The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties
Introduction
Much attention has recently been devoted to the extension of Cartesian grid flow solvers to complex geometries by immersed boundary (IB) methods (see [28], [39] for recent reviews). In these methods, the irregular boundary is not aligned with the computational grid, and the treatment of the cut-cells, cells of irregular shape which are formed by the intersection of the Cartesian cells by the immersed boundary, remains an important issue. Indeed, the discretization in these cut-cells should be designed such that: (a) the global stability and accuracy of the original Cartesian method are not severely diminished and (b) the high computational efficiency of the structured solver is preserved.
Two major classes of IB methods can be distinguished on the basis of their treatment of cut-cells. Classical IB methods such as the momentum forcing method introduced by Mohd-Yusof and co-workers [41], [14], use a finite-volume/difference structured solver in Cartesian cells away from the irregular boundary, and discard the discretization of flow equations in the cut-cells. Instead, special interpolations are used for setting the value of the dependent variables in the latter cells. Thus, strict conservation of quantities such as mass, momentum or kinetic energy is not observed near the irregular boundary. The most severe manifestations of these shortcomings is the occurrence of non-divergence free velocities or unphysical oscillations of the pressure in the vicinity of the immersed boundary [43], [30]. Numerous revisions of these interpolations are still proposed for improving the accuracy and consistency of this class of IB methods [30], [3], [49], [43].
A second class of IB methods (also called cut-cell methods or simply Cartesian grid methods, see [66], [59], [61], [32], [12], [9], [40]) aims for actually discretizing the flow equations in cut-cells. The discretization in the cut-cells is usually performed by ad hoc treatments which have more in common with the techniques used on curvilinear or unstructured body-conformal grids than Cartesian techniques. Most notable is the cell merging technique used by Ye et al. [66] and Chung [9] that merges a cut-cell with a neighboring Cartesian cell to form a new polygonal cell with more than four neighbors. The discretization stencil in this newly formed cell loses thus the five-point structure (in 2D) of Cartesian methods. Such treatments of the cut-cells generate a non-negligible bookkeeping to discretize the flow equations and actually solve them, and it is difficult to evaluate the impact of these treatments on the computational cost of the flow simulations.
The purpose of this article is to present a new IB method for incompressible viscous flows which takes the best aspects of both classes of IB methods. This method, called the LS-STAG method, is based on the symmetry preserving finite-volume method by Verstappen and Veldman [63], which has the ability to preserve on non-uniform staggered Cartesian grids the conservation properties (for total mass, momentum and kinetic energy) of the original MAC method [24]. The LS-STAG method has the following distinctive features:
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A sharp representation of the immersed boundary is obtained by using a signed distance function (i.e. the level-set function [46], [47]) for its implicit representation. Level-set methods were devised by Osher and Sethian [48] for the solution of computational physics problems involving dynamic interfaces. So far for incompressible flows, the main application areas of level-set methods have been the computation of two-phase flows [56]. In the present paper, the level-set function enables us to easily compute all relevant geometry parameters of the computational cells, reducing thus the bookkeeping associated to the handling of complex geometries.
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In contrast to classical IB methods, flow variables are actually computed in the cut-cells, and not interpolated. Furthermore, the LS-STAG method has the ability to discretize the fluxes in Cartesian and cut-cells in a consistent and unified fashion: there is no need for deriving an ad hoc treatment for the cut-cells, which would be totally disconnected from the basic MAC discretization used in the Cartesian cells.
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For building our discretization, we have required the strict conservation of global quantities such as total mass, momentum and kinetic energy in the whole fluid domain, which are crucial properties for obtaining physically realistic numerical solutions [1], [42], [63]. To achieve these preservation properties up to the cut-cells, we had to precisely take into account the terms acting on the immersed boundary in the global conservation equations, at both continuous and discrete levels. As a result, the convective, pressure and viscous fluxes have been unambiguously determined by these requirements, and the boundary conditions at the immersed boundary have been incorporated into these fluxes with a consistent manner.
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From the algorithmic point of view, one of the main consequences is that the LS-STAG discretization preserves the five-point structure of the original Cartesian method. This property allowed the use of an efficient black box multigrid solver for structured grids [62], where no ad hoc modifications had to be undertaken for taking account of the immersed boundary.
We also mention that a first attempt at constructing an energy-conserving IB method from the ideas of Verstappen and Veldman can be found in [12]. In this paper however most of the computational aspects of the method has been skipped: it appears that computation of the geometry parameters of the cut-cells, shape of the velocity control volumes, imposition of the boundary conditions at the IB surface and computation of the diffusive terms are different than in the LS-STAG method.
The paper is organized as follows. In Section 2, we recall the notations and salient properties of the staggered Cartesian mesh, and then we present the LS-STAG mesh, its extension for the handling of immersed boundaries. Section 3 presents the LS-STAG discretization in the case the immersed boundary is steady. First, we will recall the global conservation laws for total mass, momentum and kinetic energy that will be used for deriving the LS-STAG method. Then, we will present the discretization of the continuity equation, which is valid in both cut-cells and Cartesian cells. As a matter of fact, we shall observe that the consistency of the discrete continuity equation is a crucial point for building an energy and momentum preserving method for incompressible flows. In the next subsections, we will impose kinetic energy conservation upon our numerical scheme for completely characterizing the discrete pressure and convective fluxes in the cut-cells, and total momentum conservation for the determination of the viscous fluxes. We mention that the discretization of the viscous fluxes has been by far the most intricate part of the LS-STAG discretization in the cut-cells. Section 4 is devoted to numerical tests on canonical flows at low to moderate Reynolds number for assessing the accuracy and robustness of the LS-STAG method. Comparisons with an unstructured solver in terms of CPU time and accuracy will be given. Finally, we will present some results for one of the most appealing features of IB methods: the ability to compute flows with immersed moving boundaries on fixed cartesian grids, without the need for domain remeshing at each time step.
Section snippets
Preliminaries and description of the LS-STAG mesh
Let be a rectangular computational domain and its surface. The governing equations are the incompressible Navier–Stokes equations in integral form. In the following, we will consider the finite-volume discretization of the continuity equation:where is the velocity, and the momentum equations in the x and y directions, respectively:where p is the pressure and is the kinematic viscosity.
Global conservation laws for viscous incompressible flows
As early as the 1950s, it was recognized that for physically realistic integration of dynamical systems such as the fluid dynamics equations, linear and quadratic invariants of the continuous equations should be conserved by the numerical scheme [1], [34]. For the incompressible Navier–Stokes equations (1), (2a), (2b), these flow invariants are the total mass in the whole fluid domain , total momentum and, in the case of vanishing viscosity, total kinetic energy
Taylor–Couette flow
First, the spatial accuracy of the LS-STAG method is assessed on the Taylor–Couette flow between two concentric circular cylinders, as described in Fig. 7(left). The flow dynamics is governed by the Taylor number , which is the ratio between the centrifugal force and the viscous force:Below the stability threshold [23], the steady stable solution is purely orthoradial, such that its Cartesian components read for :
Concluding remarks and implementation issues
In this paper, we have developed and analyzed a new IB/cut-cell method for incompressible viscous flows. For building the LS-STAG discretization in 2D, the methodology we have followed can be roughly summarized as: enumerating the various types of cut-cells, and then discretizing the fluxes in each cut-cell such that the global conservation properties of the Navier–Stokes equations are satisfied at the discrete level.
If we adopt the terminology used for computational methods for multiphase
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