Turbulent jet theory via Lie symmetry analysis: the free plane jet

A theory of incompressible turbulent plane jets (TPJs) is proposed by advancing an improved boundary layer approximation over the limiting classical – retaining more terms in the momentum balance equations. A pressure deficit inside the jet (with respect to the ambient) must exist due to transverse turbulence (Miller & Comings, J. Fluid Mech. , vol. 3, 1957, pp. 1–16; Hussain & Clarke, Phys. Fluids , vol. 20, 1977, pp. 1416–1426). Contrary to the universally accepted invariance of the total momentum flux JT(x) (non-dimensionalized by its inlet value) as a function of the streamwise distance x , we prove that JT(x) >1 – a condition that all TPJs must satisfy; surprisingly, prior theories and most experiments do not satisfy this condition. This motivated us to apply Lie symmetry analysis with translational and dilatational transformations of the modified equations (incorporating JT>1 ), which yields scaling laws for key jet measures: the mean streamwise and transverse velocities $U(x,y)$ and $V(x,y)$ , the turbulence intensities, the Reynolds shear stress -ρ \,u'v'̄(x,y) , the mean pressure $P(x,y)$ , etc. Experiments satisfying JT(x)>1 validate our predictions for all jet measures, including, among others, the profiles of U , V and -ρ \,u'v'̄ . We further predict U ~ x-0.24 , V ~ x-0.45 , -ρ \,u'v'̄~ x-0.69 , the mass flux Qₘ ~ x0.55 , and JT increases to approximately 1.5. Contrary to the classical linear jet spread, we find sublinear spread, with the jet half-width growing like b(x)~ x0.79 , indicating a narrower jet. Our predictions differ notably from most results reported in the literature. These contradictions demand revisiting jet studies involving carefully designed facilities and boundary conditions, and highly resolved simulations.