Upstream influence and Long's model in stratified flows

Abstract

This paper describes an experimental study of a stratified fluid which is flowing over a smooth two-dimensional obstacle which induces no flow separation and in which effects of viscosity and diffusion are not important. The results are restricted to fluid of finite depth. Various properties of the flow field, in particular the criterion for the onset of gravitational instability in the lee-wave field, are measured and compared with the theoretical predictions of Long's model. The agreement is found to be generally poor, and the consequent inapplicability of Long's model is explained by the failure of Long's hypothesis of no upstream influence, which is demonstrably invalid when stationary lee waves are possible. The obstacle generates upstream motions with fluid velocities which appear to be of first order in the obstacle height. These motions have some of the character of shear fronts or columnar disturbance modes and have the same vertical structure as the corresponding lee-wave modes generated downstream. They result in a reduced fluid velocity upstream below the level of the top of the obstacle, together with a jet of increased fluid velocity above this level which pours down the lee side of the obstacle. This phenomenon becomes more pronounced as the number of modes is increased.