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The spectral model of Perry, Henbest & Chong (1986) predicts that theintegral length-scale varies very slowly with distance to the wall in theintermediate layer. The only way for the integral length scale's variation tobe more realistic while keeping with the Townsend-Perry attached eddy spectrumis to add a new wavenumber range to the model at wavenumbers smaller than thatspectrum. This necessary addition can also account for the high Reynolds numberouter peak of the turbulent kinetic energy in the intermediate layer. Ananalytic expression is obtained for this outer peak in agreement with extremelyhigh Reynolds number data by Hultmark, Vallikivi, Bailey & Smits (2012, 2013).The finding of Dallas, Vassilicos & Hewitt (2009) that it is the eddy turnovertime and not the mean flow gradient which scales with distance to the wall andskin friction velocity in the intermediate layer implies, when combined withTownsend's (1976) production-dissipation balance, that the mean flow gradienthas an outer peak at the same location as the turbulent kinetic energy. This isseen in the data of Hultmark, Vallikivi, Bailey & Smits (2012, 2013). The sameapproach also predicts that the mean flow gradient has a logarithmic decay atdistances to the wall larger than the position of the outer peak, a qualitativeprediction which the aforementioned data also support.

The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow