Д дф д дф дх р’дх + дг ‘"Hr = 0 (13.61) and д Щ – + д = 0 дх рг дх дг рг дг (13.62) These equations can be compared to the equations governing the electrical potential in a conducting medium of varying thickness. The depth of the conducting material will be proportional or inversely proportional to рг, depending on whether ф or ф is simulated. In the first case, inclined tank can be used to simulate flow around streamlined bodies of revolution, convergent wind tunnel ducts and axially symmetric air intakes. For the representation of the stream function, a tank with hyperbolic bottom can be constructed where the depth varies as 1/y, see Ref. . 13.3.3 Hodograph Tank In the hodograph plane, the velocity potential ф and the stream function ф, of compressible fluid flow are related by the following equations д 1 дф д ( р дф дд ррд (1 – M2) дд + дв р0д дв ° (13.65) and (13.66)
these equations may be compared to the equation of the electric potential in a conductor, in terms of polar coordinates
It is possible to establish the correspondence with r = r (q) and h = h(q).
The electric tank may be constructed in a circular form, where the center corresponds to the origin of the (q, $)-plane. The tank radius is limited to a value corresponding to M slightly less than one (for Mach = 1, the depth is infinite). The image of uniform flow is given by a singularity placed at a point, the distance of which from the origin corresponds to the magnitude of the velocity of the uniform stream at infinity. The singularity is represented in the tank by two electrodes, properly chosen.
To determine the form of the wing section which corresponds to the hodograph chosen, the following relations are used
cos в sin в
dx =——– d ф, dy =——— d ф (13.68)
where x and y are obtained via simple integration.