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 Um, jet half-width z1/2 and wall shear stress τ0 , follow the classical power-law scaling with the streamwise distance x: Um v /M0 ∼ (xM0/ v 2)−α, z1/2M0/ v 2 ∼ (xM0/ v 2)β and τ0 v 2/(ρ M20 ) ∼ (xM0/ v 2)−χ, where M0 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 Rem = Umzm/ v changes over from Rem < 8000 to Rem > 10 000, where zm 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 + 7γ)/(9 − γ) to (14 + 12γ)/(14 − γ), where γ ≈ 0.05 is a parameter determined from experiments. We validate the theoretical predictions against extensive experimental datasets from the literature.