Numerically stable fluid–structure interactions between compressible flow and solid structures

Abstract

We propose a novel method to implicitly two-way couple Eulerian compressible flow to volumetric Lagrangian solids. The method works for both deformable and rigid solids and for arbitrary equations of state. The method exploits the formulation of [11] which solves compressible fluid in a semi-implicit manner, solving for the advection part explicitly and then correcting the intermediate state to time t n+1 using an implicit pressure, obtained by solving a modified Poisson system. Similar to previous fluid-structure interaction methods, we apply pressure forces to the solid and enforce a velocity boundary condition on the fluid in order to satisfy a no-slip constraint. Unlike previous methods, however, we apply these coupled interactions implicitly by adding the constraint to the pressure system and combining it with any implicit solid forces in order to obtain a strongly coupled, symmetric indefinite system (similar to [17], which only handles incompressible flow). We also show that, under a few reasonable assumptions, this system can be made symmetric positive-definite by following the methodology of [16]. Because our method handles the fluidstructure interactions implicitly, we avoid introducing any new time step restrictions and obtain stable results even for high density-to-mass ratios, where explicit methods struggle or fail. We exactly conserve momentum and kinetic energy (thermal fluid-structure interactions are not considered) at the fluid-structure interface, and hence naturally handle highly non-linear phenomenon such as shocks, contacts and rarefactions.

Compressible, multiphase semi-implicit method with moment of fluid interface representation

“…We remark that Gretarsson et al [22,21] describe conservative, asymptotically preserving algorithms for fluid-structure interaction. Our work is different in that we implemented the Cell Integrated Semi-Lagrangian (CISL) approach for discretizing the advective terms and we represent deforming boundaries by the moment-of-fluid method.…”

“…Since the development of the Implicit Continuous-Fluid Eulerian (ICE) method [23] in the late 1960's, there have been many new algorithms developed for simulating compressible multiphase flow [23,18,54,44,35,60,41,32,39,19,9,22,17,11,1,12]. A summary of the key properties of algorithms that have been developed for simulating compressible multiphase flows are given chronologically in Table 1.…”

“…It is demonstrated in articles as early as 1968 [23,60,32,31,22,21,1] that numerical algorithms for multiphase flow that have the asymptotically preserving property are robust for computing solutions in which materials with widely varying properties neighbor each other. This is because the "monolithic" semi-implicit discretization for pressure leads to strong coupling between all the materials.…”

Fully conservative leak-proof treatment of thin solid structures immersed in compressible fluids

“…One of the primary research areas in solid-fluid coupling concerns the stability of the numerical methods for coupling and is essentially focused on the feedback loop where pressure is applied to the solid, the solid structure reacts and deforms, and subsequently imposes position and velocity-based boundary conditions on the fluid. While the most straightforward approach is simply to treat the coupling in an explicit way, called a partitioned method [49,41,10], researchers have focused quite a bit of attention on so-called monolithic methods that employ higher degrees of implicit coupling [42,14], in order to stabilize parts or all of this feedback loop. Another important issue regards the modifications that the Eulerian method requires to treat cells cut by the solid structure as well as those that are covered or uncovered as the structure sweeps across the Eulerian grid-especially in regards to stability and conservation.…”

“…In order to capture the fluid-structure interactions we employ the flux-split compressible coupling methodology of [14], where the fluid flux terms are split into advective terms and pressure terms. The linearly degenerate advective terms are solved independently of the structure after which an implicit, monolithic coupled system is solved for the fluid pressures, the fluid-structure impulses and the structure velocity degrees of freedom.…”