A similarity theory for the turbulent pl

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 utilized to ensure that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. For inner equations, the appropriate velocity scale is identified as the friction velocity, u∗, with the length scale being ν/u∗. When considering finite Reynolds numbers, the profiles are shown to retain a dependence on the length-scale ratio, y+1/2 = u∗y1/2/ν, where y1/2 indicates the distance from the wall at which the mean velocity is half its maximum value. In the asymptotic limit as y+1/2 approaches infinity, the familiar law of the wall is derived. For the outer equations, Um is used as the appropriate velocity scale, with length scale y1/2, retaining dependence on y+1/2 for finite values. The Reynolds shear stress scales with u∗2, while normal stresses scale with Um2. The combined analysis leads to determining functional forms of the velocity profiles in the overlap region, which must comply with power laws weakened by Reynolds number influences, yielding a theoretical friction law also expressed as a power law. Consequently, the growth of the asymptotic plane wall jet is shown to deviate from linear growth.