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National Advisory Committee for Aeronautics, Technical Notes - Laminar Boundary Layer with Heat Transfer on a Cone at Angle of Attack in a Supersonic Stream

The equations of the compressible laminar boundary layer for the
windward. streamline in the plane of symmetry (most windward streamline)
of a yawed cone are presented. Since, for a Prandtl number of l, the
energy equation resembles the momentum equation in the meridional direc-
tion (along a generator), solutions are obtained for both insulated and
cooled surfaces.

The heat-transfer rate to this most windward streamline increases
significantly with angle of attack. For a surface cooled to absolute
zero temperature, the relative increase with angle of attack is about
15 percent less than for an almost insulated surface. A supplementary
calculation shows the heat transfer to vary with the Prandtl number 'Pr
approximately as 13:60.57 , while the recovery factor is well estimated by
the square root of the Prandtl number.

The design considerations for proposed supersonic aircraft and hyper-
sonic glide vehicles indicate the use of slender fuselages. As aerody-
namic beating problems may considerably influence such a design, the
heating rates for all possible modes of vehicle operation must be esti—
mated closely. Since optimum cruise conditions or maneuvers may call
for flight at angle of attack, the aerodynamic heating loads to bodies
at angle of attack are of definite interest. With the use of laminar-
boundary-layer theory, the present report considers this problem for a
cone.

The boundary layer on a cone at angle of attack is a three—
dimensional problem. In addition to the longitudinal and normal com—
ponents of velocity that are considered at zero angle of attack, a cross-
flow velocity exists. The importance of this crossflow boundary layer
is related to the magnitude of the component of free-stream velocity
normal to the cone axis and, of course, increases with angle of attack.