Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow

Abstract

For numerical simulation of compressible viscous flow, the characteristics-based NSCBC boundary conditions proposed by Poinsot and Lele [T. Poinsot, S.K. Lele, Boundary conditions for direct simulation of compressible viscous flows, J. Comput. Phys. 101 (1992) 104–129] are frequently employed. This formulation is analyzed analytically and it is found that the linear relaxation term proposed by Rudy and Strikwerda [D.H. Rudy, J.C. Strikwerda, A nonreflecting outflow boundary condition for subsonic Navier–Stokes calculations, J. Comput. Phys. 36 (1) (1980) 55–70] to suppress slow “drift” of flow variables results in a non-zero reflection coefficient for acoustic waves. Indeed, although the NSCBC formulation of boundary conditions is often called “non-reflecting”, the magnitude of the reflection coefficient approaches unity for low frequencies. A modification of the NSCBC boundary conditions and in particular the linear relaxation term is proposed, which should appear fully non-reflecting to plane acoustic waves with normal incidence on the boundaries for all frequencies. The new formulation is implemented and successfully validated in large eddy simulation of turbulent channel flow.

Introduction

Acoustically non-reflecting boundary conditions for compressible (turbulent) flow simulation are a prerequisite for the successful application of computational fluid dynamics (CFD) to flow-acoustics problems, e.g., the numerical simulation of combustion instabilities. For linear problems, or for problems where linearization near the boundary is permissible, a variety of techniques have been developed, see the recent review of Colonius [1]. On the other hand, situations with nonlinear effects near the boundary – the prime example being the turbulent outflow problem – still pose significant problems. The Navier–Stokes characteristics boundary condition (NSCBC) developed for multi-dimensional viscous flow by Thompson, and Poinsot and Lele with the linear relaxation term proposed by Rudy and Strikwerda [2], [3], [4] has been applied successfully to many problems. However, there is evidence that this formulation, although frequently termed “non-reflecting”, is indeed partly reflecting [5], [6].

In this paper, the NSCBC boundary conditions are analyzed, and an analytical expression for the reflection coefficient of this formulation is derived. Then, a modification is proposed, which should allow the implementation of fully non-reflecting boundary conditions – at least for plane waves with normal incidence on the boundary. The proposed formulation is computationally very efficient and the restriction to plane waves is not a significant drawback for many applications in duct acoustics or combustion dynamics.

Both the original and the modified formulation for NSCBCs are tested in a large eddy simulation (LES) of compressible turbulent channel flow at low Mach number with external excitation. The numerical results agree with expectations except for very high frequencies (above the so-called cut-off frequency of the duct, i.e., the frequency of the fundamental non-plane travelling or standing acoustic mode in the duct).

In Section 2, some notation is established before characteristics based boundary conditions are reviewed briefly in Section 3. A more comprehensive discussion of the NSCBCs may be found in the original literature [2], [3], [4], [7] and in textbooks [8], [9]. Recent developments of the approach with extensions to real gases or mixtures thereof are discussed, for example in [10], [11]. The new, non-reflecting formulation is introduced and discussed in Section 4. Results of a validation study based on large eddy simulation (LES) of compressible channel flow are shown in the last section.