Incompressible Turbulent Flow Simulation Using the 
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Model and Upwind Schemes

Ferreira, V. G.; Brandi, A. C.; Kurokawa, F. A.; Seleghim Jr., P.; Castelo, A.; Cuminato, J. A.

doi:https://doi.org/10.1155/2007/12741

Mathematical Problems in Engineering

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Dynamics and Control in Sciences and Engineering

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Volume 2007 | Article ID 012741 | https://doi.org/10.1155/2007/12741

Incompressible Turbulent Flow Simulation Using the κ-ɛ Model and Upwind Schemes

V. G. Ferreira,¹A. C. Brandi,¹F. A. Kurokawa,¹P. Seleghim Jr.,²A. Castelo,¹and J. A. Cuminato¹

Academic Editor: José Manoel Balthazar

Received30 Nov 2006

Accepted25 Mar 2007

Published24 May 2007

Abstract

In the computation of turbulent flows via turbulence modeling, the treatment of the convective terms is a key issue. In the present work, we present a numerical technique for simulating two-dimensional incompressible turbulent flows. In particular, the performance of the high Reynolds κ-ɛ model and a new high-order upwind scheme (adaptative QUICKEST by Kaibara et al. (2005)) is assessed for 2D confined and free-surface incompressible turbulent flows. The model equations are solved with the fractional-step projection method in primitive variables. Solutions are obtained by using an adaptation of the front tracking GENSMAC (Tomé and McKee (1994)) methodology for calculating fluid flows at high Reynolds numbers. The calculations are performed by using the 2D version of the Freeflow simulation system (Castello et al. (2000)). A specific way of implementing wall functions is also tested and assessed. The numerical procedure is tested by solving three fluid flow problems, namely, turbulent flow over a backward-facing step, turbulent boundary layer over a flat plate under zero-pressure gradients, and a turbulent free jet impinging onto a flat surface. The numerical method is then applied to solve the flow of a horizontal jet penetrating a quiescent fluid from an entry port beneath the free surface.

References

B. P. Leonard, “Simple high-accuracy resolution program for convective modelling of discontinuities,” International Journal for Numerical Methods in Fluids, vol. 8, no. 10, pp. 1291–1318, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. Harten, “On a class of high resolution total-variation-stable finite-difference schemes,” SIAM Journal on Numerical Analysis, vol. 21, no. 1, pp. 1–23, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Varonos and G. Bergeles, “Development and assessment of a variable-order non-oscillatory scheme for convection term discretization,” International Journal for Numerical Methods in Fluids, vol. 26, no. 1, pp. 1–16, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. G. Ferreira, M. F. Tomé, N. Mangiavacchi et al., “High-order upwinding and the hydraulic jump,” International Journal for Numerical Methods in Fluids, vol. 39, no. 7, pp. 549–583, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. G. Ferreira, N. Mangiavacchi, M. F. Tomé, A. Castelo, J. A. Cuminato, and S. McKee, “Numerical simulation of turbulent free surface flow with two-equation $k - ε$ eddy-viscosity models,” International Journal for Numerical Methods in Fluids, vol. 44, no. 4, pp. 347–375, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Song, G. R. Liu, K. Y. Lam, and R. S. Amano, “On a higher-order bounded discretization scheme,” International Journal for Numerical Methods in Fluids, vol. 32, no. 7, pp. 881–897, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. A. Alves, P. J. Oliveira, and F. T. Pinho, “A convergent and universally bounded interpolation scheme for the treatment of advection,” International Journal for Numerical Methods in Fluids, vol. 41, no. 1, pp. 47–75, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. K. Kaibara, V. G. Ferreira, H. A. Navarro, J. A. Cuminato, A. Castelo, and M. F. Tomé, “Upwinding schemes for convection dominated problems,” in Proceedings of the 18th International Congress of Mechanical Engineering (COBEM '05), Ouro Preto, MG, Brazil, November 2005.
View at: Google Scholar
V. G. Ferreira, C. M. Oishi, F. A. Kurokawa et al., “A combination of implicit and adaptative upwind tools for the numerical solution of incompressible free surface flows,” Communications in Numerical Methods in Engineering, vol. 23, no. 6, pp. 419–445, 2007.
View at: Publisher Site | Google Scholar
A. Castello, M. F. Tomé, C. N. L. César, S. McKee, and J. A. Cuminato, “Freeflow: an integrated simulation system for three-dimensional free surface flows,” Journal of Computing and Visualization in Science, vol. 2, no. 4, pp. 199–210, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, vol. 6 of Course of Theoretical Physics, Butterworth-Heinemann, Newton, Mass, USA, 1975.
R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow, Springer Series in Computational Physics, Springer, New York, NY, USA, 1983.
View at: Zentralblatt MATH | MathSciNet
D. C. Wilcox, Turbulence Modeling for CFD, DCW Industries, La Caсada, Calif, USA, 1993.
D. L. Sondak and R. H. Pletcher, “Application of wall functions to generalized nonorthogonal curvilinear coordinate systems,” AIAA journal, vol. 33, no. 1, pp. 33–41, 1995.
View at: Google Scholar | Zentralblatt MATH
H. L. Norris and W. C. Reynolds, “Turbulent channel flow with a moving wavy boundary,” Tech. Rep. TR TF-7, Departament of Mechanics Engineering, Stanford University, Palo Alto, Calif, USA, 1980.
View at: Google Scholar
P. Bradshaw, Ed., Turbulence, vol. 12 of Topics in Applied Physics, Springer, Berlin, Germany, 2nd edition, 1978.
View at: MathSciNet
D. C. Wilcox, “Reassessment of the scale-determining equation for advanced turbulence models,” AIAA journal, vol. 26, no. 11, pp. 1299–1310, 1988.
View at: Google Scholar | Zentralblatt MATH
F. M. White, Viscous Fluid Flow, McGraw-Hill, New York, NY, USA, 1991.
F. Menter and T. Esch, “Elements of industrial heat transfer predictions,” in Proceedings of the 16th International Congress of Mechanical Engineering (COBEM '01), Ouro Uberlndia, MG, Brazil, November 2001.
View at: Google Scholar
B. P. Leonard, “The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection,” Computer Methods in Applied Mechanics and Engineering, vol. 88, no. 1, pp. 17–74, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
P. K. Sweby, “High resolution schemes using flux limiters for hyperbolic conservation laws,” SIAM Journal on Numerical Analysis, vol. 21, no. 5, pp. 995–1011, 1984.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
P. H. Gaskell and A. K. C. Lau, “Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm,” International Journal for Numerical Methods in Fluids, vol. 8, no. 6, pp. 617–641, 1988.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. F. Tomé and S. McKee, “GENSMAC: a computational marker and cell method for free surface flows in general domains,” Journal of Computational Physics, vol. 110, no. 1, pp. 171–186, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
V. G. Ferreira, Análise e Implementação de Esquemas de Convecção e Modelos de Turbulência para Simulação de Escoamentos Incompressíveis Envolvendo Superfícies Livres, M.S. thesis, Departament of Computer Science and Statistics, USP - University of São Paulo, São Carlos, Brazil, 2001.
S. Armfield and R. Street, “An analysis and comparison of the time accuracy of fractional-step methods for the Navier-Stokes equations on staggered grids,” International Journal for Numerical Methods in Fluids, vol. 38, no. 3, pp. 255–282, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
F. H. Harlow and J. E. Welch, “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface,” Physics of Fluids, vol. 8, no. 12, pp. 2182–2189, 1965.
View at: Publisher Site | Google Scholar
A. Chorin, “A numerical method for solving incompressible viscous flow problems,” Journal Computational of Physics, vol. 2, no. 1, pp. 12–26, 1967.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Eaton and J. P. Johnston, “Turbulent flow reattachment: an experimental study of the flow and structure behind a backward-facing step,” Tech. Rep. TR MD-39, Stanford University, Stanford, Calif, USA, 1980.
View at: Google Scholar
S. Thangam and C. G. Speziale, “Turbulent flow past a backward-facing step:a critical evaluation of two-equation models,” AIAA Journal, vol. 30, no. 5, pp. 1314–1320, 1992.
View at: Publisher Site | Google Scholar
A. C. Brandi, Estratégias “Upwind” e Modelagem $k - ε$ para Simulação Numérica de Escoamentos com Superfícies Livres em Altos Números de Reynolds, M.S. thesis, Departament of Computer Science and Statistics, USP - University of São Paulo, São Carlos, Brazil, 2005.
F. M. White, Fluid Mechanics, McGraw-Hill, New York, NY, USA, 1979.
View at: Zentralblatt MATH
E. J. Watson, “The radial spread of a liquid jet over a horizontal plane,” Journal of Fluid Mechanics, vol. 20, pp. 481–499, 1964.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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Copyright © 2007 V. G. Ferreira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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