Implementation of Turbulent Viscosity from EARSM for Two Equation Turbulence Model

Transcription

1 CFD with OpenSource software A course at Chalmers University of Technology Taught by Håkan Nilsson Project work: Implementation of Turbulent Viscosity from EARSM for Two Equation Turbulence Model Developed for OpenFOAM-2.4.x Author: Thejeshwar Sadananda Peer reviewed by: Gonzalo Montero Håkan Nilsson Disclaimer: This is a student project work, done as part of a course where OpenFOAM and some other OpenSource software are introduced to the students. Any reader should be aware that it might not be free of errors. Still, it might be useful for someone who would like learn some details similar to the ones presented in the report and in the accompanying files. The material has gone through a review process. The role of the reviewer is to go through the tutorial and make sure that it works, that it is possible to follow, and to some extent correct the writing. The reviewer has no responsibility for the contents. January 29, 2016

2 Contents Contents 1 1 Introduction 3 2 Theory 4 Algebraic Reynolds Stress Models Explicit Algebraic Reynolds Stress Model Implementation of Turbulent Viscosity 7 4 Axially Rotating Pipe Flow Case 13 Pre-Processing Simulation Post-Processing Results and Discussions 18 Results Learning during the Implementation Conclusion and Future Work 20 7 Study Questions 21 Bibliography 22 1

3 Learning outcomes The reader will learn: Why two equation turbulence model is not suited for computing Complex flows. The basic theory of Algebraic Reynolds stress Model. How to modify the existing turbulence model and test the case in OpenFOAM. Why functionobjects are required in OpenFOAM. How to implement the turbulent viscosity from EARSM in standard two equation turbulence model 2

4 Chapter 1 Introduction Standard two equation turbulence models are widely used in industries to perform CFD simulations. The two-equation turbulence models are robust and requires less computational resources and time. But these turbulence models compute the Reynolds stress based on the eddy viscosity assumption. The Reynolds stresses are computed by neglecting the rotational part of the velocity gradient tensor. Hence flow comprising of strong streamline curvature effects, adverse pressure gradient, flow separation and system rotation cannot be envisaged accurately. These features can be resolved using computational method like Direct Numerical Simulation DNS but it demands extensive computational resources and time. One of the alternative to capture these features accurately is by using turbulence models based on transport equations for the individual Reynolds stress tensor components. This has generated significant interest in obtaining algebraic approximations of the Reynolds stress transport equations which are modelled based on the Reynolds stress transport equation. In this project report, the implementation of the turbulent viscosity ν t from the Explicit Algebraic Reynolds Stress Model EARSM for the two-equation turbulence model is presented.[1] 3

5 Chapter 2 Theory This chapter provides a brief theoretical background on the Algebraic Reynolds Stress Models ARSM and the Explicit Algebraic Reynolds Stress Model EARSM. Algebraic Reynolds Stress Models The Algebraic Reynolds stress models are developed from the modelled Reynolds stress transport equation by assuming that the advection minus diffusion of the individual Reynolds stresses can be expressed as the product of the corresponding quantity for the kinetic energy K and the individual Reynolds stresses normalized by K. There is an implicit relation between the stress components and the mean velocity gradient field which replaces the Boussinesq assumption. ARSM is a systematic method for constructing a non-linear stress relationship that includes the effect of the rotational part of the mean velocity gradient tensor. This feature assists ARSM to capture the three-dimensional turbulent features accurately. It is to be noted that the implicit form of ARSM has been found to be numerically and computationally cumbersome[2]. Explicit Algebraic Reynolds Stress Model The Explicit Algebraic Reynolds stress model relates Reynolds stresses explicitly with the mean flow field. EARSM is numerically robust and requires computational resources and time which is nearly the same as that of the two-equation turbulence models. The robustness of the model is due to the Reynolds stress anisotropy term which is computed based on the normalized strain rate tensor and the rotation rate tensor.[2] The Reynolds stress anisotropy is expressed as a = β 1 S + β 2 S II SI + β 3 Ω II ΩI + β 4 SΩ ΩS + β 5 S 2 Ω Ω 2 S + β 6 SΩ 2 Ω 2 S 2 3 IV I + β 7 S 2 Ω 2 Ω 2 S V I + β 8 SΩS 2 S 2 ΩS β 9 ΩSΩ 2 Ω 2 SΩ + β 10 ΩS 2 Ω 2 Ω 2 S 2 Ω The invariants in the anisotropy term are II S = trs 2, II Ω = trω 2 ; III S = trs 3 ; IV = trsω 2 ; V = trω 2 S In equations 2.1 and 2.2 a, S, Ω represent the second rank tensors. tr represents the trace and I is the identity matrix. In addition to it, S 2, Ω 2 represent the inner product of normalized strain rate and rotation rate tensors respectively. S 2 SS ij S 2 ij S iks kj 2.3 4

6 . EXPLICIT ALGEBRAIC REYNOLDS STRESS MODEL CHAPTER 2. THEORY The normalized strain rate and rotation rate tensors are obtained by multiplying the turbulent time scale with the strain rate and rotation rate tensors. The normalized strain rate tensor is expressed as Similarly normalized rotation rate tensor is expressed as S S ij τs ij 2.4 Ω Ω ij τω ij 2.5 In the near-wall region, viscous effects are important and it is essential to limit the turbulent time scale which is limited by the Kolmogoroff time scale as proposed by Durbin 1993 [8] k µ τ = max ɛ, C τ 2.6 ρɛ where C τ = 6.0 is a model constant. The dimensional strain rate tensor and rotation rate tensor are computed using the velocity gradient as S ij = 1 Ui + U j 2 U k δ ij ; Ω ij = 1 Ui U j x j x i 3 x k 2 x j x i In this project, the turbulent viscosity ν t from EARSM is implemented for an axially rotating pipe flow case which is an application of a three-dimensional mean flow. The turbulent viscosity ν t is expressed as ν t = β 1 + II Ω β 6 kτ 2.8 where β 1, β 6 represents the coefficients used in EARSM to compute ν t. The solution of the β coefficients for the three-dimensional mean flow are β 1 = N 2N 2 7II Ω ; β 3 = 12N 1 IV ; β 4 = 2 N 2 2II Ω ; β 6 = 6N Q Q Q Q ; β 9 = 6 Q ; 2.9 The non-singular denominator is written as Q = 5 6 N 2 2II Ω 2N 2 II Ω 2.10 The β coefficients are inserted in the definition of N which is closely related to the P/ɛ ratio where P and ɛ represent production and dissipation term respectively. The non-linear equation for N is of sixth order for three-dimensional mean flow. This equation does not have a closed solution. There are two ways by which simplified solution of N can be obtained. One way is to use the solution of N for the two-dimensional mean flow. The coefficients P 1 and P 2 are computed as C 2 1 P 1 = II S 2 3 II Ω C P 2 = P 1 2 C II S II Ω 2.12 C 1 = 9 4 c where c 1 = 1.8 and C 1 For P 2 0, are model constants. N = C P P 2 + P P

7 . EXPLICIT ALGEBRAIC REYNOLDS STRESS MODEL CHAPTER 2. THEORY For P 2 < 0, N = C P P cos 3 cos 1 P 1 P1 2 P Another solution of N is to use the pertubed solution of three-dimensional non-linear N equation. In this project, equation 2.8 is implemented using the solution for two-dimensional mean flow for N. In order to further simplify the implementation process, two-dimensional mean flow solution N for P 2 0 is used. 6

8 Chapter 3 Implementation of Turbulent Viscosity The implementation of the turbulent viscosity ν t from the Explicit Algebraic Reynolds Stress Model EARSM in k ω turbulence model is presented in this chapter. Open a new terminal Initialize the OpenFOAM-2.4.x environment by typing OF24x Type the environment variable cd $WM_PROJECT_DIR Copy the komega turbulence model directory with the same parent directory structure in the user directory cp -r --parents src/turbulencemodels/incompressible/ras/komega \ $WM_PROJECT_USER_DIR cd $WM_PROJECT_USER_DIR/src/turbulenceModels/incompressible/RAS Rename the komega directory to earsmimpkomga directory mv komega earsmimpkomega Create a Make directory and two files named files and options under it. Once it is done, add the following lines in to the respective files Make/files earsmimpkomega/earsmimpkomega.c LIB = $FOAM_USER_LIBBIN/libMyIncompressibleRASModels Make/options EXE_INC = \ -I$LIB_SRC/turbulenceModels \ -I$LIB_SRC/transportModels \ -I$LIB_SRC/finiteVolume/lnInclude \ -I$LIB_SRC/meshTools/lnInclude \ -I$LIB_SRC/turbulenceModels/incompressible/RAS/lnInclude \ -I$LIB_SRC/sampling/lnInclude LIB_LIBS = 7

9 CHAPTER 3. IMPLEMENTATION OF TURBULENT VISCOSITY Remove komega.dep compiled file and rename the file name and the classname from komega to earsmimpkomega cd earsmimpkomega; rm komega.dep mv komega.c earsmimpkomega.c; mv komega.h earsmimpkomega.h sed -i s/komega/earsmimpkomega/g earsmimpkomega.* Before the implementation of turbulent viscosity ν t, it is better to compile using wmake libso initially to ensure that there are no errors in the earsmimpkomega.c and earsmimpkomega.h file. The constants c1, C1, ctau are used in EARSM to compute ν t. They are declared as dimensioned scalars. Include the data members in the declaration file earsmimpkomega.h under Model coefficients at line 96 dimensionedscalar c1_; dimensionedscalar C1_; dimensionedscalar ctau_; The β coefficients and the invariants are introduced as member functions. These member functions return volscalarfield. The member functions use an IOobject class to create an object. The object access the runtime directory information and the computes the volscalarfield at all cell centres available under mesh. The member functions used in the earsmimpkomega.h file are shown below. Add the following member functions before Reynolds stress tensor R member function in the declaration file earsmimpkomega.h at line 186. //- Return the time scale tmp<volscalarfield> time const return tmp<volscalarfield> new volscalarfield IOobject "time", mesh_.time.timename, mesh_, maxk_/epsilon, ctau_*sqrtnu/epsilon ; It is a good practice to compile after declaring every member function in earsmimpkomega.h file using wmakelibso command to easily overcome compiling errors. In addition to it, it is to be remembered that wclean command has to be used before compiling the next time. //- Return the invariant invar2s tmp<volscalarfield> invar2s const return tmp<volscalarfield> new volscalarfield 8

10 CHAPTER 3. IMPLEMENTATION OF TURBULENT VISCOSITY IOobject "invar2s", mesh_.time.timename, mesh_, trtime*symmfvc::gradu_ & \ time*symmfvc::gradu_ ; //- Return the invariant invar2r tmp<volscalarfield> invar2r const return tmp<volscalarfield> new volscalarfield IOobject "invar2r", mesh_.time.timename, mesh_, trtime*skewfvc::gradu_ & \ time*skewfvc::gradu_ ; //- Return the coefficient P1 tmp<volscalarfield> P1 const return tmp<volscalarfield> new volscalarfield IOobject "P1", mesh_.time.timename, mesh_, sqrc1_/27 + 9/20*invar2s - 2/3*invar2r*C1_ ; //- Return the coefficient P2 tmp<volscalarfield> P2 const return tmp<volscalarfield> new volscalarfield 9

11 CHAPTER 3. IMPLEMENTATION OF TURBULENT VISCOSITY IOobject "P2", mesh_.time.timename, mesh_, maxsqrp1 - powsqrc1_/9 + 9/10*invar2s + \ 2/3*invar2r,3,scalar0 ; The maxsqrp 1 powsqrc1 /9 + 9/10 invar2s + 2/3 invar2r, 3, scalar0 function is used for computing the member function P 2 such that the result does not end in terms of negative values. //- Return the coefficient N tmp<volscalarfield> N const return tmp<volscalarfield> new volscalarfield IOobject "N", mesh_.time.timename, mesh_, C1_/3 + powp1 + sqrtp2,1/3 + pow\ P1 - sqrtp2,1/3 ; //- Return the coefficient Q tmp<volscalarfield> Q const return tmp<volscalarfield> new volscalarfield IOobject "Q", mesh_.time.timename, mesh_, max5/6*sqrn - 2*invar2r*2*sqrN - invar2r \, SMALL ; The max5/6 sqrn 2 invar2r 2 sqrn invar2r, SMALL function 10

12 CHAPTER 3. IMPLEMENTATION OF TURBULENT VISCOSITY is used for computing the member function Q and this ensures that the result is neither a negative value or zero. //- Return the coefficient beta1 tmp<volscalarfield> beta1 const return tmp<volscalarfield> new volscalarfield IOobject "beta1", mesh_.time.timename, mesh_, -N*2*sqrN-7*invar2r/Q ; //- Return the coefficient beta6 tmp<volscalarfield> beta6 const return tmp<volscalarfield> new volscalarfield IOobject "beta6", mesh_.time.timename, mesh_, -6*N/Q ; The dimensioned scalars declared in the earsmimpkomega.h file are initialized in the source file earsmimpkomega.c file in the same order. In this way the compiler knows that it has allocated the required memory space to the respective data member. Include these data members in the source file earsmimpkomega.c from line 103 c1_ dimensioned<scalar>::lookuporaddtodict "c1", coeffdict_, 1.8, C1_ 11

13 CHAPTER 3. IMPLEMENTATION OF TURBULENT VISCOSITY dimensioned<scalar>::lookuporaddtodict "C1", coeffdict_, 1.8, ctau_ dimensioned<scalar>::lookuporaddtodict "ctau", coeffdict_, 6.0, Compile using wmake libso to ensure that the data members are initialized in the correct order In k ω turbulence model, the turbulent viscosity is computed as ν t = k/ω 3.1 It is to be noted that the equation 3.1 is replaced with equation 2.8 at two locations in the source file earsmimpkomega.c at line 170 and 314. nut_ = -1/2*k_*time*beta1+beta6*invar2r; Compile using wmake libso finally to confirm that the turbulent viscosity ν t from EARSM is implemented successfully. 12

14 Chapter 4 Axially Rotating Pipe Flow Case Axially rotating fully developed turbulent pipe flow is an interesting case of three-dimensional mean flow. This case is based on the pipecyclic tutorial available for the simplefoam in OpenFOAM 2.4.x. In this chapter, the pre-processing, simulation and post-processing of the case are presented. Pre-Processing The axially rotating pipe flow case is a simple three-dimensional mean flow case in which the complex flow features such as streamline curvature effects and local rotation of the streamlines can be envisaged. This case assists in demonstrating the significance of EARSM in two-equation turbulence model. As the case name indicates that the swirling inlet flow enters the pipe with a Reynolds number lying in the turbulent flow regime. The pre-processing steps for setting up the case are as follows Open a new terminal Initialize the OpenFOAM-2.4.x environment by typing OF24x Type the environment variable cd $WM_PROJECT_DIR Copy the pipecyclic tutorial from the T UT ORIALS directory and paste it under run directory. cp -r $FOAM_TUTORIALS/incompressible/simpleFoam/pipeCyclic $FOAM_RUN cd $FOAM_RUN mv pipecyclic testcaseearsm cd testcaseearsm The geometry of the axially rotating turbulent pipe flow are adopted from the pipecyclic tutorial except few minor changes have been made. sed -i "s/converttometers 1/convertToMeters 0.1/g" \ constant/polymesh/blockmeshdict sed -i "s/halfangle 45.0/halfAngle 90.0/g" constant/polymesh/blockmeshdict 13

15 . PRE-PROCESSING CHAPTER 4. AXIALLY ROTATING PIPE FLOW CASE The geometry is created using the blockmeshdict dictionary located in the constant/polymesh directory. The geometry of the pipe flow is specified in the cartesian coordinate system. The pipe of radius 0.05 m and length 1.0 m is set with a half angle of the wedge of 90 o. The coordinates along y and z axis are computed based on the radius and the half angle of the wedge. The geometry is meshed with a hexahedral mesh with the uniform grading of the mesh in all the coordinate directions. The boundary of the geometry are split into several patches. The patches are same as those patches used in pipecyclic tutorial. The patches side1 and side2 of patch type cyclicami are treated physically connected to each other. Once the computational domain is set in the blockmeshdict, the boundary conditions BC have to be applied for all the patches of the domain. This is carried out by creating certain volscalarfield and volvectorfield files under 0 directory. These files are used to initialize and specify boundary conditions for these fields. The directory name 0 suggests the starting time for the simulation. The files available under 0 directory for pipecyclic tutorial are used for setting up the case except that few changes have to be made. cp -r 0.org 0 For U volvectorfield, there is no need to modify the boundary conditions. The inlet boundary condition is codedfixedvalue. This is a generic boundary condition. It constructs on-the-fly a new boundary condition derived from fixedvaluefvpatchfield which is then used to evaluate. At the outlet of the pipe, inletoutlet boundary condition is set when it is a zerogradient condition when flow is outwards. No slip condition is used at the walls.[4] The boundary conditions for p and k volscalarfields remains the same as that of pipecyclic tutorial except that the initial value for k has to be changed. sed -i "s/uniform 1/uniform /g" 0/k The turbulentintensitykineticenergyinlet boundary condition provides turbulent kinetic energy based on the intensity level. The kqrwallfunction BC provides suitable condition for turbulence k, q and R in case of Reynolds number flow. In this Project Reynolds number is considered as 1,30,000. In order to set the boundary conditions for the specific dissipation ω, the following steps have to be executed mv 0/epsilon 0/omega sed -i "s/epsilon/omega/g" 0/omega sed -i "s/ / /g" 0/omega sed -i "s/uniform 1/uniform 29.0/g" 0/omega sed -i "s/turbulentmixinglengthdissipationrateinlet/\ turbulentmixinglengthfrequencyinlet/g" 0/omega sed -i "s/0.5/0.05/g" 0/omega The turbulentmixinglengthfrequencyinlet boundary condition at the inlet provides the specific dissipation based on the mixing length which is half the pipe diameter. The omegawall- Function BC at the walls provides the resultant of specific dissipation of viscous and log-law region.[4] 14

16 . PRE-PROCESSING CHAPTER 4. AXIALLY ROTATING PIPE FLOW CASE The physical property ν is calculated based on the inlet velocity U = 1.3m/s and Reynolds number, the kinematic viscosity ν is found to be m 2 /s The kinematic viscosity is set in the transportproperties dictionary under constant directory of the case. This ν value used in the pipecyclic tutorial is the same as computed value. Once it is done, change the turbulence model in RASProperties dictionary available under constant directory to earsmimpkomega model sed -i "s/realizableke/earsmimpkomega/g" constant/rasproperties Include the dynamic library in the controldict file under system directory "libmyincompressiblerasmodels.so" // libs; The mean velocity profile along the pipe radius is plot during the post-processing stage. The mean velocity can be obtained using fieldaverage functionobject in OpenFOAM 2.4.x. The fieldaverage functionobject calculates the time average of specified fields and writes the results in the time directories. The fieldaverage functionobject is added in the controldict dictionary.[7] functions fieldaverage1 type fieldaverage; functionobjectlibs "libfieldfunctionobjects.so"; enabled true; ; fields ; outputcontrol U mean prime2mean base outputtime; on; on; //RMS time; The post-processing is carried out using ParaView and Sample utility. The sample utility is used to extract the values of the flow variable from the axis of the pipe along the radial direction. Copy and modify the sampledict dictionary from the TUTORIALS directory. cp $FOAM_TUTORIALS/compressible/sonicFoam/\ laminar/shocktube/system/sampledict./system/. Modify the settings in the sampledict dictionary as follows //- sampledict interpolationscheme cellpoint; setformat xmgr; 15

17 . SIMULATION CHAPTER 4. AXIALLY ROTATING PIPE FLOW CASE sets line type uniform; axis z; start ; end ; npoints 10; ; fields U UMean; The cellpoint interpolation type is a linear weighted interpolation scheme which uses cell centre values and extracts the values of the mean velocity U along the z-axis at npoints.[3] Simulation The k ω turbulence model is used in simulating the axially rotating pipe flow as this model works well for wall bounded flows. In this project, the turbulent pipe flow is simulated based on ν t computed using EARSM in k ω turbulence model. The time step deltat 0.01 is set by considering the minimum cell size. Other run time information like writeinterval 10 can be set in the controldict file under system directory. It is to be noted that the default discretization schemes and the iterative solvers of the pipecyclic tutorial settings were used except few changes are made for this case. The keyword epsilon is replaced with omega in f vschemes and f vsolution dictionaries under system directory using sed command. sed -i "s/epsilon/omega/g" system/fvschemes sed -i "s/epsilon/omega/g" system/fvsolution The case is included with all the required functionobjects and utilities. Use blockmesh to mesh the computational domain. To ensure that the mesh is generated properly, check that the number of undefined boundary faces is zero. blockmesh The simplefoam solver is a steady state solver applicable for incompressible turbulent flows [3]. Run the simulation using simplefoam solver. Check that the solution is converged before post-processing the simulation. simplefoam >& log& Post-Processing Once the simulations are converged, the post-processing is carried out using the ParaView and Sample utility. Once the results are written in the time directories, it can be visualized using parafoam. As the parafoam window appears on the screen, click on the Apply button in the Properties dialog box just below the pipeline browser window. click on the slice icon and the slice1 appears on the pipeline browser. Set the origin to 0,0,0 and click z-normal axis in order to make the 16

18 . POST-PROCESSING CHAPTER 4. AXIALLY ROTATING PIPE FLOW CASE Figure 4.1: Plot of velocity field in a plane normal to z-direction slice of the domain normal to the z-axis. Select the cell center U field in the field drop down box and select surface in the adjacent dropdown box. Click on the play button to visualize the velocity field U. Use the Stream Tracer icon present in the toolbar to visualize the streamline in the flow field. After selecting the stream tracer icon, click on the Apply button and click play button to visualize the streamlines of the axially rotating turbulent pip flow. Use the sample utility to extract the mean velocity UMean field and visualize the plot using the xmgrace cp 0/U 0/UMean sample xmgrace postprocessing/sets/0.89/line_u_umean.agr 17

19 Chapter 5 Results and Discussions Results The results of the turbulent pipe flow case using the EARSM implemented and the ideal k ω turbulence model are discussed here. From the results of original k ω turbulence model, it can be seen that the mean velocity profile is not completely parabolic. This is accounted due to the exclusion of rotational rate tensor while computing Reynolds stress tensor. It is also to be noted that U velocity profile looks similar to that of the UMean velocity profile. This is due to the averaging of the momentum equation there by the velocity U corresponds to the average velocity obtained from RANS model. By plotting the mean velocity profile for the axially rotating turbulent pipe flow along z-axis for Figure 5.1: Plot of Mean and instantaneous velocity along z-axis for k ω turbulence model EARSM implemented k ω turbulence model, it is found that the mean velocity UMean is nearly constant at major part of the z-direction and drops to zero close to the wall. It may be attributed to the use of wall functions at the wall. The reason for the similar velocity profiles of U and UMean is same as that mentioned for the results of k ω turbulence model. From[1] it is learnt that the mean velocity profile has to be parabolic in nature but the computed results does not replicate the same. This is because, implementation of turbulent viscosity ν t is not sufficient to implement EARSM completely in two-equation turbulence model. The Reynolds stress anisotropy term has to be used in computing production term in turbulent kinetic energy transport equation and turbulent diffusion as suggested[1]. When the case is set up by refining the mesh to cells, it was not possible to obtain converged solution. From the residuals, it was evident that the pressure field was not converging to the required tolerance limits. 18

20 . LEARNING DURING THE IMPLEMENTATIONCHAPTER 5. RESULTS AND DISCUSSIONS Figure 5.2: Plot of Mean and Inlet velocity along z-axis for EARSM implemented k ω turbulence model Learning during the Implementation In this section, few learninga during the implementation of turbulent viscosity ν t are presented. If the data members are added, they have to be initialized under the constructor list in the source.c file in the same order as they are introduced in the declaration.h file. It is not possible to compute the volscalarfield if the computed value arises to be a complex number or negative values. The dimension set of the calling data members and/or member functions used for computing respective member functions should match. When the conditional statements are introduced in the IOobject class, a temporary dimensioned scalar has to be introduced to accept the value. The truncation error can be rectified by using the maxxyz, scalar0 function in order to get a positive value volscalarfield. Implementing the source code in small steps in a systematic way will make the user to identify and debug errors easily. 19

21 Chapter 6 Conclusion and Future Work The turbulent viscosity ν t from the Explicit Algebraic Reynolds Stress Model is implemented in k-ω turbulence model. The axially rotating pipe flow case is tested using the implemented model. It can be concluded that the turbulence model is incomplete without the implementation of the Reynolds stress anisotropy term. Considering future work, the perturbed solution of three-dimensional N equation can be considered which is a better solution of N than the solution of the two-dimensional N equation. The Reynolds stress anisotropy term must be included in the production term of the turbulent kinetic energy transport equation and turbulent diffusion term to implement EARSM in two-equation turbulence models. The EARSM implemented two-equation turbulence model results can be validated with the standard two equation turbulence models and experimental results. 20

22 Chapter 7 Study Questions Q1: What is the drawback of using eddy viscosity assumption while computing Reynolds stress in two-equation turbulence? Q2: What is ARSM? Q3: How does the cyclicami boundary condition work? Q4: How does the codedfixedvalue boundary condition work? Q5: What type of value is set to return while calculating β coefficients? Q6: What does the sample utility do? 21

23 Bibliography [1] Wallin,S. 1999, An efficient explicit algebraic Reynolds stress k ω model EARSM for aeronautical applications, FFA TN [2] Wallin,S. & Johansson, A. V. 2000, An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows, J. Fluid Mech. 403, [3] Christopher, J., Greenshields, 2015, OpenFOAM User Guide, [Online] Available: [4] OpenFOAM C++ Documentation, [Online] Available: [5] Christopher, J., Greenshields, 2015, OpenFOAM Programmer s Guide, [Online] Available: [6] Nilsson,H., How to implement a turbulence model, [Online] Available: hani/kurser/os CFD 2015/implementTurbulenceModel.pdf [7] Nilsson,H., someutilitiesandfunctionobjects.pdf, [Online] Available: hani/kurser/os CFD 2015/someUtilitiesAndFunctionObjects.pdf [8] Durbin, P.A. 1993, Application of a near-wall turbulence model to boundary layers and heat transfer, Int. J. Heat and Fluid Flow 14,