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National Advisory Committee for Aeronautics, Report - Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient

Factors that affect the development of laminar boundary
layers are pressure gradient, Mach number, and heat trans-
fer, plus the properties of the fluid under consideration.
Since mathematical complexities preclude solutions of this
problem in a completely general fashion, the literature con-
sists largely of solutions treating particular combinations of

these factors. For the flow of an ideal gas over a surface
without pressure gradient, the remaining factors have been
taken into account completely by Crocco (ref. 2) and Chap~
man and Rubesin (ref. 3). For small pressure gradients,
Low (ref. 4) has, by a perturbation analysis, treated the gen-
eral problem of the isothermal surface. With the introduc-
tion of pressure gradients of arbitrary magnitude, other
restrictions become necessary. The assumption of constant
fluid properties (density, viscosity, etc.), for example, leads

- to the greatest simplification—the separation of the momentum

and energy equations. With this assumption, for a '
special case of a decelerating stream, Howarth (ref. 5) has
obtained a series solution to the momentum equation. The
introduction of a similarity concept (that the velocity or
temperature profiles may always be expressed in terms of a
single parameter) leads to a power-law free-stream velocity
distribution. The momentum equation of this problem was
first solved by Falkner and Skan (ref. 6), whose calculations
were then improved by Hartree (ref. 7); the energy equation
was later treated by Eckert (ref. 8) and others (refs. 9 and
10). For the We problem the restriction of constant fluid
properties may be removed by,alternatively requiring that
the Mach number be essentially zero (ref. 11) or that the
Mach number and the heat transfer be limited to small
values (ref. 12).

lllingworth (ref. 13) and Stewartson (ref. 14) have dem-
onstrated that, for an insulated surface in a fluid with a
Prandtl number of 1.0, any comprensible boundary—layer
problem may be transformed to a corresponding problem in
an incompressible fluid; the earlier solutions thus become
applicable to certain compressible problems. For the case of
heat flux across the surface, the transformation of Stewartson
(ref. 14) with the concept of similarity introduced leads to a
set of nonlinear ordinary differential equations previously
quoted (ref. 14), but unsolved. Solutions to this set of
equations, which are presented herein, are applicable to
flows at arbitrary Mach number, pressure gradients of arbi—
trary magnitude (but of a form cdnsistent with the requirements of similarity), and arbitrary but constant wall temperature