Search for: All records

ABSTRACT We introduce a novel artificial compressibility technique to approximate the incompressible Navier–Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed method uses the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a method with time‐independent stiffness matrices that is suitable for pseudo‐spectral methods. The stability and temporal convergence of a semi‐implicit version of the method are established under the hypothesis that the density is approximated with a method that conserves the minimum‐maximum principle. Numerical illustrations confirm that both the semi‐implicit and explicit methods are stable and converge with order one under the classic CFL condition. Moreover, the proposed method is shown to perform better than a momentum‐based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi‐fluids in a cylinder. 

Abstract We study the stability and convergence properties of a semi-implicit time stepping scheme for the incompressible Navier–Stokes equations with variable density and viscosity. The density is assumed to be approximated in a way that conserves the minimum-maximum principle. The scheme uses a fractional time-stepping method and the momentum, which is equal to the product of the density and velocity, as a primary unknown. The semi-implicit algorithm for the coupled momentum-pressure is shown to be conditionally stable and the velocity is shown to converge inL²norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles. 

When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. ( Phys. Fluids , vol. 29, no. 5, 2017 b , 054101) and Horstmann et al. ( J. Fluid Mech. , vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.