A Theory of Turbulent Jets Via Lie Symmetry Analysis

A theory of turbulent jets (plane --TPJ, and axisymmetric -- TAJ) is developed using Lie symmetry analysis (LSA) resolving failings of current theories -- due to inappropriate application of boundary layer approximation. Applying LSA with translational and dilatational transformations to the Navier-Stokes equations yields a new family of scaling laws for key flow measures: mean streamwise and lateral velocities U and V , Reynolds shear stress R=〈-u'v'〉, mean pressure P, etc. In comparison to classical theory, we find that TPJ spreads more slowly with streamwise coordinate x, and TAJ spreads faster; and whereas classically△P=P-P_∞ = 0 (P_∞ -- ambient), our theory predicts a pressure deficit △P= -〈v' ²〉< 0 (also experimentally observed). Consequently, the total streamwise momentum flux J_T increases with x, i.e., J_T (x) >1 – a condition that all jet flows must satisfy. Trivially, the pressure deficit integral across the jet is balanced by a commensurate increase in J_T . Most experiments and theories fail this condition. Those experiments showing J_T >1 also validate many of our predictions, including profiles of U, V and R which match measurements well in both jets. In TPJ, we predict: b∼x^0.91 (half-width), Q_m ∼x^0.52 (mass flux), J_T ∼x^0.13, and centerline U_c ∼x^-0.39 ,V_c ∼x^-0.48 and R_c ∼x^-0.87. In TAJ: b∼x^1.23, Q_m ∼x^1.34, J_T ∼x^0.22, U_c ∼x^-1.12, V_c ∼x^-0.89, R_c ∼x^-2.01. Rigorously designed and accurately measured free jet experiments and simulations, and deeper explanation of the mechanisms involved, are clearly needed.