On the Importance of Geometric Conservation Law for Preserving the Temporal Accuracy of URANS Solvers

Abstract The effects of respecting the Geometric Conservation Law (GCL) to maintain the second-order temporal accuracy of flow evaluations on dynamic grids will be investigated for a FVM-based dual-time stepping URANS... [ view full abstract ]

Abstract

The effects of respecting the Geometric Conservation Law (GCL) to maintain the second-order temporal accuracy of flow evaluations on dynamic grids will be investigated for a FVM-based dual-time stepping URANS solver. For a uniform flow and on prescribed grid motions, it is shown that standard backwards difference (BDF) approaches for assessment of grid velocities alter the uniform flow condition. Besides, except for rigid grid motions, analytic velocities do not appropriately satisfy conservation laws. Only the scheme respecting GCL preserves the temporal accuracy of the solver and provides physically meaningful results. A comprehensive temporal study will be performed on a 2D flapping plate to emphasize the importance of GCL condition for unsteady flow solvers.

Keywords: Geometric Conservation Law (GCL), Finite Volume Method (FVM), Temporal Accuracy, 2D Flapping Plate


1. Introduction
The importance of respecting the geometric conservation law for dynamic mesh problems has been shown by many researchers. This law states that no disturbance should be introduced by any kind of arbitrary mesh motion for a uniform flow. Although it is mathematically proven that a scheme that satisfy GCL on dynamic grids is at least first order accurate in time, correct evaluation of velocities may preserve the temporal accuracy of solvers similar to those on stationary grids.

2. Mathematical Model

For a uniform flow condition, the Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian (ALE) formulation and conservative form are reduced into a single equation as follows;

∂/∂t ∫_Ω dΩ=∫_∂Ω V_t dS

where, Ω is the control volume, ∂Ω is the control volume surface, and V_t is the normal component of the velocity vector of the surface. This equation is known as Geometric Conservation Law or GCL equation. Since the grid locations are known functions of time, this equation implies that the cell edge velocities should be evaluated in such a way that the GCL condition is satisfied.

3. Preliminary Results
In our preliminary study, the influences of three different approaches for calculation of grid velocities on a uniform flow domain (U=U_∞=Cons.) is investigated; Analytical assessment, numerical calculation using a second-order backward difference formulation (BDF), and the velocities obtained from the GCL condition. These approaches are implemented in to the NSCODE, a FVM-based URANS solver at École Polytechnique de Montréal. Three different sinusoidal motions are investigated; rigid translation along x and y axis, rigid rotation along the center of coordinate, and a mesh deformation case, in which the cells volume are changed. The status of flow (density contours) is monitored after a completed motion period, when the grid returns into its original form. In table 1, it is shown whether the grid motion alters the uniform flow condition or not. It is clear that that only the approach that satisfy the GCL may be used for a general motion case, while the analytical methods are just applicable for the problems that the grid moves rigidly. Figure 1, presents the density and velocity contours for those cases that fail to satisfy GCL condition (table 1).