## Abstract

The starting-point for this paper lies in some results obtained by Proudman & Reid (1954) for isotropic turbulence with zero fourth-order cumulants. They showed in that case that the quantity $\overline{u^{2}}\int_{0}^{\infty}$r$^{4}$f(r) dr is not a dynamical invariant and that the first time derivative of the triple correlation k(r) is of order r$^{-4}$ when r is large. The customary assumption that all velocity cumulants in homogeneous turbulence are exponentially small for large separations, and the consequent results, about the large-scale structure of the motion and about the final period of decay of the turbulence, are thus suspect, and we have redeveloped the whole subject ab initio. The fallacy in the old assumption of exponentially small cumulants can be ascribed to the action of pressure forces, which are local in their effect but which have values determined instantaneously by the whole velocity field. If at some initial instant a finite region only of an infinite fluid is in motion, at subsequent instants pressure forces generate a surrounding irrotational velocity distribution that falls off as some integral power of the distance from the central region. Likewise, the action of pressure forces in homogeneous turbulence is to ensure the development of algebraic asymptotic forms of velocity cumulants, the analogue of the finite region of initial motion being a volume of the fluid over which the vorticity is effectively correlated. However, an essential difference between the two cases is that in homogeneous turbulence pressure forces also build up long-range statistical connexions in the vorticity distribution. Having recognized why the old assumption is wrong, it is necessary to consider what kinds of asymptotic forms of velocity cumulants (for large separation) are dynamically persistent, and to consider in particular what asymptotic forms are likely to occur when homogeneous turbulence is generated in the usual way by setting a regular array of rods across a uniform stream. We have been able to find only one kind of large-scale structure that is unchanged by dynamical action, and this is also the kind of large-scale structure that develops from a plausibly idealized initial condition representing the effect of the grid on the stream. This initial condition, according to the hypothesis on which the positive results of this paper are based, is that there is a virtual origin in time at which all integral moments of cumulants of the velocity field converge. The important consequence of this hypothesis is that the effect of pressure forces is subsequently to develop asymptotic forms that are integral power-laws. It is shown, from a consideration of all the time derivatives at the initial instant, that the velocity covariance $\overline{u_{i}u_{j}^{\prime}}$ in general becomes of order r$^{-5}$ when the separation r is large, the leading term having the property that it makes no contribution to the vorticity covariance $\overline{\omega _{i}\omega _{j}^{\prime}}$, which becomes of order r$^{-8}$. This semi-irrotational property of the asymptotic form, which arises from the fact that pressure forces act only indirectly on the vorticity, allows the asymptotic form of $\overline{u_{i}u_{j}^{\prime}}$ to be determined explicitly. By methods that are new in turbulence theory and that involve a good deal of tensor manipulation, it is found that $\overline{u_{i}u_{j}^{\prime}}$ = $\frac{1}{4}$C$_{pqmn}\left(\delta _{ip}\nabla ^{2}-\frac{\partial ^{2}}{\partial r_{i}\partial r_{p}}\right)\left(\delta _{jq}\nabla ^{2}-\frac{\partial ^{2}}{\partial r_{j}\partial r_{q}}\right)\frac{\partial ^{2}r}{\partial r_{m}\partial r_{n}}$ + O(r$^{-6}$), when r is large, where the coefficient C$_{pqmn}$ is related to the fourth integral moment of $\overline{\omega _{i}\omega _{j}^{\prime}}$ in a known way. There is a corresponding expression for the leading term in the spectrum tensor at small wave-numbers, which is now not analytic. The spectrum function giving the distribution of energy with respect to wave-number magnitude k is in general of the form E(k) = Ck$^{4}$ + O(k$^{5}$ ln k), when k is small. Corresponding expressions are found for the asymptotic forms of the various terms occurring in the dynamical equation giving the rate of change of $\overline{u_{i}u_{j}^{\prime}}$. Both the inertia and pressure terms in this equation are found to be of order r$^{-5}$, and as a consequence the coefficient C$_{pqmn}$ (and likewise C) is not a dynamical invariant. It is shown that the integral $\int \overline{u_{i}u_{j}^{\prime}}$r$_{m}$r$_{n}$ dr (which exists, despite the apparent logarithmic divergence at large r) is uniquely related to C$_{pqmn}$, and it too varies during the decay, contrary to past belief. The final period of decay is examined afresh, and it is found that the energy then varies as (t-t$_{0}$)$^{-\frac{5}{2}}$, which is also the result found experimentally; the power (-$\frac{5}{2}$) arises from the fact that the spectrum tensor is of order k$^{2}$ when the wave-number k is small, and is unaffected by the non-analytic character of that leading term. The covariance $\overline{u_{i}u_{j}^{\prime}}$ does not have a simple form in the final period; it is determined by the parameter $\int \overline{u_{i}u_{j}^{\prime}}$r$_{m}$r$_{n}$ dr alone, and this parameter depends on the previous history of the decay in a complicated way. It is rather puzzling that measurements indicate that the longitudinal correlation coefficient has a simple Gaussian form in the final period of decay, as would be the case for an analytic spectrum. We suggest this observation may be true only for turbulence of very low initial Reynolds number, for which the non-analytic part of the spectrum tensor has little time to develop. Finally, the results are specialized to correspond to turbulence which is completely isotropic. For reasons related to the symmetry, $\overline{u_{i}u_{j}^{\prime}}$ is now no larger than O(r$^{-6}$) when r is large (we have been unable to determine the exact order), and the leading term in the spectrum tensor, of order k$^{2}$, is analytic. As suggested by Proudman & Reid's work, the triple correlation k(r) is of order r$^{-4}$ when r is large and $\frac{\text{d}}{\text{d}t}\left\{\overline{u^{2}}\int_{0}^{\infty}r^{4}f(r)\,\text{d}r\right\}$ = ($\overline{u^{2}}$)$^{\frac{3}{2}}\underset r\rightarrow \infty \to{\lim}$r$^{4}$k(r).

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