Turbulence lecture 26

Turbulence lecture 26 - Turbulence Lecture 26 E (k ) 2 ( k...

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Turbulence Lecture 26 For isotropic turbulence, it is found that () ( ) 2 4 4 ij ij i j Ek kk k δ π ± k k Φ= . See Hinze for a physical derivation or Batchelor or Monin & Yaglom (group invariant theory) Similarly for isotropic turbulence. 2 2 2222 123 and are related (by continuity) 1 2 ij i j ij fg Rr u r r g r uuuu gf r f  =+   === ± functions of rr = ± Longitudinal Correlation Correlation of velocities in direction along . r ± Lateral Correlation 22 1 23 2 "The Correlation" ii uu R rf g f Rr r f = + = ± These lead to the following pair. 1
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() () () 0 0 2 sin sin Ek Rrk r k r d r kr Rr d r kr π = = Nonlinear Term: () () () ( ) ijk i j k Sr ux ux uxr =+ ±± ±±± Only functions of if homogeneous. r ± If isotropic 3 ijk Sr u = ± 3 rk 2r k ijk rrr  ++   rk 2 4r k ( ) ij k ji k rr δδ + 2r k ki j r δ kr defined by. () ( ) 2 3 2 2 pp p u + = ± ij Sr A ± has a transform 3 1 2 ik r ij ij kS r γ −∞ = i ± ± AA ± e d r For isotropic flow. 22 11 ij i j i j j i k i k kkk kk  =℘   A ± A For isotropic turbulence 2
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() 2 2 ii Ek k k π Φ= ± , Qk k d k ± Nonlinear term in equation for ( ) ii k Φ ± = ( ) 4 2 4 Tk kk k ℘= Go to equation for () () () 2 ,, 2 ii Ekt Tkt v kEkt t , is the transfer of energy to and from all wave numbers and from and to wave number
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This note was uploaded on 06/07/2011 for the course EGM 6341 taught by Professor Mei during the Spring '09 term at University of Florida.

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Turbulence lecture 26 - Turbulence Lecture 26 E (k ) 2 ( k...

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