User:Loren36/kelvin

From Wikipedia, the free encyclopedia

[edit] Theory

The dispersion relation for the Kelvin-Helmholtz instability can be derived by assuming that the fluids are incompressible flow and irrotational such that the fluid velocities u1 and u2 may be defined by a scaler potential φ1 and φ2 corresponding to the top and bottom fluid respectively, such that:

u_i = \nabla \phi_i

Using Reynolds decomposition, the scaler potential is decomposed into an ensemble of time varying perturbations superimposed upon a time independent mean state:

φi(x,y,z,t) = Φi + φi'(x,y,z,t)
\phi_i'(x,y,z,t) = \hat{\phi_i}(z) e^{i(k x + l y) + s t}

The interface between the two fluids ζ is described in a similar manner, such that

\zeta(x,y,z,t) = \hat{\zeta}(z) e^{i(k x + l y) + s t}