, where the jet impingement Chapter 6. Physical Experiments 100 point is maximally off centre to the left and right respectively. As observed by LDV, the oscillations are somewhat irregular in time, and so are the patterns observed here. Figure 6.6 is chosen as a pattern that best represents the CFD result of Figure 6.7. The asymmetric geometry of the sloping floor may have a small influence as the flow behaviour is not quite symmetrical. There may also be some slight imbalance in jet velocities due to the flow rate measurement and control difiiculties described in a previous section, possibly causing some asymmetry in the flow. The nearest neighbor jets on either side of centre exhibit similar behaviour when viewed with a vertical light sheet through their centres. In vertical planes between neighbor jets the large recirculating patterns also appear, as seen in the calculations, which tends to suggest that this part of the motion is closely in phase along the array. However, when viewed in a horizontal plane through all the jets, the impingement points do not always move together in phase, but appear to be moving, at times, up to 90° out of phase with one another. This result is not surprising. Firstly, there may be an instability between neighboring pairs of opposed jets causing spanwise asymmetry in the flow. This would not be evident in the CFD calculations that impose symmetry on either side of a central pair of jets. Secondly, the turbulence is rather large compared to the mean flow near the impingement region and this may also account for some of the eff^ ects observed. The shutter speed, giving At*g = 0.114, is comparable in duration to some of the faster flow reversal periods evident in the LDV time series plot of Figure 6.3a). The inverse of this shutter speed is equivalent to a dimensionless frequency of / ; B = ^ = 8-77 which is a little higher than the beginning of the rather flat mid-frequency range of the LDV spectrum in Figure 6.4a), suggestive of the transition region between the mean flow and turbulence. The streak-line photographs may be capturing flow patterns due in Chapter 6. Physical Experiments 101 part to the onset of turbulence as well as mean flow unsteadiness. Certainly the higher frequency turbulence is evident as a waviness and blurring of the streaks. 6.7 Particle Image Veloc imetry The Super VHS video record of the seeded flow is digitized as a series of frames for analysis of velocity fields using a particle image velocimetry (PIV) technique. Successive frame pairs are cross-correlated using the methods developed by Ketler (1993) [27] to calculate the particle displacements, and hence velocities in a grid of rectangular cells. A 30 s long record, comprising 900 frames at the rate of 30 frames/s, is analyzed here and displayed as a series of vector fields. At this frame rate one can observe the highly turbulent motions present. It is also interesting to average over a number of frames to smooth out the higher frequency motions. The 30 s record is shown in Figures 6.8 to 6.10 as a sequence of 2.5 s averages. The location of the jets is indicated on each image by two bars denoting the top and bottom edges of the orifice. Notice that the PIV method does not pick up the jet velocity as it is too high, and the particles travel too far between frames to be detected by the cross-correlation procedure. The largest velocities detectable are about one quarter of the jet velocity. This is sufficient to capture the flow patterns that occur in the cavity, which are quite variable, even over the 2.5 second averaging intervals. One can see patterns similar to those of the CFD results whereby the flow away from the impingement region is directed either to the right or left, and larger and smaller recirculation zones appear. A sequence of 3, 10 second averages, and one frame averaged over the entire 30 s are shown in Figure 6.11. Between successive 10 second averages, the patterns fluctuate only slightly, and are very clearly defined. The 30 s average frame is a very smooth and nearly symmetrical pattern. The downward fiow toward the bottom centre of the field Chapter 6. Physical Experiments 102 of view moves slightly to the left. This is probably due to the shape of the floor, which slopes downward from left to right in the view shown, and possible asymmetry in the actual jet velocities due to flow control and measurement errors. This frame is plotted again in Figure 6.12 in the same axis box as used for the CFD results. In comparing it to both the time average and enforced symmetry CFD results for the corresponding closed floor case. Figures 6.13 and 6.14 respectively, one observes very close agreement with the t ime average flow. The centres of the recirculation zones, below the jets, are much higher in the time average flow than in the enforced symmetry flow. They are about 36 below the jet axes in the former, and more than 6b below in the latter. This supports the conclusion that the mean flow in the experiments exhibits an unsteady behaviour. Chapter 6. Physical Experiments 103 Outflow 1 Figure 6.1: Schematic of Experimental Apparatus; Vertical Plane. Chapter 6. Physical Experiments 104 4 Sc,/2 Point 2 W ^^^p^^m^^m^^^^m^^z^^^^^m^/. 2B Figure 6.2: Schematic of Experimental Apparatus; Horizontal Plane. Chapter 6. Physical Experiments 105 0.75 Time [t/{B^/f'^)] 0.75 0.50 3 0.25 - | I I r I I I I I I I I I I I I I I | --0.50 0.75 I ^ 1 1 ' 1 ' ^ ' L I I I L 14 16 18 20 22 24 26 28 30 32 Time [\/(B-"/S'")] Figure 6.3: Time series for U2- a) LDV result from experiment, b) computed closed floor result. Chapter 6. Physical Experiments 106 O) a 0) Q o 0) Q. CO 1 _ (U o a. CO « c (D Q "cO -#—* o Q. CO L . 03 O a. 1.00 0.75 0.50 0.25 -0.00 Frequency f* [fB^/J^'^J Frequency f* [1B^/J"^] Figure 6.4: Fourier spectrum of LDV result from experiment: a) log-log scale over mea-surable frequency range, b) linear-log scale at low frequency. Chapter 6. Physical Experiments 107 Figure 6.5: Transient streakHne photograph from experiment, when jet impingement is maximally ofF-centre to the left. Vertical plane through jets. Chapter 6. Physical Experiments 108 Figure 6.6: Transient streakline photograph from experiment, when jet impingement is maximally off-centre to the right. Vertical plane through jets. Chapter 6. Physical Experiments 109 20 15 T 1 -I i 10 0 -r ^ I I I I i I 1 I r , , < \ I I rt t t f / / / / / ^ ^ , , ^ \ \ M M t t t f f / / / / , , , , , . - . V N W W W W V V I I 1 r ~» ^ V \ \ \ * ^ V V V ( \ I 1 ' I t I I t ) ( ^ / / I I I I ^ , , . ^ v . x \ \ \ \ \ N N s . . _ ;;;-:^ ^ \^\m :^:::::::::::: f / f / t f t t t t t \ \ \ t ( \ \ \ \ N '• •< y /• / / / I I I • t \ \ \ \ V ^ J I L. J I I I I L J L J 0 5 V 10 15 20 Jet Figure 6.7: Closed floor case transient flow when U2 = maximum {t* = 2190). Vertical plane through jets. Chapter 6. Physical Experiments 110 \ t \ f • v - \ • \ • ' \ / t \ 1 ' ( / ^ ' > ' t _L / -/ / / / /, ,^ \ -/ \ / { 1 ' \ / / \ 1 \ \ 1 —1 ^ c ' * \ ^ ' S \ V •*^  \ \ -* k \ 1 --• / ---\ X • - • • • ~-» y ' \ ' - » ^ ^ ^ / 1 , 1 1 t --\ / I --V / ' --/ y, , --~^  \ ( 1 / / 1 \ t -1 / / *-' / 1 — ' — / -\ / ->» / 1 X jr-' \ ^ -/ -1 \ 1 1 — / / --X \ \ • 1 -\ ^ • • ^ -t / -' • — ' • t -1 . ^ • 'Jet t = 0 to 2.5 s 'Jet t = 2.5 to 5.0 s / - -• ' \ -\ - \ • • -^  t i ^ -\ -^ ' 1 \ X -V -• V • -• -• \ V y /' ^ 1 1 --/ / ) / -1 \ • \ " i \ \ V ,_ / -• •V / / -\ ' 1 / \ \ \ / --1 ' ( ^ \ -**" --/ / / ' • \ X -/ -* - * . • ^ • • -' 1 1 \ -- • V • -• y • -/ / ---V - 1 — I — V --> •K d 1 / ' X -f --1—1—1 t -\ 1 .•— / ( 1 . , 1 ' • \ \ / / / \ \ ~ f ~ 1 • •^  \ / t \ \ t / V V y ^ \ \ \ t -\ ' • \ / 1 1 \ -^-t . 'Jet t = 5.0 to 7.5 s 'Jet t = 7.5 to 10.0 s Figure 6.8: PIV flow fields, 2.5 s averages between 0 and 10 s. Chapter 6. Physical Experiments 111 1 --\ / \ \ - K • X 1 \ / -• -• v -N ' 1 -»^. V -' • -> d 1 t t -^ \ ,^ \ 1 / -' -/ \ ( / / X > -• v 1 I t / J^ < ----s, / / \ ' — \ — 1 ' -1 1 • 1 -i _ . . • .^ *-\ / t ^ t \ -.^ '-t \ t • \ -1 / / ~ X / \ • -. t - / \ — -/ -• t • N --• ^ — 1 — -\ ---' V --' 1 1 ( -j ^ 1 ' • t -/ \ / -/ / T ' — --( •«*. \ \ 1 1 1 \ -— / / N. * \ \ , —r---/ • -I * f c V -/ "^ -• ' ---\ --• / -( y • t -' ( • 'Jet t = 10.0 to 12.5 s 'Jet t = 12.5 to 15.0 s • --— • . / A • V *^  • ' -/ -/ -\ / \ -^  -' ' X 1 / -/ / .z_ / -X ^ / M' / ,/, / \ > 1 1 / / / /, T ' — --/ / / / f --/ \ -\ \ 1 • -— / \ \ 1 , • V - • 1 " \ ^ t / • , / . --^  ^ ^ - . - • / -_ \ -^' t , 1 > / ^ - 1 \ / _ - *-- ^ — 1 — V - --^». -/ ^ j ^ \ \ • ' / ) / / / • y ^ / / / -1 ( / / T ' — / -/ ^ • \ t V f / -/ 1 \ \ V \ \ --f \ ' \ -V V ---y — \ / > \ • t . V X V " \ ' Je t t= 15.0 to 17.5 s 'Jet t= 17.5 to 20.0 s Figure 6.9: PIV flow fields, 2.5 s averages between 10 and 20 s. Chapter 6. Physical Experiments 112 ^ . \ ^ ^ \ t / > / \ ( \ \ •V ( —r-\ -• 1 -*-* v -u. -V -' / / y V • ---I / / / / / V \ -/ ' / \ t UJL_ V / '**. ---• \ •'—1—1 ---' 1 ~ ^ . 1 , ' - • -• • " / \ 1 / X \ — ' V . ^ - • V ' \ f \ \ \ -' \ f -t ( y " --* s k V >. / -1 I -y ' \ y > 1 • j f ' -• t -X 1 \ / --y \ -^ \ / Jf ---/ • * ^ ^ / -\ 1 / 1 / \ ' / 1 ' -' y X y -t --\ V - • -^ -/ / t -- • " 'Jet t = 20.0 to 22.5 s 'Jet t = 22.5 to 25.0 s • / --) -/ \ ~^ -\ , X. V — K ^ -^ --' — 1 — 1 1 -• -» J ^ ' -/> ^ \ ' 1 --~ ' / / ^ ' 1 / ) ( ' 1—'— \ • V \ \ \ \ \ \ 1 \ • V -• \ \ \ -* . • 1 ( -' ---f " -• / / 1 / -/ / - • --.^^ t / ( 1 ' 1 t ( ^ > V y^ t 1 / ^ \ / / -^ \ / \ ' , •^—r-V -' • ---1 \ \ ^ -*. 1 / " y / / t \ V -/ / / 1 — 1 — 1 \ • \ V / V / \ \ \ -X ' \ -\ \ \ ' — 1 — --• » ---\ \ . - J 1 / -j r ' \ \ / -/ \ -* " / ^ t / -\ -'Jet t = 25.0 to 27.5 s 'Jet t = 27.5 to 30.0 s Figure 6.10: PIV flow fields, 2.5 s averages between 20 and 30 s. Chapter 6. Physical Experiments 113 \ - ' * v \ 1 --\ --\ ' / 1 \ X -' ---A -' -• 1 / -~ • / / / • / 1 -\ / / / \ 1 1 — ^ — ' -' / / t \ \ ---• • t V V 1 / 1 • ( ' X V -• --\ ' f 1 ' -/ / " — / \ 1 -/ • -- ' V / \ \ - / \ 1 ' y \ 1 -V -- < — 1 — ----' " ' .1—1 1 V -/ 1 / 1 / / • t -I 1 \ \ / / 1 ' ( --1 1 \ \ \ / --^ / \ \ \ \ \ — 1 — ' -\ -• -\ v -' • V ' / -/ --/ "^ / • / / . 'Jet t = 0 to 10.0 S V, 'Jet t= 10.0 to 20.0 s • -- ' x" \ 1 \ --\ • X > V -^ 1 -1 \ \ -1 \ -----' -* ' \ t / ^ -• / -( ) / / / / 1—^— / ( / » I ( • 1 -' V \ \ \ \ 1 \ r ( --' ' --^  1 b /f • » , ' ' ^ » / 1 / 1 t — / -/ f , / 'Jet t = 20.0 to 30.0 s -- ' -1 t \ - \ \ • \ ^ t t \ X --1—p-X ---' ^ 1 / -\ 1 / • • / t • I ( / / / / \ -/ / 1 1 ( / \ --' \ \ \ \ ^ • -/ / -----X • / / M ' -( / " / / / -/ / -X -• -V. Jet t = 0 to 30.0 s Figure 6.11: PIV flow fields, 10 s averages between 0 and 30 s, and a 0 to 30 s average. Chapter 6. Physical Experiments 114 20 15 10 5 0 -5 --^h. \ - \ - \ - \ - \ • i t \ X "^ 1 r 1 -^r \. ' V -.«* ^ 1 1 T 1 ^ ^k . % f / • • / 1.. 1 t 4 \ / / / / / 1 1 ' V •^  1 1 1 \ "^ ^^ \ \ \ \ \ J 1 L "T -^ -«> / / ^ •^ 1 . 1 — \ — r -^ .0^ \ i / y ^ 1 L 1 -— f f ' / -/ -t — / -t -• -X -0 10 15 V Jet Figure 6.12: PIV flow field: 30 second average, axis box of closed floor CFD case. Posi-tions of recirculation centres agree with time average CFD result. Chapter 6. Physical Experiments 115 20 15 - I — 1 — I — | — 1 — r - , ^ \ \ \ t ' f t t t t f f \ t f t t f t t \ I I ( 10 0 y t / , t / , I / , I / , t t , I I , t I , I I . I \ . ^y.'.U^^^^ 1 I I I I ) t t / rt t t -t t t t t t T t t A t \ -t t 4 1 J \ \ < g « < < I I I '\ I T 1 r I L ( I ' n / 7 r::r:z::r^» ^  ^ <<-^ 'Trrr;:~; i'^ -. . • ' ' M j ' ' ' M M I ' ' M M \ ^ ' ' ' I i \ \ \ - - - ' / / / M M ^ - - ' / / / M M ^ ' ' ' ' I t \ \ \ \ '- - ' •' ' I I \ \ \ \ '• — ' ' / / / ( I » \ - ^ \ \ \ , > > t t , , I I t , , / / / . - / / / ^ - ^ / / ^ - ^ / / ^ - , / / ^ - ^ X / t 1 t f f f / / V t t t 1 t t t f T ( ^ \ t-( t-( t J I I I I I i__j ' I ' l l I I ' ' ' ' 0 5 10 15 20 ^Jet Figure 6.13: Computed closed floor time average of oscillating flow. Vertical plane through jets. Chapter 6. Physical Experiments 116 20 1 15 10 0 T W T t t t t t t t f -1—r 4-\ \ ^ '-. — ^ ^ > > > : » a j > ' ' • < ' < < g c < g < 7 /» ^ y:!^ \ \ \ i t f / 1 t f / / / J f f f / / r f t t f / -t t t f f / t t t f t ! 1 1 1 t 1 1 1 1 t -tt t M \ -t t M \ \ / ' ' - . . W ^ V \ \ M t t 1 " I " M l ' ' i \\ M \ \ \ \ V - , 1 , w \ ^ / / n n^^" ^ / V V ' ' t \ \ \ t 1 1 \ \ t ! t t-t t t t t t-t t t t t t t t f t t t / t t t t-t-/ / / M f f / / / / / f r ^ / f f t t ^ ^ ^ y / t t A-I I 0 5 V 10 15 20 Jet Figure 6.14: Computed enforced symmetry steady flow. Vertical plane through jets. Chapter 7 Conclusions and Recommendat ions 7.1 Conclusions From two dimensional numerical computations for symmetrically fired opposing jets in cross flow an instability is found at low cross flow velocities which leads to two sta-ble, asymmetric steady states. The initial guess for the solution determines which of these two states is realized. The asymmetric steady states are rather stable to transient perturbations, although a strong enough disturbance can reverse the asymmetry to the opposite state. Enforced symmetry calculations, while they reduce computational cost, are inadequate to capture this behaviour. Three dimensional computations for symmetrically fired opposing jets in cross flow also exhibit an instability of the flow at low cross flow velocities. The resulting flow, unlike the two dimensional case, is unsteady with a regular oscillation. A similar oscillatory flow is found for opposing jets in a closed floor cavity without a cross flow. The oscillations are symmetric in that the flow patterns become mirror images of themselves every 180° in phase. Instantaneously, the flow patterns are always asymmetric. Time averaged results, however, are symmetric, but significantly different from the enforced symmetry steady state results. Enforced symmetry computations are again inadequate for the three dimensional case, and a fully transient analysis is required even for the time average behaviour. The oscillatory nature resulting from the flow of opposing three dimensional turbulent jets has never been reported before to the author's knowledge. 117 Chapter 7. Conclusions and Recommendations 118 Tracer analyses done numerically for cases of opposing jets in cross flow show that the time average mixing in the unsteady flow is much more uniform than that of the enforced symmetry steady flow. The instantaneous mixing, however, is of a similar degree of non-uniformity to the enforced symmetry steady flow, so the unsteadiness does not significantly enhance the mixing between jets and cross flow. The magnitude of a turbulent Reynolds number, based on the turbulent viscosity in the numerical computations, is of the same order of magnitude as the true Reynolds number, based on molecular viscosity, at which regular oscillations occur in the laminar RIM flows reported by Wood et al. (1991) [47]. This explains the similarity in the unsteady behaviour observed in both turbulent and laminar regimes. Flow visualization and spectral measurements of velocity in physical experiments confirm the oscillatory nature of the flow for three dimensional opposing jets in a closed floor cavity. The actual flow is less regular than that computed, partly because successive jets in a row do not oscillate in phase with each other. The total root mean square velocity determined from both the mean and turbulent components of the motion in the numerical results agrees with that from the experiments very well. When three dimensional opposing jets are directed so that one jet is offset slightly with respect to the other, the nature of the results predicted numerically are essentially unchanged from the in-line case. The oscillation is no longer symmetric in that the instantaneous flow patterns never become mirror images of themselves. The oscillation frequency is nearly unchanged, however, there is a doubling of the actual period due to the asymmetry. This shows that the essential behaviour would be insensitive to asymmetries in firing due to manufacturing or operational tolerances in furnaces. The accuracy of the numerical solutions to the governing equations considered in this work has been addressed by considering the effects of t ime step size, convergence level attained at each time step, and grid dependence. Time step and convergence level Chapter 7. Conclusions and Recomniendations 119 requirements were investigated first on a preliminary grid by running computations over at least one oscillation cycle. Grid dependence was then tested by using three, successively finer grids which confirms that the results are rapidly approaching grid independence on the two finest grids. The results are quite insensitive to the exact boundary conditions used at the walls. free-slip and no-slip conditions using wall functions give very similar results. By using the multi-grid method, the computing cost required to attain the necessary level of convergence was reduced by about one half. As well, the multi-grid method is useful for grid refinement: once a periodic oscillation is reached on a given grid level, the solution can be prolongated to a finer grid and transient computations continued there to obtain a periodic result much more quickly than by starting from an arbitrary intial guess. 7.2 Recommendat ions 1. The computation of full arrays of opposing jets should be done to study the spanwise instability and asymmetry observed experimentally. 2. Experiments should be designed with only two opposing jets between side walls for better comparison with corresponding numerical computations. The conclusions regarding the very slight effects of boundary layers in the numerical predictions may not be true for these side walls, and a no-slip treatment should be tested there. 3. The k — e turbulence model used in this work has limitations due mainly to the inherent use of an eddy viscosity which does not account for the different Reynolds stress components. More sophisticated turbulence models should be applied to the ChapteT 7. Conclusions and Recommendations 120 simpler problem of two opposing jets to assess their advantages and suitability for analysis of more complex furnace flows. 4. Interlaced jets are in common use in furnaces. These should be investigated in a manner similar to this work by first considering a single jet on one wall and two offset half-jets cut by vertical planes of symmetry on the other wall. The problem could be extended to two jets on one wall interlaced between three on the other, and then longer arrays. 5. The minimum discrete jet spacing considered here is S/b = 4. It would be interest-ing to investigate the nature of the unsteadiness for narrower jet spacings which ap-proach the two dimensional limit characterized by stable asymmetric steady states. 6. The implications of gross unsteadiness for industrial furnace design and practice are significant. Transient analysis for a full furnace represents a highly increased computing cost and experimental measurements are more difficult to make in an unsteady flow. The question of whether or not unsteadiness is a desirable phe-nomenon in furnaces must therefore be answered. The results of this work show that mixing of the jet fluid with a cross flow is not improved by unsteadiness, how-ever, a flnal answer may depend on full simulations and or experiments including combustion and chemical processes. Bibliography 1] T.N. Adams and Wm. J. Frederick. Kraft Recovery Boiler Physical and Chemical Processes. The American Paper Institute, New York, 1988. 2] U. Ascher. Multigrid and multilevel methods. Lecture Notes, Department of Com-puter Science, The University of British Columbia, 1991. 3] K.N. Atkinson, Z.A. Khan, and J.H. Whitelaw. Experimental investigation of op-posed jets discharging normally into a cross-stream. Journal of Fluid Mechanics, 115:493-504, 1982. 4] J. M. M. Barata, D. F. G. Durao, and J. J. McGuirk. Numerical study of single impinging jets through a crossflow. J. Aircraft, 26(11):1002-1009, 1989. 5] B. Blackwell, C. Hastings, K. Der, J .W. Quick, and S. Hanson. Improved gas mixing in kraft recovery boilers: air jet theory. Technical Report V7582, Department of Energy, Mines and Resources Canada, Ottawa, Ontario, November 1990. 6] B. Blackwell, R. Olivier, and B Briscoe. Physical flow modelling of a vintage com-bustion engineering kraft recovery boiler. In Proceedings of the CPPA Technical Section Conference, May 1989. 7] R.D. Blevins. Applied Fluid Dynamics Handbook. Van Nostrand Reinhold Co., New York, N.Y., 1984. 8] A. Brandt. Multi-level adaptive solutions to boundary-value problems. Math. Comp., 31(138):333-390, 1977. 9] A. Brandt. Multi-level adaptive computations in fluid dynamics. AIAA J., 17(10):1165-1172, 1980. [10] T.E. Brock. Fluidics Applications: Analysis of the Literature and Bibliography. The British Hydromechanics Research Association, Cranfield, Bedford, 1968. [11] Forbes T. Brown, editor. Advances in Fluidics. The 1967 Fluidics Symposium, Chicago, Illinois, May 9-11 1967. ASME Fluidics Committee, ASME. [12] G. D. Catalano, K. S. Chang, and J. A. Mathis. Investigation of turbulent jet impingement in a confined crossflow. AIAA Journal, 27(11):1530-1535, 1989. 121 Bibliography 122 13] F. J. Davison. Nozzle scaling in isothermal furnace models. Journal of the Institute of Fuel, pages 470-475, Dec. 1968. 14] A. 0 . Demuren. Numerical calculations of steady three-dimensional turbulent jets in cross flow. Computer Methods in Applied Mechanics and Engineering, 37:309-328, 1983. 15] A. 0 . Demuren. Modeling turbulent jets in crossflow. In N. P. Cheremisinoff, G. Akay, et al., editors, Encyclopedia of Fluid Mechanics, chapter 17, pages 431-465. Gulf Pub. Co., Book Division, 1986. 16] A. 0 . Demuren and W. Rodi. Side discharges into open channels: Mathematical model. ASCE Journ. of Hydraulic Engineering, 109(12):1707-1722, Dec. 1983. 17] A.O. Demuren. False diffusion in three-dimensional flow calculations. Computers and Fluids, 13(4):411-419, 1985. 18] P. Dykshoorn. Process recovery boiler flow model tests. Technical Report RDD:87:6940-01-01:01, Babcock and Wilcox, Barberton, Ohio, July 1986. 19] W. A. Fiveland and R. A. Wessel. Numerical model for predicting performance of three-dimensional pulverized-fuel fired furnaces. Journal of Engineering for Gas Turbines and Power, 110:117-126, January 1988. 20] A. D. Gosman and R. Simitovic. An experimental study of confined jet mixing. Chemical Engineering Science, 41(7):1853-1871, 1986. 21] A.D. Gosman. TEACH-T, a computer program for the calculation of two dimen-sional turbulent recirculating flows. Department of Mechanical Engineering, Imperial College, London, England. 22] T. M. Grace. Increasing recovery boiler throughput. TAPPI Journal, pages 52-58, Nov. 1984. 23] T. M. Grace, June 22 1990. Personal communication. 24] A. K. Jones. A Model of the Kraft Recovery Furnace. PhD thesis. The Institute of Paper Chemistry and Lawrence University, 1989. 25] W.P. Jones and J.J McGuirk. Computation of a round turbulent jet discharging into a confined cross fiow. Turbulent Shear Flows II, Springer Verlag, New York, pages 233-245, 1979. 26] Y. Kamotani and I. Greber. Experiments on confined turbulent jets in crossflow. NASA Report, NASA CR-2S92, March 1974. Bibliography 123 [27] S.P. Ketler. Physical Flow Modelling of a Kraft Recovery Boiler. M.A.Sc. thesis, Department of Mechanical Engineering, The University of British Columbia, 1993. [28] Z. A. Khan, J.J. McGuirk, and J.H. Whitelaw. A row of jets in a cross flow. AGARD CP 308, paper 10, 1982. [29] Z. A. Khan and J. H. Whitelaw. Vector and scalar characteristics of opposing jets discharging normally into a cross-stream. Int. J. Heat Mass Transfer, 23:1673-1680, 1980. [30] B. E. Launder and D. B. Spalding. The numerical computation of turbulent flows. Computer Methods in Applied Engineering, 3:269-289, 1974. [31] V. Llinares, Jr. and P. J. Chapman. Combustion engineering update three level air system retrofit experience. In TAPPI Engineering Conference Proceedings, pages 629-639, 1989. [32] J. J. McGuirk and W. Rodi. A depth-averaged mathematical model for the near field of side discharges into open-channel flow. J. Fluid Mech., 86:761-781, 1978. [33] R. Mikhail, V.H. Chu, and S. B. Savage. The reattachment of a two dimensional turbulent jet in a confined cross flow. Proc. 16th lAHR Cong., Sao Paulo, Brazil, 3:414-419, 1975. [34] P. Nowak. Calculations of transonic flows around single and multi-element airfoils on a small computer. Technical Report 84-48, Delft University of Technology, Delft, 1984. [35] P. Nowak. 3d segmented multi-grid computing code. Technical report. Department of Mechanical Engineering, U.B.C., Vancouver, B.C. Canada, 1991. [36] S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere, 1980. [37] M. S. Perchanok, D. M. Bruce, and I. S. Gartshore. Velocity measurements in an isothermal scale model of a hog fuel boiler furnace. Journal of Pulp and Paper Science, 15(6):J212-J219, November 1989. [38] G. F. Robinson. A three dimensional analytical model of a large tangentially-fired furnace. Journal of the Institute of Energy, pages 116-150, Sept. 1985. [39] W. Shyy. A general coordinate system method for computing transport phenomena. Heat Transfer 86 Conference Proceedings, 2:397-401, 1986. [40] S. Sivasgaram and J. H. Whitelaw. Flow characterisstics of opposing rows of jets in a confined space. Proc Instn Mech Engrs, 200(Cl):71-75, 1986. Bibliography 124 [41] R. L. Stoy and Y. Ben-Haim. Turbulent jets in a longitudinal cross flow. ASME Journ. of Fluids Engineering, pages 551-556, Dec. 1973. [42] M. W. Thring and M. P. Newby. Combustion length of enclosed turbulent jet flames. In Fourth Symposium on Combustion, pages 789-796, Baltimore, 1952. Williams and Wilkins. [43] D. Tse. Multiple Jet Interaction with Special Relevance to Recovery Boilers. PhD thesis. Department of Mechanical Engineering, The University of British Columbia, 1994. [44] S.P. Vanka. Block implicit multigrid solution of the navier stokes equations in prim-itive variables. Journal of Computational Physics, 65:138-158, 1986. [45] A. Verloop, T. W. Sonnichsen, and 0 . Strandell. An overview of recovery boiler performance evaluations. TAPPI Journal, pages 145-152, March 1990. [46] A. Vranos and D. S. Liscinsky. Planar imaging of jet mixing in crossflow. AIAA Journal, 26(11):1297-1298, 1988. [47] P. Wood, A. Hrymak, R. Yeo, D. Johnson, and A. Tyagi. Experimental and compu-tational studies of the fluid mechanics in an opposed jet mixing head. Phys. Fluids A, 3(5):1362-1368, May 1991. [48] F.D. Yeaple. No moving parts for fluid amplifiers. Product Engineering, 31(11):17, March 14 1960. Appendix A Wall Shear Stress Effects The duct walls are treated with a free slip condition in much of this work, under the hypotheses that: 1. The boundary layers will be thin, giving little displacement effect from the walls. 2. The wall shear stress may not affect the internal pressure field appreciably over the short lengths of duct in which the jets interact. To test these hypotheses, several cases for opposing jets in a closed floor cavity are repeated using the wall function method of Launder and Spalding (1974) [30] on the two vertical walls and the bottom floor of the cavity. A . l Standard Level Two Grid For the standard level two grid, the effects of wall shear stress are investigated by restart-ing the calculation from a converged, transient result for the free slip walls, and continuing with the wall functions applied. The result is shown as a t ime series in Figure A.2 where the restart has been made from t* = 1920, and continued until t* = 3930. Figure A.2 shows only the final 1000 time units from t* — 2930 to 3930, which gives the same horizontal scale as Figure A.l (the free slip result) for comparison. The result is very nearly periodic towards the end of the time trace. The U2 oscillation amplitude differs the most, at 11 % lower than the free slip case, and the relative phases between f/i, U21 and U-i are slightly changed also. The oscillation frequency, however, is only 1 % higher 125 Appendix A. Wall Shear Stress Effects 126 at Stjg = 0.231. The flow patterns observed are also very similar, but the differences in phase between the impingement point and surrounding core flow motions are evident. Figure A.5 shows the instantaneous flow field when U2 is a maximum. The directions of the core flows above and below the impingement point are more vertical than in the corresponding flow field obtained with free slip walls (Figure A.4). An indication of the suitability of the near wall computational grid with the wall function treatment is given by examining the dimensionless wall coordinate used in the logarithmic law of the wall, y = — , (A.l) V where Ur = J— (A.2) V p is the friction velocity determined by the wall shear stress, T^,. The logarithmic law of the wall is valid in the inertial sub-layer region of the boundary layer, where the shear stress is approximately constant, given approximately by 11.63 <y+ < 2 0 0 . The numerical procedure switches to a laminar sub-layer calculation of shear stress for y'^ below 11.63, but uses the wall function method for any y'^ above this. In the present results, the minimum y"*" values adjacent to the walls are of the order of 100, and the mean values are of the order of 650 on the side walls and 1500 on the bottom floor. This suggests that the grid is not sufficiently refined near the walls to resolve the boundary layers, and therefore the wall shear stresses, accurately. It is shown in Chapter 5 that the results on this standard level two grid are somewhat grid dependent. On a more suitably refined grid, it is shown in the following section that the wall function treatment has a lesser effect on the results. Appendix A. Wall Shear Stress Effects 127 A . 2 Refined Level Two Grid The effects of wall shear stress are revisited for the refined level two grid (2R). This is done by restarting the calculation from a converged, transient result for the free slip walls, and continuing with the wall functions applied. The result is shown as a t ime series in Figure A.3, where the restart has been made from t* = 2950, and continued until t* ~ 3680. After one oscillation, the Ui time trace is very similar to the free slip result. The C/2 amplitude is less than 1 % lower, and the Strouhal number is about 1.4 % lower at Sis — -275. The instantaneous flow fields are shown for the free slip case when C/2 is a maximum [l* = 2950) in Figure A.6, and the wall function case when U2 is nearly a maximum (i* = 3670) in Figure A.7 and at a maximum (t* = 3680) in Figure A.8. Due to a slight difference in phase between the impingement point and surrounding core flow motions, with the wall functions applied, the flow field at one time step prior to U2 reaching a maximum is generally more similar to that for the free slip case when U2 is at a m.axim,um,. However, the comparison between the free slip and wall function results when U2 is a maximum, is better than that shown in a foregoing section for the coarser, standard level two grid. The dimensionless wall coordinate, T/"*", at computational nodes adjacent to the walls, has minimum values of the order of 20, and average values of the order of 500 on the side walls and 350 on the bottom wall. This is somewhat better than for the standard level two grid discussed in an earlier section, but is still not sufficient to satisfy the logarithmic law criterion of y'^ < 200. However, the results strongly suggest that the underlying instability and oscillation behaviour are mainly determined by the jet flows and duct geometry, and are only weakly affected by wall shear stress. Appendix A. Wall Shear Stress Effects 128 3 CD o _o > =) 0.50 0.25 -0.00 -0.25 -0.50 2000 2200 2400 2600 2800 Time [ t / ( b / U j ) ] Figure A. l : Closed floor case time series with free slip walls, showing periodic result. 0.50 I r -0.25 -0.50 I L 3000 3200 3400 3600 Time [ t / {b / Uj) ] 3800 Figure A.2: Closed floor case time series with wall functions, showing nearly periodic result. Appendix A. Wall Shear Stress Effects 129 3 o o o > u.ou 0.25 0.00 0.25' ' 1 ' 1 ' 1 " 1 A— 0— f \ f \ Jf 1 . 1 , 1 , 1 ' 1 ' 1 L = 2R, Free Slip L - 2R, Wall Fns. " • r ^ J -1 , 1 2400 2600 2800 3000 3200 Time [ t / { b / U j ) ] 3400 3600 Figure A.3: Closed floor case time series for U2 on the refined level two grid (2R), showing a restart with wall functions. Appendix A. Wall Shear Stress Effects 130 20 15 10 I — I — I — I — I — I — I — I — I I I — r — 1 — I — r -, , i > \ \ t r t t f t f / / / / y ^ ., _;_,» ^ \ \ \ M M t t f ( / / / / , . -, , . . N N \ \ \ \ \ \ \ \ \ \ V v . ~i , ^ .^"SVWWW \ \ V V ^ - ^ , ^ ^ ^ - . X N \ \ \ \ \ \ \ S V ^ T , J . ^ ^ ^^-.-^VVAXW \ N V -. - - ^ V V V \ \ \ > - ^ \ \ \ V \ \ I 1 ' • • • t I I ( ( I - ' ' ' ' / / ( ( I — ^ ^ / / I I I \ " y ^ / / I I I 5|fi;;;;;;;;;::;:-^\'^; -t 0 / / / -t t t t t i t t t t J M M 4- t \ \ \ 1 \ \ \ \ \ \ \ N V . ^ / t t ^ ^ f t I \ \ \ \ \ \ V -V t \ \ \ >. V V -^ -.* I I I L J - _ l l _ - L I I I L _L__I I L 0 5 V 10 15 20 Jet Figure A.4: Closed floor case transient flow when U2 — maximum (t* = 2190), with free slip walls. Standard level 2 grid. Vertical plane through jets. Appendix A. Wall Shear Stress Effects 131 T — I — I — I — I — I — I — I — I — r — r . t ( \ t ' t t t f f / / / ' / / ^ . ^ V V \ \ 20 » V V \ » ' 1 1 ) 1 ( / 15 10 V X \ \ \ \ \ \ \ \ \ \ \ M I t - . . V N W W W \ \ \ \ \ V X / - V V \ \ \ \ \ \ N X S ' ' J ^ ^ ^ ^ J ^ J f ^ 0 t / t / l-t ' t ( T f A f ^ t i t J \ 4-t -t \ ^ \ ; ; ; ; ; ; ; V ^ ^ ^ ^ ^ C N . ^ ^ - ^ - ' J ^^^^^^""^^^ - ' ' I i \ \ \ \ \ \ \ ^ -\ \ \ \ \ V v v •'' / / / / / J M * ^ " ' ' y / / / I I I I ' \ \ \ V y / / / I J I L J I I L J I I L J l _ J L 0 5 V 10 15 20 Jet Figure A.5: Closed floor case transient flow when U2 = maximum {t* = 3930), with wall functions. Standard level 2 grid. Vertical plane through jets. Appendix A. Wall Shear Stress Effects 132 20 15 10 0 —I—r-f I t I I I I I . 1 1 1 1 1 1 \ \ V \ \ V \ 1 • ^ ^ v v \ v \ v v \ \ -7777r777777777X37JIII3IT t I I I t f f / / / / / / y y ^ . — , ^ I t t t t f / / / / / f / / / y . \ \ \ \ \ \ \ \ \ \ \ \ t t I t , , v W W W W W W \ \ \ \ \ ^ , , . ^ >. v \ \ \ V \ V V \ \ \ \ • I I 1 1 I I ' I I I I I \ \ I ) t t I I I I I I I I I t t t H t t t h t t t t t h t >\\v W \ \ , ^ " M M U \ \V \ -^  ^ \ \ \ \ \ \ W » . ^ ^ ^ \ \ \ X N »^  »^ • ^ • . s . - ^ ^ , . ^ . ^ . , ^ - ^ y / / / / I I \ \ \ •—•'^>' / / / / I I I ' —-'•^^jf y / / / I I I ' ' ' ' I I ' ' ' I I l l 0 5 10 15 20 ^ V, , Figure A.6: Closed floor case transient flow when U2 = maximum [t* = 2950), with free slip walls. Refined level 2 grid (2R). Vertical plane through jets. Appendix A. Wall Shear Stress Effects 133 20 15 10 0 T — r I I t I I I X \ \ T7777r777777777X3::rZI3II 1 I t f M / / / / f / / / /y^^^^-,^^ \ WW w \ \ n 11 f / / /^. V N W W W W W W \ \ \ \ N V « . , • ^ - ^ ^ V V V \ \ \ ^ ^ V v \ \ ^ V V V \ \ ' ' t i l l ' ' I I \ \ \ \-\ \ \ \ 1 I I t I I I r I I w M t t t t t t \ t t w t \ '\ \ > I i WW \ \ \ \ N v ^ -\ \ \ \ \ \ \ w ^ . \ \ \ \ \ \ \ \ V V . .WWW •'' ^' I i www^-^-^ '''yy / / / I I -^y y y / / / I J I \ I I l__l I I I I I I L J I L 0 5 V 10 15 20 Jet Figure A.7: Closed floor case transient flow when C/2 is nearly maximura (i* = 3670), with wall functions. Refined level 2 grid (2R). Vertical plane through jets. Appendix A. Wall Shear Stress Effects 134 rTTTTTTTTTTTTTTX - 1 — I — \ — \ — r V V \ \ V \ \ \ \ \ * * » W i \ M t t I 20 0 n^ -J-i \ \ \ n t t 11 f / f / / /^^. K \ \ \ \ \ \ \ \ \ \ \ \ n t / / ^.— V X \ \ \ \ \ \ \ \ \ U t t t ( / / / , , . S N W W W W W \ \ \ \ \ \ . , , ^-^vvxNwwwwNv...,,;;;;;;;;;;;. / / M I-/ / M I / / / / T / I I \ \ / / I I I I I / r t t M \ \ \ \ \ \ J I \ \ \ \ \ \ \ \ \ \ \ \ X . \ ^ \ \ \ \ \ \ \ N V \ \ \ \ \ \ \ \ W ^ - '^ / / / 1 n \ \ ^ ^ ^ ^ " ~ ' ^-^ / / / / / t M * ^ ^ ' ^ \ \ ^. V -^ *v •* , - ^ . ' ^ - . ^ . . ^ . , — — — — * - ^ ^ ^ ^ • • • / / / I J I I L J I L J I L J I L 0 5 ^ V, 10 15 20 Jet Figure A.8: Closed floor case transient flow when f/2 = maximum (i* = 3680), with wall functions. Refined level 2 grid (2R). Vertical plane through jets. A p p e n d i x B Additional Closed Floor Results In chapter 5 the velocity fields are shown in the vertical plane through the jets for a number of cases of computational grids and time steps used. Here, a more complete set of results is presented for one case: the finest, level three (3) grid used and the refined t ime step of At* = 5, when the U velocity at point 2 is a maximum (i* = 4535). Velocity fields are shown in vertical planes through the jets, in the planes of symmetry between the jets, and in horizontal planes through the jets. The scalar fields for pressure coefficient (Cp), turbulence intensity (TI), and turbulent viscosity (i/^ *) are also shown as iso-contours in the same three planes, where C . = ^ (B.1) TI = y-^- (B.2) Ujb • (B.3) 135 Appendix B. Additional Closed Floor Results 136 TTTTTTTTTTVTTT-TTT^TrrrrTTrT I I I t I t f t f 1 f t f / / >•. , ^ - ^ ^ 20 N. ^ \ \ \ V \ ^ V \ \ \ \ V V 1 i ^ V \ \ \ » » \ \ 1 «. w ^ 15 \ \ N V ^ ^ ^ ^ . ^ - ' ' ' I I t \ \ \ 1 i M » ^ / f I I I i J l__l L J I I L J I I L J L 0 5 V 10 15 20 Jet Figure B.l : Closed floor case transient flow when TJ2 = maximum (i* = 4535). Level 3 grid, refined time step, Ai* = 5, every second vector shown. Vertical plane through jets. Appendix B. Additional Closed Floor Results 137 Level Q P O N M L K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 Cp 1.500E-1 1.400E-1 1.300E-1 1.200E-1 1.100E-1 1.000E-1 9,000E-2 8.000E-2 7.000E-2 6.000E-2 5.000E-2 4.000E-2 3.000E-2 2.000E-2 1 .OOOE-2 5.588E-9 -1.OOOE-2 -2.000E-2 -3.000E-2 -4.000E-2 -5.000E-2 -6.000E-2 -7.000E-2 -8.000E-2 -9.000E-2 -1.000E-1 0 10 15 20 Figure B.2: Closed floor case Cp field when U2 = maximum (t* = 4535). Level 3 grid, refined time step, A^* = 5. Vertical plane through jets. Appendix B. Additional Closed Floor Results 138 Level K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 TI 2.000E-1 1.900E-1 1.800E-1 1.700E-1 1.600E-1 1.500E-1 1.400E-1 1.300E-1 1.200E-1 1.100E-1 1.000E-1 9.000E-2 8.000E-2 7.000E-2 6.000E-2 5.000E-2 4.000E-2 3.000E-2 2.000E-2 1 .OOOE-2 Figure B.3: Closed fioor case TI field when U2 — maximum (t* = 4535). Level 3 grid, refined time step, Ai* = 5. Vertical plane through jets. Appendix B. Additional Closed Floor Results 139 Level L K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 vist 5.000E-2 4.800E-2 4.600E-2 4.400E-2 4.200E-2 4.000E-2 3.800E-2 3.600E-2 3.400E-2 3.200E-2 3.000E-2 2.800E-2 2.600E-2 2.400E-2 2.200E-2 2.000E-2 1.800E-2 1.600E-2 1.400E-2 1.200E-2 1.000E-2 0 10 15 20 Figure B.4: Closed floor case transient u* field when U2 = maximum (t* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane through jets. Appendix B. Additional Closed Floor Results 140 20 15 J 10 0 TTTTTTTTTTn^TTT i i t t t M / / / / / / ' ^ . . V N N W W W \ \ \ \ X V -^ ^ ^ X V N W W \ \ \ V ^ ^  ^ ^ ^ ^ ^ V V \ \ \ \ \ \ \ N >. ^ ^ ^ ^ • - > X N N N \ \ \ \ N V ^ 1—I—I—r—T— *• - * • - • —» • " * - t •«». -». T — I — r • / / / • ^ ^ V \ \ \ \ V ^ ^ V \ \ \ i * ' ' ' I I I I t -^ •< / / I I ~ — -^ — - ' • / ' / ) ^ \ ^ I -I \ 4 ^ ^ ^ ^ / / ^ y ^ yf ^ ^ ^^ : UuVU-^ I t t t t f / / y ^ \ f t t t t / / ^ , . t t t 1 t \ t t I . , . 1 \ \ \ \ \ \ \ \ \ V > / / / M / t 4 ' I i \ \ \ \ \ \ 1 \ \ \ v , \ \ V ' V V ' ^ - v . ' » » - ^ . ^ , _ „ ^ / / i I \ \ \ \ x- • • / / ( I 1 / / I I i I y J I I I I I I I I I I I l __ l I I I I L 0 5 V 10 15 20 Jet Figure B.5: Closed floor case transient flow when U2 = maximum (i* = 4535). Level 3 grid, refined time step. At* = 5, every second vector shown. Vertical plane between jets. Appendix B. Additional Closed Floor Results 141 I I I I I I I I , Level Q P O N M L K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 Cp 1.500E-1 1.400E-1 1.300E-1 1.200E-1 1.100E-1 1.000E-1 9.000E-2 8.000E-2 7.000E-2 6.000E-2 5.000E-2 4.000E-2 3.000E-2 2.000E-2 1 .OOOE-2 5.588E-9 -1.OOOE-2 -2.000E-2 -3.000E-2 -4.000E-2 -5.000E-2 -6.000E-2 -7.000E-2 -8.000E-2 -9.000E-2 -1.000E-1 0 10 15 20 Figure B.6: Closed floor case Cp field when U2 = m,axim,um, (i* refined time step, At* = 5. Vertical plane between jets. 4535). Level 3 grid, Appendix B. Additional Closed Floor Results 142 Level K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 TI 2.000E-1 1.900E-1 1.800E-1 1.700E-1 1.600E-1 1.500E-1 1.400E-1 1.300E-1 1.200E-1 1.100E-1 1 .OOOE-1 9.000E-2 8.000E-2 7.000E-2 6.000E-2 5.000E-2 4.000E-2 3.000E-2 2.000E-2 1 .OOOE-2 0 10 15 20 Figure B.7: Closed floor case TI field when U2 = maximum (i* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane between jets. Appendix B. Additional Closed Floor Results 143 Level L K J 1 H G F E D C B A 9 8 7 6 5 4 3 2 1 vist 5.000E-2 4.800E-2 4.600E-2 4.400E-2 4.200E-2 4.000E-2 3.800E-2 3.600E-2 3.400E-2 3.200E-2 3.000E-2 2.800E-2 2.600E-2 2.400E-2 2.200E-2 2.000E-2 1.800E-2 1.600E-2 1.400E-2 1.200E-2 1 .OOOE-2 0 10 15 20 Figure B.8: Closed floor case transient u^ field when U2 = maximum [t* = 4535). Level 3 grid, refined time step. At* = 5. Vertical plane between jets. Appendix B. Additional Closed Floor Results 144 4.0 2.0 0.0 I ! I 1 1 I Lil'ifcl i J i ' j J f g , S . f, ^ ,: = . ^1 ? . 0 10 15 20 ^ V Jet Figure B.9: Closed floor case transient flow when U2 = maximum (i* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets. Figure B.IO: Closed floor case Cp field when U2 = maxim,um (i* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets. Appendix B. Additional Closed Floor Results 145 Figure B . l l : Closed floor case TI field when U2 = maximum (t* = 4535). Level 3 grid, refined time step, Ai* = 5. Horizontal plane through jets. Figure B.12: Closed floor case transient 1/* field when U2 = maximum {t* = 4535). Level 3 grid, refined time step. At* = 5. Horizontal plane through jets. 

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