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National Advisory Committee for Aeronautics, Report - Method for Calculation of Laminar Heat Transfer in Air Flow Around Cylinders of Arbitrary Cross Section (Including Large Temperature Differences and Transportation Cooling)

The solution of heat-transfer problems has become, vital for
many aeronautical applications. The shapes of objects to be
cooled can often be approximated by cylinders of various cross
sections with flow normal to the acris as, for instance, heat
transfer on gas-turbine blades and on air foils heated for deicing
purposes. A laminar region always exists near the stagnation
point of such objects.

A method previously presented by E. R. G. Eckert permits
the calculation of local heat transfer around the periphery of
cylinders of arbitrary cross section in the laminar region for
flow of a fluid with constant property values with an accuracy
sufiicient for engineering purposes. The method is based on
exact solutions of the boundary-layer equations for incompres-
sible wedge—type flow and on the postulate that at any point on
the cylinder the boundary-layer growth is the same as that on a
wedge with comparable flow conditions. This method is
extended herein to take into account the influence of large tem—
perature difi’erences between the cylinder wall and the flow as
well as the influence of transpiration cooling when the same
medium as the outside flow is used as coolant. Prepared
charts make the calculation procedure very rapid. For cylinders
with solid walls and elliptic cross sections, a comparison is
made of the results of calculations based on the presented method,
the results of calculations by other known methods, and results
obtained in experimental investigations.

Calculation of the heat transferred to cylinders with
arbitrary cross sections from air flowing normal to the arm's
by a solution of the boundary-layer equations is a diflicult
problem, even when the laminar region is considered. The
problem is especially complicated by the large number of
parameters influencing heat transfer. Such parameters are:
the shape of the cross section of the cylinder, the Mach
number which determines the flow outside the boundary
layer, the temperatures on the surface of the cylinder as
well as in the stream, the stream velocity determining the
internal heat generation, and the temperature distribution
around the circumference of the cylinder. If the cylinder
is cooled by the transpiration-cooling method in which a
coolant is ejected through a porous surface into the outside
stream, the amount of coolant and its distribution around
the circumference of the cross section of the cylinder are
additional parameters. Even if a solution is obtained for
such a problem, for instance by use of an electronic computer,
the solution is very restricted because of the many parameters.
Up to the present time, therefore, the problem has been
attacked only under simplifying restrictions.