Turbulent Shock — Boundary-Layer Interaction with Control Theory and Experiment

Solvability of a problem for transonic flow with a local supersonic region

“…At small deviations, formation of shocks in the end of LSR is caused by the accumulative phenomenon for perturbations propagating in the LSR and multiply reflected from the sonic line and the airfoil [15], [16]. Recent numerical and experimental studies showed that employment of appropriate control through a perforated portion of the airfoil may prevent the accumulative phenomenon and, thus, provide smoothness of the flow [5], [6], [15]. That is why, below we shall use perforation of the bump modeling an airfoil and prescribe the corresponding boundary condition.…”

“…Then for any perturbation f ± of β ± vanishing in the small vicinities of x = 0, x = l, and having sufficiently small norm f ± W 3,2 (0,l) there exists a unique solution u ∈ W 4,2 (G) of the nonlinear problem (4), (5).…”

“…Indeed, one can get this substituting u +û for u in (5), (6), whereû is a function that satisfies the boundary conditions (5). From (14), (15), with the change of variables x = t, y = −x , we obtain problem (21), (22) of Appendix 2 in the rectangle D of the plane (x , t) with k(x , t)…”

“…Let us define the operator T g from W 3,2 (G) into W 3,2 (G) in the following way [19]: T g = u x where u is the solution of equation L g u = 0 endowed with boundary conditions (5). Then the nonlinear problem (4), (5) can be rewritten in the form T g = g. Under sufficiently small g 3 ≤ r, the operator T g is a contractive one in the norm of W 1,2 (G), since equation (6) yields…”

Effect of Dimples on Glancing Shock Wave-Turbulent Boundary Layer Interactions

“…The effect of boundary layer separation, due to strong shock waves, can distort the quality of airflow in engines inlets, reduce th effectiveness of control surfaces, and have a detrimental effect on the efficiency of lifting surfaces [1,2].…”