By now you will have experimented with different and large values of the Reynolds number \( R_m \) to discern salient
features of wall-bounded turbulent flows, especially in regions where viscous and turbulent flows dominate.
Although simplistic in its derivation, the Pai approximation can be a very useful tool in asserting simple
turbulence models (e.g. the limitations of the van Driest damping function in the mixing length model  and in
determining \( u_\tau \) in wall function calculations ).
Over the years there have been several analytical models of turbulent flows in channels or pipes. Perhaps the most
signifcant step in modelling turbulence has, in my opinion, been the development of the Lagrangian Averaged
Navier-Stokes (LANS) model which has many parallels with the Euler-Poincaré equations for geodesics in the
diffeomorphism group (see Metric Pattern Theory workboks).
For example, one can solve the corresponding boundary layer for turbulent flow past an infinite flat plate .
But we hope that the main benefit of this workbook is to get some "graphical intuition" of the physical features
of turbulent flows.
Turbulent Flows - 10 / 11
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