Lorenz_Model.nb

Lorenz_Model.nb - Lorenz Model APPH 4200 Physics of Fluids...

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Lorenz Model APPH 4200 Physics of Fluids Columbia University Introduction In 1963, Edward Lorenz modeled the nonlinear thermal convection in 2D. This is the Bénard Instability. Lorenz assumed spatial forms for the convective flow ( i.e. "rolls"), the temperature difference between rising and falling flows, and the distortion of the average temperature gradient. With this assumed spatial forms, the temporal amplitudes defined a coupled set of nonlinear ordinary differential equations. When the instability drive exceeded a threshold, the temporal solutions became chaotic. This illustrates a route to turbulence. Assummed Spatial Forms In[1]:= y @ tX_, x_, z_ D : = tX Cos @ p z D Sin @ 2 p x D ; d T @ tY_, tZ_, x_, z_ D : = tY Cos @ p z D Cos @ 2 p x D + tZ Sin @ 2 p z D Lorenz Equations The Lorenz model has three "free" parameters: pr (the Prandtl number), r (the instability drive), and b = 4/5. In[3]:= eqs = 8 D @ tX @ t D , t D ã pr H tY @ t D - tX @ t DL , D @ tY @ t D , t D ã - tX @ t D tZ @ t D + r tX @ t D - tY @ t D , D @ tZ @ t D , t D ã tX @ t D tY @ t D - b tZ @ t D< Out[3]= 8 tX £ @ t D ã pr H - tX @ t D
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Lorenz_Model.nb - Lorenz Model APPH 4200 Physics of Fluids...

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