Lecture Notes on Fluid Dynamics (1.63J/2.21J) (original) (raw)

Lecture Notes on Fluid Dynamics (1.63J/2.21J) by Chiang C. Mei, 2007 5.2 Kelvin-Helmholtz instability of flow with discontinuous shear and stratification. 5.2.1 Nonlinear formulation Consider two immiscible fluids. The upper (z > 0) and lower (z < 0) fluids have different densities ρ 1 and ρ 2. Before any perturbations appear the lower fluid is stationary, while the upper fluid moves at the steady and uniform velocityq o = U i. With gravity, this is called the Kelvin-Helmholtz instability problem. For the time being, let us assume that there is no gravity. Now let there be a disturbance on the interface whose vertical displacement is z = ζ(x, t). The corresponding disturbances of velocity and pressure are q 1 and p 1 in the upper fluid and q 2 and p 2 in the lower fluid. Let us first formulate the nonlinear problem. In the upper fluid, we have