Direct numerical simulation of a turbulent hydraulic jump: turbulence statistics and air entrainment

We present direct numerical simulation (DNS) of a stationary turbulent hydraulic jump with inflow Froude number of 2, Weber number of 1820 and density ratio of 831, consistent with ambient water air systems, all based on the inlet height and inlet velocity. A non-dissipative geometric volume of fluid (VOF) method is used to track the detailed interactions between turbulent flow structures and the nonlinear interface dynamics. Level set equations are also solved concurrent with VOF in order to calculate the interface curvature and surface tension forces. The mesh resolution is set to resolve a wide range of interfacial scales including the Hinze scale. Calculations are compared against experimental data of void fraction and interfacial scales indicating, reasonable agreement despite a Reynolds number mismatch. Multiple calculations are performed confirming weak sensitivity of low-order statistics and void fraction on the Reynolds number. The presented results provide, for the first time, a comprehensive quantitative data for a wide range of phenomena in a turbulent breaking wave using DNS. These include mean velocity fields, Reynolds stresses, turbulence production and dissipation, velocity spectra and air entrainment data. In addition, we present the energy budget as a function of streamwise location by keeping track of various energy exchange processes in the wake of the jump. The kinetic energy is mostly transferred to pressure work, potential energy and dissipation while surface energy plays a less significant role. Our results indicate that the rate associated with various energy exchange processes peak at different streamwise locations, with exchange to pressure work flux peaking first, followed by potential energy flux and then dissipation. The energy exchange process spans a streamwise length of order 10 jump heights. Furthermore, we report statistics associated with bubble transport downstream of the jump. The bubble formation is found to have a periodic nature. Meaning that the bubbles are generated in patches with a specific frequency associated with the roll-up frequency of the roller at the toe of the jump, with its footprint apparent in the velocity energy spectrum. Our study also provides the ensemble-averaged statistics of the flow which we present in this paper. These results are useful for the development and validation of reduced-order models such as dissipation models in wave dynamics simulations, Reynolds-averaged Navier-Stokes models and air entrainment models.