An energy-stable scheme for a 2D simple fluid-particle interaction problem

Highlights

• We construct an energy stable scheme for the fluid-particle interaction problems.

• We prove that the continuous model and our numerical scheme satisfy the energy law.

• We introduce an easy-to-implement algorithm to apply our numerical scheme in practice.

• Numerical tests are proposed to verify the accuracy and stability of our scheme.

Abstract

We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different time steps. In the proposed numerical scheme, the tALE mesh velocity, the incompressible NS equations, and Newton's second law are solved simultaneously. We prove that under certain conditions, the new time discretization scheme satisfies an energy law. For the space discretization, the extended finite element method (XFEM) is used to solve the problem on a fixed Cartesian mesh. The developed method is first-order accurate in time and space without being momentum conservative. To verify the accuracy and stability of our numerical scheme, we present numerical experiments including the fitting of the Jeffery orbit by rotating of an ellipse and the free-falling of an elliptic particle in water.

Introduction

Fluid-particle interaction plays an important role in scientific and engineering problems such as crude oil emulsions, fluidized suspensions and sedimentation, ship maneuvering, and underwater vehicles. In this paper, we consider interaction between a rigid body and the surrounding incompressible fluid. In the past decades, two categories of models have been proposed to describe the particle surface and the hydrodynamic interaction between the fluid and the particle, including the continuum approach [23], [36], [47], [48] and the models using the direct numerical simulation (DNS) technique. The continuum approach views solid particles and fluids as interpenetrating mixtures with different viscosities that are governed by the laws of conservation. In this approach, the motion of particles and that of fluids are combined in the same mathematical framework to form a single system of equations and the boundary conditions are imposed implicitly on the particle surface. The continuum approach is flexible and can efficiently solve various kinds of fluid-particle interaction problems. However, the false response from the viscous material used to mimic the rigid objects might produce undesirable hydrodynamic effects in the case when the particle concentration is dense, or when particle-wall and particle-particle interactions are considered. Alternatively, the DNS technique treats the fluid and the particle separately and computes hydrodynamic interactions as a part of the solution. The boundary conditions that capture precisely the momentum and pressure response between the fluid and the rigid particle are imposed explicitly on the solid surfaces. The DNS technique gives a clear understanding of the mechanisms between the fluid and the particle and is well designed for many problems involving nonlinear and geometrically complicated phenomena.

For fluid-particle interaction simulations based on the DNS technique, the fluid domain is time-dependent as the particle moves inside the fluid. In order to track the particle and its surrounding fluid, several numerical simulation techniques have been developed, including the arbitrary Lagrangian-Eulerian (ALE) method, the temporary arbitrary Lagrangian-Eulerian (tALE) method, and the fictitious domain method. For the ALE method [21], [20], [22], a boundary fitted mesh is used to describe the fluid domain. The particle surface is aligned with element boundaries. The ALE mesh is updated after each time step using the ALE mapping. Moreover, when the mesh becomes too distorted, a new mesh must be generated. In order to simulate the fluid-particle problem on a fixed Eulerian mesh, the tALE method is proposed [10]. In the tALE approach, time derivatives in the fluid region are discretized by an inverse tALE mapping. The interface of the particle is approximated with the help of the extended finite element method (XFEM) [12], [40], which is a generalization of the standard Galerkin finite element method with additional degrees of freedom near the particle surface. An alternative approach is the fictitious domain method developed by Glowinski et al. [15], [16], [17], [13]. The basic idea of this approach is to extend the problem in a geometrically complex domain to a simple and regular domain by generalizing the weak formulation of flows from the fluid domain to a fictitious domain that represents the particles. This approach enforces the constraint of rigid-body motion with a distributed Lagrange multiplier to introduce an additional body force to the interior of the region of particles. The fictitious domain method is widely used to simulate fluid-particle interaction problems, including the interaction between the fluid and a large amount of rigid particles [14], [41], [42].

The main purpose of this paper is to develop an energy-stable scheme for the simulation of fluid-particle interaction problems. Energy stability becomes crucial in numerical simulations when large time step sizes are used and long-time behavior is simulated. Ensuring numerical stability for fluid-particle interaction is far from trivial. Wood et al. [45] indicated that numerical schemes based on sequential computation of fluid and particle dynamics are often unstable if there are no sub-iteration steps between computations of the fluid and the particle. To improve the stability, certain sub-iteration steps are performed between the two sets of computations [8], [39], [43]. Efforts have been made to study the stability of numerical methods such as the immersed boundary methods [26], [27], [38], [46], the block-factorization methods [3], [35], and the weak-coupling scheme using the summation-by-parts (SBP) operators [32], [28]. The SBP-based schemes are proven to be energy-stable for both time and space discretization, and they have been applied successfully in fluid-structure interaction [31]. Other studies of the stable numerical schemes are also available [34], [44]. Recently, a stable partitioned algorithm [4], [5], [6] has been presented to simulate the fluid-particle interaction problems with light or zero-mass rigid particles.

In this paper, we develop an energy-stable time discretization scheme for simulating the fluid-particle interaction problem. The energy-stable scheme is based on a DNS-type model. The incompressible Navier-Stokes equations and Newton's second law are used to describe the fluid and the particle motion respectively. To track the time-dependent fluid domain, our numerical scheme develops a bijective mapping based on the tALE mapping [10]. To ensure energy stability, the tALE mesh velocity, the fluid equations, and Newton's second law are solved implicitly and simultaneously. We prove that under a mild condition that is usually omitted in practical implementation, the new time discretization scheme satisfies a discrete energy law. The discrete energy law is similar to the energy law of the continuous system. For the discretization in space, we use the XFEM [12], [40] to handle discontinuities near the particle surface. Only a fixed Eulerian mesh is required for the entire computation, without the need to regenerate or deform the meshes.

Numerical tests are carried out to verify the stability and accuracy of the energy-stable scheme. We simulate the Jeffery orbit of a fixed ellipse rotating in a shear flow and compare the results with those obtained from the standard tALE-XFEM method [49]. It is shown that our energy-stable scheme allows the usage of large time step sizes such as $\delta t=\delta x$, while the standard tALE-XFEM method soon blows up even with a much smaller time step size. Moreover, our energy-stable scheme remains stable in long time simulation. Tests show that our algorithm is first-order accurate in both time and space. We also simulate a thin ellipse sinking in water which displays fluttering and tumbling motions. The proposed numerical scheme is shown to be energy-stable for both cases.

The rest of the paper is organized as follows. In Section 2, we introduce the governing equations and derive the energy law for the continuous system. In Section 3 we introduce the tALE method with a bijective tALE mapping between the different time steps. The energy-stable scheme is developed in Section 4. The discrete energy law of the time discretization scheme is also proved in this section. Implementation details of the proposed scheme are described in Section 5. Numerical experiments are presented in Section 6 to support the theoretical results. The paper is concluded in Section 7.