Longitudinal velocity correlation ($Q_{LL}=u^{2}f(r)$)

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.