• Version
• 127 Downloads
• 6.88 MB File Size
• 1 File Count
• January 29, 2017 Create Date
• January 29, 2017 Last Updated

National Advisory Committee for Aeronautics, Technical Notes - An Integral Solution to the Flat Plate Laminar Boundary Layer Flow Existing Inside and After Expansion Waves and After Shock Waves Moving into Quiescent Fluid with Particular Application to the Complete Shock Tube Flow

A solution to the unsteady two-dimensional laminar boundary-layer
flow inside centered expansion waves and behind both centered. expansion
waves and. shock waves is obtained by utilizing an extension of the Karman-
Pohlhausen method. The Prandtl unsteady-boundary-layer’ equations are
integrated normal to the surface bounding the flow and are transformed
into a conical coordinate system. The resulting hyperbolic differential
equations are integrated in closed form for flow behind shock waves and
by numerical methods for the flow inside or following emansion waves.

An integral technique is applied at the discontinuities existing at the
trailing edge of the expansion fan and at contact discontinuities (entropy
discontinuities) so that the characteristic solution may proceed across
these discontinuities.

The solution to the two-dimensional unsteady laminar boundary layer
existing at all points in an air-air shock tube is obtained by this method.
A much shorter approximate method of solution is devised and. is found to
agree favorably with this method. This approximate method is used to
predict the flow in hydrogen—air and helium-air shock tubes. Plots of
wall heat-transfer rate and. skin friction in air—air, helium-air, and.
hydrogen-air shock tubes are presented.

Impetus to the study of time—dependent boundary layers has arisen
because of the increased importance of the flows initiated along the
ground and over buildings by the detonation of nuclear devices and of
the air flow over missiles in hypersonic flight. The time-dependent
nature of the nuclear-shock-initiated flows is obvious; whereas hypersonic ‘_
missile flight presents two less obvious problems, one of which is direct _
and the other, indirect. The direct problem arises because of the time- -
wise variation of the differences between conditions of the outer potential
flow at the edge of the boundary layer and of the missile skin as the mis—
sile encounters rapidly varying ambient conditions during its flight. To
date, because of the relative rapidity with which the fluid boundary layer
is able to adjust to changes, the direct problem has been treated as a
quasi-steady one; that is, for given wall and local conditions at a time
in a time—dependent flow, the boundary layer is equivalent to that in a
steady flow for the same stream and wall conditions. The main apparent
difficulty in this approach is the prediction and simulation of the cor- ‘
rect wall conditions.since they are in turn dependent on the time history
of the boundary layer.