Algebraic Reynolds Stress Models

Modeling Turbulence Anisot ropy using Algebraic Reynolds Stress Models

Jyoti Sankar Bose

A thesis submitted to the Department of Mechanical Engineering in conformity with the requirements for the degree of Master of Science (Engineering)

Queen's University Kingston, Ontario, Canada August, 1997

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Turbulence anisotropy has an important role to play in the transport of momentum and heat. Thus a correct representation of anisotropy is an important element in turbulence modeling. Linear eddy viscosity models are popular for solving various flow problems of engineering importance. The inherent inability of these models to capture the effects arising from Reynolds stress anisotropy cdls for the development of nonlinear turbulent models. This thesis is focused on rnodeling turbulence anisotropy using algebraic Reynolds Stress models. which are nonlinear eddy viscosity models. Literature overview is con- centrated upon the potential of algebraic Reynolds stress models to capture turbulence anisotropy. The problem of numerical instability is an important drawback for Rodi's algebraic Reynolds Stress model. A new regularization technique has been developed to prevent any possible singularity in Rodi's algebraic Reynolds Stress model. Rodi's algebraic Reynolds Stress model has been modified to predict inhornogeneous flows correctly. The modified algebraic Reynolds Stress model has also been iniproved to closely represent the original Reynolds Stress mode1 over the entire range of tirne scales for the turbulence and the mean flow strain field. The necessary conditions for an algebraic Reynolds Stress model to predict secondary flow in a square duct has been studied and a general self-consistent algebraic Reynolds stress model lias been proposed for square duct flow. This model has then been extended for inhomogeneous flows. The algebraic Reynolds stress models have been incorporated into an originaliy written general-purpose three-dimensional flow modeling computer program. A new block-colocated numerical scheme has been used as the algorithm. To improve the performance of the solver multigrid acceleration has been used. A higher-order discretization scheme has been applied to improve numerical accuracy. An apparent viscosity approach has been irnplemented to ensure velocity/Reynolds-stress coupling when Rodi's algebraic Reynolds stress model has been used. The assumptions of Rodi's algebraic Reynolds Stress model have been validated using available experimental, direct numericd simulation and large eddy simulation results. The role of the pressure-strain models on the predictive abilities of Rodi's algebraic Reynolds stress model and its modification has been studied by evaluating a fully-developed channel flow and comparing the results with experimental data. Four algebraic Reynolds stress models have been used to solve a fully-developed square duct flow. The results are compared against simulation results for the purpose of calibrating the model constants. Acknowledgement

1 would like to express my sincere gratitude and appreciation to my superviser. Dr. Miodrag D . Matovic, for his guidance, continuous interest. constant helpful criticism and encouragement. His help, cooperation and thoughtful advice during the development of the flow modeling computer program is highly appreciated. 1 also want to thank him for introducing me to the advanced topics in numerical modeling and giving me an opportunity to explore this challenging field. 1 wish to thank Dr. Daniel Ewing for his help, cooperation and valuable discussions during the early stage of my research program. It is a pleaçure to acknowledge my graditude to Dr. Fengshan Liu for innumerable helpful discussions on different issues of CFD. 1 also want to thank him for his tirnely assistance and cooperation during the development of the software. My special thanks are directed towards the Ministry of Education and Training of Ontario and Qiieen's University for financial support through scholanhips and bursars. 1 also want to thank Mr. Hongyi Xu for numerous debates on turbulence modeling and close cooperation on wide range of research tasks. 1 would like to thank Mr. Stuart McIlwain for his efforts in computer system support. 1 am grateful to my friends Mr. Govin Varadrajan, Mr. Yogesh Aurora. Mr. Ptineet Chauhan, Mr. Sanjay Verma, Mr. Saugata Datta. Mr. Ka1 Prasad. Mr. George John and Mr. Raj Maitra for their help. continuous encouragement and support during the research program. Lastly 1 wish to thank my parents for their love, understanding and constant encouragement.

iii Contents

Abstract i .. Acknowledgement III.

Contents iii

List of Figures vii

List of Tables x

Nomenclature xi

1 Introduction 1 1.1 Objectives of the study ...... 2 1.2 Motivation of the study ...... 3 1.3 Literature overview ...... 5

2 Governing Equations and Mathematical Models 8

2.1 The Fundamental Equations ...... 8 2.2 Two-Equation Models ...... 10 2.3 Reynolds Stress Model ...... 13 2.4 Algebraic Reynolds Stress Model ...... 16 2.5 Regularized Implicit Rodi's Algebraic Reynolds Stress Model . . . . . 19 2.6 Modified Rodi's Algebraic Reynolds Stress Model ...... 21 2.7 Irnproved Modified Rodi's Algebraic Reynolds Stress Model ...... 22 2.8 Turbulence Modeling of Square Duct Flow using Algebraic Reynolds Stress Models ...... 24 2.8.1 Introduction ...... 24 2.8.2 Origin of the Secondary flow in a Square Duct ...... 24 2.8.3 Turbulence modeling of a Square Duct flow: Previous work . . 26 2.8.4 Turbulence modeling of a Square Duct flow using Algebraic Reynolds Stress Models: An Analytical study ...... 28

3 The Multigrid code for solving turbulent flow problems 33 3.1 Introduction ...... 33 3.2 The BLOC0 algorithm ...... 33 3.3 The Additive Correction Multigrid method ...... 36 3.4 The SIMPLER-based BLOCO with Additive Correction Multigrid nlethod 35 3.5 The SMARS scheme and its implementation ...... 40 3.6 Implementation of Rodi's Algebraic Reynolds Stress mode1 ...... 43 3.7 Solution Strategy for flow solving developing flow through a Square Duct and a Plane Channel ...... 47 3.5 Boundary Conditions and t heir Implementation ...... 48 3.8.1 The Inlet Boundary Conditions ...... 19 3.8.2 The Outlet Boundary Conditions ...... 50 3.8.3 The Wall Boundary Conditions ...... 50 3.5.1 The Wall Boundary Conditions in a Square Duct ...... 52

4 Presentation and Discussion of Results 54 4.1 Introduction ...... 54 4.2 Evaluation of the Rodi's ASM assumptions using experimental data . 55 4.3 Test of the performance of the solver using Lid Driven Cubic Cavity . 64 4.4 Test of the SMART scheme using Lid-Driven Cubic Cavity flow problem 66 4.5 Numerical Evaluation of a fully-developed Plane Channel Flow using ASMs ...... 61 4.6 Numerical evaluation of a fully-developed flow through a Square Duct using ASMs ...... 73

5 Summary and conclusions 92 5.1 Original contributions ...... 92 5.2 Scope for further research ...... 95

Bibliograp hy 98

Curriculum Vitae 108 List of Figures

Square Duct Bow ...... Typical secondary-flow streamline pattern in a square duct ...... Distribution of (P&) from the DNS results. The minima at y+ = r' r' = 50 is (P&) = 0.2. The maximas at y+ = 50.zi = 12 and yi = 12,zC =50 is (P&) =2.25 ...... - ......

Staggered grid arrangement ...... 2 x 2 grid forms 1 fine grid ce11 ...... Additive Correction Multigrid Cycles. S - Smoothing. R - Restriction.

P - Prolongation, E - Exact solution ...... The locations (x) and variable values (4)along the x-axis for upstream node (U), central node (C) and downstream node (D).Cu, - convective

flux at the west face (w) ...... Staggered arrangement of Reynolds stresses ...... Solution strategy for solving square duct flow, L = n x 1 ......

Axisymmetric turbulent jet ...... - Kinetic energy balance for vw2 = O, Normalized kinetic energy balance - 3 - Kinetic energy equation /(LIc /(x - 20)) ...... - - Kinetic energy balance for vw2 = u3, Normalized kinetic energy balance - Kinetic energy equation /(üC3/(z- IO)) ...... - VU balance for vw2 = O, Normalized W balance - balance equation

(ü3(- O)) ......

vii - - 4.5 TE balance for uw2= v3? Nomalized VÜ balance - vii balance equation

- 4.6 ww balance for vw2 = O, Normalized W balance - balance equa- -3 tion /(LIc /(x - 20)) ...... - - 4.7 W balance for vw2 = v3, Normalized uiui balance - uiur balance equa- -3 tion /(Uc /(x - x0)) ...... - 4.8 Convective transport of aij for vw2 = O. Normalized kinetic energy balance - Kinetic energy balance /(~:/(s - s0)) ...... - . . . - 4.9 Diffusive transport of Reynolds stress anisotropy. v~~(~).for vw2 = 0. Normalized kinetic energy equation - Kinetic energy equation / (t/,3/(r-

- - 4.10 Diffusive transport of Reynolds stress anisotropy, v,(").for vw2 = v3. Norrnalized kinetic energy equation - Kinetic energy equation / (Ü: / (r-

4-11 Lid-driven 3-dimensional cavity problem ......

4.12 Solver Performance ......

4.13 w velocity profiles at centrai plane. x=0.5. z=0.5 ......

4.14 v veloci ty profiles at central plane, x=0.5. y=0.5 ...... 4.15 Geometry and coordinate system for plane channel flow ...... -- 4.16 U/Uc profiles for fully-developed channel flow ......

4.17 - al* (= ~/k)for fuliy-developed channel flow ...... , ......

4.18 ail (= [(m/k)- (2/3)] ) for fully-developed channel flow ......

4.19 as2 (= [(~lk)- (2/3)] ) for fully-developed channel flow ......

4.20 as3 (= [(m/k)- (2/3)] ) for fully-developed channel flow ...... 4.21 (Ü/oc)Velocity profiles in the wall-bisector of square duct . . . . . -- 4.22 (Vr/Uc)Velocity profiles along the diagonal-bisector of square duct .

4.23 - al2 (= wk)for fully-developed square duct ......

4.24 al1 (= [(~/k)- (2/3)] ) for fully-developed square duct ......

S.. Vlll 4.25 a22 (= [(~lk).(2/3)] ) for fully-developed square duct ...... 78

4.26 a33 (= [(ww/~). (2/3)] ) for fully-developed square duct ...... 75 4.27 contours predicted by Model 1 ...... 80 4.28 Ü contours predicted by Mode1 2 ...... 80

4.29 Ü contours from LES results(Kajishima & Miyake. 1992) ...... S1 4.30 Mean secondary velocity vectors predicted by Model 1. maximum = 0.01& ...... 81 4.31 Mean secondary velocity vectors predicted by Model 3 . maximum = 0.054 Uc ...... 82

4.32 MeansecondaryvelocityvectorsfromtheDNSresults ...... 82 4.33 DistributionofkpredictedbyModel1 ...... 83 4.34 Distribution of k from the LES results ...... 83 4.35 Distribution of k predicted by Mode1 4 ...... S4 4.36 DistributionofWpredictedby Mode11 ...... 84 4.37 DistributionofvivpredictedbyModel3 ...... Sj 4.38 Distribution of from LES results ...... 8.5

4.39 Distribution of (UV - uiv) predicted by Mode1 1 ...... 86 4.40 Distribution of (E- m)frorn the LES results ...... 86 4.11 Distribution of (FV - TE) predicted by Mode1 4 ...... 87 4.42 Distribution of (-P&) predicted by Model 1 ...... 87 4.43 Distribution of (%le) predicted by Mode1 2 ...... 88 List of Tables

4.1 Table comparing the parameters in the three-dimensional lid-driven cavity problem ...... 6.5 4.2 Models tested for plane channel flow ...... 69 4.3 Under-relaxation factors in channel flow cdculations ...... 69

4.4 Models tested for the fully-developed square duct flow ...... CLia Nomenclature

Latin Symbols

coefficients of a discretized equation stress invariant (= ailau) -Ac streamwise vorticity convection by mean motion -4 production of strearnwise vorticity by normal stresses .As production of streamwise vorticity by shear stresses A, viscous diffusion of streamwise vorticity aij anisotropic Reynolds stress tensor ai jmi fourth rank tensor of 'rapid' part of pressure-strain mode1 source term of a discretized equation total convection through the volume interface model constants of 'slow' part of pressure-strain model C, . C+, mode1 constants of 'rapid' part of pressure-strain mode1 ckl. ck2.ck3. Ck5.Ck6, Cki. Ck8 mode1 coefficients of square duct flow models

CP mode1 constant for eddy viscosity cc I- cc2 k - E model constants CI': c?' mode1 constants of wall-reflection part of pressure-strain mode1

1 II cko . ck ! c mode1 coefficients of Gessner and Emery square duct flow mode1 D total diffusion through the volume interface DA. diffusive transport of kinetic energy [ Nm-2s-1 ] diffusive transport of dissipation rate of kinetic energy [ Nrn-*s-l ]

~,(a) diffusive transport of anisotropic Reynolds stress tensor [ Nrn-'s-l j

D additional terms in low Reynolds number h - c model [ rn?~-~]

Ë additional terms in low Reynolds number k - E mode1 [ kg m-l.~-~j F control volume interface, ce11 face area [ m2 ]

Fij functional over space and time model constant of low Reynolds number k - E model damping function of low Reynolds number k - c model wall-damping function in wall-reflection part of pressure-strain model dimensionless constant scalar functions of the irreducible invariants of Sij and w, heigth [ m ] turbulent kinetic energy [ m 2 s- 2 ] length [ m ] model coefficients of algebraic Reynolds Stress models turbulent length scale [ m ] wall-normal unit vector in i direction instantaneous pressure [ Pa ] mean pressure [ Pa ] production of dissipation rate of turbulent kinetic energy [ :Vrn-'s-' i production of turbulent kinetic energy [ Nrn%-' ] production of Reynolds stress tensor [ Nrn-'s-' ] fluctuating pressure [ Pa ] pressure correction [ Pa ] Reynolds number local turbulent Reynolds stress radial distance [ rn ] mean strain-rate tensor [ s-* ] source term of discretized equation excluding the pressure gradient part source term of pressure equation turbulent diffusion correlation of kinetic energy [ Nm-3s-L] turbulent diffiision correlation of Reynolds stress [ Nm-3s-L ] matrix functions of Sij and wij time [s]

xii mean velocities in x,y and z directions [ m/s ] mean centerline velocity [ rn/s ] mean velocity in the i-th coordinate direction [ m/s ] fluctuating velocity in the 2-th coordinate direction [ mls ] wall-friction velocity [ m/s ] volume [ m3 ] secondary flow velocity in a square duct along the diagonal [ mls ] instantaneous velocity in the i-th coordinate direction [ m/s ] coordinates distance of z-th control volume [ m ] wall-normal distance of P-th point [ m ] virtual origin of the jet from the jet exit [ rn ] distance of i-th control volume face [ rn ] dirnensionless wall distance

Greek Symbols

model constants of 'rapid' part of pressure-strain model model constants of 'rapid' part of pressure-strain model under-relaxation factor of pressure Kronecker delta function dissipation rate of turbulent kinetic energy [ ~rn-*s-'] dissipation rate of Reynolds stress tensor [ Nrn-*s-' j destruction of dissipation rate of turbulent kinetic energy [ Nrn-%-l j approximate dissipation rate of turbulent kinetic energy used in

low Reynolds number k - e model [ Nm-*s-l ] von Karman constant variable used in Rodi's algebraic Reynolds Stress model (Eq. 2.94)

xiii kinematic rnolecular viscosity [ m2/s ] apparent viscosity tensor [ m2/s ] diffusion coefficient tensor in discretized mornentum equations [ m2,/s turbulent eddy viscosity [ m2/s] density [ kg/m3 ] mean-strain invariant [ (SijSij)'I2]

k - 6 mode1 constants turbulent time scale [s] Reynolds stress tensor [Pa] Boussinesq part of Reynolds stress tensor [Pa] deviatoric part of Reynolds stress tensor [Pa] non-Boussinesq part of Reynolds stress tensor [Pa] initial time [s] solution variable of a discretized equation pressure-strain term [ N~z-~s-'] 'slow' part of pressure-strain term [ Nrn-*s-' ] 'rapid' part of pressure-strain term [ Nm-2s-L] wall-reflection part of pressure-strain term [ NmA2s-l ]

terms of wall-reflection part of pressure-strain term [ Ah%- l 1

mean streamwise vorticity [ s-' ] mean vorticity tensor [ s-' ]

Subscripts

bottom nodal point of control volume bottom control volume face central nodal point of control volume deferred correction

xiv east nodal point of control volume east control volume face face value indices for tensor notation inlet conditions north nodal point of control volume n north control volume face .new value at current iteration old value at previous iteration radial component south nodal point of control volume south control volume face top nodal point of control volume top control volume face azimuthal component west nodal point of control volume west controI volume face coordinates shear component axial component

Superscripts

guessed value bottom nodal point of control volume bottom control volume face central nodal point of control volume east nodal point of control volume east control volunie face north nodal point of control volume north control volume face pressure pressure correction south nodal point of control volume south control voiume face top nodal point of control volume t top control volume face UDS Upwind Differencing scheme mean streamwise velocity east face velocity west face velocity north face velocity south face veiocity west nodal point of control volume west control volume face bottom face velocity top face velocity

xvi Chapter 1 Introduction

Most of the flows of engineering interest are turbulent in nature. They are three- dimensional. inhomogeneous and anisotropic. Due to these complexities. the turbulent motion and the momentum. heat and mass transfer phenornena associated with it are difficult to decscribe and thus to predict theoretically. As predictions by ex- perimentation are usually very expensive, calculation methods are of great practical interest.

The t hree principal computational routes by which turbulence and/or i ts pffects can be solved are: direct numerical simulation (DNS), large eddy simulation (LES) and single-point closure models. It is true that DNS or LES can be used to gain a better insight into the physics of turbulence, but projected advances in cornputer capacity make it highly unlikely that the complex turbulent flows would be solved on a routine bais by DNS or LES at least in the next decade. Even though it is well known that single-point closure models do not capture the coherent structures (George. 1989) and that they fail when the spectrum is very complex (Mathieu. 1990). it appears that singie-point closure rnodels are likely to remain the widely employed predictive tool for the solution of the turbulence problems of engineering importance.

Among the single-point turbulence models the 'standard' (linear) k - E mode1 (Launder and Spalding, 1974) is the most widely used. The standard k - E mode1 is based on the assumption of isotropic turbulence with an equilibrium spectruni. thus it fails to predict the level of anisotropy in most flows of practical relevance. The

weaknesses and shortcomings of standard k - E in predicting complex turbulent flow fields prompted development of nonlinear models. There is a belief that second-order closure models represent the best hope for reliable predictions of turbulence anisotropy. It is often argued that the second-order closure models should perform better than the first-order models because in these the third-order correlation is modeled, while in the first-order models the second-order term is approximated. By this logic. third-order closures would perform even better. and so on. In reality, however. model effectiveness does not necessarily increase with increasing model complexity. In any higher order closure mode1 the number of terms which need to be modeled increases dramatically. while our experimental knowledge about their dynamics and variability in different flows diminishes by the sarne rate. Also. higher order closure models require solution of larger number of partial differential equations which is not a trivial task and increases computational cos t significantly. Algebraic Reynolds Stress models (MM) are an approximation of the full Reynolds stress transport rnodels by the algebraic relationships. Thus in effect these are nonlinear k - E models and have a more physical description of the stress-strain relations of the turbulence than the k - 6 model itself.

1.1 Objectivesofthestudy

The objectives of the present study are as follows:

1. To verify the assumptions of ASMs and to investigate their effects on

2 the prediction of correct Reynolds stress anisotropy in different turbulent flow fields.

2. To investigate the role played by the different modeled terms in an ASM on the performance of this turbulence model.

4. To calibrate the model coefficients of ASMs against three-dimensional inhomogeneous flows (as these coefficients are usually approxirnated by cornparison with experimental data for homogeneous flows).

5. To investigate the problem of numerical instability of the popularly used -4SM developed by Rodi (1976).

1.2 Motivation of the study

The inherent inability of the eddy viscosity models to predict turbulence anisotropy. and the complexity and computational cost of full Reynolds Stress models (RSM). gave inipetus in the turbulence modeling community to the formulation and the application of rational extensions of the two-equation eddy viscosity models. Algebraic Reynolds stress models rest on a stronger theoretical basis than any other nonlinear eddy viscosity model. Although ASMs have been able to predict levels of anisotropy close to that of RSM for different flow configurations (Rodi, 1980), their superiority in predictive ability has not been demonstrated in a consistent and systematic rnanner. Again the hope that ASMs would replace k - e as the 'engineering tool' in industry has not been fulfilled. With the goal of establishing the capabilities and inadequa- cies of ASMs there is a need for systematic study in which the assumptions of these models have to be verified and the models have to be applied to flows with increasing complexity which are of practical importance. Researchers have criticized ASMs for their failures to predict correctly several Bow problems (Craft et. al., 1993). However, it should be remembered that in -4SMs the pressure-strain and dissipation terms have to be modeled. Even in the domain of RSM these terms still remain a controversial issue. Thus it is necessary to investigate the role of the pressure-strain and dissipation models in the predictions of ASM. The coefficients in the pressure-strain models are usually calibrated with respect to experimental data of homogeneous or near-homogeneous flows. Proper calibration of these models for three-dimensional inhomogeneous flows have to be done in order to improve the predictive abilities of the pressure-strain models. The first step in this ambitious project would be to compare the predictions of an ASM. with different pressure-strain terms, at the same Reynolds number against a database which is extensive. since terrn-by-term cornparison is necessary. Proper wall treatment has to be done and a higher-order discretization scherne has to be used to reduce nunierical inaccuracies. It has been reported by several researchers that the popularly used ASM developed by Rodi poses serious numerical difficulties (Taulbee, 1992). It was believed that this problem could be circumvented by solving the equation set explicitly. How- ever. the 'explicit' ASMs developed are either internally inconsistent. or have been developed with the assumption of two-dimensional Bow (Girimaji. 1996). As the development of a self-consistent explicit version of Rodik ASM for three-dimensional Rows is almost impossible it is necessary to work with 'implicit' versions. which do not exhibit the problem of interna1 inconsistency. Numerical stability of a turbulence mode1 is always difficult to investigate. especially when boundary conditions and nonlinearities are present. Rodi's ASM may have inherent instability or numerical instability problems may arise in solving the set of algebraic equations coupled with momentum and continuity equations and k and E equations. Again none of the stabilization practices for Rodi's ASM suggested in literature have been used for three-dimensional problems. Thus arises a need to investigate the problem of numerical instability in Rodi's ASM in detail and to develop an effective stabilization scheme which can be used for three-dimensional flows where mult igrid is used to accelerate the solver.

1.3 Literature overview

The deficiencies of the 'standard' k - E model have been reported by Kanjalic (1994). Speziale (1991). Rodi (1980). Speziale (1987). Leschziner and Lien (1995). Lien and

Leschziner (1994) and others. Two major problems associated with k - E mode1 are (Speziale. 1991): (a) it cannot account for Reynolds stress relaxation effects. and (h) it is oblivious to the presence of rotational strains.

The ASMs promise to combine at least partially the economy of the k - e model wit h the universality of the RSM. In the words of Lien and Leschziner (1994): "4SM rests on a stronger fundamental basis which prescribes formally the physical coupling between stresses and strains" . The most popular ASM was developed by Rodi (19'76). ASMs have been applied successfully to predict the partition of normal stresses in a number of shear layers (e-g. Rodi, 1976. Taulbee. 1992. Girimaji. 1996). to accourir for the variation of angular velocity in the time evolution of kinetic energy in rotating homogeneous shear flow (Gatski and Speziale, 1993. Girimaji. 1996. Speziale and .Yu. 1996). to produce turbulence-driven secondary flows in square ducts and rectangular ducts (e-g. Launder and Ying (Demuren and Rodi, 1984), Gessner and Emery. 1976. Nakayama et. al., 1983, Demuren and Rodi. 1984. Rodi et. al. (Nallasamy. 1987). Coustiex et. al. (Nallasamy. 1987)): to describe the influence of buoyancy on varioiis vertical and horizontal shear layers (e.g. Gibson and Launder, 1976. Shabbir and Taulbee, 1990), to predict Reynolds stress profiles in an axisymmetric round jet with swirl (Wall. 1993), to predict swirling, recirculating flows in cornbustors (e.g. Sloan et. al.. 1986. Liou and Hwang, 1989, Sharif and Wong, 1995). to estimate strongly swirling flows in cyclone chambers (Zhang et. al., 1992)' to calculate turbulent wall jets (Ljuboja and Rodi, 1980), etc. In ASM the isotropic mode1 of the dissipation term is commonly used in turbulence modeling. The pressure-strain term can be broken into .slow' term. mpid' term and wall-reflection term. Models for the 'slow' term were developed by Rotta (Launder, 1990) and Lumley (1978). Models for the 'rapid' part of the pressure-strain term include Hanjalic and Launder's model (1972), Naot et. al. (Launder et. al.. 1975), Shih and Lumley (Launder? 1990) and Craft et. al. (Launder. 1990). Shir (Launder. 1990) and Gibson and Launder (Launder, 1990) have developed models for the wall-reflection term of pressure-strain tensor. There are models where the wall-effect term has been absorbed into the 'slow' and 'rapid' terms. e.g. Launder et. al. model (1975) and Speziale et. al. (1991) model. The absence of any diffusion or damping in Rodi's ASM is particularly troii- blesome when applied to non-equilibrium flow fields (Taulbee. 1992. Huang and Leschziner, 1985). Problems related to solving these algebraic Reynolds stress equations implicitly have been mentioned by several authors (e.g. Rodi. 1980. Gatski and Speziale. 1993. Craft et. al., 1993). The first attempt of developing 'explicit' ASM was carried out by Pope (1975). The explicit ASMs developed by Gatski and Speziale (1993j: Taulbee (1992), Taulbee et. al. (1994), Speziale and Xu (1996). and Xu and Speziale (1996) suffer from the problem of interna1 inconsistency (Girimaji. 1996). Recently. a 'self-consistent' explicit ASM has been developed by Girimaji (1996) for two-dimensional flows. However. the extensions to three-dimensional formulation is nearly impossible. Numerical instability of Rodi's ASM might arise from velocity/Reynolds-stress decoupling. Huang and Leschziner (1985), Liou and Hwang (1989) and Sloan et. al. (1986) have developed stabilization schemes for Rodi's ASM to prevent the velocity/Reynolds-stress decoupling in two-dimensional flows. Complications arise when proper velocity/Reynolds-stress coupling is applied to the popularly used staggered grid arrangement for three-dimensional problems (Lien and Leschziner. 1996)

(especially when mult igrid/multiblock method is used to accelerate the solution scherne). The use of non-staggered grids with fully colocated storage alleviates these complexities at the expense of weaker pressure/velocity coupling. Some of the colocated grid schemes reported in literature are: the Chorin (1968) scheme. the Rhie and Chow (1983) scheme. the Armfield (1991) scheme and BLOC0 developed by Matovic (1993). In complex flows like that of a square duct flow numerical diffusion can smear the predictions thus introducing significant numerical inaccuracies. Thus higher-order discretization schemes have to be implemented. Some of the popularly used higher- order schemes used in literature are: QUICK (Leonard. 1979). second-order upwintl scheme (Shyy. 1985). skew-upwind scheme (Raithby, 1976). SMART (Gaskell and Lau. 1988) and COPLA (Choi et. al.. 1995). Darwish and Moukalled (1994) extenderi the third-order SMARS scherne for non-uniform grids. Binder (1996) applied SMART for three-diniensional turbulence modeling problems. Chapter 2 Governing Equations and Mat hemat ical Models

2.1 The Fundament al Equations

The governing equations for the flow of a viscous incompressible fluid with constant properties are the continuity equation and the Navier-Stokes equation. given by

where vi and P are instantaneous velocity and pressure. They can be decomposed into the mean and fluctuating components as follows,

From the equations (2.1). (2.2) and (2.3). the mean continuity and the Reynolds- averaged Navier-Stokes (UNS) equations can be obtained, which take the forni

- where Tij UiUj, is the Reynolds stress tensor. Combining (2.2), (2.3) and (2.5) the Reynolds stress transport equation can be obt ained

where

are. respectively. the production term, pressure-strain correlation. dissipation rate tensor and turbulent diffusion correlation. Contraction of (2.6) gives the transport equation of the turbulent kinetic energy. k (EL

where

are. respectively, the production. dissipation rate and kinetic energy diffusion correlation.

If the Reynolds stress anisotropic tensor, aij, is defined as

then combining (2.6) and (2.7) the transport equation of a, can be obtained:

where. Da,/Dt is the material derivative of a, and vij(")is the diffusive transport of aij For stationary processes,

i.e. Da,/Dt is the convective transport of aij- .4 transport equation can also be obtained for the turbulent dissipation rate and is given by

where

DE= are. respect ively, the production, diffusion and destruction of dissipation.

2.2 Two-Equation Models

The two-equation models are commonly used to solve engineering flows. Of the tliree popular two-equation models (k -E, k-I and k-w), k-E is the most often used. In this mode1 the II and t transport equations have to be solved coupled with the Reynolds- averaged Navier-Stokes (RANS) and the mean continuity equations. As the equation set involves more unknowns than the nurnber of equations it is necessary to mode1 a few terms, namely the Reynolds stress term of the RANS, the diffusion term of the k-equation (2.7) and the production. destruction and diffusion terrns of the E-equation (3.12). The Reynolds stress tensor depends on the global history of the mean velocity field. ri-Generally. where Fij[.]denotes a functional over space and time, V is the volume of the fluid and ri, is the initial time. In (2.13) it is understood that there is an implicit dependenc~ of the fluctuating velocity, ui, on the initial and boundary conditions. For turbulent flows that are sufficiently far from solid boundaries and where the memory of the initial condition has faded away it is not unreasonable to assume that the initial and boundary conditions on the fluctuating velocity merely set the length and time scales of turbulence. Hence with this assumption

where 1, and T, are turbulent length scale and turbulent time scale. respectively. If the right hand side (RHS) of (2.14) is expressed as a Taylor series expansion. then. neglecting the higher order terms?

where

is the mean strain-rate tensor and y- - l:/rOis the eddy viscosity. Of course. thr assumption that Tij can be characterized by a single length scale and time scale is ail idealization. The deviatoric part of Tij is known as the Boussinesq hypothesis

The time scale is defined as T, - Z,/u and the dissipation, E, is defined by the relation e - kt/l,, which can be obtained for isotropie turbulence with an equilibriiini spectrurn. Thus the eddy viscosity can be expressed as

where C, is assumed to be a universal constant for high Reynolds number flows. In the k-equation (2.7) the turbulent diffusion term is modeled using a gradient diffusion hypothesis:

Thus the modeled k-equation becomes

In the E equation (2.12) the rnodeling of P, follows the physical reasoning that

the production of dissipation is governed by the level of anisotropy in ~i, and the mean velocity gradients (Speziale. 1991). Thus

where G1is the dimensionless constant.

By assuniing the Bow to be isotropic, the destructive term. E, in (2.12) cm bt. determined by the turbulent length and time scales

where Cc* is ais0 the dimensionless constant. The diffusion term. De.of (2.12) is modeled using the gradient diffusion hypothesis

where a, is a constant. Thus the E equation takes the final form

Many of the defects of the linear ('standard') k - c mode1 lie in its use of Boussi- nesq approximation for the Reynolds-stress deviatoric part. To overcome some of the limitations of linear eddy-viscosity models researchers have proposed aniso tropic generalizations of the eddy-viscosity concept. In the last twenty years great efforts have been spent on improving k - E model using different approximations for the Reynolds-stress term. Nonlinear eddy-viscositp models were developed by Speziale (1987), Yoshizawa (1984): Rubenstein and Barton (1990). Nisizima and Yoshizawa (1987), Huang and Rajagopal (1996) and others.

2.3 Reynolds Stress Mode1

A natural next step in turbulence rnodeling is modeling each Reynolds stress conipo- nent. Although Reynolds Stress Modeiing (RSM) is not the present subject of study

an overview of this method is given as a basis for later discussion of ASM. To close the Reynolds stress transport equation (2.6) the pressure-strain. diffusion and dissipation terms have to be modeled. Supported by the Kolmogorov theory of small scale universal equilibrium. which postulates local isotropy for dissipating scales at high enough Reynolds numbers. the dissipation term is often rnodeled as being isotropie. i.e.

However. many authors have challenged the validity of this assumption (e.g. Durbiri and Speziale. 1991) in the near wall region where length scales are generally sniall and anisotropy is large. Speziale (1990) modeled the dissipation term as

where f, is a dimensionless constant or a function of the invariants of a,. The diffusive transport term in (2.6) is difficult to model. Accurate measurements of this term are few and its physical nature is not well understood either. Somc experimental evidence suggests that the pressure-velocity correlations may be snialler in magnitude than the triple-velocity correlation. Sol as a first approximation the influence of the pressure fluctuations on diffusion is neglected. The triple-velocity correlation is typically modeled using a gradient diffusion hypothesis

Some of the popular models are according to: Haajalic and Launder (1972)

and Daly and Harlow (1970)

where C, z 0.11 is a constant.

Qij in (2.6) is also known as the redistribvtion term because it drives turbiilence towards isotropy. Modeling Qij is perhaps the most controversial topic in second- moment closure modeling for the 1st two decades. This term is usually expressed as a sum of three terrns

where (the 'slow' term) represents the interaction between turbulent quantities only. and Qij2 (the *rapidoterm) governs the interaction between mean strain rate and turbulence fluctuations. Sonie rnodels of the pressure-strain term require another term (aijw.the wall-reflection term) to model the near-wall phenomena.

The general form for Qijl is (Lumley. 1978)

where A2 (= aüaij)is the stress invariant and the coefficients are functions of A? and the local turbulent Reynolds number, Rt, ( k2/ue). The popular model for aijl is the Rotta model (Launder, 1990) where Cl is a constant. It has been found that the value of Cl differs froni one flow to another but usually it is positive and it may vaq between 1 and 3 (Hanjalic and Launder. 1972). A general form of the model for Qij2 is:

where aljmi is a fourth rank tensor. According to Hanjalic and Launder (1972)

where a. ,8. 7 and q are functions of C*2. Naot et. al. (Launder et. al.. 1975) used a different form of aijmi which reduces Qij2 to the form

where C2 is a constant (C2 = 0.6). In order to remove some of the deficiences of the above first-order model for Qij2 Shih and Lumley (Launder. 1990) proposed n second-order model

2 Tmj71i Qij2 = -C2(Pij - -dijPk)+ C2eaG (+) -0.2 [-srnl - 3 k + Tjm3)]?.;j6)azl

However. the above model fails to predict the trend in a simple shear flow in local equilibriiim. It gives values of üü greater than îïïïü which is opposite to experimental evidence (Launder. 1990). Researchers have also explored the possibility of adding cubic term to the above model. The wall-reflection term, is usually broken down into two parts which were modeied by Shir and Gibson and Launder (Launder, l99O), respectively.

@ijwl

@ijw2 = where ni is the wall-normal unit vector in the direction i and fi is the wall-damping function. Launder et. al. (1975) proposed a hear relationship for the fi

F where

xp is the wall-normal distance and n is the von Karman constant (rc=0.42). Deniuren and Rodi(l984) modified fi and proposed a quadratic relationship:

There is a growing trend among turbulence modeler~to include the wall-reff ection part into Qij1 and Qij2. Launder et. al. (1975) modeled aij2 ~O~~OWS

a. ,3 and r are empirical constants which are functions of fl. They used the Rotta model for CDijl and expressed Cl as a function of fi. Speziale et. al. (1991j proposecl a quadratic mode1 of aijin which the need for special near-wall corrections have been eliminated. They rnodified CLin the Rotta mode1 for Qij1. They modeled Qij2 as

are constants and 02 is a function of A*.

2.4 Algebraic Reynolds Stress Mode1

Algebraic Reynolds Stress models (ASM) are a group of nonlinear eddy-viscosity models. They attempt to model al1 the Reynolds stresses without the need to solve corresponding transport equations. Rodi's 4SM (1976) is the starting basis of many other ASMs developed later. Rodi postulated that

Dij Tij Dk mil Tij fl~ (2) - -- (ii) -ry --. Dt - k Dt ' dxl - k dzl Thus Rodik ASM can be formulated by assuming (i) The structural form of Reynolds stress is self-similar in space and tirne. Le.. Daij/Dt O. Daij/Dt is exactly equal to zero for homogeneous flows at equilià- rium and is a reasonable approximation for slowly-evolving flows. This is a .weak- equilibrium' assumption (Girimaji. 1996). (ii) The diffusive transport of the Reynolds stress anisotropy. z)$'~. is sniall. For

hornogeneous flow ~i,(") is exactly zero. u'sing the assumptions of RodiosASM the transport equation of ai,. (2.9) reduces to

Rodi (1976) proposed a set of simultaneous algebraic equations for modeling rij using

the Rotta model for Qijl(2.32). the Naot. et. al. model of ai,,(2.35) and th

isotropic model for EQ (2.25)

Rearranging the terms

This equation was developed for high Reynolds number flow not too close to the boundary.

As +ij appears on both sides of (2.47) it is 'implicit' in nature and is prone to instability in complex flows. For steady problems very small under-relaxat ion factor ha to be applied and for time-dependent problems very small time-step has tc bc applied in order to niaintain stability in ASM (Taulbee. 1992). These increase the computational effort to the point where the advantage of the two-equation model is lost . Pope (1975) attempted to circumvent the limitations of Rodi's ASM by searching

for an explicit formulation. Using the isotropic model for eij and any quasi-linear model for a,. (2.45) takes the form

where

is the mean vorticity. T (E (k/c))is the turbulent time scale and Lidepend on the

model chosen for QG. If the term within the square brackets on the left hand sicle (LHS) of (2.48) is treated implicitly the solution to this equation can be written in the general form using matrix notation

where f is a tensor function of its arguments. Using the Cayley-Hamilton theoreni Pope determined f. Thus a can be written as

where G(") are scalar functions of the irreducible invariants of S and w. and Tt"' are matrix functions of S and W. For a two-dimensional flow O 5 X 5 2. aiid for a three-dimensional flow 1 5 X 5 10. Pope solved G(X)for two-dimensional flows. This approach of explicitly expressing a, results in an effective turbulent viscosity that is a function of (Pkle),the mean strain rate and the mean vorticity

Pope's ASM formulation is semi-explicit because differential equations have to be solved to determine (p&. Gatski and Speziale (1993) and Taulbee (1992) tried to linearize the problem by treating (Pk/e)as a known quantity to be specified externally. This ratio is typically set at its equilibrium value. While Taulbee set (?+) to unity Gatski and Speziale assumed

The above expression can be obtained in the equilibrium limit from the transport equation of the turbulent time scale (k/~)(Taulbee. 1992)

where DI; is the diffusive transport of k. The açsumption of the equilibrium value of (P&) is reasonable if the flow is near equilibrium. However. when used away from equilibrium there is no guarantee that (P&) calculated form ASM will even be close to the equilibrium value assumed to calculate uturbij.Thus explicit ASMs developed by Gatski and Speziale (1993) and Taulbee (1992) can be internally inconsistent when used away from equilibriuin (Girimaji. 1996). Girimaji (1996) developed a fully explicit. self-consistent algebraic expression for Reynolds stresses. He retained the term within the square brackets on the Ieft hand side (LHS) of (2.48) in its explicit form. Girimaji's 'explicit' ASM was derived for two-dimensional flows, and a similar formulation for three-dimensional flows is impossible.

2.5 Regularized Implicit Rodi's Algebraic Reynolds Stress Mode1

Rodi's ASM is ill-behaved. particularly for complex turbulent flows where the hypothesis of weak-equilibrium no longer holds. It has been noticed in this work that the second term of (2.47) may become singular, when its denominator. [Ci + (P& - l)]. approaches zero. In order to prevent numerical instability this term of (2.47) may be regularized using a Pade-type approximation. The main idea behind the regularization technique is to replace an infinite summation by an approximate function to prevent the probiem of boundedness (Rosenau' 1989). Let

(2.55)

According to the weak equilibrium assumption, the difference between Pk and E is srnaIl. and.# so is the vaiue of 7. If 7 is small and positive

Using a Pade-type approximation this infinite sum can be regularized as

and it follows that for 7 > 0:

Similarly. when q is srnall and negative the infinite sum

can be regularized as

giving

Using (2.58) and (2.61), Rodi7sASM can be expressed as " It can be shown that the denominator of the second term of (2.47) can never be zero (regardless of the sign of v) when this regularization technique is used. The denominator of the second term of (2.47) can be written in the form

For positive value of 17 using (2.58)

For D* in (2.65) to be zero

.4s CI > 1. î) in (2.66) becomes negative. which contradicts the starting assumptiori. Thus when rl > 0. D* # O. Using similar arguments it can also be proved that for a negative value of q, D* cannot be equal to zero.

2.6 Modified Rodi's Algebraic Reynolds Stress Mode1

An investigation of both the assumptions of Rodi's ASM using experimental data has been carried out (Bose et. al., 1996) for a turbulent axisymmetric jet (details of the results are in Chapter 4) and it has been concluded that the second approximation of .ASM. i.e. vij'"'= O. is incorrect. A modification to Rodi's ASM (2.46) is sought through the inclusion of a modeled diffusion term:

where z),(~) is modeled using the gradient diffusion hypothesis. The addition of Dili"' rnakes (2.67) highly nonlinear. Thus 2)ij(") obtained from the previous time-step or previous iteration for steady flows has to be used to maintain numerical stability. The effect of solid boundaries can be incorporated by adjusting the pressure-strain terni accordingly .

2.7 Improved Modified Rodi's Algebraic Reynolds Stress Mode1

From (2.48) it is clear that any ASM is a system of equations to be solved for aij in terms of rSij and TWij- Thus for Daij/Dt O, Rodi's ASM represents an asyniptot ic state in the solution of the differential equation for Reynolds stresses with riegligi- ble diffusion. As Daij/Dt z O when the flow is in an equilibrium state. therefore constancy of Uij wodd require that (P&) be constant. Since

then 7Sij or Ta. where O = (SijSij)''*,would be constant. Hence for Daij/ Dt i 0.

TU would assume a certain unique value associated with the asymptotic state of the modeled Reynolds stress and dissipation transport equations. It has been proved bu Taulbee (1992) that in Rodi's ASM. Daij/Dt = O when ro is relatively large. and for srnall values of ra Rodi's ASM may be expected to be in error. Thus the modified Rodi's ASM (2.67) developed in the previous section would only approximately mimic the solution of RSM for a large value of 70. Here an attempt is made to obtain an algebraic formulation that would closely represent the solution to RSM for the entire range of TC. A transport equation of ra can be obtained from (2.54)

Again using the Rotta model for Gijl (2.32), the Naot. et. al. model of Qij2 (3.35) and the isotropic model for c, (2.25) in the transport equation of a, (2.9)

Combining (2.69) and (2.70) a transport equation of (aij/ra)can be obtained

Taulbee (1992) showed that for a small value of rcr (2.71) can accurately miniic the solution of RSM even if D(aij/~~)/Dtis assumed to be zero. D(aij/~0)/Di = O also represents the asymptotic state Daij/Dt = O. Thus an improved Rodi's .\SM can be formulated

In (3.72) a quasi-non-local convective effect has been incorporated through the inch- sion of Do/Dt. It may be noted that (2.72) is not only suitable for inhomogeneoiis flows. it also promises to agree with the RSM solution for al1 values of 70.

If any quasi-linear model is used for Qij the improved ASM takes the general form

where Li depends on the model of ai,.In fact (2.73) is the most general formulation of ASM. Thus Taulbee's ASM can be obtained from (2.73) by assurning y (normal)

direction

Figure 2.1: Square Duct flow

2.8 Turbulence Modeling of Square Duct Flow using Algebraic Reynolds Stress Models

2.8.1 Introduction

Turbulent flows in straight square ducts are always accompanied by secondary motion in the plane perpendicular to the streamwise direction. This is secondary motion of Prandtl's second kind. Here the secondary flow velocity is about 2-3% of the bulk mean flow velocity. Different turbulence models have been used to predict secondary flow in a square duct. An analytical study has been conducted to obtain the most general ASM for solving square duct flow.

2.8.2 Origin of the Secondary flow in a Square Duct

For a fully-developed incompressible constant-property flow through a straight square duct. the transport equation for the streamwise vorticity, Figure 2.2: Typical secondary-flow streamline pattern in a square duct

has the following form (in the coordinate system given by Fig 2.1)

where Ac is the streamwise vorticity convection by the mean motion. Ax and As are the production of vorticity terms. AN is the effect of the normal Reynolds stresses and As is the effect of the shear stresses. -4, accounts for the viscous diffusion of the st reamwise vorticity. In a laminar flow no secondary flow is observed and in absence of Reynolds stresses Ac and A, balance each other. Hence, secondary flow cannot occur in a uniform flow when the flow mechanisms serve only to transport and destroy vorticity. So. secondary flow in a square duct is solely due to the imbalance of

Starting form (2.77) a simple criterion concerning the structure of the normal Reynolds stresses can be deduced (Speziale. 1982) which will serve as a sufficient condition for the development of secondary flow in a square duct. From symmetry arguments there cannot be an unidirectional mean turbulent flow (Le. an absence of secondary flow) in a square duct unless Hence. secondary flow will develop if the condition

is violated. or in other words secondary flow will develop if - - w2 - ,u2 # castant. (2-80 )

-4s Reynolds stresses vanish at the solid wall of the duct, the value of this constant 1s zero.

2.8.3 Turbulence modeling of a Square Duct flow: Previous work

For a square duct flow the two-equation eddy-viscosity models (e-g. il.-c. k-w and k-1) predict (Speziale. 1982) 7 = 2 = (2/3)k, and vzU = O. Thus linear eddy-viscosity models cannot predict secondary Aow in a square duct and nonlinear turbulenci. models have to be used. The first Aow calculation in a straight square duct was carried out by Launcier - - and Ying (Demuren and Rodi, 1984). They derived algebraic expressions for (w2- u2) and viu by simpli&ing the transport equations for these stresses obtained from RSM of Hanjalic and Launder (1972). The other Reynolds stresses were calculated from a standard eddy-viscosi ty model. The model developed by Gessner and Emery (1976) was an improvemerit of the

Launder and Ying model. Gessner and Emery neglected the convection and tlir diffusion terms in the Reynolds stress transport equation (2.6) thus obtaining

Gij was modeled using the models proposed by Rotta and Hanjalic and Launder. From experimental data it is known that the cross-strearn velocities. V and W.are -- two orders of magnitude les~than the streamwise velocity, Ü. Thus V. W and their derivatives were neglected. Again in the developed region al1 derivatives of velocitics

and Tij with respect to x are zero. According to the boundary layer theory such

approximations can also be made in the developing flow. To express TG explicitly Gessner and Emery assumed the flow to be in a state of equilibrium (Le. Pr, = 4. The final form of the six algebraic Reynolds stress expressions were

where cka CÇ and c' are constants. The assumptions made were

and

This mode1 developed by Gessner and Emery (1976) was used by Gessner and Ernery (1980) and Nakayama et. al. (1983) for square duct calculations. However. Nakayama et. al. modified the model constants. It may be noted that and a expressions are exactly the same as that of the k - E model. Again as k is related to the normal Reynolds stress components, U'11 can be obtained from

Demuren and Rodi (1984) used a modified Rodi's ASM to predict square duct flow and rectangular duct flow. They assumed local equilibrium condition (2.81). Launder et. al. model (1975) for the and solved the three independent Reynolds stress components (w~uul and W) from implicit relations coupled with the momen-

tum! continuity, k and E equations. This makes the solution process computationally costly and nurnerically unstable. Speziale (1987) used his nonlinear eddy-viscosity rnodel and Nisizima (1990) used

Nisizima and Yoshizawa's (1987) anisotropic k - e model to solve square duct flow. Myong and Kobayashi (1991) made square duct flow predictions using the anisotropic k - c model developed by the first author. Niami and Gessner (1995) used a RSPVI to predict flow in square duct. Using the direct numerical simulation (DNS) data of Gavrilakis (N92). Mompean et. al. (1996) carried out an a - primi test of the nonlinear eddy-viscosity models of Gatski and Speziale (1993). Rubenstein and Barton (l99O), Speziale (1987) and Shih et. al.. They aIso did a posteriori test for the above mentioned models. However. their calculations may not be strictly considered as turbulence modeling as they used the DNS data of Gavrilakis at a y+ of 10 as their wall-boundary conditions.

2.8.4 Turbulence modeling of a Square Duct flow using Al- gebraic Reynolds Stress Models: An Analytical study

For a developed square duct flow Daij/Dt sz O. By assuming '~~j'")to be negligible the transport equation of aij, (2.9), can be sirnplified as (2.45). If cij is assumed to be isotropic the only term to be modeled is Qu. Rodi's ASM, (2.46), can be obtained by using the Rotta model and the Naot et. al. rnodel for aij1and

Using the Shir mode1 for aijWI,the Gibson and Launder model for aijw2,Rodi's AShI for wall-bounded flow becomes - UV and cm be accurately predicted using isotropic assumptions (Nakayama et. al.. 1983) and üù can be obtained from (2.90). Using the boundary layer assumptions as mentioned in (2.88), m. uiw and E of Rodi's ASM (2.47) can be expressed explicitly - - 2 2 au- au- UV = -k + -A (-uV + -WU) ! 3 3 (:) ay 82 - - 9 7 au- du- ww = =k3 + =A3 (:) (-UV + pu). 3~

where

-4s oii and W are equal, no secondary flow can be produced by Rodi's MM if the wall-reflection terms are neglected. When the wall effects are included in Rodi's ASM W. and of (2.91) beconie

w here (2.96)-(2.98) form a set of simultaneous equations? which when solved gives vii = m. Thus (2.91) cannot predict any secondary flow in a square duct. Let the Hanjalic and Launder (1972) model for be used to replace the Naot et. al. model in RodiTsASM. If the boundary layer assumptions (2.88) are used and an equilibrium state (i-e. RIE= 1) assurned,

where Ckl:Ck2 and Ck3are model coefficients. This mode1 can predict secondary flow if Ckl# Ck2and Ck3 # O. Most of the ASMs for square-duct flow used in literature can be derived in the form (2.100-2.102). Using Rotta model for

This tiras the mode1 developed by Gessner and Ernery (1976). The ASM used by Nakayama et. al. (1984) is similar, but the constants are different.

If Launder et. al.'s model (1975) for Qij is used then

where the a. 0 and Clare functions of fi. Even Gatski and Speziale's (1993) explicit ASM. where Speziale et. a1.k (1991) model for aijwas used, can be expressecl in the form (2-100-2.102) with Figure 2.3: Distribution of (%le) from DNS (Gavrilakis, 1992). The minima at yf = rf = 50 is (P&) = 0.2. The maximas at y+ = 50, zf = 12 and y+ = 12, zf = 50 is (Pk/c)= 2.25

Mompean et. al. (1996) investigated (Pk/t)field (Fig.2.3) for a square duct obtaiiied froin DNS calculations of Gavrilakis (1992). It is quite clear frotii tlieir study that the local equilibrium condition is valid over a small Fraction of the flow cross section only and therefore cannot serve in modeling this flow. Thus there is a possibility that the ASMs reported in literature for the square duct flow svould be internally inconsistent in regions of the flow which are far from equilibrium. To develop a self-consistent ASM for the square duct flow the equilibrium assumption has to be abandoned. A self-consistent ASM for the squareduct flow can be written as where Ck5,Ck6. Ckii Ck8 and Ckgdepend on the mode1 used for ai,'and Ck5$ Ck8 and CM # O. Again developed flow through a square duct is homogeneous in the strearnwise direction. but in the other two directions vij(")= O may not be true. Thus

represents the most general ASM for a square duct flow. Chapter 3 The Multigrid code for solving turbulent flow problems

3.1 Introduction

The BLOCO algorithm of Matovic (1993) has been modified and a SIMPLER-based BLOCO has been proposed. This has been implemented in a general-purpose three- dimensional flow modeling program that includes multigid acceleration. The rniilti- grid data structure of the 'Recursive multigrid cycle' developed by Gjesdal (Gjesdal. 1995) has been used. To increase the accuracy of the discretization scheme SMART has been used. The velocity/Reynolds-stress decoupling problem of Rodi's -4SM Ilas been discussed and a stabilization scheme has been implemented (Section 3.6). -4 solution strategy used to solve developing flow calculations has also been discussed.

3.2 The BLOCO algorithm

Majority of the popular finite-volume methods for solving incompressible Ruid flow problems use staggered grid arrangement. In staggered grid arrangement al1 the Figure 3.1: Staggered grid arrangement velocity components are displaced by half a mesh width in their respective coordinate direction from pressure and other scalar variables (Fig. 3 In a colocated niesh schenie al1 the variables are stored at the same set of grid points. The BLOCO (Block Colocated) algorithm (Matovic. 1993) uses a fully-colocatrd storage scheme and solves the discretized equations in a block fashion (i.e. al1 variables are solved and updated before moving to the next control volume). To ensure pressure/velocity coupling in BLOCO the face-velocities are obtained by solving momentum equations for the inter-nodal points (which are in the same location as the staggered velocities in staggered grid arrangement) e-g. the east face velocity. u,. is obtained from

where Aisue are the coefficients for the east face velocity. Fe, is the east cell-face area. s'". is the source term excluding the pressure gradient part, and pc and p~ are the pressures at the central and east nodal point, respectively. The substitution of expression (3.1) and those for the other faces into a discretized continui ty equation gives the pressure correction equation

where

In BLOC0 the dependence of p' on its neighbouring values has been neglected. Thiis p' equation is explicit in nature. The face velocities are corrected by p'. e-g. for east face the corrected face velocity.

The corrected face velocities are used to calculate the coefficients for the nodal velocities and other scalars. The pressure field is updated by p' underrelaxed by the parameter a,:

P=P*+L~~P', where p* is the guessed pressure field. Figure 3.2: 2 x 2 grid forms 1 fine grid ce11

3.3 The Additive Correction Multigrid method

Multigrid methods are used to accelerate the convergence of basic iterative nieth- ods (called smoothers in the context of multigrid methods). The essential multigrid principle is to approximate the smooth (long wavelength) part of the error on the coarser grids. The non-smooth part of the error is reduced with a small number of iterations with a basic iterative method on the fine grid. The important advantagc of the multigrid methods is the fast grid-independent convergence. Here the Additive Correction Multigrid (ACM) rnethod (Hutchinson and Raithby. 1986) has been used. Let the solution of a partial differential equation be discretized by a finite-volume scheme on a structured two-dimensional grid. The set of algebraic equations is of the form:

where (P is the solution variable, a's are the coefficients and b is the source terrn. This equation will not be satisfied during the iteration. Let the value of the current iterate Figure 3.3: Additive Correction Multigrid Cycles, S - Smoothing. R - Restriction. P - Prolongation. E - Exact solution

be denoted by 8 and the current residual by r. Thus

Let the coarse grid be made of blocks of 2x2 fine grid cells (Fig.3.2). After relaxation on the fine grid the residuals are restricted to the coarse grid where tliey add up to become the source term. Recognizing that the off-diagonal coefficients represent fluxes through the ce11 faces the coarse grid coefficients are calculated by restricting the fine grid coefficients. So. on the coarse grid

On the coarsest grid the unknowns are solved exactly. These are then prolonged to the fine grid cells, to become corrections. The fine grid variables are corrected and updated and more relaxation sweeps are performed. This cycling is then carried out until a converged solution is obtained. In a summary of a two-grid Additive Correction Muitigrid cycle consists of the following steps:

Pre-smoothing on the fine grid Restriction of residuals and coefficients frorn the fine grid to the coarse grid

Solve on the coarse grid for the corrections (exactly if this is the coarsest

gr id

Prolongate the corrections to the fine grid

Update the fine grid variables

Post-smoothing on the fine grid

When more grids are involved. the same algorithm is applied recursively.

3.4 The SIMPLER-based BLOCO with Additive Correction Multigrid method

The BLOCO algorithm has been extended and a SIMPLER-based BLOCO lias beeii introduced. In SIMPLER-based BLOCO pseudovelocity. û. have to be calculated froni the monientum equations for the inter-nodal points. e.g. for the east face

Using the above expression for u, and sirnilar expressions for the other faces in the coiitinuity equation the pressure equation can be obtained

where and the coefficients are the same as in BLOCO. This pressure equation is solved to obtain a correct pressure field. The absence of under-relaxation factor for pressure makes this very efficient and also stable. The coefficients for the momentum equations and the equations of the other scalars are obtained from the face velocities. The pressure, momentum and scalar equations are solved and updated before rnoving to the next cell. To enhance numerical stability for turbulent flow problems in the SIMPLER-based BLOCO the momentum and pressure equations are solved over the entire domain keeping the turbulence quantities unchanged. After this is completed the turbulent quantities are solved while the velocities and pressure remain fixed. The SIMPLER-based BLOCO algorithm used to solve a turbulent flow probleni can be summerized as follows:

For each control volume, do

1. Discretize the momenturn equations around each cell face using closest nodal velocities and pressures.

2. Calculate the face velocities by solving the momentum equations for the face velocities.

3. Calculate the pseudo-face velocities from the momentum equations for the face velocities.

4. Discretize the momenturn equations at the ce11 node.

5. Solve the momentum and pressure equations to obtain the nodal velocities and pressure and update them.

Move to the next control volume and repeat the procedure for the entire domain.

Again for each control volume. do

1. Discretize the momentum equations around each ce11 face using closest nodal velocities and pressures. 2. Calculate the face velocities by solving the momentum equations for the face velocities.

3. Discretize the equations for turbulence quantities.

4. Solve the equations for turbulence quantities and update them.

O Move to the next control volume and repeat the procedure for the entire domain.

To accelerate the solution Additive Correction Multigrid has been applied. As this solver has been developed for turbulent flows the turbulence quantities have been kept constant during the multigrid cycle (Ferziger and Peric. 1996). It has been observed that the performance of the solver does not change significantly when the multigrid cycle is used for momentum equations. So, in the present solver the multigrid cycle is applied on the pressure equation only. Smoothers have a very important role to play in multigrid methods. Different smoothers: the Gauss-Seidel. the red-black Gauss-Seidel. the Tridiagonal Matrix AI- gorit hm and the Stone's S trongly Implicit solver (SIP) extended to three-diniensional probiems (Leister Sr Peric. 1994) have been implemented. The modified SIP was found to be the most efficient. The modified SIP has also been used as the soiver in the coarsest grid. In the finest grid Gauss-Seidel has been used as the smoother.

3.5 The SMART scheme and its implementation

In finite voiume method the straightfonvard approximation for calculating the coefficients is the Central Difference Scheme (CDS). However, it produces unphysical oscillations in convection dominated flow regions. The Upwind Difference Schenie (UDS) and the Hybrid Difference Scheme (HDS) (Patankar. 1980) were developeci to remedy these difficulties encountered by CDS. According to UDS the coefficients of the discretized equat ions are

where D and C are the diffusion and convection fluxes at the ce11 faces. Although one-sided schemes (UDS or HDS) are unconditionally bounded and highly stable. they are highly diffusive when the flow direction is skewed relative to the grid lines. To increase the accuracy of the numerical results. higher-order schemes have been developed. eg, the QUICK (Quadratic Upwind Interpolation for Convective Kinematics. Leonard. 1979) scheme. the second-order upwind schenie (Shyy. 1985). the skew-upwind scherne (Raithby. 1976). the SMART (Sharp ancl Monotonic Algorithm for Realistic Transport. Gaskell and Lau. 1988) schenie. tlic COPLA (Combination of Piecewise Linear Approximation, Choi et. al.. 1995) schenir. etc. According to Choi et. al. (1995) the QUICK scheme, the second-order upwind scheme and the skew-upwind scheme suffer from boundedness problem. While the COPLA scheme is difficult to implement. implementation of SMART becomes simple when the deferred correction method is used. In the present work the SMART schenie has been implemented in the SIMPLER-based BLOC0 algorithm. Consider grid nodes at upstream. center and downstream with values dl-. dc and dD and distances xu. zc and XD, respectively (Fig. 3.4). The value of tlir transport variable at the interface between adjacent control volume. 4,. needs to be interpolated from the known values of neighbouring nodes. Let two dimensionless variables be introduced:

In the SMART scheme for a non-uniform grid (Darwish and Moukalled. 1994) Figure 3.4: The locations (x) and variable values (4) along the x-axis for upstream node (U). central node (C) and downstream node (D), C, - convective flux at the west face (w )

J at #c < O and & > 1 . (3.13)

After $f is calculated the convective flux at the control volume face can be reformu- Iated into

where f = W.e. s, n:b. t and denotes values obtained using the UDS and is obtained using the SMART scheme. Cf is the mass flow rate at the ce11 face which has been obtained using the UDS scheme. In SIMPLER-based BLOCO. Cf is calculated using the face velocities which are obtained by solving the momentum equations for the internodal points. If SMART is used the discretization equation becomes

Usually the term in brackets in (3.16) makes the solution scheme unstable. To improve numerical stability this term has been calculated from the last iteration. Figure 3.5: Staggered arrangement of Reynolds stresses

3.6 Implementation of Rodi's Algebraic Reynolds Stress mode1

For nonlinear turbulence modeling the tendency towards instability arises from the absence of diffusion related strain terms in the momentum equations. coupled with the occurrence of negative normal stresses in the iterative solution sequence (Huang and Leschziner. 1985). This is also known as the velocity/Reynolds-stress decou pling problem (Lien and Leschziner. 1996). To circumvent the problem of velocity/Reynolds-stress decoupling of Rodi's ASSI (using staggered grids) Huang and Leschziner (1985) further staggered the Reynolds stress locations with respect to the rnean velocities (Fig.3.5). This practice works well for two-dimensional cases. but becomes impractical in three-dimensional flow field. especially when multigrid/multiblock method is used to accelerate the solver. Huang and Leschziner prevented negative values of the normal components of Reynolds stresses by substantial modification of Rodi's algebraic equations. However, these modifications are unrelated to the physics of the flow. To prevent numerical instability of Rodi's ASM. Liou and Hwang (1989) decom-

posed rij into a Boussinesq part and a non-Boussinesq part

In the k - c model (T~~)~introduces the eddy viscosity, y., which adds to the lam-

inar viscosity. Y: and makes the diffusion coefficient quite large and unconditionally positive. thus irnproving the numerical stability. Liou and Hwang modified the nio-

mentum equations of the k - E mode1 by adding additional source terms for (Ï~~)~. was calculated from algebraic equations derived from Rodi's ASM. Liou and Hwang's method has been implemented in the present study and it has been noticed that in regions of fluid flow which are highly anisotropic large source terrns are produced which deteriorate the numerical stability of the solution scheme. Again it niiist be remembered that Rodi's ASM is an anisotropic model, thus the assuniption of isotropic diffusion coefficient is unrealistic. Sloan et. al. (1986) implemented a numerically stable modified Rodi's ASM for a staggered grid (for cylindrical coordinates) by deriving a set of apparent viscosities. vaij.(which are unconditionally positive) from the algebraic Reynolds st,ress qia- tions. These apparent viscosities add to Y to form the diffusion coefficients in the discretized momentum equations

and make the system stable. In the present study an approach similar to that of Sloan et. al. has been adopted. Lien and Leschziner (1996) have also used a similar apparent viscosity approach (where they derived the apparent viscosities froni the Reynolds stress differential equations) to implement a stable RSM for solving a complex three-dimensional flow. qj for Rodi's ASM (2.47) can be expressed in the form where X has been expressed in (2.95). Substitution into the momentum equation leads to

For instance the Ü-momentum equation can be written as

Thus the t hree components of the apparent viscosity obtained from the &nonient ilni equation are

Using these apparent viscosities the Ü-momentum equation can be reforniulated as follows

where Sirnilarly the 7-momentum and the W-mornentum equations can be reformu- lated. The apparent viscosity obtained from the r-momentum and the W-moment urn equations are

Sloan et. al. have also extracted diffusion coefficients for k and c equations using the apparent viscosity technique. However. this makes the source terms of thesc equations bulky and it becomes difficult to assure that k and É remain positive ail over the domain during the iterative solution. In the present study k and E equations are solved using Boussinesq approximation (as in k - É model). This method not only is numerically stable. but also is computationally inexpensive. It may be noted that the algebraic Reynolds stress equations of Rodi's ASM are impiicit equations. Following Sloan et. al.'s approach these equations are rearrangetl so that they become more explicit, thus easier to solve. The final form of Rodi's aigebraic Reynolds stress equations take the form

4 k aÜ- [l+3A(;)5;]~~=~k+~A(~)[(--2-3 3 ax ay WU t

- - - -. au a~ dV- au- du- [1+ X (z+ %)] m=-A wu+-vu + -vu (f) (f) [gm+dz 3~ dz

46 Figure 3.6: Solution strategy for solving square duct flow. L = n x 1

- aw aü k au- au- aw- = A ()[%VW + -ww + -UV + [1+ X (f) ( + ) dz dy

3.7 Solution Strat egy for flow solving developing flow through a Square Duct and a Plane Chan- nel

In the present study turbulent Bow tlirough a square duct and a plane channel have been investigated. It has been observed from experimental data that for high Reynolds number fully-developed profiles for Reynolds stresses can be obtained at about 80 H downstream. where H is the height of the square duct (Gessner and Emery. 1981). and the computational dornain (L) needs to match that ratio. The large value of (L/H) increases the ratio between the coefficients in the flow direction and the cross-flow direction. This deteriorates the performance of the solver and the problem becomes extremeiy unstable. The rate of deterioration of the solver performance has been found to be proportional to (LIH).

The flow through a square duct is a three-dimensional parabolic flow. The popii- lar method to solve such a flow problem is the pressure 'decoupling' method (Patankar. 1980). An important demerit of this method is that a special-purpose solver has to be developed. One of the objectives of the present study is development of a general- purpose solver which can be used for solving both two and three-dimensional flows on different 0ow domains. Hutchinson and Raithby (1986) developed an Additive Cor- rect ion Multigrid scheme to improve the performance for three-dimensional problenis with anisotropic coefficients. It has been observed that this scheme fails to work in the case of a turbulent square duct flow when the (LIH)> 50. To circurnvent the problem the length of the domain. L, has been divided into several subdomains. each having length 1 and these subdomains are solved separatel? (Fig 3.6). This reduces the degree of anisotropy of the coefficients. thus problems of numerical stability and deterioration of solver performance are eliminated. In this method the outlet profile of a variable obtained from the (n- 1)-th subdomain is used as the inlet profile of the variable for solving the equations on the n-th subdoniain. Again it has been obse~edthat larger (l/H) ratios are permissible without loss of solver performance for laminar flow problems than turbulent flow problems with k - e niodel. For solving ASM (IIH) ratio has to be further reduced in order to niaintain the solver performance. The same strategy has also been applied to channel Aow calculations.

3.8 Boundary Conditions and t heir Implementa- tion

Boundary conditions have to be specified to solve the set of partial differential equations. The boundary conditions applied to the computational domain used in the present work can be classified into the following types: inlet, exit. and solid walls. Special treatment also has to be done for wall boundary conditions in a square diict when turbulence models are used. Unlike the staggered grid schemes in colocated grid schemes boundary conditions for pressure have to be specified. However, the boundary conditions for pressure arc similar for al1 types of boundaries and they are of Neumann type.

3.8.1 The Inlet Boundary Conditions

The inlet boundary conditions are of Dirichlet type, i.e. al1 variables are specified at the inlet. As discussed earlier for channel flow and square duct calcufations the physical domain bas been divided into several subdomains. Inlet boundary conditions have to be specified for calculating the first subdomain. The profiles obtained at the outlet for this subdomain for al1 variables have been used as the inlet conditions for the next subdomain. and so on. For channel flow and square duct flow calculations the mean velocities at the inlet of the first subdomain are specified as a plug profile, Le. is constant. C and - W are zero. For k - E calculations inlet values of the k and E have been estiniateci such that eddy viscosity. UT, is about 100 times the molecular viscosity. Y. This has been done by assuming at the inlet (Martinuzzi, 1985)

Al1 ASM calculations have been started with efully-developed k - E results where irilet values of Tij have been derived from algebraic Reynolds stress equations (e.g. froni (3.28)-(3.33)for Rodi's ASM). 3.8.2 The Outlet Boundary Conditions

At outlet boundaries. Neumann type boundary condition have been specified for a11 the variables. with the first derivative set to zero. Although this is not strictly valid for the initial subdomains. it is appropriate in the fully-developed region. which is of main interest in this study.

3.8.3 The Wall Boundary Conditions

Al1 the velocity components are zero at the wall. There are two widely used tech- niques of implementing the wall-boundary conditions in turbulence modeling: the wall

function approach and the low Reynolds number k - E modeling (daniping function).

3.8.3.1 Wall Functions

As the flow characteristics near the wall are dorninated by viscous effects the flow in this region can be represented by the Couette flow profiles based on the logarithmic law of the wall. Therefore, by using the Couette flow profiles for the near wall region.

Pk and E (using the assumption of local equilibrium. c = Pk)near the wall can be calculated and supplied as the boundary conditions for solving the k atid E equations.

After the mean velocities. k and c are determined for the near-wall point the different components of rij can be obtained by solving ASM equations.

3.8.3.2 Low Reynolds Number k - E modeling method

hlthough the wall-function approach is widely used in turbulence modeling it does not give very accurate results for flows where the solid wall plays an important role. The low Reynolds number k - e mode1 is used to improve modeling the near-wall region. In al1 these models wall damping functions are used to damp out the effects of turbulence in low Reynolds number regions. The popular low Reynolds number k - e models used in Iiterature are the models of Launder and Sharrna, Lam and Bremhorst and Chien (Patel et. al.. 1985). It has been noted by Chen and Patel (1988) that the Lani and Bremhorst model demonstrates a strong tendency of divergence. Chien's moclel was found to be computationally expensive (Heyerichs, 1995). As the predictions of the Launder and Sharma model for a channel flow was found to be in close agreement with experimental data (Heyerichs. 1995), it has been used in the present stiidy.

The equations that comprise the low Reynolds number k - c model of Launder and Sharma are (Patel et. al.. 1985)

and y is the normal distance from the wall. At the wall a11 the velocities. k and < are al1 set to zero at the wall. f, is introduced to incorporate the influence of the wall. In the fully developed region of the flow D and Ë approach zero and E beconles approximately equal to E. To improve the performance of the Launder and Sharma model in the nonequilibrium flow Yap introduced an additional source term S, (Launder. 1990) into the E-equation

Most of the low Reynolds number models have been developed for isotropie models (e.g. k - E). When these have to be applied to anisotropic models special extensions have to incorporated. However. any special treatment to the anisotropic model may be avoided by determining the turbulence effects in the semi-viscous siib-

layer (y+ < 50) (where y+ = yu,/v and ZL~= J=) from low Reynolds number k - e model to which the anisotropic model can be coupled at y+ - 50 (Lien and Leschziner, 1996). For ASM at IJ+ -- 50, k and e have been obtained from the low

Reynolds number k - 6 model and the Reynolds stresses have been calculated from

the algebraic Reynolds stress equations (e-g. from (3.28)-(3.33) for Rodi 's ASM ) .

3.8.4 The Wall Boundary Conditions in a Square Duct

The wall function and the low Reynolds number turbulence modeling approach have been derived for two-dimensional boundary layer flows. None of these are directly applicable for modeling the near-wall region of corner flows. In al1 the predictions of turbulent square duct flow using wall-functions k was estimated incorrectly near

the corners. In the present study k - E mode1 calculation has been carried out us in^ t raditional wall-function approach. and the results show excessive k concent rat ion near the corners. This can be attributed to the violation of the local equilibriiini assumption in the log-law region at the corner (Fig.2.3).

In the low Reynolds number k - E models C, is modified by f,. For a corner flow f, is influenced by both the walls. It has been noticed in the present study that Launder and Sharma low Reynolds number model with Yap correction does not remedy the defect of overprediction of k in the corner region. Nisizirna (1990) used a modification of the Van Driest damping function to pre-

dict square duct flow j, = [1 - exp (-:+)]- [l - exp (31 where A = 12.5. The wall boundary conditions for k and c were

It has been observed in the present study that Van Driest's damping function produces reasonable predictions of k and c in the corner region. Thus Van Driest's damping function has been used for calculating square duct flows. Chapter 4 Presentation and Discussion of Results

4.1 Introduction

The objectives of numerical experiments in this study are: (a) To numerically evaluate the higher-order accurate SIMPLER-based BLOC0 aigorithm and its multigrid implementation. (b) To validate the assumptions of Rodi's ASM. (c) To study the ability of Rodi's ASM and the modified Rodi's ASM to predict the Levels of anisotropy in a 'building-block' inhomogeneous flow. (d) To test the effectiveness of the regularization technique and the apparent c?isco.stty stabilization scheme developed for Rodi's ASM. (e) To investigate the role of pressure-strain models in the predictions of Rodi's ASM and its variants. (f) To calibrate the coefficients of pressure-strain models using an inhomogeneoiis flow fieId. The performance of the multigrid solver and the effectiveness of the higher-order discretization scherne has been tested by evaluating the flow in a lid-driven cubic cavity. The assumptions of Rodi's ASM have been evaluated using experimental data of Hussein et. al. (1994) for an axisymmetric turbulent jet. Fully-developed turbulent flow through a plane channel has been calculated to study the predictive ability of Rodi's ASM and the modified Rodi's ASM. This test case shows the effectiveness of the regularization technique and the apparent uiscosit!j stabilization scheme for Rodi's ASM. The role of the pressure-strain models on Rodi's ASM can also be noticed. For the purpose of calibrating the model constants in ASMs the Nakayama et. al. model. the Gatski and Speziale model, the Gessner and Emery model and a self- consistent modified Gessner and Emery mode1 (as developed in Section 2.8.4) have been evaluated for a fully developed square duct flow and the the results obtainecl have been cornpared against DNSJLES data. This test also aims at demonstrating the effect of interna1 inconsistency on mode1 predictions.

4.2 Evaluation of the Rodi's ASM assumptions using experimental data

As discussed in Chapter 2 Rodi's ASM has been formulated by neglecting the convective and diffusive transport terms of aij. Fu et. al. (1988) reported that Rodi's

-4SM introduces serious errors in the prediction of an axisymmetric turbulent jet flow. They made an attempt to isolate the origin of the problem and argued that the defect lies in the diffusion of shearing component of the anisotropic Reynolds stress tensor. The accuracy of both the assumptions of Rodi's ASM have been examined (Bose et. al.. 1996) using experimental data of a high Reynolds number turbulent jet (shorvri schematically in Fig. 4.1) in an infinite environment, investigated by Hussein et. al. (1994). The database of Hussein et. al. (1994) contains velocity-moments to the third

- W* balance (azimuthal)

balance (shear)

\ Y Y I II III

where I. II. III?IV and V are the convection, turbulent diffusion. pressure diffusion

production and pressure-strain of Tij.

For a11 the Reynolds stress components I and IV can be computed from th^ experimental data. The dissipation terms have been provided by Hussein et. al.. Using the experimental data II can be also be calculated for 2 balance and ET balance equations. vw2 present in term II of 3 balance and 2 balance could not - be rneasured by Hussein et. al. with LDA or FHW. So. they approxiniated ou!? - - - with the assumption aw2 2: v3. The rneasurements of uw2obtained with SHW show - that vw2 Lies between O and 3. Thus in the present study two different sets of - - calculations have been carried out: one using vw2 = v3, and the other. assuming - vw2 = O. Having obtained the convection, turbulent diffusion and production terms. the pressure diffusion and the pressure-strain terms can now be backed out.. In the 2 balance and the balance equations the pressure-diffusion ternis caii be assumed to be sufficiently srnaIl compared to the pressure-strain terrns Thus the pressure-strain terms can be computed from the balances of the two equations. To compute the pressure terms in the 3 balance and the 2 balance equation the k transport equation have to be considered. The k transport equation for an axisymmetric jet at high Reynolds number is as follows:

v Wk where Ik. IIk7 IIIk and IVk are the convection. turbulent diffusion. pressure diffusion and production terms.

As E profile has been provided and Ik. IIk, IIIk and IVk can be calculatetl. tlitt pressure diffusion term can be obtained from the k balance as follows:

d - (T7) = -(& +II, + IV, CC), - (-)p + Assuming that the axial component of the diffusion of k is negligible compared to the radial component

(z)2 -(Ik + II* + IV* + L). T dr

Thus the pressure diffusion of 2 equation can be calculated

The pressure diffusion of 3 can be computed form (4.9) as follows:

Thus the pressure-strain terms of 3 and 2 can be backed out. -- The pressure-~trainterms of u2, v2 and 2 balance each out to within machine - errer for both the assumptions of vw2.This verifies the calculations. Mean convection - Turbulent diffusion ------Pressure diffusion ------Production . - Dissipation ----.-

- Figure 4.2: Kinetic energy balance for uw2 = O. Normalized kinetic energy balance - Kinetic energy equation /(Li,-3 /(x - x0))

- - The plots for k and Tij balances for uw2 = u3 have been compared with those provided by Hussein et. al. and discrepancies have been observed. Further investigation shows that Hussein et. al. incorrectly calculated the radial components of the turbulent diffusion terms for k and Tij balance equations. (l/r)a(r@/8r. (l/r)~(r&)/&. - - (l/r)d(rT)/8~(ï/r)a(rz)/dr and (~/?-)~(Tuu*)/~T~as a(x)/&. a(u2t~)/&. 8(7) jar. - B(u.w2)/ar and 8(uv?)/&. respectively. - As tnu? is present in IIk in k equation, and in II in 7 and 2 equations. the k. - - u? and 3 balances are influenced by the vw2 assumptions. The effect of tliese two assumptions can be observed in Fig.4.2, Fig.4.3, Fig.4.4, Fig.4.5. Fig.4.6 and Fig.4.7. - Lic is the centerline velocity and XO is the virtual origin of the jet. - The components of Daij/Dt (Fig.4.8) for vw2 = O compared with the production of the k suggest that the first assumption of Rodi's ASM, i.e. Da,/Dt z O, is satisfied - - for the the present flow situation. Similar observation can be found for vw2 = v3. In Fig.4.9 and Fig.4.10 the radial and azimuthal components of v~~(~'change - drastically with the two different assumptions of the vw2 term. In both cases the peak 1 I b 1 Mean convection - 0.8 - Turbulent diffusion --- - Pressure diffusion ------0.6 - Produdon --- - Dissipation ------

. .--. .-----.------.,.y-;----.-- O - +.-.---...... _ a _.--- - - _.--- # - -0.2 - - * - -0.4 - cc - - C. - -0.6 - -. - - _-- _._.---.- - - -0.8 -

- 1 1 I 1 O 0.05 O. 1 0.1 5 0.2 r/(x-x0) - - Figure 4.3: Kinetic energy balance for vw2 = f,Normalized kinetic energy balance - Kinetic energy equation /(Ü:/(x - zO))

Mean convection - Turbulent diffusion ------Production - -.-- _ - _ Dissipation Velocity-pressure gradient ------5 _ -.- -'c -. - 5.- ______---.--f-.~~.-. ---+ - .;.>~

.-- -'

- Figure 4.4: üü balance for vu2 = O, Normalized Vv balance - üü balance equation /(Z3/(x- 50)) Mean convection - Turbulent diffusion ------Production ------Dissipation - Velocity-pressure gradient -.-.-- - -.-. - - - -_ --- -.- - -. - --

- - Figure 4.5: Vv balance for vw2 = v3. Normalized üü balance - Vv balance equation m3/(~- XO))

Mean convection - Turbulent diffusion ------Production -----.. Dissipation . -- - Velocity-pressure gradient ------. -.------. ---_-.- _ -.5_-.- -.-.- _ _- -.-___-.-.

- Figure 4.6: ww balance for vw2 = O, Normalized utui balance - w balance equation /(~c3/(x- ~0)) Mean convection - --- - _ - Turbulent diffusion ------Production ------Dissipation - - _ - _ - _ Velocity-pressure gradient ------

- - Figure 4.7: ww balance for vw2= v3. Normalized zuui balance - ÜE balance equation /(cC3/(x- 20))

Axial convection - Radial convection ------Azimuthal convection ------. Shear convection ------.-__- Production of kinetic energy - - - -

- Figure 4.8: Convective transport of aij for vu2 = O, Normalized kinetic energy balance - Kinetic energy balance /(z/(z - IO)) Axiai diffusion - Radiai diffusion ------Admuthal diffusion ------Shear diffusion - _---_ - - Production of kinetic energy -- - -

- Figure 4.9: Diffusive transport of Reynolds stress anisotropy. vi,(').for v-w2 = 0. Norrnalized kinetic energy equation - Kinetic energy equation /(üC3/(z- z0))

0.3 - I I I Axial diffusion - Radial diffusion ------0.2 - Azimuthal diffusion ------_ Shear diffusion - *.- -.-. _. -S. Production of kinetic energy ----.- -..

0.1 - -,.

/---*-.-..--./- O -

-.____-----______---_------A------0.1 -

-0.2 -

-0.3 I 1 I O 0.05 0.1 0.1 5 0.2 r/(x-xO) - - Figure 4.10: Diffusive transport of Reynolds stress anisotropy, z),(~)! for vw* = 03. Norrnalized kinetic energy equation - Kinetic energy equation / (ûc3/(1- 10)) -. u=v=O,w=I

Figure 4.11: Lid-driven 3-dimensional cavity problem values of the different components are comparable in magnitude to the production of k. Thus for a turbulent axisymmetric jet flow it may not be reasonable to make the second assumption of Rodi's ASM.

4.3 Test of the performance of the solver using Lid Driven Cubic Cavity

The three-dimensional lid-driven cavity problem (Fig. 4.11) for laminar flow conditions is a common test case for three-dimensional flow modeling (e.g. Gaskell et. al.. 1988). Thus it has been chosen for testing the performance of the SIMPLER-based BLOC0 algorit hm wit h Additive Correction Multigrid method. The Reynolds number. defined here as Re = WH/qin this test case is 100, where W is the velocity of the south surface, H is the length of one side of the cubic cavity. and v is the viscosity of the cavity fluid. HDS scheme has been used to discretize the momentum equations Finest-grid size Number of Number of Under-relaxation factors grid levels finest-grid In finest grid 1 Fine-to-coarse-grid ! relaxations ' U, V' W P Restriction 8x8~8 3 16 0.5 1.O 0.5 I

Table 4.1: Calculation parameters of the three-dimensional lid-driven cavity problem

10 20 30 40 50 60 Iteration number

Figure 4.12: Solver Performance for the nodal points and the inter-nodal points (i.e. to calculate face velocities). The calculations have been performed on three different grids: 8 x 8 x 8. 16 x 16 x 16 and

32 x 32 x 32. The calculation parameters are given in Table 4.1. Here the under- relaxation factors have not been optimized. The Mass Residual Ratios (Le. Mass Residual / Initial Mass Residual) for the three grids are plotted against the number of iterations (number of V-cycles) of the multigrid (Figure 4.12). The convergence history indicates that true grid independence (optimal mult igrid efficiency) has not been achieved, but the iteration count ratio of approximately 2 between two successive refinement levels is less than expected single grid ratio of S (proportional to the node count), thus providing significant savings in computational cost. Two main reasons for this sub-optimal performance are seen in relatively explicit, nature of additive correction and the use of Gauss-Seidel smoother at the finest level. Figure 4.13: w velocity profiles at central plane. x=0.5. z=0.5

4.4 Test of the SMART scheme using Lid-Driven Cubic Cavity flow problem

The [id-driven cubic cavity flow problem (Fig. 4.11) is also used to compare the discretization error of the HDS scheme and the SMART schenie against the QUICK scheme as calculated by Gaskell et. al. (1988) for the same grid size and Reynolds number. The calculations were carried out over half of the domain imposing symmetry

boundary conditions at x = 0.5. on a 16 x 32 x 32 mesh and Reynolds number of 1000. The results are compared at x = z = 0.5 (Fig.4.13) and x = y = 0.5 (Fig.4.14) profiles. The results obtained using the SMART scheme are comparable with those predicted by Gaskell et. al.. As expected the HDS scheme is more diffusive. thus it cannot correctly capture the peaks in the velocity profiles. In the present code SMART scheme has been implernented for discretizing the momentum equations of the nodal points only. The cell-face momentum equations have been discretized using HDS. This may be the possible reason behind the small discrepancies at (Fig.4.13). ll]Gaskell et. al.

Figure 4.14: v velocity profiles at central plane. x=0.5. y=0.5

Figure 4.15: Geometry and coordinate system for plane channel flow

4.5 Numerical Evaluation of a fully-developed Plane Channel Flow using ASMs

The fully-developed plane channel flow at high Reynolds number is one of the niost t horoughly investigated flows both by turbulence modelers and by experimentalists. Not surprisingly, al1 the linear eddy-viscosity models fail to accurately predict the Reynolds stress anisotropy for such a flow field. It has thus been selected to test the ability of Rodi's ASM and the modified Rodih ASM to predict the mean and turbulence characteristics of this flow. It is difficult to find complete, highly accurate and consistent experimental data for the high Reynolds number channel flow having high aspect ratio and long development length. It is true that the experimental data of Hussain and Reynolds (1975) are generally considered to be more accurate. however. their data set were incomplete since only the longitudinal normal stress component was presented. In the present study the experimental data of Laufer (Speziale. 1987) will be used for the purpose of cornparison. This data set has also been used by Speziale (1987) to test his nonlinear eddy-viscosity mode1 and by Demuren and Sarkar (1992) to compare the predictive abilities of different RSMs. The test case is a plane channel flow at Reynolds number based on the centerline velocity. Üc, and the half-width of the channel (=H/2), Rec = 30800. The plane channel flow has been studied using a domain with 34 grid points across the channel. The grid is non-uniform. The non-uniform control volume coordinates. Yi. are generated according to Pollard (Heyerich, 1995)

where.

the two parameters al (O < al < 0.5) and a (O1 > 1) control the uniformity of gridding, n is the number of control volumes and L is the length of the domain in the coordinate direction under consideration. aland pl have been chosen as 0.5 and 1.03. respectively. such that fine grids are present near the two walls at y. = O and y, = 1 and coarse grid near the center of the duct. In the present study three versions of regularized Rodi's ASM (3.47) and two versions of regularized modified implicit ASM (2.67) developed in Section 2.6 have been tested. The three versions of Rodi's ASM are Model 1. Model 2 and Model 3 (Table 4.2). Model 2a and Model 3a are extensions of Model 2 and Model 3. respectively. where v~~'"),have been added. Mode1 no. @ij2 fi v,'"' Mode1 1 Naot et. al. Launder et. al. O Mode1 2 Naot et. al. Demuren & Rodi O Model 3 Shih & Lumley Demuren & Rodi O Model 2a Naot et. al. Demuren St Rodi Present Model 3a Shih & Lumley Demuren t Rodi Present Table 4.2: Models tested for plane channel flow

Mode1 - --Under-relaxat - ion factors- number U V,W P k, c uiaj vaij Model 1 1.0 0.5, 0.5 1.0 0.5, 0.5 0.5 0.5 I Model 2 1.0 0.5, 0.5 1.0 0.5, 0.5 0.5 0.5 Model 3 1.0 0.3, 0.3 1.0 0.5, 0.5 0.4 0.5 Model 2a 1.0 0.5. 0.5 1.0 0.5, 0.5 0.5 1 0.5 , Model 3a 1.0 0.3' 0.3 1 1.0 0.5. 0.5 0.3 0.5 Table 4.3: Under-relaxation factors in cbannel flow cakulations

In al1 the models Eij has been assumed to be isotropie. Different models for 80H. As the flow is parallel to the grid lines HDS has been used to discretize the differential equations. The under-relaxation factors for the different variables are shown in Table 4.3. For any iteration v~~(~)has been under-relaxed to maintain numerical stability

where ûr is the under-relaxation factor. For both Model 2a and 3a this uncier- Mode1 1 - Model 2 ------_ Mode[ 3 .-.-... Model2a Model 3a ----- , Laufer 0 - - -

0.2 t

O + O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Y/H -- Figure 4.16: U/Ucprofiles for fully-developed channel flow

Model 1 - Model 2 ------Model 3 ------Model2a Model 3a ---- - Laufer 0

Figure 4.17: - al? (= m/k)for fully-developed channel flow relaxation factor was set to 0.5. It must be noted that the under-relaxation factors used in the present study are quite high for implicit ASMs. This dernonstrates the effectiveness of regdariration technique (developed in Section 2.5) and the apparent viscosit y stabilization scheme (as discussed in Section 3.6). The under-relaxation factors have not been optimized and it is believed that for Models 1 and 2 calculations could have been perforrned with even higher values. Lower values of under-relaxation factors have been used For Models 3 and 3a due to the quadratic terms present in Shih and Lumley mode1 for

Qij2- Figure 4.18: al1 (= [(îlulk)- (2/3)] ) for fully-developed channel flow

Modd 21 ------Mode1 3 ...... Model2a Model 3a ---.- - Laufer 0

Figure 4.19: a22 (= [(filk)- (2/3)] ) for fully-developed channel flow

Model 1 - - Mode1 2 ------Mode1 3 .--.- Model2a - Model3a - - - - - Laufer - - -. ---6--____ -----___------__ ------z**--- O 1,-SI ---.-- O O O --'*-....7.~0- yr.?.*p - ' .I.:.?*-.- c- :-5 - .--7 ------O --.:......

O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Y/H

Figure 4.20: as3 (= [(nlk)- (2/3)] ) for fully-developed channel flow The fully-developed streamwise velocity profiles (Ü),normalized by the centerline velocity, Üc,are compared with the experimental results of Laufer (Fig. 1.16). Al1

these predictions agree well with experimental data. k - c model also gives similar - LT estimation. Thus it can be concluded that the mean streamwise velocity profile prediction does not depend on the different levels of anisotropy estirnated by different. models. The different components of ai, have been cornpared with experimental data in

Fig. 4.18, 4.19. 4.20 and 4.17. It rnay be noted that ail, a22 and a33 are zero over

the entire channel for k - E model. Experimental data of Laufer exhibit high levels of anisotropy in the near-wall region (i.e. (ylH) < 0.1) of the duct. In the relaxation region near the duct centerline. (0.35 < (ylH) < OS), the relaxation towards isotropy corresponds to the relaxation in the shear stress. Between the two regions anisotropy gradua1 decreases. Judged by the anisotropy level in the near-wall region al1 mode1 predictions are in good agreement with experiniental data only for Vv (Fig. 4.19). For al1 the models estimate the trend of increasing anisostropy. Models 3 and 3a being close to the experirnental results. However. for Îuw none of the models calculate the level of anisot ropy in the near-wall region correctly. Models 2 and 2a are close to the experimental data in the imer layer outside the near-wall region (i.e. 0.1 < (yjH) < 0.35). In this region Models 3 and 3a overpredict the levels of anisotropy for u7i by about 35%, Vv by about 20%, and ÜEü by about

70%. Model 1 shows reasonable predictions for uzL and vlvl in this region. however. it overestimates the level of anisotropy for by about 33%.

In the relaxation region for ûü and Vv al1 the models show the trend of relaxation towards isotropy. For the models underpredict the level of isotropy considerably. a22 is overpredicted by Models 2, 2a. 3 and 3a, while Model 1 underestimates it. For -ww contrary to the experimental data a11 the models show a trend towards increased aniost ropy. For ai* (Fig. 4.17) the predictions of Models 1, 2 and 2a are comparable to the experimental data of Laufer. Models 3 and 3a calculate a12 correctly only in the near-wall region. while they overestirnate al2 in the rest of the channel. The present test case makes it clear that if the dissipation rate tensor. eij. is assumed to be isotropic. the predictions of Rodi's ASM are very sensitive to the model for

4.6 Numerical evaluation of a fully-developed flow through a Square Duct using ASMs

A fully-developed turbulent flow through a square duct habeen selected as a three- dimensional test case due to its rich flow structure, availability of experimental. DNS and LES data and the notorious inability of k - r model to predict secondary flows. As discussed in Section 2.8.3 different ASMs were used to predict the secondary flow in a square duct. However, none of the models were able to predict the turbulence quantities associated with the secondary Bow generation correctly. The disagreement is probably due to the empiricism involved in modeling the various correlations in the transport equations- The predictive ability of the turbulence models are also seriousiy affected by improper treatment of the solid walls (Section 3.8.4), especially the corner region. Other than the work done by Mompean et. al. (1996) al1 predictions of square duct flow using turbulence models were compared with respect to experimental data. The reliability of experimental data in the near-corner region for higher-order moments is questionable. Both direct numerical simulation (DM)(Gavrilakis. 1992. and Huser and Biringen. 1993) and large eddy simulation (LES) (e-g. Madabhushi and Vanka. 1991. and Kajishima and Miyake, 1992) results were reported in literature for fully developed low Reynolds number turbulent Row in a square duct. The availability of DNS and LES results for this flow configuration helped researchers to investigate in detail the different mechanisms responsible for the secondary fiow pattern and could also be used to calibrate the model constants. Usually turbulence models are applied to high Reynolds nurnber Aows. However. it is believed that it would be better to calibrate the model constants with respect to data for low Reynolds number inhomogeneous flows where DNS/LES data are available, rather than approximating them by cornparison with experimental data for high Reynolds number homogeneous flows. This test case can be considered as the first step of the calibration process, to be extended to higher Reynolds numbers as new and better experimental data become available. Four ASM variants (Table 4.4) have been compared with DNS/LES data for a fully-developed square duct flow. The Nakayama et. al. (1984) rnodel. Gatski and Speziale (1993) model, and Gessner and Emery (1976) model are based on the equilibrium assumption and have general equations (2.100-2.102). Mode1 4 is an extension of the Gessner and Emery (1976) model. It has been developed by omitting Mode1 number Mode1 name (Pk/€)

Mode1 I Nakavama et. al. (19841 -1

Mode1 2 ' ~atsk& Speziale i1993) .- (Cc*- l)/(Ccl - 1) ' Mode1 3 Gessner & Emew (1976) 1 1 -. I 1 Model 4 1 Modified Gessner & Emery 1 # constant - -- Table 4.4: Models tested for the fully-developed square duct flow

the equilibrium flow assumption and has general equations (2.107-2.109). Thus in the present test case not only have the predictive abilities of different ASMs with different rnodels for @,, been investigated, the validity of the equilibrium assumption has also been tested. An attempt has also been made to solve the regularized Rodi's ASM (2.47) and the regularized modified Rodi's ASM (2.67) for the square duct flow configuration. No secondary flow pattern could be seen which in agreement with the analysis in Section 2.8.4. The self-consistent mode1 (Model 4) has also been extended by including D,,'"' ternis in it. However. problems with numerical instability have been noticed ancl this attempt has been aborted. The Reynolds number based on the bulk velocity, Üb,and the hydraulic diameter of the square duct (equal to the duct height, H) has been selected to be Reb = 5000. The experimental study of Gessner and Emery (1980) showed that for a Reynolds number of 50.000 the streamwise velocity for a square duct reached a fully developed state at x = 50H. It is expected that the Reynolds stresses take longer distance to becorne fully developed. The development length reduces for a lower Reynolds number. Although no experimental study on flow development has been reported at the Reynolds number of 5000. it can be safely hssumed that for x > 50H the flow is fully developed for both streamwise velocity and Reynolds stresses. In the present test case al1 the results are shown for x > 50H. To incorporate the effects of the solid walls and the corner regions in the flow field a low Reynolds number mode1 employing a modified Van Driest damping function (3.44) habeen adopted. Al1 turbulent length scales present in the rnodels have been Mode11 - Mode1 2 ------Mode1 3 .-.---- 1.2 Mode1 4 DNS -

-- Figure 4.21: (Cr/&) Velocity profiles in the wall-bisector of square duct

Model 1 - Modd 2 ------Model 3 -----.

-- Figure 4.22: (V1/Uc)Velocity profiles along the diagonal-bisector of square duct

modified by multiplying them with fp to obtain low Reynolds number ASMs for a square duct. The calculations have been carried out on a 18 x 18 non-uniform grid over the cross-section (al = 0.5, Pl = 1.008 in (4.12)). For a square-duct flow the region near the corner is of major importance both in terms of gradient resolution and the impact on secondary flow development. Thus the grid has been arranged in such a way that there are 6 grid-points in the region O - 0.1H from the wall in the near-corner region and coarse grid in the center of the square duct in the y directions. The first grid point in the y direction has been taken at a distance of y+ = 0.03 from the wall. To Model 1 - Modd 2 ------Model 3 .--.--- Mode14 DNS -

- O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Y/H

Figure 4.23: - al* (= ~/k)for fully-developed square duct

minimize numerical diffusion SMART scheme has been used. Al1 the ASMs used in this test case are explicit in nature. Thus the system of equations is more stable than that of Rodi's ASM. Under-relaxation factors are only

applied to the velocities, k and E and they have been chosen as 0.5. -- Fig. 4.2 1 compare the calculated streamwise velocity distributions. (Lr/Uc) (gc - the centerline velocity). along the wall-bisector with the DNS result of Gavrilakis (for Reb = 4410). Unlike a plane channel flow. in a square duct flow Ü predicted - by different models of ASM are not the same. U estimated by Model 1 is in good agreement wi th the DNS result. The relatively smaller flow velocities obtained from Model 2, 3 and 4 near (y/H) = 0.125 can be attributed to the overprediction of çecondary flow velocity along the diagonal, (=-(r2+ w~)'/~), by these rnodels (as shown in Fig. 4.22). Fig. 4.22 also shows that is underpredicted by Model 1 in the region (y/H) < 0.25. The worst prediction for is returned by Model 3. The different components of ai, along the wall-bisector have been cornpared with the DNS results of Gavrilakis in Fig. 4.24, Fig. 4.25, Fig. 4.26 and Fig. 4.23. .\gain. when k - e mode1 is used the normal components of the anisotropic tensor i.e. ail. a22 and are zero over the entire duct. Along the wall-bisector of a square duct the flow shows similarities with the two- Model 3 - Model 2 ------Mode1 3 ...... Mode14 DNS O

Figure 4.24: al1 (= [(~/k)- (2/3)] ) for fully-developed square duct

Model 1 - Model 2 ------Mode1 3 ...... Mode14 DNS 0

______*_---______i __-----__*----- .

%. . ______-_C----- eoOOO . __----- tL/.- W. . .-- - *-.?-, . - . ., .-,oI+P.*.o.--*-

_.' ...... oooO ...... OOoO --. . . O** oO oO

Figure 4.25: a22 (= [(~lll)- (2/3)] ) for fully-developed square duct

Figure 4.26: as3 (= [(vziilk)- (2/3)] ) for fully-developed square duct dimensional channel flow. In the near-wall region (i.e. (y/H) < 0.15) high ievels of anisotropy are exhibited. while near the centerline of the duct (or in the relaxation region, 0.4 < (y/H) < 0.5) there is relaxation towards isotropy. The flow gradually becomes isotropic from the near-wall region to the relaxation region. Calculations obtained from the four models are in qualitative agreement with the DNS result in the relaxation region. In this region Model 2 and Model 4 underpredict the level of anisotropy for the three normal components of Reynolds stresses. While Model 3 overpredicts the level of anisotropy near the centerline! Model 1 shows good good agreement with the simulation results for ail, a22 and a33. The level of anisotropy is severely underpredicted by al1 the models in the near- wall region. In the inner layer outside the near-wall region (0.15 < (y/H) < 0.4) Model 2 and Model 4 predict the trend of gradua1 reduction in the level of anisotropy for TE. and m. Calculations obtained from Model 1 show almost constant values of al,. a22 and a3j in this region. While Model 3 has qualitatively shown the correct trend for a11 and a22, it has returned the opposite trend for a33 in 0.15 < (y/H) < 0.4. al* profiles predicted by al1 the models are close to each other. WhiIe the models overestimate ail in (y/H) < 0.15. they underpredict al2 in the rest of the duct along the wall-bisector. Thus in summary al1 the models fail to capture the anisotropy evident in the DNS resul ts along the wall-bisector. -- The contours of the normalized mean streamwise velocity, (U/Uc),obtained froni the LES results of Kajishima and Miyake (1992) (for Reb = 6200) (Fig.4.29) show distortion of the Û-isovels near the corner. This takes place due to the momentum transfer by the secondary flow towards the corner. Model 1 underestimates this distortion of Ü-isovels (Fig.4.27). This discrepancy is consistent with the underpredic t ion of by this model. The Ü-isovels predicted by Model 2 show excessive distortion (Fig.4.28). This can also be noticed in the predictions obtained from Model 3 and Model 4. however, the level of the distortions of Ù-isovels depend on the prediction of P. Figure 4.28: contours predicted by Mode1 2 Figure 4.29: Ü contours from LES (Kajishima & Miyake, 1992)

-Figure 4.30: Mean secondary velocity vectors predicted by Mode1 1, maximum = 0.01 uc Figure 4.31: Mean secondary velocity vectors predicted by Model 3, maximum = 0.054 Tc

I 1

i r II) yh Figure 4.32: Mean secondary velocity vectors from the DNS (~avrilaEis,1992) Figure 4.33: Distribution of k predicted by Mode1 1

Figure 4.34: Distribution of k hom LES (Kajishima & Miyake, 1992) Figure 4.35: Distribution of k predicted by Model 4

Figure 4.36: Distribution of predicted by Model 1 Figure 4.37: Distribution of Vw predicted by Mode1 3

O = X2

Figure 4.38: Distribution of from LES (Kajishima & Miyake, 1992) Y/H

Figure 4.39: Distribution of (m- W) predicted by Mode! 1

Ir!! i i 1

a s - x2 Hl2 Figure 4.40: Distribution of (W - W) from LES (Kajishima & ~i~aEe,1992) Y/H

Figure 4.41: Distribution of (vii - îuui) predicted by Mode1 4

Figure 4.42: Distribution of (P&) predicted by Mode1 1 Figure 4.43: Distribution of (P,/E)predicted by Model 2

Fig.4.30 and Fig.4.31 display the secondaq flow velocity vectors predicted by Model 1 and Model 3. The predictions of the secondary flow velocity vectors by Model 2 and Model 4 are close to that of Model 3. The two strearnwise, counter- rotating vortices in each corner region have been predicted by al1 the models, however, the intensity of the vorticeç depends on the magnitude of 7.Unlike the DNS result of Gavrilakis (Fig.4.32), the vortex centers in the mode1 predictions are closer to the diagonal-bisector and to the center of the duct. The distortion in the secondary flow pattern can be attributed to the incorrect predictions of the secondary flow velocities in the near-wall region by the turbulence models. Similar phenomena can be observed in the predictions of Nisizirna (1990), who also used Van Driest darnping funetion. Thus underprediction of near- wall secondary velocities may be related to the insufficiency of the Van Driest damping Function. Fig.4.34 delineates k (normalized by vc2)distribution obtained from the LES results of Kajishima and Miyake. To understand k-contour pattern with maxima near the wall along the wall-bisector and minima along the diagonal-bisector. we need to take a closer look at the production term in k transport equation:

As the primary shear stresses (mand üjü) are proportional to the Ü gradients. reduced values of these gradients along the diagonal-bisector result in lower production of k. This explains the distortion of the k contours near the diagonal-bisector. The high shear region combined with the high velocity gradients near the wall results in a high Pk and this corresponds to the location of the peak value of k. As argued by Prandtl it is this presence of high and low shear regions in the corner of the square duct that generates the convective transport of secondary velocity. The contour maps of k (normalized by vc2)predicted by Model 1 and Model 4 are shown in Fig.4.33 and Fig.4.35. The predictions by Model 2 and Model 3 are similar to that of Model 4. Significant reduction of k values near the diagonal bisector exhibited by Models 2, 3 and 4 is consistent with the high value of predicted by these rnodels. Model 1's inability to predict the distortion of the k contours near the diagonal-bisector can be attributed to the absence of the low shear zone along the diagonal-bisector. Thus in the prediction of Model 1 the mechanism of convection of streamwise velocity has been disturbed resulting in the underprediction of the secondary flow velocity. It is interesting to note that none of the models have correctly predicted the low values of k along the diagonal-bisector as obtained from the LES results of Kajishima and Miyake (Fig.4.34). While the maxima of k shown by the LES result is 0.012. those calculated by Models 1, 2, 3 and 4 are 0.019, 0.02, 0.021, 0.021, respectively. The location of the maximas predicted by Model 1 agree with that of the LES result (at (y/H) = 0.5, near the z-wall). Model 2, Model 3 and Model 4 have predicted the location of the peak value around (y/H) = 0.25 (near the z-wall).

Al1 models have correctly predicted the distinct negative peak of ÜÜJ (normalized - 2 by Clc ) in the corner region along the diagonal-bisector (as shown in the LES result of Kajizawa and Miyake, Fig.4.38). However, the exact location of the peak depends on the model. The magnitude of the negative peak of predicted by Model 1 (Fig.4.36) is -0.00015, while the LES result shows this negative peak to be -0.0002. Al1 other models overestimate this value. Both Model 2 and Model 4 overpredict the magnitude of the peak by about four times. Model 3 overestimates this peak value by almost one order of magnitude (Fig.4.37) (the peak value of = -0.0015). This

explains the excessively high moment um transfer dong the diagonal bisector O btained in the prediction of Mode1 3. The (m- ~OW)(normalized by Üc2)distribution calculated by Model 1 shows qualitative agreement with the LES results of Kajizawa and Miyake (Fig.4.40). Model 2. Model 3 and Model 4 (Fig.4.41) show similar trends of anisotropy of normal stress components, however. they incorrectly predict the location of the positions of maxima and minima. The magnitude of the maxima in the LES result is 0.0015. The magnitudes of the maxima predicted by Model 2, Model 3 and Model 4 are 0.0021. 0.006 and 0.0025. respectively. while Model 1 underestimate this peak value by almost one order of magnitude (=0.0005). Thus underestimation of the degree of anisotropy near the walls results in a lower value of 7predicted by Model 1. Again overprediction of - V' by Model 3 can thus be attributed to the overprediction of the magnitudes of the

maxima and minima of (W - îuu) (it overestimates the magnitude of the maxima by four times).

Thus ü?ü and (37 - W) distributions estimated by the models confirm that the secondary normal stress difference is the major driving force of the secondary flow pattern while the secondary shear stress term (G) plays an important role in t ransporting t his secondary motion (Perkins, 1970). To test the self-consistency of the models (P&) distributions predicted by the four models are compared with that obtained from the DNS results of Gavrilakis (Mompean et. al., 1996) (Fig.2.3). Al1 the models have successfully estirnated the high value of (P&) near the wall, although they fail to predict the exact location of the maximas. Model 2 (Fig.4.43). Model 3, and Model 4 predict the magnitude of the

local minima along the diagonal-bisector conectly (model predictions are ri 0.3. while according to DNS data = 0.2), but fail to predict the exact location. The minima of (P&) correspond to the location of the low shear zone in the flow field. The shift of the low shear zone dong the diagonal towards the center of the duct in the predictions of Models 2. 3 and 4 exptain the shift in the vortex centers as observed before. From the distributions of (P&) predicted by Models 1, 2 and 3 it can also be concluded that these models fail to satisfy the criterion of self-consistency. To investigate the discrepancies in the predicted results let al1 models be expressed in the general form (2.100)-(2.102). (However. it was not possible to find the model constants Ckl and Ck2for Mode1 1.) For Models 2, 3 and 4 CkS= Ckl - &.

Again. for Models 2, 3 and 4, (41- CkZ)are

respectively. It is interesting to observe that in Models 2 and 4. Ckland Ck2depend on the velocity distribution, while in Model 3 they are constant. Thus poor predictions exhibited by Model 3 can be attributed to the constant value of (Ckl- &). In summary from this test case it has been observed that the four models are in qualitative agreement with the DNS and LES results. The predictions of Model 2 and Model 4 have been found to be close to the simulation results. The cause of discrepancies of Model 1 and Model 3 have been investigated in detail. Froni cornparison of predictions of Model 3 and Model 4, it can be observed that interna1 inconsistency of a mode1 deteriorates its predictive ability. Chapter 5 Summary and conclusions

5.1 Original contributions

Original contributions of this work can be summarized as follows:

1. The two assumptions of Rodi's ASM have been validated using experimental data of Hussein et. al. (1994) for a high Reynolds number axisymmetric jet in an infinite environment. In the self-preserving state of an axisymmetric jet the convective transport of the aiiisotropic Reynolds stress tensor is negligible. However. the diffusive transport of the anisotropic Reynolds stress tensor has an important effect on the flow field. Thus the second assumption of Rodi's ASM is not correct and a modified Rodi's ASM has been proposed.

2. The modified Rodi's ASM has been improved to closely represent the original Reynolds Stress mode1 equation over the range of time scales for the turbulence and mean flow strain field. Quasi-non-local convective effects have also been included in this formulation. A general improved ASM has been proposed for any quasi-linear mode1 of the pressure-strain term. 3. The BLOCO algorithm introduced by Matovic (1993) has been extended and a SIMPLER-based BLOCO has been developed for non-staggered grids and fully-colocated storage. In this algorithm the pressure equation is solved to produce a correct pressure field. The absence of under- relaxation factor for pressure makes this algorithm efficient and stable. The pressure and velocities remain well coupled; rnoreover, during the nu- rnerous validation tests' the pressure/velocity decoupling (a checkerboard pressure field) was never encountered.

1. To increase the accuracy of the numerical results the SMART scheme has been implemented. This scheme has been tested using a lid-driven cubic cavity. The predictions of the SMART scheme are in good agreement with the results obtained from the QUICK scheme of Gaskell et. al. (1988). Improvement of the SMART scheme over the HDS scheme can also be noticed.

5. The Additive Correction Multigrid has been successfully implemented into an originally written general-purpose three-dimensional flow modeling code using the rnultigrid data structure of the 'Recursive rnultigrid V- cycle' developed by Gjesdal in Fortran90. The code has been developed to solve laminar flows, turbulent flows with k - E mode1 and turbulent flows with Rodi's ASM and its modifications. three other ASMs and their modifications. The performance of the multigrid solver is fairly good.

6. A regulariration technique has been introduced that makes Rodi's ASM well-behaved by removing any possible singularity.

7. The apparent viscosity method of Sloan et. al. (1986) has been modified and it has been applied to a colocated scheme to ensure velocity/Reynolds- stress coupling in when Rodi's ASM and its modifications are used. The high values of under-relaxation factors used for the channel flow test case (Table 5.3) demonstrate the effectiveness of the stabilization practices. 8. Three versions of regulaxized Rodi's ASM and two versions of regularized modified Rodi's ASM have been solved for flow through a plane channel at Reynolds number of 30800, and the results have been compared with Laufer's cross-wire measurements. None of the models could predict the levels of anisotropy in the experimental data accurately. It has been observed that the quadratic expression of fi, developed by Demuren and Rodi. performs better than the linear expression, developed by Launder et. al.. Naot et. al.'s linear model of Qij2 gave better predictions of the levels of anisotropy than the quadratic model of Shih and Lumley. The predictions obtained from the modified Rodi's ASM for this test case did not show any significant improvement over Rodi's ASM. The importance of the pressure-strain model on the predictive ability of Rodi's ASM has been demonstrated.

9. An analytical study has been conducted to obtain a general ASM that can predict secondary flow in a straight square duct. It was proved theoretically that Rodi's ASM is unable to predict secondary flow in a square duct even when wall-effects are introduced for the pressure-strain term. It has also been showed that the model used for the pressure-strain term determines whether an ASM would predict secondary flow in a square duct. A self-consistent ASM has been developed for a square duct and it also has been extended for solving inhornogeneous flow.

10. A study has been carried out to obtain a suitable implementation technique of the corner boundary conditions. The traditional wall-function approach and the Launder and Sharma low Reynolds number approach with Yap correction have been found unsuitable in the corner region of a square duct. The Van Driest damping function approach gives reasonable predictions. However, incorrect predictions of secondary flow velocities in the near-wall region indicate the limitations of this method. 11. The predictions obtained from the Nakayama et. al. model. Gatski and Speziale model, Gessner and Emery model and self-consistent Gess- ner and Emery model for a low Reynolds number flow (Rea = 5000) have been compared with the results of DNS/LES calculations. Although qualitatively the mode1 predictions are in fairly good agreement with the numerical simulation results, none of the models have been able to predict the level of anisotropy correctly. While the Nakayama et. al. model underpredicted the secondary flow along the diagonal bisector al1 the other models overestimated it. The model predictions of Gatski and Speziale model and the modified Gessner and Emery model are close. The results of this test also prove that the Nakayama et. al. model. Gatski and Speziale model and Gessner and Emery model are internally inconsistent. It can also be concluded that interna1 inconsistency of a model deteriorates its predictive capability. This test case tried to investigate the mechanisms respoosible for the discrepancies of the turbulent models to accurately predict the square duct flow.

5.2 Scope for further research

There are many areas for future work suggested by the present study. Sorne of them are:

1. Numerical accuracy of the SMART scheme can be irnproved by discretizing the cell-face momentum equations using SMART.

2. To improve the performance of the Additive Correction Multigrid a better smoother should be used in the finest grid. Modified SIP is the most promising candidate and it has already been used as a srnoother in al1 O t her grids levels.

3. The regularizatia technique developed for Rodi's ASM does not guarantee that the predictive capabilities of the model would be retained in flows far-from equilibrium. This deficiency can be overcomed by imple- rnenting a Padetype approximation that establishes some limited consistency wit h the Rapid Distortion Theory (RDT) which applies to strongly sheared turbulent flows that are far from equilibrium. Speziale and Xu (1996) used such a technique to regularize Gatski and Speziale's explicit ASM.

4. The ability of the regularized modified Rodi's ASM in predicting levels of anisotropy can be tested by applying it to different three-dimensional inhomogeneous flows.

5. The regularized modified Rodios ASM can be used to solve the flow through a plane channel using other models for the diffusive transport term.

6. There is a need tu calibrate model constants of pressure-strain models for inhomogeneous flows. The calibration exercise can be carried out by incorporating these models in a self-consistent ASM model for a square duct and the results can be compared with DNS/LES data for a low Reynolds number fully-developed square duct flow.

7. A selfconsistent ASM has been developed for a square duct which include the diffusive transport of the anisotropic Reynolds stress tensor. The predictive ability of this model can be tested using different models for the diffusive transport term and different pressure-strain term models.

S. The improved Rodi's ASM that has been developed can be regularized. Shen it can be applied to different flow fields (where the convective transport of the anisotropic Reynolds stress tensor is not negligible) to test its predictive ability.

9. The channel flow and square duct calculations can also be carried out with more complex models of the pressure-strain terms. It would be interesting to see how Lumley mode1 for QG1 and cubic models for 0ij2 (e-g. Fu et. al. (Launder, 1990)) effect the predictive abilities of ASMs.

10. In the present study the dissipation tensor has been modeled using the isotropic assumption. Turbulent flows near solid boundaries - or at low Reynolds numbers - can exhibit significant anisotropies in the dissipation tensor (Mansour et. al.. 1988). It would be interesting to see how different anisotropic dissipation rnodels improve the predictions of ASMs in two- dimensional and three-dimensional flows. Bibliography

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