Solutions of Laminar-Boundary-Layer Equations Which Result in Specific- Weight-Flow Profiles Locally Exceeding Free-Stream Values.

摘要

Solutions of the laminar boundary layer equations when large temperature changes in the boundary layer and large pressure changes in the main stream occur simultaneously were found to be very sensitive to the behavior of the third-order derivative of the boundary-layer stream function as the specific weight flow approached its free-stream value. (The specific weight flow is proportional to the first derivative of the stream function). Theoretically, all derivatives of the specific weight flow should vanish at the outer edge of the boundary layer; however, in numerical solutions, only a restricted number of these conditions can be applied. Under assumed constant-wall temperature and small Mach numbers, solutions of the laminar-boundary-layer equations for stream-to-wall temperature ratios of 2 and 4, Euler numbers of 0.5 and 1, and rates of cooling-air flow through the porous wall signified by values of the coolant flow parameter of 0, -0.5, and -1 previously reported did not fulfill the condition that the third-order derivative of the stream function vanish at the outer edge of the boundary layer. New solutions which not only fulfilled this condition but also which resulted in very small values of higher-order derivatives were therefore obtained. The resulting specific-weight-flow, velocity, and temperature distributions and the local heat-transfer coefficients are tabulated and are compared with those determined previously. Friction coefficients and dimensionless displacement, momentum, convection, and thermal boundary-layer thicknesses are also tabulated. The new solutions resulted in specific weight flows which exceeded the free-stream values. These excesses ranged from 2 percent for the impermeable wall, a stream-to-wall temperature ratio of 2, and an Euler number of 0.5 to 15 percent for the permeable wall with a coolant flow parameter of -1, a stream-to-wall temperature ratio of 4, and an Euler number of 1, the most severe case considered.