Contribution to the theory of stationary separation zones

Abstract

Experiment shows that the stationary flow pattern about a bluff body with closed separation zone, in the case of laminar flow about the body and in the separation zone, breaks down for a subsonic stream velocity in the Reynolds number range from 10¹ to 10². However, experiment shows that for a supersonic stream velocity a stable stationary flow pattern is observed with the existence of laminar stagnant zones adjacent to the body (the stagnant zone behind an aft-facing step on the body surface, the stagnant zone ahead of a gradual forward-facing step on the body surface, the forward separation zone formed by the tip of a spike, the stagnant zone formed when a shock impinges on a body surface) at high Reynolds numbers of the order of 10⁴−10⁶.

Thus, experiments indicate that in certain ranges of variation of M and R, under certain boundary condition, stationary solutions of the viscous fluid equations of motion exist and are stable. Outside these ranges and under other boundary conditions the flow about a body with a closed separation zone has a more (Karman vortex street for M≪1) or less (pulsating flow in the near wake behind the body for M>1) marked unsteady nature, indicating instability of the stationary solutions of the equations of motion under these conditions. To date no theoretical justification has been presented for the existence of stable stationary flows with separation zones in the ranges indicated.

In the following an attempt is made to find the region of existence of possible stationary flows with a closed separation zone in that range of Reynolds numbers in which the flow in the viscous mixing region may be described by the Prandtl equations. In so doing the boundary conditions for the flow within the separation zone are selected so that the flow pattern within the zone is significantly simplified and use of the analysis methods applicable in hydrodynamics becomes possible. In the first part (§§1–4) we study the field of possible stationary flows for the case of an incompressible fluid. It is shown that only under special boundary conditions within the separation zone (ideal dissipator) does the flow about a flat plat as R→∞ approach the Kirchhoff flow with fluid at rest within the zone. In this case the drag coefficient of the system consisting of the plate plus the ideal dissipator c_x→π/(π^{+ +}4), i.e., it approaches a value which is half that obtained by Kirchhoff for an ideal fluid.

A qualitative study of the field of possible stationary flows in the c_xR plane made it possible to discover the existence of a region, having an upper bound at R≈10², which degenerates into a line. In this region the stationary flows have a singular flow configuration with inviscid vortical-type attachment.

The existence of a connection between the flow configuration in the inviscid vortical attachment region and the stability of the stationary solutions is investigated in the second part (§§6–7), both for the case of individual solutions obtained by the method of linear hydrodynamic stability theory and on the basis of the available experimental data obtained over a wide range of Reynolds numbers for both subsonic and supersonic flow velocities. This investigation makes it possible to formulate a rule for finding stable stationary flows with separation zones and to apply this rule to analyze separation-type flows, both laminar and in certain special cases turbulent.