On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries
Introduction
In recent years, the immersed-boundary method that is based on the structured mesh has gained considerable popularity in computational fluid dynamics for solving complex and moving-boundary problems. Despite its wide applications, so far there is not a unified definition of the method, possibly because there are many variations in the existing implementations. Here we follow the classification approach by Mittal and Iaccarino [1] where the immersed-boundary method is in general classified into two types. One type involves a diffused boundary whose effect on the flow field is incorporated as a volumetric force spread into the bulk fluid, typically within the distance of a few grid cells from the physical boundary [2], [3]. The volumetric force may be determined from the constitutive law in case of an elastic boundary [2], [4], or by a feedback mechanism in which the force depends on the difference between the interpolated velocity at the interface and the desired boundary condition [3]. The other type of the immersed-boundary method retains the singular representation of the physical boundary and thus the nature of the surface force exerted by the boundary on the adjacent fluid. This type of “sharp-interface” methods can typically achieve higher order of accuracy than the “diffuse-interface” methods. Several distinct sharp-interface approaches have been formulated in the past to treat the boundary conditions at the fluid–solid interface. For example, in the “cut-cell” approach [5], a finite-volume scheme is designed to represent the conservation equations for the irregular cells cut through by the boundary, whereas the bulk flow is discretized using the standard finite-difference method. In the method presented by Li and coworkers [6], [7], the solution experiences discontinuities across the physical interface immersed in the domain, and the finite-difference formulae involving the nodes across the interface are corrected by taking into consideration of the discontinuities.
In another type of sharp-interface methods, an unknown forcing term is introduced only at the nodal points immediately next to the fluid–solid interface, whose direction and magnitude are such that the boundary conditions at the location of the fluid–solid interface are satisfied. The forcing does not have to be explicitly calculated but can be incorporated through a local flow field reconstruction around the forcing points. To reconstruct the flow locally, an interpolation scheme is applied, and the pressure and velocity information at the fluid–solid interface are included as input data in the scheme. Therefore, the boundary conditions at the interface are enforced through the interpolation, and actual evaluation of the forcing is never needed. Since there is no feedback iteration involved, this method is also termed “direct forcing” approach. Many existing implementations fall into this category [8], [9], [10], [11], [12], [13], [14], [15].
In the direct-forcing approach, the construction of the interpolation stencil is flexible and may take several topological forms. Fig. 1 shows some of the examples of the stencil. For simplicity, we only use a non-staggered grid for illustration. The interpolation points may be located either on the fluid side of the interface (Fig. 1a and c), or on the solid side (Fig. 1b and d). In the latter case, the values of the flow variables at the points inside the solid body can be considered a smooth extrapolation of the physical flow field (and thus, no discontinuity across the interface is involved). In Fig. 1a and b, the interpolation is carried out along the direction of one coordinate. Given boundary conditions at the body-intercept with the coordinate line (unfilled circle in the figure), the fluid velocity at the node marked by a filled circle or square is interpolated from the flow field, and for the rest of the nodes on the fluid side, a standard finite-difference stencil can be applied to discretize the Navier–Stokes equation. Examples of previous works that adopted this strategy include Fadlun et al. [8] and Berthelsen and Faltinsen [13] among others.
In Fig. 1c and d, a two-dimensional local region around the interpolation point is chosen, and the normal intersect of the point with the interface is used to determine the region of support in the stencil (correspondingly, a three-dimensional region is chosen for a 3D problem). This strategy, used by several previous works [10], [11], [12], has been more popular compared to the unidirectional interpolation shown in Fig. 1a and b since using the closest point on the interface in the interpolation would reduce the numerical error. In addition to these examples, other flow reconstruction strategies have also been adopted, e.g., the least squares fitting [15] where the reconstruction is independent of the mesh topology.
Compared to the other sharp-interface methods such as the cut-cell [5] and the discontinuity methods [6], [7], the direct-forcing or flow-reconstruction approach is much simpler in formulation and implementation. In addition, the reconstruction procedure does not incur significant computational cost, and like the other methods, it maintains the order of accuracy of the finite-difference discretization of the bulk flow. Given its advantages, the direct-forcing approach is particularly attractive and has been applied in many problems, especially in biological flows [11], [12] where the boundaries are typically highly complex and a boundary-conforming mesh is difficult to generate. However, one drawback of the method is that it is prone to temporal oscillations when the boundary is moving [16], [13], [14], [17]. Specifically, pressure fluctuations may happen when a boundary moves across the nodal points on the fixed volumetric mesh and the numerical description of the boundary nodes changes instantaneously between the standard finite-difference formula and the flow reconstruction. To illustrate the problem, we use the interpolation stencil shown in Fig. 2 as an example and provide a brief explanation. As shown in Fig. 2a, when the boundary advances into the fluid region, some of the interpolated nodes may become occupied by the solid body, and nearby nodes in bulk fluid region will thus be defined as the new locations of interpolation. Correspondingly, the stencil at the latter nodes and the numerical description associated with the stencil changes immediately from those for the discrete Navier–Stokes equation to those for the flow field interpolation. Similarly, the immediate switch of the stencil may occur for some of the nodes when the immersed boundary retreats from the fluid region, as shown in Fig. 2b. Such instantaneous change of the numerical description at the boundary nodes creates a temporal discontinuity in the velocity. The discontinuity is further amplified by a factor of 1/Δt for the right-hand side of the pressure Poisson equation when solving an incompressible flow, thus causing the force to oscillate significantly. From this perspective, the artificial oscillations as seen previously are caused by the inconsistent treatments between the boundary nodes and the bulk flow, and sudden change of the numerical descriptions from one time step to next has created the temporal jump. In Section 2.2, we will give more detailed discussion of this problem.
It has been limited study about the numerical oscillation associated with the direct-forcing approach. Uhlmann [16] pointed out that the methods of Kim et al. [9] and Fadlun et al. [8] had led to strong force oscillations when simulating flows interacting with rigid particles, and thus he adopted a diffuse-interface approach instead. Berthelsen and Faltinsen [13] dealt with stationary-boundary flows and only pointed out the potential problem with moving boundaries. Pan and Shen [14] illustrated the force oscillations that appeared in their simulation for a moving-cylinder problem, but reduced the oscillations by increasing the size of time step. In another work, Liao et al. [17] introduced a forcing term within the solid body when solving the momentum equation. The treatment appears to suppress the force oscillations in their numerical tests. However, it is not clear why the treatment would work or how the treatment could be extended to other direct-forcing implementations.
Although the situation of emergence or disappearance of the fluid nodes near a moving boundary was previously discussed, e.g., in Yang and Balaras [11] and Mittal et al. [12], the discussions were only limited to the closure problem of the discrete equation system. For example, the velocity history at the newly emerged fluid nodes is not available but may be needed when computing the explicit terms in the momentum equation. To close the equations, one can either calculate the velocity of previous time steps at those nodes through a field extrapolation [11], or modify the temporal scheme for the momentum equation at the nodes [12]. These special treatments do not address the present numerical oscillation issue caused by the instantaneous change of the computational stencils as the grid points move in or out of the solid body. To suppress such oscillations, one could increase the grid resolution to reduce the numerical error in both discretization of the Navier–Stokes equation and the flow field reconstruction, so that the two approximations would be closer to each other and both are closer to the exact solution. As a result, the temporal discontinuity caused by the stencil transition is also reduced. However, the computational cost would rise inevitably as the grid is refined.
The present discussion is limited to the direct-forcing immersed-boundary methods for viscous incompressible flows. We point out that there are several other methods using fixed and structured grids for moving-boundary problems. For example, a finite-element method combined with a fictitious-domain formulation was developed to simulate particle-laden flows, where the rigid-body motion inside the particle volume is enforced through Lagrange multipliers; a penalty/fictitious-domain method was designed to handle solid surfaces and to simulate particle-laden or multiphase flows [18], [19], [20]; and a ghost-fluid approach was developed to solve compressible flows [21].
The goal of this paper is to propose a remedy approach that avoids the instantaneous change of the numerical description near the immersed boundary. Such an approach can be implemented together with previous direct-forcing immersed-boundary methods regardless the interpolation stencil being used. We will show that in our formulation, the numerical oscillations in the moving boundary problems associated with the sudden change of the stencils are effectively reduced and meanwhile, the computational cost does not increase significantly. We will describe the details of the immersed-boundary formulation in Section 2 and present the simulation results for both two- and three-dimensional (2D and 3D) problems in Section 3. Conclusions and further discussions are given in Section 4.
Section snippets
Governing equations and finite-difference discretization
The flow considered here is governed by the three-dimensional, viscous incompressible momentum equation and the continuity equation,where ui is the velocity, ρ and ν are the constant density and viscosity, and p is the pressure. The governing equations are discretized on a nonuniform Cartesian grid using a cell-centered, non-staggering arrangement (Fig. 3) of the primitive variables, ui and p. The incompressible momentum equation is integrated in time
Results and discussions
The immersed-boundary method described in Section 2 has been implemented in both 2D and 3D moving-boundary problems. We would like to point out that several important algorithms are based on those described by Mittal et al. [12], who formulated a versatile immersed-boundary method for 3D complex boundaries such as biomimetic swimming and flying problems. For example, the algorithms for determining inside and outside of a 3D body, projection of a grid point onto a general immersed boundary, and
Conclusions
A hybrid formulation has been presented for a family of immersed-boundary methods which retain the sharp-interface representation of the solid body surface and employ local flow reconstruction to facilitate the finite-difference discretization near the immersed boundary.
The present formulation combines both flow reconstruction and discretization of the Navier–Stokes equation at the same fluid nodes immediately next to the solid body, which is different from the previous “direct forcing”
Acknowledgements
This work is partially supported by the NSF under Grant No. CBET-0954381. PFS would like to acknowledge a fellowship support from the FCT, Portugal. We gratefully thank Prof. Eldredge of UCLA for providing simulation data for the 2D hovering flight.
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