Turbulence lecture 23

# Turbulence lecture 23 - Turbulence Lecture 23 Plugging in...

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Turbulence Lecture 23 Plugging in the Fourier series. Multiply by e and integrate over x . im x i ±± () 2 2 ˆˆ L im x ik x ii k L eu k e x u tt ∂∂ ∂= ± m By orthogonality, non-zero when k=m 2 3 2 1 1 ˆ 0 L ik m x i k L if k m uk e x otherwise tL = =∂ = i ±± ± ± ........... ˆ j u u tx =− ± ²³³´ ³ ³µ j / ( ) ........... jj i ik u k u −− + A ± AA ± ± since () () ˆ ikx k k ax ake = ˆ ikx k k a ika k e x = Momentum Equation is Spectral Space. () ( ) () () () 2 ij j i i i k u i kpk kuk t A ± ± i ± Continuity ˆ 00 i i u ik u k x == = ± 1

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() ˆ 0 ii ku k = ( i=1, 2, 3 ) ± / not individually The wave number amplitude is perpendicular to k . Non-linear information is in convolution sum. () () ˆ ik x k ux uke = i ±± ± Triad interactions ,, kk AA A sum over all the possible triad interactions. Have a coupled system of equations. If the process is linear, non-linear term is zero.
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Turbulence lecture 23 - Turbulence Lecture 23 Plugging in...

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