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

A new theory for the turbulent plane wall jet without external stream is pr oposed based on a similarity analysis of the governing equations. The asymp totic invariance principle (AIP) is used to require that properly scaled pr ofiles reduce to similarity solutions of the inner and outer equations sepa rately in the limit of infinite Reynolds number. Application to the inner e quations shows that the appropriate velocity scale is the friction velocity , u(s),, and the length scale is v/u(*),. For finite Reynolds numbers, the profiles retain a dependance 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 velocit y has dropped to 1/2 its maximum value. In the limit as y(1/2)(+) --> infin ity, the familiar law of the wall is obtained. Application of the AIP to th e outer equations shows the appropriate velocity scale to be U-m, the veloc ity maximum, and the length scale y(1/2); but again the profiles retain a d ependence on y(1/2)(+) for finite values of it. The Reynolds shear stress i n the outer layer scales with u(*)(2), while the normal stresses scale with U-m(2). Also U-m similar to y(1/2)(n) where n < -1/2 and must be determine d from the data. The theory cannot rule out the possibility that the outer how may retain a dependence on the source conditions, even asymptotically. The fact that both these profiles describe the entire wall jet for finite v alues 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 theoret ical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, alth ough the difference from linear growth is small. It is hypothesized that the inner part of the wall jet and the inner part o f the zero-pressure-gradient boundary layer are the same. It follows immedi ately that all of the walljet and boundary layer parameters should be the s ame, except for two in the outer flow which can differ only by a constant s cale factor. The theory is shown to be in excellent agreement with the expe rimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B-1 =(U(m )v / M-o)/(y(1/2)M(o)/v(2))(n) = constant where n approximate to -0.528, ap pears to be required to determine the entire flow for a given source.