2. Stability Investigation of a Difference Scheme for Incompressible Navier—Stokes Equations D. Chibisov, V. Ganzha, E.W. Mayr, E.V. Vorozhtsov

3. Difference scheme of Kim and Moin (J. Comp. Phys., Vol. 59 (1985), p. 308-323 The Navier—Stokes equations for 2D incompressible fluid flows: The staggered grid in two dimensions The 2nd fractional step: D  div 2 Governing Equations and Difference Method

4. The available empirical stability condition of the Moin-Kim scheme where 3 Fourier Symbol Linearization of Navier—Stokes equations: Linearized difference scheme:

5. 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

6. Case 3: all kappa’s 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:

7. The particular case ξ = η : Root of equation

8. Fig. 4. The surface τ = τ(a,b) Another particular case: (high Reynolds numbers) 5 The Method of Discrete Perturbation The behavior of α and β agrees with that obtained by the Fourier method.

9. 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 The new formula for time step: 30  30 grid

10. 6 Verification of Stability Conditions 6.2 Lid-Driven Cavity Problem a) Re = 1, θ = 3: stable for b) Re = 400, θ < 0.1: stable for from 33 to 58 30  30 grid