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National Advisory Committee for Aeronautics, Report - Average Properties of Compressible Laminar Boundary Layer on Flat Plate with Unsteady Flight Velocity

In many current problems of aerodynamics, unsteady
motions of a surface are important. The accelerating and
decelerating phases of missile flight and the intermittent
flow in an engine during unstable combustion are examples.
The nature of the boundary layer of these unsteady flows
may be studied with a view to determining friction drag,
surface temperature, and rate of heat transfer through the
surface. Usually, the boundary layer is so thin that' it
responds almost instantly to temporal changes in flow con—
ditions. Thus, the time history of the boundary layer is a
succession of steady states, and such a boundary layer is
called quasi-steady. If the motion involves accelerations
which are particularly rapid, the quasi-steady description
may require correction.

In any case, it is often desired to assess the average effect
of fluctuations in flow conditions. For example, if an insu-
lated body is in motion with a speed that varies in time about
some average value, the average friction drag and the aver-
age surface temperature may differ from values appropriate
to steady motion at the average speed. Another example is
furnished by the speculation that the net performance of a
heat exchanger could be altered by imparting an unsteady
motion to the wall.

In the present study, as an idealized special case of the
foregoing type of problem, a semi-infinite flat plate is assumed
to be in motion parallel to its sinface and normal to its
leading edge with a flight velocity U(t) that is always in
the same direction but has a magnitude that fluctuates with
time. The resulting boundary layer is assumed laminar and
compressible, and the surface is either insulated or at con-
stant temperature. The assumed Prandtl number is 0.72,
which is appropriate for air under normal conditions.

The boundary layer is assumed to be nearly quasisteady.
The basic boundary-layer analysis is already available for
this problem in references 1 and 2, which treat the insulated
plate and the constant plate temperature cases, respectively.