Code Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes. Subrahmanya P. Veluri

Transcription

1 Code Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes Subramanya P. Veluri Dissertation submitted to te faculty of te Virginia Polytecnic Institute and State University in partial fulfillment of te requirements for te degree of Doctor of Pilosopy In Aerospace Engineering Cristoper J. Roy, Cair William Mason Leig S. McCue-Weil Danes K. Tafti August 6, 00 Blacksburg, Virginia Keywords: Code Verification, Metod of Manufactured Solutions, Discretization Error, Order of Accuracy, Computational Fluid Dynamics Copyrigt 00, Subramanya P. Veluri

2 Code Verification and Numerical Accuracy Assessment for Finite Volume CFD Codes Subramanya P. Veluri ABSTRACT A detailed code verification study of an unstructured finite volume Computational Fluid Dynamics (CFD) code is performed. Te Metod of Manufactured Solutions is used to generate exact solutions for te Euler and Navier-Stokes equations to verify te correctness of te code troug order of accuracy testing. Te verification testing is performed on different mes types wic include triangular and quadrilateral elements in D and tetraedral, prismatic, and exaedral elements in 3D. Te requirements of systematic mes refinement are discussed, particularly in regards to unstructured meses. Different code options verified include te baseline steady state governing equations, transport models, turbulence models, boundary conditions and unsteady flows. Coding mistakes, algoritm inconsistencies, and mes quality sensitivities uncovered during te code verification are presented. In recent years, tere as been significant work on te development of algoritms for te compressible Navier-Stokes equations on unstructured grids. One of te callenging tasks during te development of tese algoritms is te formulation of consistent and accurate diffusion operators. Te robustness and accuracy of diffusion operators depends on mes quality. A survey of diffusion operators for compressible CFD solvers is conducted to understand different formulation procedures for diffusion fluxes. A patc-wise version of te Metod of Manufactured Solutions is used to test te accuracy of selected diffusion operators. Tis testing of diffusion operators is limited to cell-centered finite volume metods wic are formally second order accurate. Tese diffusion operators are tested and compared on different D mes topologies to study te effect of mes quality (stretcing, aspect ratio, skewness, and curvature) on teir numerical accuracy. Quantities examined include te numerical approximation errors and order of accuracy associated wit face gradient reconstruction. From te analysis, defects in some of te numerical formulations are identified along wit some robust and accurate diffusion operators.

3 Acknowledgements I would like to express my sincere gratitude to Dr. Cristoper J. Roy for giving me an opportunity to work on tis researc. I am tankful to im for is guidance, encouragement, and is continuous financial support during my PD. I tank my committee members Dr. William Mason, Dr. Leig S. McCue-Weil, and Dr. Danes K. Tafti for serving on my committee. Tis work is supported by te National Aeronautic and Space Administration s Constellation University Institutes Program (CUIP) wit Claudia Meyer and Jeffrey Ryback of NASA Glenn Researc Center serving as program managers and Kevin Tucker and Jeffrey West of NASA Marsall Space Fligt Center serving as tecnical monitors. I would like to tank Edward Luke from Mississippi State University for saring is ideas and elping me during my researc. I tank my parents, broter, and sister for encouraging me and giving me te confidence required to complete my PD successfully. I tank Ravi Duggirala for is tougtful ideas and useful researc discussions wenever needed during my researc. I also want to tank system administrators James Clark, Sannon Price from Auburn University and Steve Edwards from Virginia Polytecnic Institute and State University for teir elp wen required. Finally, I want to tank all my fellow researc students at Auburn University and Virginia Polytecnic Institute and State University for teir valuable suggestions and providing me wit ideal work environment. I tank all my friends for understanding me and providing te necessary elp. iii

4 Table of Contents Nomenclature... viii List of Figures... x List of Tables... xv Introduction.... Motivation.... Literature Review..... Code Verification..... Diffusion Operators... 8 Basic Concepts of Verification Verification Code Verification Order of Accuracy Formal Order of Accuracy Observed Order of Accuracy Metod of Manufactured Solutions Procedure for MMS Attributes of a Good Manufactured Solution Advantages of Metod of Manufactured Solutions Mes Generation D Mes Topologies D Structured Meses D Unstructured Meses D Hybrid Meses D Mes Topologies D Structured Meses... 3 iv

5 3.. 3D Unstructured Meses D Hybrid Meses Systematic Mes Refinement Verification of a Finite Volume CFD Code Finite Volume Code Verification of Baseline Governing Equations Flow Equations Verification of te Euler Equations Verification of te Navier-Stokes Equations Effect of Mes Quality Verification of Transport Models Verification of Boundary Conditions Mes Topologies No-slip Wall No-slip Adiabatic Wall No-slip Isotermal Wall Slip Wall Isentropic Inflow Outflow Extrapolation Farfield Verification of Turbulence Models Verification of Time Accuracy for Unsteady Flows Issues Uncovered During Code Verification Coding Mistakes/Algoritm Inconsistencies... 7 v

6 4.7. Grid Sensitivities Finite Volume Diffusion Operators Robustness and Accuracy Testing Framework Cell-Centered Finite Volume Diffusion Operators Node Averaging Metod Unweigted and Weigted Least Squares Loci-CHEM Diffusion Operator FAMLSQ Metod wit Compact Stencil Mes Topologies Effect of Mes Quality D Structured Meses Uniform Mes Aspect Ratio Stretcing Skewness Curvature D Unstructured Meses Mes wit Isotropic Cells Skewness wit Aspect Ratio Skewness wit Curvature Conclusions Summary of Results Main Contributions of tis Work Future Work and Recommendations References... 0 vi

7 Appendix A Appendix B Appendix C... 0 vii

8 Nomenclature C p, C v = specific eats, J/Kg K DE = discretization error e = energy, J g p = coefficient of te leading error term = normalized grid spacing; entalpy, J/Kg K f = eat of formation, J/Kg K F = blending function f = general solution variable k = turbulent kinetic energy p = spatial order of accuracy; pressure, N/m q = temporal order of accuracy q L = laminar eat flux q T = turbulent eat flux R = gas constant, J/Kg K r = refinement factor T = temperature, K t = time, sec V i = Volume of cell i u, v, w = Cartesian velocity components, m/s x, y, z = Cartesian coordinates, m viii

9 Greek Symbols ε = error µ = viscosity, N sec/m µ T = turbulent viscosity, N sec/m ρ = density, Kg/m 3 ω = turbulent dissipation rate Superscripts n = time level Subscripts exact = exact continuum value k = mes level,,, 3, etc.; fine to coarse ref = reference value ix

10 List of Figures Figure : D structured meses: a) Cartesian, b) stretced Cartesian, c) curvilinear, and d) skewed curvilinear... Figure : D unstructured meses: a) general unstructured, b) uni-directional diagonal, and c) alternate diagonal... Figure 3: D ybrid meses: a) skewed ybrid and b) igly skewed ybrid...3 Figure 4: 3D structured meses: a) Cartesian and b) skewed curvilinear...4 Figure 5: 3D unstructured meses: a) unstructured mes wit tetraedral cells, unstructured mes wit prismatic cells, c) igly skewed unstructured mes wit tetraedral cells, and d) igly skewed unstructured mes wit prismatic cells...5 Figure 6: 3D ybrid meses: a) skewed ybrid and b) igly skewed ybrid...6 Figure 7: a) Cube split into five tetraedral cells, b) 3D unstructured mes wit tetraedral cells generated from a skewed and stretced curvilinear mes...8 Figure 8: a) Manufactured Solution of u-velocity and b) mass source term...33 Figure 9: Order of accuracy results for Euler equations on a 3D skewed ybrid mes using a) L norm of te discretization error, b) L norm of te discretization error, and c) L norm of te discretization error...34 Figure 0: Order of accuracy results for Navier-Stokes equations on a D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...36 Figure : Order of accuracy results for Navier-Stokes equations on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...36 Figure : Order of accuracy results using L norm of te discretization error for Navier- Stokes equations on a D skewed curvilinear mes using a) original diffusion operator and b) new diffusion operator...38 Figure 3: Order of accuracy results using L norm of te discretization error for Navier- Stokes equations on a D alternate diagonal unstructured mes using a) old diffusion operator and b) modified diffusion operator...39 Figure 4: Order of accuracy results for Navier-Stokes equations on a D igly skewed ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...40 x

11 Figure 5: Order of accuracy results for Navier-Stokes equations on a 3D igly skewed ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...4 Figure 6: Comparison of L norm discretization error on 3D skewed ybrid mes and 3D igly skewed ybrid mes...4 Figure 7: Order of accuracy results for Suterland s law of viscosity on a 3D skewed ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...44 Figure 8: Wavy surface used as a well-defined boundary for 3D meses...45 Figure 9: Meses used for boundary condition verification a) D ybrid mes and b) 3D ybrid mes...46 Figure 0: Temperature contours wen te bottom boundary is defined as a no-slip adiabatic wall...48 Figure : Order of accuracy results for adiabatic no-slip wall boundary on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...49 Figure : Temperature contours wen te bottom boundary is defined as a no-slip isotermal wall...50 Figure 3: Order of accuracy results for isotermal no-slip wall boundary on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...5 Figure 4: Order of accuracy results for slip wall boundary wit te Euler equations on a 3D skewed ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...54 Figure 5: Order of accuracy results for slip wall boundary wit te Navier-Stokes equations on a 3D ybrid mes wit straigt boundaries using a) L norm of te discretization error and b) L norm of te discretization error...54 Figure 6: Order of accuracy results for isentropic inflow boundary on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...56 xi

12 Figure 7: Order of accuracy results for subsonic outflow boundary on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...58 Figure 8: Order of accuracy results for extrapolate boundary on a 3D ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...60 Figure 9: Order of accuracy results for farfield boundary on a 3D skewed ybrid mes using a) L norm of te discretization error and b) L norm of te discretization error...6 Figure 30: Ratio of production term to te destruction term in turbulent kinetic energy equation in a D rectangular domain...63 Figure 3: Order of accuracy results on te 3D skewed ybrid mes for k-ω turbulence model (F=) using a) L norm of te discretization error and b) L norm of te discretization error...64 Figure 3: Order of accuracy results on te 3D skewed ybrid mes for k-ε turbulence model (F=0) using a) L norm of te discretization error and b) L norm of te discretization error...65 Figure 33: Order of accuracy results on te D ybrid mes for k-ε turbulence model (F=0) using a) L norm of te discretization error and b) L norm of te discretization error...66 Figure 34: Order of accuracy results on te 3D igly skewed curvilinear (i.e., structured) mes wit exaedral cells for k-ε turbulence model (F=0) using a) L norm of te discretization error and b) L norm of te discretization error...66 Figure 35: Order of accuracy results on te 3D unstructured mes wit tetraedral cells for k-ε turbulence model (F=0) using a) L norm of te discretization error and b) L norm of te discretization error...67 Figure 36: Order of accuracy results for time accuracy of te unsteady flows on te D ybrid grid using a) L norm of te discretization error and b) L norm of te discretization error...70 Figure 37: Order of accuracy results for time accuracy of te unsteady flows on te 3D ybrid grid using a) L norm of te discretization error and b) L norm of te discretization error...70 xii

13 Figure 38: Srinking of a mes over a focal point for a D unstructured mes...76 Figure 39: An illustration of te reconstruction of a centered control volume about face F c as sown in te saded region...78 Figure 40: Structured mes topologies in D: a) uniform mes, b) mes wit aspect ratio cells, c) stretced mes, and d) mes wit skewed cells...80 Figure 4: D structured mes curvilinear mes...8 Figure 4: Unstructured mes topologies in D: a) mes wit isotropic cells, b) mes wit aspect ratio cells, and c) curvilinear mes...8 Figure 43: Order of accuracy results for gradients at face center on a D structured uniform mes using a) node averaging metod and b) weigted least squares metod...83 Figure 44: Order of accuracy results for gradients at face center on a D structured mes wit aspect ratio using a) node averaging metod and b) weigted least squares metod...84 Figure 45: Order of accuracy results for gradients at face center on a D structured mes wit stretced cells using a) node averaging metod and b) weigted least squares metod...85 Figure 46: Order of accuracy results for gradients at face center on a D structured mes wit skewed cells using a) node averaging metod and b) weigted least squares metod...86 Figure 47: Order of accuracy results for gradients at face center on a D structured mes wit skewed cells using a) node averaging metod and b) weigted least squares metod...87 Figure 48: Order of accuracy results for gradients at face center on a D unstructured mes wit isotropic triangular cells using a) node averaging metod and b) weigted least squares metod...88 Figure 49: Order of accuracy results for gradients at face center on a D unstructured mes wit isotropic triangular cells using Loci-CHEM approac...89 Figure 50: Comparison of error in te gradients at te face center for te tested numerical metods on a D unstructured mes wit isotropic triangular cells...90 xiii

14 Figure 5: Order of accuracy results for gradients at face center on a D unstructured triangular mes wit skewed and aspect ratio cells using a) node averaging metod and b) weigted least squares metod...9 Figure 5: Order of accuracy results for gradients at face center on a D unstructured triangular mes wit skewed and aspect ratio cells using Loci-CHEM approac...9 Figure 53: Comparison of error in te gradients at te face center for te tested numerical metods on a D unstructured triangular mes wit skewed and aspect ratio cells...9 Figure 54: Order of accuracy results for gradients at face center on a D unstructured triangular mes wit skewed and curved cells using a) node averaging metod and b) weigted least squares metod...94 Figure 55: Order of accuracy results for gradients at face center on a D unstructured triangular mes wit skewed and curved cells using Loci-CHEM approac...95 Figure 56: Order of accuracy results for gradients at face center on a D unstructured triangular mes wit skewed and curved cells using FAMLSQ approac...96 Figure 57: Comparison of error in te gradients at te face center for te tested numerical metods on a D unstructured triangular mes wit skewed and curved cells...96 Figure A: Verification of different options in te finite volume Loci-CHEM CFD code.08 xiv

15 List of Tables Table. Different mes levels and mes types... 8 Table. Order of accuracy of te governing equations on igly skewed cells in 3D Table B. Quantification of cell quality..09 Table C. Constants for D Manufactured Solutions...0 Table C. Constants for 3D Manufactured Solutions.. xv

16 . Introduction. Motivation Computational Fluid Dynamics, CFD, is playing an ever-increasing role in te design, analysis, and optimization of many engineering problems. CFD is used across all disciplines were fluid flow is important. Apart from te CFD applications in aerospace engineering, oter applications of CFD are in automobile engineering, industrial manufacturing, civil engineering, environmental engineering, naval arcitecture and many more. It is terefore very important tat decision makers ave confidence in te correctness of te CFD predictions. Wit increases in computational power, CFD practitioners often focus on solving more complex and difficult problems rater tan demonstrating accuracy of teir current problems, wic can lead to a decrease in te quality of te simulations. Wit te numerical problems involving fluid flows getting more complicated, mistakes made by te developers in building a CFD code for solving tose numerical problems will increase. Hatton conducted a study on te reliability of te scientific software. According to Hatton, approximately 40% of te software failures are caused due to static faults wic are mistakes made by te developers during building a code. Some examples of static faults include dependency on uninitialized variables, too many arguments passed into a subroutine, use of nonlocal variables in functions, etc. Hatton classified is study into two parts wic are collectively called te T Experiments. In te first part of te study, e examined codes from a wide range of scientific disciplines using static testing were testing can be done witout running a code and in te second part of te study e examined codes in a single disciple using dynamic testing. During te static testing, Hatton examined 00 different codes in 40 different application areas written in C or FORTRAN. He observed tat te C codes contained approximately 8 serious static faults for every 000 lines of executable code, wile te FORTRAN codes contained approximately serious faults per 000 lines wic can lead to te failure of software meaning software giving a wrong solution. In is dynamic testing, were te testing is done by actually running te code, Hatton examined codes in te area of seismic data processing. In tis study, e examined 9 independent, mature commercial codes developed independently by different companies wic employed te same algoritm, te same programming language, te same user

17 defined parameters, and te same input data and te agreement between te codes was only witin 00% (i.e., a factor of two). Tese experiments done by Hatton provide motivation to work towards building confidence in CFD solutions. So it is necessary to find any coding mistakes or bugs tat may ave been made during te development stage of a code. Engineers and oter CFD practitioners wo rely on te numerical solutions from CFD codes need assurance tat te codes are free from coding mistakes. Te procedure of finding mistakes in te code is called code verification.. Literature Review Topics investigated include te code verification of finite volume CFD codes using te Metod of Manufactured Solutions and also te robustness and accuracy of diffusion operators. Te numerical formulations of te diffusion terms in te Navier-Stokes equations are termed as diffusion operators. During te code verification process of an unstructured finite volume CFD code, it was observed tat te formulation of an accurate and robust viscous operator is a callenging task wic motivated te study and testing of different diffusion operators... Code Verification Verification of CFD codes as been te subject of many studies in recent years. Abanto et al. demonstrated an approac to test te accuracy of some of te most widespread commercial codes. Tey presented grid convergence studies on atypical CFD cases using some commercial CFD packages. Teir verification test cases include an incompressible laminar Poiseuille flow, a manufactured incompressible laminar boundary layer flow, an incompressible re-circulating flow and an incompressible annular flow. Different types of structured and unstructured meses were used during te study. Tey observed non-monotonic grid convergence for all teir test cases. Iterative convergence of te discrete equations to macine zero did not guaranty accurate flow field predications wic meant tat te codes converged to wrong solutions. From teir study, tey recommended tat users perform te verification of commercial CFD codes and be cautious wen using te commercial codes on industrial problems. Kleb and Wood 3 pointed out tat te computational simulation community is not routinely publising independently verifiable tests to accompany new models or algoritms. Tey mentioned te importance of conducting component-level verification tests before attempting system-level verification and also publising tem wen introducing a new

18 component algoritm. Tey proposed a protocol for te introduction of new metods and pysical models tat would provide te computational community wit a credible istory of documented, repeatable verification tests tat would enable independent replication. Roace 4 discussed te verification of codes and calculations along wit some definitions and descriptions related to confidence building in computational fluid dynamics. Verification was described as solving te equations rigt and validation as solving te rigt equations. Different aspects discussed in te paper include te distinction between code verification and validation, grid convergence and iterative convergence, truncation error and discretization error. Also discussed were verification of calculations, error taxonomies, code verification via systematic grid convergence testing, te Grid Convergence Index (GCI) and sensitivity of grid convergence testing. According to te autor, verification does not include all aspects of code quality assurance like te important concerns of version control or arciving of input data. In te books by Roace 5 and Knupp and Salari, 6 te autors compreensively discussed code verification, te Metod of Manufactured Solutions (MMS) used to obtain exact solutions for code verification purposes, and order of accuracy verification. Recently, Oberkampf and Roy 7 also in teir book discussed in detail te concepts of verification. Teir detailed focus was on fundamental concepts of code verification and software engineering, solution verification, validation, and predictive capability of scientific computing. Te first application of MMS for code verification was by Roace and Steinberg in In teir work, tey used te MMS approac to verify a code for generating treedimensional transformations for elliptic partial differential equations. Additional discussions of te MMS procedure for code verification ave been presented by Roace. 5,9 More recently, Oberkampf and Roy 7 discussed te use of MMS for generating exact solutions along wit te order of accuracy testing for code verification. In prior work, MMS as been used to verify two compressible CFD codes 0- : Premo 3 and WIND. 4 Te Premo code employed a node-centered approac using unstructured meses and te Wind code employed a similar sceme on structured meses. Bot codes used Roe s upwind metod wit MUSCL extrapolation for te convective terms and central differences for te diffusion terms making te codes second-order accurate. Te form of Manufactured Solution 3

19 was cosen to be a smoot, infinitely differentiable and cosen suc tat all te terms in te governing equations were exercised. Te Manufactured Solutions were tested on different grid levels and bot te codes demonstrated second-order special accuracy as te mes was refined, tus giving a ig degree of confidence tat te codes were free from coding mistakes. Te MMS was found to be an invaluable tool for finding coding mistakes. In teir work, te autors successfully verified bot te inviscid Euler equations and te laminar Navier-Stokes equations. An alternative statistical approac to MMS was proposed by Hebert and Luke 5 for te Loci- CHEM combustion CFD code. 6 In teir approac, tey employed a single mes level wic was srunk down and used to statistically sample te discretization error in different regions of te domain of interest. Teir work successfully verified te Loci-CHEM CFD code for te 3D, multi-species, laminar Navier-Stokes equations using bot statistical and traditional MMS. Anoter similar approac to statistical MMS was te downscaling approac to order verification 7,8 wic also employed a single mes wic was scaled down about a single point in te domain instead of statistically sampling te smaller meses in te domain of interest. Bot te statistical MMS and te downscaling approaces ave an advantage of being relatively inexpensive because tey do not require mes refinement. But one of te disadvantages of using tese metods is tat tey neglect te possibility of discretization error transport into te scaleddown domain. Tus, it is possible to pass a code order of accuracy verification test using statistical MMS or te downscaling approac for a case tat would fail te test using traditional MMS. MMS was also applied to verify te boundary conditions. Bond et al. 9,0 documented te development of a Manufactured Solution capable of testing te governing equations and boundary conditions commonly implemented in CFD codes. Along wit te Euler, Navier- Stokes, and Reynolds-averaged Navier-Stokes equations, verification of boundary conditions including slip wall, no-slip wall (adiabatic and isotermal) boundary and outflow (subsonic, supersonic and mixed) boundary conditions was performed. Te derived Manufactured Solution was applicable not only to Premo for wic te Manufactured Solution was used, but also to general CFD codes. Te verification of te Premo code was done on skewed, non-uniform, tree-dimensional meses and te sequence of meses were designed to obtain asymptotic results wit reasonable computational cost. Te Manufactured Solution used ad identified a number of formulation weaknesses wit boundary conditions and gradient reconstruction 4

20 metods. In teir study, te reason for te inconsistency in te order of accuracy results for te Euler slip wall boundary was found to be not because of te coding mistakes but instead caused by a problem wit te weak enforcement of te slip condition. A newly implemented caracteristic formulation eliminated te problems and second order convergence was observed. Te implementation of Green-Gauss gradient calculation in Premo was found to be strictly valid only for tetraedral element types because te implementation neglected some additional support necessary for non-cartesian exaedral element types and ence sowed poor order performance. Tere ave been tree coordinated efforts to apply MMS to turbulent flows. Pelletier and co-workers ave summarized teir work on D incompressible turbulent sear layers using a finite element code wit a focus on a logaritmic form of te k-ε two-equation RANS model., Tey employed Manufactured Solutions wic mimic turbulent sear flows, wit te turbulent kinetic energy and te turbulent eddy viscosity as te two quantities specified in te Manufactured Solution. For te cases examined, tey were able to verify te code by reproducing te formal order of accuracy. More recently, Eca and co-workers ave publised a series of papers on Manufactured Solutions for te D incompressible turbulent Navier-Stokes equations. 3-5 Tey also employed pysically-based Manufactured Solutions, in tis case mimicking wall-bounded turbulent flow. Tis group looked at bot finite-difference and finitevolume discretizations, and examined a number of turbulence models including te Spalart- Allmaras one-equation model 6 and two two-equation models: Menter s baseline (BSL) version k-ω model 7 and Kok s turbulent/non-turbulent k-ω model. 8 Wile successful in some cases, teir pysically-based Manufactured Solution often led to numerical instabilities, a reduction in te observed mes convergence rate, or even inconsistency of te numerical sceme (i.e., te discretization error did not decrease as te mes was refined). In order to independently test different aspects of te governing equations, in some cases tey replaced certain discretized terms (or even wole equations) wit te analytic counterpart from te Manufactured Solution. For te Spalart-Allmaras model tey specified te turbulent viscosity, wile for te two equation models tey specified bot te turbulent eddy viscosity and te turbulent kinetic energy. Te cases tey examined employed a Reynolds number of 0 6 and used Cartesian meses wic were clustered in te y-direction towards te wall. 5

21 Roy et al. 9 presented a different approac for verifying RANS turbulence models in CFD codes. Teir approac used smoot, non-pysical Manufactured Solutions wic were cosen suc tat tey provided contributions from all te terms in te turbulence transport equations including convection, diffusion, production, and destruction terms. Te turbulence model verified was te baseline version of Menter s k-ω model and te code used was te Loci-CHEM CFD code. Special attention was paid to te blending function wic allowed te model to switc between a k-ω and a transformed k-ε model. In teir approac, Manufactured Solutions were selected suc tat tey activate only one branc of te min and max functions in te turbulence model. Certain terms were turned off in bot te numerical code and te Manufactured Solution to focus on different parts of te turbulence equations. Te turbulence transport equations along wit te RANS equations were verified in te Loci-CHEM CFD code by computing te observed order of accuracy on a series of consistently refined two-dimensional meses. For te structured mes topologies examined, i.e. Cartesian and skewed curvilinear meses, te observed order matced te formal order of two. But for te unstructured meses examined, te observed order of accuracy was found to be first order. Testing te Euler equations and Navier-Stokes equations on te unstructured meses suggested tat te source for te reduced order of accuracy was te diffusion operator on te unstructured meses. Tomas et al. 8 presented a new metodology for verification of finite volume computational metods using unstructured meses. Te discretization-order properties were studied in computational windows, constructed witin a collection of meses or a single mes and tests were performed witin eac window to address a combination of problem-, solution-, and discretizaton/mes-related features affecting discretization error convergence. A computational window was used to improve te verification of unstructured mes computational metods intended for large-scale applications and te integral norms do not provide sufficient information to isolate te source of errors. A downscaling tecnique was also used, in addition to a traditional mes refinement. Meses witin te windows were constrained to be consistently refined, enabling a meaningful assessment of asymptotic error convergence on unstructured grids. Two types of finite volume scemes were considered: node centered scemes and cell centered scemes. Demonstration of te metodology was sown wic included a comparative accuracy assessment of commonly used scemes on general mixed grids and te identification of local accuracy deterioration at boundary intersections. Te autors provided recommendations 6

22 on coosing relevant tests to verify a code for large-scale computations. Tey recommended tat eac problem-related feature like boundary conditions etc. to be addressed by coosing an appropriate computational window and eac solution related features like socks, boundary layer, flow separation etc. to be addressed by coosing an appropriate Manufactured Solution. Proper mes refinement between different mes levels used for code verification is important to acieve correct order of accuracy. Salas 30 presented te necessary conditions to properly establis mes convergence. He demonstrated is ideas using a teoretical model and a numerical example and sowed tat anomalously low or ig observed rates can be exibited by oterwise well-beaved algoritms because of improper use of mes refinement ratios in different directions. In is study, e sowed tat as te mes aspect ratio increased, finer meses are required to enter te asymptotic range. Etienne et al. 3 demonstrated tree approaces to verification of temporal accuracy of unsteady time accurate flow solvers. Wen iger order time-stepping scemes are used wit second order special discretization, verifying te time-stepping scemes requires very fine meses tat are often impractical to perform calculations on due to limits in memory size and CPU times. Te tree approaces presented to verify time accuracy in a flow solver included te direct approac in wic bot te mes size and time step were refined simultaneously in a consistent and coerent fasion, te decoupled approac in wic te mes size was so small tat te spatial discretization error could be neglected so tat only a time-step refinement study was done and te Iterated Ricardson Extrapolation metod wic allowed verification of te time-integrators by time step refinement on a coarse mes. In te direct approac wit a iger order time-stepping sceme, every time te time step was refined by a factor of two, te mes needs to be refined by a iger factor of n/p were n and p are te formal orders of time and space, respectively, requiring tis approac to use very fine meses for verifying iger order time-stepping scemes. In te case of decoupled approac, a mes size sufficiently small was used so tat te spatial contribution to te error was neglected compared to te temporal contribution and again tis approac required a very fine mes wic makes tis approac along wit te direct approac very expensive or impractical for code verification. In te case of te Iterated Ricardson Extrapolation metod, for a fixed mes size, a time step refinement study was performed, and Ricardson extrapolation was applied repeatedly to te data to yield an 7

23 approximation to te exact solution to te ordinary differential equations in time resulting from spatial discretization wit te given fixed mes size. Performing Ricardson extrapolation decoupled te time step refinement from te mes size refinement for estimating te temporal contribution to te global error in te final time. If te Ricardson extrapolation in time was feasible, ten time-step refinement on any mes owever coarse was sufficient to acieve verification of time integrators of arbitrarily ig order of accuracy. Tremblay et al. 3 presented te use of MMS for fluid-structure interactions code verification and debugging. MMS was used to generate exact solutions and ten test te code using systematic mes refinement. Manufactured Solutions were presented for two-dimensional incompressible flow strongly coupled wit an isotropic structure undergoing large displacements. Teir results illustrated te power and cost-effectiveness of te approac... Diffusion Operators Since te callenge of formulating a diffusion operator for an unstructured mes is more complicated wen compared to a structured mes, te literature survey is concentrated mostly on te diffusion operator formulations on unstructured meses. Knigt 33 developed an implicit algoritm for te two-dimensional, compressible, laminar, Navier-Stokes equations using an unstructured grid consisting of triangular cells. A cell centered data structure was employed wit te flow variables stored at te cell centroids. For te viscous fluxes, te values on te faces were calculated by applying Gauss s teorem to te quadrilateral defined by te cell centroids of te cells adjacent to te face and te two nodes defining te end points. Te solution variables at te nodes were obtained by te second order interpolation of te conserved variables from tose cells saring te node. A similar formulation of viscous fluxes on te cell faces was used by Ollivier-Gooc et al. 34 Tey presented a new approac for a ig-order-accurate finite-volume discretization for diffusive fluxes tat was based on te gradients computed during solution reconstruction. In teir analysis, teir scemes based on linear and cubic reconstruction acieved second and fourt order accuracy, respectively, wile te scemes based on quadratic reconstruction were second order accurate. Tey examined bot vertex centered and cell centered control volumes for linear, quadratic and cubic reconstructions. For cell centered control volumes, gradients were calculated by using Green-Gauss integration around a diamond connecting te end points of an edge and te centroids of te cells tat sare te edge. Te 8

24 solution at te end points of te edge was estimated by averaging data in incident control volumes. Te numerical experiments conducted sowed tat nominal accuracy was attained in all cases for te advection-diffusion problems. Vaassen et al. 35 presented a finite volume cell centered sceme for te solution of te tree-dimensional Navier-Stokes equations were tey used a conservative and consistent discretization approac for te diffusive terms based on an extended version of Coirier s diamond pat. 36 A Gauss formula was employed to perform te integration of fluxes on te cell surfaces. To obtain a consistent discretization of te diffusion terms regardless of te irregularity of te mes, a second order approximation of te gradient was used. For te calculation of gradients on te face centers, a diamond pat was built by connecting eac vertex of te face to te left and rigt neigboring cell centroids, forming a polyedron on wic a Green-Gauss formula was applied. Numerical results using te solver were sown to ave good accuracy even on igly distorted meses. Grismer et al. 37 developed an implicit, unstructured Euler/Navier-Stokes finite volume solver wic is based on a cell centered sceme. In teir procedure for te calculation of viscous fluxes, a piecewise linear reconstruction was used to approximate te solution variables witin cells. Te linear reconstruction was derived from te cell values and gradients by applying a second-order Taylor-series expansion. Te gradient at te cell centroid was evaluated by minimizing te weigted error between te reconstructed function and te neigboring cell values. Te gradient on te face was sown as a vector sum of te components normal and tangential to te face. Te tangential component of te gradient vector at te cell face was te average of tangential components of te gradients in te two cells saring te face. Luke 39 used te same formulation for te viscous fluxes wit some canges in te normal gradient calculation. Te canges were made after te formulation for te diffusive fluxes was sown to be inconsistent on a skewed curvilinear mes. 38 Te diffusion operator formulations used by Grismer 37 and Luke 39 are different from te approaces discussed earlier. Instead of directly calculating te gradients at te face centers, te gradients were calculated at te cell centriod and ten interpolated to te face centers. Mavriplis 40 examined te accuracy of te various gradient reconstruction tecniques on unstructured meses. He demonstrated tat te unweigted least-squares gradient construction 9

25 severely under-estimated normal gradients for igly stretced mesed in te presence of surface curvature. It was sown tat te above beavior could be expected for vertex based discretizations and cell-centered discretizations operating on triangular and quadrilateral meses (tetraedral and prismatic meses in 3D). Te use of inverse distance weigting in te leastsquares reconstruction could be used to recover good accuracy for te vertex and cell-centered discretizations on quadrilateral meses and for vertex discretizations on triangular meses; owever, and te tecnique was sown to be ineffective for cell-centered discretizations on triangular meses. Te Green-Gauss construction tecnique produced adequate gradient estimate in all te cases. Te autor concluded tat te use of inverse distance weigted least-squares gradients and Green-Gauss gradients in te discretization of convective and viscous terms was a prudent strategy, but te approaces were sown to be less robust tan upwind scemes and often required gradient limiting to acieve stable solutions. Delanaye et al. 4 presented a second-order finite volume cell-centered tecnique for computing steady-state solutions of te full Euler and Navier-Stokes equations on unstructured meses. Te sceme was designed suc tat its accuracy was only weakly sensitive to mes distortions. An original quadratic reconstruction wit a fixed stencil and ig-order flux integration by te Gauss quadrature rule was employed to compute te advective term of te equations. For te diffusive term, te derivatives were obtained at te midpoint of an edge by using a linear interpolation between te rigt and left neigbors. Te sceme s weak sensitivity to mes distortions was demonstrated for inviscid flow calculation wit te computation of Ringleb s flow and for te viscous flow calculation wit te computation of te viscous flow around an isotermal flat plate. Since te ig-order sceme produces oscillations in te vicinity of discontinuities leading to instability, te full quadratic reconstruction was used in smoot flows and te sceme was automatically switced to a linear reconstruction at te vicinity of discontinuities. Two inviscid flow computations over te NACA00 airfoil, a supersonic flow at a free stream Mac number of. wit zero angle of attack and a transonic flow at free stream Mac number of 0.8 wit angle of attack of.5 were performed to illustrate te application of teir quadratic reconstruction sceme. Tey also investigated te supersonic flow over a compression ramp by using a ybrid grid. 0

26 Haselbacer et al. 4,43 presented an upwind flow solution metod for mixed unstructured meses in two dimensions consisting of triangular and quadrilateral meses. Te discretization of te viscous fluxes on general unstructured meses was studied and a positive discretization was developed for Laplace s equation and extended to Navier-Stokes equations. A positive discretization in tis case was defined as te discretization of te Laplacian were te weigts in te discretized equation were positive. It is desirable for any discretization tecnique to ensure positivity since turbulence models are extremely sensitive to non-positive weigts. An approximate form of te viscous fluxes was devised tat was independent of te cell topology, an approac tey referred to as mes transparent. Te autors constructed control volumes from dual cells and te solution variables were stored at mes vertices. Te autors investigated te viscous fluxes to obtain a positive and second order accurate sceme on arbitrary mixed meses tat was compatible wit te edge based data structure along wit mes transparency. Te investigation was simplified by studying Laplace s equation. Tey presented numerical results for transonic inviscid flow, laminar flow about an airfoil wit large separation and turbulent transonic flow over an airfoil. A mes refinement study was conducted to assess te accuracy of different control constructions and te influence of triangular, quadrilateral and mixed meses for viscous flows. Median-dual control volumes on triangular meses resulted in iger numerical error tan te containment dual control volumes on quadrilateral and triangular meses. Diskin et al. 44 compared te node-centered and cell-centered scemes for unstructured finite-volume discretization of Poisson s equation as a model of te viscous fluxes. Accuracy and efficiency were studied for six nominally second-order accurate scemes wic include a nodecentered sceme, cell-centered node-averaging scemes wit and witout clipping, and cellcentered scemes wit un-weigted, weigted and approximately mapped least-squares face gradient reconstruction. Among te considered scemes, te node centered sceme as te lowest complexity and te cell-centered node-averaging sceme as te igest complexity. Tey tested te scemes on grids wic ranged from structured regular grids to irregular grids composed of arbitrary mixtures of triangles and quadrilaterals, including random perturbations of te grid points to examine te worst possible beavior of te solution. Two classes of tests were considered. Te first class of tests involved smoot manufactured solution on bot isotropic and igly anisotropic grids wit discontinuous metrics, typical of tose encountered in unstructured

27 grid adaptation and te second class of tests were performed on consistently refined stretced grids generated around a curved body representative of ig Reynolds number turbulent flow simulations. From te first class of tests, tey observed tat te face least-square metods, nodeaveraging metod witout clipping, and te node centered metod demonstrated second-order convergence of discretization errors. From te second class of tests, tey observed tat te nodecentered sceme was always second order accurate and cell-centered node-averaging scemes were less accurate and also failed to converge to te exact solution wen clipping of te nodeaveraged values was used. Cell-centered node averaging scemes are second-order accurate and stable wen te coefficients of te pseudo-laplacian operator are close to and on igly stretced and deformed grids, some coefficients of pseudo-laplacian become negative or larger tan, wic as a detrimental effect on stability and robustness (Bart 45 ). Holmes and Connell 46 proposed to enforce stability by clipping te coefficients between 0 and. In tis paper, it was sown tat clipping seriously degrades te solution accuracy. Te cell-centered scemes using least square face gradient reconstruction ad more compact stencils wit a complexity similar to tat of te node centered scemes. For simulations on igly anisotropic curved grids, te least square metods ad to be amended by modifying te sceme stencil to reflect te direction of strong coupling. Te autors concluded tat te accuracies of te nodecentered and best cell-centered scemes were comparable at equivalent number of degrees of freedom. Many autors also employ finite-element metods for computing viscous flows governed by te Navier-Stokes equations. Sun et al. 47 extended a spectral volume metod to andle viscous flows. Using te spectral volume metod ig-order accuracy was acieved troug ig-order polynomial reconstructions witin spectral volumes. Tey developed a formulation similar to te Local Discontinuous Galerkin (LDG) approac to discretize te viscous fluxes. Gauss s teorem was used to integrate te gradients in te control volume. Kannan et al. 48 improved te Navier-Stokes solver developed by Sun et al. 47 based on te spectral volume metod wit te use of a new viscous flux formulation. Instead of a LDG-type approac, a penalty approac based on te first metod of Bassi and Rebay 49 was used. Te advantage of te penalty approac over te LDG approac was te speed up of te convergence wit te implicit metod and indicated tat te approac ad a great potential for 3D flow problems. Fidkowski and Darmofal 50 also used a finite-element approac for ig order discretizations of te

28 compressible Navier-Stokes equations. Te viscous flux terms were discretized using te second form of Bassi and Rebay. 49 3

29 . Basic Concepts of Verification. Verification Verification addresses te matematical correctness of numerical simulations and it plays an important role in building confidence in CFD solutions. Tere are two fundamental aspects to verification: code verification and solution verification. Code verification is te process of ensuring, tat tere are no mistakes (bugs) in a computer code or inconsistencies in te numerical algoritm. Solution verification is te process of estimating te tree types of numerical error tat can occur in numerical simulations: round-off error, iterative error, and discretization error.. Code Verification Code verification 7 is te process of ensuring, tat tere are no mistakes (bugs) in a computer code or inconsistencies in te numerical algoritm. By performing code verification, a code can be tested for mistakes and can be followed by code debugging to identify and remove te mistakes. During te development of scientific software, tere is a ig possibility of making a number of coding mistakes. Tese mistakes need to be removed before using te code on real applications.. Coding mistake: A simple example of a coding mistake in a FORTRAN program is sown below. FORTRAN Code: TEMPNORM = 0 DO J =, JMAX - DO I =, JMAX - RES = RES + ((T(I,J) - TOLD(I,J))/DT )** TEMPNORM = TEMPNORM + T(I,J)** ENDDO ENDDO Te underlined part of te code is a mistake in FORTRAN code sown ere. Te wrong term JMAX is used instead of te rigt term IMAX. Tis kind of coding mistake is a 4

30 possibility wile developing a code. In te process of code verification, mistakes like te one sown above can be identified and removed.. Algoritm Inconsistency: For a consistent numerical algoritm, te discretized equations must approac te original partial differential equations in te limit as mes size ( x, t) approaces zero. A famous example of an inconsistent algoritm is te DuFort-Frankel 5 differencing of a D unsteady eat equation. 5 0 (.) For te D unsteady eat equation sown above, te DuFort-Frankel differencing is given by (.) for wic te leading terms in te truncation error are sown below. 6 (.3),,, If x and t approac zero at te same rate suc tat t/ x = β, ten te discretized equations will not approac original partial difference equations, i.e., te D unsteady eat equation, instead tey converge to a different partial difference equation sown below wic is a yperbolic equation. (.4) In te process of code verification, numerical inconsistencies can also be identified and corrected..3 Order of Accuracy Order of accuracy 5,6,7 is te rate of decrease of discretization error wit mes refinement. Order of accuracy test, wic is equivalent to testing weter te formal order of accuracy matces te observed order of accuracy, is te most exercising code verification test. Tis test not only determines weter te solution is converging, but also weter or not te discretization error is reduced at te expected rate. 5

31 .3. Formal Order of Accuracy Te formal order of accuracy of a numerical sceme is te rate at wic te discretized equations approac te original partial differential equations. For all discretization approaces (finite difference, finite volume, finite element, etc.) te formal order of accuracy is obtained from a truncation error analysis of te discrete algoritm. Te formal order will be te power of x or t in te leading terms in te truncation error. Any partial differential equation can be written as te sum of te finite difference equation and te truncation error. As an example, for a D unsteady eat equation, applying Taylor series and discretizing te D unsteady eat equation wit a forward difference in time and a central difference in space, it can be written as: (.5), From te above expression, te formal order of accuracy of te finite difference sceme is first order in time and second order in space since te leading terms in te truncation error contain te factors t and ( x), respectively..3. Observed Order of Accuracy Te observed order of accuracy is computed directly from te code output for a given set of simulations on systematically refined grids. Te observed order of accuracy will not matc te formal order of accuracy due to mistakes in te computer code, defective numerical algoritms, and wen numerical solutions are not in te asymptotic grid convergence range. For calculating te observed order of accuracy using two mes levels, it is required to ave an exact solution to find te discretization error. Assuming tat exact solution to te partial differential equations is known, let us now consider te metod for calculating te observed order of accuracy. Te discretization error is formally defined as te difference between te exact solution to te discrete equations and te exact solution to te governing partial differential equations. Since te exact solution to te discrete equations (wic will be different on different mes levels) is generally not known, te numerical solution on te same mes level is substituted in its place, tus neglecting iterative 6

32 error and round-off error. Te observed order of accuracy can be evaluated eiter locally witin te solution domain or globally by employing a norm of te discretization error or for a global system response quantity. Consider a series expansion of te discretization error in terms of k, a measure of te element size on te mes level k, (.6) were f k is te numerical solution on mes k, g p is te coefficient of te leading error term, and p is te formal order of accuracy. Neglecting te iger order terms, we can write te discretization error equation for a fine mes (k=) and a coarse mes (k=) in terms of te observed order of accuracy as Since te exact solution is known, tese two equations can be solved for te observed order of accuracy. Introducing r, te ratio of coarse to fine mes element spacing (r = / > ), te observed order of accuracy becomes (.7) Tus, wen te exact solution is known, only two solutions are required to obtain te observed order of accuracy..4 Metod of Manufactured Solutions Code verification using order of accuracy testing requires an exact solution to te governing equations wic are tested in te code. Traditional exact solutions exist only for simple governing equations. For complex governing equations wic can andle complex pysics, complex geometries, and significant nonlinearities, it is difficult to find exact solutions. Te Metod of Manufactured Solutions, (MMS) is a general approac for obtaining exact solutions for code verification purposes..4. Procedure for MMS Te procedure for applying MMS wit order of accuracy verification is demonstrated below for a simple example problem. 7 (.8) 7

33 . Coose te specific form of te governing equations; ere we coose te linear D eat equation.. Coose te Manufactured Solution. 0 (.9), (.0) 3. Operate te governing equations on te cosen solution, resulting in analytic source terms. (.) 4. Solve te modified governing equation (original equation plus source terms) on various mes levels. (.) 5. Compute te observed order of accuracy and compare it wit te formal order of accuracy..4. Attributes of a Good Manufactured Solution Te cosen Manufactured Solution sould be smoot, analytic functions wit smoot derivatives. Te coice of smoot derivatives will allow te formal order of accuracy to be acieved on relatively coarse meses. Trigonometric and exponential functions are selected as Manufactured Solutions since tey are smoot and infinitely differential. Tere is an advantage of using trigonometric functions as tey can be adjusted to include only a fraction of an oscillation period over te domain wic makes it easier to acieve te asymptotic range. Te Manufactured Solutions need to be selected suc tat no derivatives vanis, including cross derivatives if tey are present in te governing equations. Even toug te Manufactured Solution is not required to be pysically realistic, since we are only testing te matematics, tey sould be cosen to give realizable pysical states. For example, if te code requires tat te 8

34 temperature to be positive for te calculation of speed of sound, ten te Manufactured Solution sould be cosen suc tat te temperature values are only positive. A Manufactured Solution sould be selected suc tat all te terms in te governing equation ave similar magnitudes meaning one term in te governing equation sould not dominate oter terms. For example, a Manufactured Solution sould be selected suc tat te convective terms and diffusive terms in Navier-Stokes equations ave te same order of magnitude. It can be cecked weter all te terms in te governing equations are rougly te same order of magnitude, by examining te ratios of tose terms in te considered domain..4.3 Advantages of Metod of Manufactured Solutions Tere are several advantages of using MMS. Tis procedure can be applied to all discretization scemes (finite-difference, finite-volume, and finite element). MMS can also be applied witout trouble wen dealing wit nonlinear equations and multiple equations like te Navier-Stokes equations. Roace 5 sowed tat by using MMS for code verification, very sensitive mistakes in te discretization can be determined. Te sensitivity of MMS procedure for code verification is explained wit tis example were during te testing of a compressible Navier-Stokes code, te reason for non-convergence of global norms of te discretization error was found to be because of a small discrepancy in te 4 t significant digit for te termal conductivity between te governing equations and te numerical implementation in te code. 9 9

35 3. Mes Generation A finite volume CFD code is verified on different mes topologies in te verification process. To verify all mes transformations are coded correctly, te code needs to be run on te most general mes types 9,38 wic include meses wit mild skewness, aspect ratio, curvature, and stretcing. Te most general mes types are termed as ybrid meses wic contain different mes topologies. However, te code needs to be run on simpler meses if te code verification fails on te general meses. Simpler meses include meses wit isolated mes topologies and mes qualities. Te different mes topologies considered for code verification in bot D and 3D are classified as structured, unstructured, and ybrid meses wic are a combination of structured and unstructured meses. 3. D Mes Topologies Different D structured and unstructured mes topologies are used during te code verification of te finite volume CFD code. Te most general mes topology used for D verification is a D ybrid mes wic includes quadrilateral and triangular cells wit curvilinear boundaries, skewed cells, and stretced cells. 3.. D Structured Meses Te D structured meses used for code verification are te Cartesian mes, te stretced Cartesian mes, te curvilinear mes, and te skewed curvilinear mes. Te D structured meses are sown in Figure. Te D structured meses considered ere contain quadrilateral cells. By testing te code on tese meses, te beavior of te code on te quadrilateral cell topology along wit its cell quality effect can be studied. All tese D structured meses are generated using te mes generation tool GRIDGEN. 57 Te stretced Cartesian mes can be used to isolate te effects of grid stretcing and aspect ratio, te curvilinear mes can be used to test te effects of curved boundaries witout te presence of skewness or stretcing, and te skewed curvilinear mes tests all te effects on a single structured mes type. 0

36 a) b) c) d) Figure : D structured meses: a) Cartesian, b) stretced Cartesian, c) curvilinear, and d) skewed curvilinear 3.. D Unstructured Meses Selected D unstructured meses used for code verification are sown in Figure. Te D unstructured meses considered ere contain triangular cells. Te general unstructured mes is generated by automatic mes generation in GRIDGEN. Oter unstructured meses, i.e., te unidirectional diagonal and bidirectional diagonal unstructured meses sown in Figure, are generated by starting from a structured mes and ten adding diagonals in te quadrilateral cells.

37 Te D unstructured meses generated by adding diagonals in a structured mes are used in te verification study to acieve a uniform and consistent refinement between different mes levels. By testing te code on tese meses, te beavior of te code on te triangular cell topology along wit its cell quality effect can be studied. Te concept of uniform and consistent mes refinement and its necessity for code verification purposes are discussed in Section 3.3. a) b) Figure : D unstructured meses: a) general unstructured, b) uni-directional diagonal, and c) alternate diagonal c)