A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue

Abstract

We propose a theory of a steady circular hydraulic jump based on the shallow-water model obtained from the depth-averaged Navier–Stokes equations. The flow structure both upstream and downstream of the jump is determined by considering the flow over a plate of finite radius. The radius of the jump is found using the far-field conditions together with the jump conditions that include the effects of surface tension. We show that a steady circular hydraulic jump does not exist if the surface tension is above a certain critical value. The solution of the problem provides a basis for the hydrodynamic stability analysis of the hydraulic jump. An analogy between the hydraulic jump and a detonation wave is pointed out.

Nonlinear Dynamics of Self-Sustained Supersonic Reaction Waves: Fickett’s Detonation Analogue

“…The results presented are for parameters, Q = 5, K i = 1, K r = 2 and ν = 0.5. Below the critical value α = 5.7, the steady solution is stable, and propagated with the steady wave structure given above at its constant CJ speed given by (6). Above this critical value the travelling wave solution is unstable and develops a stable limit-cycle, as shown in Fig.…”

Accurate Numerical Simulation and Analysis of Multi-Dimensional Shock and Detonation Waves

“…This equation has a regular solution crossing the sonic point c = u provided / = 0 at the same point, which is precisely Figure 9: Left -comparison between our theoretical prediction of the hydraulic jump structure [9] and experiment. Right -jamiton profile in the theory of traffic shocks [7J.…”

“…Two of such phenomena, a circular hydraulic jump [9] and a traffic jam [7], have been investigated in detail. It was shown that both in traffic jam and the hydraulic jump, the theory of the steady solution is similar to the Zel'dovich-von Neumann-Doering theory of detonation [6].…”

“…This appears to be a major mathematical challenge with wide-ranging implications for a large number of applications where self-sustained shocks are possible, not only in shallow-water systems and traffic flows, but also in certain astrophysical flows (accretion shocks,) flows in flexible tubes, two-phase flows, and others. The analogy is seen in the main properties of the shallow-water and traffic-flow systems [9,7]:…”

“…The discovery of the analogy allowed us to develop the first rational theory of a classical phenomenon in fluid mechanics, the hydraulic jump [9], and to find a simple analytical solution to another textbook problem in gasdynamics/partial differential equations, namely the traffic jam [7] described within the framework of a continuum model.…”