On overset interpolation strategies and conservation on unstructured grids in OpenFOAM

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Abstract

In this paper, the effects of interpolation methodology on the accuracy of computed solutions are discussed in the context of unstructured overset grids. An in-house implementation of the overset method for OpenFOAM, OPERA (Chandar, 2015; Chandar et al., 2015) has been used. Several verification and validation studies reported previously used a simple interpolation technique such as inverse distance weighting and this was found to be adequate for prediction of forces. For closed problems however, the force history is plagued with unphysical oscillations. Also, a quantitative assessment of interpolation errors is lacking. The current work aims to understand the effects of interpolation error on conservation and how this error can be minimized by enforcing flux correction and/or better interpolation techniques. Apart from enforcing conservation, the pressure equation needs to be adjusted as well in an implicit manner to guarantee the transfer of fluxes correctly between overlapping regions. Importance of conservation and higher order interpolation are also discussed in the context of multiphase flows.

Introduction

The overset mesh/grid approach needs no introduction to the CFD community and has been around for several years [1], [2]. Its application to moving meshes, mesh refinement and part addition/removal are quite well known. The overset method is convenient to treat problems that involve large displacements through translation and rotation with the exception of large deformation. This can be achieved by wrapping each object of interest with a mesh of its own, and have it overlap with other stationary or moving meshes as shown in Fig. 1. Each of the meshes shown in the figure exists as separate entities and is generated individually. The overset mesh algorithm classifies cells that need interpolation from other meshes (

), cells that solve ( ) and cells that are inside the solid body ( ). are mandatory when they lie next to or near an outer boundary and are explicitly specified. At least two layers are required for interpolation of field quantities so that the gradients are also represented accurately. fringe cells are automatically flagged depending on the overlap strategy, i.e. if a cell of a given mesh overlaps a cell from a different mesh that has a smaller cell volume, the cell becomes an implicit

cell. Well known codes such as OVERFLOW [3], CFL3D [4], Fun3D [5] and HELIOS [6] have overset capability. Commercial codes such as STAR-CCM+ [7] and ANSYS Fluent [8] also have introduced overset meshing in 2012 and 2016 respectively. On the open-source front, Overture [9] a structured mesh overset grid library is available, however without turbulence model implementations. For OpenFOAM [10], there have been developments earlier such as FoamedOver [11] and naoeFOAM [12] and they rely on the commercial overset grid library SUGGAR++ [13]. Not much of the information is available on the capabilities of FoamedOver and naoeFOAM, but it has been proven effective for the problems considered in their work. ESI-OpenCFD [14] released the native overset implementation to the public [10] and is currently receiving lots of attention for improving the computational speed and accuracy.A crucial aspect of the overset method is the choice of interpolation and the conservative properties of the overall method. The purpose of interpolation is to transfer information between meshes accurately. At the end, if the solution is well represented collectively on all meshes, the accuracy of the method would be as good as that of a single mesh. Unfortunately, this is not the case always and errors due to interpolation impact overall accuracy of the solution. When the solution is not well represented on overset meshes, there is a global mass imbalance or conservation error. For flows that involve only a single phase, unsteady forces have a high frequency oscillation component [15] and are usually related to the unsteady mass defect. These high frequency oscillations may be detrimental to fluid–solid interaction problems such as vortex induced vibrations. Quon and Smith [16] using Fun3D and SUGGAR++ demonstrated the usage of radial basis functions (RBF) for interpolation and showed that conservation errors reduce for certain RBF basis functions. Theoretical assessment of non-conservative algorithms on structured meshes was dealt in detail by Tang et al. [17]. From their work and also from Tang et al. [18], it is clear that conservation errors decrease with mesh size. Recently Rinaldi et al. [19] demonstrate the use of a super mesh to ensure less conservation errors on overset meshes. For higher order flow solvers based on Discontinuous Galerkin (DG) and Flux Reconstruction (FR) methods, higher order overset meshes based on Local Galerkin Projection (LGP) and higher order polynomials have been explored [20], [21], [22].For single phase flows, the effect of non-conservative interpolation methods manifests itself as unphysical oscillations or small deviations from the corresponding single mesh cases. However, for two-phase flows that have large density differences such as air and water, the effect of interpolation is readily visible and manifests itself as a disappearance of one of the phases, air or water over time. This aspect is dealt in detail later on in the paper.To summarize, one is usually presented with two choices to improve the quality of the solution without having to refine or stitch the mesh and they are as follows: •Improve interpolation accuracy using better interpolation methods •Enforce conservation at the interfaceBoth approaches are dealt in this paper in the context of single and multiphase flows and conservation errors with respect to different interpolation methods are presented.OpenFOAM is used as the flow solver in conjunction with the overset grid library OPERA [23], [24]. OPERA is an in-house unstructured overset connectivity library that interfaces with OpenFOAM at the matrix level thereby ensuring tight coupling between overset meshes. The reminder of the paper is organized as follows: In Section 2, the problem of conservation is introduced using the Poisson equation on overset meshes followed by Section 3 where the governing fluid equations are described in the OpenFOAM framework. Two computational test cases are then demonstrated using different interpolation techniques in Section 5, and finally conclusions are presented in Section 6.

Section snippets

Poisson equation on overset meshes Since the pressure equation that is derived from the incompressible Navier–Stokes equations is a Poisson equation, we first analyze its discretization on an pseudo-overset mesh such as in Fig. 2. A pseudo overset mesh is nothing but a single mesh which mimics an overset mesh. Artificial overset boundaries are marked out to represent actual overset boundaries on disconnected meshes.Consider the Poisson equation

where and

are scalar and vector fields respectively. In Fig. 2, two

Incompressible Navier–Stokes and the pressure equation The pressure equation in incompressible flows represents a constraint to the flow so that the velocity field is divergence free. The pressure equation in OpenFOAM has minor variations compared to other implementations. To recap, the governing incompressible flow equations are:

where , and

represent the velocity vector, pressure and the velocity of the moving mesh respectively. In OpenFOAM, the above equations are solved over multiple outer and inner corrector

Interpolation errors To quantify interpolation errors for different interpolation methods, polynomials of up to order three are considered for interpolating the discrete data between meshes (e.g. Eq. (31)). A polynomial of the form

with is used as a test function, and the

volume weighted errors are computed on interpolating cells for two- and three-dimensional configurations as shown in Figs. 6, 7 respectively. The two-dimensional case is a simple NACA airfoil with two overlapping meshes.

Results and discussions To evaluate the characteristics of different interpolation schemes, several cases are considered and are described below:

Concluding remarks Various interpolation schemes in conjunction with flux correction were discussed in the context of overset grids. Taking the example of the Poisson equation, the conservation equations were derived and the conditions for global conservation were stated. With a small rearrangement of the equations, it was cast into a conservation equation with a source term. Due to the non-unique property of this source term on different overlapping meshes, conservation cannot be guaranteed. Flux based

Acknowledgments The author wishes to acknowledge the suggestions made by Bill Henshaw, Rensselaer Polytechnic Institute on the interpolation techniques, Bharathi Boppana, Institute of High Performance Computing Singapore for aiding in the STAR-CCM+ computations, and Nguyen Vinh-Tan, Institute of High Performance Computing for generating the swimmer mesh.

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• A flux correction approach for the pressure equation in incompressible flows on overset meshes in OpenFOAM 2022, Computer Physics CommunicationsCitation Excerpt :All of the methods that alter the fluxes to satisfy global conservation in incompressible flows do this flux correction step prior to the solution of the pressure equation while leaving the pressure equation uncorrected. In the context of incompressible flows, especially methods that use some form of pressure correction such as SIMPLE [31] or PISO [32], the pressure equation alone determines the extent to which conservation of mass is satisfied [22]. Thus solving the pressure equation appropriately is of utmost importance if conservation of mass is required.

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Overset meshes for incompressible flows: On preserving accuracy of underlying discretizations

AsheshSharmaa,∗, ShreyasAnanthana, JayanarayananSitaramanb, StephenThomasa, Michael A.Spraguea

aNational Renewable Energy Laboratory, Golden, CO, USA

bParallel Geometric Algorithms, LLC, Sunnyvale, CA, USA

a r t i c l e i n f o

a b s t r a c t

Article history:

Received 27 March 2020

Received in revised form 19 September 2020

Accepted 30 October 2020

Available online xxxx

Keywords:

Computational fluid dynamics

Unstructured meshes

Chimera grid

Overset

Additive Schwarz

Wind energy

This study on overset meshes for incompressible-flow simulations is motivated by accurate prediction of wind farm aerodynamics involving large motions and deformations of components with complex geometry. Using first-order hyperbolic and elliptic equation proxies for the incompressible Navier-Stokes (NS) equations, we investigate the influence of information exchange between overset meshes on numerical performance where the underlying discretization is second-order accurate. The first aspect of information exchange surrounds interpolation of solution where we examine Lagrange and point-cloud-based interpolation for creating constraint equations between overset meshes. To maintain overall second-order accuracy, higher-order interpolation is required for elliptic problems, but linear interpolation is sufficient for hyperbolic problems in first-order form. Higher-order point-cloud-based interpolation provides a pathway to maintaining accuracy in unstructured meshes, but at higher complexity. The second aspect of information exchange focuses on comparing the approaches of overset single system (OSS) and overset Additive Schwarz (OAS) for coupling the linear systems of the overlapping meshes. While the former involves a single linear system, in the latter the discrete linear systems are solved separately, and solving the global system is accomplished through outer iterations and sequential information exchange in a Jacobi fashion. For the test cases studied, accuracy for hyperbolic problems is maintained by performing two outer iterations, whereas many outer iterations are required for elliptic systems. The order-of-accuracy studies explored here are critical for verifying the overset-mesh coupling algorithms used in engineering simulations. Accuracy of these simulations themselves is, however, quantified using engineering quantities of interest such as drag, power, etc. Consequently, we conclude with numerical experiments using NS equations for incompressible flows where we show that linear interpolation and few outer iterations are sufficient for achieving asymptotic convergence of engineering quantities of interest.

©2020 Published by Elsevier Inc.

Though I am not familiar with the overset mesh, I would not recommend you to use an incompatible mesh for DES and LES unless you deeply trained yourself on such methods.