Elsevier

Computers & Fluids

Volume 106, 5 January 2015, Pages 79-92
Computers & Fluids

Capturing matched layer at absorbing boundary with finite volume schemes

https://doi.org/10.1016/j.compfluid.2014.09.040Get rights and content

Highlights

  • The matched layer is automatically captured at absorbing boundary.

  • The absorbing boundary condition is hence extremely simple but robust.

  • It is embedded in finite volume schemes, not related to other nonreflecting conditions.

  • It is established via Fourier analysis and plane waves, theoretically, reflection = 0.

  • Non-trivial test examples in 1D–3D spaces are demonstrated.

Abstract

In compressible flow computations, an important treatment for absorbing boundary condition (ABC) is to create a matched layer at the boundary, as in the well-known PML (perfectly matched layer) method. In the present paper, it is shown that with cell-centered finite volume (FV) schemes, the matched layer can be captured directly as a discontinuity across the absorbing boundary, rendering an extremely simple yet robust ABC. The new ABC is inherently embedded in FV schemes of cell-centered type, and often associated with the captured matched layer, which serves to match the flow variables across the boundary. It has been used empirically for years and was found to have no direct relation to any existing nonreflecting boundary condition (NRBC). Instead, it is attributed to the shock-capturing capability of the FV scheme, as well as a nonreflecting (NR) observation that for any scheme, no spurious reflection is generated at any interior point of the domain. A Fourier analysis with plane waves is employed to the local Euler solution to justify the NR observation and the ABC is consequently established.

The ABC performs perfectly with zero reflection in one dimensional space. In multi-dimensional spaces, the phase error and reflection due to discretization are discussed. With appropriate grid resolution at the boundary, one can always suppress the spurious reflection to any designated level. Several non-trivial numerical examples in one and multi-dimensional spaces are tested to demonstrate its robustness.

Introduction

Nonreflecting boundary conditions (NRBCs) play a key role in fluid dynamics computations while remaining as a challenging topic in the current research areas in engineering and applied mathematics. Spurious reflections resulting from an inappropriate numerical boundary condition contaminate the flow field and may eventually spoil the entire computed flow. There have been a huge number of literatures dealing with the topic of NRBC. Herein, we focus only on the related issues of the absorbing boundary condition (ABC).

For a boundary with flow conditions specified, an ABC plays a dual role: to enforce the given boundary conditions and to absorb the wave or disturbance propagating from the domain interior. Usually, there are two ways to absorb the outgoing waves and to avoid spurious reflections at the boundary. In one way, the given flow boundary condition is modified to an admissible numerical boundary condition, as in the characteristics-based NRBC (cf. e.g., [2]). In the other way, a matched layer (or absorbing layer) is artificially created at the boundary, which smoothly matches the inconsistent flow data across the boundary. A typical work along this track is the recent perfectly matched layer (PML) method (cf. e.g., [3], [4], [5]).

In recent years, another interesting ABC for finite volume (FV) schemes was discovered empirically, e.g., Leveque [6] and Loh et al. [7], [8], [9], [10]. When the flow conditions prescribed at the ghost cell centers (GCCs) do not match the flow within the domain, a matched layer similar to the one in the PML method is automatically captured at the boundary to do the matching, saving the work of creating a matched layer or matching the flow data. The ABC is hence extremely simple yet robust. The purpose of the present paper is to examine and establish this new ABC for FV schemes.

Despite its attractive advantages, investigations show that the new ABC does not seem to directly relate to any existing NRBC, but should be credited to the shock-capturing capability of FV schemes, as well as a well-accepted empirical observation that no spurious reflection occurs at any interior point of the domain. For brevity, hereafter, this observation will be called the nonreflecting observation or NR observation.

The contents are arranged as follows. First of all, Sections 2 Simple concept of absorbing, 3 Fourier analysis of the local Euler solution present the theoretical work to prove the NR observation. In Section 2, We begin with the introduction of a simple but unconventional concept of absorbing. Based on this concept, Fourier analysis is conducted to the local Euler solution in Section 3 to justify the NR observation. If, by his/her own experience, the reader would accept the NR observation for granted, Sections 2 Simple concept of absorbing, 3 Fourier analysis of the local Euler solution may be temporarily skipped, as they are tedious and require concepts in real analysis.

Section 4 shows that, due to a boundary treatment in cell-centered FV schemes, the NR observation can be extended and applied to the boundary itself. The new ABC is then established based on the shock-capturing capability of the FV schemes. With a one dimensional example, the section illustrates how the ABC works and the automatic capturing/creation of the associated absorbing layer (matched layer).

The ABC was found working perfectly in one dimensional space with zero reflection, but in multi-dimensional computations, lack of sufficient grid resolution on the boundary may still introduce spurious reflection. Section 5 is devoted to the phase error analysis at a boundary element (a line segment or a surface element) in multi-dimensional spaces, and the discussion of reflection coefficient. So that the spurious reflection can be suppressed to any designated level by choosing an appropriate grid resolution at the boundary. In Section 6 the new ABC is tested in several non-trivial examples in multi-dimensional spaces. Finally, the paper is concluded with remarks in Section 7.

Throughout the paper, V(x,t)=(ρ,u,v,w,p)T is employed to represent the primitive flow variables in the solution of the three dimensional Euler equations, where x,t,u,v,w,ρ,p are respectively the coordinates (x,y,z), time, the three velocity components, density, and pressure. W(x,t)=(ρ,ρu,ρv,ρw,ρe)T represents the conservative flow variables, with the energye=pρ(γ-1)+12(u2+v2+w2),γ=1.4.U(x,t) represents a numerical solution or an approximation of V(x,t).

Section snippets

Simple concept of absorbing

Our past investigations show that the new ABC cannot be inferred from any existing NRBC. For further examination, a simple but unconventional concept of absorbing is put forward as our definition for absorbing. It is based on the plane waves in the theory of linear partial differential equations [1], and is introduced below step by step in a logical way by mathematical analysis. The term “local Euler solution” refers to the local analytical solution of the Euler equations in a small

Fourier analysis of the local Euler solution

As in Section 2, the proof in this section is still theoretical, using the concepts in real analysis. No numerical treatment is involved yet. It will be shown theoretically that, at any interior point of the computational domain, the local Euler solution, V(x,t0), and its numerical approximation, U(x,t0), can be expressed as a superposition of plane waves. Hence they are absorbed, and the NR observation is proved. Here, by interior point, we mean a point completely lying within the domain.

Absorbing boundary condition for FV schemes

The theoretical concepts and proofs in Sections 2 Simple concept of absorbing, 3 Fourier analysis of the local Euler solution are now combined with the actual FV schemes to construct the new ABC. A brief review of the FV schemes is first sketched to find out the features that are essential to the ABC. In this section, by “boundary”, it is always meant the absorbing boundary, Γ, of the domain, R, other boundaries, such as solid walls, are not involved.

Phase error in multi-dimensional spaces

In real multi-dimensional computations, if the grid on Γ lacks of adequate resolution, the discretization may introduce phase errors and spurious reflection. To investigate the error, recall that the local numerical solution U(x,t0) near a boundary point is no other than a superposition of plane waves (cf. (10), Section 3). For a given wave number k, one needs only to consider the phase error for a single plane wave on a boundary element, and the suppression of the subsequent reflection.

Multi-dimensional test examples

As demonstrated in Section 4.3, in one dimensional space, the new ABC performs perfectly and achieves its theoretical prediction – zero reflection. In multi-dimensional spaces, due to discretization on the boundary, spurious reflection may appear but is controllable (cf. Section 5.2). In this section, the inherent ABC will be tested in several non-trivial, multi-dimensional examples, to demonstrate its simplicity, versatility, and capability of handling complicated nonreflecting situations.

Concluding remarks

The present paper reveals a hidden capability of the cell-centered FV schemes – the inherent ABC. It works with schemes in such class and differs from any existing NRBCs in that the absorbing (matched) layer is automatically captured as a discontinuity at the absorbing boundary, rather than artificially fit or created. The new ABC plays a dual role as both the given flow BC and the ABC, there is no further implementation to follow. As such, it is extremely simple, yet versatile and robust, as

Acknowledgements

This work was partially sponsored by NASA Glenn Research Center, Cleveland, Ohio, USA. The NASA scientists helped to generate the grids and to compute several examples in Section 6. Their contributions are cordially acknowledged.

The author is obliged to the reviewers for helpful suggestions and to Professor L.S. Luo of Old Dominion University, Virginia, USA for fruitful discussions.

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