An integral solution to the flat-plate laminar boundary-layer flow existing inside and after expansion waves and after shock waves moving into quiescent fluid with particular application to the complete shock-tube flow

Robert L. Trimpi, Nathaniel B. Cohen
Jun 1957

A solution to the unsteady two-dimensional laminar boundary-layer flow inside centered expansion waves and behind both centered expansion waves and shock waves is obtained by utilizing an extension of the Karman-Pohlhausen method. The Prandtl unsteady-boundary-layer equations are integrated normal to the surface bounding the flow and are transformed into a conical coordinate system. The resulting hyperbolic differential equations are integrated in closed form for flow behind shock waves and by numerical methods for the flow inside or following expansion waves. An integral technique is applied at the discontinuities existing at the trailing edge of the expansion fan and at contact discontinuities (entropy discontinuities) so that the characteristic solution may proceed across these discontinuities. The solution to the two-dimensional unsteady laminar boundary layer existing at all points in an air-air shock tube is obtained by this method. A much shorter approximate method of solution is devised and is found to agree favorably with this method. This approximate method is used to predict the flow in hydrogen-air and helium-air shock tubes. Plots of wall heat-transfer rate and skin friction in air-air, helium-air, and hydrogen-air shock tubes are presented.

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