By John Horvath

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In the case of stationary jet flows we can take multiple copies of the flow at different times and compute the average [6]. The average flow will be a smooth function with a better hope for an analytical description (Fig. 1). 3 Methods to Study Turbulence 49 theory is to be able to make predictions without a priori knowledge of the exact flow. In other words, we would like to start with the full NS equation and try to model the behavior of the averaged flows from it. 1) where U is the average flow and u are fluctuations.

3) we obtain: R A ({∂ (k)}) = 1 2 dk→ k→ (2α )d − 2zbd 2 ∂ → (k→ )∂ → (−k→ ) cos δb1−d/2 dk→ ik→ ·x→ → → e ∂ (k ) dx→ . (2α )d By considering an infinitesimal transformation and taking the limit of τφ ∀ 0 yields the differential rescaling of A lim τφ∀0 RA − A = − 2zd τφ + 2z cos δ∂ → (x→ ) dx→ sin δ∂ → (x→ ) δ∂ → x→ 1− d 2 dx→ . 2 Differential Scale Transformations 33 Thus, the total change in A is ξA 1 2 6 = 2z δ ξφ + 2z I1 K d +B 2 2 ∗∂(x) dx − 2z d − sin δ∂(x) δ∂ (x) 1 − d 2 δ2 K d 2 cos δ∂(x) dx dx .

13) where n 0 is the average particle density. 14) which is the standard result for the Debye length [6]. The validity of this results rests on the assumption that the cos(αξ) is a slow varying function of position, which is certainly true at high temperatures. References 1. B. Kogut, An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 4 (1979) 2. M. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C: Solid State Phys. 6, 1181 (1973) 3.