Turbulence

Turbulence is irregular fluid motion with interacting velocity fluctuations over a range of scales. It can mix a particle suspension and promote fluid entrainment into a turbulent plume.

Reynolds averaging separates a resolved mean from zero-mean fluctuations. The idealized averaging operation is linear and idempotent, commutes with the derivatives under consideration and treats resolved mean factors as constant inside its averaging scale. Products then satisfy $\overline{fg}=\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}$. Actual finite spatial filters need scale separation or corrections when these rules are only approximate.

A fixed observation point alternates between turbulent and non-turbulent fluid because a fluctuating interface passes it. This is distinct from internal intermittency, which concerns variations within fluid that remains turbulent.

Wave turbulence describes the statistical transfer of energy between interacting waves. The wave period, nonlinear interaction time, phase correlations and resonances determine the transfer rate. A weak wave cascade treats the change in one interaction as small; critical balance instead makes the nonlinear interaction rate comparable to the linear wave frequency.

If a wave packet experiences a fractional nonlinear change $\chi ={\tau }_w/{\tau }_{\mathrm{nl}}\ll 1$ per interaction, and successive interaction phases are decorrelated, $N$ changes add as a random walk of size $\sqrt{N}\chi$. Order-one transfer needs $N\sim {\chi }^{-2}$ interactions, giving the displayed cascade time. Phase coherence or resonant constraints can alter this estimate; weakness alone does not justify random addition.

Magnetohydrodynamic turbulence is irregular nonlinear motion of an electrically conducting fluid coupled to a magnetic field. Alfvén waves and magnetosonic waves give propagation times in addition to the nonlinear mixing time. A strong guide field distinguishes directions parallel and perpendicular to it, so the resulting cascades need not be isotropic.

In a local isotropic hydrodynamic inertial-range model with only $ϵ$ and $k$, dimensional analysis fixes the Kolmogorov 1941 theory spectrum. A mean magnetic field supplies the independent Alfvén speed, permitting the arbitrary dimensionless function in the displayed formula even before anisotropy adds parallel scales. Cascade-time assumptions such as weak decorrelated interactions or critical balance are therefore necessary to select an MHD spectrum.

An electron-magnetohydrodynamic cascade transfers magnetic fluctuations through the nonlinear electron magnetohydrodynamics induction equation. Writing $b_l=\delta B_l/B_0$, its perpendicular nonlinear interaction time is ${\tau }_{\mathrm{nl}}\sim l_{\perp }^2/(v_A{d}_i{b}_l)$. The anisotropic whistler wave time is ${\tau }_w\sim l_{\parallel }{l}_{\perp }/(v_A{d}_i)$. Scaling predictions depend on locality, phase decorrelation and the assumed balance of these times.

Critical balance sets ${\tau }_{\mathrm{nl}}\sim {\tau }_w$, hence $l_{\parallel }\sim l_{\perp }/b_l$. A local strong cascade with constant energy flux has $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$, giving the displayed exponents and perpendicular magnetic energy spectrum $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The anisotropy becomes stronger towards small perpendicular scales.

For $b_l{l}_{\parallel }/l_{\perp }\ll 1$, the weak wave cascade estimate gives ${\tau }_{\mathrm{cas}}\sim l_{\perp }^3/(v_A{d}_i{b}_l^2{l}_{\parallel })$. Constant energy flux $ϵ\sim v_A^2{b}_l^2/{\tau }_{\mathrm{cas}}$ yields the displayed amplitude. An additionally isotropic local cascade has $b_l\propto l^{1/2}$ and shell-integrated magnetic energy spectrum $E_B(k)\propto k^{-2}$. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.

Critical balance is a scale-dependent ordering in which the wave-propagation time and nonlinear interaction time are comparable. For strong Alfvénic turbulence, $k_{\parallel }{v}_A\sim k_{\perp }\delta u_{\perp }$. For electron magnetohydrodynamics, the linear rate is $v_A{d}_i{k}_{\parallel }{k}_{\perp }$ and the nonlinear rate is $v_A{d}_i{k}_{\perp }^2\delta B/B_0$, so critical balance instead gives $\delta B/B_0\sim k_{\parallel }/k_{\perp }$. This assumption is an estimate of decorrelation dynamics, not an exact equality at every point.

Balance the inertial wave rate against horizontal nonlinear advection. The anisotropy $ϵ=|k_{\parallel }|/k_{\perp }$ is then of the same order as the Rossby number $k_{\perp }U/(2\Omega )$. Horizontal geostrophic flow gives kinematic pressure amplitude $\delta p\sim 2\Omega U/k_{\perp }\sim U^2/ϵ$. The first ageostrophic flow is $O(ϵU)$ and supplies the small horizontal divergence balancing $-{\partial }_z{u}_{\parallel }$. Rapid inertial oscillations are excluded by this slow-time ordering.

Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables ${\mathbf{z}}^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

For balanced anisotropic Alfvénic turbulence, suppose ${\tau }_A=l_{\parallel }/v_A\ll {\tau }_{\mathrm{nl}}=l_{\perp }/\delta z$ and resonant interactions preserve the parallel scale during the perpendicular cascade. Random encounter accumulation gives ${\tau }_{\mathrm{cas}}=v_A{l}_{\perp }^2/(\delta z^2{l}_{\parallel })$. Constant energy flux then gives $\delta z\propto l_{\perp }^{1/2}$ and the displayed perpendicular turbulent energy spectrum. Unlike isotropic phenomenology, the fixed parallel scale is independent of the decreasing perpendicular scale.

In a weak Alfvénic cascade at fixed $l_{\parallel }$, $\chi ={\tau }_A/{\tau }_{\mathrm{nl}}\propto l_{\perp }^{-1/2}$ grows as the perpendicular scale shrinks. If its outer value is ${\chi }_0\ll 1$ at $L_{\perp }$, it reaches order one at the displayed scale. Beyond it the weak-interaction ordering fails and a critical balance model may become appropriate.

The Iroshnikov-Kraichnan spectrum phenomenology assumes balanced, isotropic local Alfvénic turbulence with $\delta z_l\ll v_A$, decorrelated counterpropagating encounters, ${\tau }_A=l/v_A$ and ${\tau }_{\mathrm{nl}}=l/\delta z_l$. A weak wave cascade takes ${\tau }_{\mathrm{cas}}\sim {\tau }_{\mathrm{nl}}^2/{\tau }_A=v_{A}l/\delta z_l^2$. Constant energy flux gives $\delta z_l^4\sim ϵv_{A}l$ and the displayed shell-integrated turbulent energy spectrum. Isotropy is an additional assumption, not a consequence of a strong guide field.

Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass $\mathcal{E}$. With ${\tau }_{\mathrm{nl}}\sim {\ell }_{\perp }/\delta u_{\ell }$ and $\mathcal{E}\sim \delta u_{\ell }^2/{\tau }_{\mathrm{nl}}$, one obtains $\delta u_{\ell }\sim (\mathcal{E}{\ell }_{\perp }{)}^{1/3}$. Defining the one-dimensional perpendicular turbulent energy spectrum by $\delta u_{\ell }^2\sim k_{\perp }E(k_{\perp })$ gives $E(k_{\perp })\sim {\mathcal{E}}^{2/3}{k}_{\perp }^{-5/3}$. Critical balance also gives $k_{\parallel }\sim {\mathcal{E}}^{1/3}{k}_{\perp }^{2/3}/v_A$. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

A turbulence closure approximates the unclosed higher-order moments in evolution equations for lower-order turbulent statistics. For example, an equation for the velocity correlation tensor involves triple correlations, whose evolution involves fourth-order correlations and pressure. A closure must be checked against conservation laws, realizability and boundary conditions; matching an energy spectrum alone does not establish all these properties.

This turbulence closure assumes the velocity increment skewness is the same throughout the equilibrium range, with its value fixed by the Kolmogorov two-thirds law and Kolmogorov four-fifths law. It yields a simple crossover between differentiable and inertial-scale increments, but suppresses the scale dependence of skewness and has no independent spectral-transfer dynamics.

Set $S_2=\sqrt{15}{\beta }^{3/2}{v}_{\eta }^{2}h$ and $r=(15\beta {)}^{3/4}\eta x$. The Kolmogorov equation for structure functions and the constant-skewness turbulence closure give the displayed ordinary differential equation, with $h(0)=0$. Its small-separation balance is $h\sim x^2$; its large-separation balance is $h\sim x^{2/3}$. The first matches differentiable velocity increments, and the second recovers the assumed Kolmogorov two-thirds law.

EDQNM closure models the evolution of the turbulent energy spectrum through spectral triad interactions. Fourth-order correlations are approximated using products of second-order correlations, an eddy damping limits the resulting triple correlations, and their memory integrals are replaced by expressions using current spectra. The modeled damping times are additional physical input. This allows evolving spectra and scale-dependent transfer, but does not automatically recover coherent structures or internal intermittency. Bertoglio, Squires and Ferziger's simulation comparison explains the damping assumptions and tests their limitations.

A two-equation closure transporting turbulent kinetic energy $k$ and its turbulent kinetic energy dissipation rate $ϵ$. Its eddy viscosity is $C_{\mu }{k}^2/ϵ$. In a high-Reynolds-number wall-normal reduction, turbulent diffusion has coefficients ${\nu }_T/{\sigma }_k$ and ${\nu }_T/{\sigma }_{ϵ}$; the energy equation has shear production $P$ and destruction $ϵ$, while the dissipation equation has modeled production $C_{ϵ1}Pϵ/k$ and destruction $C_{ϵ2}{ϵ}^2/k$. The dissipation production/destruction terms are closure terms, not a second conservation of energy.

A stationary turbulence edge with no production has an advection–turbulent-diffusion leading balance. For $k\propto x^p$, $ϵ\propto x^q$ and ${\nu }_T=C_{\mu }{k}^2/ϵ$, this balance gives $2p-q=1$ and $q/p={\sigma }_{ϵ}/{\sigma }_k=\sigma$. The integral length $k^{3/2}/ϵ$ and turnover frequency $ϵ/k$ both tend to zero when $1<\sigma <3/2$. The front viscosity is linear, with ${\nu }_T^{\prime }(0^{+})=(2{\sigma }_k-{\sigma }_{ϵ})U_0$. This is formal high-Reynolds-number consistency, not a uniform justification for neglecting molecular viscosity at the exact edge.

In the constant-stress overlap layer, the law of the wall gives $U_y=u_{\ast }/(\kappa y)$ and the Reynolds stress gives $-\overline{u^{\prime }{v}^{\prime }}\simeq u_{\ast }^2$. Thus ${\nu }_T=u_{\ast }\kappa y$ and local balance gives $P\simeq ϵ=u_{\ast }^3/(\kappa y)$. Combining with the K-epsilon model viscosity formula gives the displayed constant $k$. The energy diffusion term then vanishes. The dissipation diffusion term does not: it is $u_{\ast }^4/({\sigma }_{ϵ}{y}^2)$. Balancing it against the dissipation-equation source terms yields the displayed Von Kármán constant relation. Neglecting dissipation diffusion would incorrectly require equal production and destruction constants.

Turbulent mixing redistributes a transported scalar, such as temperature or particle concentration, through irregular fluid motions and eventual molecular diffusion. A top-hat plume model assumes this mixing keeps the scalar approximately uniform within a selected cross section; that uniformity is an additional closure, not a consequence of thin geometry alone.

Mixing across surfaces of constant density, or isopycnals. In the deep ocean, breaking internal waves can supply the mechanical energy needed to mix against stable stratification. The divergence of the turbulent buoyancy flux determines local water-mass transformation.

A planar turbulent mixing layer develops between two parallel streams of different mean velocities. Their shear flow generates turbulence that transports streamwise linear momentum across the interface. A thin-layer eddy viscosity model with ${\nu }_T=C\Delta U\delta (x)$ has thickness proportional to downstream distance. For $\Delta U/U_c\ll 1$, its leading mean velocity transition is an error function.

For a planar turbulent mixing layer, let $U_c=(U_a+U_b)/2$, $\Delta U=U_a-U_b$, ${\nu }_T=C\Delta U\delta$, $\delta =S\Delta Ux/U_c$, and $\xi =\sqrt{S/C}y/\delta$. The stream function $\psi =U_c[y+\delta \sqrt{C/S}F(\xi )]$ enforces mean incompressibility. Substitution into $U{U}_x+V{U}_y={\nu }_T{U}_{yy}$ gives the displayed ordinary differential equation, with $F^{\prime }(\pm \infty )=\pm \Delta U/(2U_c)$. When $\Delta U/U_c$ is small, its linear approximation is $F^{\prime \prime \prime }+\xi F^{\prime \prime }=0$, so $F^{\prime \prime }$ is a Gaussian function and $F^{\prime }$ an error function.

Richardson pair dispersion concerns relative separation of two marked fluid elements, rather than their centre-of-mass motion. In an inertial range, scale-local relative diffusivity is of order ${ϵ}^{1/3}{r}^{4/3}$. A self-similar closure gives $d⟨r^2⟩/dt=A{ϵ}^{1/3}⟨r^2{⟩}^{2/3}$ and hence a $t^3$ regime after loss of memory of initial separation. Stationarity and inertial-range separation alone do not imply this law at arbitrarily short times.

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.

At separations well above the integral scale of turbulence, two fluid velocities can decorrelate spatially. If the relative-velocity autocorrelation $R_{\mathrm{rel}}$ is stationary and integrable, the displacement identity used in Taylor turbulent dispersion gives the displayed long-time linear mean-square growth. With speed $U$ and Lagrangian integral time of order $L/U$, its relative diffusivity is of order $UL$. The characteristic separation grows as $t^{1/2}$, unlike inertial-range Richardson pair dispersion.

The dimensionless Richardson constant is the coefficient in the mean-square pair-separation law. Under the simple self-similar diffusivity closure, $g=(A/3{)}^3$. Its universality requires scale locality, adequate scale separation and loss of initial-condition memory; a finite-time fit need not isolate this asymptotic coefficient.

Taylor turbulent dispersion describes the absolute displacement of a marked fluid element in stationary homogeneous turbulence. Here $R_L$ is its Lagrangian velocity autocorrelation. Continuous correlation at zero gives ballistic spreading at short times, and an integrable correlation gives diffusive spreading at long times. This is distinct from shear-induced Taylor dispersion and from Richardson pair dispersion.

For a zero-mean stationary velocity process of variance ${\sigma }_v^2$ and normalized autocorrelation $(1+|s|{)}^{-\alpha }$, integrating velocity twice gives the displayed expression, exactly rather than just asymptotically. For $\alpha >1$ the long-time diffusion coefficient is ${\sigma }_v^2/(\alpha -1)$. At $\alpha =1$, the displacement variance is asymptotic to $2{\sigma }_v^{2}t\log t$. For $0<\alpha <1$, it is asymptotic to $2{\sigma }_v^2{t}^{2-\alpha }/[(1-\alpha )(2-\alpha )]$. Linear variance growth does not by itself imply a central limit theorem or a normal distribution for displacement.

For stationary statistics along a Lagrangian trajectory $\mathbf{X}$, the velocity autocorrelation depends on the lag $s$. Integrating velocity twice gives the mean-square displacement $2{\int }_0^t(t-s)R_L(s)ds$. If $R_L$ is integrable, the long-time displacement is diffusive. Its normalized one-sided area is the Lagrangian integral time.

The Lagrangian integral time is the area under a normalized velocity autocorrelation along a Lagrangian trajectory. If this correlation is integrable, Taylor turbulent dispersion has asymptotic mean-square displacement $2R_L(0)t_{L}t$. The integral time need not equal every short-time decorrelation scale, particularly when the autocorrelation oscillates.

For $A=U_0^2(1/L_f-1/L_{\rho })>0$, define $K_{\infty }=A{L}_v$ and $y=\sqrt{K/K_{\infty }}$. The positive-energy solution is $y(t)=[y_0+\tanh (\sqrt{K_{\infty }}t/(2L_v))]/[1+y_0\tanh (\sqrt{K_{\infty }}t/(2L_v))]$. Energies above and below $K_{\infty }$ approach it from their respective sides. The zero-energy equation is not Lipschitz and admits both permanent rest and delayed startup, so a unique positive steady limit needs a positive seed or an additional startup prescription. With negligible available and internal energy exchange, the long-time instantaneous mixing efficiency fraction is $L_f/L_{\rho }$, independent of $L_v$.

With boundary fluxes suppressed, let $B$ transfer kinetic energy to potential energy, $M$ transfer available potential energy irreversibly to background potential energy, and $W$ transfer internal energy to potential energy. Then $\dot{P}=B+W$, ${\dot{P}}_b=M+W$, $\dot{A}=B-M$, and $\dot{K}=F-B-ϵ$. The internal reservoir has $\dot{I}=ϵ-W$, conserving total energy apart from forcing $F$. The distinction between reversible buoyancy exchange and irreversible mixing is essential outside a quasi-steady negligible-$A$ regime.

Instantaneous mixing efficiency is the fraction of instantaneous irreversible mechanical-energy loss that raises background potential energy, after subtracting internal-to-potential exchange. The ratio $\Gamma =M/ϵ$ is a different quantity, related by $\eta =\Gamma /(1+\Gamma )$. Replacing $M$ by buoyancy exchange $B$ additionally assumes negligible change of available potential energy.

Cumulative mixing efficiency is the ratio of total irreversible density-mixing conversion to total irreversible mechanical-energy loss over an interval. It is a loss-rate-weighted average of instantaneous mixing efficiency, not an unweighted time average. Stored kinetic energy, changes in available potential energy and boundary work must be accounted for before replacing its denominator by integrated forcing.

For a slender plane jet with conserved specific momentum flux $M_0=\int U^{2}dy$ and a mixing length proportional to transverse position, dimensional balance gives width proportional to $x$ and centreline speed proportional to $\sqrt{M_0/x}$. The streamfunction $\psi =\sqrt{M_{0}x}f(\eta )$, $\eta =y/x$, yields $U=\sqrt{M_0/x}{f}^{\prime }$ and $V=\sqrt{M_0/x}(\eta f^{\prime }-f/2)$. With $l=C_1|y|$, the upper-half Reynolds-averaged momentum equation integrates to $f{f}^{\prime }=2C_1^2{\eta }^2(f^{\prime \prime }{)}^2$ for a finite centreline velocity and zero centreline stress. The momentum normalization is $2{\int }_0^{{\eta }_w}(f^{\prime }{)}^{2}d\eta =1$ for a compact jet, or the same integral to infinity for a decaying jet. These formal equations do not justify an independently imposed edge condition.

For a symmetric localized turbulent plane jet similarity profile, integrated momentum balance gives $f{f}^{\prime }=2C_1^2{\eta }^2(f^{\prime \prime }{)}^2$ on $\eta >0$. At a finite free edge ${\eta }_w>0$, $f^{\prime }({\eta }_w)=0$ and finite $f({\eta }_w)$ force $f^{\prime \prime }({\eta }_w)=0$. Thus the mixing length model $l=C_1|y|$ cannot simultaneously impose zero velocity and a nonzero maximum velocity gradient at the edge. Its eddy viscosity vanishes there. Moreover, a nonzero finite centreline speed $f^{\prime }(0)=a$ forces $|f^{\prime \prime }|\sim a/(\sqrt{2}{C}_1\sqrt{\eta })$ near the centre, displaying a singular-shear defect of that local closure. The molecular-viscosity neglect cannot be uniform in this core.

A mixing length is the characteristic transverse displacement over which a turbulent parcel retains its streamwise momentum before mixing with its surroundings. Taylor expansion of a mean velocity profile gives a fluctuation of magnitude $l|U_y|$. Combining comparable transverse fluctuations with down-gradient Reynolds stress transport gives the closure $\overline{u^{\prime }{v}^{\prime }}=-l^2|U_y|U_y$. This is a phenomenological approximation, not a consequence of the Navier-Stokes equation. In a free jet, a prescribed local mixing length must also be compatible with centreline regularity and vanishing outer momentum flux.

A mixing-length closure uses a prescribed mixing length $\ell$ to approximate turbulent transport. For mean shear ${\partial }_{z}U$, a standard model gives eddy viscosity ${\nu }_t={\ell }^2|{\partial }_{z}U|$ and down-gradient Reynolds stress. Scalar-diffusion versions replace momentum transport by an appropriate scalar flux. The choice of length and correlations is a modelling assumption rather than an exact closure of the Navier-Stokes equation.

Kinetic energy per unit mass in velocity fluctuations. Mean shear, buoyancy and transport supply or redistribute it, while molecular viscosity dissipates it. Its local budget motivates the local turbulent kinetic energy balance when storage and transport are negligible.

The work of the Reynolds stress against the mean velocity gradient transfers kinetic energy from the mean flow into fluctuations. In a plane shear flow $U(y)$ this is $P=-\overline{u^{\prime }{v}^{\prime }}{U}_y$. The eddy viscosity closure gives $P={\nu }_T{U}_y^2\ge 0$. Buoyancy production and transport are separate terms in the turbulent kinetic energy budget.

In stationary locally equilibrated turbulence, mean-shear production and buoyancy production balance viscous dissipation after transport terms are neglected. Stable buoyancy flux is negative and consumes part of the shear input. This approximation gives $ϵ=u_{\ast }^3/(\kappa z)$ in a neutral logarithmic layer and $ϵ\sim B_0$ in buoyancy-dominated free convection.

Let $S=U_z>0$, $N^2=-(g/{\rho }_0){\overline{\rho }}_z>0$, $q^2=\overline{u_i^{\prime }{u}_i^{\prime }}$, and constant correlations $C_u=-\overline{u^{\prime }{w}^{\prime }}/q^2$ and $C_{\rho }^2={\overline{{\rho }^{\prime }{w}^{\prime }}}^2/(q^2\overline{{\rho }^{\prime 2}})$. Using $ϵ=q^3/L_u$ and $\chi =\overline{{\rho }^{\prime 2}}q/L_{\rho }$, the scalar-variance balance implies $\overline{{\rho }^{\prime }{w}^{\prime }}=-C_{\rho }^2{L}_{\rho }q{\overline{\rho }}_z$. Substituting in the energy budget gives the displayed quadratic for $q>0$.

Write $a=C_{\rho }^2{L}_{\rho }{N}^2/(C_{u}S)$ and $b=1/(L_u{C}_{u}S)$. The ratio of total sinks to production is $R=a/q+bq$. Positive equilibria exist only for $4ab\le 1$, equivalent to $\mathrm{Ri}\le L_u{C}_u^2/(4L_{\rho }{C}_{\rho }^2)$. The larger-$q$ root has flux Richardson number $(1-\sqrt{1-4ab})/2$; the smaller root has the plus sign. Under the local kinetic-energy evolution the larger root is stable and the smaller root unstable. The critical value is closure-dependent and is not the universal inviscid linear-stability threshold.

Velocity-fluctuation covariances transport mean momentum. For positive wall shear, $-\overline{u^{\prime }{w}^{\prime }}$ is the downward momentum-transport magnitude and contributes to the total shear stress. Some stress conventions include the minus sign in the definition of the tensor itself; the momentum equation must use the same convention.

Rotating anisotropic convection can produce an azimuthal-meridional velocity correlation even without a gradient of the mean rotation rate. Its Reynolds stress transports angular momentum and can drive differential rotation, whereas a purely viscous closure only erases shear. Energy comes from the thermally driven convection; the Coriolis force changes the correlations but does no work on a parcel.

For shear ${\mathbf{u}}_0=-2Ax{\mathbf{e}}_y$, the spatially averaged perturbation kinetic energy obeys $dE/dt=2A\rho ⟨u_x^{\prime }{u}_y^{\prime }⟩-\rho \nu ⟨|\nabla {\mathbf{u}}^{\prime }{|}^2⟩$. Pressure and the Coriolis force do no net work. Thus positive radial-azimuthal velocity correlation extracts mean-flow energy. A leading two-dimensional shearing wave has $\mathrm{Re}(v_x{v}_y^{\ast })=-k_x{k}_y|Z{|}^2/k^4>0$.

// Target: astrophysics.bigb

For a stationary, horizontally homogeneous mean flow without a horizontal pressure gradient, the averaged horizontal Navier-Stokes equation is $\nu U_{zz}-(\overline{u^{\prime }{w}^{\prime }}{)}_z=0$. Integrating gives the displayed physical shear stress. The derivative of the Reynolds covariance belongs in the stress-divergence equation, not inside the stress itself. The kinematic stress is ${\tau }_d/\rho$, which must be used when an eddy viscosity with units of kinematic viscosity multiplies the mean gradient.

An eddy viscosity parametrizes turbulent momentum transport by $\overline{u^{\prime }{v}^{\prime }}=-{\nu }_T{U}_y$. The mixing length model gives ${\nu }_T=l^2|U_y|\ge 0$. This coefficient depends on the mean flow and position; it is distinct from molecular kinematic viscosity. It describes momentum transport rather than molecular energy dissipation.

For an incompressible mean flow, an eddy viscosity can model the non-isotropic part of the Reynolds stress through the strain-rate tensor $S_{ij}=(U_{i,j}+U_{j,i})/2$. The isotropic part is fixed separately by the turbulent kinetic energy $e_t=\overline{u_k^{\prime }{u}_k^{\prime }}/2$ in $d$ fluctuating dimensions. Taking the trace proves that the isotropic term is necessary: $S_{ii}=0$, whereas $-\rho \overline{u_i^{\prime }{u}_i^{\prime }}=-2\rho e_t$. In the mean momentum equation the isotropic term can be absorbed into $P_{\mathrm{eff}}=P+2\rho e_t/d$.

An Eulerian time average of an incompressible constant-density Navier-Stokes equation gives the displayed mean momentum equation. Decompose velocity into its mean and a zero-mean fluctuation. In conservative form, the averaged momentum flux is $\overline{u_i{u}_j}=U_i{U}_j+\overline{u_i^{\prime }{u}_j^{\prime }}$; mean incompressibility then turns its first term into mean advection. The fluctuation covariance is the Reynolds stress divided by density, using the positive covariance convention. Closing this extra unknown requires a turbulence model.

Internal intermittency is the fluctuating concentration of gradients, transfer and viscous dissipation inside turbulent fluid. It is distinct from an interface between turbulent and non-turbulent regions intermittently passing an observer. Its scale dependence is reflected in non-Gaussian increments and coarse-grained viscous dissipation moments.

Refined similarity writes $\Delta v=(r{ϵ}_r{)}^{1/3}V$ with universal normalized conditional statistics at high local Reynolds number. Then $S_p=C_p{r}^{p/3}⟨{ϵ}_r^{p/3}⟩$. The distribution of coarse-grained energy dissipation need not be universal. Refined similarity itself does not prescribe that distribution.

If $\log ({ϵ}_r/ϵ)$ is Gaussian with variance ${\sigma }_r^2$, normalization gives $⟨{ϵ}_r^q⟩/{ϵ}^q=\exp [q(q-1){\sigma }_r^2/2]$. Taking $\exp ({\sigma }_r^2)=B(\ell /r{)}^{\mu }$ predicts ${\zeta }_p=p/3-\mu p(p-3)/18$. Its eventual negative high-order exponents show that this extrapolation cannot hold for arbitrarily large orders with finite uniform velocity moments.

Coarse-grained energy dissipation is a spatial average of the local rate $2\nu S_{ij}{S}_{ij}$ over a region of size $r$. Its random value ${ϵ}_r$ has mean $ϵ$ in homogeneous turbulence. Refined similarity uses this local averaged viscous dissipation to scale velocity increments.

Integral-scale intermittency is modulation of the energy-containing motions and the energy supply on large lengths or long turnover times. Averaging velocity statistics over regions or realizations with differing viscous dissipation rates can invalidate universal coefficients based only on the global mean viscous dissipation.

Mixing turbulent populations with differing viscous dissipation gives $S_p={\beta }_p{r}^{p/3}⟨{ϵ}_a^{p/3}⟩$. Relative to the same power of the mean viscous dissipation, the coefficient changes with the intensity distribution except at $p=3$. Thus universality of all unconditional coefficients is incompatible with arbitrary integral-scale intermittency.

An energy cascade transfers kinetic energy among scales through nonlinear motion. A forward cascade carries energy toward smaller spatial scales, where kinematic viscosity can dissipate it. Conservation and a mean scale-space flux do not uniquely prescribe the geometry or locality of the transfer mechanism.

The quadratic nonlinear term of the Navier-Stokes equation couples three Fourier modes whose wave vectors close to a triangle. These spectral triad interactions transfer kinetic energy between wave numbers. A turbulence closure can model their triple correlations while retaining more scale information than a prescribed local power law.

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.

Although the local viscous dissipation is $2\nu S_{ij}{S}_{ij}$, increasingly intense velocity gradients can compensate for vanishing kinematic viscosity. A Burgers vortex illustrates this mechanism: its radius shrinks while its excess integrated viscous dissipation remains finite. Such a local model does not by itself prove the zeroth law of turbulence for a whole flow.

The eddy turnover time estimates how long a velocity difference $\delta u_l$ takes to distort a structure of size $l$. In a local strong energy cascade, the transfer time is assumed comparable to it. A weak wave cascade can transfer energy much more slowly because a single encounter makes only a small fractional change.

A two-dimensional forward two-dimensional enstrophy cascade has the idealized spectrum $E(k)\propto k^{-3}$. The energy integral above a wavenumber scales as its inverse square, whereas the cumulative enstrophy integral $\int k^{2}E(k)dk$ grows logarithmically. The spectrum used here omits possible further logarithmic corrections.

For a wide $C{k}^{-3}$ range and separation well below its outer scale, the second-order longitudinal structure function is controlled by ${\int }_{k_0}^{\pi /r}{k}^{2}E(k)dk$. The low-wavenumber filter term gives $S_2(r)\simeq C{r}^2\log [\pi /(k_{0}r)]/4$, larger than the order-$r^2$ energy above the cutoff. This is a marginal dependence on larger-scale enstrophy.

The Townsend-Betchov cascade cartoon organizes inertial range vorticity into sheets by bi-axial strain, then uses sheet instability and roll-up to generate finer vortex tubes. Dissipative tube cores balance stretching with kinematic viscosity. Such differing mechanisms across scales challenge literal identical-step, memoryless cascade models without contradicting a mean energy flux.

Kolmogorov 1941 theory assumes locally universal small-scale velocity statistics set by mean viscous dissipation and kinematic viscosity at large Reynolds number. In its inertial range dimensional form, $S_p(r)\sim {ϵ}^{p/3}{r}^{p/3}$. Large-scale modulation and internal intermittency require qualifying this unconditional scaling.

In the inertial range, velocity increment statistics also become independent of kinematic viscosity. Combined with the Kolmogorov first similarity hypothesis, this gives $S_p(r)={\beta }_p(ϵr{)}^{p/3}$ by dimensional analysis. Signed integer moments and absolute noninteger moments must be distinguished; the signed third-order coefficient is negative in a forward energy cascade.

The second-order longitudinal structure function has the displayed form in the idealized inertial range of Kolmogorov 1941 theory. It predicts that the typical longitudinal velocity increment grows as the cube root of separation. Unlike the Kolmogorov four-fifths law, its exponent and coefficient are not exact consequences of the Navier-Stokes equation.

In the locally isotropic equilibrium range, the small-scale velocity increment statistics depend on the separation, mean viscous dissipation and kinematic viscosity, independently of large-scale forcing details. Dimensional analysis leaves the Kolmogorov microscales as normalization scales and gives the displayed second-order longitudinal structure function. Universality of $F$ is part of the hypothesis, not something proved by dimensional analysis.

The equilibrium range consists of scales below the energy-containing motions whose statistics can adjust rapidly to the local energy supply. It includes the inertial range and dissipative scales. Approximate equilibrium concerns mean transfer and viscous dissipation, not absence of intermittent spatial fluctuations.

The dissipation range contains scales at which kinematic viscosity appreciably affects velocity gradients and converts kinetic energy into heat. In a high-Reynolds-number three-dimensional cascade its characteristic length and time are the Kolmogorov microscales. Intense vortex cores can be intermittent within this range.

The Kolmogorov length, velocity and time are $\eta =({\nu }^3/ϵ{)}^{1/4}$, $v_{\eta }=(\nu ϵ{)}^{1/4}$ and ${\tau }_{\eta }=(\nu /ϵ{)}^{1/2}$. With $ϵ\sim u^3/\ell$, their ratios to the integral scales of turbulence are powers of the Reynolds number. They describe a mean scale balance; intermittent dissipative structures can have other local scales.

The Kolmogorov length scale is the length among the Kolmogorov microscales. Kinematic viscosity $\nu$ and mean energy dissipation per unit mass $ϵ$ give the displayed dimensional scale. The associated speed is $(\nu ϵ{)}^{1/4}$, so the Reynolds number formed at this length is one.

The inertial range lies between the energy-containing and viscous scales. Direct forcing and viscous dissipation are small there, and nonlinear transfer carries an approximately constant mean energy flux in a three-dimensional forward cascade. Scale similarity and universality are additional statistical hypotheses.

Homogeneous turbulence has velocity statistics unchanged by translating all observation points. Mean spatial divergences vanish when the moments exist and averaging commutes with derivatives. Homogeneity is distinct from isotropy: it does not require equal statistics in different directions.

The integral scale of turbulence characterizes energy-containing velocity correlations; in isotropic turbulence one definition is $\ell ={\int }_0^{\infty }f(r)dr$. Its turnover time is of order $\ell /u$. Large-scale intensity fluctuations can change the energy supplied to the equilibrium range.