A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

Abstract

In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid.We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat [89], which justifies the use of a path-conservative approach for treating the non-conservative products.The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion.The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.Key words: diffuse interface model, compressible flows over fixed and moving solids, immersed boundary method for compressible flows, arbitrary high-order discontinuous Galerkin schemes, a posteriori sub-cell finite volume limiter (MOOD), path-conservative schemes for hyperbolic PDE with non-conservative products,

High Order ADER Schemes for Continuum Mechanics

“…Yet, before to be applied to practical problems, the GPR model may require a coupling with an interface tracking/capturing technique for the modeling of moving material boundaries such as in free surface flows or solid body motion. In particular, in this paper, we couple the GPR model with a simple diffuse interface approach, see [173,59,91,126]. For example, very interesting computational results with similar diffuse interface approaches and level set techniques for compressible multi-material flows have been obtained for example in [95,85,84,144,51,139,119,12].…”

An urban large-eddy-based dispersion model for marginal grid resolutions: CAIRDIO v1.0

“…p I is the interface pressure, reflecting Newton's third law of motion near a fixed wall. As in Kemm et al (2020), it will be assumed that it is in equilibrium with the surrounding fluid pressure p. T is the stress tensor, which in LES averaging contains the contributions from subscale and surface fluxes. In this model, the implicit approach is used for spatial filtering (Schumann, 1975).…”

“…On the other hand, an equivalent boundary forcing can be more rigorously deduced from a two-phase flow model (Drew, 1983) when neglecting the restoring source terms. Treating the second phase as an indeformable solid, Kemm et al (2020) derived a diffusive-interface (DI) model for compressible fluids. One of the main advantages of their approach is the algorithmic simplicity, as the boundary-forcing term is analytically coupled to the DI, which is advected as a scalar.…”

An urban large-eddy-simulation-based dispersion model for marginal grid resolutions: CAIRDIO v1.0

“…The influence of the urban canopy on the PBL in NWP models can be considered through sophisticated canopy parameterizations (Martilli et al, 2002;Schubert et al, 2012). The improvements of these so-called urbanized NWPs are substantially reflected in the modeled pollutant concentrations (Kim et al, 2015;Wang et al, 2019). Nevertheless, as the model domain does not allow for an explicit representation of buildings, the pollutants emitted at street level are diluted over the entire grid cell, which can considerably deviate from real conditions, where a large part of the physical volume may be impervious.…”