Elsevier

Journal of Computational Physics

Volume 329, 15 January 2017, Pages 141-172
Journal of Computational Physics

An enhanced FIVER method for multi-material flow problems with second-order convergence rate

https://doi.org/10.1016/j.jcp.2016.10.028Get rights and content

Abstract

The finite volume (FV) method with exact two-material Riemann problems (FIVER) is an Eulerian computational method for the solution of multi-material flow problems. It is robust in the presence of large density jumps at the fluid–fluid interfaces, and the presence of large structural motions, deformations, and even topological changes at the fluid–structure interfaces. To achieve simplicity in implementation, it approximates each material interface by a surrogate surface which conforms to the control volume boundaries. Unfortunately, this approximation introduces a first-order error of the geometric type in the solution process. In this paper, it is first shown that this error causes the original version of FIVER to be inconsistent in the neighborhood of material interfaces and degrades its global order of spatial accuracy. Then, an enhanced version of FIVER is presented to rectify this issue, restore consistency, and achieve for smooth problems the desired global convergence rate. To this effect, the original definition of a surrogate material interface is retained because of its attractive simplicity. However, the solution at this interface of a two-material Riemann problem is enhanced with a simple reconstruction procedure based on interpolation and extrapolation. Next, the extrapolation component of this procedure is equipped with a limiter in order to achieve nonlinear stability for non-smooth problems. In the one-dimensional inviscid setting, the resulting FIVER method is also shown to be total variation bounded. Focusing on the context of a second-order FV semi-discretization, the nonlinear stability and second-order global convergence rate of this enhanced FIVER method are illustrated for several model multi-fluid and fluid–structure interaction problems. The potential of this computational method for complex multi-material flow problems is also demonstrated with the simulation of the collapse of an air bubble submerged in water and the comparison of the computed results with corresponding experimental data.

Introduction

FIVER (finite volume method with exact two-material Riemann problems) is a finite volume (FV) method for the solution of multi-material, fluid and fluid–structure interaction (FSI) problems. Its underlying semi-discretization procedure is based on the solution of local, one-dimensional, two-material Riemann problems. It was originally developed in [1], in the context of explicit time-discretizations, for the solution of compressible, inviscid, two-phase flow problems characterized by simple equations of state (EOS) but large contact discontinuities (density jumps). However, it is equally applicable to the solution of incompressible, multi-material, fluid or FSI problems. In [2], the concept of a one-dimensional fluid–fluid Riemann problem was extended to that of a local fluid–structure Riemann problem, and FIVER was transformed into an Eulerian embedded boundary method (EBM) — or immersed boundary method — for the solution of highly nonlinear fluid and FSI problems. These include those FSI problems which cannot be handled by Arbitrary Lagrangian Eulerian (ALE) methods [3], [4], [5] because of large structural motions, finite deformations, and/or topological changes that challenge the performance of mesh motion schemes [6], [7]. Unlike most other EBMs which operate exclusively on Cartesian grids (for example, see [8], [9], [10] and [11]), FIVER can operate on arbitrary grids. This is noteworthy because even in the pure Eulerian setting, the ability of an EBM to perform on arbitrary, unstructured grids is still particularly advantageous for complex geometries and viscous flows.

For multi-material problems with complex fluid EOS, the analytical solution of a local, one-dimensional, two-material Riemann problem may be either impossible to obtain explicitly, or computationally intensive to evaluate numerically. To address this issue, a computationally efficient sparse grid tabulation technique was developed in [12] for accelerating the numerical solution of arbitrary fluid–fluid and fluid–structure Riemann problems, and thereby enabling the generalization of FIVER to multi-material problems with complex fluid EOS. In [13], the generalized FIVER method was validated for the solution of failure-induced FSI problems. Specifically, it was applied to the simulation of two experiments on the dynamic underwater implosion of cylindrical shells. In both cases, it reproduced with high-fidelity the large deformations of the collapsing structure and the compression waves emanating from it.

FIVER was also extended to implicit time-discretizations in [14], and to viscous, multi-material fluid and FSI problems in [15].

For turbulent viscous flows, Eulerian EBMs typically suffer from the fact that they do not track the boundary layers around dynamic rigid or flexible bodies. Consequently, the application of these methods to such problems requires either high mesh resolutions in a large part of the computational fluid domain, or adaptive mesh refinement. Unfortunately, the first option is computationally inefficient, and the second one is labor intensive to implement. To address these issues, an ALE variant of FIVER which maintains all moving boundary layers resolved during turbulent FSI computations was presented in [16].

Most recently, a generic, comprehensive, and yet effective approach for representing a fractured fluid–structure interface was developed in [17] and incorporated in FIVER. Specifically, this approach enables the coupling of FIVER with many finite element based fracture methods for the solution of multi-material FSI problems with dynamic fracture. These methods include, among others, the interelement fracture [18] and remeshing [19] techniques, the extended finite element method (XFEM) [20], and the element deletion method [21]. Equipped with this fractured fluid–structure interface representation, FIVER was further validated for highly nonlinear FSI problems with crack propagation, flow seepage through narrow cracks, and structural fragmentation [17].

Because of the large scope of applications outlined above, FIVER is today more than a computational method. It is a genuine computational framework for multi-material, inviscid or viscous, laminar or turbulent, fluid and FSI problems. This framework is driven by two simple ideas:

  • Keeping the underlying semi-discretization scheme as close as possible to a Godunov-type scheme in order to facilitate its implementation in a large body of existing flow solvers.

  • Solving at each material interface local, one-dimensional, two-material Riemann problems in order to avoid traversing explicitly this interface during the semi-discretization process and therefore achieve robustness with respect to strong contact discontinuities.

To achieve simplicity in the implementation of both of these ideas, all instances of FIVER reviewed above approximate a material interface by a surrogate surface which conforms to the boundaries of the Eulerian cells associated with the discretization of a given computational fluid domain into finite volumes (Fig. 2.3). Unfortunately, if h denotes the size of a generic cell in the CFD (Computational Fluid Dynamics) grid, this approximation introduces in the solution process an interface position error of the order of h/2. Numerical experiments reveal that consequently, when FIVER is equipped with a second-order FV semi-discretization scheme based on Roe's approximate Riemann solver [22] and a MUSCL (Monotonic Upwinding Scheme for Conservation Laws) [25] technique, it delivers for relatively smooth multi-material problems where no limiter is needed a zero-order convergence rate at the material interfaces, and a first-order global convergence rate. Such flaws are typical of Eulerian methods for multi-material problems. Nevertheless, the main objective of this paper is to correct these issues and present an enhanced FIVER method which retains the simplicity afforded by the surrogate material interface of the original FIVER method, without violating consistency at the material interfaces or degrading the expected global convergence rate. To this effect, the remainder of this paper is organized as follows.

In Section 2, the governing equations of interest, the original FIVER framework and its underlying FV semi-discretization, the chosen interface capturing technique, and the time-discretization of the resulting multi-material semi-discrete fluid or FSI problem are overviewed. In Section 3, the undesirable effect of the geometric error associated with FIVER's surrogate material interface is highlighted using a mathematical analysis of a simple one-dimensional FSI problem. Following some of the ideas presented in [23], a simple approach based on extrapolation and interpolation is presented in Section 4 for eliminating this detrimental effect. In Section 5, an extrapolation component of this approach is equipped with a limiter in order to achieve nonlinear stability for non-smooth problems. In this section, it is also shown that in the one-dimensional inviscid setting, the resulting FIVER method is total variation bounded (TVB) and therefore nonlinearly stable. In Section 6, the numerical properties of the enhanced FIVER framework presented in this paper are illustrated and its performance is assessed for smooth and non-smooth model multi-material problems. In particular, it is shown numerically that when the new FIVER framework is equipped with a second-order FV semi-discretization and applied to the solution of smooth multi-material problems, it delivers a second-order global convergence rate. The robustness of this framework and its potential for complex multi-material problems are also demonstrated in this section with the simulation of the collapse of an air bubble submerged in water and the comparison of the obtained numerical results with the corresponding experimental data [24]. Finally, Section 7 concludes this paper.

Section snippets

Background

As stated in the introduction, FIVER is a FV method for the solution of multi-material problems based on the solution of local, one-dimensional, two-material Riemann problems. When all materials are fluid materials, FIVER is essentially a Godunov-type method for multi-phase flow computations. When at least one material is a solid material, it becomes a genuine EBM for CFD. Here, the basic FIVER framework is reviewed in order to set the stage for the main contributions of this paper which are:

Detrimental effect of the surrogate material interface

Using as a backdrop a simple one-dimensional FSI problem, it is shown here for the first time that the surrogate material interface (2.9) that simplifies the FIVER computations also degrades their accuracy. To this effect, attention is focused on the one-dimensional version of Equations (2.2) whose spatial discretization on a uniform grid of cell size h can be written asW˙i+1h(Φi(i+1)Φi(i1))=0 where here and throughout the remainder of this paper, the dot symbol designates a time derivative

Enhancements of spatial accuracy for smooth problems

Here, two aspects of FIVER are revisited in order to eliminate the detrimental effect of the surrogate material interface highlighted in the previous section and enhance the overall spatial accuracy of this computational framework:

  • The solution of a two-material Riemann problem is performed at the captured location of the true material interface ΓM, in order to avoid introducing in the FIVER solution process a local geometric error of the order O(h/2).

  • The constant extrapolation procedures

Nonlinear stability

It is well-known that, in principle, an extrapolation step such as (4.7) or (2.3) should be equipped with a limiter in order to avoid generating, for non-smooth problems, spurious oscillations in the numerical solution. This issue has been addressed in great details in the thesis work [42] underlying this paper. In particular, it was concluded in [42] that to achieve nonlinear numerical stability, the enhanced version of FIVER proposed here requires a limiter only for the extrapolation step

Preliminaries

Both the original FIVER method and its enhanced counterpart presented in this paper were implemented in the massively parallel flow solver AERO-F [43], [44]. In this CFD code, the semi-discretization of the convective fluxes can be performed using a second-order accurate upwind scheme based on any of the Roe [22], HLLE [45], or HLLC [46] approximate Riemann solvers. Time-discretization can be performed using a first- or second-order implicit time-integrator, or a first-, second-, or

Summary and conclusions

In this paper, attention focuses on enhancing the convergence rate of the original finite volume (FV) method with exact two-material Riemann problems (FIVER). This Eulerian computational method for the solution of multi-material flow problems is robust with respect to large density jumps at fluid–fluid interfaces. It is also equally robust with respect to large structural motions, deformations, and even topological changes at fluid–structure interfaces. It is particularly simple to implement

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