 Structure function (turbulence)

A structure function measures moments of a field's differences across a separation. For a velocity field $u$, define the longitudinal incrementThe 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.^[ref]

An independent smooth-field example is $u(x)=U\sin (kx)$ in one dimension, averaged over a whole period. Direct integration givesThe 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,with $\epsilon$ 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.^[ref]