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The symmetries and similarities of the zero-pressure-gradient turbulentboundary layer (ZPGTBL) are investigated to derive the full set of similarityvariables, to derive the similarity equations, and to obtain a higher-orderapproximate solution of the mean velocity profile. Previous analyses have notresulted in all the similarity variables. We perform a symmetry analysis of theequations for ZPGTBL using Lie dilation groups, and obtain local, leading-ordersymmetries of the equations. The full set of similarity variables were obtainedin terms of the boundary layer parameters. The friction velocity was shown tobe the outer-layer velocity scale. The downstream evolution of the boundarythickness and the friction velocity is obtained analytically. The dependentsimilarity variables are written as asymptotic expansions. By asymptoticallymatching the expansions, an approximate similarity solution up to the thirdorder in the overlapping layer are obtained. These results are obtained fromfirst principles without any major assumptions and a turbulence model. Thesimilarities and differences between ZPGTBL and turbulent channel flows interms of the similarity equations, the gauge functions and the approximatesolutions are discussed. In particular, the leading-order expansions areidentical for ZPGTBL and channel flows, supporting the notion of universalityof the near-wall layer. In addition, the logarithmic friction law for ZPGTBL isaccurate to all orders while it is only accurate at the leading order inchannel flows. The results will help further understand ZPGTBL and the issue ofuniversality of the near-wall layer in wall-bounded turbulent flows.

Symmetries and similarities of the zero-pressure-gradient turbulent boundary