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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.

Turbulence is fluid flow with irregular, interacting velocity fluctuations over a range of lengths and times. An unsteady laminar flow also has a time-dependent velocity field; the important distinction is the organization and interaction of the fluctuations. In many flows a large Reynolds number allows nonlinear advection to distort the motion before kinematic viscosity smooths it, but transition depends on geometry and disturbances as well as the chosen Reynolds number.

The Reynolds pipe-and-dye experiment makes the distinction visible. A narrow band of colored water can remain coherent in a glass pipe, then break into irregular structures and spread across the pipe as the discharge increases. The inlet and pre-existing disturbances matter, so the observation does not establish a universal transition value.^[ref]

In a three-dimensional forward energy cascade, nonlinear interactions transfer kinetic energy towards smaller scales until viscous dissipation converts it into heat. This transfer helps explain why a turbulent flow mixes a passive scalar efficiently, although final molecular homogenization still requires molecular diffusion. After Reynolds averaging, correlations of fluctuating velocities produce the Reynolds stress, an additional momentum flux. Predicting that flux from mean quantities is the turbulence closure problem; a model for it is an extra physical assumption.

Turbulence intensity expresses the size of a measured velocity fluctuation relative to a specified mean speed. For a streamwise velocity record $u(t)$, defineHere the symbol in the square-root definition is $u_{\mathrm{rms}}$, the root mean square fluctuation, and angle brackets denote the averaging operation. The dimensionless intensity is often reported as $100I$ percent. The averaging interval and measured velocity component are part of the definition.

For four equally weighted observations $9,11,9,11\mathrm{m}{\mathrm{s}}^{-1}$, the mean is $10\mathrm{m}{\mathrm{s}}^{-1}$ and each fluctuation is $\pm 1\mathrm{m}{\mathrm{s}}^{-1}$. Their record root mean square is therefore $1\mathrm{m}{\mathrm{s}}^{-1}$, giving $I=0.10$, or 10 percent. This calculation describes that record; it does not estimate every fluctuation timescale from four observations.

A primary wind-tunnel investigation used hot-wire anemometry, recording mean and fluctuating signals to characterize the wind-tunnel turbulence intensity. Its bandwidth comparison demonstrates that integrating only part of the measured fluctuation spectrum changes the recovered root mean square.^[ref] An instrument's temporal response, filtering and record duration therefore affect the reported intensity. Slow changes of the mean can also contribute to a measured fluctuation unless the averaging convention separates them.

A three-component convention instead useswhere $k$ is the turbulent kinetic energy per unit mass. This agrees with the streamwise definition when the three fluctuation variances are equal, as in isotropic turbulence. A nonzero root mean square alone does not establish turbulence: a periodic laminar velocity oscillation also has a nonzero value. The ratio also becomes ill-conditioned near zero mean speed.

Turbulence can raise dust random speeds, increase layer thickness and mix density enhancements by eddy diffusion, delaying secular gravitational instability. On sufficiently long wavelengths, weak-drag growth is proportional to $|k|$, whereas simple diffusive damping is proportional to $k^2$. Thus diffusion alone need not remove all long-wave growth in an infinite idealization, although finite size and lifetime can make it ineffective. Coherent turbulent concentration can also increase local surface density, so the net effect requires a stochastic transport model.

Ugo Piomelli researches computational models of turbulence, including large eddy simulation. Queen's University's institutional account describes work aimed at predicting complicated flows with tractable computational cost and identifies his former Canada Research Chair in turbulence simulation and modeling.^[ref]

Piomelli coauthored the dynamic subgrid-scale model work with Massimo Germano, Parviz Moin and William H. Cabot.^[ref] In a retrospective with Moin, he describes incorporating the proposed model into a channel-flow code. An initially delayed transition proved to reflect an incorrectly entered perturbation amplitude; correcting it changed the comparison with reference data.^[ref]

The episode makes a methodological point: a discrepancy can test the closure, the numerical implementation or the specified physical problem. If one simulation starts with a disturbance one hundred times smaller than intended, equal times no longer represent equal stages of its growth. Checking initial conditions therefore precedes attributing every difference to the subgrid-scale model. The numerical factor in this illustration is hypothetical; the retrospective does not quantify the mistyped factor.

Victor S. L'vov is a researcher in theoretical physics whose work includes turbulence, wave turbulence and nonlinear dynamical systems. His Weizmann Institute of Science research homepage identifies wave and hydrodynamic turbulence as central interests.^[ref]

In 1998 he and collaborators introduced an improved shell model of turbulence designed to reduce spurious correlations and oscillations between model scales. A shell model of turbulence replaces a spatially resolved velocity field by amplitudes $u_n$, which are complex numbers attached to wavenumbers $k_n=k_0{\lambda }^n$, $\lambda >1$. Nonlinear interactions transfer energy between shells, while forcing and viscosity supply and dissipate it. With $\lambda =2$, ten shell steps cover a factor of $1024$ in wavenumber: a small collection of variables can investigate an energy cascade spanning widely separated scales.^[ref]

This research makes a useful distinction between reproducing scale statistics and reproducing the full spatial geometry of a turbulent flow. The paper tests scaling and shell correlations in a deliberately reduced dynamical system; that success does not make it an exact replacement for the Navier-Stokes equations in arbitrary apparatus.

William H. Cabot coauthored the dynamic subgrid-scale model developed with Massimo Germano, Ugo Piomelli and Parviz Moin. Their 1990 report compared modeled unresolved stresses with data from direct numerical simulation and tested large eddy simulation of transitional and turbulent channel flow.^[ref] A participant retrospective identifies Cabot as a Stanford research associate during the collaboration.^[ref]

The numerical comparison illustrates an important issue in turbulence research: matching the objects being compared. A coarse simulation describes resolved motion, while a finely resolved reference contains additional fluctuations. The report filtered the reference data before some comparisons, so the comparison did not automatically count intentionally omitted small scales as errors in a resolved field.^[ref]

For an original arithmetic illustration, fine-grid velocities $0$ and $2$ have mean $1$ and mean square $2$. The square of their mean is only $1$. Replacing both samples by the mean preserves that mean but loses a fluctuation contribution of $1$. This elementary difference is why an averaged velocity and an averaged momentum flux contain different information, and why subgrid-scale stress requires explicit treatment.

At fixed outer velocity and integral scale of turbulence, the mean viscous dissipation per unit mass approaches a positive finite value as the Reynolds number becomes large. The dimensionless coefficient $C_{ϵ}=ϵ\ell /u^3$ is then of order one. This is a phenomenological property of established three-dimensional turbulence, not a consequence of taking kinematic viscosity to zero in an arbitrary smooth flow.