A novel iterative direct-forcing immersed boundary method and its finite volume applications

Abstract

We present a novel iterative immersed boundary (IB) method in which the body force updating is incorporated into the pressure iterations. Because the body force and pressure are solved simultaneously, the boundary condition on the immersed boundary can be fully verified. The computational costs of this iterative IB method is comparable to those of conventional IB methods. We also introduce an improved body force distribution function which transfers the body force in the discrete volume of IB points to surrounding Cartesian grids totally. To alleviate the demanding computational requirements of a full-resolved direct numerical simulation, a wall-layer model is presented. The accuracy and capability of the present method is verified by a variety of two- and three-dimensional numerical simulations, ranging from laminar flow past a cylinder and a sphere to turbulent flow around a cylinder. The improvement of the iterative IB method is fully demonstrated and the influences of different body force distribution strategies is discussed.

Introduction

Fluid–solid interaction (FSI) is of great importance in many disciplines ranging from civil engineering (sediment entrainment near river bed) to medical science (drug transportation in blood cell flow). Recently, the immersed boundary (IB) method has provoked enormous interest amongst scientists and engineers for its potential capability of tackling such complex FSI problems in an easy and straightforward manner. In this paper, we describe a novel iterative IB approach which improves the solution accuracy considerably with little additional computational cost.

The IB method was first introduced by Peskin [1] in the simulation of blood flow around the flexible leaflet of a human heart. In the framework of the IB method, the fluid motion equations are discretized on a fixed Cartesian grid which, generally, does not conform to the geometry of moving solids. As a result, the boundary conditions on the fluid–solid interface which manifest the interaction between fluid and solid cannot be imposed straightforwardly. Instead, an extra singular body force is added into the momentum equation to take such interaction into account. After the pioneering work of Peskin [1], numerous IB approaches and results have been reported; readers are referred to the reviews by Peskin [2] and Mittal and Iaccarino [3].

Traditionally, the IB approach has been divided into two main categories: the feedback-forcing approach and the direct-forcing approach. The feedback-forcing IB approach, developed by Goldstein et al. [4], determines the singular body force using the differences between the calculated displacement and velocity and the expected ones on the IB points with two negative empirical constants. These two empirically determined constants, acting as spring coefficients, are determined on a case-by-case strategy and should be large enough to suppress the disturbance of the motion of solids. Although this feedback-forcing approach was proven to be capable of handling problems of flows past stationary and moving cylinder (refer to [5]), it suffers severely from the instability caused by the inherent feedback nature of the method. To tackle this difficulty, the direct-forcing IB approach was introduced by Mohd-Yusof [6] and Fadlun et al. [7]. In this approach, the body force term is directly deduced from the momentum equation by setting the velocity at IB points to the desired velocity using interpolation/distribution functions. In this manner, the boundary conditions are ‘exactly’ imposed.

The direct-forcing IB method has been used successfully by many researchers in various applications. Uhlmann [8] presented a direct-forcing IB approach using the high-order regularized delta function suggested by Peskin [2] in the velocity interpolation and the force distribution. The body force is distributed to adjacent Cartesian grids both inside and outside of the immersed boundary within a width of several grid spacings to give a smooth force transfer between the IB points and the Cartesian grids. Various applications, such as laminar flow over stationary and oscillating cylinders, sedimentation of two-dimensional circular discs and spheres in a laminar regime, have been investigated. Good agreement has been observed between numerical results and the published data. Contrary to Uhlmann’s efforts to represent the immersed boundary smoothly, Zhang and Zheng [9] (referred as Z&Z hereafter) used a simple linear delta function instead and distributed the force only to the adjoining Cartesian grids inside the immersed boundary for the purpose of generating a sharp fluid–solid interface. Their results have been verified for laminar flows past fixed and moving cylinders and also a stationary sphere. Shu et al. [10] pointed out that the forcing process of interpolating velocity and distributing force ‘is actually an iterative procedure, trying to satisfy the no-slip boundary conditions on the solid wall’. In his work, he suggested correcting the velocity on the Cartesian grids near the immersed boundary directly to the boundary conditions on the fluid–solid interface, instead of using the interpolation/distribution functions. Simplicity, robustness and ease of implementation were reported. Other applications of the direct-forcing IB method include the application of a hybrid IB and lattice Boltzmann method [10], [11] and a turbulence wall-layer model for large eddy simulation (LES) using the IB method [12], [13], [14].

As we know, the correct calculation of the body force, which represents the fluid–solid boundary conditions, is a key issue for all type of IB approaches. In the direct-forcing IB approach, to calculate the body force at time level $n+\frac{1}{2}$ (second-order time discretization), we need know the pressure pⁿ first. Unfortunately, pⁿ is unknown until we solve the Navier–Stokes equations. A common treatment is using the old pressure pⁿ⁻¹. This inevitably introduces errors into the body force term especially in the simulation of turbulence flows in which the fluctuation of pressure is intensive. On the other hand, the widely-used conjugated gradient pressure solver for the pressure Poisson equation (PPE) is iterative. In each iteration, the updated pressure p^n,k is closer to the real solution than those before. Here, subscript k is the index of pressure iteration. So, it is a better approximation comparing with pⁿ and can generate a more accurate body force. In this study, we utilize this iterative character of the PPE solver and a novel iterative direct-forcing IB method is presented. In this iterative method, the body force is updated within the pressure iterations using the intermediate pressure p^n,k. Higher accuracy of the solution is therefore expected to be achieved.

On the aspect of efficiency, this iterative IB method does not cost much extra computational cost. On one hand, about 90% of the computational time, even higher for parallel computing, is consumed by the PPE solver. On the other hand, it is not necessary to solve the PPE for the intermediate pressure iteratively until convergence reached. Instead, the iterative PPE solver is only iterated once in each force-pressure iteration. So, the iteratively updated body force would not undermine the total computational efficiency significantly.

A new body force distribution function and a simple wall-layer model (WLM) for turbulence near-wall modelling are also introduced in this paper.

This paper is organized as the follow. The methodology of the novel iterative IB method will be introduced in Section 2.2. Then, the body force distribution function and the WLM will be presented in Sections 2.3 Interpolation and distribution strategies, 2.4 Wall-layer model, respectively. In Section 3, we apply the iterative method together with the body force distribution function and the WLM to a series of numerical tests. The merits and drawbacks of the present method will be discussed in Section 4.