Once again on the supersonic flow separation near a corner

Abstract

Laminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous-inviscid interaction concept. The 'triple-deck model' is used to describe the interaction process. The governing equations of the interaction may be formally derived from the Navier-Stokes equations if the ramp angle θ is represented as θ = θ 0 Re −1/4 , where θ 0 is an order-one quantity and Re is the Reynolds number, assumed large. To solve the interaction problem two numerical methods have been used. The first method employs a finite-difference approximation of the governing equations with respect to both the streamwise and wall-normal coordinates. The resulting algebraic equations are linearized using a Newton-Raphson strategy and then solved with the Thomas-matrix technique. The second method uses finite differences in the streamwise direction in combination with Chebychev collocation in the normal direction and Newton-Raphson linearization.Our main concern is with the flow behaviour at large values of θ 0 . The calculations show that as the ramp angle θ 0 increases, additional eddies form near the corner point inside the separation region. The behaviour of the solution does not give any indication that there exists a critical value θ * 0 of the ramp angle θ 0 , as suggested by Smith & Khorrami (1991) who claimed that as θ 0 approaches θ * 0 , a singularity develops near the reattachment point, preventing the continuation of the solution beyond θ * 0 . Instead we find that the numerical solution agrees with Neiland's (1970) theory of reattachment, which does not involve any restriction upon the ramp angle.

Instability of supersonic compression ramp flow

“…For instance, for α = 4.5, we find separation at x sep ≈ −13.67 and the downstream reattachment at x reat ≈= 9.245. This is consistent with the data presented in [5], whereas in Smith & Khorrami [12] the corresponding values are x sep ≈ −5 and x reat ≈ 4. However, for α = 7.5, the results in [5] give the separation point to be x sep ≈ −52, with reattachment at x reat ≈ 18 and the minimal skin friction at x ms ≈ 12.…”

“…Smith & Khorrami [12] suggest that the solutions cannot be obtained beyond some critical ramp angle because of a singularity which develops as this critical angle is approached. Our results compare well, see also [11], with those of [5] for angles up to α = 5, although there are slight differences for α = 7.5, which may be due to the differences in the grid sizes used. For instance, for α = 4.5, we find separation at x sep ≈ −13.67 and the downstream reattachment at x reat ≈= 9.245.…”

“…A direct hybrid finite difference and spectral method similar to that described in [5] is used for the computation of the basic flow, and the technique is robust and can compute the flow accurately for large ramp angles. Two distinct approaches are used to investigate the instability of the computed flow.…”

“…Two methods were used for computing the basic flow. The first is as described in [5] and referred to as Method B in that paper. In the second approach, we worked with primitive variables rather than with streamfunction vorticity variables.…”

Subsonic flow past localised heating elements in boundary layers

“…In Sychev et al (1998), different approaches are described to solve these kind of problems. Our method consists of using finite differences in the x-direction (stream-wise direction) and Chebyshev collocation method in the y-direction (wall normal direction) using the technique as described in Korolev et al (2002), Logue (2008). Hereafter, the subscript b in the variables is omitted.…”