Turbulent plane jet similarity ($\psi =\sqrt{M_{0}x}f(y/x)$)

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.