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

A rapid and sufficiently accurate method, for most.practical purposes,
of determining laminar-boundary-layer characteristics in flow with a given
free—stream.Mach number and given velocity distribution at the edge of the
boundary layer is presented. The method can be easily applied to flow with
zero pressure gradient for any (constant) Prandtl number of the order of
unity and any given temperature distribution along the wall. Numerical
examples are given to illustrate the method and the satisfactory accuracy
obtained. For flow in an axial pressure gradient, the method can be
applied for a Prandtl number of unity and any given uniform wall temper-
ature. The methods developed here are based on an application of the
Karman integral method to both the momentum and energy equations, in
conjunction with a sixth-degree velocity profile and a seventh-degree
stagnation—enthalpy profile. A single boundary-layer thickness and one
of the coefficients in the thermal profile are the parameters in this
two—parameter method.

The aim of this report is to present a relatively simple method,
sufficiently accurate for most practical purposes, of calculating the
laminar—boundary-layer characteristics in the compressible flow over a
given object with heat transfer at the wall.

The method is based on the extension of the Karman-Pohlhausen method
to sixth-degree velocity profiles and seventh-degree stagnation-enthalpy
profiles. The use of sixth—degree velocity profiles is in accordance with
the conclusions of reference 1, wherein it was found that such profiles
can usually be expected to lead to results of adequate accuracy without
much increase in computational work. Such profiles have been applied
with satisfactory results for compressible flow over a flat plate with
heat transfer (ref. 2) and for compressible flow in an axial pressure
gradient without heat transfer (refs. 5 and h).