Stability of Supersonic Boundary Layers on a Cone at an Angle of Attack

The stability and receptivity of three-dimensional supersonic boundary layers over a 7ϒ sharp tipped straight cone at an angle of attack of 4.2ϒ is numerically investigated at a free stream Mach number of 3.5 and at two high Reynolds numbers, 0.25 and 0.50*10 6 /inch. The generation and evolution of stationary crossflow vortices are also investigated by performing simulations with three-dimensional roughness elements located on the surface of the cone. The flow fields with and without the roughness elements are obtained by solving the full Navier-Stokes equations in cylindrical coordinates using the fifth-order accurate weighted essentially non-oscillatory (WENO) scheme for spatial discretization and using the thirdorder total-variation-diminishing (TVD) Runge-Kutta scheme for temporal integration. Stability computations reveal that the azimuthal wavenumbers are in the range of m ~ 25-50 for the most amplified traveling disturbances and in the range of m ~ 40-70 for the stationary disturbances. The N-Factor computations predicted that transition would occur further forward in the middle of the cone compared to the transition fronts near the windward and the leeward planes. The simulations revealed that the crossflow vortices originating from the nose region propagate towards the leeward plane. No perturbations were observed in the lower part of the cone. I. Introduction Three-dimensional boundary layers exist when the inviscid streamlines are curved in the spanwise direction. When the inviscid streamlines are curved, there exists a pressure gradient in the direction normal to the inviscid streamlines. Inside the boundary layer, due to the viscous effect, the velocity is smaller than that in the inviscid region. Hence, this pressure gradient causes a velocity component, called crossflow velocity, inside the boundary layer that is perpendicular to the inviscid-velocity vector. This crossflow velocity contains an inflection point in its profile and causes a new instability called crossflow instability. The crossflow instability is unstable to three-dimensional traveling and stationary disturbances. The stationary disturbances originate from isolated roughness elements and appear as corotating vortices. The stationary crossflow vortices dominate the transition process in most of the cases except in high turbulence environments. This phenomenon is observed in several incompressible and compressible flows including swept wings, rotating disks, rotating cones and cones at angles of attack. The linear and nonlinear crossflow instability in incompressible flow is well explained by the pioneering work of Gergory et al. (1955), Dehyle and Bippes (1996), Saric et al. (1998) and Malik et al. (1994). The major findings about the linear and nonlinear crossflow instability in incompressible flows are summarized in a review paper by Saric et al. (2003). Early investigations of the stability characteristics of supersonic boundary layers 6-8 revealed the important finding that the unstable disturbances in supersonic boundary layers are three-dimensional. They also found that the wave angles of the most amplified disturbances are inclined around 60-65 degrees from the inviscid streamlines in a boundary layer with an edge Mach number of 3.5. The linear instability of axi-symmetric three-dimensional compressible boundary layers for a rotating cone was numerically investigated by Balakumar and Reed (1991). Their calculations showed that the growth rate of the traveling disturbances is increased by a factor of 2 to 4 due to the presence of the crossflow compared with the two-dimensional flow over a non-rotating cone and this increase decreases with increasing Mach number.