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National Advisory Committee for Aeronautics, Report - The Compressible Laminar Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient

An approximate method for the calculation of the compressible laminar boundary layer with heat transfer and
arbitrary pressure gradient, based on Thwaites’ correlation
concept, is presented. The method results from the application
of Stewartson’s transformation to Prandtl’s equations, which
yields a nonlinear set of two first—order differential equations.
These equations are then expressed in terms of dimensionless
parameters related to the wall shear, the surface heat transfer,
and the transformed free-stream velocity. Il'hwaites’ concept
of the unique interdependence of these parameters is assumed.
The evaluation of these quantities is then carried out by utilizing
exact solutions recently obtained.

With the resulting relations, methods are derived for the
calculation of the two-dimensional and axially symmetric
laminar boundary layer with arbitrary free-stream velocity
distribution, Mach number, and swim temperature level.

The combined effect of heat transfer and pressure gradient
is demonstrated by applying the method to calculate the characteristics of the boundary layer on thin supersonic surfaces, and in a highly cooled, convergent—divergent, axially symmetric
rocket nozzle.

In recent years, with the advent of laminar airfoils and
with the observation of laminar boundary layers at Reynolds
numbers as high as 50X10° (ref. 2), the ability to reliably
estimate viscous flow and heat-transfer effects for a laminar
boundary layer has become increasingly important. More
over, with high—altitude flight becoming more common, the
subsequent lower Reynolds numbers encountered should
more frequently produce a laminar boundary layer. Sta-
bility calculations based on the theory of Lees and Lin
(ref. 3) have also emphasized the possibilities of maintaining
a laminar boundary layer through cooling of aerodynamic
surfaces. The efiect of favorable pressure gradients in
increasing the stability of laminar‘ boundary layers may
also make solutions to the laminar problem applicable to
the design of nozzles and turbine blades.

Solutions of the laminar-boundary—layer equations that
include effects of compressibility, pressure gradient, and
heat transfer have been quite limited in number. Of the
exact solutions, most have restrictions of range or applica-
tion, or both. The solutions of references 4 and 5 are re-

stricted to zero pressure gradient, while those of reference 6
allow small pressure gradients. The developments of refer-
ence 7 are restricted to small heat transfer and low Mach
number. Solutions obtained by assuming that fluid proper-
ties are constant or that the Mach number is essentially zero
are obtained in references 8 to 10. Those solutions of
references 11 to 13 that are for a Prandtl number of 1 are
not restricted in range of compressibility, pressure gradient,
or heat transfer. However, they apply to specific types of
free-stream velocity distribution that are inappropriate for
general practical problems.