Thesis: Interaction of a Shock Wave with the Wake Behind a Flat Plate in Supersonic Flow
Abstract
This thesis is intended as the first step toward understanding the complex phenomenon of supersonic vortex breakdown which occurs when shock waves and vortices interact. Experiments confirm that when vortices produced by a wing tip in uniform supersonic flow pass through a strong enough normal shock, a conical shock is formed upstream of the original shock, whose axis is aligned with that of the vortex. Inside the cone a recirculation region is observed with reversed flow along its axis.
To study the phenomenon we use, high Reynolds number, asymptotic analysis of the Navier-Stokes equations. Our model of the problem uses the wake of a flat plate, in uniform supersonic flow aligned with the free stream, to mimic the vortices. An oblique shock is provoked by a wedge situated above the wake and symmetry is imposed in the plane of the plate. Our analysis shows that the shock-wake interaction may be treated as inviscid, provided that the velocity on the wake axis, immediately before the interaction region, is larger than Re−1/8.
Assuming the flow is unperturbed upstream of the shock allows us to use Laplace Transforms to solve the linearised governing equations. We find the shock is reflected at the sonic line much like in the problem of a shock impinging on the boundary layer of a flat plate considered by Lighthill. To solve the nonlinear problem we write the unsteady equations in conservative form, allowing us to use a modification of the Lax-Wendroff method. We discover the existence of a critical value of the minimum initial longitudinal velocity profile, below which the solution no longer converges to a steady state. For this critical value we find that at the axis of symmetry the pressure tends to a constant and the longitudinal velocity decays to zero downstream of the interaction region. Using this known behaviour, we perform asymptotic analysis of the governing equations to study the region in which viscosity becomes important. We present an analytical solution for this region which exhibits reverse flow.
British Applied Mathematics Colloquium 2006 Presentation: Shock Wave Vortex Interaction
Keele, April 24-27, 2006
Abstract
We investigate a phenomenon which occurs when shock waves and vortices interact leading to the so called vortex breakdown phenomenon. Supersonic vortex breakdown can be a violent event. It has been observed when the vortices generated by the wing tips of an aircraft collide with the shock waves produced in supersonic flight and is one of the reasons why we do not see multiple aircrafts travelling at supersonic speeds simultaneously.
The phenomenon was investigated experimentally in the paper ‘Supersonic vortex breakdown during vortex/cylinder interaction’ [I.M.Kalkhoran, M.K.Smart and F.Y.Wang J.Fluid Mech vol.369 pp.351-380] in 1998. It was shown that when vortices produced by a wing tip in uniform supersonic flow pass through a normal shock, a conical shock is formed upstream of the original shock, whose axis is aligned with the vortex stream. Inside the cone a recirculation region is observed with reversed flow along the axis of the cone. The size of the cone and hence extent of the recirculation region was found to depend on both the strengths of the shock and vortices. Since upstream influence is not characteristic in supersonic flows this behaviour is unexpected.
To study the phenomenon we use high Reynolds number asymptotic analysis of the Navier-Stokes equations. Our model of the problem uses the wake of a flat plate in uniform supersonic flow to mimic the vortices. The shock is assumed to be produced by a wedge situated above the wake. We assume the flow is unperturbed upstream of the shock allowing us to use Laplace Transforms to solve our governing equations. It is of interest to note that the asymptotic analysis of the Navier-Stokes equations shows that the shock-wake interaction may be treated as inviscid provided that the velocity on the wake axis is larger than Re^{-1/8}.
