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Figure - 1. Dynamic adaptive grid around NACA0012 airfoil

ADA is a three dimensional unstructured grid adaptation code developed by Dr. Erdal Oktay. It is capable of refining the appropriate regions of the flow by detecting the local flow features. Dominant flow features may be shock waves and expansion waves. The starting grid and flow solution is obtained from any grid generator and from any unstructured flow solver. ADA is an individual program, which is independent from the grid generator and flow solver. To refine the mesh, no man-hour and new grid generator is needed. With this feature it can easily be used together with the CFD programs simultaneously as an interface. Effective search algorithms and data structures make this grid adaptation program very fast and reliable.

Adaptation algorithm senses the flow features at different regions and embed these regions if the grid spacing is not sufficient for desired flow feature. Grid adapter may also unrefine the grid locally in the region where the feature alters its position as is met in unsteady flow solutions. The feature detector uses the gradient of the flow parameter to sense the flow feature. Velocity, Mach number, pressure and density can be used as flow parameter. By defining a uniform flow distribution on the wall and setting these parameters to zero at the inner flow domain, it is also possible to refine boundary layer grid for Navier-Stokes calculations. The threshold values for the parameters are determined with respect to the distribution of the parameters, which is characterized by their averages (Qave) and standard deviations (Qsd), where Q is any detection parameter. The following relations are used to set the threshold parameter for refinement.

Qref. th = Qave + a Qsd

The average and the standard deviation are defined as


The value of the parameter a can be selected by experience between zero and one. If detection parameter value greater than the threshold value than that edge is marked as to be divided. Each division of edges means a new node creation on the middle of the edge. Edges of faces are checked before deciding to divide the cell. If two of the edges of the face are divided, than third edge has also to be divided for the appropriate adaptation of the grid. A cell may only be divided into two, four or eight children cells, as shown in Fig. 1-4. To obey this rule, divisions of all edges are controlled. When all edges of the cell are divided, in addition to four corner child cells (tetrahedrons) an internal octahedron is occurred (5-6-7-8-9-10 in Fig. 4). To find four internal tetrahedrons, this internal octahedron is divided into four tetrahedrons by the shortest diagonal (7-8 in Fig.4). The algorithm of the grid adaptation technique is as follows:

1. Read grid and flow variables

2. Construction of lines using nodes

3. Construction of  line - face connectivity

4. Construction of line - cell connectivity

5. Calculation of threshold value

6. Normalization of flow features

7. Marking edges to be divided

8. Dividing edges from their midpoints (create new nodes)

9. Checking of face division. If two edges of the face divided, divide third edge too.

10. Create new cells

11. Rearrangement of interconnectivity of cells and renumbering of nodes

12. Write new grid

In the step 3 and 4 of the algorithm a fast and efficient search algorithm is used. This new algorithm makes grid adaptation task very fast.

Fig. 5 shows a non-adaptive grid and Fig. 6 shows an adapted grid. Mach contours obtained by using non-adaptive and adaptive grids are given in Fig. 7 and Fig.8 respectively.

The present three-dimensional unstructured adaptive grid technique is very useful tool with its fast and good adaptation capability in the computational fluid dynamics (CFD) applications. Its valuable contribution to the precision of the aerodynamic coefficients is promising in the design and analyses.

a) Two children cells are created when one edge of a face is dividedb) Four children cells are created when three edges of a face are dividedc) Four children cells are created when two edges which are in opposite faces are dividedd) Eight children cells are created when all edges of a cell are divided
Figure - 2. Enrichment strategies used by subdividing a cell into children.

a) Before adaptation on body surface
b) After adaptation on body surface
Figure - 3. Effect of adaptation on the computational mesh

a) Before adaptation
b) After adaptation
Figure - 4. Effect of adaptation on flow solution