TY - GEN A1 - Schneider, Thomas A1 - Botta, Nicola A1 - Geratz, Karl Josef A1 - Klein, Rupert T1 - Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow N2 - When attempting to compute unsteady, variable density flows at very small or zero Mach number using a standard finite volume compressible flow solver one faces at least the following difficulties: (i) Spatial pressure variations vanish as the Mach number $M \rightarrow 0$, but they do affect the velocity field at leading order; (ii) the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux; (iii) violation of this constraint would crucially affect the transport of mass, thereby disabling a code to properly advect even a constant density distribution. A previous companion paper derived the above observations from a single time - multiple length scale asymptotic analysis for $M \ll 1$, applied to the conservation form of the governing equations and assuming an ideal gas with constant specific heats. The paper then restricted to weakly compressible one-dimensional flows and introduced a semi-implicit extension of a compressible flow solver, designed to handle the interaction of long wavelength acoustics with small scale, large amplitude density fluctuations. In the present paper we concentrate on the limit of zero Mach number for multi-dimensional, variable density flows. The construction of numerical fluxes for all conserved quantities involves: An explicit upwind step (1) yielding predictions for the nonlinear convective flux components. This procedure still neglects the influence of pressure gradients on the convective fluxes during the time step. Suitable corrections are applied in step (2), which guarantees compliance of the convective fluxes with the divergence constraint. This step requires the solution of a Poisson-type equation to obtain the relevant pressure gradients. Step (3), which requires the solution of a second Poisson-type equation, yields the yet unknown (non-convective) pressure contribution to the total flux of momentum. The final, cell centered velocity field exactly satisfies a discrete divergence constraint consistent with the asymptotic limit. Notice that step (1) can be done by any standard finite volume compressible flow solver and that the input to steps (2) and (3) involves solely the fluxes from step (1), but is independent on how these were obtained. Thus, we claim that our approach allows any such solver to be extended to simulate incompressible flows. Extensions to the weakly compressible regime $0 < M \ll 1$, reactive flows and more complex equations of state will be addressed in follow-up publications. T3 - ZIB-Report - SC-98-31 KW - Incompressible variable density flows KW - conservation laws Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3749 ER - TY - GEN A1 - Oevermann, Michael A1 - Klein, Rupert A1 - Berger, Marsha A1 - Goodman, Jonathan T1 - A Projection Method for Two-Phase Incompressible Flow with Surface Tension and Sharp Interface Resolution N2 - We present a fully second order projection method for the simulation of two-phase incompressible flow with surface tension. The Navier-Stokes equations are solved with a projection method on a fixed Cartesian grid. The free interface between the two fluids is tracked with a level set approach. The conditions at the interface for the pressure, the pressure gradient, and the velocity are explicitly incorporated into the scheme leading to a sharp representation of the pressure discontinuity and the interfacial force. The scheme in the presented form does not introduce additional points in the standard finite difference stencils. Computational results are compared with analytic solutions for a static round bubble, damped surface waves, and Rayleigh-Taylor instabilities. T3 - ZIB-Report - 00-17 KW - Two-phase flow KW - immersed interface method KW - incompressible flow KW - projection method KW - level set KW - multi grid Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5851 ER -