A ghost-cell immersed boundary method for flow in complex geometry

Abstract

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.

Introduction

In computational fluid dynamics, including geophysical fluid dynamics (GFD), the primary issues are accuracy, computational efficiency, and, especially, the handling of complex geometry. All large-scale geophysical flows involve complex three-dimensional geometry and turbulence. Accurate representation of multi-scale, time-dependent physical phenomena is required. A grid that is not well suited to the problem can lead to unsatisfactory results, instability, or lack of convergence.

The development of accurate and efficient methods that can deal with arbitrarily complex geometry would represent a significant advance. The immersed boundary method (IBM) has recently been demonstrated to be applicable to complex geometries while requiring significantly less computation than competing methods without sacrificing accuracy [8], [44]. The IBM specifies a body force in such a way as to simulate the presence of a surface without altering the computational grid. The main advantages of the IBM are memory and CPU savings and ease of grid generation compared to unstructured grid methods [44]. Bodies of almost arbitrary shape can be dealt with. Furthermore, flows with multiple bodies or islands may be computed at reasonable computational cost.

The IBM was first introduced by Peskin [36]. More recently, Goldstein et al. [15] and Saiki and Biringen [38] published extensions. They employed feedback forcing to represent the effect of solid body. The feedback force is added to the momentum equation to bring the fluid velocity to zero at the desired points. However, this technique may induce spurious oscillations and restricts the computational time step [15], effectively limiting the technique to two-dimensions. Mohd-Yusof [33] suggested an approach that introduces a body-force f such that the desired velocity distribution V is obtained at the boundary $\text{Ω}$. In principle, there are no restrictions on the velocity distribution V or the motion of $\text{Ω}$. He implemented the method for a complex geometry in a pseudo-spectral code while avoiding the need of a small computational time step. The method costs no more than the base computational scheme. Fadlun et al. [8] applied this approach to a three-dimensional finite-difference method on a staggered grid and showed that the approach was more efficient than feedback forcing.

A number of other immersed boundary methods have been applied to problems of irregular geometry. Calhoun and LeVeque [3], [4] proposed a streamfunction-vorticity method to model irregular shapes in a Cartesian grid. The irregular boundary is represented by discontinuity conditions. They extended the method developed by McKenney et al. [32] to solve a Poisson equation on an irregular region using a Cartesian grid. Pember et al. [35] presented an adaptive Cartesian mesh solver for the Euler equations. Their method treats the boundary cells as regular cells, thus avoiding instability problems. Almgren et al. [2] developed a Cartesian grid projection method for the incompressible Euler equations in complex geometry. The same group also proposed a second-order accurate method for solving the Poisson equation on two-dimensional Cartesian grids with embedded boundaries [21]. McCorquodale et al. [31] extended this approach to the solution of the time-dependent heat equation. On the other hand, Udaykumar et al. [42] and Ye et al. [46] have presented a finite-volume Cartesian method without momentum forcing. They reshaped the immersed boundary cells to fit the local geometry and used quadratic interpolation to calculate the fluxes across the cell faces while preserving second-order accuracy. They showed that the method is applicable to moving geometry problems. However, the above studies mainly focus on two-dimensional applications. Particularly, the streamfunction-vorticity method is hard to extend to three dimensions. Kirkpatrick et al. [24] present a second-order accurate IBM on a non-uniform, staggered three-dimensional Cartesian grid. The approach requires truncating the Cartesian cells at the boundary to create new three-dimensional cells which conform to the shape of the surface. The reshaped-cell method developed by Kirkpatrick et al. [24], Udaykumar et al. [42], and Ye et al. [46] is very similar to the shaved cell approach in MIT General Ocean Model (MITGCM) [1]. However, the implementation for the reshaped cell approach is complicated and only no-slip boundary conditions can be applied. Kim et al. [22] developed an immersed boundary method that uses both momentum forcing and mass sources/sinks. An extensive review of the immersed boundary methods for turbulent flow simulations can be found in [19].

In this paper, we extend the idea of Fadlun et al. [8] and Verzicco et al. [43] via a ghost cell approach. In Fadlun et al. [8], the velocity at the first grid point outside the body (u_i in Fig. 1) is obtained by linearly interpolating the velocity at the second grid point (u_i+1) and the velocity at the body surface (V), see Fig. 1. This approach applies momentum forcing within the flow field. The interpolation direction (the direction to the second grid point) used by Fadlun et al. [8] is either the streamwise (x) or the transverse (y) direction. They also successfully implemented the immersed boundary algorithm in large-eddy simulation (LES) of turbulent flow in a motored axisymmetric piston-cylinder assembly [44]. This approach does not reduce the stability of the underlying time-integration scheme and very good quantitative agreement with experimental measurements was obtained. For comparable accuracy, the computational requirements for the IBM approach are much lower than simulations on an unstructured, boundary-fitted mesh as given in previous published paper [17], [44].

The current approach attempts to achieve higher-order representation of the boundary using a ghost zone inside the body. The ghost cell method is very popular for treating two-phase flows, obtaining accurate discretization across the interface [9], [10], [45]. The particular method proposed by Fedkiw et al. [9], [10] is known as the Ghost Fluid Method (GFM) and was developed to capture discontinuities such as shocks, detonations and deflagrations. They also used the technique to solve a variable coefficient Poisson equation on an irregular domain using a Cartesian grid [14], [28]. Forrer and Jeltsch [12] provided a higher-order wall treatment based on Cartesian grids using the ghost-cell idea. However, the method has been implemented only for two-dimensional compressible inviscid flows with symmetry boundary conditions. In GFD, the ghost cell method promises to not only represent realistic complex geometry but also provide the flexibility needed to impose various boundary conditions including a log-law boundary condition.

We describe the systematic treatment of various boundary conditions in Section 2. The approach imposes the specified boundary condition by extrapolating the variable to a ghost node inside the body. High-order extrapolation is used to preserve the overall accuracy. The present approach is more flexible with respect to the incorporation of boundary conditions. In order to verify the accuracy of the IBM, flow over a circular cylinder and a three-dimensional turbulent flow over wavy boundary are simulated using LES. Both results are compared with published experiments and boundary-fitted grid simulations. We also extend the current approach to an existing ocean model and compared the IBM results with previous stair-step and partial-cell ones. The main advantage of the current approach is the ease of programming, which requires only that an immersed boundary module be added to an existing code.

The current method can readily be implemented in any existing Cartesian grid code. This paper is organized as follows. Section 2 introduces the governing equations and numerical implementation of the method. The generalized ghost cell method and the polynomial reconstruction schemes are laid out. Different boundary conditions and the implementation to both non-staggered and staggered grids are discussed. Section 3 validates the approach for flow over a cylinder and evaluates the accuracy. We also extend the new method to LES of three-dimensional turbulent flow over wavy boundary using a Cartesian grid and compares the results with a well-resolved boundary-fitted grid simulation. Section 4 illustrates the implementation to MIT global circulation model (MITGCM) and compares with previous methods for a geophysical flow. Finally, conclusions are drawn in Section 5.