A preconditioning for the spectral solution of incompressible variable-density flows

Abstract

In the present study, the efficiency of preconditioners for solving linear systems associated with the discretized variable-density incompressible Navier-Stokes equations with semiimplicit second-order accuracy in time and spectral accuracy in space is investigated. The method, in which the inverse operator for the constant-density flow system acts as preconditioner, is implemented for three iterative solvers: the General Minimal Residual, the Conjugate Gradient and the Richardson Minimal Residual. We discuss the method, first, in the context of the one-dimensional flow case where a top-hat like profile for the density is used. Numerical evidence shows that the convergence is significantly improved due to the notable decrease in the condition number of the operators. Most importantly, we then validate the robustness and convergence properties of the method on two more realistic problems: the two-dimensional Rayleigh-Taylor instability problem and the three-dimensional variable-density swirling jet.

Figures (16)

• Figure 1: Condition number of the viscous operators versus the Chebyshev collocation points number $N$ with $s = 2$ and $a = 0.01$.
• Figure 2: Condition number of the viscous operators versus the density ratio $s$ with $a = 0.01$ and $N = 256$ Chebyshev collocation points.
• Figure 3: Condition number of the viscous operators versus $a$ with $s=2$ and $N=256$ Chebyshev collocation points.
• Figure 4: Convergence history for the five preconditioned iterative solvers to solve $\operatorname{\mathbf{V}}_\rho(\mathbfit{u}) = \mathbfit{b}$ with $a = 10^{-3}$ and $s = 2$ with Fourier differentiation (left side) and Chebyshev differentiation (right side) and $N = 256$ collocation points. Convergence history for different parameters can be found in \ref{['apdx:precond.visc.conv']}.
• Figure 5: Condition number of the pressure operators versus Chebyshev collocation point number $N$ with $s = 2$.