A direct-forcing fictitious domain method for particulate flows

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Abstract

A direct-forcing fictitious domain (DF/FD) method for the simulation of particulate flows is reported. The new method is a non-Lagrange-multiplier version of our previous DLM/FD code and is obtained by employing a discrete δ-function in the form of bi(tri-) function to transfer explicitly quantities between the Eulerian and Lagrangian nodes, as in the immersed boundary method. Due to the use of the collocation-point approach for the rigidity constraint and the integration over the particle domain, the Lagrangian nodes are retracted a little from the particle boundary. Our method in case of a prescribed velocity on the boundary is verified via the comparison to the benchmark results on the flow over a fixed cylinder in a wide channel and to our spectral-element results for a channel with the width of four cylinder diameters. We then verify our new method for the case of the particulate flows through various typical flow situations, including the sedimentation of a circular particle in a vertical channel, the sedimentation of a sphere in a vertical pipe, the inertial migration of a sphere in a circular Poiseuille flow, the behavior of a neutrally-buoyant sphere in Couette flow, and the rotation of a prolate spheroid in Couette flow. The accuracy and robustness of the new method are fully demonstrated, in particular for the case of relatively low Reynolds numbers and the neutrally-buoyant case.

Introduction

The non-boundary-fitted (or Cartesian grid) method has become increasingly popular for the solution of the fluid flow problems in a complex geometry or with moving boundaries. A variety of non-boundary-fitted approaches have been developed, and they can be roughly classified into two families [1], [2]: the body-force based method (e.g., [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]) and the non-body-force based method (e.g., [2], [14], [15], [16], [17]). For the former, a body-force (or momentum forcing) is introduced into the momentum equation. The fractional step scheme is often used in the body-force based methods to simplify the computation in the following way: the Navier–Stokes equations are solved for the known body-force obtained at the previous time level, with the boundary condition on the immersed boundary disregarded, and subsequently the boundary condition is used to determine the body-force. There exist various body-force based methods in the literature that differ in the way the body-force is calculated. For example, for the Lagrange-multiplier based fictitious domain method (e.g. [3], [4], [5], [6], [7]), the body-force is introduced as a Lagrange multiplier in a weak formulation, and solved from the boundary condition in an iterative way. For the elastic-force based immersed boundary method (e.g., [8], [9], [10]), the boundary (or the body) moves with the local fluid and the body-force is explicitly calculated from the displacement or deformation of the boundary (large stiffness constant as an approximation to the rigid-boundary problems). For the direct-forcing based immersed boundary method (e.g. [11], [12], [13]), the body-force is computed explicitly from the boundary condition in two typical manners: for one [11], the body-force is defined at the Lagrangian points on the boundary, and a discrete δ-function is used to interpolate the fluid velocity from the Eulerian nodes to the Lagrangian nodes for the calculation of the body-force and then to distribute the body-force from the Lagrangian nodes to the Eulerian nodes for the solution of the fluid momentum equation; for the other, the body-force is directly defined at the Eulerian nodes located in the immediate vicinity of the boundary in either the real fluid region [12] or the fictitious domain [13] (thus there is no need for the transfer of the body-force), and the target velocities at the forcing points for the calculation of the body-force is obtained by interpolation of the prescribed velocity at the boundary and other fluid velocities at the Eulerian nodes. Unlike the body-force based method, the non-body-force based method does not introduce a body-force, instead, it accounts for the boundary condition by either transforming it into independent equations [14] or using it to modify the expressions of differential operators for the Eulerian nodes in the immediate vicinity of the boundary ([15], [16], [17]). One advantage of the non-body-force based method is that the jump boundary conditions on the surface can be conveniently handled.

We are concerned with the simulation of particulate flows (fluid–structure interactions) in the present study. In principle, all aforementioned methods can be applied to the particulate flows, however, the body-force based method has been predominantly used so far. A common feature for the body-force based method is that the hydrodynamic force on the particles can be calculated from the body-force and is not necessary to be explicitly computed in order to determine the particle velocities. For the distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) method proposed by Glowinski et al. [5], the particle velocities and the body-force (Lagrange multiplier) are solved together with the implicit scheme, while for the immersed boundary method (e.g., [11], [12], [13]), they are obtained explicitly. The DLM/FD method has been successfully applied to a wide range of particulate flow problems (e.g., [6], [18], [19], [20], [21], [22]). However, its calculation of the particle velocity and body-force is a little more involved and expensive compared to the direct-forcing immersed boundary (DF/IB) method. The aim of the present study is to present a non-Lagrange-multiplier version of our previous DLM/FD code without sacrificing the accuracy and robustness by employing a discrete δ-function in the form of bi(tri-) function to transfer explicitly quantities between the Eulerian and Lagrangian nodes, as in the immersed boundary method. In our previous DLM/FD code, the bi(tri)-linear function is actually already used to interpolate the quantities from the Eulerian nodes to the Lagrangian nodes and to distribute the body-force from the Lagrangian nodes to the Eulerian nodes, however, it is not defined as a discrete δ-function to explicitly transfer (or equate) the quantities between the two frames, and therefore the body-force needs to be determined implicitly from the constraint of the velocities in the two frames being equal in the solid domain or on the boundary. For convenience, we refer to our new method as direct-forcing fictitious domain (DF/FD) method, since on one hand it is resulted from a modification over our previous DLM/FD code and on the other hand it deals with the body-force in essentially the same way as the DF/IB method. Compared to the DF/IB method for the particulate flows proposed by Uhlmann [11], in which the body-force is only distributed on the particle boundary, our body-force is distributed over the particle inner domain for the constraint that all inner fluids move as a rigid-body so that the time acceleration term of the inner fluids involved in the calculation of the hydrodynamic torque on the particles does not need to be calculated explicitly, thus circumventing the difficulty in dealing with the nearly neutrally-buoyant case in [11]. We note that other non-Lagrange-multiplier versions of the FD method have been developed by Sharma and Patankar [23] and Veeramani et al. [24]. Compared to these two methods, our method has the same feature that the constraint of the rigid-body motion for the inner fluids is used to derive an explicit expression for the particle velocities, but the form of the explicit expression and the computational schemes used are different.

The rest of the paper is organized as follows: we describe the new scheme for the explicit solution of the particle velocities and the body-force in detail in the following section. In Section 3, we first validate the method in case of prescribed velocity on the immersed boundary via the benchmark problem of the flow over a fixed cylinder, and it is shown that for the case of the singular body-force distributed only on the boundary the discrete nominal support area for the body-force is not important to the results and the Lagrangian nodes should be retracted a little from the boundary if one wants to use the body-force to calculate the hydrodynamic force on the body. We then verify our new method for the case of the particulate flows through various typical flow situations, including the sedimentation of a circular particle in a vertical channel, the sedimentation of a sphere in a vertical pipe, the inertial migration of a sphere in circular Poiseuille flow, the behavior of a neutrally-buoyant sphere in Couette flow, and the rotation of a prolate spheroid in Couette flow. Concluding remarks are given in the final section.

Section snippets

Numerical model

The method proposed in the present study is an improved version of our DLM/FD code, and only differs in that the Lagrange multiplier (i.e., body-force or momentum forcing) and particle velocities are solved in a non-iterative way. Hence, the description of the new algorithm can be started from the DLM/FD formulation proposed by Glowinski et al. [5]. However, we will derive the new algorithm directly from the primitive governing equations, since such a derivation is straightforward and can make

Flow over a fixed cylinder

We first validate the method in case of prescribed velocity on the immersed boundary via the benchmark problem of the flow over a fixed cylinder, considering that our method in this case differs from any previously reported immersed boundary method in the implementation details regarding the solution of the body-force. In addition, through this example, we aim to demonstrate that for the case of the singular body-force being distributed only on the boundary the discrete nominal support area for

Conclusions

We have presented the direct-forcing fictitious domain (DF/FD) method for the simulation of particulate flows. The new method is a non-Lagrange-multiplier version of our previous DLM/FD code and is obtained by employing a discrete δ-function in the form of bi(tri-) function to transfer explicitly quantities between the Eulerian and Lagrangian nodes, as in the immersed boundary method. Our method in case of a prescribed velocity on the boundary is verified via the comparison to the benchmark

Acknowledgements

The authors would like to acknowledge the support of National Natural Science Foundation of China (Grant Nos.: 10472104 and 10602051), and thank the anonymous referees for helpful suggestions.

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