Turbulence intermittency

In a turbulent fluid, nearby points often move at similar speeds, but small regions can contain a very sharp change in motion. These rare intense changes are called intermittency. Can the fluid equations predict how their statistics change as the distance between the points changes? For example, compare two points on a horizontal line: their velocity can differ along that line and also across it. Both changes matter, and their extreme events need a joint explanation.[1] [2]

Turbulent motion contains rare regions where velocity changes sharply across a short distance. Can the incompressible fluid equations predict the coupled statistics of those changes along and across their separation? The scope is three-dimensional turbulence with homogeneous statistics—no preferred position—and isotropy—no preferred direction—at high Reynolds number, where a wide range of scales separates large-scale stirring from viscous smoothing.

Take two points separated along x by a distance r. Their x-velocity difference is longitudinal; their y-velocity difference is one transverse component. Averaging powers of these increments gives structure functions. In one chosen velocity unit, an illustrative set of absolute increments 1,1,1,1 has mean 1 and mean fourth power 1. The synthetic set 0,0,0,4 has the same mean, but mean fourth power 64: higher powers expose the burst. These numbers explain the statistic, not a measured turbulent dataset.

For positive even powers p and q, a normalized mixed moment is $S_{p,q}(r)=\langle(\delta u/U)^p(\delta v/U)^q\rangle$, with U a characteristic velocity. In the intermediate range between forcing and viscous scales, its power-law description is $S_{p,q}(r)\sim(r/L)^{\zeta_{p,q}}$, where L is a large-scale length. Doubling r within that range multiplies the scaling expression by $2^{\zeta_{p,q}}$. The unknown exponents describe the joint statistics, not just one typical speed.[1] [2]

The question concerns an ideal three-dimensional fluid with no compression and no preferred place or direction in its statistics, at strong turbulence. A recent framework relates the along-and-across statistics to even smaller-scale changes. It still needs their scaling numbers as input rather than predicting all of them from the fluid motion.[1] [2]

The September 2026 joint multifractal framework relates increment scaling to velocity-gradient statistics and tests the relations against direct numerical simulations. It respects exact third-moment constraints but still requires the full mixed-exponent set as input. A closed prediction must account for coupled stretching and rotation and the pressure field’s nonlocal influence. Transverse increments are not simply pure rotation. Finite resolution and sampling of rare events also limit tests of high-order statistics; numerical agreement is not a derivation of every exponent from the dynamics.[1] [2]

Sharp local changes matter for mixing and for where fluid motion is converted into heat. Two flows with similar average motion can therefore have very different intense patches. Their statistics supply information that an average speed leaves out.[1]

Viscous energy loss depends on local deformation, and mixing can be dominated by concentrated small-scale motion. Explaining the joint extremes would connect these consequences to the mechanisms that amplify or attenuate velocity gradients, rather than only fitting their probability tails. This supplies a physical basis for models of unresolved scales; transferring it to flows with walls, stratification or preferred directions requires additional work.[1] [2]