Isotropic turbulence

Isotropic turbulence has statistics invariant under rotations as well as translations. With no mean velocity, its two-point velocity correlation tensor depends on separation length and on longitudinal/transverse directions, and its one-component velocity variances are equal.

Small-scale joint velocity increment statistics become approximately invariant under rotations of the observation configuration, even when the energy-containing flow is anisotropic. This is a scale-restricted hypothesis; it does not make the entire velocity field isotropic or remove internal intermittency.

The Loitsyansky integral is the second spatial moment of the trace velocity correlation tensor, with the displayed sign convention. If correlations decay sufficiently rapidly, it is the coefficient of $k^4/(24{\pi }^2)$ in the turbulent energy spectrum. In terms of the normalized longitudinal velocity correlation it is $8\pi u^2{\int }_0^{\infty }{r}^{4}f(r)dr$ when the boundary terms vanish; conventions omitting $8\pi$ also occur. Its evolution is $\dot{I}=8\pi [r^4{u}^{3}K{]}_{\infty }-12\nu L$, where $L$ is the Saffman integral. Thus it is invariant when $L=0$ and the longitudinal triple correlation is $o(r^{-4})$, not under arbitrary turbulent conditions.

For a confined incompressible flow with no velocity flux across the boundary, an integrated divergence identity gives $⟨|\int \mathbf{x}\times \mathbf{u}dV{|}^2⟩=-\iint r^2⟨\mathbf{u}\cdot {\mathbf{u}}^{\prime }⟩dVd{V}^{\prime }$. In a large approximately homogeneous region with short correlations, the latter is approximately $VI$. Conservation of angular momentum in an isolated torque-free cloud then motivates conservation of the Loitsyansky integral. A fixed subvolume of infinite turbulence is not an isolated cloud: advective transport, pressure torques, wall stresses and long-range correlations can invalidate the argument.

Assume a nonzero invariant Loitsyansky integral, self-preserving large-scale correlations, and high-Reynolds number freely decaying isotropic turbulence. Then $I\propto u^2{\ell }^5$ and $-d(u^2)/dt\propto u^3/\ell$ imply the displayed decay and growth laws. The assumptions on correlation shape and the dissipation coefficient are needed in addition to conservation of the integral.

The Saffman integral is $L=\int ⟨\mathbf{u}(\mathbf{x})\cdot \mathbf{u}(\mathbf{x}+\mathbf{r})⟩d^{3}r$. With integrable trace correlation it equals the large-volume limit of $⟨|{\int }_V\mathbf{u}dV{|}^2⟩/|V|$. It is nonnegative and measures momentum-fluctuation density. Under appropriate unforced far-field flux conditions it is invariant.

With nonzero invariant $L$, a self-preserving integral scale of turbulence gives $L\propto \mathcal{E}{\ell }^3$. Combining this with freely decaying high-Reynolds-number viscous dissipation $ϵ\propto {\mathcal{E}}^{3/2}/\ell$ gives $\mathcal{E}\propto (t-t_{\ast }{)}^{-6/5}$ and $\ell \propto (t-t_{\ast }{)}^{2/5}$. The closure assumptions, not the invariant alone, determine this exponent.

A finite nonzero Saffman integral gives $E(k)=L{k}^2/(4{\pi }^2)+o(k^2)$ in the standard isotropic spectrum normalization. This follows by expanding the radial spectral transform at small wavenumber. A further analytic term requires additional correlation moments.

The velocity correlation tensor is $Q_{ij}(\mathbf{r})=⟨u_i(\mathbf{x})u_j(\mathbf{x}+\mathbf{r})⟩$. For isotropic turbulence it splits into longitudinal and transverse parts. Incompressibility constrains them, while the Fourier transform gives the turbulent energy spectrum. Its trace is the scalar velocity-dot-product correlation.

The Kármán-Howarth equation evolves two-point correlations in homogeneous isotropic turbulence. In trace form it is ${\partial }_{t}C=r^{-2}{\partial }_r[r^{-1}{\partial }_r(r^4{u}^{3}K)]+2\nu r^{-2}{\partial }_r(r^{2}C\prime )$. Integrating with radial volume weight reduces conservation of the Saffman integral to vanishing nonlinear and viscous boundary fluxes.

For unforced homogeneous turbulence with isotropic turbulence statistics, put $F=u^{2}f=u^2-S_2/2$ and $S_3=6u^{3}K$. The longitudinal Kármán-Howarth equation is $F_t=r^{-4}{\partial }_r[r^4(S_3/6-\nu S_2^{\prime })]$. Since ${\partial }_t{u}^2=-2ϵ/3$, regularity at zero givesNeglecting the last term in a locally equilibrated small-scale range gives the displayed equation. The Kolmogorov four-fifths law follows where the viscous term is negligible as well.

The signed third-order longitudinal structure function has this leading inertial range form under homogeneity, isotropy and the usual local-equilibrium scale separation. Its coefficient follows from the Kármán-Howarth equation and the mean kinetic energy budget, without assuming the Kolmogorov two-thirds law. Its negative sign encodes a forward energy cascade; it does not establish every higher-order similarity exponent.

For a separation parallel to a unit vector $\mathbf{n}$, the longitudinal velocity correlation is $Q_{LL}=⟨u_L(\mathbf{x})u_L(\mathbf{x}+r\mathbf{n})⟩=u^{2}f(r)$. In isotropic incompressible turbulence the transverse normalized correlation is $g=f+rf\prime /2$ and the trace is $C=u^2(3f+rf\prime )$.

The order-$p$ longitudinal velocity structure function is $S_p(r)=⟨[u_L(\mathbf{x}+r\mathbf{n})-u_L(\mathbf{x}){]}^p⟩$; absolute increments are used for noninteger orders. At second order $S_2=2u^2[1-f(r)]$. It suppresses a smooth common velocity but does not exactly remove all contributions from motions larger than the separation.

A velocity increment is the difference of a velocity field between two positions. Its longitudinal component is the projection onto their separation direction. Moments of increments define longitudinal structure functions; the Kolmogorov 1941 theory estimates a typical inertial-range magnitude as $(ϵr{)}^{1/3}$.

A smooth velocity varying on a scale larger than $r$ contributes $\Delta U_L\simeq r{\partial }_L{U}_L$ to a velocity increment. Higher powers produce mixed moments with smaller-scale increments. longitudinal structure functions therefore attenuate, but do not exactly filter out, large-scale strain and intensity modulation.

In isotropic incompressible three-dimensional turbulence, $3S_2(r)/4=\int E(k)H(kr)dk$, with $H(x)=1+3\cos x/x^2-3\sin x/x^3$. It is smooth at zero, where $H=x^2/10+O(x^4)$, and tends to one at large argument. A sharp wavenumber cutoff omits the finite large-scale-gradient contribution.

The turbulent energy spectrum $E(k)$ distributes kinetic energy over wavenumber shells, normalized by ${\int }_0^{\infty }E(k)dk=⟨|\mathbf{u}{|}^2⟩/2$. For isotropic incompressible flow its tensor form is ${\Phi }_{ij}=E(k)({\delta }_{ij}-k_i{k}_j/k^2)/(4\pi k^2)$. Its low-wavenumber behavior records large-eddy momentum statistics.

For isotropic incompressible flow, directional averaging gives $⟨({\partial }_x{u}_x{)}^3⟩=(8/105)⟨\mathrm{tr}{S}^3⟩=-(2/35)⟨{\omega }_i{\omega }_j{S}_{ij}⟩$. Positive mean vortex stretching therefore implies negative longitudinal derivative skewness. Statistical homogeneity alone does not determine the sign in a fixed direction.