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National Advisory Committee for Aeronautics, Technical Notes - Method for Calculation of Heat Transfer in Laminar Region of Air Flow Around Cylinders of Arbitrary Cross Section (Including Large Temperature Differences and Transpiration 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 axis as, for instance, heat transfer on gas-turbine blades
and on airfoils 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 calcu-
lation of local heat transfer around the periphery of cylinders of arbi—
trary cross section in the laminar region for flow of a fluid with con-
stant property values with an accuracy sufficient for engineering
purposes. The method is based on exact solutions of the boundary—layer
equations for incompressible wedge-type flow and on the postulate that
on any location of 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 temperature differ-
ences between the cylinder wall and the flow as well as the influence of
transpiration cooling when the same medium as in the outside flow is used
as coolant. Prepared charts make the calculation procedure very rapid.
For cylinders with solid walls and elliptic cross section, a comparison
is made between the results of calculations based on the presented method
and the results of calculations by other known methods as well as of
Vexperimental investigations.

A calculation of heat transfer to cylinders with arbitrary cross
section in an air flow normal to the axis by‘a solution of the boundary-
layer equations is a difficult prdblem, even for the laminar region. 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 Out-
side 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 circumfer-
ence of'the cylinder. If the cylinder is cooled by the transpiration—
cooZL'Lng method in which a coolant is ejected through the porous surface
into the outside stream, the amount of coolant and its distributiOn
around the circumference of the cross section of the cylinder are addi—
tional parameters.