Abstract
Code verification is mainly concerned with programming errors. Once a code is free of such errors, the verification of results can characterize the code accuracy, in particular the truncation error is of interest. Accuracy is an important feature of any code and the verification results provide some measure that can be used to compare code performance. Code verification requires an exact solution to which the numerical solution can be compared and the error precisely quantified. In the computational fluid dynamics community the method of manufactured solutions (MMS) is recommended as it can produce an exact solution sufficiently complex to test all code routines. However, it requires the addition of source terms in the equations of motion, a possibility that is almost never available in commercial codes. Exact solutions are also employed, but they represent very simplified flows that cannot test all code routines. Other verification tests exist, but they are also limited in comparison with MMS. Yet, even some of these limited tests are impossible to perform in many commercial codes. This paper presents a test based on linear stability theory. It is shown that the test is very demanding. It is also shown that the test could be performed even on codes that are very restrictive on what a user is allowed to do. It is not as complete as the MMS, but it is substantially more general than simple exact solutions of the Navier–Stokes equations. For instance it can account for three-dimensionality and compressibility effects, among other generalizations. The results enable a comparison of several codes in terms of refinement necessary for a grid-independent solution and the accuracy of the converged solution.
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References
Betchov R, Criminale WO (1967) Stability of parallel flow. Academic press, San Diego
Betchov R, Szewczyk A (1978) Numerical study of the Taylor-Green vortices. Phys Fluids 21(6):871–875
Bond R, Ober C, Knupp P (2007) Manufactured solution for computational fluid dynamics boundary condition verification. AIAA J 45(9):2224–2236
Brady P, Herrmann M, Lopez J (2012) Code verification for finite volume multiphase scalar equations using the method of manufactured solutions. J Comput Phys 231(7):2924–2944
Criminale WO, Jackson TL, Joslin RD (2003)Theory and computation in hydrodynamic stability. Cambridge University Press, UK
Drazin PG, Reid WH (1981) Hydrodynamic stability. Cambridge University Press, UK
Eça L, Hoekstra M (2009) Evaluation of numerical error estimation based on grid refinement studies with the method of the manufactured solutions. Comput Fluids 38:1580–1591
Eça L, Hoekstra M, Hay A, Pelletier D (2007) A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow. Int J Comput Fluid Dyn 21(3-4):175–188
Eça L, Hoekstra M, Hay A, Pelletier D (2007) On the construction of manufactured solutions for one and two-equation eddy-viscosity models. Int J Numer Methods Fluids 54(2):119–154
Gennaro EM, Colaciti AK, Medeiros MA (2013) On the controversy regarding the effect of flow shear on the strouhal number of cylinder vortex shedding. Aerosp Sci Technol. doi:10.1016/j.ast.2013.04.002
Germanos RA, Medeiros M (2008) Numerical simulation of a strongly spanwise modulated wavetrain in a compressible mixing layer (AIAA Paper-2008-4390)
Germanos RA, Souza LF, Medeiros MAF (2009) Numerical investigation of the three-dimensional secondary instabilities in the time-developing compressible mixing layer. J Braz Soc Mech Sci Eng 31(2):125–136
Huerre P, Monkewitz PA (1990) Local and global instabilities in spatially developing flows. Annu Rev Fluid Mech 22:473–573
Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier–Stokes equations. J Comp Phys 59(2):308–323
Lesieur M (2008) Turbulent in fluids. Springer, Berlin
Lin CC (1955) The theory of hydrodynamic stability. Cambridge University Press, UK
Mack LM (1984) Boundary layer linear stability theory. In: AGARD-R-709, pp 3.1–3.81
Obertkampf WL, Trucano TG (2002) Verification and validation in computational fluid dynamics. Prog Aerosp Sci 38(3):209–272
Obertkampf WL, Trucano TG, Hirsch C (2004) Verification, validation and predictive capability in computational engineering and physics. Appl Mech Rev 57(5):345–384
Roache P (1998) Verification of codes and calculations. AIAA J 36(5):696–702
Roache PJ (1997) Quatification of uncertainty in computational fluid dynamics. Annu Rev Fluid Mech 29:123–160
Roache PJ (2002) Code verification by the method of manufactured solutions. Trans ASME 124(1):4–10
Roache PJ (2009) Fundamentals of verification and validation. Hermosa Publishers, New Mexico
Roy CJ (2005) Review of code and solution verification procedures for computational simulation. J Comput Phys 205:131–156
Roy CJ, Nelson CC, Smith TM, Ober CC (2004) Verification of Euler/Navier–Stokes codes using the method of manufactured solutions. Int J Numer Methods Fluids 44(6):599–620
Sandham ND, Reynolds WC (1991) Three-dimensional simulations of large eddies in the compressible mixing layer. J Fluid Mech 224:133–158
Shunn L, Ham F, Moin P (2012) Verification of variable-density flow solvers using manufactured solutions. J Comput Phys 231(9):3801–3827
Silva HG, Souza LF, Medeiros MAF (2010) Verification of a mixed high-order accurate DNS code for laminar turbulent transition by the method of manufactured solutions. Int J Numer Meth Fluids 64(3):336–354
Taylor GI, Green AE (1937) Mechanism of the production of small eddies. Proc R Soc Lond A 158:499–521
Theofilis V (2003) Advances in global linear instability of nonparallel and three-dimensional flows. Prog Aerosp Sci 39(4):249–315
Theofilis V (2011) Global linear instability. Annu Rev Fluid Mech 43:319–352
Vedovoto J, Neto AS, Mura A, Silva L (2011) Application of the method of manufactured solutions to the verification of a pressure-based finite-volume numerical scheme. Comput Fluids 51(1):85–99
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The authors would like to acknowledge CNPq, FAPESP and EMBRAER for the financial support provided to this study.
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Technical Editor: Fernando Alves Rochinha.
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Gennaro, E.M., Simões, L.G.C., Malatesta, V. et al. Verification and accuracy comparison of commercial CFD codes using hydrodynamic instability. J Braz. Soc. Mech. Sci. Eng. 36, 59–68 (2014). https://doi.org/10.1007/s40430-013-0057-3
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DOI: https://doi.org/10.1007/s40430-013-0057-3