Implicit finite element schemes for stationary compressible particle-laden gas flows

Abstract

a b s t r a c tThe derivation of macroscopic models for particle-laden gas flows is reviewed. Semiimplicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust.

Implicit finite element schemes for the stationary compressible Euler equations

“…This is enforced by and the diffusion tensor has vanishing row and column sums. The application of the discrete diffusion operator to node i results in the low‐order operator which has positive semi‐definite off‐diagonal blocks L ij 31.…”

“…The transformation to characteristic variables is performed to determine how much artificial diffusion may be safely removed, while the artificial limited antidiffusion is inserted into the residual in conservative variables. The proposed algorithm for the computation of the antidiffusive correction is based on 8, 10, 31 and has to be applied for each space dimension d : In a loop over edges { ij }: Compute the Roe‐averaged eigenvalues and matrices of left and right eigenvectors of the uni‐directional Roe matrices. Transform the solution differences on the edge to the characteristic variables. Determine the upwind node I and the downwind node J for each characteristic field k : If : If a λ<0: Compute the raw diffusive/antidiffusive fluxes on the current edge For each characteristic field k update the sums of positive and negative edge contributions Reverse the sign of the flux and add it to the upper/lower bounds In a loop over nodes: Evaluate the nodal correction factors In a loop over edges { ij }: …”

“…Free stream boundary conditions can be implemented as follows 31, 32. Assume that a set of free stream invariants is specified.…”

“…In the case of a subsonic outlet, the outlet pressure may be prescribed, while at a subsonic inlet e.g. density, tangential velocity, and pressure may be fixed 31.…”

Failsafe flux limiting and constrained data projections for equations of gas dynamics

“…Many practical aspects (matrix assembly, defect correction, weak imposition of characteristic boundary conditions) of developing an unstructured mesh finite element code for systems of conservation laws are addressed in [7,12].…”