High Resolution Solutions for the Supersonic Formation of Shocks in Transonic Flow

Abstract

We present numerical solutions of two problems for the unsteady transonic small disturbance equations whose solutions contain shocks. The first problem is a twodimensional Riemann problem with initial data corresponding to a slightly supersonic flow hitting the corner of an expanding duct at t = 0. The second problem is a boundary value problem that describes steady transonic flow over an airfoil. In both cases, the solutions contain regions of supersonic and subsonic flow, and an expansion wave interacts with a sonic line to produce a shock. We use high resolution methods, together with local grid refinement, to investigate the nature of the solution in the neighborhood of the point where the shock forms. We find that the shock originates in the supersonic region as originally proposed by Guderley, and very close to, but not at, the sonic line.

A Continuous, Two-Way Free Boundary in the Unsteady Transonic Small Disturbance Equations

“…Based on our solution of the linearized problem in Section 3, we conjectured that the solution of the nonlinear problem has a shock at the sonic line. The numerical results we have presented appear to confirm the existence of this shock, and in addition (see [18]) show that the shock forms inside the supersonic region. In the formulation of a free boundary problem on the domain shown in Figure 11, therefore, part of the free boundary is continuous (the continuous part of the sonic line) and the remainder consists of a transonic shock.…”

“…The exact location of this point is difficult to capture numerically, but it appears to be very close to or at the sonic line. Careful numerical calculations [18] show that the shock forms strictly inside the supersonic region.…”

On the Self-similar Diffraction of a Weak Shock into an Expansion Wavefront

“…Our use of a grid continuation procedure, with grid refinement continuing until evidence of grid convergence is obtained, suggests that the picture of the disappearance of a shock at the sonic line is not likely to change under further grid refinement. The disappearance of a diffracting shock at a sonic point contrasts with the formation of a shock at a supersonic point due to coalescence of compression waves that are reflected from a sonic line, as originally proposed by Guderley [2] and recently confirmed in the computations in [14].…”

“…The disappearance of a diffracting shock at a sonic point differs from the formation of shocks in two-dimensional Riemann problems and transonic flows that is caused by the focusing of characteristics reflected off a sonic line. Such shocks typically form at supersonic points [14]. See [10] for an analysis of related problems.…”

“…The "pseudo-velocities"ū andv are recovered from the potential in the standard way, and the physical velocity perturbations u and v are then obtained through (7.1). See [14,15] for full details. For "pseudo-steady" solutions, the potential ϕ(r, θ) is related to the potential Φ(ρ, θ) in (6.4) by…”