Shock diffraction in channels with 90° bends

Abstract

A study has been made of how initially planar shocks in air propagate around 90° bends in channels of nearly rectangular cross-section. In shallow bends for which the radius of curvature R is much greater than the radius r of the channel, the shock recovers from a highly curved profile at the start of the bend to regain planarity towards the end of the bend. This occurs on account of the acceleration of the triple point across the channel following its interaction with the expansion waves generated at the convex wall. In sharp bends the shock profiles retain their pronounced curvature for some distance downstream of the bend.At the start of a shallow bend (R/r ≈ 6) the shock at the concave wall, initial Mach number M0, accelerates to Mw = 1.15M0 and remains at this value until towards the end of the bend it begins to attenuate. At the convex wall, shocks of M0 > 1.7 attenuate to Mw = 0.7M0 and propagate at this value for some distance around the bend. In the early stages of a sharper bend (R/r ≈ 3) the shock at the concave wall strengthens to Mw = 1.3M0, remaining at this value for some distance downstream of the bend. At the convex wall the shock decelerates to 0.6M0.Whitham's (1974) ray theory is shown to predict with reasonable accuracy the Mach numbers of the wall shocks at both surfaces for both bends tested and the range of incident shock velocities used, 1.2 < M0 < 3. The agreement between the theory and experimental results is particularly close for stronger shocks propagating along the inner bend. Predictions from 3-shock theory (Courant & Friedrichs 1948) of the Mach number at the outer wall are consistently higher than those from Whitham's analysis. In turn, the latter tends to slightly overestimate the strength of the wall shock.A model is developed, based on an extension of Whitham's analysis, and is shown to predict the length of the Mach stem produced by shocks of M0 > 2 over the initial stages of the bend.