Problem 5: Rayleigh–Taylor instability

Suppose that a heavy fluid of constant density \(\rho_1\) is placed above a light fluid of another constant density \(\rho_2\), and separated by a surface S in a vertically downward gravitational field of acceleration \(g\). In the unperturbed state, the surface S was a horizontal plane located at \(y = 0\) (with the \(y\) axis taken vertically upward) and the fluid was at rest, and the density was \[ \rho_1 \quad \text{for } y > 0; \qquad \rho_2(<\rho_1) \quad \text{for } y < 0 \] Suppose that the surface S is deformed in the form \[ y = \zeta(x,t) \] Both above and below S, the flow is assumed to be irrotational and the velocity potential \(\phi\) is expressed as \[ \phi = \phi_1(x,y,t) \quad \text{for } y > \zeta; \qquad \phi_2(x,y,t) \quad \text{for } y < \zeta \]

(1) Derive linear perturbation equations for small perturbations \(\zeta\), \(\phi_1\), and \(\phi_2\) from the dynamic boundary condition and kinematic boundary condition.

(2) Expressing the perturbations in the following forms with a growth rate \(\sigma\) and a wavenumber \(k\), \[ \zeta = Ae^{\sigma t} e^{ikx}, \qquad \phi_i = B_i e^{\sigma t} e^{ikx - ky}, \qquad (i = 1, 2) \] derive an equation to determine the growth rate \(\sigma\), where \(A, B_1, B_2\) are constants. State whether the basic state is stable or unstable.

(3) Apply the above analysis to the case where a lighter fluid is placed above a heavy fluid, and derive a conclusion that there exists an interfacial wave (the internal gravity wave). State what is the frequency.

RayleighTaylorWav

1 This problem set contains 6 problems, and your final score will be based on the 4 problems on which you earned the highest scores.