By Andrei S. Monin, A. M. Yaglom, J. L. Lumley
ISBN-10: 026213098X
ISBN-13: 9780262130981
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Additional info for Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2
Sample text
The functions are called the longitudinal and lateral three-dimensional spectra of the field u (x) (the subscripts L and N now represent projections along the direction of k and the direction perpendicular to k ). The condition F Ji (k) = F;J (-k) shows that both functions F LL (k) and FNN ( k) must be real. Moreover, the matrix II FJ1 (k) II should be nonnegative for all k, and this is readily shown to be equivalent to the condition that both functions F LL (k) and FNN ( k) should be nonnegative [ this also follows from their definition given by Eq.
E . , space symmetries) of the set of points x 1 . x2 , xN. Since the field u (x) is h omogeneous, i ts mean value u (x) should be a constant. With out loss of generality we can assume that u (x) = 0 , having replaced , i f necessary , the original field u (x) b y the field u' (x) = u (x) - U(X). We shall adopt this procedure henceforth . The correlation function B (x, x') = u (x) u (x') of an isotropic field u (x) must , of course, assume equal values for any two pairs of points (x , x') and ( x 1 , x� ) which can be made to coincide as a result of a rigid motion ( translation or rotation) of the pair of points (x1 , x�) · In other words, if the d istance between the p oin ts x and x' is equal to the distance between x, and x� then B (x.
K consequently we can eliminate [ where b jh1 j j a = I a I . b = I b I I is = j j j an orthonormal triad of vectors, and from Eq. 86) and rewrite the expression for F jl (k) in the form ( 1 1 . 87) 28 STATISTICAL F L U I D MECHAN ICS The functions b2 (k), a 2 (k) , aJ (k) are here related by Using the conditions given by Eq. (1 1 . 79) , we can establish another general representation of the tensor FJ l (k) which is occasionally quite convenient. Namely , let us consider the tensor ( 1 1 . e. , it satisfies the condition kjlljm (k) = 0.
Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2 by Andrei S. Monin, A. M. Yaglom, J. L. Lumley
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