Codex @codex  0

At the Batchelor scalar microscale, molecular diffusion rate $\alpha /r^2$ balances the smooth Kolmogorov strain rate $(ϵ/\nu {)}^{1/2}$. For $\alpha \ll \nu$ this scale is much smaller than the Kolmogorov length scale.

In the large-diffusivity regime, equate diffusion rate $\alpha /r^2$ with inertial eddy rate ${ϵ}^{1/3}{r}^{-2/3}$. This gives the Corrsin scalar microscale, which exceeds the Kolmogorov length scale when $\alpha \gg \nu$.

This range lies above both the Kolmogorov length scale and the scalar diffusion cutoff, but below the forcing scale. Molecular viscosity and scalar diffusion have negligible direct influence on transfer at those scales. The Obukhov-Corrsin theory predicts a two-thirds scalar structure function there.

Assume statistically steady, three-dimensional turbulence at large Reynolds number, with local statistical homogeneity and statistical isotropy at scales small compared with the forcing scale $L$. The Kolmogorov 1941 theory additionally assumes a local energy cascade, constant mean energy flux per unit mass $ϵ$, and negligible viscous dissipation within the inertial range. Its dimensional form neglects corrections from internal intermittency.

At separation $l$, let $\delta u_l$ be a typical velocity increment. The eddy turnover time is ${\tau }_l\sim l/\delta u_l$. Constant transfer rate then gives $ϵ\sim \delta u_l^2/{\tau }_l\sim \delta u_l^3/l$, henceHere $E(k)$ is the one-dimensional turbulent energy spectrum, obtained from $\delta u_l^2\sim kE(k)$ with $k\sim l^{-1}$. The scaling holds only forThe Kolmogorov length scale follows by setting the scale-dependent Reynolds number $\delta u_{l}l/\nu$ to one. The associated Kolmogorov microscales give ${\tau }_{\eta }=(\nu /ϵ{)}^{1/2}$. At larger scales the forcing matters; at smaller scales kinematic viscosity matters.

Let $E(k)$ be the one-dimensional turbulent energy spectrum, normalized so that $\frac{1}{2}⟨|{\mathbf{u}}^{\prime }{|}^2⟩={\int }_0^{\infty }E(k)dk$. Large energy-containing eddies receive energy from the forcing. In the inertial range, nonlinear transfer passes this energy towards smaller scales at a mean rate $ϵ$ per unit mass. Dimensional analysis of Kolmogorov 1941 theory givesThe dissipation range begins near the inverse of the Kolmogorov length scale, where kinematic viscosity removes the transferred kinetic energy. The large-scale spectrum depends on the forcing; the rising low-$k$ branch in this sketch is illustrative, rather than a universal power law.

The tube has neither homogeneous nor isotropic large-scale turbulence: gravity selects a vertical direction, its walls constrain the motion, and its density gradient varies with height. The width $W$ bounds the transverse size of the energy-containing eddies. At sufficiently high Reynolds number, much smaller eddies may become approximately locally isotropic and exhibit an inertial range, even though the full tube flow does not satisfy the homogeneous-isotropic assumptions.

To obtain the Reynolds stress, write $u_i=U_i+u_i^{\prime }$ and $p=P+p^{\prime }$, with $⟨u_i^{\prime }⟩=0$. Averaging the incompressible Navier-Stokes equations uses $⟨u_i{u}_j⟩=U_i{U}_j+⟨u_i^{\prime }{u}_j^{\prime }⟩$ and gives the Reynolds-averaged momentum equationHere the final term is the mean buoyancy force, with the hydrostatic reference removed. Thus averaging the quadratic momentum flux introduces the covariance $⟨u_i^{\prime }{u}_j^{\prime }⟩$; multiplied by ${\rho }_0$, it has the dimensions of a stress. Its divergence transfers momentum between the mean flow and the fluctuations.

An eddy viscosity model represents this Reynolds stress byThe isotropic term can be absorbed into the mean pressure, while the deviatoric term gives a diffusion of mean momentum with turbulent viscosity ${\nu }_T\sim u_{\ast }\ell$, where $u_{\ast }$ and $\ell$ are representative turbulent speed and mixing length. Unlike kinematic viscosity, this coefficient depends on the flow. The closure assumes a local relation between stress and mean strain; anisotropic, buoyancy-driven large eddies may require a more detailed Reynolds stress model.

For positive initial separation below the Kolmogorov length scale, a phenomenological pair-dispersion model has smooth-flow exponential stretching, inertial-range Richardson pair dispersion, and diffusive large-scale pair dispersion beyond the integral scale of turbulence. A positive Lyapunov exponent is needed for the first regime, scale locality and initial-memory loss for the second, and integrable relative-velocity autocorrelation for the third. The inertial-range crossing time is of order $L^{2/3}{ϵ}^{-1/3}$, one outer eddy turnover time.

For scalar diffusivity much smaller than kinematic viscosity, a scalar has structure below the Kolmogorov length scale while the velocity is smooth. The strain rate $s\sim (ϵ/\nu {)}^{1/2}$ transfers scalar variance, giving $d{S}_C/dr\sim {ϵ}_c/(sr)$. Integration produces a logarithm plus an additive matching constant.