Fresh water in a stream is flowing at U1 2 msec The wind abo

Fresh water in a stream is flowing at U_1 =2 m/sec. The wind above the stream is blowing in the same direction at U_2 = 15 m/sec. Determine the critical wave length lambda_crit[in m] where you expect the onset of the Kelvin-Helmholtz instability due to a small perturbation of the water surface. Neglect surface tension effects. Assume rho_H2O = 1000 kg/m^3 and rho_air = 1.2 kg/m^3. Do you consider this a small perturbation? Why or why not?

Solution

For Kelvin Helmholtz instability onset for air moving over a stationary lake, we have

k² * rho₁ * rho₂ (U₁ - U₂)² > k*g (rho₁² - rho₂²)

k*1000*1.2 (2 - 15)² > 9.81*(1000² - 1.2²)

k > 48.372

Wave length = 1/k = 1/48.372 = 0.02067 m or 20.67 mm