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Local Collocation Approach for Solving Turbulent Combined Forced and Natural Convection Problems

  • Robert Vertnik (a1) and Božidar Šarler (a2)

Abstract

An application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) in solution of incompressible turbulent combined forced and natural convection is for the first time explored in the present paper. The turbulent flow equations are described by the low-Re number к – ε model with Launder and Sharma and Abe et al. closure coefficients. The involved temperature, velocity, pressure, turbulent kinetic energy and dissipation fields are represented on overlapping 5-noded sub-domains through the collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second order partial derivatives of the fields are calculated from the respective derivatives of the RBF’s. The involved equations are solved through the explicit time stepping. The pressure-velocity coupling is based on Chorin’s fractional step method. The adaptive upwinding technique, proposed by Lin and Atluri , is used because of the convection dominated situation. The solution procedure is represented for a 2D upward channel flow with differentially heated walls. The results have been assessed by achieving a reasonable agreement with the direct numerical simulation of Kasagi and Nishimura for Reynolds number 4494, based on the channel width, and Grash of number 9.6×105. The advantages of the represented mesh-free approach are its simplicity, accuracy, similar coding in 2D and 3D, and straightforward applicability in non-uniform node arrangements.

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Corresponding author

Corresponding author. URL: www.ung.si, Email: robert.vertnik@ung.si

References

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[1] Abe, K., Kondoh, T. and Nagano, Y., A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows-II, thermal field calculations, Int. J. Heat. Mass. Trans., 38 (1995), pp. 14671481.
[2] Ampofo, F. and Karayiannis, T. G., Experimental benchmark data for turbulent natural convection in an air filled square cavity, Int. J. Heat. Mass. Trans., 46 (2003), pp. 35513572.
[3] Atluri, S. N. and Shen, S., The Meshless Method, Tech Science Press, Encino, 2002.
[4] Atluri, S. N., The Meshless Method (MLPG) for Domain and BIE Discretization, Tech Science Press, Forsyth, 2004.
[5] Atluri, S. N. and ŠArler, B., Proceedings of the 5th ICCES Symposium on Meshless Methods, University of Nova Gorica Press, Nova Gorica, 2009.
[6] Billard, F., Uribe, J. C. and Laurence, D., A new formulation of the v2 - f model using elliptic blending and its application to heat transfer prediction, Proceedings of 7th International Symposium on Engineering Turbulence Modelling and Measurements, Editors: Leschziner, M. A., pp. 8994, Limassol, Cyprus, June 4-6, 2008.
[7] Bredberg, J., On two-equation Eddy-Viscosity models, Internal Report 01/8, Department of Thermo and Fluid Dynamics, Chalmers University of Technology, Göteborg, Sweden, 2001.
[8] Breitkopf, P. and Huerta, A., Meshfree and Particle Based Approaches in Computational Mechanics, Kogan Page Science, London, 2003.
[9] Buhmann, M. D., Radial Basis Functions, Cambridge University Press, Cambridge, 2000.
[10] Chen, Y. T., Nie, J. H., Armaly, B. F. and Hsieh, H. T., Turbulent separated convection flow adjacent to backward-facing step-effects of step height, Int. J. Heat. Mass. Trans., 49 (2006), pp. 36703680.
[11] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Com-put. Phys., 2 (1967), pp. 1226.
[12] Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishers, Singapore, 2007.
[13] Ferreira, A. J. M., Kansa, E. J., Fasshauer, G. E. and Leitāao, V. M. A., Progress on Meshless Methods, Computational Methods in Applied Sciences, Vol. 11, Springer, Berlin, 2009.
[14] Divo, E. and Kassab, A. J., An efficient localized RBF meshless method for fluid flow and conjugate heat transfer, ASME J. Heat. Trans., 129 (2007), pp. 124136.
[15] Gu, Y. T. and Liu, G. R., Meshless technique for convection dominated problems, Comput. Mech., 38 (2005), pp. 171182.
[16] Henkes, R. A. W. M., Vlugt, F. F.Van der and Hoogendoorn, C. J., Natural-convection flow in a square cavity calculated with low-Reynolds-number turbulence models, Int. J. Heat. Mass. Trans., 34 (1991), pp. 377388.
[17] Hsieh, K. J. and Lien, F. S., Numerical modelling of buoyancy-driven turbulent flows in enclosures, Int. J. Heat. Fluid. Flow., 25 (2004), pp. 659670.
[18] Kansa, E. J., Multiquadrics-a scattered data approximation scheme with application to computational fluid dynamics, I-surface approximations and partial derivative estimates, Comput. Math. Appl., 19 (1990), pp. 127145.
[19] Kansa, E. J., Multiquadrics-a scattered data approximation scheme with application to computational fluid dynamics, II-solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19 (1990), pp. 147161.
[20] Kasagi, N., and Nishimura, M., Direct numerical simulation of combined forced and natural turbulent convection in a vertical plane channel, Int. J. Heat. Fluid. Flow., 18 (1997), pp. 8899.
[21] Keshmiri, A., Addad, Y., Cotton, M. A., Laurence, D. R. and Billard, F., Refined eddy viscosity schemes and large eddy simulations for ascending mixed convection flows, Proceedings of CHT-08 ICHMT International Symposium on Advances in Computational Heat Transfer, CHT-08-407, Marrakesh, Morocco, May 11-16, 2008.
[22] Kosec, G. and Šarler, B., Solution of heat transfer and fluid flow problems by the simplified explicit local radial basis function collocation method, Int. J. Numer. Methods. Heat. Fluid. Flow., 18 (2008), pp. 868882.
[23] Launder, B. E. and Sharma, B. I., Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc, Lett. Heat. Mass. Trans., 1 (1974), pp. 131138.
[24] Le, H., Moin, P. and Kim, J., Direct numerical simulation of turbulent flow over a backward-facing step, J. Fluid. Mech., 330 (1997), pp. 349374.
[25] Lee, C. K., Liu, X., and Fan, S. C., Local multiquadric approximation for solving boundary value problems, Comput. Mech., 30 (2003), pp. 396409.
[26] Li, S. and Liu, W. K., Meshfree Particle Methods, 2nd Corrected Printing, Springer Verlag, Berlin, 2007.
[27] Lin, H. and Atluri, S. N., Meshless local Petrov-Galerkin (MLPG) method for convection diffusion, Comput. Model. Eng. Sci., 1 (2000), pp. 4560.
[28] Liu, G. R. and Gu, Y. T., An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht, 2005.
[29] Liu, G. R., Mesh Free Methods, Second Edition, CRC Press, Boca Raton, 2009.
[30] Šarler, B. and Vertnik, R., Meshfree local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 51 (2006), pp. 12691282.
[31] Šarler, B., From global to local radial basis function collocation method for transport phenomena, in: Leitao, V. M. A., Alves, C. J. S., and Armando-Duarte, C., Advances in Meshfree Techniques (Computational Methods in Applied Sciences, Vol. 5), Springer Verlag, Dordrecht, 2007, pp. 257282.
[32] Vertnik, R. and Šarler, B., Meshless local radial basis function collocation method for con-vective diffusive solid-liquid phase change problems, Int. J. Numer. Methods. Heat. Fluid. Flow., 16 (2006), pp. 617640.
[33] Vertnik, R. and Šarler, B., Solution of incompressible turbulent flow by a mesh-free method, Comput. Model. Eng. Sci., 44 (2009), pp. 6695.
[34] Vertnik, R. and Šarler, B., Simulation of turbulent flow and heat transfer in continuous casting of billets by a meshless method, in: Ludwig, A., editor, Proceedings of the 3rd Steelsim Conference, September 8-10, 2009, Leoben, Austria, ASMET, CC2, CD-ROM.
[35] Wang, B. C., Yee, E., Yin, J. and Bergstrom, D. J., A general dynamic linear tensor-diffusivity subgrid-scale heat flux model for large-eddy simulation of turbulent thermal flows, Numer. Heat. Trans., 51B (2007), pp. 205227.
[36] Wilcox, D. C., Turbulence modeling for CFD, DCW Industries, Inc., California, 1993.
[37] Yilmaz, T. and Fraser, S. M., Turbulent natural convection in a vertical parallel-plate channel with asymmetric heating, Int. J. Heat. Mass. Trans., 50 (2007), pp. 26122623.
[38] Yin, J. and Bergstrom, D. J., LES of combined forced and natural turbulent convection in a vertical slot, Computational Fluid Dynamics 2004, Springer Berlin Heidelberg, pp. 567572.
[39] Yoder, D. A. and Georgiadis, N. J., Implementation and validation of the Chien k - ε turbulence model in the Wind Navier-Stokes code, NASA Technical Memorandum 209080, 1999.
[40] Zhou, Y., Zhang, R., Staroselsky, I. and Chen, H., Numerical simulation of laminar and turbulent buoyancy-driven flows using a lattice Boltzmann based algorithm, Int. J. Heat. Mass. Trans., 47 (2004), pp. 48694879.

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