• Version
• 95 Downloads
• 1.98 MB File Size
• 1 File Count
• January 17, 2017 Create Date
• January 17, 2017 Last Updated

National Advisory Committee for Aeronautics, Technical Notes - Reflection of a Weak Shock Wave from a Boundary Layer Along a Flat Plate

The present paper is concerned with the phenomena encountered when
a plane oblique shock wave is incident upon the boundary layer of a flat
plate. In an effort to simplify the prdblenb the flow field was divided
into a viscous layer near the wall and a supersonic potential outer flow.
The pressure disturbances due to the incident wave would be propagated
upstream and downstream in the subsonic portion of the boundary layer,
thus giving rise to perturbations of the boundary layer. By restricting
the study to infinitesimal incident compression waves, only small per-
turbations were encountered and hence the ordinary linearized theory
could be applied to the outer flow. In the laminar case, the boundary—
layer treatment was based upon a momentumpintegral equation previously
derived by Howarth. The two flows must be compatible; hence, the deflec-
tion of the streamlines near the boundary layer was expressed in terms
of the vertical velocity component along the edge of the boundary layer
and this relation was used as a boundary condition for the outer flow.
The boundary condition determined the form of solution upstream and down-
stream of the point of incidence. Determination of the constants of
integration was accomplished by a consideration of conditions at infinity
and a matching of the two flows at the point of incidence. With the
outer flow thus determined, boundary-layer growth and pressure distribu-
tion were computed and results for the laminar case were obtained as
follows:

(a) The pressure disturbance along the wall decreased exponentially
from a definite value at the point of incidence to zero far upstream of
the point of incidence. Downstream of the point of incidence, the pres-
sure rose to a maximum value and then dropped off to the value corre-
sponding to regular reflection.

(b) The disturbances produced by the interaction decayed exponen-
tially upstreang for a free-stream Mach number of approximately 2 and a
Reynolds number of approximately 1500 in the undisturbed boundary~layer
displacement thickness the upstream influence was of the order of 30
boundary-layer displacement thicknesses.

(c) The "self-induced" pressure gradient along the wall was such
that the boundary layer might separate ahead of the point of incidence.
If separation occurred, the separation point moved upstream as the shock
strength was increased. With increasing Reynolds number, the separation
point also moved upstream, whereas for increasing Mach number, the sepa-
ration point moved downstream.

In the turbulent case the upstream influence was quite small and the
incident wave must be reflected as a shock wave.