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Next: Boundary conditions Up: The stability problem Previous: The linearized equations

Scaling of the equations

We use $ d$ as length scale, $ d^2/k$ as time scale and $ T_b-T_u=\beta d$ as the scale for temperature. Let $ \boldsymbol {u},p,\theta$ are the nondimensional variables. The linear stability equations become

$\displaystyle \nabla\cdot \boldsymbol {u}=0$ (10)

$\displaystyle \frac{\ensuremath{\partial}\boldsymbol {u}}{\ensuremath{\partial}t} = -\nabla p + R_a P_r \theta \hat{k} + P_r \Delta\boldsymbol {u}$ (11)

$\displaystyle \frac{\ensuremath{\partial}\theta}{\ensuremath{\partial}t} - w = \Delta\theta$ (12)

Here the nondimensional Rayleigh number and Prandtl number are given by

$\displaystyle R_a = \frac{g \alpha \beta d^4}{\kappa \nu},\quad P_r=\frac{\nu}{k}$ (13)

The boundary conditions are to be applied on the boundary at $ z=0$ and $ z=1$. Note that Rayleigh number is positive when $ \beta > 0$ and is the ratio of buoyancy force and viscous force. On the other hand, Prandtl number depends on the properties of the fluid: ratio of molecular diffusion due to momentum and heat.

We now eliminate all the dependent variables except $ w=u_3$. The curl of equation (11) gives

$\displaystyle \frac{\ensuremath{\partial}\boldsymbol {\omega}}{\ensuremath{\par...
..._r (\nabla\theta \boldsymbol {\times}\hat{k}) + P_r \Delta\boldsymbol {\omega},$ (14)

where we $ \boldsymbol {\omega}=\nabla\boldsymbol {\times}\boldsymbol {u}$ is the vorticity of the flow. In the above equation we have used the vector identity

$\displaystyle \nabla\boldsymbol {\times}(\phi \boldsymbol {A}) = \phi (\nabla\b...
...bol {\times}\boldsymbol {A}) + \nabla\phi
\boldsymbol {\times} \boldsymbol {A}
$

Taking the curl of equation (14) again and using the vector identities

$\displaystyle \nabla\boldsymbol {\times}(\nabla\boldsymbol {\times}\boldsymbol {A}) = \nabla(\nabla\boldsymbol {\cdot}\boldsymbol {A}) -
\Delta\boldsymbol {A}
$

$\displaystyle \nabla\boldsymbol {\times}(\boldsymbol {A}\boldsymbol {\times}\bo...
... {A}(\nabla\cdot\boldsymbol {B}) - \boldsymbol {B}(\nabla\cdot\boldsymbol {A})
$

we get

$\displaystyle \frac{\ensuremath{\partial}}{\ensuremath{\partial}t}\Delta\boldsy...
...{\partial}\theta}{\ensuremath{\partial}z}\right) + P_r \Delta^2\boldsymbol {u},$ (15)

In particular for the vertical velocity $ w$ we have

$\displaystyle \frac{\ensuremath{\partial}}{\ensuremath{\partial}t}\Delta w = R_a P_r \Delta_1\theta+ P_r \Delta^2 w,$ (16)

where $ \Delta_1=\ensuremath{\partial}^2/\ensuremath{\partial}x^2 + \ensuremath{\partial}^2/\ensuremath{\partial}y^2$ is the horizontal laplacian. This equation can be written is

$\displaystyle \left(\frac{1}{P_r}\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\Delta w = R_a \Delta_1 \theta
$

Similarly the heat equation (12) can be written as

$\displaystyle \left(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\theta = w
$

If we eliminate $ \theta$ then we get

$\displaystyle \left(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}-\Delta...
...remath{\partial}}{\ensuremath{\partial}t}-\Delta\right)\Delta w = R_a\Delta_1 w$ (17)

If we eliminate $ w$ then we get same equation (17) satisfied by $ \theta$.

From the vertical component of $ \boldsymbol {\omega}$ we have

$\displaystyle \frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}t}=P_r \Delta \omega_3$ (18)

from which we can determine the vertical vorticity component. From the definition of $ \omega_3=\ensuremath{\partial}v/\ensuremath{\partial}x - \ensuremath{\partial}u/\ensuremath{\partial}y$ and the equation of continuity, we get

$\displaystyle \Delta_1 u = -\frac{\ensuremath{\partial}^2 w}{\ensuremath{\parti...
...remath{\partial}z}-\frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}y}$ (19)

$\displaystyle \Delta_1 v = -\frac{\ensuremath{\partial}^2 w}{\ensuremath{\parti...
...remath{\partial}z}+\frac{\ensuremath{\partial}\omega_3}{\ensuremath{\partial}x}$ (20)

and these equation can be used to find $ u$ and $ v$.
next up previous
Next: Boundary conditions Up: The stability problem Previous: The linearized equations
A/C for Homepage of Dr. S Ghorai 2003-01-16