On boundary conditions and numerical methods for the unsteady incompressible Navier-Stokes equations. (English) Zbl 0977.76062
Summary: We discuss various numerical methods and boundary conditions to solve the unsteady incompressible Navier-Stokes equations. It is shown, via the analysis of governing partial differential equations and discretized algebraic equations, that boundary conditions necessary to solve the incompressible Navier-Stokes equations are conditions either for normal and tangential components of velocities, or alternatively for normal velocity and tangential vorticity components. In an explicit formulation, the solution of the resulting system of algebraic equations is more efficiently by solving the Poisson equation either for pressure or for velocity. The boundary conditions necessary to solve these Poisson equations are provided from velocity boundary conditions and momentum equations. Finally, we present numerical simulations of selected unsteady flows.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
unsteady incompressible Navier-Stokes equations; boundary conditions; Poisson equation; pressure; velocityReferences:
| [1] | Fletcher, C. A.J., Computational techniques for fluid dynamics 2 (1991), Springer-Verlag: Springer-Verlag New York, Berlin · Zbl 0706.76001 |
| [2] | Gresho, P. M.; Sani, R. L., On pressure boundary conditions for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 7, 1111-1145 (1987) · Zbl 0644.76025 |
| [3] | Gresho, P. M., Incompressible fluid dynamics: some fundamental formulation issues, Annual Review of Fluid Mechanics, 23, 413-453 (1991) · Zbl 0717.76006 |
| [4] | Orszag, S. A.; Israeli, M., Numerical simulation of viscous incompressible flows, Annual Review of Fluid Mechanics, 6, 281-318 (1974) · Zbl 0295.76016 |
| [5] | Lighthill, M. J., Introduction. Boundary layer theory., (Rosenhead, L., Laminar boundary layers (1963), Clarendon Press: Clarendon Press Oxford), Chapter II · Zbl 0038.11504 |
| [6] | Lamb, H., Hydrodynamics (1945), Dover: Dover New York · JFM 26.0868.02 |
| [7] | Gatski, T. B.; Grosch, C. E.; Rose, M. E., A numerical study of the two-dimensional equations in vorticity-velocity variables, Journal of Computational Physics, 48, 1-22 (1982) · Zbl 0502.76040 |
| [8] | Gatski, T. B.; Grosch, C. E.; Rose, M. E., The numerical solution of the Navier-Stokes equations for 3-dimensional, unsteady, incompressible flows by compact schemes, Journal of Computational Physics, 82, 298-329 (1989) · Zbl 0667.76051 |
| [9] | Koh Y-M. Numerical solution of three-dimensional vorticity transport equations. PhD thesis, Department of Aeronautics, Imperial College, University of London, 1987; Koh Y-M. Numerical solution of three-dimensional vorticity transport equations. PhD thesis, Department of Aeronautics, Imperial College, University of London, 1987 |
| [10] | Koh, Y.-M.; Bradshaw, P., Numerical solution of two-dimensional or axi-symmetric incompressible flow using the vorticity equations, KSME Journal, 8, 264-276 (1994) |
| [11] | Gresho PM, Sani RL. Incompressible flow and the finite element method. Wiley, New York 1998; Gresho PM, Sani RL. Incompressible flow and the finite element method. Wiley, New York 1998 · Zbl 0941.76002 |
| [12] | Koh, Y.-M., The pressure distribution around unsteady boundary layers, Journal of Fluid Mechanics, 255, 437-446 (1993) · Zbl 0784.76023 |
| [13] | Jeffreys, H. H.; Jeffreys, B. S., Methods of mathematical physics (1978), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0037.31704 |
| [14] | Batchelor, G. K., An introduction to fluid dynamics (1970), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0152.44402 |
| [15] | Koh, Y.-M., Pressure distribution and flow development in unsteady incompressible laminar boundary layers, KSME Journal, 7, 213-222 (1993) |
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.
