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 J(T)(x) (non-dimensionalized by its inlet value) as a function of the streamwise distance $x$, we prove that J(T)(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 J(T)>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 -p (u'v') over bar (x,y), the mean pressure P(x,y), etc. Experiments satisfying J(T)(x)>1 validate our predictions for all jet measures, including, among others, the profiles of U, V and p-(u'v') over bar. We further predict U similar to x(-0.24), V similar to x(-0.45),p-(u'v') over bar similar to x(-0.69), the mass flux Q(m) similar to x(0.55), and J(T) 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)similar to x(0.79), 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.