Stationary solutions of incompressible viscous flow in a wall-driven semi-circular cavity

Authors

DOI:

https://doi.org/10.14311/AP.2021.61.0516

Keywords:

driven semi-circular cavity flow, large Reynolds number flow, numerical solutions, 2-D incompressible viscous flow

Abstract

Stationary numerical solutions of incompressible viscous flow inside a wall-driven semicircular cavity are presented. After a conformal mapping of the geometry, using a body-fitted mesh, the Navier-Stokes equations are solved numerically. The stationary solutions of the flow in a wall-driven semi-circular cavity are computed up to Re = 24000. The present results are in good agreement with the published results found in the literature. Our results show that as the Reynolds number increases, the sizes of the secondary and tertiary vortices increase, whereas the size of the primary vortex decreases. At large Reynolds numbers, the vorticity at the primary vortex centre increases almost linearly stating that Batchelor’s mean-square law is not valid for wall-driven semi-circular cavity flow. Detailed results are presented and also tabulated for future references and benchmark purposes.

References

U. Ghia, K. Ghia, C. Shin. High-re solutions for incompressible flow using the navier-stokes equations and a multigrid method. Journal of Computational Physics 48(3):387–411, 1982. https://doi.org/10.1016/0021-9991(82)90058-4.

E. Erturk. Discussions on driven cavity flow. International Journal for Numerical Methods in Fluids 60(3):275–294, 2009. https://doi.org/10.1002/fld.1887.

B. Bastl, M. Brandner, J. Egermaier, et al. Numerical simulation of lid-driven cavity flow by isogeometric analysis. Acta Polytechnica 61(SI):33–48, 2021. https://doi.org/10.14311/AP.2021.61.0033.

I. Demirdžić, Ž. Lilek, M. Perić. Fluid flow and heat transfer test problems for non-orthogonal grids: Bench-mark solutions. International Journal for Numerical Methods in Fluids 15(3):329–354, 1992. https://doi.org/10.1002/fld.1650150306.

E. Erturk, B. Dursun. Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity. ZAMM - Journal of Applied Mathematics and Mechanics 87(5):377–392, 2007. https://doi.org/10.1002/zamm.200610322.

C. Ribbens, L. Watson, C.-Y. Wang. Steady viscous flow in a triangular cavity. Journal of Computational Physics 112(1):173–181, 1994. https://doi.org/10.1006/jcph.1994.1090.

P. Gaskell, H. Thompson, M. Savage. A finite element analysis of steady viscous flow in triangular cavities. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 213(3):263–276, 1999. https://doi.org/10.1243/0954406991522635.

E. Erturk, O. Gokcol. Fine grid numerical solutions of triangular cavity flow. The European Physical Journal - Applied Physics 38(1):97–105, 2007. https://doi.org/10.1051/epjap:2007057.

W. McQuain, C. Ribbens, C.-Y. Wang, L. Watson. Steady viscous flow in a trapezoidal cavity. Computers & Fluids 23(4):613–626, 1994. https://doi.org/10.1016/0045-7930(94)90055-8.

R. Glowinski, G. Guidoboni, T.-W. Pan. Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity. Journal of Computational Physics 216(1):76–91, 2006. https://doi.org/10.1016/j.jcp.2005.11.021.

F. Yang, X. Shi, X. Guo, Q. Sai. MRT lattice Boltzmann schemes for high Reynolds number flow in two-dimensional lid-driven semi-circular cavity. Energy Procedia 16:639–644, 2012. https://doi.org/10.1016/j.egypro.2012.01.103.

F. Yang, L. Liu, X. Shi, X. Guo. Lattice BGK study on flow pattern in two-dimensional wall-driven semi-circular cavity. Advanced Materials Research 354-355:549–598, 2011. https://doi.org/10.4028/www.scientific.net/AMR.354-355.594.

L. Ding, W. Shi, H. Luo, H. Zheng. Investigation of incompressible flow within ½ circular cavity using lattice Boltzmann method. International Journal for Numerical Methods in Fluids 60(8):919–936, 2009. https://doi.org/10.1002/fld.1925.

P. Yu, Z. Tian. A compact scheme for the streamfunction-velocity formulation of the 2D steady incompressible Navier-Stokes equations in polar coordinates. Journal of Scientific Computing 56:165–189, 2013. https://doi.org/10.1007/s10915-012-9667-7.

H. Mercan, K. Atalık. Vortex formation in lid-driven arc-shape cavity flows at high Reynolds numbers. European Journal of Mechanics - B/Fluids 28(1):61–71, 2009. https://doi.org/10.1016/j.euromechflu.2008.02.001.

C. Migeon, A. Texier, G. Pineau. Effects of lid-driven cavity shape on the flow establishment phase. Journal of Fluids and Structures 14(4):469–488, 2000. https://doi.org/10.1006/jfls.1999.0282.

E. Erturk, O. Gokcol. Numerical solutions of steady incompressible flow around a circular cylinder up to Reynolds number 500. International Journal of Mechanical Engineering and Technology 9(10):1368–1378, 2018.

C. Christov, R. Marinova, T. Marinov. Identifying the stationary viscous flows around a circular cylinder at high Reynolds numbers. Lecture Notes in Computer Science 4818:175–183, 2008. https://doi.org/10.1007/978-3-540-78827-0_18.

H. Schlichting, K. Gersten. Boundary Layer Theory (8th revised and enlarged edn). Springer-Verlag Berlin Heidelberg, New York, USA, 2000.

F. Smith. A structure for laminar flow past a bluff body at high Reynolds number. Journal of Fluid Mechanics 155:175–191, 1985. https://doi.org/10.1017/S0022112085001768.

D. Peregrine. A note on the steady high-Reynolds-number flow about a circular cylinder. Journal of Fluid Mechanics 157:493–500, 1985. https://doi.org/10.1017/S0022112085002464.

B. Fornberg. A numerical study of steady viscous flow past a circular cylinder. Journal of Fluid Mechanics 98(4):819–855, 1980. https://doi.org/10.1017/S0022112080000419.

B. Fornberg. Steady viscous flow past a circular cylinder up to Reynolds number 600. Journal of Computational Physics 61(2):297–320, 1985. https://doi.org/10.1016/0021-9991(85)90089-0.

J. Son, T. Hanratty. Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200 and 500. Journal of Fluid Mechanics 35(2):369–386, 1969. https://doi.org/10.1017/S0022112069001169.

S. Tuann, M. Olson. Numerical studies of the flow around a circular cylinder by a finite element method. Computers and Fluids 6(4):219–240, 1978. https://doi.org/10.1016/0045-7930(78)90015-4.

S. Dennis, G. Chang. Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. Journal of Fluid Mechanics 42(3):471–489, 1970. https://doi.org/10.1017/S0022112070001428.

C. Christov, R. Marinova, T. Marinov. Does the stationary viscous flow around a circular cylinder exist for large Reynolds numbers? a numerical solution via variational imbedding. Journal of Computational and Applied Mathematics 226(2):205–217, 2009. https://doi.org/10.1016/j.cam.2008.08.022.

H. Lomax, J. Steger. Relaxation methods in fluid mechanics. Annual Review of Fluid Mechanics 7(1):63–88, 1975. https://doi.org/10.1146/annurev.fl.07.010175.000431.

G. Batchelor. On steady laminar flow with closed streamlines at large Reynolds number. Journal of Fluid Mechanics 1(2):177–190, 1956. https://doi.org/10.1017/S0022112056000123.

O. Burggraf. Analytical and numerical studies of the structure of steady separated flows. Journal of Fluid Mechanics 24(1):113–151, 1966. https://doi.org/10.1017/S0022112066000545.

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2021-08-31

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