This paper presents an evolutionary step in sharp capturing of shocked, high acoustic impedance mismatch (AIM) interfaces in an adaptive mesh refinement (AMR) environment. The central theme which ...guides the present development addresses the need to optimize between the algorithmic complexities in advanced front capturing and front tracking methods developed recently for high AIM interfaces with the simplicity requirements imposed by the AMR multi-level dynamic solutions implementation. The paper shows that we have achieved this objective by means of relaxing the strict conservative treatment of AMR prolongation/restriction operators in the interfacial region and by using a natural-neighbor-interpolation (NNI) algorithm to eliminate the need for ghost cell extrapolation into the other fluid in a characteristics-based matching (CBM) scheme. The later is based on a two-fluid Riemann solver, which brings the accuracy and robustness of front-tracking approach into the fast local level set front-capturing implementation of the CBM method. A broad set of test problems (including shocked multi-gaseous media, bubble collapse, underwater explosion and shock passing over a liquid drop suspended in a gaseous medium) was performed and the results demonstrate that the fundamental assumptions/approximations made in modifying the AMR prolongation/restriction operators and in using the NNI algorithm for interfacial treatment are acceptable from the accuracy point of view, while they enable an effective implementation and utility of the structured AMR technology for solving complex multiphase problems in a highly compressible setting.
A high-order numerical method is developed for the computation of compressible multiphase flows. The model we use is based on the Baer and Nunziato type systems 4. Among all the other available ...models in the literature, these systems present the advantage to be able to simulate either interface or mixture problems. Nevertheless, they still raise some issues, mainly based on their non-conservative feature. The numerical method we propose is a discontinuous Galerkin type. In this work, the interior side integrals are computed thanks to 2. Robustness and high order of accuracy of the method are proved on classical interface problems, but also on suitably derived analytical solutions.
Shock waves interacting with multi-material interfaces in compressible flows result in complex shock diffraction patterns involving total or partial reflection, refraction and transmission of the ...impinging shock wave. To simulate such complicated interfacial dynamics problems, a fixed Cartesian grid approach in conjunction with levelset interface tracking is attractive. In this regard, a unified Riemann solver based Ghost Fluid Method (GFM) is developed to accurately resolve and represent the embedded solid and fluid object(s) in high speed compressible multiphase flows. In addition, the Riemann solver based GFM approach is augmented with a quadtree (octree in three-dimensions) based Local Mesh Refinement (LMR) technique for efficient and high fidelity computations involving strong shock interactions. The paper reports on a conservative formulation for accurate calculation of ENO-based numerical fluxes at the fine-coarse mesh boundaries. The numerical examples presented in this paper clearly demonstrate that the methodology produces accurate benchmark solutions and effectively captures fine structures in the flow field.
These notes give an overview on how the relative entropy stability framework can be employed to derive a posteriori error estimates for semi-(spatially)-discrete discontinuous Galerkin schemes ...approximating systems of hyperbolic conservation laws endowed with one strictly convex entropy. We also show how these methods can be extended as to cover a related, higher order, model for compressible multiphase flows with non-convex energy.
A general framework is developed for solving high-speed and high-intensity multi-material interaction problems on adaptively refined Cartesian meshes. The framework is applicable for interfaces ...separating materials with very different properties and in the presence of strong shocks. A sharp interface treatment is maintained through a modified Ghost Fluid Method. The embedded boundaries are tracked and represented with level sets. A tree-based Local Mesh Refinement scheme is employed to efficiently resolve the desired physics. Results are shown for situations that cover varied combination of materials (fluids, rigid solids and deformable solids) with careful benchmarking to establish the validity and the versatility of the approach.
Computing compressible two-phase flows that consider different materials and physical properties is conducted. A finite volume numerical method based on Godunov approach is developed and implemented ...to solve an Euler type mathematical model. This model consists of five partial differential equations in one space dimension and it is known as the reduced model. A fixed Eulerian mesh is considered and the hyperbolic problem is tackled using a robust and efficient HLL and HLLC Riemann solvers. The performance of the two solvers is verified against a comprehensive suite of numerical case studies in one and two dimensional space. Computing the evolution of interfaces between two immiscible fluids is considered as a major challenge for the present model and the numerical technique. The achieved numerical results display a good agreement with all reference data.
The simulation of compressible multiphase problems is a difficult task for modelization and mathematical reasons. Here, thanks to a probabilistic multiscale interpretation of multiphase flows, we ...construct a numerical scheme that provides a solution to these difficulties. Three types of terms can be identified in the scheme in addition to the temporal term. One is a conservative term, the second one plays the role of a nonconservative term that is related to interfacial quantities, and the last one is a relaxation term that is associated with acoustic phenomena. The key feature of the scheme is that it is locally conservative, contrarily to many other schemes devoted to compressible multiphase problems. In many physical situations, it is reasonable to assume that the relaxation is instantaneous. We present an asymptotic expansion of the scheme that keeps the local conservation properties of the original scheme. The asymptotic expansion relies on the understanding of an equilibrium variety. Its structure depends, in principle, on the Riemann solver. We show that it is not the case for several standard solvers, and hence this variety is characterized by the local pressure and velocity of the flow. Several numerical test cases are presented in order to demonstrate the potential of this technique.
PDF
