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National Advisory Committee for Aeronautics, Technical Notes - Method for Calculation of Compressible Laminar Boundary Layer Characteristics in Axial Pressure Gradient with Zero Heat Transfer

The Karman-Pohlhausen method is extended primarily to sixth-degree
velocity profiles for determining the characteristics of the compreSsible
laminar boundary layer over an adiabatic wall in the presence of an axial
pressure gradient. It is assumed that the Prandtl number is unity and
that the coefficient of viscosity varies linearly with the temperature.

A general approximate solution which permits a rapid determination of
the boundary-layer characteristics for any given free-stream Mach number
and given velocity distribution at the outer edge of the boundary layer
is obtained. Numerical examples indicate that this solution will in
practice lead to results of satisfactory accuracy, including the critical
Reynolds number for stability. For the special purpose of calculating
the location of the separation point in an adverse pressure gradient, a
short and simple method, based on the use of a seventh-degree velocity
profile, is derived. The numerical example given here indicates that
this method should in practice lead to sufficiently accurate results.
For the special case of flow near a forward stagnation point it is shown
that the Karman-Pohlhausen method with the usual fourth-degree profiles
leads to results of adequate accuracy, even for the critical Reynolds
number.

In reference 1 it was concluded that from the viewpoint of both
accuracy and convenience of calculation a suitable method for deter-
mining.the characteristics of a compressible laminar boundary layer is
that based on an extension of the Karmén—Pohlhausen integral method to
velocity Profiles of higher degree than the fourth, especially sixth
degree. An ordinary differential equation for general types of flow was
derived, but only the flow over a flat plate at zero incidence was
investigated in detail. The purpose of the present investigation is to
apply explicitly this method to flows with axial pressure gradients.