= Structure function
{disambiguate=turbulence}
A <structure function (turbulence)> measures moments of a field's differences across a separation. For a velocity field $u$, define the longitudinal increment
$$
\delta u_L(x,r)=\bigl[u(x+r\widehat r)-u(x)\bigr]\cdot\widehat r,
\qquad S_p(r)=\langle|\delta u_L(x,r)|^p\rangle.
$$
The average may be over an ensemble or over a suitable homogeneous sample. Stating the component, separation direction and absolute-value convention matters: signed odd moments contain information that absolute moments erase.<Dissipation of Energy in the Locally Isotropic Turbulence by Andrei N. Kolmogorov>[Opening definitions and longitudinal second/third moments, translated pages 15–16]{ref}
An independent smooth-field example is $u(x)=U\sin(kx)$ in one dimension, averaged over a whole period. Direct integration gives
$$
S_2(r)=U^2[1-\cos(kr)]\simeq\frac{U^2k^2r^2}{2}\quad(kr\ll1).
$$
The quadratic small-separation behavior records differentiability. A constant offset added to the velocity disappears from every increment.
The signed third moment has a special physical role. In the inertial range of high-Reynolds-number, locally isotropic, incompressible turbulence under the usual stationary energy-balance assumptions,
$$
\langle(\delta u_L)^3\rangle=-\frac45\varepsilon r,
$$
with $\varepsilon$ the mean dissipation rate per unit mass. This relation concerns a signed cubic increment, not $S_3$ as defined with an absolute value above. Its sign records a directed transfer that a second-order spectrum cannot determine. At finite viscosity the energy-balance relation has an additional derivative term; anisotropic, transient or compressible data need their own justification before this inertial-range formula is applied.<Dissipation of Energy in the Locally Isotropic Turbulence by Andrei N. Kolmogorov>[Equations relating third moments, viscous derivative term and mean dissipation, translated pages 15–16]{ref}
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