Absolute Equilibrium Ensembles

Abstract

In this chapter, we will consider a turbulence within a box (that is to say which can be expanded into an infinite serie of discrete wave vectors ${\vec k_n}$ with velocity amplitudes $\underline {\hat u} \left( {{{\vec k}_n},t} \right)$,as introduced earlier). Let us first consider the Navier Stokes equations in Fourier space, relating this infinite set of modes. The equation is truncated by retaining the modes lower than a cutoff wave number k _max , and properly normalized:

$$(\frac{\partial }{{\partial t}} + v{k^2}){\hat u_i}(\overrightarrow k ,t)=-\frac{i}{2}{P_{ijm}}(\overrightarrow k )\sum {} {\hat u_j}(\overrightarrow p ,t){\hat u_m}(\overrightarrow q ,t)$$

((X - 1 - 1))

where the sum $\sum {}$only keeps the modes such that $\overrightarrow k$= $\overrightarrow p$+ $\overrightarrow p$and whose modulus is smaller than k _max . If k_max is superior or equal to the Kolmogorov wave number (in three dimensions), it is expected that the truncated equations will correctly represent the turbulence, and this is precisely what is done in the so-called direct-numerical simulations of turbulence. If this condition is not fulfilled, and if we start initially with an energy spectrum sharply peaked at an initial energetic wave number k₀(in a freely decaying turbulence), eq. (X-1-1) will properly describe the evolution of turbulence in the early stage, when the cascade has not yet reached k_max. But as soon as the Kolmogorov energy cascade forms (that is, from Chapter VII, at a time of about 5 initial large-eddy turnover times), energy will tend to go beyond k_max in the dissipative scales, which is not permitted by the truncated equation (X-1-1).