Numerical simulations of stratified inviscid flow over a smooth obstacle

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Abstract
Results of numerical simulations of the flow of a non-rotating, inviscid, Boussinesq fluid over smooth two-dimensional obstacles are described. The fluid has finite depth and a rigid lid. Far upstream of the obstacle the horizontal velocity and buoyancy frequency are uniform. Comparisons with linear theory for small-amplitude obstacles are made and the long-time behaviour is compared with steady-state Long's model solutions. Comparisons with the time-dependent results of Baines (1979) are done. For Froude numbers between ½ and 1 the obstacle amplitude is varied in order to determine the amplitudes needed to initiate wave breaking. These results are compared with the predictions of Long's model and with the experimental results of Baines (1977) showing good agreement with the latter. It is found that wave breaking occurs for amplitudes significantly lower than Long's model predicts for a large range of Froude numbers. This is shown to be the result of the generation of large-amplitude lee waves with wavelengths longer than that of stationary lee waves, but not long enough to propagate upstream. The behaviour of these waves is coupled to the generation of both longer mode-one waves which do propagate upstream from the obstacle and to mode-two waves which propagate against the flow as they are advected downstream. It is also coupled to oscillations in the wave drag. The periods of the wave drag oscillations are compared to experimental results showing good agreement with cases for which oscillations have been observed. The behaviour of these large transient lee waves is compared with the theoretical results contained in Grimshaw & Yi (1991), showing some similarities. As the Froude number approaches 0.5 the breaking behaviour is no longer due to these large waves, with the result that wave breaking occurs much later.