Nonlinear shock–structure interaction in a hypersonic flow

This investigation takes an analytical approach to the oscillations of a cantilevered plate immersed in a hypersonic flow with shock impingement. In particular, we derive a mathematical model keyed in on the fact that (1) the shock impingement point moves along the structure as it oscillates and (2) the local curvature of the structure changes the shock reflection angle and thus the compressible flow properties. For cantilever boundary conditions, large motion at the free end my render both effects significant. We show that the movement of the shock impingement point varies approximately with the cotangent of the oblique shock angle and this implicit relationship generates surprisingly strong nonlinear effects in the equation governing the first mode of vibration. A new, geometrically modified third-order Piston Theory is adapted to accurately capture structural curvature induced changes in flow properties as well as possible expansion wave interaction. Our model takes the form of a nonlinearly damped Duffing oscillator with quadratic and cubic nonlinearities stemming from the nonlinear aerodynamic generalized forces as well as geometric and inertial structural nonlinearities. Despite inviscid flow modeling, a perturbation solution to the governing equation shows good agreement with structural oscillations predicted by high-fidelity turbulent computational fluid dynamic simulations.