Abstract
First-principle-based prediction of mean-flow quantities of wall-bounded turbulent flows (channel, pipe and turbulent boundary layer (TBL)) is of great importance from both physics and engineering standpoints. Here we present a symmetry-based approach which yields analytical expressions for the mean-velocity profile (MVP) from a Lie-group analysis. After verifying the dilatation-group invariance of the Reynolds averaged Navier–Stokes (RANS) equation in the presence of a wall, we depart from previous Lie-group studies of wall turbulence by selecting a stress length function as a similarity variable. We argue that this stress length function characterizes the symmetry property of wall flows having a simple dilatation-invariant form. Three kinds of (local) invariant forms of the length function are postulated, a combination of which yields a multi-layer formula giving its distribution in the entire flow region normal to the wall and hence also the MVP, using the mean-momentum equation. In particular, based on this multi-layer formula, we obtain analytical expressions for the (universal) wall function and separate wake functions for pipe and channel, which are validated by data from direct numerical simulations (DNS). In conclusion, an analytical expression for the entire MVP of wall turbulence, beyond the log law or power law, is developed in this paper and the theory can be used to describe the mean turbulent kinetic-energy distribution, as well as a variety of boundary conditions such as pressure gradient, wall roughness, buoyancy, etc. where the dilatation-group invariance is valid in the wall-normal direction.
Highlights
Canonical wall-bounded flows (channel, pipe and turbulent boundary layer (TBL))are widely seen in engineering applications and in nature (Smits & Marusic 2013)
Note that the current analysis focuses on the group invariants of the stress length function instead of the mean velocity in earlier works by Oberlack (2001), Lindgren, Osterlund & Johansson (2004) and Marati et al (2006), explained as follows
These postulated solutions should be validated by direct numerical simulations (DNS) data, which we present in detail
Summary
Are widely seen in engineering applications and in nature (Smits & Marusic 2013). Turbulent channel and pipe are internal flows driven by a pressure gradient, which fully determines the mean velocity profile (MVP) and the. The constant dilatation invariant for the mean velocity, as assumed in previous works, is only valid in part of the viscous sublayer very close to the wall where U+ ≈ y+; and this constancy is lost except in a restricted region beyond the log layer where Barenblatt argued it is the power law (Barenblatt 1993). Note that the current symmetry analysis is significantly different from previous works modelling the mean velocity (Nickels 2004; Del Alamo & Jimenez 2006; Monkewitz et al 2007; Panton 2007; L’vov, Procaccia & Rudenko 2008) by two features: a unified description of the mean velocities of all three canonical flows (channel, pipe and TBL) is obtained for the first time and the current parameters adequately characterize the physical multi-layer structure in the flow. In appendix B, we discuss the main features of a current symmetry-based approach and its generality to other wall flows
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