A similarity theory for the turbulent plane wall jet without external stream

Abstract

A new theory for the turbulent plane wall jet without external stream is proposed based on a similarity analysis of the governing equations. The asymptotic invariance principle (AIP) is used to require that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. Application to the inner equations shows that the appropriate velocity scale is the friction velocity, u∗, and the length scale is v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio, y⁺_1/2 = u∗y_1/2/v, where y_1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its maximum value. In the limit as y⁺_1/2 → ∞, the familiar law of the wall is obtained. Application of the AIP to the outer equations shows the appropriate velocity scale to be U_m, the velocity maximum, and the length scale y_1/2; but again the profiles retain a dependence on y⁺_1/2 for finite values of it. The Reynolds shear stress in the outer layer scales with u²_*, while the normal stresses scale with U²_m. Also U_m ∼ yⁿ_1/2 where n < −1/2 and must be determined from the data. The theory cannot rule out the possibility that the outer flow may retain a dependence on the source conditions, even asymptotically.

The fact that both these profiles describe the entire wall jet for finite values of y⁺_1/2, but reduce to inner and outer profiles in the limit, is used to determine their functional forms in the ‘overlap’ region which both retain. The result from near asymptotics is that the velocity profiles in the overlap region must be power laws, but with parameters which depend on Reynolds number y⁺_1/2 and are only asymptotically constant. The theoretical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, although the difference from linear growth is small.

It is hypothesized that the inner part of the wall jet and the inner part of the zero-pressure-gradient boundary layer are the same. It follows immediately that all of the wall jet and boundary layer parameters should be the same, except for two in the outer flow which can differ only by a constant scale factor. The theory is shown to be in excellent agreement with the experimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B₁ = (U_mv/M_o)/ (y⁺_1/2M_o/v²)ⁿ = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.