On the significance of the geometric conservation law for flow computations on moving meshes

doi:10.1016/S0045-7825(00)00173-0

Computer Methods in Applied Mechanics and Engineering

Volume 190, Issues 11–12, 8 December 2000, Pages 1467-1482

https://doi.org/10.1016/S0045-7825(00)00173-0 Get rights and content

Abstract

The objective of this paper is to establish a firm theoretical basis for the enforcement of discrete geometric conservation laws (D-GCLs) while solving flow problems with moving meshes. The GCL condition governs the geometric parameters of a given numerical solution method, and requires that these be computed so that the numerical procedure reproduces exactly a constant solution. In this paper, we show that this requirement corresponds to a time-accuracy condition. More specifically, we prove that satisfying an appropriate D-GCL is a sufficient condition for a numerical scheme to be at least first-order time-accurate on moving meshes.

Introduction

In most modern computational fluid dynamics applications, the fluid domain is enclosed by moving boundaries. Typical examples include flows in reciprocating engines, and aeroelasticity. When the fluid domain boundaries experience motions of large amplitude, it becomes necessary to solve the time-dependent governing equations on a moving mesh. This incurs the computation of some geometric quantities that involve the grid positions and velocities. A useful guideline for evaluating these quantities is provided by the enforcement of the so-called discrete geometric conservation law (D-GCL). This law states that the computation of the geometric parameters must be performed in such a way that the resulting numerical scheme preserves the state of a uniform flow, independently of the mesh motion. In Section 2, we show that this essentially implies that the following differential identity $d d t ∫_{K(t)} d x=∫_{∂K(t)} κ (s,t)· n (s,t) d s,$ where $κ$ denotes the velocity of the volume K, ∂K the boundary of this volume, and $n$ the exterior unit normal to ∂K, cannot be approximated in an arbitrary manner.

The idea of exploiting the above differential identity to derive schemes for computing the discrete mesh velocities and other geometric parameters goes back to the early days of computational fluid dynamics (CFD). The terminology `geometric conservation law' was coined by Thomas and Lombard [1], and the computational method proposed by Godunov [2] already incorporated this requirement.

However, there are recurrent assertions in the literature stating that in practice, enforcing the D-GCL when computing on moving meshes is unnecessary. Furthermore, while there exists sufficient numerical evidence showing that satisfying the D-GCL improves considerably the time-accuracy of numerical computations [3], [4], [5], the theoretical status of this requirement is unclear. After all, why should one pay special attention to a uniform flow field? Why this particular solution of the Navier–Stokes equations must be computed exactly, while the other solutions are only approximated by a given numerical scheme?

It is quite clear, however, that the requirement of computing exactly a uniform field is closely related to an accuracy condition. For steady elliptic equations, the Ciarlet–Raviart lemma [6] shows that preserving a uniform field (zero-order polynomial) implies first-order accuracy. One may suspect that such a result is also true in some sense for unsteady hyperbolic equations. Our aim is to prove that this is indeed the case. More specifically, in this paper we prove that

For a given scheme that is p-order time-accurate on a fixed mesh, satisfying the corresponding p-discrete geometric conservation law is a sufficient condition for this scheme to be at least first-order time-accurate on a moving mesh.

Note that for p>1, the result stated above does not imply that the resulting scheme will be p-order time-accurate on a moving mesh. Nevertheless, it implies that the more accurate is a given scheme on a fixed mesh, the more important it is for this scheme to satisfy the D-GCL when applied to a flow computation on a moving mesh. In practice, the enforcement of the D-GCL seems to be mandatory for high-order implicit schemes.

The remainder of this paper is organized as follows. In Section 2, we expose the role of the D-GCL in the design of numerical algorithms for flow computations on moving meshes. For this purpose, we adopt the framework of finite volume methods, and the mathematical formalism of [3]. A similar but more general presentation can also be found in [4]. The reader interested in the application of the D-GCL to other numerical methods may consult [4] for finite element methods, and [1] for finite-difference schemes. In Section 3, we present an analysis of the truncation error of first-order time-accurate finite volume schemes on a moving mesh. This analysis shows that the first-order D-GCL is a sufficient condition for annihilating the additional low-order terms of the truncation error that arise in the case of a moving mesh, and therefore for preserving the first-order time-accuracy of the given scheme on a moving mesh. In Section 4, we extend this result to second-order time-discretizations based on the backward differentiation formula, and show that satisfying the second-order D-GCL is a sufficient condition for annihilating the lower-order terms of the truncation error. In Section 5, we discuss two aerodynamic simulations that highlight the potential of the D-GCLs on the accuracy and performance of flow solvers for problems with moving boundaries. In Section 6, we conclude this paper.

Section snippets

The discrete geometric conservation law

We consider the approximation in a domain $Ω(t)$ whose boundaries vary in time of the conservation law $∂ u ∂ t +∇· F (u)=0,$ where $u∈ R^{m}$ and $F$ is a smooth function from $R^{m}$ into $R^{m}$ . We denote by $x =ϕ(a)$ the map connecting the domain $Ω(0)$ at time t=0 to its position $Ω(t)$ at time t, and by $κ (x,t)$ the velocity field of the domain $Ω(t)$ at $x ∈Ω(t)$ $κ (x,t)= ∂ ϕ ∂ t (ϕ^{−1} (x,t)).$ We discretize the computational domain $Ω(0)$ by non-overlapping control volumes denoted by K. We denote by ∂K the boundary of K, and assume that ∂K

Error analysis of first-order schemes

Next, we turn to the analysis of the truncation error of the family of numerical schemes (4). First, we consider the first-order schemes $|K^{n+1} |u_{K}^{n+1} −|K^{n} |u_{K}^{n} +τ∑_{e∈∂K} | e ̄ |Φ(u^{m}_{e_{L}},u^{m}_{e_{R}}, n ̄_{e}, κ ̄_{e})=0.$ In addition to consistency and conservativity, we assume that the flux function also satisfies the following property.

Assumption A.2

$∑_{e∈∂K} |e|Φ(u_{e_{L}} (t),u_{e_{R}} (t), n_{e}, κ_{e})=∑_{e∈∂K} |e|(F (u(g_{e},t)− κ_{e} u(g_{e},t))· n_{e} + O (|e_{h} |× O (h^{q})).$ Here, h denotes the mesh spacing and |e_h| a typical measure of a face e of the triangulation.

Error analysis of second-order schemes

On fixed grids, second-order time-accurate schemes are often constructed using the backward differentiation formula. On moving grids, the specification of these schemes to problem (2) can be written as $β_{+} (|K^{n+1} |u_{K}^{n+1} −|K^{n} |u_{K}^{n})−β_{−} (|K^{n} |u_{K}^{n} −|K^{n−1} |u_{K}^{n−1})+τ∑_{e∈∂K} | e ̄ |Φ(u^{n+1}_{e_{L}},u^{n+1}_{e_{R}}, n ̄_{e}, κ ̄_{e})=0,$ where β₊=3/2 and β₋=1/2. The second-order D-GCL associated with these schemes is given by (11). Again, for notational simplicity, we set tⁿ=0.

In general, a second-order time-accurate scheme is employed together

Numerical examples

In order to illustrate the mathematical results presented in this paper, we consider in this section the solution of two three-dimensional transonic flow problems with moving boundaries around the AGARD Wing 445.6 [12]. This wing is an AGARD standard aeroelastic configuration with a 45^∘ quarter-chord sweep angle, a panel aspect ratio of 1.65, a taper ratio of 0.66, and a NACA 65A004 airfoil section. The model selected here is the so-called 2.5-ft weakened model 3 whose measured modal

Concluding remarks

For each numerical method designed for the solution of unsteady flow problems on moving grids, there exists a D-GCL that governs its geometric parameters. From a physical viewpoint, this law ensures that the given numerical method reproduces exactly a uniform flow. From a mathematical viewpoint, it is a sufficient condition for ensuring that the given numerical method is at least first-order time-accurate on moving grids. A numerical scheme that satisfies its D-GCL delivers in general a higher

Acknowledgements

The authors acknowledge partial support by the Air Force Office of Scientific Research under Grant F49620-97-1-0059.

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On the significance of the geometric conservation law for flow computations on moving meshes

Abstract

Introduction

Section snippets

The discrete geometric conservation law

Error analysis of first-order schemes

Error analysis of second-order schemes

Numerical examples

Concluding remarks

Acknowledgements

Comput. Methods Appl. Mech. Engrg.

Finite Elements Analysis Design

Geometric conservation law and its application to flow computations on moving grids

AIAA J.

A difference scheme for numerical computation of discontinuous solutions of equations of fluids dynamics

Math. Sb.

The Finite Element Method for Elliptic Problems

Hyperbolic Systems of Conservation Laws