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National Advisory Committee for Aeronautics, Technical Notes - A Power Series Solution for the Unsteady Laminar Boundary Layer Flow in an Expansion Wave of Finite Width Moving Through a Gas Initially at Rest

The equations of motion and energy for the laminary boundary-layer
flow in an expansion wave of finite width moving into undisturbed fluid,
such as in a shock tube, were considered. Solutions in the form of
infinite power series for velocity and local enthalpy functions were
indicated, and the first three terms of each series were numerically
evaluated. Validity of the numerical results was restricted to the
region near the leading edge of the expansion wave. Skin friction and
heat transfer were compared with values given by a solution which con-
sidered the expansion wave as equivalent to a line discontinuity across
which existence of isentropic expansion relations was assumed. These
solutions were shown to be very different, qualitatively as well as
quantitatively. Singularities in the flow field.were discussed in
regard to both the finite-width expansionawave and the line-expansion-
wave solutions.

Extensive use of shock tubes for aerodynamic research has placed
emphasis upon the effects of using a real, rather than a perfect, gas.
The flow phenomena for an ideal fluid are easily derived and are given,
for example, in references 1 to 5. Deviations from the ideal flow may
occur because the working fluid is imperfect and because fluid viscosity
and thermal conductivity introduce effects of the shock—tube walls upon
the flow. For moderate shock—pressure ratios, imperfect gas effects may
be neglected, but the wall effects may remain important; the present
analysis considers only the fluid viscosity and conductivity.

Deviations from ideal shock-tube flow are most easily seen experi-
mentally through measurements of the shock-wave velocity. Experimental
attenuations in shock strength were investigated in references 1, 2, 4,
5,-and 6. Reference 6 also presented meaSurements of static—pressure
variations with time at a fixed position after passage of the shock wave.
Experimental timewise density variations in the flow through the use of
a chrono—interferometer were shown in reference 7.

Various theoretical studies have been'carried out as well. Refer-
ences l and h considered a reduction in mass flow at the entrOpy discon-
tinuity computed from the boundary-layer displacement thickness and the
free stream.corresponding to the unattenuated shock. Attenuation in the
shock strength was then found by setting the mass flow through the shock
wave equal to the reduced mass flow at the entrepy discontinuity and com-
puting the new shock strength.