×

Third-order upwind finite element solutions of high Reynolds number flows. (English) Zbl 0845.76045

Summary: A full upwind finite element scheme based on the Petrov-Galerkin formulation is presented for accurate solutions of high Reynolds number flows, and then a simplified finite scheme with third-order upwinding is developed to obtain a practical algorithm for numerical simulation, because structures of the full upwind finite element scheme are very complicated. Numerical dissipation added in the proposed upwind scheme is represented by high-order derivatives of flow velocities and pressure. Computational results for the subcritical and the supercritical Reynolds number flows around a circular cylinder and for flows around twin circular cylinders are shown to illustrate the effectiveness and robustness of the simplified upwind finite element scheme.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kondo, N.; Tosaka, N.; Nishimura, T., Numerical simulation of viscous flows by the third-order upwind finite element method, (Theoretical and Applied Mechanics, 39 (1990), University of Tokyo Press), 237-250 · Zbl 0745.76039
[2] Kondo, N.; Tosaka, N.; Nishimura, T., Third-order upwind finite element formulations for incompressible viscous flow problems, Comput. Methods Appl. Mech. Engrg., 93, 169-187 (1991) · Zbl 0747.76066
[3] Kondo, N.; Tosaka, N.; Nishimura, T., Computation of incompressible viscous flows by the third-order upwind finite element method, Internat. J. Numer. Methods Fluids, 15, 1013-1024 (1992) · Zbl 0762.76056
[4] Christie, L.; Griffiths, D. F.; Mitchell, A. R.; Zienkiewicz, O. C., Finite element methods for second order differential equations with significant first derivative, Internat. J. Numer. Methods Engrg., 10, 1389-1396 (1976) · Zbl 0342.65065
[5] Heinrich, J. C.; Huyakorn, P. S.; Zienkiewicz, O. C.; Michell, A. R., A ‘upwind’ finite element scheme for two-dimensional convective transport equations, Internat. J. Numer. Methods Engrg., 11, 131-143 (1977) · Zbl 0353.65065
[6] Zienkiewicz, O. C.; Heinrich, J. T., The finite element method and convective problems in fluid mechanics, (Gallagher, R. H.; etal., Finite Elements in Fluids, 3 (1978), Wiley: Wiley New York), 1-22
[7] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with particular emphasis of the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[8] Devloo, P.; Oden, J. T.; Strouboulis, T., Implementation of an adaptive refinement technique for the SUPG algorithm, Comput. Methods Appl. Mech. Engrg., 61, 339-358 (1987) · Zbl 0596.73066
[9] Tezduyar, T. E.; Ganjoo, D. K., Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 59, 49-71 (1986) · Zbl 0604.76077
[10] Donea, J., A. Taylor-Galerkin method for convective transport problem, Internat. J. Numer. Methods Engrg., 20, 101-119 (1984) · Zbl 0524.65071
[11] Donea, J.; Giuliani, S.; Laval, H.; Quartapelle, L., Time-accurate solution of advection-diffusion problems by finite elements, Comput. Methods Appl. Mech. Engrg., 45, 123-145 (1984) · Zbl 0514.76083
[12] Selmin, V.; Donea, J.; Quartapelle, L., Finite element methods for nonlinear advection, Comput. Methods Appl. Mech., 52, 817-845 (1985) · Zbl 0573.76005
[13] Laval, H.; Quartapelle, L., A fractional-step Taylor-Galerkin method for unsteady incompressible flows, Internat. J. Numer. Methods Engrg., 11, 501-513 (1990) · Zbl 0711.76019
[14] Kawamura, T.; Kuwahara, K., Computation of high Reynolds number flow around a circular cylinder with surface roughness, (AIAA-84-0340, AIAA 22nd Aerospace Science Meeting (1984))
[15] Leonard, B. P., A survey of finite differences with upwinding for numerical modeling of the incompressible convective diffusion equation, (Taylor, C.; Morgan, K., Comput. Tech. Trans. Turb. Flow (1981), Pineridge: Pineridge Swansea), 1-35
[16] Leonard, B. P., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Engrg., 19, 59-98 (1979) · Zbl 0423.76070
[17] Leonard, B. P., Simple high-accuracy resolution program for convective modelling of discontinuities, Internat. J. Numer. Methods Fluids, 8, 1291-1318 (1988) · Zbl 0667.76125
[18] Donea, J.; Giuliani, S.; Laval, H.; Quartapelle, L., Finite element solution of the unsteady Navier-Stokes equations by a fractional step method, Comput. Methods Appl. Mech. Engrg., 30, 53-73 (1982) · Zbl 0481.76037
[19] Thompson, J. F.; Marsi, Z. U.A.; Mastin, C. W., Numerical Grid Generation, Foundation and Application (1985), North-Holland: North-Holland Amsterdam
[20] Tamura, T.; Kuwahara, K., Direct finite difference computation of turbulent flow around a circular cylinder, (Proc. Internat. Symp. on Computational Fluid Dynamics (1989)), 701-706
[21] Cantwell, B.; Coles, D., An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J. Fluid Mech., 136, 321-374 (1983)
[22] Tsuboi, K.; Tamura, T.; Kuwahara, K., Numerical study for vortex induced vibration of a circular cylinder in high-Reynolds-number flow, (AIAA-89-0294, AIAA 27th Aerospace Science Meeting (1989))
[23] Fletcher, C. A.J., (Computational Techniques for Fluid Dynamics, Vols. 1 and 2 (1988), Springer: Springer Berlin) · Zbl 0706.76001
[24] Kondo, N., Direct third-order upwind finite element simulation of high Reynolds number flows around a circular cylinder, (Proc. 1st Internat. Symp. on Computational Wind Engineering. Proc. 1st Internat. Symp. on Computational Wind Engineering, J. Wind Engrg. (1992)), 321-326
[25] Donea, J.; Quartapelle, L., An introduction to finite element methods for transient advection problems, Comput. Methods Appl. Mech. Engrg., 95, 169-203 (1992) · Zbl 0772.76035
[26] Kakuda, N.; Tosaka, N., Numerical simulation of high Reynolds number flows by Petrov-Galerkin finite element method, (Proc. 1st Internat. Symp. on Computational Wind Engineering. Proc. 1st Internat. Symp. on Computational Wind Engineering, J. Wind Engrg. (1992)), 315-320
[27] Chang, K. S.; Song, C. J., Interactive vortex shedding from a pair of circular cylinders in a transverse arrangement, Internat. J. Numer. Methods Fluids, 11, 317-329 (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.