Investigation of imploding shock waves using the hydraulic analogy

Abstract

Imploding hydr aulic jumps in water are created by reservoir-type (RT) and gate-type (GT) water tables. After comparison with the Chestner-Chisnell-Whitham (CCW) approximate solution of imploding hydraulic jumps, the critical rang e of oper ation where the dynamics of the imploding water waves are similar to imploding gaseous shock waves is identified. Possible details are identified that enable water waves in this critical range to behave analogously to gaseous shock waves. 1 Nomenclature = Propagation Velocity of Infinitesimal Disturbances = Velocity / Characteristi c Velocity = Froud e Number = Gravitational Const ant Fr c 9 h = Height of Water He, HI = Initial Test Section Water Height , Initial Reservoir Water Height TJ = HI/Ho >. = Wavelength R = Imploding Hydraulic Jump Radius Ro = Radius of t he Wat er Table Test Section u , u = Velocity Vector, Velocity Component ~ = Defined in Text = Initial Conditions 2 Introduction A cylindrical or spherical imploding shock wave amplifies as its radius approaches zero [1]. As the study of shock waves typic ally involves measurements of phenomena propagationg at several hundred meters per second , it is of interest to study a physically analogous phenomenon which requires less sophisticated diagnosti cs and apparatus. The "hydraulic analogy" has the advantage that gravity waves in water propagate at a velocity three orders of magnitude slower than shock waves in gases, greatly facilita ting visualization of t he wave dynamics. In the 1940's and 50's, this technique was extensively used to study st eady, compressible flows involving supersonic airfoils, inlets, and nozzles. Unsteady phenomena such as regular and Mach reflection of hydraulic jumps were also invest igat ed [2-4]. The st udy of t he amplification process of imploding surface waves, however, has not been reported in t he lit erature. The mathematical similarity between t he equations governing compressible fluid dynamics and shallow water waves permits shock wave dynamic s to be st udied using hydraulic jumps, a