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Normal modes

Rayleigh was able to take normal mode of the following form (in view of the symmetry of the (16), (17) and the boundary conditions (26),27)

$\displaystyle w(x,y,z,t) = W(z) f(x,y)e^{\sigma t},\quad \theta(x,y,z,t)=\Theta(z) f(x,y) e^{\sigma t}$ (28)

with the requirement that

$\displaystyle \Delta_1 f + a^2 f=0$ (29)

Here $ a$ is called the horizontal wavenumber. In general, a disturbance excites components for each real value of $ a$. Now equations (12), (16) and (17) becomes

$\displaystyle (D^2-a^2-\sigma)\Theta=-W$ (30)

$\displaystyle (D^2-a^2)(D^2-a^2-\frac{\sigma}{P_r})W = a^2 R_a \Theta$ (31)

$\displaystyle (D^2-a^2)(D^2-a^2-\sigma)(D^2-a^2-\frac{\sigma}{P_r})W = - a^2 R_a W$ (32)

subject to boundary conditions

$\displaystyle \left. \begin{array}{c} W = DW = \Theta = 0 \quad \mbox{(rigid)}\\ W=D^2W=\Theta=0\quad \mbox{(free)} \end{array} \right\}$   at $\displaystyle z=0,1$ (33)

which can be written as using (31)

$\displaystyle \left. \begin{array}{llr} W& = DW = D^4W-\left(2a^2+\frac{\sigma}...
...\quad &\mbox{(rigid)}\\ W&=D^2W=D^4W=0\quad &\mbox{(free)} \end{array} \right\}$   at $\displaystyle z=0,1$ (34)

This gives three conditions at each of the end point for the sixth order equation (32) to determine the countable infinity of eigen values $ \sigma_j(a,R_a,P_r)$ and associated eigen functions $ W_j (j=1,2,\cdots)$. Let $ \sigma=\sigma_r + i \sigma_i$. For a given values of $ a,R_a$ and $ P_r$, a complete set of solutions $ W_j$ satisfying the boundary conditions (BCs) is needed to represnt an arbitrary initial disturbance. These $ W_j$ are called modes of the solution. For given $ R_a,P_r$, the flow is unstable if $ \sigma_r > 0$ for any mode with any real value of $ a$ and stable if $ \sigma_r \le 0$ for all modes. Hence the critical value of $ R_a$ denoted by $ R_c$ is such that $ \sigma_r(a,R_a,P_r)>0$ for some $ a$ whenever $ R_a > R_c$ and $ \sigma_r(a,R_a,P_r)\le 0$ for all $ a$ whenever $ R\le R_c$.
next up previous
Next: General stability characteristics Up: The stability problem Previous: Boundary conditions
A/C for Homepage of Dr. S Ghorai 2003-01-16