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 Incoming links: Dissipation of Energy in the Locally Isotropic Turbulence by Andrei N. Kolmogorov

u

, define the longitudinal increment

δ u_{L} (x, r) = [u (x + r r) - u (x)] \cdot r, S_{p} (r) = ⟨ ∣ δ u_{L} (x, r) ∣^{p} ⟩ .

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.^[ref]

An independent smooth-field example is

u (x) = U sin (k x)

in one dimension, averaged over a whole period. Direct integration gives

S_{2} (r) = U^{2} [1 - cos (k r)] ≃ \frac{U ^{2} k ^{2} r ^{2}}{2} (k r ≪ 1) .

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,

⟨(δ u_{L})^{3} ⟩ = - \frac{4}{5} ε r,

with

ε

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]