An implicit ghost-cell immersed boundary method for simulations of moving body problems with control of spurious force oscillations

Abstract

A fully-implicit ghost-cell immersed boundary method for simulations of flow over complex moving bodies on a Cartesian grid is presented. The present immersed boundary method is highly capable of controlling the generation of spurious force oscillations on the surface of a moving body, thereby producing an accurate and stable solution. Spurious force oscillations on the surface of an immersed moving body are reduced by alleviating spatial and temporal discontinuities in the pressure and velocity fields across non-grid conforming immersed boundaries. A sharp-interface ghost-cell immersed-boundary method is coupled with a mass source and sink algorithm to improve the conservation of mass across non-grid conforming immersed boundaries. To facilitate the control for the temporal discontinuity in the flow field due to a motion of an immersed body, a fully-implicit time-integration scheme is employed. A novel backward time-integration scheme is developed to effectively treat multiple layers of fresh cells generated by a motion of an immersed body at a high CFL number condition. The present backward time-integration scheme allows to impose more accurate and stable velocity vectors on fresh cells than those interpolated. The effectiveness of the present fully-implicit ghost-cell immersed boundary method coupled with a mass source and sink algorithm for reducing spurious force oscillations during simulations of moving body problems is demonstrated in a number of test cases.

Introduction

In recent years, immersed boundary methods have received special attention for simulations of fluid flow in complex configurations or over moving bodies [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [18], [19], [20]. Compared to methods using body-conforming grids, immersed boundary methods usually utilize a simple Cartesian grid and impose necessary boundary conditions on non-grid conforming surfaces of immersed bodies by assigning proper body forces. The use of an immersed boundary method significantly reduces the computational cost and time by avoiding repeated generation of body-fitted grids during moving-body simulations.

Immersed boundary methods are categorized into two major groups [21]. Methods in the first group utilize continuous body forces [2], [3], [4], [5], [6], [7], [8], which are added to the non-discretized Navier–Stokes equations. The continuous forcing immersed boundary methods have been popular for simulations of the interaction between fluid flow and elastic immersed structures. However, it is reported that representation of sharp boundaries and rigid objects are difficult for immersed boundary methods based on continuous forcing because of the restriction imposed by numerical stability [21]. In order to maintain the numerical stability, the continuous forcing methods model interfaces between the fluid and objects with diffuse boundaries and allow a relatively small time-step size for integration of the governing equations.

Methods in the second group correspond to discrete forcing immersed boundary methods and impose discrete body forces on the computational cells close to the surfaces of immersed bodies to realize necessary boundary conditions [1], [9], [10], [11], [13], [14], [15], [16], [18], [19], [20]. Immersed boundary methods in this category allow to use a larger time-step size than that allowed by continuous forcing immersed boundary methods and, with a proper treatment of discrete forcing, permit sharp representation of immersed boundaries [18], [20].

However, discrete forcing immersed boundary methods are generally known to suffer from the generation of spurious force or pressure oscillations on the surface of a moving immersed body [1], [15], [16], [20]. Uhlmann [15] attempted to reduce spurious force oscillations by combining a direct forcing method of Fadlun et al. [10] and discrete delta functions of Peskin [2]. To control the spurious force oscillations during a moving body simulation, Yang and Balaras [16] proposed an elaborate extrapolation technique, which assigns the velocity and pressure to grid cells where solid becomes fluid due to a body motion. Later, Yang and Balaras’ extrapolation procedure was simplified by Yang and Stern [17]. Recently, Lee et al. [1] identified two major sources of spurious force oscillations through a numerical experiment. One source is related to the spatial discontinuity in the pressure field across the immersed boundary where grid cells, which were previously inside an immersed body, become fluid cells due to a body motion. The other source of spurious force oscillations is associated with the temporal discontinuity in the velocity field where fluid cells turn into solid cells with a body motion. Lee et al. also found that spurious force oscillations tend to be reduced by decreasing the grid spacing and increasing the time-step size. Seo and Mittal [20] identified the primary cause of spurious force oscillations as the non-conservation of fluid mass near the immersed boundary, which is a result of the violation of geometric conservation across the immersed boundary. Seo and Mittal reported that a combination of a sharp-interface immersed boundary method and a cut cell method significantly improves the geometric conservation, thereby reducing spurious force oscillations.

Motivated by the recent understanding of sources of spurious force oscillations, in the present study, a new immersed boundary method which provides highly enhanced capability for controlling the generation of spurious force oscillations in moving-body simulations, is developed. The present method is based on a ghost-cell method [18] coupled with a mass source/sink algorithm [11]. A mass source/sink method is used to better conserve the fluid mass in the vicinity of immersed boundaries, thereby reducing one of the sources of spurious force oscillations [1], [20].

Unlike the original ghost-cell method [18], in the present method, the governing equations are integrated in time using a fully implicit method. A fully-implicit time integration of the governing equations allows to use a broad range of ratios of the time-step size to the grid spacing, thereby providing enhanced control of spurious force oscillations. However, a special treatment for multiple layers of fresh cells, due to the use of a large time-step size, is required. To effectively and accurately deal with multiple layers of fresh cells, a novel fresh-cell treatment algorithm based on a backward time-integration scheme is also developed. As will be shown in the following sections, the present fully-implicit ghost-cell immersed boundary method coupled with a mass source/sink algorithm is found to effectively suppress spurious force oscillations during simulations of flow over moving bodies.

In the following section, the present numerical methods including a fully-implicit time-integration method, a ghost-cell immersed boundary method with mass sources/sinks, and a backward time-integration method for assigning velocity vectors to fresh cells, are described in detail. In Section 3, results from test simulations for demonstrating the spatial and temporal accuracy of the method and the effectiveness of the new methodology for reducing spurious force oscillations, are presented. Finally, concluding remarks are followed in Section 4.