By Dr. A. C. Thompson
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Extra resources for Basic Hydrodynamics
At points in the interior of the grid this equation is used to form a set of simultaneous equations giving the value of the potential function at a point in terms of values at nearby points. At the edges of the grid the value of the function is fixed, or a special form of the finite difference equation is used, to give the required gradient of the function. 1 Finite difference grid for a simple flow • / / / • S s / / / / / / / / / / / / / / s ssssssss y=0 6X δψ y=h =0 = 5m ce 3 g BL3 Ci. 3 Finite difference equations In a regular rectangular grid the finite difference form of Laplace's equation is particularly simple.
The calculation could be made stable for any values of Ax} Δψ, etc. by using velocities halfway along the Ax step instead of at the beginning. This would be an implicit as opposed to an explicit calculation scheme and require some iteration as the velocities ί/Ι+1/2, k = ( Uj, κ + UJ+K)/2 etc. are not known at the start of the step. However, this could be built into the program quite readily. 8) The effect of a pressure gradient on the boundary layer calculated can be tested by including the term (—dp/dx) (1/pi/j K).
The subroutine contains the major changes. (2) In subroutine 3000 the pressure coefficients and force coefficients are calculated from the velocities and the force coefficient printed. Lines 3010 to 3020 input the angle of flow and circulation as in TRANCl. In line 3025 a message 'FINDING FORCES' is printed, and the screen cleared for printing further results by line 3026. (3) Lines 3030 to 3060 find the velocity potential at points on the transformed shape. Line 3065 sets initial values for the integration which will calculate force coefficients CD (drag) and CL (lit).