The compressible laminar boundary layer with heat transfer and small pressure gradient Page: 3 of 69
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NACA TN 3028
Solutions of the compressible laminar-boundary-layer equations for
the special case of zero pressure gradient have been obtained by several
authors. The theory of Chapman and Rubesin (ref. 3), for example, pre-
sents a very simple method for calculating boundary-layer characteristics
over a flat plate with arbitrary heat transfer. The more recent, and
in general more exact, studies of Klunker and McLean (refs. 4 and 5),
Van Driest (ref. 6), Young and Janssen (ref. 7), and Moore (ref. 8)
have demonstrated that the theory of Chapman and Rubesin yields excellent
results for reasonably low ambient air temperatures at Mach numbers up a
to about 5.
Solutions for the more general case of arbitrary heat transfer and
arbitrary pressure gradient are still in an early stage of development.
Tani, in a little known paper (ref. 9), used a perturbation procedure
to obtain direct solutions of the boundary-layer differential equations
with a Falkner-Skan type external velocity distribution (ue " xm) and
heat transfer. Results are easily obtainable from tabulated functions,
but are limited to a Prandtl number of 1, small Mach numbers, and small
rates of heat transfer. Furthermore, the Falkner-Skan type of external
velocity distribution is not appropriate for supersonic flow over thin
wings. Ginzel (ref. 10), Kalikhman (ref. 11), and Libby and Morduchow
(extension of ref. 12) have obtained solutions of the compressible
laminar-boundary-layer equations by an extended Pohbhausen method. How-
ever, the accuracy of the Pohlhausen method under conditions of heat
transfer at high speeds has not been determined. In addition, the amount
of work required in a particular application of references 10 and 11 is
prohibitive because the simultaneous numerical solution of two differen-
tial equations is required. Libby and Morduchow avoid this difficulty by
the additional assumption that certain variable quantities remain constant
over the entire length of boundary-layer development.
The purpose of the present report is to present a method of solu-
tion developed at the NACA Lewis laboratory that is free of many of the
limitations of references 9 to 12. An accurate method for calculating
velocity and temperature profiles and skin-friction and heat-transfer
characteristics for the compressible laminar boundary layer with heat
transfer and a small pressure gradient is derived. The permissible
pressure gradient may be of a form and magnitude usually encountered
over thin aerodynamic shapes in supersonic flight. The solution is
obtained by a method of perturbation on the flat-plate solution of
Chapman and Rubesin; it constitutes the first two terms of a Maclaurin
series expansion in terms of the free-stream velocity gradient parameter.
The method involves the direct solution of the boundary-layer differen-
tial equations. Although the theory applies for any'constant Prandtl
number, tabulated results presented in this report apply, in general,
for a Prandtl number of 0.72. For the case of heat transfer, results
are limited to an isothermal wall.
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Low, George M. The compressible laminar boundary layer with heat transfer and small pressure gradient, report, October 1953; (digital.library.unt.edu/ark:/67531/metadc56987/m1/3/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.