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

TL;DR

This paper uses symmetry analysis to derive comprehensive similarity variables and equations for zero-pressure-gradient turbulent boundary layers, providing new insights into their structure and universality compared to channel flows.

Contribution

It derives the full set of similarity variables and equations for ZPGTBL using Lie symmetry analysis, including higher-order approximate solutions without turbulence models.

Findings

Full set of similarity variables obtained analytically.

Boundary layer evolution described explicitly.

Logarithmic friction law accurate to all orders.

Abstract

The symmetries and similarities of the zero-pressure-gradient turbulent boundary layer (ZPGTBL) are investigated to derive the full set of similarity variables, to derive the similarity equations, and to obtain a higher-order approximate solution of the mean velocity profile. Previous analyses have not resulted in all the similarity variables. We perform a symmetry analysis of the equations for ZPGTBL using Lie dilation groups, and obtain local, leading-order symmetries of the equations. The full set of similarity variables were obtained in terms of the boundary layer parameters. The friction velocity was shown to be the outer-layer velocity scale. The downstream evolution of the boundary thickness and the friction velocity is obtained analytically. The dependent similarity variables are written as asymptotic expansions. By asymptotically matching the expansions, an approximate similarity solution up to the third order in the overlapping layer are obtained. These results are obtained from first principles without any major assumptions and a turbulence model. The similarities and differences between ZPGTBL and turbulent channel flows in terms of the similarity equations, the gauge functions and the approximate solutions are discussed. In particular, the leading-order expansions are identical for ZPGTBL and channel flows, supporting the notion of universality of the near-wall layer. In addition, the logarithmic friction law for ZPGTBL is accurate to all orders while it is only accurate at the leading order in channel flows. The results will help further understand ZPGTBL and the issue of universality of the near-wall layer in wall-bounded turbulent flows.