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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 [3] and in determining $$u_\tau$$ in wall function calculations [8]).

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 [12].

But we hope that the main benefit of this workbook is to get some "graphical intuition" of the physical features of turbulent flows.

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# References

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1. Hussain, A.K.M.F. and Reynolds, W.C. "Measurements in fully developed turbulent channel flow", ASME J. Fluids Engng, Vol 97, 1975, pp. 568-580.
2. Pai, S.I. "On turbulent flow between parallel plates", ASME J Appl Mech, Vol 2, 1953, pp 109-114.
3. Ratnanather, J.T. "Studies in simple turbulence models", Oxford University Computing Laboratory - Numerical Analysis Group, Report 87/7, 1986.
4. Clauser, F.H. "The turbulent boundary layer", Adv Appl Mech, Vol 4, 1956, pp. 1-51.
5. Chapman, D.R. and Kuhn, G.D. "The limiting behaviour of turbulence near a wall", J Fluid Mech, Vol 170, 1986, pp. 265-292.
6. Klebanoff, P.S. "Characteristics of turbulence in a boundary layer with zero pressure gradient", NACA TR 1247, 1955.
7. Patel, V.C., Röaut;di, W. and Schuerer, G. "Turbulence models for near-wall and low Reynolds number flows: a review", AIAA J., Vol 23, Sept. 1985, pp. 1308-1319.
8. Ratnanather, J.T. Numerical analysis of turbulent flow. D.Phil. thesis, University of Oxford, 1989.
9. Tennekes, H. and Lumley, J.L. "A first course in turbulence", MIT Press, 1972.
10. Phillips, W.R.C. "The inner region of a turbulent boundary layer", Phys Fluids, Vol 30, 1987, pp. 2354-2360.
11. Phillips, W.R.C. and Ratnanather, J.T. "The outer region of a turbulent boundary layer", Phys Fluids A, Vol 2, 1990, pp. 427-434.
12. Cheskidov, A. Boundary layer for the Navier-Stokes-alpha model of fluid turbulence, Archive for Rational Mechanics and Analysis 172, pp. 333-362.
13. Laufer, J. (1951) Investigation of turbulent flow in a two-dimensional channel. NACA Report 1053, NACA Technical Note 2123.