An immersed boundary–lattice-Boltzmann method for the simulation of the flow past an impulsively started cylinder

Abstract

We present a lattice-Boltzmann method coupled with an immersed boundary technique for the simulation of bluff body flows. The lattice-Boltzmann method for the modeling of the Navier–Stokes equations, is enhanced by a forcing term to account for the no-slip boundary condition on a non-grid conforming boundary. We investigate two alternatives of coupling the boundary forcing term with the grid nodes, namely the direct and the interpolated forcing techniques. The present LB–IB methods are validated in simulations of the incompressible flow past an impulsively started cylinder at low and moderate Reynolds numbers. We present diagnostics such as the near wall vorticity field and the drag coefficient and comparisons with previous computational and experimental works and assess the advantages and drawbacks of the two techniques.

Introduction

A number of important flow phenomena in science and engineering, ranging for example from hemodynamics [3] and fish swimming [35] to robotic insects [11] and prosthetic heart valves [29], involve flows past complex geometries. The computational study of these flows requires the development of numerical techniques capable of handling complex unsteady geometries. A number of computational methods, such as finite elements, finite volume and unstructured mesh finite differences (see [23, and references therein]) have been developed that can handle complex geometries for systems such as blood flow past endovascular devices [5] or multiphase turbulent flows in realistic combustors [26]. More recently, lattice-Boltzmann methods [34] have been advocated as effective computational tools for the simulation of complex flows ranging from car aerodynamics to flows in porous media. The results of these simulations may depend however on the grid distribution [31], [18], while intensive meshing requirements may adversely affect the efficiency of the simulations for flows with time dependent, complex geometries. In the early 1970s motivated by problems in biological flows, Peskin [28], [30] pioneered the use of immersed boundary methods that require a cartesian grid and a suitable modification of the governing equations to take into account the effects of the boundaries. The simplicity in meshing and the relative ease of implementation of these techniques has led to intensive research efforts (see [24, and references therein]) making the IB technique a potent alternative for simulations of flows with complex, unsteady geometries.

In this work, we present the coupling of immersed boundary (IB) and lattice-Boltzmann (LB) methods. LB models [34] solve the Navier–Stokes equations of incompressible fluids by following the evolution of distribution functions on a lattice. Boundaries are approximated on the regular lattice in a staircase fashion making them suitable for simulations of flows past grid conforming geometries. The accuracy of the method depends on the detail of the boundary representation and efficiency requirements dictate the local refinement of the lattice [12]. A number of works have addressed the coupling of IB and LB methods. Feng and Michaelides [14], [15] report on continuous and discrete forcing models applied to sedimentation problems. Peng et al. [27] couple a discrete forcing model to a multi-relaxation time LB model considering variable resolution. Shi and Phan-Tien [32] propose a model that integrates distributed Lagrange multipliers and fictitious domain methods in the framework of LB models. In this paper, we present an alternative model coupling an LB model of the incompressible Navier–Stokes equations with an IB technique. The algorithmic novelty of our work consists of the simple way in which the LB model incorporates the IB forcing term. In addition, we investigate two variants of coupling the boundary force with the Eulerian grid namely: a direct forcing (DF) and an interpolated forcing (IF) distinguished by the way in which the forcing term from the boundary is computed on the grid nodes.

The present method is validated on simulations of the flow past an impulsively started circular cylinder at moderate Reynolds numbers. The flow past an impulsively started cylinder is a well-established benchmark problem and the subject of detailed experimental investigations for Reynolds numbers ranging from 40 to 10,000 [9], [10], [4]. The simulation of this flow is challenging as it requires the accurate tracking of the initial sharp vorticity generation on the cylinder surface and the precise identification of the subsequent regions of the disruption of the primarily shedded vortices by their induced secondary vortical fields. There is a plethora of results from simulations using different numerical methods such as finite difference schemes [8], [2], vortex methods [20], wavelet collocation schemes [19], and lattice-Boltzmann models [22]. In these simulations, diagnostics such as streamlines, vorticity contours and in particular the drag coefficient have been used to elucidate the advantages and drawbacks of the different methodologies. We present such diagnostics and use the value of the asymptotic drag coefficient to establish the order of accuracy of our method.

The paper is organized as follows: we first present in Section 2 the hybrid scheme and discuss the LB model and the two variants of the IB algorithm. In Section 3, we present simulations of the flow around an impulsively started cylinder at low and moderate Reynolds numbers. We quantify the convergence of both IB approaches and conclude in Section 4.