A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Considering droplet phenomena at low Mach numbers, large differences in the magnitude of the occurring characteristic waves are presented. As acoustic phenomena often play a minor role in such applications, classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction. In this work, a novel scheme based on a specific level set ghost fluid method and an implicit-explicit (IMEX) flux splitting is proposed to overcome this timestep restriction. A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface. In this part of the domain, the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases. It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method. Applications to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.

High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

Numerical dissipation of flux splitting schemes for contact discontinuities

The contact discontinuity is simulated by three kinds of flux splitting schemes to evaluate and analyse the influence of numerical dissipation in this paper. The numerical results of one-dimensional contact discontinuity problem show that if the flow velocity on both sides of the contact discontinuity is not simultaneously supersonic, the non-physical pressure and velocity waves may occur when the initial theoretically contact discontinuity is smeared into a transition zone spanning several grid-cells caused by numerical dissipations. Since these non-physical waves have no effect on the corresponding density dissipation, this paper considers these fluctuations as only numerical errors and are not part of the numerical dissipation. In addition, for two-dimensional flow field, the characteristics of high-order accuracy difference schemes, i.e. low dissipation and high resolution, may induce the multi-dimensional non-physical waves that interfere with each other to produce more complex non-physical flow structures, so the fluctuations in the calculated results should be treated with caution.

Characteristic-Based Flux Splitting to Solve Compressible Flow over an Airfoil

Current paper presents the Characteristics-based Flux Splitting to improve the numerical behaviour for the inviscid compressible flows. NACA 0012 airfoil is chosen as the test case at angles of attack in the steady state for subsonic, transonic, and supersonic regimes. To calculate the convective fluxes of the Euler equations by the finite volume approach, the Characteristics-based Flux Splitting (CFS) based on the local Mach number is introduced and compared with the conventional Jameson Averaging Method (JAM). To overcome the numerical oscillations especially at the shocked region, artificial dissipation is implemented for the JAM scheme while there is no need to such dissipations for the CFS scheme. For the time integration the Runge–Kutta method is applied. Consistent boundary condition based on characteristics is employed. The results prove the excellence of the CFS scheme regarding the accuracy, stability and convergence. The results are validated and compared with those of others available in the literature.