Stability Investigation of a Difference Scheme

1
Stability Investigation of a Difference Scheme
for Incompressible NavierStokes Equations
D. Chibisov, V. Ganzha, E.W. Mayr, E.V. Vorozhtsov
2
2 Governing Equations and Difference Method

• Difference scheme of Kim and Moin (J. Comp.
  Phys., Vol. 59 (1985), p. 308-323

The NavierStokes equations for 2D incompressible
fluid flows
The staggered grid in two dimensions
The 2nd fractional step
D ? div
3
The available empirical stability condition of
the Moin-Kim scheme
where
3 Fourier Symbol
Linearization of NavierStokes equations
Linearized difference scheme
4
The von Neumann necessary stability condition
4 Analytic Investigation of Eigenvalues
Case 1
The scheme is absolutely stable
Case 2
The scheme is weakly unstable
5
Case 3 all kappas are different from zero.
Möbius transformation
Implementation of the above mathematical
procedure with Mathematica
As a result, the following formula for the
resultant was obtained
6
The particular case ? ?
Root of equation
7
Another particular case
(high Reynolds numbers)
Fig. 4. The surface t t(a,b)
5 The Method of Discrete Perturbation
The behavior of a and ß agrees with that obtained
by the Fourier method.
8
6 Verification of Stability Conditions 6.1 The
TaylorGreen Vortex
The analytic solution of the NavierStokes
equations, with ? ? 1, is given by formulas
30?30 grid
The new formula for time step
9
6 Verification of Stability Conditions 6.2
Lid-Driven Cavity Problem
30?30 grid
from 33 to 58
a) Re 1, ? 3 stable for
b) Re 400, ? lt 0.1 stable for