Implementation of a stable high-order overset grid method for high-fidelity simulations

doi:10.1016/j.compfluid.2020.104449

Computers & Fluids

Volume 211, 30 October 2020, 104449

https://doi.org/10.1016/j.compfluid.2020.104449 Get rights and content

Abstract

The implementation of an overset grid method into a compressible in-house computational fluid dynamics solver is described. The framework employs a novel algorithm for the generation of the composite grid, i.e. to identify the discretisation, interpolation and non-physical points. This algorithm was designed to minimise and simplify the user input, while maintaining the flexibility to handle complex setups. An explicit fourth-order Lagrange interpolation scheme is used, so that the formal order of accuracy of the finite difference scheme used in the flow solver is matched. An instability linked to the possible presence of uncoupled numerical solutions on separate grids sharing the same physical location is discussed, and a modification of the composite grid generation algorithm that prevents this instability is introduced. The final overset grid method is validated with two test cases: the convection of an isentropic vortex and the Taylor-Green vortex.

Introduction

Thanks to their ability to accurately represent high wavenumber phenomena on relatively coarse grids, high-order schemes are of particular interest for Computational Fluid Dynamics (CFD) solvers. High-order schemes are especially attractive in the context of Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), where the smallest scales need to be resolved accurately. Finite difference methods on structured grids are particularly well suited for high-order schemes, as they are often easily generalisable to an arbitrary order. However, because of their stringent requirements regarding the spatial discretisation, those methods suffer from a lack of flexibility when applied to complex geometries [1], and it is often unavoidable to use a multi-block setup [2]. Particular attention must then be paid to the treatment of the interfaces between the different blocks, especially if accurate acoustic predictions are desired [3].

The simplest interface treatment is halo exchange: at the interfaces the grid of each block is extended with halo regions where the solution obtained in the adjacent block is copied, so that the same finite difference scheme as used for the interior points can be applied at the interface [4]. In consequence, no additional numerical error is generated at the interface, and the grid is subjected to the same constraint across the interface than inside the block, i.e. the grid cannot be singular across the interface (the metrics cannot be discontinuous). While this method would give the best accuracy, the constraint on the grid makes this method only suited to simple geometries.

In order to increase the flexibility of finite difference solvers using multi-block structured grids, Kim and Lee [5] developed the ‘characteristic interface condition’. In this approach, one-sided finite difference schemes are employed for the near-interface points on each side of the interface, so that the differencing stencils do not cross the interface. The characteristics of the Navier-Stokes equations are computed across the interface, and communicated from one block to the other according to their physical direction of propagation. As the differencing stencils do not cross the interface, the grid is allowed to be singular across the interface. In consequence, this method is more flexible than halo exchange, and allows to handle more complex geometries. Nonetheless, the interface points are still required to match exactly in the two adjacent blocks. In addition, the authors have noticed that the accuracy of the characteristic interface condition sometimes decreases significantly when the component of the convection velocity normal to the interface is arbitrarily small (which happens either if the flow velocity is close to zero or if the velocity vector is aligned with the interface).

Alternatively, complex geometries can be dealt with using an overset grid method. The overset grid methods can be seen as generalisations of the halo exchange, which eliminate the requirement to exactly match the grid of the halo region of one block to the near boundary grid of the adjacent block. Instead, in the overset grid approaches, the domain is divided in multiple structured grids that are completely independent of each other, and that overlap close to their boundaries. Next to the interfaces, rather than simply copying the solution between two matching points from different grids, an interpolation scheme is used to make the link between the overlapping regions of the overlapping grids (in case the points of the overlapping regions of two blocks match exactly, the interpolation simply becomes a copy of the solution between the matching points, so that the overset grid method naturally converges to a halo exchange). A specific algorithm is used for the generation of the composite mask in order to identify the discretisation, interpolation and exterior points, so that the same finite difference scheme can be used for all interior points [6], as for halo exchange. An example of an overset grid setup for the simulation of aerofoil flow is illustrated in Fig. 1.

To the authors’ knowledge, the application of overset grid methods to CFD simulation started in 1970 with an investigation of the inviscid transonic flow over aerofoils by Magnus and Yoshihara [7]. The method has then been further developed in the following years [1], [8], [9]. More recently, overset grid methods have been successfully used by different authors to handle aeroacoustics simulations of complex geometries with high-order finite difference methods [10], [11], [12], [13]. The main advantage of overset grid methods lies in their ability to provide great flexibility on the grid topology, especially as it allows to cut holes in a grid [10], while at the same time maintaining the accuracy of the simulation by using high-order interpolation [14]. This flexibility makes it possible to have nearly optimal grid quality in different regions of the domain dominated by different physics. This is particularly attractive for computational aeroacoustics: close to solid surfaces, body-fitted grids are used to accurately resolve the hydrodynamics fluctuations, whereas a single Cartesian background grid is used where the flow is nearly uniform to accurately predict the acoustic waves propagation [11].

The present article describes the implementation of a new overset grid method into the in-house CFD solver HiPSTAR (High Performance Solver for Turbulence and Aeroacoustics Research) [15]. This overset grid method, briefly presented in Deuse and Sandberg [16], has already been used for different simulations of the flow around aerofoils, and the resulting self-noise [16], [17]. However, the subject of this paper is to provide the details of the implementation and introduce the novel elements needed to ensure accurate and stable solutions. First, the algorithm used for the generation of the composite grid is described, and the method for computation of the interpolation weights is presented. Next, the stability of the method is addressed: an explanation is given for an instability that arises due to the overlapping grids, and a modification of the method is provided in order to avoid this issue. Finally, the newly implemented overset grid method is validated with two different test cases, namely the convection of an isentropic vortex, and the Taylor-Green vortex.

Section snippets

Generation of the composite grid

The algorithm used to generate the composite grid is perhaps the most critical part of an overset grid method. It consists of the identification of the discretisation and interpolation points, and the hole cutting procedure. Indeed, the quality of the composite grid has a major impact on the accuracy and the stability of the method. This part is also the most complicated to implement in a CFD code. There exist several codes that are able to produce composite grids [18], [19], [20], [21], [22];

Computation of the interpolation weights

An explicit Lagrange interpolation scheme is used in the present overset grid method. Let us consider a function f of two variables ξ and η, discretised as $f_{i j} = f (ξ_{i}, η_{j})$ where $ξ_{i} = i Δ ξ$ and $η_{j} = j Δ η$ . In the present application, f denotes any flow variable, and ξ and η denote the computational coordinates in the block from which f is interpolated. The interpolated field $\tilde{f}$ is constructed as a linear combination of the discrete values f_ij, associated to the interpolation weights w_ij: $\tilde{f} (ξ, η) = \sum_{(i, j) \in S} f_{i j} w$

Improvement of the stability

It is well known that unless some sort of regularisation is employed (such as filtering [10] or artificial dissipation [6]), the overset grid methods are prone to numerical instabilities, especially when they are combined with low-dissipative, high-order methods [2], [24], [25], [26]. In general, it cannot be guaranteed that the interpolation will not increase the energy of the solution, and so instabilities may arise next to overlapping interfaces. In its original version, the present overset

Validation

In order to validate the newly implemented overset grid method, two test cases have been selected: the convection of an isentropic vortex and the Taylor-Green vortex. The governing equations are the full compressible Navier-Stokes equations, in non-dimensional form. In the following, all quantities have been made non-dimensional, using a given length, velocity, density and temperature as the reference variables. The dimensionless continuity, momentum and energy equations are: $\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x_{j}} (ρ u_{j}) = 0,$ $\frac{\partial}{\partial t}$

Conclusions

An overset grid method framework has been implemented into the in-house CFD solver HiPSTAR. The novel composite grid generation algorithm has been described. It consists of a method that classifies each point of the grid into discretisation, interpolation and non-physical points. The variables are communicated between the different grids using an explicit fourth-order Lagrange interpolation scheme. The interpolation is performed in the computational space, and the interpolation offsets are

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia.

Acknowledgements

This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia.

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Implementation of a stable high-order overset grid method for high-fidelity simulations

Abstract

Introduction

Section snippets

Generation of the composite grid

Computation of the interpolation weights

Improvement of the stability

Validation

Conclusions

Declaration of Competing Interest

Acknowledgements

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