Computer simulation of thermal convection in Rayleigh-Bénard cell ...
Computer simulation of thermal convection in Rayleigh-Bénard cell ...
Computer simulation of thermal convection in Rayleigh-Bénard cell ...
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Hubert Jopek<br />
<strong>Computer</strong> <strong>simulation</strong> <strong>of</strong> <strong>thermal</strong> <strong>convection</strong> <strong>in</strong> <strong>Rayleigh</strong>-<strong>Bénard</strong> <strong>cell</strong><br />
5.4. Boundary Conditions<br />
The boundary conditions for the temperature are as follows:<br />
z = 0<br />
z = 1<br />
→<br />
→<br />
θ = 0<br />
. (5.27)<br />
θ = 0<br />
It is so because <strong>of</strong> the fact that the temperature at the top and the bottom is fixed.<br />
Boundary conditions for the streamfunction - let the shear forces at the top and at<br />
the bottom be neglected:<br />
z = 0<br />
z = 1<br />
→<br />
→<br />
∂υ<br />
x<br />
∂z<br />
∂υ<br />
x<br />
∂z<br />
= 0<br />
= 0<br />
. (5.28)<br />
The follow<strong>in</strong>g expressions satisfy assumed conditions:<br />
Ψ(<br />
x,<br />
z,<br />
t)<br />
= ψ ( t)s<strong>in</strong>(<br />
πz)s<strong>in</strong>(<br />
ax)<br />
θ(<br />
x,<br />
z,<br />
t)<br />
= T1(<br />
t)s<strong>in</strong>(<br />
πz)cos(<br />
ax)<br />
−T2(<br />
t)s<strong>in</strong>(2πz)<br />
where the parameter a is to be determ<strong>in</strong>ed.<br />
, (5.29)<br />
The function Ψ is this part <strong>of</strong> model which is responsible for aris<strong>in</strong>g convective<br />
rolls which can be observed <strong>in</strong> real experiment. The second equation is the<br />
temperature deviation function which consists <strong>of</strong> two parts. The former part<br />
T1<br />
describes the temperature difference between the upward and downward mov<strong>in</strong>g<br />
parts <strong>of</strong> a convective <strong>cell</strong>, while the latter is the description <strong>of</strong> the deviation from the<br />
l<strong>in</strong>ear temperature variation <strong>in</strong> the centre <strong>of</strong> a convective <strong>cell</strong> as a as a function <strong>of</strong><br />
vertical position z .<br />
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