Supersonic Separated Flow Downstream of a Backward Facing Step


Book Description

The method of integral relations is applied to the study of the viscous separated flow field downstream of a backward facing step in laminar supersonic flow. The boundary layer equations are reduced to a set of coupled ordinary differential equations in a formulation which incorporates a two parameter velocity profile. The neighborhood of the corner and the reattachment point are examined in detail, taking into account the proper interaction between the viscous and inviscid flow. The complete flow field in these regions is found from the analyses of the limiting forms of the equations. In particular, it is found that the pressure gradient and thickness of the boundary layer at the reattachment point are related and cannot be arbitrarily specified. (Author).




SUPERSONIC FLOW SEPARATION ON A BACKWARD FACING STEP.


Book Description

The paper describes an experimental survey of flow separation on backward facing steps having different heights at a Mach number of 2.4. Reattachment and critical points are found for three regimes, laminar, transitional, and turbulent. Reattachment occurs at a point where the pressure is 35% of the free stream value for turbulent flow, and 60% of this value in the laminar case. The length of the free shear layer is found to be one-half that of the separating streamline, a result which emphasizes the importance of the reattachment region. The flow downstream of the critical point is found to be relatively independent of the base flow. Disturbances in the spanwise direction are always observed in laminar flow but do not affect the base pressure.







Computational Study of Supersonic Flow Over Backward-Facing Steps at High Reynolds Number


Book Description

The problem of laminar separated flow over a backward-facing step is solved numerically for the limiting case of infinite Reynolds number. The flow model adopted is that deduced by Batchelor for incompressible flow: an inviscid rotational eddy confined within the recirculation portion of the separated shear layer. It is argued that the reattachment process is essentially inviscid in the limit R approaches infinity, and the flow field in the reattachment zone is computed on this basis. The computed results support Chapman's model of the reattachment process as the correct limit case, to second-order accuracy, as the flow angle entering the reattachment zone approaches zero. For the recirculation zone, the boundary-layer equations are simplified by assuming a constant pressure eddy. The Dorodnitsyn transformation is applied and numerical solutions obtained using an implicit finite-difference scheme. The solutions are carried out from station-to-station in the direction of flow over a complete cycle of recirculation. Iteration yields the unique value of the vorticity in the inviscid layer upstream of separation, including the effects of corner expansion and recirculation on the development of the separated shear layer. (Author).




Separation and Reattachment of Supersonic Turbulent Boundary Layer Behind Down Stream Facing Steps and Over Cavities


Book Description

Surface pressure measurements were obtained for the flow regimes created when a turbulent boundary layer in supersonic flow separates over the classical downstream-facing step and over a cavity. The experiments were conducted in the Mach number range 2.0 to 4.0 on a model having a ducted, one-foot diameter, axisymmetric body. The model provided step heights of 0.250, 1.020, and 1.675 inches, cavity lengths of 0.837 and 1.675 inches, and a cavity depth of 1.675 inches. The recompression shoulder of the cavity had either a square or a rounded shape. In general, the pressure rise at reattachment is found to be in good agreement with Korst's theory. The shape and scale of the reattachment pressure distribution are influenced considerably by Mach number. Reattachment occurs at a point where the pressure rise is about 50% complete. A point of flow reversal is found to be located upstream of the reattachment point. (Author).










The Flow Field and Heat Transfer Downstream of a Rearward Facing Step in Supersonic Flow


Book Description

An experimental investigation of the flow field, and the model pressure and steady-state heat transfer distributions for a rearward-facing step in supersonic flow is described. Tests were conducted using a water-cooled model with a step height adjustable to 0.443 and 0.750 inches at free stream Mach numbers of 2.5, 3.5, and 5.0, and at Reynolds numbers based on length of surface ahead of separation of approximately 250,000 to 1,800,000. It was found that the Reynolds number based on step height is an important parameter and that both the base pressure and the maximum heat transfer at reattachment may be predicted as a function of this parameter. Several representative flow fields are presented along with analyses of the various regions of these fields. It was found that the depressed base pressure is communicated upstream of the step through the subsonic portion of the attached boundary layer resulting in a pressure gradient immediately upstream of the step. It is shown that the rapid corner expansion is not the commonly used Prandtl-Meyer expansion, but rather is accurately described by the method of inviscid rotational characteristics which accounts for both the entropy gradient in the boundary layer and the pressure gradient upstream of the step. This description of the corner expansion also accurately predicts the position of the lip shock associated with the rapid expansion.