Turbulent mixing and wave radiation in non-Boussinesq internal bores

Bores, or hydraulic jumps, appear in many natural settings and are useful in many industrial applications. If the densities of the two fluids between which a bore propagates are very different (i.e., water and air), the less dense fluid can be neglected when modeling a bore analytically-a single-layer hydraulic model will accurately predict a bore's speed of propagation. A two-layer model is required, however, if the densities are more similar. Mass is conserved separately in each layer and momentum is conserved globally, but the model requires for closure an assumption about the loss of energy across a bore. In the Boussinesq limit, it is known that there is a decrease of the total energy flux across a bore, but in the expanding layer, turbulent mixing at the interface entrains high speed fluid from the contracting layer, resulting in an increase in the flux of kinetic energy across the expanding layer of a bore. But it is unclear if this finding will extend to non-Boussinesq bores. We directly examine the flux of energy within non-Boussinesq bores using two-dimensional direct numerical simulations and find that a gain of energy across the expanding layer only occurs for bores where the density ratio, defined as the ratio of the density of the lighter fluid to the heavier fluid, is greater than approximately one half. For smaller density ratios, undular waves generated at the bore's front dominate over the effects of turbulent mixing, and the expanding layer loses energy across the bore. Based on our results, we show that if one can predict the amount of energy radiated by bores through undular waves, it is possible to derive an accurate model for the propagation of non-Boussinesq bores. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4745478]