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

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

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.

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

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... 4. Verification... 4. Code Verification... 4.3 Order of Accuracy... 5.3. Formal Order of Accuracy... 6.3. Observed Order of Accuracy... 6.4 Metod of Manufactured Solutions... 7.4. Procedure for MMS... 7.4. Attributes of a Good Manufactured Solution... 8.4.3 Advantages of Metod of Manufactured Solutions... 9 3 Mes Generation... 0 3. D Mes Topologies... 0 3.. D Structured Meses... 0 3.. D Unstructured Meses... 3..3 D Hybrid Meses... 3 3. 3D Mes Topologies... 3 3.. 3D Structured Meses... 3 iv

3.. 3D Unstructured Meses... 4 3..3 3D Hybrid Meses... 6 3.3 Systematic Mes Refinement... 6 4 Verification of a Finite Volume CFD Code... 9 4. Finite Volume Code... 9 4. Verification of Baseline Governing Equations... 30 4.. Flow Equations... 30 4.. Verification of te Euler Equations... 3 4..3 Verification of te Navier-Stokes Equations... 34 4..4 Effect of Mes Quality... 39 4.3 Verification of Transport Models... 43 4.4 Verification of Boundary Conditions... 44 4.4. Mes Topologies... 46 4.4. No-slip Wall... 46 4.4.. No-slip Adiabatic Wall... 46 4.4.. No-slip Isotermal Wall... 49 4.4.3 Slip Wall... 5 4.4.4 Isentropic Inflow... 55 4.4.5 Outflow... 56 4.4.6 Extrapolation... 58 4.4.7 Farfield... 60 4.5 Verification of Turbulence Models... 6 4.6 Verification of Time Accuracy for Unsteady Flows... 67 4.7 Issues Uncovered During Code Verification... 7 4.7. Coding Mistakes/Algoritm Inconsistencies... 7 v

4.7. Grid Sensitivities... 7 5 Finite Volume Diffusion Operators... 73 5. Robustness and Accuracy... 73 5. Testing Framework... 75 5.3 Cell-Centered Finite Volume Diffusion Operators... 76 5.3. Node Averaging Metod... 77 5.3. Unweigted and Weigted Least Squares... 77 5.3.3 Loci-CHEM Diffusion Operator... 77 5.3.4 FAMLSQ Metod wit Compact Stencil... 78 5.4 Mes Topologies... 79 5.5 Effect of Mes Quality... 8 5.5. D Structured Meses... 8 5.5.. Uniform Mes... 8 5.5.. Aspect Ratio... 83 5.5..3 Stretcing... 84 5.5..4 Skewness... 85 5.5..5 Curvature... 86 5.5. D Unstructured Meses... 87 5.5.. Mes wit Isotropic Cells... 88 5.5.. Skewness wit Aspect Ratio... 90 5.5..3 Skewness wit Curvature... 9 6 Conclusions... 98 6. Summary of Results... 98 6. Main Contributions of tis Work... 99 6.3 Future Work and Recommendations... 00 References... 0 vi

Appendix A... 08 Appendix B... 09 Appendix C... 0 vii

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

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

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

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

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

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

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

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... 43 Table B. Quantification of cell quality..09 Table C. Constants for D Manufactured Solutions...0 Table C. Constants for 3D Manufactured Solutions.. xv

. 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

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

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 984. 8 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

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

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

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

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

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

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

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

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

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

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

. 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

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

.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

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

. 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

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

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

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.

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)

3..3 D Hybrid Meses A D ybrid mes contains bot quadrilateral and triangular cell topologies. Te D skewed ybrid mes and te igly skewed ybrid mes considered for code verification are sown in a) b) Figure 3. Te D ybrid meses are also generated using te mes generation tool GRIDGEN. By performing code verification on tese meses, te beavior of te code on bot te triangular and quadrilateral cell topologies wit all te different cell quality effects is tested. a) b) Figure 3: D ybrid meses: a) skewed ybrid and b) igly skewed ybrid 3. 3D Mes Topologies Different 3D structured and unstructured mes topologies are used during te code verification of te finite volume CFD code. Te most general mes topology used for 3D verification is a 3D ybrid mes wic includes exaedral cells, tetraedral cells, and prismatic cells wit curvilinear boundaries, skewed cells, and stretced cells. 3.. 3D Structured Meses Te 3D structured meses used during code verification include te Cartesian mes and te skewed curvilinear mes. Te 3D structured meses are sown in Figure 4. Te 3D structured meses contain exaedral cells. By testing te code on tese meses, te beavior of te code on te exaedral cell topology along wit its cell quality effect can be studied. All te 3

3D structured meses are also generated using GRIDGEN. Te skewed curvilinear mes is generated to test te effects of aspect ratio, skewness, stretcing and te effect of curved boundaries on te code. a) b) Figure 4: 3D structured meses: a) Cartesian and b) skewed curvilinear 3.. 3D Unstructured Meses Different 3D unstructured meses used for code verification are sown in Figure 5. Te 3D unstructured meses considered ere contain tetraedral cells and prismatic cells. By testing te code on tese meses, te beavior of te code on te tetraedral and prismatic cell topologies along wit teir cell quality effects can be studied. Te 3D unstructured meses used during code verification include te unstructured meses wit tetraedral and prismatic cells as sown in Figure 5a and Figure 5b, respectively, and tese meses ave cells close to isotropic cells. Te igly skewed unstructured meses wit tetraedral and prismatic cells are sown in Figure 5c and Figure 5d, respectively, and tese meses contain igly skewed and stretced cells wic can be used to study te cell quality effect wile testing te code. Te unstructured mes wit prismatic cells is generated in GRIDGEN by starting wit an unstructured D domain and projecting in te tird direction normal to te D domain. For generating all te oter 3D unstructured meses, particularly for te unstructured tetraedral meses, a mes generation 4

code is developed using Fortran. Te reason for developing a Fortran code to generate tese 3D unstructured meses is discussed in Section 3.3. a) b) c) d) 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

3..3 3D Hybrid Meses Te skewed 3D ybrid mes and te igly skewed 3D ybrid mes used for code verification are sown in Figure 6. Te 3D ybrid meses considered contain exaedral, tetraedral, and prismatic cells. By testing te code on tese meses, te beavior of te code on te exaedral, tetraedral and prismatic cell topologies along wit teir cell quality effect can be studied. To isolate te cell quality effects, 3D ybrid meses wic ave cells close to isotropic can be used to test te code. Again, for generating all te 3D ybrid meses, a mes generation code is developed using Fortran. a) b) Figure 6: 3D ybrid meses: a) skewed ybrid and b) igly skewed ybrid 3.3 Systematic Mes Refinement Systematic mes refinement 7 is defined as uniform and consistent refinement over a spatial domain. A mes is said to be uniformly refined if te mes is refined in all te coordinate directions equally and it is said to be consistently refined if te mes quality stays constant or improves wit mes refinement. For te purpose of code verification, it is necessary to ave a systematic mes refinement. 7 In te case of structured meses, refinement/coarsening of te meses for te verification purpose are straigtforward. A coarse mes is generated from a fine mes by removing every alternate mes point or mes line to produce mes levels wit a refinement factor of two. In tis process, te mes quality is maintained for te structured 6

meses. But in te case of unstructured meses refinement/coarsening of meses wit a uniform refinement factor trougout te domain preserving te mes quality is more callenging, particularly in 3D. In te D case, systematic mes refinement can be acieved by generating an unstructured mes from a structured mes by splitting quadrilaterals into triangles using diagonals. 38 To te autor s knowledge, generation of 3D unstructured meses wit uniform refinement preserving te mes quality as not yet been acieved using commercial software. Terefore, for code verification purposes, a mes generation code is developed to generate 3D unstructured meses from a 3D structured mes wit exaedral elements. In te process, a cube will be split into five tetraedral cells as sown in Figure 7a. Te central tetraedral cell, wic does not sare a surface boundary wit te parent cube, is isotropic and te oter four tetraedral cells ave te same topology wit good cell quality. An unstructured mes wit tetraedral elements generated using te mes generation code is sown in Figure 7b. By generating unstructured meses in tis fasion, a uniform and consistent refinement can be acieved by a uniform refinement factor and maintaining te cell quality between te mes levels. Based on tis concept of generating an unstructured mes from a structured mes, anoter code for generating 3D ybrid meses wic contain exaedral, tetraedral, and prismatic cells wit proper connectivity between different cell types was also developed. Te prismatic cells in te 3D ybrid mes are generated by diagonally splitting a exaedral cell into two prismatic cells. To obtain a ybrid mes from a structured mes of exaedral cells, 5 percent of te exaedral cells are split into prismatic cells, 50 percent of te exaedral cells are split into tetraedral cells and te oter 5 percent are left as exaedral cells. Te 3D ybrid meses are generated to satisfy te uniform and consistent refinement criteria between te mes levels required for code order of accuracy verification. 7

a) b) Figure 7: a) Cube split into five tetraedral cells, b) 3D unstructured mes wit tetraedral cells generated from a skewed and stretced curvilinear mes Te different mes levels and mes types used for code verification are given in Table. A maximum of seven mes levels are used for D mes topologies and a maximum of five mes levels are used for 3D mes topologies wit systematic mes refinement between consecutive mes levels. D Mes Topologies Table. Different mes levels and mes types 3D Mes Topologies Structured Unstructured Hybrid Structured Unstructured Hybrid 8 8 8 96 8 8 8 30 664 6 6 5 384 6 6 6 560 33 3 3 04 536 3 3 3 0480 06496 64 64 4096 644 64 64 64 63840 85968 8 8 6384 4576 8 8 8 3070 685744 56 56 65536 98304 5 5 644 3936 8

4. Verification of a Finite Volume CFD Code During te verification of a finite volume CFD code, different code options need to be tested wic include baseline steady-state governing equations, Suterland s law for viscosity, equation of state, boundary conditions, turbulence models, time accuracy of unsteady flows, etc. Te options in te code are verified by comparing te observed order of accuracy calculated for te CFD solutions for multiple systematically-refined meses to te formal order of accuracy of te numerical metod. An option in te code is considered fully verified if it passes te order of accuracy test on te D ybrid mes and 3D ybrid mes wit all cell topologies wic include skewness, aspect ratio, curvature, and mes stretcing. Wen a verification test fails on te ybrid meses, ten te governing equations are tested on oter simpler meses to find weter te discrete formulation of te governing equations is inconsistent on a particular mes topology or due to te cell quality attributes or coding mistakes. During te code verification process, te observed order of accuracy is calculated using te L, L and L norms of te discretization error. Normally te observed order of accuracy calculated using L and L norms of te discretization error sowed similar beavior, but te observed order of accuracy calculated using L norm of te discretization error asymptoted to te formal order of accuracy at a slower rate requiring more mes levels to see te asymptotic beavior. Te observed order of accuracy results are sown for te different options verified on different meses using L and L norms of te discretization error. Also, te verification test results of different options in te finite volume code are sown mostly on D and 3D ybrid meses. Te verification test results on oter meses are sown for some interesting cases wen te code options ad problems during testing. A summary of te options verified in te finite volume CFD code is sown in Appendix A. 4. Finite Volume Code Te code verification procedure can be applied to any scientific computing code in general, but in te current work, te finite volume CFD code verified is Loci-CHEM. 6,58 Loci- CHEM was developed at Mississippi State University using te Loci framework 59,60 and can simulate tree-dimensional flows of turbulent, cemically-reacting mixtures of termally perfect gases. Te Loci framework provides a ig-level programming environment for numerical 9

metods tat is automatically parallel and utilizes a logic-based strategy to detect or prevent common software faults (suc as errors in loop bounds or errors caused by subroutine calling sequences being inconsistent wit data dependencies). Te code uses an unstructured, edge based metod wit a formal order of accuracy of two. 4. Verification of Baseline Governing Equations Te baseline governing equations include te 3D steady state Euler and te Navier- Stokes equations. Removing te viscous terms from te 3D Navier-Stokes equations lead to 3D Euler equations and removing te z-direction variables ( / z = 0 and w = 0) converts te equations in tree dimensions to two dimensions. 4.. Flow Equations Te 3D, steady state, Farve-averaged Navier-Stokes equations 6 can be written as 0 (4.) 0 (4.) 0 (4.3) 0 (4.4) (4.5) were t ij is te laminar stress tensor given by 0 3, 3 3,,, (4.6) 30

and τ ij is te turbulent stress tensor given by 3, 3, 3,,,. Te turbulent and laminar eat flux terms are,,, and te total energy and entalpy are, (4.7) (4.8) (4.9) were Te perfect gas equation of state is assumed as. (4.0) (4.) and te eat of formation and excited energy mode parameter can be calculated as were ref = 0, T ref = 98K, and n = 5/ are used. (4.) 4.. Verification of te Euler Equations In our current work, we adere to te pilosopy tat code verification is simply a matematical test to ensure te numerical solution truly represents te solution to te continuum matematical equations tat are being solved. As suc, we specifically coose te Manufactured Solutions wic are not pysically realistic, but wic are simple, smoot, and exercise all terms in te governing equations. Te steady Manufactured Solutions employed take te following general form 3

,, (4.3) were φ = [ρ, u, v, w, p, k, ω] T represents any of te primitive variables and te f s ( ) functions represent sine or cosine functions. After selecting te constants in te above equation, te D Manufactured Solution used for te verification of Euler equations is given as 0.5 0.75 0. 0.08.5 70 5.5 5.5 7 0.6 90 5.5 0 0.9 00000 0000 50000.5 5000 0.75 Te 3D Manufactured Solution used for te verification of Euler equations is given as (4.4) 0.5 0.75 0. 0.45 0. 0.8 0.08 0.65 0.05 0.75 0. 0.5 70 5 0.5 5 0.85 0 0.4 7 0.6 4 0.8 4 0.9 90 5 0.8 0 0.8 5 0.5 (4.5) 0.9 5 0.4 5 0.6 80 0 0.85 0 0.9 0.5 0.4 0.8 5 0.75 00000 0000 0.4 50000 0.45 0000 0.85 5000 0.75 0000 0.7 0000 0.8 Tese Manufactured Solutions are smoot and te source terms generated after applying tem to te governing equations also vary smootly over te considered domain. As an example, a Manufactured Solution for te x-component of velocity is sown in Figure 8a and a smoot analytic source term in 3D domain is sown in Figure 8b. 3

a) b) Figure 8: a) Manufactured Solution of u-velocity and b) mass source term Te Euler equations are discretized and solved on different mes levels to calculate te observed order of accuracy. Te code is ten tested on te structured, unstructured and ybrid meses in bot D and 3D and is verified successfully to be second order accurate. Te order of accuracy results for te verification of Euler equation are sown for te 3D ybrid case only. Te observed order of accuracy results calculated using L, L and L norms of te discretization error on te 3D skewed ybrid mes are sown in Figure 9. In te plots sown, te observed order of accuracy p is calculated for all te conserved variables and is plotted on te y-axis against te normalized mes size on te x-axis. Five mes levels are used for te verification of Euler equations in 3D wit a uniform mes refinement of two between consecutive mes levels. Te coarsest skewed ybrid mes is generated starting from a structured mes containing 8 8 8 cells and te finest skewed ybrid mes is generated starting from a structured mes containing 8 8 8 cells. Te value of = means te finest mes and a iger value represents a coarser mes. It is seen in te plot tat as te mes is refined, te observed order of accuracy asymptotes to two wic is te formal order of te code. As mentioned previously, te observed order of accuracy results using L and L norms of te discretization error are similar and for te subsequent verification results, only te observed order of accuracy results calculated using L and L norms are presented. 33

Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t 3 4 5 6 7 8 3 4 5 6 7 8 a) b) Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t 3 4 5 6 7 8 c) 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 4..3 Verification of te Navier-Stokes Equations Successful verification of te Euler equations on a particular mes and failure of te order of accuracy test for te Navier-Stokes equations means tat tere is a problem wit te formulation of te diffusion operator on tat particular mes, i.e., an algoritm inconsistency, a 34

coding mistake, or simply mes quality sensitivity. Te Navier-Stokes equations are tested on te structured, unstructured and ybrid meses in bot D and 3D and in te process of testing, several issues in te formulation of te diffusion operator in te code were uncovered and corrected. Te Manufactured Solution used for te Euler equations is also used for te Navier- Stokes equations. In te process of selecting te Manufactured Solution for code verification purposes, it is required tat different terms in te governing equations be rougly same order of magnitude. Tis prevents te larger magnitude errors from masking errors in oter terms of smaller magnitude. In te current work, during te verification of Navier-Stokes equations, a constant viscosity value of 0 N/m is used suc tat tere is an approximately equal contribution from te inviscid and viscous terms. Te Navier-Stokes equations are verified successfully to be second order accurate on te D ybrid and 3D ybrid meses after correcting te formulation of te diffusion operator eac time te code failed on a particular mes. Te observed order of accuracy results calculated using L and L norms of te discretization error for te Navier- Stokes equations on te D ybrid mes are sown in Figure 0. Te observed order of accuracy results calculated using L and L norms of te discretization error for te Navier-Stokes equations on te 3D skewed ybrid mes are sown in Figure. 35

4 Order of Accuracy, p 3 ro ro*u ro*v ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*e t 0 5 0 5 0 5 0 5 0 a) b) 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 Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t 3 4 3 4 a) b) 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 Te code was tested on different mes types and failed initially on simpler meses. Eac time te code failed on a particular mes, a problem wit te numerical formulation of diffusion operator was detected and a correction in te numerical formulation was made by te code 36

developers. Initially, te Navier-Stokes equations were successfully verified on D rectangular Cartesian and stretced Cartesian meses, but on te skewed curvilinear meses, te discretization error did not decrease wit mes refinement. Since te code was already verified wen Euler equations were tested, te problem or te error was found to be in te diffusion operator formulation. A modification was made to te diffusion operator by Luke 39 and te new diffusion operator rectified te problem. An observed order of accuracy of two was attained wit mes refinement after te modification. Te code verification results on te skewed curvilinear mes wit te original diffusion operator and te new diffusion operator are sown in Figure. Te improvement in te observed order of accuracy results wit te use of modified diffusion operator for te structured meses is explained ere. In te cell-centered finite volume diffusion operator, gradients are required at te cell faces to compute te viscous fluxes. Since unstructured finite-volume CFD codes typically compute and store gradients at te cell centers a mecanism for obtaining gradients at te faces is required. Te Loci-CHEM code calculates te gradient at te face by computing bot normal and tangential components of te gradient. Te original formulation for calculating te normal component of te face gradient utilized te strategy suggested by Strang et al. 6 for te Cobalt 60 code. Tis approac effectively neglects a term in te normal gradient wic can lead to stencils wit negative weigts (wic affects te code s stability). Luke 39 modified te normal gradient calculation suc tat a limiter is applied to te offending term tat bot maintains second order accuracy in smoot regions of te flow and ensures a positive stencil. 37

Order of Accuracy, p 0 ro ro*uvel ro*vvel ro*e t Order of Accuracy, p 4 3 ro ro*uvel ro*vvel ro*e t 5 0 5 0 5 0 5 0 5 0 5 a) b) 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 Later on D unstructured mes wit triangular cells, preliminary investigations sowed tat te observed order of accuracy was approacing one wit mes refinement. To determine te reason for tis reduction in order of accuracy, te Navier-Stokes equations were tested on different types of unstructured meses (triangular cells generated from a structured topology). On all tese D unstructured meses wit triangular cells, te beavior was te same. Te Euler equation were successfully verified to be second order accurate on tese D unstructured meses wic explained tat te problem was wit te diffusion operator formulation on te D unstructured meses. Again, Luke 63 came up wit a modification in te formulation of te diffusion operator wic rectified te problem. Te estimation of gradients at te face centers is te cornerstone of te diffusion flux computations wic are required to be second order accurate. In te old diffusion operator used in Loci-CHEM, gradients at face centers were computed by splitting te gradients into tangential and normal components and te diffusion operator was using te inviscid operator (gradients at cell centers). Wile te inviscid operator retained second order accuracy, te diffusion operator degenerated to first order as stencils departed symmetry. In order to more reliably obtain second order gradients at te face centers, a modified metod was considered. In tis metod, a new control volume centered about te face 38

was constructed and a second order function reconstruction was used to compute te nodal values of a control volume centered about te face center. Greens teorem was employed to compute second order gradients at te face center in te control volume (diamond cell). Tis form of reconstruction retained second order gradients at face centers even wen te gradients at cell centers degenerate to first order. Te modified diffusion operator formulation was successfully verified to be second order accurate on all D unstructured meses wit triangular cells. Te code verification results on te D alternate diagonal unstructured mes wit te old diffusion operator and te modified diffusion operator are sown in Figure 3. Order of Accuracy, p 3 ro ro*u ro*v ro*e t 0 Order of Accuracy, p 4 3 ro ro*u ro*v ro*e t 5 0 5 0 5 0 5 0 a) b) 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 Tese two modifications in te formulation of te diffusion operator enabled te Navier- Stokes equations to be successfully verified as second order accurate on te D ybrid mes and te 3D skewed ybrid mes, tus covering te effect of all cell topologies wic include some skewness, aspect ratio, curvature, and mes stretcing. 4..4 Effect of Mes Quality Te Navier-Stokes equations are also tested on te D igly skewed ybrid meses (a) b) 39

Figure 3b) to look at te effect of cell quality of quadrilateral cells and triangular cells on te discrete formulation of te governing equations. On tis mes, te code is successfully verified and te observed order of accuracy approaces two wit mes refinement. Te observed order of accuracy results calculated using L and L norms of te discretization error for te Navier-Stokes equations on te D igly skewed ybrid mes are sown in Figure 4. Tis sows tat te finite volume code works fine on igly skewed quadrilateral and triangular cells in D. 4 4 Order of Accuracy, p 3 ro ro*u ro*v ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*e t 5 0 5 0 5 0 5 0 a) b) 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 Te Navier-Stokes equations are also tested on a 3D igly skewed ybrid mes (Figure 6b) to look at te effect of cell quality of exaedral, prismatic, and tetraedral cells on te discrete formulation of te governing equations. Te 3D igly skewed ybrid mes is generated starting from a 3D skewed curvilinear mes sown in Figure 4b and using te mes generation code wic generates te ybrid mes (containing exaedral, tetraedral, and prismatic cells) from te structured mes. Te Navier-Stokes equations are successfully verified to be second order accurate on te 3D skewed curvilinear mes wit exaedral cells sown in Figure 4b, but te verification test failed on te 3D igly skewed ybrid mes. Te observed order of accuracy results calculated using L and L norms of te discretization error for te Navier-Stokes 40

equations on te 3D igly skewed ybrid mes are sown in Figure 5. From te plot, te observed order of accuracy seems to approac a value less tan one wit mes refinement. Te difference between te 3D skewed ybrid mes (Figure 6a) and te 3D igly skewed ybrid mes (Figure 6b) is only te quality of te cells in te mes; oterwise bot te meses ave te same mes topology and connectivity. Tis indicates a problem in te discrete formulation of te governing equations wen te cells ave a comparatively lower quality. Te cell quality is quantified wit some parameters and quantification of cell quality of te 3D skewed ybrid mes and te 3D igly skewed ybrid mes is sown in Table B in Appendix B. Order of Accuracy, p 3 0 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 3 0 - ro ro*u ro*v ro*w ro*e t - - 3 4 3 4 a) b) 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 Te L norms of te discretization error for bot te 3D skewed ybrid mes and te 3D igly skewed ybrid mes are compared and it is observed tat te error is iger for te igly skewed ybrid meses relative to te skewed ybrid meses for te same number of cells and similar mes structure. Te comparison of te L norm of te discretization error is sown in Figure 6. In te plot, errors sown in te solid lines correspond to te 3D skewed ybrid mes and te errors sown in dased lines correspond to te 3D igly skewed ybrid mes. Tis plot gives te effect of cell quality on te error in te solution. Te error in te solution eiter decreases slowly or does not decrease wit mes refinement for lower quality meses. 4

0 4 st order-slope LNorm 0 3 0 0 0 0 0-0 - nd order-slope ro - S ro*u - S ro*v - S ro*w - S ro*e t - S ro - HS ro*u - HS ro*v - HS ro*w - HS ro*e t - HS 0-3 0-4 0-5 4 6 8 Figure 6: Comparison of L norm discretization error on 3D skewed ybrid mes and 3D igly skewed ybrid mes To furter study te discrete formulations of te inviscid and viscous terms in te governing equations, te Euler and Navier-Stokes equations are tested separately on meses wit a particular mes topology, i.e., eiter igly skewed tetraedral cells or igly skewed prismatic cells alone. Te above analysis explains ow MMS along wit te order of accuracy test can be used to find sensitivities to mes quality by testing on different meses wit different cell topologies and different cell quality. From te study, it is observed tat te code is successfully verified to be second order accurate wile testing Euler and Navier-Stokes equations on igly skewed exaedral and prismatic cells but failed te order of accuracy test wile testing Euler equations and Navier-Stokes equations on igly skewed tetraedral cells and ence on te 3D igly skewed ybrid mes. Te Euler equations are only first order accurate and te Navier- Stokes equations are less tan first order accurate on igly skewed tetraedral cells wic led to te same effect on 3D igly skewed ybrid mes. Te order of accuracy of te governing equations on te igly skewed cells in 3D is tabulated in Table. Te reason for failure of code verification on te 3D igly skewed ybrid mes is because of te instability in te inviscid operator. Testing only te eat equation (similar to testing te diffusion operator) on te igly skewed 3D ybrid mes produced second order accuracy. On te igly skewed mes, a limiter is needed to acieve iterative convergence. Running te Euler equations wit te limiter, te code converges to only first order because of a 4

stability problem caused by te geometry of te gradient stencil at te boundary. Since tere is instability in te inviscid operator, te limiter is forced to be active and it affects te diffusion operator wic also incorporates te limiter. Te diffusion operator incorporates te limiter to ensure tat it does not introduce new solution extrema. Table. Order of accuracy of te governing equations on igly skewed cells in 3D Euler Equations Navier-Stokes Equations Hexaedral Cells nd Order nd Order Prismatic Cells nd Order nd Order Tetraedral Cells st Order Less tan st Order Hybrid Cells st Order Less tan st Order 4.3 Verification of Transport Models Oter equations verified in te finite volume code are te equation of state, termally perfect termodynamic model and Suterland s law of viscosity. Verifying te Navier-Stokes equations in te Loci-CHEM code automatically verifies equation of state. For verifying te termally perfect termodynamic model and Suterland s law of viscosity, termal conductivity and viscosity are defined as functions of temperature. Bot te equations are tested in 3D skewed ybrid mes and are successfully verified to be second order accurate wit te observed order of accuracy approacing two wit mes refinement. Te Manufactured Solution used for te verification of te transport equations is same as te Manufactured Solution used for te Euler and Navier-Stokes equations. Te order of accuracy results for te Suterland s law of viscosity on a 3D skewed ybrid grid are sown in Figure 7. 43

Order of Accuracy, p 4 ro ro*u ro*v ro*w 3 ro*e t Order of Accuracy, p 4 ro ro*u ro*v ro*w 3 ro*e t 0 3 4 0 3 4 a) b) 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 4.4 Verification of Boundary Conditions In order to verify te implementation of a boundary condition in a code, te Manufactured Solution can be tailored to exactly satisfy a given boundary condition on a domain boundary. A general approac for tailoring Manufactured Solution to ensure tat a given boundary condition is satisfied along a domain boundary was developed by Bond et al. 0 Te approac is explained wit a simple example in D. 7 Te standard form of te Manufactured Solution for te D steady-state solution can be written as were,, (4.6), (4.7) A boundary in D can be represented by a general curve F(x,y) = C, were C is a constant. Te new Manufactured Solution for verifying boundary conditions can be found by multiplying te φ (x,y) term wit te function [C-F(x,y)] m as sown below:,,,. (4.8) 44

Tis procedure will ensure tat te Manufactured Solution is equal to a constant φ 0 satisfying a Diriclet boundary condition φ(x,y) = φ 0 along te specified boundary for m =. For m =, it ensures tat te Manufactured Solution will satisfy bot Diriclet and Neumann (zero normal gradient) boundary conditions along te specified boundary. To test a boundary condition option were te boundary is curved, a well-defined curved boundary for D meses, i.e., F(x,y) = C and a well defined curved surface for 3D meses, i.e., F(x,y,z) = C are considered as one side of a domain in D and 3D, respectively, and te mes is built along wit tat boundary. Te analytic definition of te curved boundaries considered for te boundary condition verification in bot D and 3D are defined as given below., 5 80 0.05 0,, 5 80 6 80 0.06 0.05 0 (4.9) Tese boundaries are used bot in te Manufactured Solution and te grid generation. Te curved surface used for te verification of boundary conditions in 3D is sown in Figure 8. Te boundary conditions are also tested wen te boundary is a straigt boundary (straigt line in D and a flat surface in 3D) instead of a curved boundary. Figure 8: Wavy surface used as a well-defined boundary for 3D meses Te different boundary condition options verified in te finite volume code include te no-slip adiabatic wall, no-slip isotermal wall, slip wall wit te Euler equations and te Navier-Stokes equations, isentropic inflow, outflow, extrapolation, and farfield. 45

4.4. Mes Topologies Te meses used for verification of boundary conditions are different and a separate set of meses are generated for boundary condition verification in bot D and 3D. Te D ybrid mes and te 3D ybrid mes wit a curved boundary used for te boundary condition verification are sown in Figure 9. In te figure, for bot meses te bottom boundaries are te well defined curved boundaries wic are tested for different boundary condition options. Wile generating tese meses for boundary condition verification, te mes is generated to be normal to te tested boundary. a) b) Figure 9: Meses used for boundary condition verification a) D ybrid mes and b) 3D ybrid mes 4.4. No-slip Wall Te no-slip wall boundary is verified as an adiabatic boundary and an isotermal boundary. In bot te cases, te no-slip wall boundary is tested as a straigt wall boundary and also as a curved wall boundary. 4.4.. No-slip Adiabatic Wall By testing te no-slip wall as an adiabatic boundary, a Neumann boundary condition for temperature (dt/dn = 0) is verified along wit te no-slip condition (V r = 0) on a particular boundary. Te no-slip wall is tested as an adiabatic boundary on meses wit bot straigt and 46

curved boundaries in D and 3D. Te D Manufactured Solution used for te verification of noslip wall as an adiabatic boundary is of te form,,,,,, (4.0),, and te function F(x,y) is defined for a straigt boundary and a curved boundary as, 0, 5 80 0.05 0 (4.) Te 3D Manufactured Solution used for te verification of no-slip wall as an adiabatic boundary is of te form,,,,,,,,,,,, (4.),,,,,,,, and te function F(x,y,z) is defined for a straigt boundary and a curved boundary as,, 0,, 5 80 6 80 0.06 0.05 0 (4.3) Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. By selecting te Manufactured Solution as sown above, all te velocity components become zero on te F = 0 boundary and te normal derivatives of density and pressure become zero at F = 0 boundary wic makes te normal derivative of temperature also zero at tat boundary satisfying bot te no-slip condition and te adiabatic condition. Te temperature contours in a 3D domain wen te bottom boundary is defined as an adiabatic no-slip wall are sown in Figure 0. 47

Figure 0: Temperature contours wen te bottom boundary is defined as a no-slip adiabatic wall Te no-slip wall is tested as an adiabatic boundary on te D ybrid mes and 3D ybrid mes wit curved boundaries and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown ere for te 3D ybrid mes wit curved boundaries and te order of accuracy results using L and L norms of te discretization error for te adiabatic no-slip wall boundary on a 3D ybrid mes are sown in Figure. 48

Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 ro ro*u ro*v ro*w 3 ro*e t 3 4 5 6 7 8 0 3 4 5 6 7 8 a) b) 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 4.4.. No-slip Isotermal Wall By testing te no-slip wall as an isotermal boundary, a Diriclet boundary condition for temperature (T = constant) is verified along wit te no-slip condition (V r = 0) on a particular boundary. Te no-slip wall is tested as an isotermal boundary on meses wit bot straigt and curved boundaries in D and 3D. Te D Manufactured Solution used for te verification of noslip wall as an isotermal boundary is of te form,,,,,, (4.4),,,, Te 3D Manufactured Solution used for te verification of no-slip wall as an adiabatic boundary is of te form 49

,,,,,,,,,,,,,,,, (4.5),,,,,,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. By selecting te Manufactured Solution as sown above, all te velocity components become zero on te F = 0 boundary and te temperature becomes constant at tat boundary satisfying bot te no-slip condition and te isotermal condition. Here, te Manufactured Solution for temperature is not used, but a constant temperature boundary needs to be satisfied. Hence, te Manufactured Solution for density is selected in terms of pressure and temperature wic satisfies te needed conditions for isotermal boundary. Te temperature contours in a 3D domain wen te bottom boundary is defined as an isotermal noslip wall are sown in Figure. Figure : Temperature contours wen te bottom boundary is defined as a no-slip isotermal wall Te no-slip wall is tested as an isotermal boundary on te D ybrid mes and 3D ybrid mes wit curved boundaries and te observed order of accuracy calculated from te 50

numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown ere for te 3D ybrid mes wit curved boundaries and te order of accuracy results using L and L norms of te discretization error for te isotermal no-slip wall boundary on a 3D ybrid mes are sown in Figure 3. Order of Accuracy, p 4 ro ro*u ro*v ro*w 3 ro*e t Order of Accuracy, p 4 ro ro*u ro*v ro*w 3 ro*e t 4 6 8 4 6 8 a) b) 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 4.4.3 Slip Wall Testing te slip wall is verifying te slip condition V n = 0 on a particular boundary were V n is te velocity component normal to te surface. Also, on a slip wall boundary, te viscous terms need to be zero. Tis boundary condition is defined wit impermeable or reflecting options available in te Loci-CHEM code. Te slip wall boundary condition is tested wit bot te Euler and te Navier-Stokes equations. Te D Manufactured Solution used for te verification of slip wall boundary is of te form,,,, (4.6),, 5

,, Te 3D Manufactured Solution used for te verification of slip wall boundary is of te form,,,,,,,,,,,, (4.7),,,,,,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. Te above Manufactured Solutions for slip wall boundary condition verification will work only for testing on straigt boundaries since te normal velocity component of zero cannot be acieved for a curved boundary wit tis Manufactured Solution. Tese Manufactured Solutions are used wile testing slip wall along wit te Navier-Stokes equations. For te curved boundaries, selecting te Manufactured Solutions wile testing slip wall along wit te Navier-Stokes equations is callenging and testing of slip wall wit te Navier-Stokes equations on curved boundaries is not performed in tis work. Hence, te slip wall wit te Navier-Stokes equations is tested on D rectangular mes wit straigt boundary and 3D ybrid mes wit straigt boundary. In te case of verification of slip wall wit te Euler equations, selecting te Manufactured Solution for testing on curved boundaries is relatively easier since tere is no need to deal wit te viscous terms on te curved boundary. Te velocity component normal to te slip wall boundary is zero and can be derived using te expression 3,, 0 0 0 (4.8) Te D Manufactured Solution for verification of slip wall wen using curved boundaries and Euler equations is of te form 5

,,,,, (4.9),, Te 3D Manufactured Solution used for te verification of slip wall wen using curved boundaries and Euler equations is of te form,,,,,,,,,,,,,,,, (4.30),,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. Te slip wall wit te Euler equations is tested on D ybrid mes and 3D skewed ybrid mes wit curved boundaries also and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown ere for te 3D skewed ybrid mes wit curved boundaries and te order of accuracy results using L and L norms of te discretization error for te slip wall boundary wit te Euler equations on a 3D skewed ybrid mes are sown in Figure 4. Te slip wall wit te Navier-Stokes equations is tested on D rectangular mes and 3D ybrid mes wit straigt boundaries also and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results sown ere are for te 3D ybrid mes wit straigt boundaries and te order of accuracy results using L and L norms of te discretization error for te slip wall boundary wit te Navier-Stokes are sown in Figure 5. 53

Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t 3 4 3 4 a) b) 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 Order of Accuracy, p 4 3 ro ro*uvel ro*vvel ro*wvel ro*e t Order of Accuracy, p 4 3 ro ro*uvel ro*vvel ro*wvel ro*e t 3 4 3 4 a) b) 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

4.4.4 Isentropic Inflow Tis boundary condition is an inflow boundary tat requires te total temperature and total pressure to be constant on te inflow boundary. Tis boundary condition is applicable to only subsonic inflow conditions in te code. Te isentropic inflow boundary is tested on D rectangular mes and 3D ybrid mes wit straigt boundaries. Te governing equations used wile testing isentropic boundary condition are te Euler equations. Te D Manufactured Solution used for te verification of isentropic inflow boundary condition is of te form,,,,,, (4.3),, Te 3D Manufactured Solution used for te verification of isentropic inflow boundary condition is of te form,,,,,,,,,,,, (4.3),,,,,,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. Te above Manufactured Solutions for isentropic inflow boundary condition verification are developed for testing on straigt boundaries. For curved boundaries, te Manufactured Solutions need to be selected very carefully and verification of isentropic inflow on a curved boundary is not performed in tis work. By selecting te Manufactured Solution as above, all te variables are constant on te F = 0 inflow boundary and also te derivatives of te variables are zero normal to te boundary. Te u- velocity and w-velocity are zero on te F = 0 boundary and wit only v-velocity defined on tat boundary, te velocity is normal to te inflow boundary. Wit density, pressure, and te velocity components constant on te boundary, te stagnation pressure and stagnation temperature are 55

also constant wic satisfies te isentropic boundary condition on te inflow boundary. Te isentropic inflow boundary condition is successfully verified on a D rectangular mes wit straigt boundaries and 3D ybrid mes wit straigt boundaries and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown ere for te 3D ybrid mes wit straigt boundaries and te order of accuracy results using L and L norms of te discretization error for te isentropic inflow boundary on a 3D ybrid mes are sown in Figure 6. Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t 3 4 3 4 a) b) 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 4.4.5 Outflow Tis boundary condition is a caracteristic based outflow condition. Te outflow boundary condition is tested as a subsonic boundary. For tis outflow boundary condition, a static pressure is imposed at te boundary. Te outflow boundary is tested on D rectangular mes wit straigt boundary and 3D ybrid mes wit straigt boundary. Te governing equations used wile testing outflow boundary condition are te Euler equations. Te D Manufactured Solution used for te verification of te outflow boundary condition as a subsonic outflow is of te form 56

,,,,,, (4.33),, Te 3D Manufactured Solution used for te verification of outflow boundary condition as a subsonic outflow is of te form,,,,,,,,,,,, (4.34),,,,,,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. Te above Manufactured Solutions for outflow boundary condition verification work for testing on straigt boundaries. For te curved boundaries, te Manufactured Solutions need to be selected very carefully and verification of outflow on a curved boundary is not performed ere. By selecting te Manufactured Solution as above, te u- and w-velocities are zero on te y = 0 boundary and only a negative v-velocity is defined on tis boundary wic ensures tat te flow is going out and is normal to tat boundary. Te outflow boundary condition is tested as a subsonic outflow on a D rectangular mes wit straigt boundaries and 3D ybrid mes wit straigt boundaries and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown for te 3D ybrid mes wit straigt boundaries and te order of accuracy results using L and L norms of te discretization error for te outflow boundary tested as a subsonic boundary on a 3D ybrid mes are sown in Figure 7. 57

Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t 3 4 3 4 a) b) 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 4.4.6 Extrapolation Te extrapolation-based boundary condition is useful for supersonic outflow conditions. Te outflow boundary condition is tested wit supersonic flow conditions. Te extrapolation boundary condition is tested on a D rectangular mes and a 3D ybrid mes wit straigt boundaries. Te governing equations used wile testing outflow boundary condition are te Euler equations. Te D Manufactured Solution used for te verification of te extrapolation boundary condition wit supersonic flow is of te form,,, 700, 900,, (4.35) 00000,, Te 3D Manufactured Solution used for te verification of extrapolation boundary condition wit supersonic flow is of te form 58

,,,,,, 700,, 900,,,, (4.36),, 800,, 00000,,,, Te constants used in te D and 3D Manufactured Solutions are presented in Appendix C in Table C and Table C, respectively. Te above Manufactured Solutions for extrapolation boundary condition verification are only for testing on straigt boundaries. For te curved boundaries, te Manufactured Solutions need to be andled carefully and verification of extrapolation boundary condition on a curved boundary is not performed ere. By selecting te Manufactured Solution as above, te u- and w-velocities are zero on te y = 0 boundary and only a negative v-velocity is defined on te y = 0 boundary wic ensures tat te flow is going and te flow is normal to tat boundary. Te Manufactured Solution also ensures tat te derivatives of all te variables at te boundary are zero. Te extrapolation boundary condition is tested on a D rectangular mes and a 3D ybrid mes wit straigt boundaries and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown for te 3D ybrid mes wit straigt boundaries and te order of accuracy results using L and L norms of te discretization error for te extrapolation boundary wit supersonic flow on a 3D ybrid mes are sown in Figure 8. 59

Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*w ro*e t 3 4 5 6 7 8 3 4 5 6 7 8 a) b) 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 4.4.7 Farfield Te farfield boundary condition is an inflow-outflow caracteristic based boundary condition. It is mostly suitable for a farfield condition used in external flows. It is suitable for bot subsonic and supersonic flow conditions. Here, te farfield boundary condition is tested wit subsonic flow conditions on D ybrid and 3D skewed ybrid meses wit curved boundaries. Wile testing te farfield boundary condition, te Navier-Stokes equations are used as te governing equations. Te same Manufactured Solution used for testing te baseline governing equations, i.e., te Euler and te Navier-Stokes equations are used for testing te farfield boundary condition. Te farfield boundary condition is successfully verified on te D and 3D ybrid meses and te observed order of accuracy calculated from te numerical solutions approaced two wit mes refinement for te meses considered. Te results are sown for te 3D skewed ybrid mes and te order of accuracy results using L and L norms of te discretization error for te farfield boundary condition on a 3D skewed ybrid mes are sown in Figure 9. 60

Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t 3 4 5 6 7 8 3 4 5 6 7 8 a) b) 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 4.5 Verification of Turbulence Models Verification of turbulence models provides additional callenges for MMS for different reasons. 9 One of te reasons is tat te turbulence models often employ min or max functions to switc from one beavior to anoter, tus causing te source terms to no longer be continuously differentiable. Te turbulence models tested in te finite volume code are te basic k-ω turbulence model and te k-ε turbulence model wic are part of te baseline version of te Menter s k-ω model 7 and te Menter s Sear Stress Transport k-ω model. 7 In bot models, te k-ω turbulence model gets activated in te boundary layer region and te k-ε turbulence model gets activated away from te wall boundaries in te free sear layers. Te general form of te turbulent kinetic energy equation and te turbulent frequency equation for compressible flow are: 0 (4.37) 6

Te turbulence production terms is given by 0 were te full compressible stress tensor is used: (4.38) (4.39) 3, 3, 3,, (4.40) Te blending function F in te turbulent dissipation rate equation can be used to activate/deactivate te cross diffusion term wic sifts te turbulence equations between te k- ω turbulence model and te k-ε turbulence model. By setting te blending function F to zero, te cross diffusion term in te turbulent dissipation rate equation is activated and te k-ε turbulence model is tested. By setting te blending function F to unity, te cross diffusion term in te turbulent dissipation rate equation is deactivated and te k-ω turbulence model is tested. Te constants used in te D and 3D Manufactured Solutions for te verification of turbulence equations are presented in Appendix C in Table C and Table C, respectively. In te process of selecting te Manufactured Solution for code verification purposes, it is required tat different terms in te governing equations are rougly te same order of magnitude suc tat te contribution from eac term in te governing equation is of same order of magnitude. Tis prevents te larger magnitude terms from masking errors in oter terms of smaller magnitude. 6

During te verification of turbulence models, te Manufactured Solutions are generated suc tat all terms in te turbulence models are rougly same order of magnitude over te domain considered for verification. For example, wen te source terms are generated for te turbulent kinetic energy equation, te ratios of te convection, diffusion, and production terms to te destruction terms are calculated to ceck tat all te terms in te turbulent kinetic energy equation are of similar orders of magnitude. As an example, te ratio of production term to te destruction terms in te turbulent kinetic energy equation in a D rectangular domain is sown in Figure 30. From te figure, te ratio of te production term to te destruction term is maintained suc tat te production term and te destruction term are of similar orders of magnitude in te complete domain. Similar source term ratios for te turbulent dissipation rate equation are calculated to ceck all te terms are of similar orders of magnitude. 9 Figure 30: Ratio of production term to te destruction term in turbulent kinetic energy equation in a D rectangular domain Te k-ω turbulence model is tested on te 3D skewed ybrid mes and te observed order of accuracy approaces two wit mes refinement. Te observed order of accuracy result for te k-ω turbulence model on te 3D skewed ybrid mes is sown in Figure 3. During te testing of k-ε turbulence model, a problem wit te turbulent dissipation rate equation was observed and te discretization error for tat equation did not decrease at expected rate. Te observed order of accuracy of te ρω discretization error norms dropped to zero wit mes 63

refinement, but all te oter conserved variable discretization norms approaced two wit mes refinement. Te beavior of te observed order of accuracy wit mes refinement for te k-ε turbulence model on te 3D skewed ybrid mes is sown in Figure 3. Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t ro*k ro*ε Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t ro*k ro*ε 0.5.5 3 3.5 4 0.5.5 3 3.5 4 a) b) 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

Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t ro*k ro*ω Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t ro*k ro*ω 0 0.5.5 3 3.5 4.5.5 3 3.5 4 a) b) 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 To explore te reason for failure of te verification test for te k-ε turbulence model on te 3D skewed ybrid mes, it is tested on simpler meses. Initially te k-ε turbulence model is tested on te D ybrid mes and it is observed tat te verification is successful wit all te norms of te discretization errors approacing two wit mes refinement. Te order of accuracy result for te k-ε turbulence model on te D ybrid mes is sown in Figure 33. Te above test is also done on a igly skewed 3D curvilinear mes wit exaedral cells and te k-ε turbulence model is successfully verified wit all te norms of te discretization error approacing two wit mes refinement. Te order of accuracy result for te k-ε turbulence model in te 3D igly skewed curvilinear mes is sown in Figure 34. 65

Order of Accuracy, p 3 ro ro*u ro*v.5 ro*e t ro*k ro*ω.5 Order of Accuracy, p 3.5 3.5.5 ro ro*u ro*v ro*e t ro*k ro*ω 0.5 0 0 30 40 0.5 0 0 30 40 a) b) 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 Order of Accuracy, p 4 3.5 3.5.5 ro ro*u ro*v ro*w ro*e t ro*k ro*ω Order of Accuracy, p 4 3.5 3.5.5 ro ro*u ro*v ro*w ro*e t ro*k ro*ω 0.5 4 6 8 0.5 4 6 8 a) b) 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 In addition, te k-ε turbulence model is tested on a 3D unstructured mes wit tetraedral cells and it is successfully verified wit all te norms of te discretization error approacing two 66

wit mes refinement. Te order of accuracy result for te k-ε turbulence model on 3D unstructured mes wit tetraedral cells is sown in Figure 35. By testing on different meses, it can be concluded tat tere is an issue in te discrete formulation of some of te terms in te turbulent dissipation rate equation as it works correctly only on D mes topologies, 3D structured mes topologies, and 3D unstructured mes wit tetraedral cells but it fails on 3D unstructured mes topologies wit skewed cells. From te above analysis, it is determined tat te issue is isolated to te cross-diffusion term in te turbulent dissipation rate equation on 3D unstructured mes wit skewed tetraedral cells. Order of Accuracy, p 4.5 4 3.5 3.5.5 ro ro*u ro*v ro*w ro*e t ro*k ro*ε Order of Accuracy, p 4.5 4 3.5 3.5.5 ro ro*u ro*v ro*w ro*e t ro*k ro*ε 0.5 0.5 4 6 8 4 6 8 a) b) 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 4.6 Verification of Time Accuracy for Unsteady Flows It is more difficult to apply te verification procedure using te order of accuracy test to problems tat involve bot spatial and temporal discretization, especially wen te spatial order is different from te temporal order. A combined spatial and temporal order verification metod was developed by Kamm et al. 64 In teir approac, tey use te Newton-type iterative procedure to solve a coupled, non-linear set of algebraic equations to calculate te coefficients and observed order of accuracies for te spatial and temporal terms in te discretization error 67

expansion. In tis present work, a simpler approac for spatial and temporal order verification is proposed. Neglecting te iger order terms, te discretization error for a sceme wit spatial and temporal terms, can be written as (4.4) were and are te observed orders of accuracy in space and time, respectively, g x and g t are te coefficients of spatial and time terms, respectively, and x and t are normalized spatial and temporal discretizations respectively. Similarly, wit different set of coefficients, te norm of te discretization error can be found as (4.4) Initially, a spatial mes refinement study is performed wit a fixed time step to calculate and g x using tree mes levels wic makes te discretization error equation (4.43) were φ = g ˆ is te fixed temporal error term. Using tree mes solutions, refined by te factor q t t r x, coarse ( r x ), medium ( r x x x calculated 7 as ), and fine ( ), te observed order of accuracy can be x (4.44) were r is te spatial refinement factor between two mes levels and te coefficient of te x spatial term g x can be calculated as 68

(4.45) Similarly, a temporal refinement study is performed on a fixed mes to calculate and g t using tree temporal discretizations, coarse ( r t ), medium ( t r tt ), and fine ( t ). Wit all te coefficients calculated, te spatial step size and te temporal step size can be cosen suc tat te spatial discretization error term as te same order of magnitude as te temporal discretization error term. It is required tat te spatial and temporal discretization error terms ave te same order of magnitude suc tat bot te terms ave te same effect on te discretization error of te sceme. Wit one error term muc smaller tan te oter error term, it makes it difficult to verify te order of accuracy of te smaller error term. If te temporal discretization error term is so small wen compared to te spatial discretization error term, ten mistakes in te temporal discretization will not be seen on very fine meses. Once tese two terms are approximately te same order of magnitude, combined spatial and temporal order verification is conducted by coosing te temporal refinement factor suc tat te temporal error term drops by te same order of magnitude as te spatial error term wit refinement, i.e., r t r x pˆ qˆ =. Here r t is te temporal refinement factor, r x is te spatial refinement factor, is te spatial order and is te temporal order. In our case, te formal order is in bot space and time, i.e., = =. Using tis procedure, te unsteady time term is verified on te D ybrid mes and te 3D ybrid mes for Navier-Stokes equations. Te observed order of accuracy on bot te meses approaced two wit mes refinement. Te order of accuracy results using L discretization error for te time accuracy of te unsteady flows on D ybrid mes and 3D ybrid mes are sown in Figure 36 and Figure 37, respectively. 69

Order of Accuracy, p 3 ro ro*u ro*v ro*e t Order of Accuracy, p 3 ro ro*u ro*v ro*e t 0 4 6 8 0 0 4 6 8 0 a) b) 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 Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t Order of Accuracy, p 4 3 ro ro*u ro*v ro*w ro*e t 3 4 3 4 a) b) 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

4.7 Issues Uncovered During Code Verification During te verification of te finite volume code, several coding mistakes and algoritm inconsistencies were uncovered. Some code options failed te order of accuracy test for te first time and te verification studies elped in learning more about te code or te algoritm and ultimately removing some of te mistakes in te code or algoritm. Because te verification procedure is so sensitive to minor issues like te mes topology or mes quality, tey can often uncover sensitivities to tese issues. Issues uncovered during te code verification are documented ere. 4.7. Coding Mistakes/Algoritm Inconsistencies A coding mistake was found and corrected in te formulation of te diffusion operator wile testing te Navier-Stokes equations in te finite volume code on te D skewed curvilinear mes. Initially, te discretization error did not decrease wit mes refinement and a modification was done to te diffusion operator by Luke. 39 Te new diffusion operator rectified te problem and an observed order of accuracy of two was attained wit mes refinement. A coding mistake was found and corrected in te formulation of te diffusion operator wile testing te Navier-Stokes equations on D unstructured mes wit triangular cells. Initially, te diffusion operator was found to be only first order accurate on D unstructured meses and after modifying te diffusion operator formulation for unstructured grids, te diffusion operator tested was found to be second order accurate. Testing te k-ε turbulence model on different mes topologies, it was found tat tere was an issue in te discrete formulation of te cross-diffusion term in te turbulent dissipation rate equation. Te present formulation works fine on D mes topologies, 3D structured mes topologies, and 3D unstructured mes wit tetraedral cells but failed on 3D unstructured mes topologies wit skewed cells. 4.7. Grid Sensitivities Systematic mes refinement was found to be important for te verification process. Lack of systematic mes refinement on unstructured meses led to te failure of code verification test. As an example, initially 3D unstructured meses containing tetraedral cells were generated using GRIDGEN. Using tis mes generation tool, a domain was filled wit tetraedral cells 7

automatically. Generating different mes levels in GRIDGEN, te meses were not systematically refined. Wen Navier-Stokes equations were tested using tese meses generated in GRIDGEN, te code verification test failed were te observed order of accuracy did not matc te formal order. Later, te 3D unstructured mes wit tetraedral cells was generated using te developed mes generation code and meses from tis code were systematically refined between te mes levels. Using tese meses, te Navier-Stokes equations were successfully verified to second order accurate, ence explaining te importance of systematic mes refinement for code verification. During te verification of te no-slip wall boundary conditions it was found tat te mes sould be normal to te wall and te order of accuracy test failed if te mes is not normal to te wall. To explain tis, initially wile testing te adiabatic no-slip wall boundary condition in D, te meses wic were used for verification did not ave mes normal to te boundary. Wile testing te adiabatic no-slip wall boundary on tese meses, te calculated observed order of accuracy did not matc te formal order and error was found to be near te boundary. Later, wen te meses are generated again suc tat te mes is normal to te boundary, te adiabatic no-slip wall boundary was successfully verified. Te discrete formulation of te governing equations was found to be sensitive to igly skewed tetraedral and prismatic cells. Te baseline governing equations were successfully verified on te igly skewed D ybrid mes and igly skewed 3D curvilinear mes wit exaedral cells, but failed te verification test on a igly skewed 3D ybrid mes. 7

5. Finite Volume Diffusion Operators For well over a decade, tere as been significant work on te development of algoritms for te compressible Navier-Stokes equations on unstructured grids. Tis work is motivated by te fact tat unstructured meses allow more automation of te mes generation process and tus less of te analyst s time relative to structured meses. One of te callenging tasks during te development of tese algoritms is te formulation of consistent and accurate diffusion operators. A robust approac is desired for te treatment of diffusion operators suc tat tey are at least second order accurate on various structured and unstructured cell topologies. In addition, te accuracy of diffusion operators also depends on mes quality. A survey of diffusion operators for compressible CFD solvers was conducted to understand different formulation procedures for diffusion fluxes. A patc-wise version of te Metod of Manufactured Solutions was used to test te accuracy of some of te selected diffusion operators. Tese diffusion operators were tested and compared on different D mes topologies to study te effect of mes quality, mes resolution, and te solution beavior. For D structured meses, mes quality includes stretcing, aspect ratio, skewness, and curvature, but for D unstructured meses, tese mes quality attributes cannot be isolated. As an example, a triangular mes wit curvature, aspect ratio or stretcing will also ave skewness. Quantities examined include te numerical approximation errors and order of accuracy associated wit face gradient reconstruction. In te present work, te testing of diffusion operators is limited to cellcentered finite volume metods in D wic are formally second order accurate. 5. Robustness and Accuracy In te development of a numerical diffusion operator, one generally desires te creation of an accurate and robust operator. Te consistent and accurate treatment of diffusion fluxes are particularly callenging for finite volume solvers. Generally, it is required for te discrete operator to sare important properties wit te original continuous operator. For diffusion operators tere are tree properties tat we would like to satisfy and tese are as follows: 39. Te operator sould be numerically conservative. If te operator is defined as a sum of fluxes over all of te faces of a cell, ten te operator is conservative if te sared faces utilize te same flux. 73

. Te operator sould not generate new extrema. Tat is, te maximum and minimum solution values for diffusion processes sould occur at te boundaries. 3. Te operator sould be linearity preserving. Tat is, te operator sould evaluate identically to zero wen given a linear function. Satisfaction of te first property of conservation wen using a finite-volume sceme is easily acieved. If te operator is defined as a sum of fluxes over all of te faces of a cell, and te sared faces utilize te same flux, ten conservation is automatically acieved. Te callenge ten turns to te formulation of te cell interface fluxes. Te second property can be evaluated simply for linear scemes. For linear scemes te operator wic is formed from te sum of interface fluxes is written as a weigted sum of neigboring cell values. If all of te weigts in tis sum are positive ten te operator will not produce new extrema as eac cell value is an average of neigboring values and tus will be bounded by tem. For tis argument to old, te weigts of immediately adjacent cells must be non-zero, oterwise a degenerated rotated Laplacian could result. An operator tat does not satisfy tis property could result in solutions were tere is unbounded growt in te solution. As a result, te second property is required to produce a robust sceme. It as been sown for finite-element scemes tat a positive stencil will be generated for Delaunay simplex meses. For mixed element meses, suc as tose found in typical unstructured viscous meses, no suc guarantee as been sown. For finite-volume meses, te most typical strategy for formulating te viscous fluxes is to employ a directional derivative tecnique wereby te derivative in te direction of te vector connecting cell centroids is computed using te cell values on eiter side of te face, wile gradients in oter directions are computed troug an average of te cell centered gradients. Tis averaging tecnique avoids te rotated Laplacian stencil tat will result if a simple average of cell centered gradients were used. Strang et al. 6 sowed tat for VGRID generated meses up to te tirty percent of mes cells ad negative coefficients in te diffusion operator stencil. Haselbacer et al. 4 sowed tat for a linear sceme, one could not in general satisfy all of te tree conditions listed above; terefore one must eiter relax tese requirements or resort to some form of non-linear limiting procedure to create a robust diffusion operator witin te context of a finite volume sceme. Wen te stencil positivity property is violated, it usually doesn t result in failure of te sceme; owever te robustness of suc a sceme is not satisfactory for production use. Finally, linearity preserving is generally required to acieve a 74

second order reconstruction. At te very least, failure to satisfy te linearity preservation property guarantees tat te solution will be igly sensitive to te local quality and distribution of mes elements. 5. Testing Framework Te accuracy of a diffusion operator in te Navier-Stokes equations can be tested using approaces tat are similar to tose from te Metod of Manufactured Solutions 9 often employed for code verification. During tis procedure, a smoot analytic solution is selected to assess te diffusion operator. For te diffusion operator to be verified, te observed order of accuracy is required to matc te formal order of accuracy. An analytic solution is used to find te exact solution wic is used for te calculation of te error for te numerical diffusion operator. Te difference in te procedure applied ere wen compared to te MMS is tat te analytic solution is not applied to any governing equation to generate a source term. During te formulation of te diffusion operator, te common procedure is to calculate te gradients on te cell faces and tese solution gradients on te cell faces are tested for te accuracy of te diffusion operator. For testing te diffusion operators in finite volume CFD codes, initially, te numerical solution values at te cell centers are obtained from te analytic solution selected for testing and te solution values at te cell centers are used to calculate te numerical solution gradients at te face centers. Te values of te solution gradients may be different depending on te type of formulation used for te calculation of te gradients. Te analytic solution can also be used to calculate te exact solution gradients at te face centers. Ten te error in te numerical formulation of te gradient can be calculated as te difference between te numerical solution gradients and te exact solution gradients at te face centers. Te procedure is repeated on multiple mes levels to get te error from eac mes level wic are used to calculate te observed order of accuracy. If te observed order of accuracy calculated from multiple mes levels matces te formal order, te diffusion operator can be considered verified. Since te mes quality can affect te accuracy of te diffusion operator, it can be altered during te testing process to build an accurate and consistent diffusion operator. In te process of calculating te observed order of accuracy, different mes levels are required to obtain te discretization error on different systematically refined meses. By doing a traditional way of refinement as already discussed in te order of accuracy verification using 75

MMS, te computations become expensive particularly for 3D problems were eac level of refinement is 8 times larger. So, particularly for testing te diffusion operators a different approac is used in terms of mes refinement. Tis alternate metod used is similar to te downscaling approac to order verification 7,8 wic employs a single mes wic is scaled down about a single point in te domain. In te downscaling approac, several parts of te mes in a domain are selected and tey are repeatedly srunk towards a focal point. In our approac, only te stencil or te mes to be scaled is only considered for testing te numerical formulation instead of te complete domain. An example of srinking a mes over a focal point for a D unstructured mes is sown in Figure 38. In te figure, te mes is srunk over te cell centroid of te central triangle wic is te focal point and te wole mes is scaled by a factor of two. By srinking te mes in tis fasion, a systematic mes refinement is acieved wic means tat te refinement is uniform and te mes quality remains same wit refinement. Similarly, te mes can be furter scaled down several times to obtain multiple mes levels for testing te formulation of a diffusion operator. Figure 38: Srinking of a mes over a focal point for a D unstructured mes 5.3 Cell-Centered Finite Volume Diffusion Operators After a brief literature survey, a few commonly used cell centered finite volume diffusion operator formulations in te literature are selected and tested for teir accuracy on different mes topologies wit different cell quality aspects. Te different numerical formulations tested include te node-averaged metod, unweigted least squares, weigted least squares, te numerical 76

formulation used by te Loci-CHEM CFD code, and te Face Approximate Mapped Least Squares (FAMLSQ) metod 44 wic uses a compact stencil. Tese metods are briefly discussed below. 5.3. Node Averaging Metod For cell centered formulations, a common metod for calculating te solution gradients at te face centers is te node averaging metod. 44 In tis metod, te solution values are reconstructed at te nodes from te surrounding cell centers. Te solution reconstruction is proposed in [65, 66] and used in [67] as an averaging procedure tat is based on a constrained optimization to satisfy te Laplacian properties. For te calculation of gradients on te face center, te derivative tangent to te face is computed as te divided difference between te solution values reconstructed at te nodes and te derivative normal to te face is computed using te solution values at te centroids of te cells saring te face. Te gradient is resolved from te derivative along te face and te derivative along te line connecting te cell centers across te face. 5.3. Unweigted and Weigted Least Squares In te weigted least squares metod, 44 te contributions from te adjacent cells to te minimized functional are weigted wit weigts inversely proportional to te distance from te central point, i.e., te face center wen calculating te gradients on te face. In te unweigted least squares metod, 44 all contributions from te adjacent cells are equally weigted. 5.3.3 Loci-CHEM Diffusion Operator In order to more reliably obtain second order gradients at te face, an alternative metod for computing tese gradients is introduced in te Loci-CHEM code. In tis metod, a new control volume centered about te face is constructed. Initially, a least squares approac is used to calculate te solution gradients at te cell center. 39 Ten, a second order function reconstruction is used to compute te nodal values of a control volume centered about a face. Tis is constructed by computing two points tat are projected above and below te face centroid in te normal direction by a factor of 0.5 to form points 0.5, and 0.5 as illustrated in Figure 39. is te face normal vector and is te vector tat connects te left and rigt cell centroids. 77

Figure 39: An illustration of te reconstruction of a centered control volume about face F c as sown in te saded region Te cell gradients are ten used in te respective cells to reconstruct te function values at te points P l and P r wile at te function values at te face nodes are reconstructed using te average of te left and rigt cell function reconstructions. Te gray area in Figure 39 illustrates tis reconstructed diamond cell. Te left and rigt points of te reconstruction along wit te edges of te face form triangles tat bound tis volume. Green s teorem is ten employed to compute a second order face gradient. 5.3.4 FAMLSQ Metod wit Compact Stencil A more general approximate mapping (FAMLSQ) metod 44 tat is based on te distance function, defined as te distance to te nearest boundary is explained ere. Te FAMLSQ metod useful for curved boundaries, applies a least-square minimization in a locally constructed coordinate system. f r = f 0 + κ ξ + λ η (5.) Te local coordinates (ξ, η ) are constructed using te distance function, wic provides information on te closest boundary point. First te distance function is reconstructed at te face center. Te coordinate vectors at te face center are defined as a unit η -directional vector pointing ing in te direction opposite to te closest boundary point and its ortonormal ormal ξ -directional vector. Te η -coordinate is taken as te distance from te boundary and te ξ -coordinate is te projection of te vector connecting te stencil point wit te face center onto te ξ -direction. Te stencil selected for calculating te solution gradients at face centers is called te compact 78

stencil. 44 Te compact stencil typically involves two prime cells and two auxiliary cells; one for eac prime cell. An auxiliary cell is cosen from te pool of te cells saring te nodes of te face as te cell closest to te prime cell, but not its face neigbor. Te compact stencil is important for discretizations on ig-aspect-ratio grids to represent correctly te direction of te strong coupling. 5.4 Mes Topologies Te different formulations of diffusion operators are tested on different mes types wic can isolate te mes quality parameters including cell aspect ratio, cell stretcing, cell skewness, and curvature. All te above types of cells are considered in te evaluation of te diffusion operators. In te process of evaluation, only a minimum grouping of cells required (i.e. te stencil) are considered for te calculation of te solution gradients on te face centers of a single cell. For D structured meses, only 9 cells are used and te solution gradients are calculated on te face centers of te central cell. Te D structured meses wit different cell quality aspects considered for testing te different formulations of diffusion operators are sown in Figure 40 and Figure 4. A uniform mes is sown in Figure 40a, a mes wit aspect ratio cells is sown in Figure 40b, a stretced mes is sown in Figure 40c, a mes wit skewed cells is sown in Figure 40d, and a curvilinear mes is sown in Figure 4. For D unstructured meses, te stencil contains 6 cells and only 3 cells are required for te calculation of solution gradients on te face centers of te central cell. Te triangular cells in te tree corners of te domain are not required for te calculation of solution gradients. Te D unstructured meses wit different cell quality aspects considered for testing are sown in Figure 4. An isotropic triangular mes is sown in Figure 4a, a triangular mes wit aspect ratio cells is sown in Figure 4b were te cells also ave skewness associated wit tem, and a curvilinear triangular mes is sown in Figure 4c. 79

0.3 Aspect Ratio 5: 0. 0. 0.06 0.04 0.0 0 0 0. 0. 0.3 0 0 0. 0. 0.3 a) b) Stretcing Factor.5 Skew Angle 60º 0.4 0.3 0.3 0. 0. 0. 0. 0 0 0. 0. 0.3 0.4 0 0 0. 0. 0.3 0.4 c) d) 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

θ = 30º Figure 4: D structured mes curvilinear mes Aspect Ratio.5: a) b) θ = 80º c) Figure 4: Unstructured mes topologies in D: a) mes wit isotropic cells, b) mes wit aspect ratio cells, and c) curvilinear mes 8

5.5 Effect of Mes Quality Te evaluation criteria for diffusion operators are te calculation of te solution gradients on te face centers to compare tem wit te exact gradients for errors in te solution gradients and calculation of te observed order of accuracy wit mes refinement. Te different numerical formulations considered are tested for accuracy on D structured meses and D unstructured meses wic ave some cell quality aspects. Te analytic solution used wile testing te different numerical formulations on D meses is of te form given below: 5.5. D Structured Meses, On D structured meses, tree numerical formulations of diffusion operators are tested. Tey include te node-averaged, unweigted least squares, and weigted least squares metods. 5.5.. Uniform Mes All tree numerical formulations are tested on te D structured uniform mes (Figure 40a). Te solution gradients u/ x and u/ y are calculated on te four faces of te cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions. On a uniform mes all tree numerical formulations metods produced similar errors. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested on D structured uniform mes. Te observed order of accuracy results for te solution gradients at te face centers for te left, rigt, top, and bottom faces of te cell on a D structured uniform mes using node averaging metod and weigted least squares metod are sown in Figure 43. (5.) 8

3.5 3.5 3 3 Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b 5 0 5 0 5 0 5 0 a) b) 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 5.5.. Aspect Ratio Te effect of cell quality on te numerical formulations of te diffusion operators is tested. Te numerical formulations of diffusion operators are tested on D structured mes wic as cells wit a constant aspect ratio (Figure 40b). Meses wit aspect ratios of 5:, 0: and 50: are used wile testing te numerical formulations. Te solution gradients u/ x and u/ y are calculated on te four faces of te cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions. All tree numerical formulations metods produced similar errors on te mes wit aspect ratio cells. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested on te D structured mes wit aspect ratio cells even for cell wit a ig aspect ratio of 50:. Te cange in aspect ratio as no effect on te accuracy and a second order beavior is observed for all aspect ratio cells considered. Te observed order of accuracy results for te solution gradients at te face centers on a D structured mes wit aspect ratio cells of 50: using te node averaging metod and te weigted least squares metod are sown in Figure 44. 83

3.5 3.5 3 3 Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b 5 0 5 0 5 0 5 0 a) b) 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 5.5..3 Stretcing Te numerical formulations of diffusion operators are tested on D stretced mes wit a constant stretcing factor (Figure 40c). Stretced meses wit several stretcing factors between and.5 in bot te coordinate directions are used wile testing te numerical formulations. One important note ere is tat te cell centers of te stretced cells in te mes are not considered as te geometric cell centers, but tey are calculated depending on te stretcing involved in te mes. Te solution gradients u/ x and u/ y are calculated on te four faces of te cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions. All tree numerical formulations metods produced similar errors on te D stretced mes. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested on stretced mes wit stretcing factors up to.5. Te cange in stretcing factor as no effect on te accuracy and a second order beavior is observed for all stretcing factors considered. Te observed order of accuracy results for te solution gradients at te face centers on a D stretced mes wit a stretcing factor of.5 using node averaging metod and weigted least squares metod are sown in Figure 45. 84

3.5 3.5 3 3 Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b Order of Accuracy, p.5.5 0.5 0-0.5 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b 5 0 5 0 5 0 5 0 a) b) 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 5.5..4 Skewness Te numerical formulations of diffusion operators are tested on a D structured mes wit skewed cells (Figure 40d). Skewness in te structured mes can be measured in many possible ways, but ere skewness is measured as te minimum angle of te cell. Te skewed meses considered for testing ave all te cells wit same skewness. Te D structured meses wit a minimum angle between two sides of a cell of 60º, 30º, 5º, and º are used wile testing te numerical formulations. Te solution gradients u/ x and u/ y are calculated on te four faces of te cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions, tus maintaining te cell quality wit mes refinement. All tree numerical formulations metods produced similar errors on te mes wit skewed cells. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested on D structured mes wit skewed cells. Te cange in skewness of te mes as no effect on te accuracy and a second order beavior is observed for all skewed cells considered. Te observed order of accuracy results for te solution gradients at te face centers on a D structured mes wit igly skewed cells wit a minimum 85

angle of º using node averaging metod and weigted least squares metod are sown in Figure 46. Order of Accuracy, p du/dx-l du/dy-l 0 du/dx-r du/dy-r du/dx-t 8 du/dy-t du/dx-b 6 du/dy-b 4 0 - Order of Accuracy, p du/dx-l du/dy-l 0 du/dx-r du/dy-r du/dx-t 8 du/dy-t du/dx-b 6 du/dy-b 4 0 - -4-4 5 0 5 5 0 5 a) b) 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 5.5..5 Curvature Te effect of curvature as a mes quality aspect is also examined for te different numerical formulations of diffusion operators. Te mes considered for testing te effect of curvature as an inner radius of 5 wit different angles made by te stencil range from 5º to 90º. Te solution gradients u/ x and u/ y are calculated on te four faces of te cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions wic maintains te quality of te mes. Te difference in te errors in te solution gradients is small for all te tested numerical formulations metods on te D structured curvilinear mes. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested on D structured curvilinear mes (Figure 4). Te cange in te mes curvature as no effect on te accuracy. Te observed order of accuracy results for te solution gradients at te face centers on a D structured curvilinear mes using node averaging metod and weigted least squares metod are sown in Figure 47. 86

Order of Accuracy, p du/dx-l du/dy-l 0 du/dx-r du/dy-r du/dx-t 8 du/dy-t du/dx-b 6 du/dy-b 4 0 - Order of Accuracy, p 4 0 du/dx-l du/dy-l du/dx-r du/dy-r du/dx-t du/dy-t du/dx-b du/dy-b -4 5 0 5 5 0 5 a) b) 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 As a conclusion, all te numerical formulations tested on D structured meses are second order accurate on every type of mes wic as a particular mes quality aspect associated to it. Te effect of cell quality on te errors in te solution gradients calculated using any particular numerical metod is negligible as te errors did not vary muc between structured meses wit different cell quality. Tis approac of testing te numerical formulation can be extended to 3D witout muc effort in terms of mes generation. Also, te srinking aspect of mes refinement for calculating te order of accuracy from te errors from different mes levels can be applied easily in 3D saving computational resources. Te accuracy of tese numerical formulations is tested under more difficult situations wen tey are examined on D unstructured meses. 5.5. D Unstructured Meses On D unstructured meses, te four different numerical formulations of diffusion operators are tested. Tey include te node-averaged metod, weigted least squares metod, numerical formulation of diffusion operator used by te Loci-CHEM code, and te numerical formulation wic uses a compact stencil and applied as a least squares procedure to calculate te solution gradients at te face centers wic is called te Face Approximate Mapping Least 87

Squares (FAMLSQ). 44 curvature. Te FAMLSQ metod is tested only for unstructured meses wit 5.5.. Mes wit Isotropic Cells Apart from te FAMLSQ metod, oter numerical formulations are initially tested on te D unstructured mes wit isotropic cells (Figure 4a). Te solution gradients u/ x and u/ y are calculated on te tree cell faces of a triangle wic is te central cell of te stencil considered. Te mes refinement is again acieved by fixing te cell centroid of te central cell as te focal point and srinking te remaining domain equally in all directions. Tree numerical formulations wic include te node averaged metod, te weigted least squares metod, and te numerical formulation in te Loci-CHEM code are tested on a D unstructured mes wit equilateral triangles. Te same analytic solution used for te D structured meses is used ere. Te solution gradients calculated at te face centers sowed second order convergence wit mes refinement for all te numerical formulations tested. Te observed order of accuracy results for te solution gradients at te face centers for te tree sides of te triangular cell using node averaging metod and weigted least squares metod are sown in Figure 48. Order of Accuracy, p 4 3 0 - - du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B 0 0 0 Order of Accuracy, p 3.5.5 du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B 0 0 0 a) b) 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

In te process of testing te diffusion operator in te Loci-CHEM code, te solution gradients are calculated at te face center of only one side of te triangular cell. Te observed order of accuracy results for te solution gradients at one of te face centers of te triangular cell using numerical formulation of diffusion operator in Loci-CHEM code are sown in Figure 49. Te error in te solution gradients calculated at te face center on a D unstructured mes wit isotropic cells using te tree numerical metods (node averaging, weigted least squares and Loci-CHEM formulation) is compared and sown in Figure 50. Te errors in te solution gradients are almost similar from te tree metods and Loci-CHEM approac produced sligtly iger errors. Order of Accuracy, p 3 du/dx - L du/dy - L 0 0 0 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

DE du/dx (NA) 0 du/dy (NA) 0 du/dx (WLSQ) du/dy (WLSQ) 0 0 du/dx (LC) du/dy (LC) 0 - st-order-slope nd-order-slope 0-0 -3 0-4 0-5 0-6 0 0 0 0 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 5.5.. Skewness wit Aspect Ratio For D unstructured meses, triangular cells wit ig aspect ratios will also be igly skewed. Tese two aspects of mes quality cannot be tested separately wen testing te numerical formulations on triangular cells. Te aspect ratio of a triangular cell in tis study is defined as te minimum value of te base to eigt ratio and te skewness is defined as te smallest angle made by any two sides in a triangle. Te skewed meses wit a minimum angle of 30º, 0º, 5º, º, 0.º, and 0.0º are used wile testing te numerical formulations. Te igest aspect ratio acieved using tese meses are approximately 7000:. Te numerical formulations of diffusion operators are tested on igly skewed triangular unstructured meses. Te skewed meses considered for testing ave all te cells wit same skewness. Te solution gradients u/ x and u/ y are calculated on te tree faces of te triangular cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions, tus maintaining te cell quality wit mes refinement. For te node averaging metod and te weigted least squares metod te solution gradients sowed second order convergence wit mes refinement for bot te numerical formulations tested on D unstructured triangular mes wit skewed and aspect ratio cells (Figure 4b). Te cange in skewness/aspect ratio of te mes as no effect on te accuracy and a second order beavior is observed even for te igly skewed meses. Te 90

observed order of accuracy results for te solution gradients at te tree face centers on a igly skewed triangular mes wit an aspect ratio of 70: and a minimum angle between sides of º using node averaging metod and weigted least squares metod are sown in Figure 5. Wile testing te diffusion operator in te Loci-CHEM code, te solution gradients are calculated at te face center of only one side of te triangular cell and te solution gradients sowed second order convergence wit mes refinement for te Loci-CHEM formulation of te diffusion operator. Te observed order of accuracy results for te solution gradients at one of te face centers of te triangular cell on a igly skewed mes wit an aspect ratio of 70: and a minimum angle between sides of º using numerical formulation of diffusion operator in Loci-CHEM code are sown in Figure 5. Te error in te solution gradients calculated at te face center on te unstructured mes wit igly skewed and aspect ratio cells using te tree numerical metods is compared and sown in Figure 53. Of te tree numerical metods tested, Loci-CHEM approac produced iger errors and te weigted least squares approac produced least errors on unstructured mes wit igly skewed and aspect ratio cells. Order of Accuracy, p.5.5 du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B Order of Accuracy, p 3.5.5 du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B 0 0 0 0 0 0 a) b) 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

Order of Accuracy, p 3 du/dx - L du/dy - L 0 0 0 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 DE 0 7 du/dx (NA) du/dy (NA) 0 5 du/dx (WLSQ) du/dy (WLSQ) du/dx (LC) 0 3 du/dy (LC) st-order-slope 0 nd-order-slope 0-0 -3 0-5 0 0 0 0 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 5.5..3 Skewness wit Curvature Testing te effect of curvature for D unstructured meses wit triangular cells will also ave skewness and possibly stretcing in te radial direction associated wit te D unstructured mes. Tese mes quality aspects are tested togeter wile concentrating on te effect of curvature on te numerical formulations of te diffusion operators. Te numerical formulations of diffusion operators are tested on meses wit ig curvature wic ave an inner radius of 0 9

and te angle made by te stencil ranging from 8º to 60º. For te node averaging metod and te weigted least squares metods, te solution gradients u/ x and u/ y are calculated on te tree faces of te triangular cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te cell center of te central cell as te focal point and srinking te remaining domain equally in all directions, tus maintaining te cell quality wit mes refinement. For te node averaging metod, it is observed tat te solution gradient sowed zerot order accuracy wit mes refinement wile testing on curved meses. Te errors in te calculation of te solution gradients do not decrease wit mes refinement. Te inconsistency started sowing up wen te angle made by te stencil is above 0º. Certainly, te node average metod is inconsistent wile testing it on te D unstructured curvilinear mes (Figure 4c). Te observed order of accuracy results for te solution gradients at te tree face centers using node averaging metod on a D unstructured mes wit curvature (wen angle made by te stencil is 60º) is sown in Figure 54a. For te weigted least squares metod, it is observed tat te solution gradient converged to first order accuracy wit mes refinement wile testing on curved meses. Again, te first order beavior started sowing up wen te angle made by te stencil is 0º and above. Te weigted least squares metod is considered to be first order accurate wile testing it on te D unstructured curvilinear mes. Te observed order of accuracy results for te solution gradients at te tree face centers weigted least squares metod on a D unstructured mes wit curvature (wen angle made by te stencil is 60º) is sown in Figure 54b. So bot te node averaging and weigted least squares metods fail to be second order accurate and an accuracy drop is observed wile testing tese metods on D unstructured meses wic curved boundaries. 93

Order of Accuracy, p du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B 0 - - 0 0 0 0 Order of Accuracy, p.5 0.5 0-0.5 - -.5 - du/dx - L du/dy - L du/dx - R du/dy - R du/dx - B du/dy - B 0 40 60 80 a) b) 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 Te next metod tested on D unstructured curvilinear meses is te Loci-CHEM formulation of diffusion operator. Te solution gradients u/ x and u/ y are calculated on one of te tree faces of te triangular cell wic is te central cell of te stencil considered. Te mes refinement is done by fixing te face center at wic te solution gradients are calculated as te focal point and srinking te remaining domain equally in all directions. Wit te Loci- CHEM formulation tested on curved meses, it is observed tat te solution gradients calculated at te face center sowed second order beavior wit mes refinement. Te Loci-CHEM formulation is tested on a curved mes were te angle made by te stencil is 80º. Also, te cange in mes curvature as no effect on te accuracy. Te observed order of accuracy results for te solution gradients on te face center using Loci-CHEM formulation of diffusion operator on a D unstructured mes wit curvature is sown in Figure 55. 94

Order of Accuracy, p 3 du/dx - L du/dy - L 0 0 0 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 Te oter metod tested on D unstructured curvilinear meses is te Face Approximate Mapping Least Squares (FAMLSQ) metod wic uses a compact stencil to formulate te solution gradients at te face center. Te testing is done in a sligtly different way for tis formulation were te mes is srunk suc tat te radial cell size and te angle made by te stencil are reduced by a factor of two for every refinement instead of refinement in te x and y coordinate directions. Systematic mes refinement is satisfied even for tis kind of refinement were te quality of te mes improves instead of staying constant wit every refinement. Using tis formulation, te solution gradients u/ x and u/ y are calculated on one of te tree faces of te triangular cell wic is te central cell of te stencil considered. Tis formulation used for calculating te solution gradients at te face center sowed second order accuracy wit mes refinement. Te observed order of accuracy results for te solution gradients on te face center using te FAMLSQ formulation wit a compact stencil on a D unstructured mes wit curvature is sown in Figure 56. Te error in te solution gradients calculated at te face center on te unstructured mes wit igly skewed and curved cells using te four numerical metods tested is compared and sown in Figure 57. From te plot, te errors in te solution gradients using node averaging metod do not reduce wit mes refinement and te errors in te solution gradients using weigted least squares metod reduce at first order wit mes refinement. Loci- CHEM approac and FAMLSQ approac produced similar errors and te errors reduced at 95

second order wit mes refinement for bot te approaces on D unstructured mes wit skewed and curved cells. 4 Order of Accuracy, p 3 0 - - du/dx - L du/dy - L -3 0 0 0 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 0-0 - DE 0-3 0-4 0-5 0-6 du/dx (NA) du/dy (NA) du/dx (WLSQ) du/dy (WLSQ) du/dx (LC) du/dy (LC) du/dx (FAMLSQ) du/dy (FAMLSQ) st-order-slope nd-order-slope 0 0 0 0 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 From te tests done on D unstructured meses wit different mes qualities, it can be concluded tat te node averaging metod and te weigted least squares metod are second order accurate on skewed unstructured mes wit aspect ratio cells, but teir accuracy drops 96

wen te formulations are tested on unstructured curvilinear meses. Te numerical formulation of diffusion operator in Loci-CHEM code and te FAMLSQ metod using compact stencil are second order accurate on unstructured meses wit all mes qualities and can be considered as consistent formulations for calculation of diffusion terms on D unstructured meses from tis analysis. Tis approac of testing te numerical formulation can be extended to 3D, since te srinking aspect of mes refinement for calculating te order of accuracy from different mes levels can be applied easily in 3D saving significant computational power and time. 97

6. Conclusions 6. Summary of Results A detailed code verification study of an unstructured finite volume code was performed. Te finite volume code, Loci-CHEM, is used for reactive flow simulations of rocket engines. Te MMS was used to generate exact solutions for complex governing equations. Systematic mes refinement wic is important for code verification was explained. Different options in te finite volume CFD code were verified wic included te baseline steady-state governing equations, transport models, different boundary condition options, turbulence models, and time accuracy of unsteady flows. All te options were determined to be verified wen te observed order of accuracy matced te formal order on te D ybrid and 3D ybrid mes wic contained all cell topologies (triangular, quadrilateral, exaedral, tetraedral, and prismatic cells). Wen te verification process failed on any one of tese complex ybrid meses, ten simpler meses were considered to isolate te problem. Coding mistakes, algoritm inconsistencies, and mes quality sensitivities uncovered during code verification were presented. By testing te finite volume code on different meses, te effect of cell quality and cell topology on te order of accuracy of te code was also assessed. A survey of diffusion operators for compressible CFD solvers was conducted to understand te different formulation procedures for diffusion terms. Some of te diffusion operators extensively used in CFD solvers and some of te recently developed diffusion operators were evaluated by testing tem on different structured and unstructured mes topologies wit different mes quality aspects for accuracy. A patc-wise version of te Metod of Manufactured Solutions for code verification was used for te evaluation of te diffusion operators. Te numerical formulations were tested on D structured and D unstructured meses wit different cell quality aspects like skewness, aspect ratio, stretcing and curvature. During te testing of te numerical formulations, te observed order of accuracy calculated from multiple mes levels was compared wit te formal order of accuracy. Multiple mes levels were acieved by srinking te mes over a focal point rater tan a conventional mes refinement. All te numerical formulations tested on D structured meses were accurate and consistent as tey sowed second order convergence on every type of mes considered. All te numerical formulations tested on D unstructured meses were accurate wen tested on skewed 98

mes wit aspect ratio cells, but te node averaging metod and weigted least squares metod were not consistent as teir accuracy dropped to less tan second order wile testing on D unstructured curved meses. Te diffusion operator formulation in te Loci-CHEM CFD code and te FAMLSQ approac using a compact stencil were found to be consistent and accurate diffusion operators wile testing te formulations on D unstructured meses as tey acieved second order convergence on D unstructured meses wit all cell quality aspects. 6. Main Contributions of tis Work Tere are some new contributions to te field of Computational Fluid Dynamics from tis researc work.. Te requirements for generating 3D unstructured meses wit systematic mes refinement were identified and FORTRAN codes were developed to generate multiple mes levels for 3D unstructured meses and 3D ybrid meses for code verification purposes. To our knowledge, present commercial mes generation software cannot generate different mes levels of 3D unstructured meses or 3D ybrid meses wic satisfy systematic mes refinement.. Te effect of mes quality on numerical accuracy was studied by testing governing equations and different numerical formulations on different mes topologies wit varying mes qualities. 3. A unique way of verifying te time accuracy of unsteady flows was identified. In te process of verifying te spatial and time terms in a governing equation simultaneously, a metod for te selection of spatial and temporal step sizes suc tat te discretization error in bot spatial and temporal terms are of similar magnitudes was developed. 4. Te srinking of meses of over a focal point for generating multiple mes levels was used for testing of different formulations witout actually running te code. Tis procedure was applied successfully to decrease te computational effort in code verification. 5. Troug te code verification work discussed erein, Loci-CHEM is currently te most verified compressible CFD code. We know of no oter compressible CFD codes presently wic are verified as extensively as Loci-CHEM. 99

6.3 Future Work and Recommendations Some future recommendations are mentioned ere. One of tem is generating Manufactured Solutions for verification of boundary conditions wen te boundaries ave curvature. It is required to come up wit an intelligent way to design te Manufactured Solutions for some of te boundary condition options wen applied to curved boundaries to satisfy constraints on normal derivatives. Anoter recommendation is te application of verification of different numerical formulations in 3D using te srinking tecnique for generating multiple mes levels wic makes it easier in terms of computational resources for code verification purposes. In tis case, te numerical formulations are tested witout running te code. 00

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Appendix A A summary of several options verified in te finite volume Loci-CHEM CFD code is sown in Figure A. Figure A: Verification of different options in te finite volume Loci-CHEM CFD code 08

Appendix B After te Navier-Stokes equations were verified to be second order accurate on a 3D skewed ybrid mes but sowed a reduced accuracy on te 3D igly skewed ybrid mes, te mes quality was quantified wit some parameters. Te cell quality aspects were quantified and tabulated for te 65 65 65 meses. Tese numbers wic quantify te cell quality did not cange muc for different mes levels and improved wit mes refinement for a particular mes type wic satisfied systematic mes refinement. Te cell quality is quantified for te 3D skewed ybrid mes and te 3D igly skewed ybrid mes are sown in Table B. Table B. Quantification of cell quality 3D skewed ybrid mes 3D igly skewed ybrid mes Mes Size 65 65 65 65 65 65 Maximum Cell Angle 37.453 58.545 Maximum Twist 0.00087 0.0048055 Maximum Sear Twist 0.00030653 0.00008 In te above table, maximum cell angle is defined as te maximum cell to face angle and it is te angle between te face normal and te cell centroids. Tis angle provides an indication of mes isotropy. Lower te maximum cell angle better is te mes quality. For non-triangular faces, te faces can be non-planar i.e. twisted. Te twist metric is te measure of te non-planar component of te face geometry. Twist is te deviation of te face in te direction away from te plane of te face and sear twist is te deviation of te face in te plane of te face itself. A value of 0. indicates tat te face geometry deviates from te planar description by 0 percent. 09

Appendix C For boundary conditions and turbulence models, te constants and te trigonometric functions used in te Manufactured Solutions in D and 3D are in Table C and Table C, respectively. Table C. Constants for D Manufactured Solutions Equation, φ φ 0 φ x φ y φ xy ρ (kg/m 3 ) 0.5-0. 0.08 u (m/s) 70 7-8 5.5 v (m/s) 90-5 0 - p (N/m ) 0 5 0. 0 5 0.75 0 5-0.5 0 5 T (K) 350 0-30 5 k (m /s ) 780 60-0 80 ω (/s) 50-30.5 40 Equation, φ a φx a φy a φxy ρ (kg/m 3 ) 0.75.5 u (m/s).5.5 0.6 v (m/s).5 0.9 p (N/m ).5 0.75 T (K) 0.75.5.5 k (m /s ) 0.65 0.7 0.8 ω (/s) 0.75 0.875 0.6 Equation, φ f s (x) f s (y) f s (xy) ρ (kg/m 3 ) cos sin cos u (m/s) sin cos cos v (m/s) sin cos cos p (N/m ) cos sin sin T (K) cos sin cos k (m /s ) cos sin cos ω (/s) cos sin cos 0

Table C. Constants for 3D Manufactured Solutions Equation, φ φ 0 φ x φ y φ z φ xy φ yz φ zx ρ (kg/m 3 ) 0.5-0. 0. 0.08 0.05 0. u (m/s) 70 7-5 -0 7 4-4 v (m/s) 90-5 0 5 - -5 5 w (m/s) 80-0 0-5 p (N/m ) 0 5 0. 0 5 0.5 0 5 0. 0 5-0.5 0 5-0. 0 5 0. 0 5 T (K) 350 0-30 0 0-5 k (m /s ) 780 60-0 80 80 60-70 ω (/s) 50-30.5 0 40-5 5 Equation, φ a φx a φy a φz a φxy a φyz a φzx ρ (kg/m 3 ) 0.75 0.45 0.8 0.65 0.75 0.5 u (m/s) 0.5 0.85 0.4 0.6 0.8 0.9 v (m/s) 0.8 0.8 0.5 0.9 0.4 0.6 w (m/s) 0.85 0.9 0.5 0.4 0.8 0.75 p (N/m ) 0.4 0.45 0.85 0.75 0.7 0.8 T (K) 0.75.5 0.5 0.65 0.75 0.5 k (m /s ) 0.65 0.7 0.8 0.8 0.85 0.6 ω (/s) 0.75 0.875 0.65 0.6 0.75 0.8 Equation, φ f s (x) f s (y) f s (z) f s (xy) f s (yz) f s (zx) ρ (kg/m 3 ) cos sin sin cos sin cos u (m/s) sin cos cos cos sin Cos v (m/s) sin cos cos cos sin cos w (m/s) cos sin cos sin sin cos p (N/m ) cos cos sin cos sin cos T (K) cos sin sin cos sin cos k (m /s ) cos cos sin cos cos sin ω (/s) cos cos sin cos cos sin