A Study of Physical and Numerical Effects of Dissipation on Turbulent Flow Simulations

Junqueira-Junior, Carlos; Azevedo, Joao Luiz F.; Scalabrin, Leonardo C.; Basso, Edson

doi:10.5028/jatm.v5i2.179

ABSTRACT:

The present work is primarily concerned with studying the effects of artificial dissipation and of certain diffusive terms in the turbulence model formulation on the capability of representing turbulent boundary layer flows. The flows of interest in the present case are assumed to be adequately represented by the compressible Reynolds-averaged Navier-Stokes equations, and the Spalart-Allmaras eddy viscosity model is used for turbulence closure. The equations are discretized in the context of a general purpose, density-based, unstructured grid finite volume method. Spatial discretization is based on the Steger-Warming flux vector splitting scheme and temporal discretization uses a backward Euler point-implicit integration. The work discusses in detail the theoretical and numerical formulations of the selected model. The computational studies consider the turbulent flow over a flat plate at 0.3 freestream Mach number. The paper demonstrates that the excessive artificial dissipation automatically generated by the original spatial discretization scheme can deteriorate boundary layer predictions. Moreover, the results also show that the inclusion of Spalart-Allmaras model cross-diffusion terms is primarily important in the viscous sublayer region of the boundary layer. Finally, the paper also demonstrates how the spatial discretization scheme can be selectively modified to correctly control the artificial dissipation such that the flow simulation tool remains robust for high-speed applications at the same time that it can accurately compute turbulent boundary layers.

KEYWORDS:
Computational fluid dynamics; Turbulence modeling; Flux vector splitting scheme; Artificial dissipation

Full text is available only in PDF.

REFERENCES

Anderson, J.D., 1991, “Fundamentals of Aerodynamics”, McGraw-Hill International Editions, New York, NY, USA.
Anderson, W.K., Thomas, J.L. and van Leer, B., 1986, “A Comparison Of Finite Volume Flux Vector Splittings For The Euler Equations”, AIAA Journal, Vol. 24, No. 9, pp. 1453-1460.
Azevedo, J.L.F., 1988, “Transonic Aeroelastic Analysis Of Launch Vehicle Configurations”, Ph.D. Thesis, Stanford University, Stanford, California, USA.
Basso, E., 1997, “Numerical Analysis of Cascade Flow Problems for All Speed Regimes”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil (in Portuguese, original title is Análise Numérica de Escoamentos em Grades Lineares de Perfis para Vários Regimes de Velocidade).
Batina, J., 1990, “Three-Dimensional Flux-Split Euler Schemes Involving Unstructured Dynamic Meshes”, AIAA Paper No. 90-1649, 21st AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Seattle, WA.
Batten, P., Craft, T.J., Leschziner, M.A. and Loyau, H., 1999, “Reynolds-Stress-Transport Modeling for Compressible Aerodynamics Applications”, AIAA Journal, Vol. 37, No. 7, pp. 785-797.
Bibb, K.L., Peraire, J. and Riley, C.J., 1997, “Hypersonic Flow Computations On Unstructured Meshes”, AIAA Paper No. 97-0625.
Bigarella, E.D.V., 2007, “Advanced Turbulence Modeling for Complex Aerospace Applications”, Ph.D. Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil .
Bigarella, E.D.V. and Azevedo, J.L.F., 2009, “Turbulence Modeling for Aerospace Applications, Turbulence”, Vol. 6 of Coleção Cadernos de Turbulência, Chapter 4, ABCM, Rio de Janeiro, RJ, 1st ed., pp. 215296 (in Portuguese, original title is Modelagem de Turbulência para Aplicações Aeroespaciais).
Bigarella, E.D.V. and Azevedo, J.L.F., 2012, “A Study Of Convective Flux Schemes For Aerospace Flows”, Journal of the Brazilian Society for Mechanical Sciences and Engineering, Vol. 34, No. 3, pp. 314-329.
Bigarella, E.D.V., Azevedo, J.L.F. and Scalabrin, L.C., 2007, “Centered and Upwind Multigrid Turbulent Flow Simulations of Launch Vehicle Configurations”, Journal of Spacecraft and Rockets, Vol. 44, No. 1, pp. 52-65.
Breviglieri, C., 2010, “High-Order Unstructured Spectral Finite Volume Method for Aerodynamic Applications”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil .
Breviglieri, C., Azevedo, J.L.F. and Basso, E., 2010a, “An Unstructured Grid Implementation Of High-Order Spectral Finite Volume Schemes”, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 32, No. 5 (Special Issue), pp. 419-433.
Breviglieri, C., Azevedo, J.L.F., Basso, E. and Souza, M.A.F., 2010b, “Implicit High-Order Spectral Finite Volume Method For Inviscid Compressible Flows”, AIAA Journal, Vol. 48, No. 10, pp. 2365-2376.
Chapman, D., 1981, “Trends and Pacing Items in Computational Aerodynamics, Seventh International Conference on Numerical Methods in Fluid Dynamics”, edited by W. Reynolds and R. MacCormack, Lecture Notes in Physics, Vol. 141, Springer, Heidelberg, pp. 1-11.
Coles, D.E. and Hirst, E.A., 1969, “Computation Of Turbulent Boundary Layers”, Proceedings of the 1968 U.S. Air Force Office of Scientific Research-Internal Flow Program-Stanford Conference, Vol. 2, Stanford University Press, Palo Alto, CA, USA.
Gnoffo, P.A., 2003, “Computational Aerothermodynamics In Aeroassist Applications”, Journal of Spacecraft and Rockets, Vol. 40, No. 3, pp. 305-312.
Hellsten, A. and Laine, S., 2000, “Explicit Algebraic Reynolds-Stress Modelling In Decelerating And Separating Flows”, AIAA Paper No. 2000-2313.
Hirsch, C., 1990, “Numerical Computation of Internal and External Flows, Vol. II: Computational Methods for Inviscid and Viscous Flows”, Wiley, New York, USA.
Jameson, A., 1995a, “Analysis And Design Of Numerical Schemes For Gas Dynamics 1 - Artificial Diffusion, Upwind Biasing, Limiters And Their Effect On Accuracy And Multigrid Convergence”, International Journal of Computational Dynamics, Vol. 4, No. 3-4, pp. 171-218.
Jameson, A., 1995b, “Analysis And Design Of Numerical Schemes For Gas Dynamics 2 - Artificial Diffusion And Discrete Shock Structure”, International Journal of Computational Dynamics, Vol. 5, No. 1-2, pp. 1-38.
Jameson, A., Schmidt, W. and Turkel, E., 1981, “Numerical Solution Of The Euler Equations By Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes”, AIAA Paper No. 81-1259.
Jawahar, P. and Kamath, H., 2000, “A High Resolution Procedure For Euler And Navier-Stokes Computation On Unstructured Grids”, Journal of Computational Physics, Vol. 164, No. 1, pp. 165-203.
Junqueira-Junior, C.A., 2012, “A Study On The Extension Of An Upwind Parallel Solver For Turbulent Flow Applications”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo, Brazil.
Junqueira-Junior, C.A., Azevedo, J.L.F., Basso, E. and Scalabrin, L., 2011, “Study Of Robust And Efficient Turbulence Closures For Aerospace Applications”, 32nd Iberian Latin American Congress on Computational Methods in Engineering, Ouro Preto, Minas Gerais, Brazil.
Lomax, H., Pulliam, T.H. and Zingg, D.W., 2001, “Fundamentals of Computation Fluid Dynamics”, Springer, New York, USA.
MacCormack, R.W. and Candler, G.V., 1989, “The Solution of the Navier Stokes Equations Using Gauss-Seidel Line Relaxation”, Computer and Fluids, Vol. 17, No. 1, pp. 135-150.
Mavriplis, D.J., 1988, “Accurate Multigrid Solution of the Euler Equations on Unstructured and Adaptive Meshes”, AIAA Journal, Vol. 28, No. 2, pp. 213-221.
Ramalho, M.V.C., Azevedo, J.H.A. and Azevedo, J.L.F., 2011, “Further Investigation into the Origin of the Carbuncle Phenomenon in Aerodynamic Simulations”, AIAA Paper No. 2011-1184, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA.
Roe, P.L., 1981, “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”, Journal of Computational Physics, Vol. 43, No. 2, pp. 357-372.
Scalabrin, L.C., 2007, “Numerical Simulation of Weakly Ionized Hypersonic Flow Over Reentry Capsules”, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, USA.
Scalabrin, L.C. and Boyd, I.D., 2007, “Numerical Simulations of the FIRE-II Convective and Radiative Heating Rates”, AIAA Paper No. 2007-4044.
Schlichting, H., 1978, “Boundary-Layer Theory”, 7th ed., McGraw Hill, New York, NY, USA.
Schwartzentruber, T.E., Scalabrin, L.C. and Boyd, I.D., 2007, “A Modular Particle-Continuum Numerical Method For Hypersonic Non-Equilibrium Gas Flows”, Journal of Computational Physics, Vol. 225, No. 1, pp. 1159-1174.
Schwartzentruber, T.E., Scalabrin, L.C. and Boyd, I.D., 2008, “Hybrid Particle-Continuum Simulations Of Non-Equilibrium Hypersonic Blunt-Body Flowfields”, Journal of Thermophysics and Heat Transfer, Vol. 22, No. 1, pp. 29-37.
Spalart, P.R. and Allmaras, S.R., 1992, “A One-Equation Turbulence Model For Aerodynamic Flows”, AIAA Paper No. 92-0439.
Spalart, P.R. and Allmaras, S.R., 1994, “A One-Equation Turbulence Model For Aerodynamic Flows”, La Recherche Aerospatiale, Vol. 1, No. 1, pp. 5-21.
Steger, J.L. and Warming, R.F., 1981, “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to the Finite-Difference Method”, Journal of Computational Physics, Vol. 40, No. 2, pp. 263-293.
Tennekes, H. and Lumley, J.L., 1972, “A First Course In Turbulence”, The MIT Press, Cambridge, MA, USA.
Turkel, E. and Vatsa, V.N., 1990, “Effect of Artificial Viscosity on Three-Dimensional Flow Solutions”, AIAA Journal, Vol. 32, No. 1, pp. 39-45.
van Albada, G.D., 1982, van Leer, B. and Roberts, W.W., 1982, “A Comparative Study of Computational Methods in Cosmic Gas Dynamics”, Astronomy and Astrophysics, Vol. 108, No. 1, pp. 76-84.
van Leer, B., 1979, “Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method”, Journal of Computational Physics, Vol. 32, No. 1, pp. 101-136.
Venkatakrishnan, V., 1995, “Implicit Schemes and Parallel Computing in Unstructured Grid CFD”, VKI Lecture Series, Rhode-Saint-Genèse, Belgium.
von Karman, T., 1934, “Turbulence and Skin Friction”, Journal of the Aeronautical Sciences, Vol. 1, No. 1, pp. 1-20.
Wallin, S. and Johansson, A.V., 2000, “An Explicit Algebraic Reynolds Stress Model For Incompressible And Compressible Turbulent Flows”, Journal of Fluid Mechanics, Vol. 403, pp. 89-132.
Wolf, W.R., 2006, “Simulation of Compressible Aerodynamic Flows Using Unstructured Grid, High-Order, Non-Oscillatory Schemes”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil (in Portuguese, original title is Simulação de Escoamentos Aerodinâmicos Compressíveis Utilizando Esquemas Não Oscilatórios de Alta Ordem de Precisão em Malhas Não Estruturadas).
Wolf, W.R. and Azevedo, J.L.F., 2006, “High-Order Unstructured Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Aerodynamic Flows”, AIAA Journal , Vol. 44, No. 10, pp. 2295-2310.
Wolf, W.R. and Azevedo, J.L.F., 2007, “High-Order ENO and WENO Schemes for Unstructured Grids”, International Journal for Numerical Methods in Fluids, Vol. 55, No. 10, pp. 917-943.
Wright, M.J., 1997, “A Family of Data Parallel Relaxation Methods for the Navier-Stokes Equations”, Ph.D. Thesis, University of Minnesota, Minneapolis, MN, USA.

Publication Dates

Publication in this collection
Apr-Jun 2013

History

Received
04 Oct 2012
Accepted
28 Jan 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[1] Anderson, J.D., 1991, “Fundamentals of Aerodynamics”, McGraw-Hill International Editions, New York, NY, USA.

[2] Anderson, W.K., Thomas, J.L. and van Leer, B., 1986, “A Comparison Of Finite Volume Flux Vector Splittings For The Euler Equations”, AIAA Journal, Vol. 24, No. 9, pp. 1453-1460.

[3] Azevedo, J.L.F., 1988, “Transonic Aeroelastic Analysis Of Launch Vehicle Configurations”, Ph.D. Thesis, Stanford University, Stanford, California, USA.

[4] Basso, E., 1997, “Numerical Analysis of Cascade Flow Problems for All Speed Regimes”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil (in Portuguese, original title is Análise Numérica de Escoamentos em Grades Lineares de Perfis para Vários Regimes de Velocidade).

[5] Batina, J., 1990, “Three-Dimensional Flux-Split Euler Schemes Involving Unstructured Dynamic Meshes”, AIAA Paper No. 90-1649, 21st AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Seattle, WA.

[6] Batten, P., Craft, T.J., Leschziner, M.A. and Loyau, H., 1999, “Reynolds-Stress-Transport Modeling for Compressible Aerodynamics Applications”, AIAA Journal, Vol. 37, No. 7, pp. 785-797.

[7] Bibb, K.L., Peraire, J. and Riley, C.J., 1997, “Hypersonic Flow Computations On Unstructured Meshes”, AIAA Paper No. 97-0625.

[8] Bigarella, E.D.V., 2007, “Advanced Turbulence Modeling for Complex Aerospace Applications”, Ph.D. Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil .

[9] Bigarella, E.D.V. and Azevedo, J.L.F., 2009, “Turbulence Modeling for Aerospace Applications, Turbulence”, Vol. 6 of Coleção Cadernos de Turbulência, Chapter 4, ABCM, Rio de Janeiro, RJ, 1st ed., pp. 215296 (in Portuguese, original title is Modelagem de Turbulência para Aplicações Aeroespaciais).

[10] Bigarella, E.D.V. and Azevedo, J.L.F., 2012, “A Study Of Convective Flux Schemes For Aerospace Flows”, Journal of the Brazilian Society for Mechanical Sciences and Engineering, Vol. 34, No. 3, pp. 314-329.

[11] Bigarella, E.D.V., Azevedo, J.L.F. and Scalabrin, L.C., 2007, “Centered and Upwind Multigrid Turbulent Flow Simulations of Launch Vehicle Configurations”, Journal of Spacecraft and Rockets, Vol. 44, No. 1, pp. 52-65.

[12] Breviglieri, C., 2010, “High-Order Unstructured Spectral Finite Volume Method for Aerodynamic Applications”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil .

[13] Breviglieri, C., Azevedo, J.L.F. and Basso, E., 2010a, “An Unstructured Grid Implementation Of High-Order Spectral Finite Volume Schemes”, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 32, No. 5 (Special Issue), pp. 419-433.

[14] Breviglieri, C., Azevedo, J.L.F., Basso, E. and Souza, M.A.F., 2010b, “Implicit High-Order Spectral Finite Volume Method For Inviscid Compressible Flows”, AIAA Journal, Vol. 48, No. 10, pp. 2365-2376.

[15] Chapman, D., 1981, “Trends and Pacing Items in Computational Aerodynamics, Seventh International Conference on Numerical Methods in Fluid Dynamics”, edited by W. Reynolds and R. MacCormack, Lecture Notes in Physics, Vol. 141, Springer, Heidelberg, pp. 1-11.

[16] Coles, D.E. and Hirst, E.A., 1969, “Computation Of Turbulent Boundary Layers”, Proceedings of the 1968 U.S. Air Force Office of Scientific Research-Internal Flow Program-Stanford Conference, Vol. 2, Stanford University Press, Palo Alto, CA, USA.

[17] Gnoffo, P.A., 2003, “Computational Aerothermodynamics In Aeroassist Applications”, Journal of Spacecraft and Rockets, Vol. 40, No. 3, pp. 305-312.

[18] Hellsten, A. and Laine, S., 2000, “Explicit Algebraic Reynolds-Stress Modelling In Decelerating And Separating Flows”, AIAA Paper No. 2000-2313.

[19] Hirsch, C., 1990, “Numerical Computation of Internal and External Flows, Vol. II: Computational Methods for Inviscid and Viscous Flows”, Wiley, New York, USA.

[20] Jameson, A., 1995a, “Analysis And Design Of Numerical Schemes For Gas Dynamics 1 - Artificial Diffusion, Upwind Biasing, Limiters And Their Effect On Accuracy And Multigrid Convergence”, International Journal of Computational Dynamics, Vol. 4, No. 3-4, pp. 171-218.

[21] Jameson, A., 1995b, “Analysis And Design Of Numerical Schemes For Gas Dynamics 2 - Artificial Diffusion And Discrete Shock Structure”, International Journal of Computational Dynamics, Vol. 5, No. 1-2, pp. 1-38.

[22] Jameson, A., Schmidt, W. and Turkel, E., 1981, “Numerical Solution Of The Euler Equations By Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes”, AIAA Paper No. 81-1259.

[23] Jawahar, P. and Kamath, H., 2000, “A High Resolution Procedure For Euler And Navier-Stokes Computation On Unstructured Grids”, Journal of Computational Physics, Vol. 164, No. 1, pp. 165-203.

[24] Junqueira-Junior, C.A., 2012, “A Study On The Extension Of An Upwind Parallel Solver For Turbulent Flow Applications”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, São Paulo, Brazil.

[25] Junqueira-Junior, C.A., Azevedo, J.L.F., Basso, E. and Scalabrin, L., 2011, “Study Of Robust And Efficient Turbulence Closures For Aerospace Applications”, 32nd Iberian Latin American Congress on Computational Methods in Engineering, Ouro Preto, Minas Gerais, Brazil.

[26] Lomax, H., Pulliam, T.H. and Zingg, D.W., 2001, “Fundamentals of Computation Fluid Dynamics”, Springer, New York, USA.

[27] MacCormack, R.W. and Candler, G.V., 1989, “The Solution of the Navier Stokes Equations Using Gauss-Seidel Line Relaxation”, Computer and Fluids, Vol. 17, No. 1, pp. 135-150.

[28] Mavriplis, D.J., 1988, “Accurate Multigrid Solution of the Euler Equations on Unstructured and Adaptive Meshes”, AIAA Journal, Vol. 28, No. 2, pp. 213-221.

[29] Ramalho, M.V.C., Azevedo, J.H.A. and Azevedo, J.L.F., 2011, “Further Investigation into the Origin of the Carbuncle Phenomenon in Aerodynamic Simulations”, AIAA Paper No. 2011-1184, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA.

[30] Roe, P.L., 1981, “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”, Journal of Computational Physics, Vol. 43, No. 2, pp. 357-372.

[31] Scalabrin, L.C., 2007, “Numerical Simulation of Weakly Ionized Hypersonic Flow Over Reentry Capsules”, Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan, USA.

[32] Scalabrin, L.C. and Boyd, I.D., 2007, “Numerical Simulations of the FIRE-II Convective and Radiative Heating Rates”, AIAA Paper No. 2007-4044.

[33] Schlichting, H., 1978, “Boundary-Layer Theory”, 7th ed., McGraw Hill, New York, NY, USA.

[34] Schwartzentruber, T.E., Scalabrin, L.C. and Boyd, I.D., 2007, “A Modular Particle-Continuum Numerical Method For Hypersonic Non-Equilibrium Gas Flows”, Journal of Computational Physics, Vol. 225, No. 1, pp. 1159-1174.

[35] Schwartzentruber, T.E., Scalabrin, L.C. and Boyd, I.D., 2008, “Hybrid Particle-Continuum Simulations Of Non-Equilibrium Hypersonic Blunt-Body Flowfields”, Journal of Thermophysics and Heat Transfer, Vol. 22, No. 1, pp. 29-37.

[36] Spalart, P.R. and Allmaras, S.R., 1992, “A One-Equation Turbulence Model For Aerodynamic Flows”, AIAA Paper No. 92-0439.

[37] Spalart, P.R. and Allmaras, S.R., 1994, “A One-Equation Turbulence Model For Aerodynamic Flows”, La Recherche Aerospatiale, Vol. 1, No. 1, pp. 5-21.

[38] Steger, J.L. and Warming, R.F., 1981, “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to the Finite-Difference Method”, Journal of Computational Physics, Vol. 40, No. 2, pp. 263-293.

[39] Tennekes, H. and Lumley, J.L., 1972, “A First Course In Turbulence”, The MIT Press, Cambridge, MA, USA.

[40] Turkel, E. and Vatsa, V.N., 1990, “Effect of Artificial Viscosity on Three-Dimensional Flow Solutions”, AIAA Journal, Vol. 32, No. 1, pp. 39-45.

[41] van Albada, G.D., 1982, van Leer, B. and Roberts, W.W., 1982, “A Comparative Study of Computational Methods in Cosmic Gas Dynamics”, Astronomy and Astrophysics, Vol. 108, No. 1, pp. 76-84.

[42] van Leer, B., 1979, “Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov’s Method”, Journal of Computational Physics, Vol. 32, No. 1, pp. 101-136.

[43] Venkatakrishnan, V., 1995, “Implicit Schemes and Parallel Computing in Unstructured Grid CFD”, VKI Lecture Series, Rhode-Saint-Genèse, Belgium.

[44] von Karman, T., 1934, “Turbulence and Skin Friction”, Journal of the Aeronautical Sciences, Vol. 1, No. 1, pp. 1-20.

[45] Wallin, S. and Johansson, A.V., 2000, “An Explicit Algebraic Reynolds Stress Model For Incompressible And Compressible Turbulent Flows”, Journal of Fluid Mechanics, Vol. 403, pp. 89-132.

[46] Wolf, W.R., 2006, “Simulation of Compressible Aerodynamic Flows Using Unstructured Grid, High-Order, Non-Oscillatory Schemes”, Master’s Thesis, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil (in Portuguese, original title is Simulação de Escoamentos Aerodinâmicos Compressíveis Utilizando Esquemas Não Oscilatórios de Alta Ordem de Precisão em Malhas Não Estruturadas).

[47] Wolf, W.R. and Azevedo, J.L.F., 2006, “High-Order Unstructured Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Aerodynamic Flows”, AIAA Journal , Vol. 44, No. 10, pp. 2295-2310.

[48] Wolf, W.R. and Azevedo, J.L.F., 2007, “High-Order ENO and WENO Schemes for Unstructured Grids”, International Journal for Numerical Methods in Fluids, Vol. 55, No. 10, pp. 917-943.

[49] Wright, M.J., 1997, “A Family of Data Parallel Relaxation Methods for the Navier-Stokes Equations”, Ph.D. Thesis, University of Minnesota, Minneapolis, MN, USA.

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