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Multiplying each side by k , , the vanishing of the LHS follows trivially from (88). 2. Homogeneous uelocity-field moments. In order to develop a formalism based on equation (91), we need to know something about the general properties of the moment hierarchy in wavenumber space. We begin by considering the implications of homogeneity, meaning that moments are translationally invariant in configuration (x) space. (x)up(x’)) exp(-ik - x - i k ’ . x ’ ) = ( l / L ) 6 1 1 d3xdd)r(u,(x)up(x-r)) exp(-i(k+k’)’ x) exp(ik“r).

2. Fourier analysis of the turbulent uelocityfield The introduction of Fourier analysis leads to three main benefits. It converts differential operators into multipliers; it gives us a relatively simple picture of the physics of turbulence; and it allows us to define the degrees of freedom of the turbulent system. We begin by considering the turbulent fluid to be occupying a cubic box of side L. x) k where the wavevector k is given by k=(2n/L)(nl,nl, n3 (85) and n l , n2, and n3 are integers, each of which is summed over the range from minus to plus infinity.

129) Clearly this allows us to replace k, in the first term on the right of (127) by ky-/,= J,, and so find As each triple moment is symmetric under interchange of k and;, it follows that the above integrand is antisymmetric under interchange of k and j , and therefore vanishes when integrated over all space with respect to these variables. From which we conclude that Jox T(k, I ) dk = 0. (131) It will be seen later that, not only is this an important result, but also the form of the proof will be found to be very helpful later on.

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A retrospective and prospective survey of Monte Carlo Method by Halton

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