Elsevier

Journal of Computational Physics

Volume 217, Issue 2, 20 September 2006, Pages 277-294
Journal of Computational Physics

Short note
A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids

https://doi.org/10.1016/j.jcp.2006.01.031Get rights and content

Abstract

A Mach-uniform finite volume scheme for solving the unsteady Euler equations on staggered unstructured triangular grids that uses linear reconstruction is described. The scheme is applied to three benchmark problems and is found to be considerably more accurate than a similar scheme based on piecewise constant reconstruction.

Introduction

Numerical schemes to solve the Euler or Navier–Stokes equations on staggered unstructured triangular grids can be found in [1], [2], [3], [4], [5], [6], [7], [8]. In [8], we presented a scheme for solving the unsteady incompressible Navier–Stokes equations on staggered unstructured triangular grids. This method uses linear reconstruction to achieve second-order accuracy. In [7], a scheme was presented that is first-order accurate, and can handle both incompressible and compressible flow in a unified way. The aim of the present paper is to increase the accuracy of the scheme presented in [7] using the linear reconstruction of staggered vector fields as described in [8]. Linear reconstruction of scalar quantities is also necessary and will be presented. Our new method solves the Mach-uniform Euler equations with second-order accuracy.

We briefly review related work on unstructured staggered schemes and on Mach-uniform methods. The covolume method described in [1], [2], [3], [4] uses Delaunay grids and a dual mesh obtained by connecting the circumcenters. Scalar variables are located in the circumcenters, and as in all other schemes mentioned here except in [6], velocity components normal to the faces of the primal grid are stored. A similar scheme is described in [5], where conservation of kinetic energy and momentum are emphasised.

Schemes that use circumcenters require grids to be of Delaunay type. In our method, scalar variables are stored in centroids, and the Delaunay property is not required.

The scheme presented in [6] uses centroids instead of circumcenters, and stores momentum vectors in faces or vertices. The scheme described in [7] also uses centroids, but it stores the normal velocity components in faces. This is also the approach taken in [8].

Except for [8], the schemes described are first-order accurate. Our scheme is second-order accurate.

Except for [2], [7], these papers are restricted to incompressible flows. Staggered schemes have an advantage over collocated ones in the incompressible limit, because they do not require numerical stabilization to couple velocity and pressure. In general, there is no reason to use staggered schemes for fully compressible flow. However, in certain applications, as for example flow in internal combustion engines or flow around aircraft in take-off or landing conditions, compressible and incompressible regions occur simultaneously. Standard compressible schemes suffer from efficiency and accuracy loss when the Mach number becomes small (below 0.2). Such problems require Mach-uniform methods that can handle flows at all speeds. Staggering of the grid is attractive for Mach-uniform methods, because of superior properties in the incompressible case. Mach-uniform methods that use structured staggered grids were presented in [9], [10], and more recently in [11], [12]. The only Mach-uniform method that uses staggered unstructured grids that we know of is presented in [7]. This method we extend here to second-order accuracy.

In Section 2 a dimensionless form of the Euler equations is presented that is convenient for Mach-uniform schemes. Section 3 discusses a Mach-uniform pressure-correction method. Spatial discretization is presented in Section 4. Linear reconstruction is discussed in Section 5, and in Section 6 we consider flux limiting methods. Finally, numerical results are presented in Section 7.

Section snippets

Governing equations

The Euler equations are given by:ρt+·(uρ)=0,mαt+·(umα)=-p,α,ρEt+·(uρH)=0,where ρ is the density, p is the pressure, m=ρu is the momentum, u is the velocity, E=e+12u·u is the total energy and H=h+12u·u is the total enthalpy. Greek indices indicate coordinate directions; Latin indices will refer to locations in a computational grid. We have h = γe with γ the ratio of specific heats, and the equation of state for a perfect gas can be written as p = (γ  1)ρe. By using the equation of

Pressure-correction method

The Mach-uniform pressure-correction sequential update procedure to be used has been presented in [7], [11]. In this section we discuss only the time discretization.

Initial conditions determine the starting solution vectors m0, ρ0, and p0. First the new density is computed from the discretized mass conservation equationρn+1-ρnΔt+·(unρn+1)=0,where the superscripts indicate the time level, and Δt is the time step.

Next, a prediction of the momentum field mα is computed from the momentum

Discretization on unstructured triangular staggered grids

An unstructured triangular grid will be used. We choose the normal momentum components averaged over grid edgesme=1lele(m·Ne)dlas primary momentum unknowns, where Ne is one of the two possible unit normal vectors in edge e, end le is the length of edge e. Scalar variables are associated with cells, as shown in Fig. 1.

Linear reconstruction of scalars

A convected scalar quantity ψ is reconstructed in face e (see the left part of Fig. 2) by using the formulaψe=ψ1+ψ1,e·(re-r1)ifuenl¯e0,ψ2+ψ2,e·(re-r2)ifuenl¯e<0,where r is the position vector. The gradient ∇ψ is determined by least squares approximation either from the cells surrounding the upwind cell (1 or 2) or from the cells surrounding the upwind vertex (A or B), as illustrated in Fig. 3. These two methods will be called cell-based and vertex-based reconstruction, respectively.

Monotonicity considerations

In the vicinity of steep gradients or discontinuities, spurious oscillations may occur. Not only are these oscillations non-physical, but if they become too large, the density may get close to zero. The convecting velocity is computed by dividing the momentum by the density, and therefore it may become very large, so the solution procedure breaks up.

These spurious wiggles are a well-known phenomenon, which was investigated by Godunov in [15]. His order barrier theorem shows that linear

Numerical results

The purpose of the numerical experiments described here is to show improved accuracy compared to the first-order upwind method described in [7].

Conservation properties

As shown in [26], since the momentum field on an unstructured staggered grid consists of incomparable normal components, it cannot be explicitly shown that the discretization of the momentum equation is conservative. Hence, the Lax–Wendroff theorem does not apply. Nevertheless, as shown in by numerical experiments [26], numerical solutions satisfy the Rankine–Hugoniot conditions. Our scheme conserves mass locally, which is a direct consequence of our choice of the control volumes. The kinetic

Conclusion

A novel Mach-uniform staggered unstructured scheme for solving Euler equations that uses linear reconstruction has been presented. It has been demonstrated that this scheme is second-order accurate for smooth solutions if the cell-based reconstruction is used to evaluate the convected quantities in the Mach-uniform pressure correction equation, and the node-based reconstructions are used in the density and the momentum equation. The estimate of the accuracy was based on the global truncation

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    Supported by the Technology Foundation STW, applied science division of NWO and the technology program of the Ministry of Economic Affairs (Project DWI.5552).

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