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 />
Next coefficients T & 1,T &<br />
2<br />
are found:<br />
T&<br />
1<br />
T&<br />
2<br />
2<br />
= aψ<br />
− ( π<br />
+ a<br />
πa<br />
2<br />
= ψT1<br />
− 4π<br />
T<br />
2<br />
2<br />
) T −πaψT<br />
1<br />
2<br />
. (5.33)<br />
F<strong>in</strong>ally some new variables will be <strong>in</strong>troduced <strong>in</strong> order to simplify the notation, the<br />
first <strong>of</strong> them is new time variable:<br />
2 2<br />
t '' = ( π + a ) t'<br />
. (5.34)<br />
Us<strong>in</strong>g this variable and neglect<strong>in</strong>g aga<strong>in</strong> primes, the follow<strong>in</strong>g expressions are set:<br />
X(<br />
t)<br />
=<br />
( a<br />
rπ<br />
Y(<br />
t)<br />
= T1(<br />
t)<br />
2<br />
Z(<br />
t)<br />
= πrT<br />
( t)<br />
r =<br />
( a<br />
2<br />
a<br />
2<br />
2<br />
2<br />
2<br />
+ π )<br />
2<br />
4π<br />
b =<br />
2 2<br />
a + π<br />
aπ<br />
2<br />
+ π )<br />
3<br />
R<br />
ψ ( t)<br />
2<br />
. (5.35)<br />
Hav<strong>in</strong>g all these parameters def<strong>in</strong>ed the Lorenz model can be written <strong>in</strong> the follow<strong>in</strong>g<br />
form [2]:<br />
X& = σ ( Y − X )<br />
Y&<br />
= rX − XZ −Y<br />
Z&<br />
= XY − bZ<br />
. (5.36)<br />
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