The Computational Grid

Before the Navier-Stokes equations can be applied to something, such as an aircraft, computer specialists must represent the aircraft's surface and the space around it in a form usable by the computer. So they represent the airplane and its surroundings as a series of regularly spaced points, known as a computational grid. They then supply the coordinates and related parameters of the grid to the software that applies the Navier-Stokes equaitions to the data. The computer calculates a value for the parameters of interest - air velocity and pressure - for each of the grid points.

In effect, the computational grid breaks up (the technical term is "descretizes") the computational problem in space; the calculations are carried out at regular intervals to simulate the passage of time, so the simulation is temporally discrete as well. The closer together - and therefore more numerous - the points are in the computational grid, and the more often they are computed (the shorter time interval), the more accurate and realistic the simulation is. In fact, for objects with complex shapes, even defining the surface and generating a computational grid can be a challenge.

Since the usual methods of discretization, finite differences, finite volumes and finite elements, use neighboring points to calculate derivatives, the nature of the mesh or grid on which the computation is performed is important. There exist two mesh types, structured and unstructured, characterized by the connectivity of the points. Below is a two-dimensional example of each type of grid.

Structured meshes have a regular connectivity, meaning each point has the same number of neighbors (for some grids a small number of points will have a different number of neighbors).

Unstructured meshes have irregular connectivity: each point can have a different number of neighbors. In some cases part of the grid is structured and part unstructured (e.g., in viscous flows where the boundary layer could be structured and the rest of the flow unstructured).