Physics – Fluid Dynamics
Scientific paper
2011-12-29
Physics
Fluid Dynamics
8 pages, 2 figures
Scientific paper
A novel Lie-group analysis of the (unclosed) mean momentum equation (MME) for turbulent channel and pipe flows yields an analytic multi-layer formula for the mixing length (hence the mean velocity profile) as a candidate invariant solution. Two kinds of local invariant solutions are proposed: one describing viscous sublayer, buffer layer, log-layer, and a newly discovered central core, and the other describing the bulk zone where a quasi-balance is established between turbulent production and dissipation. In particular, a simple form, 1-r^m, is derived for describing the mixing length in the bulk zone of channel (m=4) and pipe (m=5) flows. Furthermore, an ansatz based on a relation between prolonged Lie-group invariants is proposed to derive composite invariant solutions to the MME, yielding the entire mixing length profile. The theory proposes a thorough quantification of commonly recognized multi-layer structure using three kinds of parameters: scaling, layer thickness and transition sharpness. All the above assumptions are validated by direct numerical simulation (DNS) of channel and pipe flows. A general framework extending this Lie-group analysis to other wall-bounded turbulent flows is proposed.
Chen Xi
Hussain Fazle
She Zhen-Su
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