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Turbulence lecture 26

# Turbulence lecture 26 - Turbulence Lecture 26 E(k 2 k ij ki...

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Turbulence Lecture 26 For isotropic turbulence, it is found that ( ) ( ) ( ) 2 4 4 ij ij i j E k k k 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 2 2 2 2 1 2 3 and are related (by continuity) 1 2 ij i j ij f g R r u rr g r u u u u f g g f rf δ = + = = = = + ± functions of r r = ± Longitudinal Correlation Correlation of velocities in direction along . r ± Lateral Correlation ( ) ( ) ( ) ( ) 2 2 1 2 3 2 2 2 "The Correlation" ii u u R r f g f R r rf = + = + = ± These lead to the following pair. 1

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( ) ( ) ( ) ( ) 0 0 2 sin sin E k R r kr krdr kr R r E k dr kr π = = Nonlinear Term: ( ) ( ) ( ) ( ) ijk i j k S r u x u x u x r = + ± ± ± ± ± Only functions of if homogeneous. r ± If isotropic ( ) 3 ijk S r u = ± 3 rk 2r k i j k rr r + + rk 2 4r k ( ) i jk j ik r r δ δ + 2r k k ij r δ ( ) k r defined by.
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Turbulence lecture 26 - Turbulence Lecture 26 E(k 2 k ij ki...

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