Stability Investigation of a Difference Scheme
for Incompressible Navier—Stokes Equations
D. Chibisov, V. Ganzha, E.W. Mayr, E.V. Vorozhtsov
2 Governing Equations and Difference Method
•
The Navier—Stokes equations
for 2D incompressible fluid flows:
The staggered grid in two dimensions
Difference scheme of Kim and Moin (J.
Comp. Phys., Vol. 59 (1985), p. 308-323
H (vn )  (v)v, G  grad, L  
The 2nd fractional step:
D  div
p n 1  p n  p
The available empirical stability condition of the Moin-Kim scheme
1   2

 3
 1,
Cconv
Cdiff
h
u
v


where 1  ,  2  ,  3 
  3 1   42 ,  3 
, 4  1
h1
h2
h2
h12  h22
h12


3 Fourier Symbol
Linearization of Navier—Stokes equations: u  U   u, v  V   v, p  P.
Linearized difference scheme:
The von Neumann necessary stability condition:
4 Analytic Investigation of Eigenvalues
Case 1: 1   2  0 The scheme is absolutely stable
Case 2:  3   4  0, 1  0,  2  0
  1   2
The scheme is weakly unstable
Case 3: all kappa’s are different from zero.
Möbius transformation:   (  1) /(  1).
|  j | 1  Re  j  0
  i
Implementation of the above mathematical procedure with Mathematica:
As a result, the following formula for the resultant was obtained:
The particular case ξ = η :
Root of equation R  ,  ,    0 ( 5  1   2 ) :
Another particular case:
0   6   3   4 1 (high Reynolds numbers)
t 10
2
5
1.5
1 b
0.5
1
a
0.5
1.5
2
Fig. 4. The surface τ = τ(a,b)
5 The Method of Discrete Perturbation
The behavior of α and β agrees with that obtained by the Fourier method.
6 Verification of Stability Conditions
6.1 The Taylor—Green Vortex
The analytic solution of the Navier—Stokes equations, with ν = ρ = 1, is given by formulas
30
30
25
25
25
20
20
20
15
15
15
10
10
10
5
5
5
5
  min
j,k
10
15
20
25
30
30
5
10
15
20
25
30
3030 grid
5
0.5
u
v

h1 h2
The new formula for time step:
10
15
20
25
30
6 Verification of Stability Conditions
6.2 Lid-Driven Cavity Problem
3030 grid
(2)
from 33 to 58
a) Re = 1, θ = 3: stable for  n / 
(2)
5
b) Re = 400, θ < 0.1: stable for  n / 
 (2)  min
j,k
0.5
u v

h1 h2