Scaling transitions in plane turbulent wall jets

In the fully developed region of a plane turbulent wall jet, the key jet parameters, including the jet velocity U m , jet half-width z 1/2 and wall shear stress $ \tau_{0}$ , follow the classical power-law scaling with the streamwise distance x : U m $v$ / M 0 ∼ ( xM 0 / $v$ 2 ) − α , z 1/2 M 0 / $v$ 2 ∼ ( xM 0 / $v$ 2 ) β and $ \tau_{0}$ $v$ 2 /( ρ $M_{0}^{2}$ ) ∼ ( xM 0 / $v$ 2 ) − χ , where M 0 is the source kinematic momentum flux, $v$ is the coefficient of kinematic viscosity of fluid, ρ is the mass density of fluid and α , β and χ are the positive scaling exponents. We present a theoretical framework to determine these exponents. Our framework reveals that each jet parameter exhibits a scaling transition. This transition is driven by a shift in the scaling law of the skin-friction coefficient as the Reynolds number Re m = U m z m / $v$ changes over from Re m < 8000 to Re m > 10 000, where z m is the wall-normal location corresponding to the jet velocity. Specifically, α transitions from 4(1 + γ )/(9 − γ ) to 13(1 + γ )/[2(14 − γ )], β from 8/(9 − γ ) to 13/(14 − γ ) and χ from (9 + 7 γ )/(9 − γ ) to (14 + 12 γ )/(14 − γ ), where γ ≈ 0.05 is a parameter determined from experiments. We validate the theoretical predictions against extensive experimental datasets from the literature.