Numerical Study of Strong Slot Injection into a Supersonic Laminar Boundary Layer

Abstract

This paper addresses the problem of strong slot injection into a supersonic laminar boundary layer within the framework of triple-deck theory. A previous numerical technique for the same problem presented a pressure sensitivity or instability due to the ineffective imposition of the downstream boundary condition and could not account for separation ahead of the slot. The present study removes these two limitations. Solutions to the triple- deck equations are presented for a wide range of wall injection velocities, and the onset of separation is indeed found to occur ahead of the slot. HE general problem of predicting the influence on the flowfield of injecting a secondary fluid into a well- developed boundary layer is of great practical significance. For example, fluid injection is widely used in the aircraft engine industry for reducing heat transfer across turbine blades and is an effective means of controlling transition or separation (blowoff) of boundary layers over airplane control surfaces. This paper addresses the particular case of strong slot injection into a high Reynolds number supersonic laminar boundary layer over a flat plate, within the framework of the triple-deck theory of Stewartson et al. M The injection velocity and the slot length are therefore assumed to be respectively of order Re-3/8Uw and Re~3/8x0. 0^ is the freestream velocity, x0 the length of the plate preceding the slot, and Re the asymptotically large Reynolds number, RC^U^XQ/V, v being the kinematic viscosity and the superscript ~ indicating physical quantities. For these conditions three regions or decks are identified around the slot: the lower, middle, and outer decks, characterized by viscous, inviscid rotational, and inviscid irrotational flow conditions and having characteristic heights of order Re-5/8x0, Re~1/2x0, and Re~3/8x0 respectively. For a detailed review of the triple-deck theory and its most im- portant applications, see Ref. 2. Smith and Stewartson3'4 consider the problem of slot as well as plate injection into a supersonic laminar boundary layer. They provide a comprehensive review of the literature, formulate a rigorous asymptotic analysis for the strong in- jection problem, and analyze the connections of their theory with the problem of weak and massive blowing. For the case of strong blowing by slot injection, of interst here, Ref. 3 provides analytical solutions for the linearized equations, valid when the injection velocity is small in the triple-deck scaling, as well as a numerical technique for the full nonlinear problem. However, such a numerical technique presents a pressure sensitivity, or instability, due to, as the authors