Velocity correlation tensor ($Q_{ij}(\mathbf{r})$)

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.