Computational grids

Chapter 8 Computational grids

8.1 Grid types

The computational grids are subdivided into: structured grids (see Fig. 8.1a), block structured grids (see Fig. 8.1b), unstructured grids (see Fig. 8.1c).

Disadvantage of the structured grid is shown in Fig. 8.2. Refinement of the grid close to the wall results in the refinement in areas where this refinement is not necessary. This disadvantage can be overcome by use of block-structured (Fig. 8.1b) and unstructured grid (Fig. 8.1c).

Figure 8.1: Samples of a) structured grid for an airfoil, b) block structured grid for cylinder in channel and c) unstructured grid for an airfoil.

The quality of the grid has a strong impact on the accuracy of numerical prediction. The change of the cell topology within the computational domain

Figure 8.2: Illustration of structured grid disadvantage. should be smooth especially at the border between different grid blocks. The

grid resolution should be high especially in areas of boundary layers and close to the free surface. For this sake the special refinement is used in these areas. To increase the accuracy of the computations in boundary layers one uses special grid boundary layers close to walls.

8.2 Overset or Chimera grids

For complicated objects one uses overset or Chimera grids. The idea of chimera or overset grids is to generate the grids separately around each geo- metrical entity in the computational domain. After that the grids are com- bined together in such a way that they overlap each other where they meet. The crucial operation is an accurate transfer of quantities between the dif- ferent grids at the overlapping region. The most important advantage of the overset or Chimera grid is the possibility to generate high quality structured particular grids separately for different body elements completely indepen- dent of each other, without having to take care of the interface between grids.

8.3 Morphing grids

Very efficient way of CFD body simulation is the use of moving or morphing grids [6]. The idea is the computational grid is moved in accordance with the displacement of the body by using an analytical weighted regridding which is a type of extrapolation of rigid transformation. The possible problem of morphing grid is poor quality caused by its motion. Consequently if the mesh surrounding the body is allowed to deform the elements around the body deform. This can quickly lead to poor quality elements if care is not taken. An alternative method is to replicate the motion of the body with the fluid domain split into an inner and outer region. The outer domain remains fixed in space while the inner domain containing the body moves laterally to replicate the motion. The mesh in the inner sub domain remains locked in

position relative to the lateral motion of the body. This prevents deformation of the detailed mesh around the body. The outer mesh is deformed due to the motion of the inner region.

If moving grids are used the Navier Stokes should be transformed to take the

velocity of grid faces E U g into account,

Thomas and Lombard have shown that the function E U g can not be arbi- trary rather than they have to be found from the Geometric Conservation Law

dU

U E g ndS D 0 E (8.2)

@t

Where U and S are respectively volume and surface of cells. The equa- tion (8.2) is derived from the condition that the computation of the control volumes or of the grid velocities must be performed in a such a way that the resulting numerical scheme preserves the state of the uniform flow, in- dependently of the deformation of the grid. The equation (8.2) is satisfied automatically if the control volumes don’t change their shape. The Geomet- ric conservation law (8.2) should solve coupled with other fluid flow equations using the same discretizations schemes.

More detailed information about grid generation can be found in [3], [7] and [8].

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Part II

Mathematical modelling of

turbulent flows