The hydraulic jump in circular jet impingement and in other thin liquid films

The circular hydraulic jump exhibits behavior quite different from that commonly observed in planar jumps. Here we examine experimentally some of the causes and consequences of those differences. We suggest that surface tension plays a dominant role in establishing the shape of the circular jump for impinging jets. The importance of surface tension is a direct result of the thinness of the liquid films normally encountered in circular jump configurations. A sequence of instabilities appears in the jump's structure as the subcritical liquid film becomes thicker and surface tension effects decrease. These conclusions are corroborated by experiments on thin planar films which result in unusual jump structures, like those seen in circular jumps. In addition, we show that the standard momentum balance for the circular jump is effective only at relatively low supercritical Froude numbers or at low ratios of downstream to upstream depth. Typical values of those parameters for circular jumps are often quite large relative to the usual values for planar open-channel flows.List of Symbols d jet diameter - D fictitious downstream drag force - Fr_d jet Froude number, u_f/ gd - Fr_h supercritical film Froude number, u_fd²/8r_j gh³ - Fr_s subcritical film Froude number, u_fd²/8r_s gs³ - g gravitational body force - h local thickness of liquid sheet - p hydrostatic pressure - r radius measured from jet stagnation point - r_j radius at which hydraulic jump begins - r_s radius at which subcritical depth equals s - R radius of curvature of jump interface - Re_d Reynolds number of the jet, u_fd/v. - s liquid sheet thickness after hydraulic jump - u(r,y) radial velocity distribution in liquid film - u_f velocity of impinging jet - _h depth average velocity for sheet of thickness h, u_fd²/8rh - y distance normal to the wall - We Weber number of jump, s pg/Greek letters v liquid kinematic viscosity - liquid density - surface tension