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National Advisory Committee for Aeronautics, Technical Notes - Critical Study of Integral Methods in Compressible Laminar Boundary Layers

A number of the most promising integral methods for solving approxi—
mately the compressible-laminar—boundary—layer equations are investigated
in order to determine a computationally convenient and sufficiently
accurate method of calculating boundary—layer characteristics. The chief
methods considered are: (a) The one—parameter Karman—Pohlhausen method,
with three different assumptions for the velocity profiles, and (b) the
two-parameter method, first applied by Sutton, with two different assump-
tions for the velocity profiles. After the methods are explicitly
described in general terms for the case of zero pressure gradient and
for the case of a pressure gradient in the direction of flow with zero
heat transfer, they are applied to the calculation of the compressible
laminar boundary layer over a surface with zero pressure gradient, with
and without heat transfer at the surface, for the purpose of establishing
the accuracy of the methods. Comparison of the results is made with those
of known exact solutions for skin-friction and heat— transfer coefficients,
velocity profiles, velocity derivatives, and especially laminar-boundary—
layer stability. From this comparison it is found that the Karman-
Pohlhausen method with a sixth— -degree polynomial as the velocity profile
is the most suitable for many practical purposes.

It is well-known that the differential equations of two—dimensional
compressible-laminar4boundary-layer flow are difficult to solve exactly.
Stewartson (reference 1) and Illingworth (reference 2) have recently shown
that if the Prandtl number is unity and the viscosity coefficient is
proportional to the temperature, then the equations for the compressible
heat-insulated boundary layer with a given pressure gradient can be trans-
formed into the equations for an incompressible boundary layer with a
different pressure gradient; however, this principle appears at present
tedious to apply in practice.

The most frequently used and most fruitful methods of solving the
boundary-layer equations approximately are the integral methods, in
which the partial differential equations are integrated over the boundary—
layer thickness, and are hence satisfied only "in the average." By
assuming definite forms for the velocity profiles as functions of the
normal distance, ordinary differential equations are obtained, with
distance along the surface as the independent variable. Any integral
method may be regarded as either of two types: (a) The single—integral
type, in which the partial differential equations are integrated once
across the boundary-layer thickness, and the profiles contain a single
parameter to be determined by the resulting ordinary differential equa-
tion; (b) the multiple—integral type, in which several (say m) integral
equations are used, and the assumed velocity profiles contain m param—
eters to be determined by the m resulting ordinary differential equations.