= Turbulent plane jet similarity
{title2=$\psi=\sqrt{M_0x}\,f(y/x)$}
For a slender plane jet with conserved specific momentum flux $M_0=\int U^2dy$ 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> $\psi=\sqrt{M_0x}f(\eta)$, $\eta=y/x$, yields $U=\sqrt{M_0/x}f'$ and $V=\sqrt{M_0/x}(\eta f'-f/2)$. With $l=C_1|y|$, the upper-half <Reynolds-averaged momentum equation> integrates to $ff'=2C_1^2\eta^2(f'')^2$ for a finite centreline velocity and zero centreline stress. The momentum normalization is $2\int_0^{\eta_w}(f')^2d\eta=1$ 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.
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