Geodesic
On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers
Matematičeskoe modelirovanie i čislennye metody, no. 8 (2015), pp. 92-109 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The research explored questions of the convergence of iterative processes and correctness of the solutions on the example of the problem about a steady-state flat square lid-driven cavity flow of incompressible viscous liquid. The problem is solved for Reynolds numbers of $15\,000< Re < 20\,000$ and steps of grid $1/128 > h > 1/2048$. The findings of the research illustrate that not for all relationships between Re and h the convergence of iterative processes is stable and the resulting steady-state solutions are qualitatively correct. We conducted a qualitative analysis of the solutions of the problem in the coordinate system $(Re, 1/h)$ in terms of the convergence of iterative process, solution correctness and the required computing time. According to the literature and the results of systematic calculations we conclude that the stability of the convergence of iterative process on the coarse grid depends on the degree of influence of the artificial viscosity and/or the condition number of the matrix of difference elliptical linear algebraic equations, and on the detailed grid it depends on the grid Reynolds number. At high Reynolds numbers steady calculations can be carried out either on very coarse grids, or on very detailed ones. The width of the zone of instability in terms of parameter $1/h$ increases with increasing Reynolds number. Since the coarse grid solution is incorrect, and the use of detailed grid leads to very high costs of computer time, the further increase of the Reynolds number in the problem is associated with increasing the order of approximation of the differential equations.
Keywords: Navier–Stokes equations, a lid-driven cavity flow, a convergence of iterative process. 
@article{MMCM_2015_8_a5,
     author = {A. A. Fomin and L. N. Fomina},
     title = {On stationary solution of the problem of an incompressible viscous fluid at high {Reynolds} numbers},
     journal = {Matemati\v{c}eskoe modelirovanie i \v{c}islennye metody},
     pages = {92--109},
     year = {2015},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMCM_2015_8_a5/}
}
TY  - JOUR
AU  - A. A. Fomin
AU  - L. N. Fomina
TI  - On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers
JO  - Matematičeskoe modelirovanie i čislennye metody
PY  - 2015
SP  - 92
EP  - 109
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/MMCM_2015_8_a5/
LA  - ru
ID  - MMCM_2015_8_a5
ER  - 
%0 Journal Article
%A A. A. Fomin
%A L. N. Fomina
%T On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers
%J Matematičeskoe modelirovanie i čislennye metody
%D 2015
%P 92-109
%N 8
%U http://geodesic.mathdoc.fr/item/MMCM_2015_8_a5/
%G ru
%F MMCM_2015_8_a5
A. A. Fomin; L. N. Fomina. On stationary solution of the problem of an incompressible viscous fluid at high Reynolds numbers. Matematičeskoe modelirovanie i čislennye metody, no. 8 (2015), pp. 92-109. http://geodesic.mathdoc.fr/item/MMCM_2015_8_a5/

[1] Burggraf O.R., “Analytical and numerical studies of the structure of steady separated flows”, J. of Fluid Mechanics, 24 (1966), 113–151

[2] Ghia U., Ghia K.N., Shin C.T., “High-Re solution for incompressible flow using the Navier — Stokes equations and a multigrid method”, J. of Computational Physics, 48 (1982), 387–411

[3] Bruneau C.-H., Jouron C., “An efficient scheme for solving steady incompressible Navier Stokes equations”, J. of Computational Physics, 89 (1990), 389–413 | DOI

[4] Barragy E., Carey G.F., “Stream function-vorticity driven cavity solution using p finite elements”, Computers Fluids, 26:5 (1997), 453–468

[5] Marinova R.S., Christov C.I., Marinov T.T., “A fully coupled solver for incompressible Navier-Stokes equations using operator splitting”, Int. J. of Computational Fluid Dynamics, 17:5 (2003), 371–385 | DOI

[6] Bruneau C.-H., Saad M., “The 2D lid-driven cavity problem revisited”, Computers Fluids, 35 (2006), 326–348

[7] Kumar D.S., Kumar K.S., Kumar M.D., “A fine grid solution for a lid-driven cavity flow using multigrid method”, Engineering Applications of Computational Fluid Mechanics, 3:3 (2009), 336–354

[8] Erturk E., Corke T.C., Gokcol C., “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers”, Int. J. for Numerical Methods in Fluids, 48 (2005), 747–774

[9] Cardoso N., Bicudo P., “Time dependent simulation of the Driven Lid Cavity at High Reynolds Number”, physics.fly-dyn, 2009, 1–20 arxiv.org/pdf/0809.3098.pdf

[10] Erturk E., Gokcol C., “Fourthorder compact formulation of Navier — Stokes equations and driven cavity flow at high Reynolds numbers”, Int. J. for Numerical Methods in Fluids, 2006, no. 50, 421–436

[11] Wahba E.M., “Steady flow simulation inside a driven cavity up to Reynolds number 35000”, Computers Fluids, 66 (2012), 85–97

[12] Basarab M.A., Mathematical Modeling and Computational Methods, 2014, no. 1, 18–35

[13] Fomin A.A., Fomina L.N., Computer Research and Modeling, 7:1 (2015), 35–50

[14] Belotserkovskiy O.M., Gushchin V.A., Shchennikov V.V., Math. Math. Phys., 15:1 (1975), 197–207

[15] Patankar S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Co, 1980, 197 pp.

[16] Fomin A.A., Fomina L.N., Tomsk State University J. of Mathematics and Mechanics, 2011, no. 14, 45–54

[17] Erturk E., “Discussions on driven cavity flow”, Int. J. for Numerical Methods in Fluids, 60 (2009), 275–294

[18] Roache P.J., Computational Fluid Dynamics, Hermosa Publs, Albuquerque, 1976, 446 pp.