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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 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.

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.