A second-order-accurate immersed boundary ghost-cell method with hybrid reconstruction for compressible flow simulations

Highlights

• A bilinearly complete extrapolation scheme is developed for the reconstruction of the ghost-cell.

• A hybrid GCM based on both baseline GCM and improved GCM is proposed and constructed.

• A hybrid GCM applied in compressible flow is validated.

Abstract

This study presents an improved ghost-cell immersed boundary method for geometrically complex boundaries in compressible flow simulations. A bilinearly complete extrapolation scheme is developed for the reconstruction of the ghost-cell. The second-order accuracy of the improved ghost-cell method (GCM) is shown in unit test cases and is also theoretically proven. A hybrid GCM based on both baseline GCM and improved GCM is proposed and constructed. The hybrid GCM applied in compressible flow is validated against five test cases: (a) Stationary rotating vortex, (b) Prandtl–Meyer expansion flow, (c) Double Mach reflection, (d) Moving-shock/obstacle interaction, (e) Blunt body shock-induced combustion. This paper provides a comprehensive comparison of their performance in terms of various accuracy and computation time measurements. The simulation results demonstrate that the hybrid GCM has higher accuracy and convergence than the remaining two GCMs in all cases. By directly comparing the primitive variables along the boundary, it can be concluded that the hybrid GCM has significant advantages in compressible flow simulations. The results of CPU time show that the hybrid GCM can provide more accurate results while ensuring the efficiency of the calculation.

Introduction

There has been extensive research on computational fluid dynamics (CFD) simulations employing complex geometries within the computational domain in recent years. Traditionally, the solid boundary is approximated with body-fitted grids, structured or unstructured. The main disadvantage of body-fitted grids is that much effort must be put into the pre-processing stage. In recent years, researchers have shown an increased interest in using Cartesian grids for simulations. One of the main obstacles is that the complex geometries do not coincide with the Cartesian grid. Immersed boundary (IB) methods are among the most widely used Cartesian grid methods for dealing with complex boundaries. The importance of immersed boundary methods is indisputable.

The immersed boundary method (IBM) proposed by Peskin [1], [2], [3] is employed to handle elastic boundaries for simulating blood flow in the heart. A recent systematic literature review [4] refers to this method as the “continuous forcing method”. The forcing is incorporated into the continuous equations before discretization. This method is suitable for the case where the boundary is elastic, and the Dirac delta function used for the elastic boundary does not fit well with the rigid boundary in general [4].

The discrete forcing approach is proposed to overcome the problem that the elastic force source term cannot be given precisely in the Navier–Stokes (NS) governing equations [4]. The discrete forcing approach is implemented by modifying the computational stencil localized at the difference scheme boundary and imposing the boundary conditions on the immersed boundary. The discrete forcing approach can reconstruct the boundary accurately. Previous research has established two approaches to modify the computational stencil: cut-cell methods (CCM) and ghost-cell methods (GCM).

Cut-cell methods based on finite volume IB methods – in comparison to finite difference ghost-cell methods – are attractive as they enforce strict conservation of mass, momentum and energy [5], [6], [7]. The primary issue however is the presence of small cells, which results in an excessively small time step in case of an explicit scheme or a poorly conditioned matrix in case of an implicit scheme [7].

The concept of GCM is most commonly associated with the works of Mohd-Yusof [8] and Fadlun [9]. The boundary condition on the IB is enforced through the ghost-cells. Ghost-cells are defined as cells in the solid that have at least one neighbor in the fluid. For each ghost-cell, an interpolation scheme that implicitly incorporates the boundary condition on the IB is then devised [4]. Different interpolation procedures for the mirror point [10] and extrapolation procedures for the ghost point [11] can be utilized to obtain a second or even higher order accuracy [12], [13], [14], [15]. A simple GCM interpolation scheme [16] is bilinear interpolation using the information of surrounding fluid nodes. However, at high Reynolds numbers, when the resolution is marginal, linear reconstruction could lead to erroneous predictions [4]. An improved method [17] employs a linear interpolation in the tangential and a quadratic interpolation in the normal direction. Felten et al. [18] demonstrated that interpolation errors do not significantly affect the results as long as the simulation has a sufficiently fine grid and short time steps. A seminal study in this area was the work of Tseng et al. [10] who proposed methods to solve the incompressible NS equation by an implicit pressure-corrected finite volume method on Cartesian grids. Tseng et al. [10] studied the effects of the local stencil near the boundary for the problem when too few points are available for interpolation. Pan et al. [19], [20], [21] proposed a simple and stable ghost-cell method which was developed to treat the boundary condition for the immersed bodies in the flow field. Pan et al. demonstrated that the spatial accuracy of the method is second-order-accurate in the $L_2$ norm for both velocity and pressure. The GCM has shown large potential to handle different fluid–solid interaction problems, including those with moving objects [22], [23], [24] and highly complex geometries [15], [25], [26], [27]. Ji et al. [28] tested the hypersonic blunt-body shock-induced combustion phenomenon employing the cut-cell method with second-order accuracy and proved the accuracy by merging the cut cells to cope with the small-cell problem. Combining the immersed boundary method with adaptive mesh refinement (AMR) improves accuracy of numerical simulations considerably. Deiterding et al. [29] used the ghost-cell method with AMR techniques to improve computational efficiency in the case of hypersonic blunt-body flows. These studies [30], [31], [32] showed good results regarding the robustness and accuracy of the ghost-cell method.

However, the simple GCM method [18] suffers from an issue by reconstructing the value at ghost-cells that are too close to the boundary. Chi et al. [33], [34] proposed an improved method whose ghost-cells are mirrored through the boundary to farther image points. In this paper, a general boundary condition treatment based on the idea of Chi et al. [33], [34] is developed and validated. The concept of the image point is adopted and modified to construct a simple and stable bilinear reconstruction scheme. A new reconstruction stencil is carried out to eliminate some of the assumptions of the method as given in previous publications [33], [34]. The main advantage of the current approach is the ease of programming, which requires only that an improved reconstruction stencil be added to an existing code.

This paper is organized as follows: Section 2 introduces the computational model, including governing equations, numerical methods, improved ghost-cell method and hybrid method. Section 3 compares and verifies the applicability of the hybrid method for a number of cases: unit test, stationary rotating vortex, Prandtl–Meyer expansion wave, double Mach reflection, moving-shock/obstacle interaction and blunt body shock-induced combustion. All GCM methods in this section are implemented in the AMROC framework [35], [36], [37]. The CCM method used for comparison is implemented in the AMReX framework [38]. Section 4 concludes the paper.