By Suzuki H.
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Additional resources for 2-designs over GF(2^m)
Still taking the case of Brownian motion as an example, we choose the random stirring forces to be highly uncorrelated in time. This means that the function w(k, t - t') should be very much peaked near f = t',and we approximate such behaviour by a delta function, thus w ( k ; t-f') = W(k)G(t- t'). (138) This assumption has the virtue that if there are no initial time correlations, then any subsequent correlations will be due to the nonlinear coupling of the Navier-Stokes equation. In order to specify W(k), we have to evaluate the last term on the RHS of equation (136).
Altematively, we can envisage a transfer from one range of eddy sizes to another: a process known as the energy cascade. Our starting point is equation (1 18) for the single-time correlation. We specialize the LHS to the isotropic case by using (102) to reduce the spectral tensor to its isotropic form. Then we can form the equation for the energy spectrum E(k, r), as defined by (107), but now generalized to the time-dependent case. (j, d%Qpy,(j,k-j, - k ; 4 -k-j,k; 01 (122) and the last step follows from the requirement of homogeneity, that the triple moment cannot be affected by changing k to - k , whereas the inertial transport operator Ma&) changes sign.
Therefore the energy spectrum at high wavenumbers may take a universal form. These ideas were first formalized by Kolmogorov in two famous hypotheses (Kolmogorov 1941a, b). The Kolmogorov hypotheses are essentially similarity principles for the energy spectrum and may be expressed in the following way. 1162 W D McComb First, at high enough wavenumbers, the spectrum can only depend on the fluid viscosity, the dissipation rate and the wavenumber itself. Then, on dimensional grounds, the energy spectrum may be written E ( K ) = u2q f ( k q ) = v’’~&‘/y(k q) (149) where the second line follows from (146) and (147), f is an unknown function of universal form, and the dissipation length scale q is given by equation (145).