Abstract
We propose a first minimal theory for boundary layer turbulence that captures very well the profile of the mean-square velocity fluctuations in the streamwise direction and give a quantitative prediction of the Townsend-Perry constants. Our theory is based on connecting all moments of velocity fluctuations as a function of the distance to the wall with the turbulent energy spectrum. A similar spectral theory was proposed in G. Gioia and P. Chakraborty [Phys. Rev. Lett. 96, 044502 (2006)] to explain the friction factor and the von K\'arm\'an law in G. Gioia, N. Guttenberg, N. Goldenfeld, and P. Chakraborty [Phys. Rev. Lett. 105, 184501 (2010)]. We generalized it by including fluctuations in the wall-shear stress and the streamwise velocity. The theoretical predictions for the mean velocity and mean-square fluctuations reproduce the shape of the velocity profiles in the buffer and inertial layer obtained from wind tunnel experiments.
Highlights
Turbulence is a ubiquitous phenomenon encountered in very diverse natural systems, from the large-scale atmosphere [1] and oceans [2] all the way down to quantum fluids [3], as well as in engineered systems, such as pipelines, heat exchangers, wind turbines, etc
We propose a first minimal theory for boundary layer turbulence that captures very well the profile of the mean-square velocity fluctuations in the streamwise direction and give a quantitative prediction of the TownsendPerry constants
We typically differentiate between four flow regions as moving away from the wall [6,7]: (i) The viscous region is closest to the wall and dominated by viscous flows, (ii) the buffer layer makes the transition from viscous to turbulent flows and is where detached eddies initially form, (iii) the inertial layer where the turbulent eddies form from the attached eddies and the log laws of the wall applies, and (iv) the wake, the fully developed energetic region where turbulent fluctuations can be described by homogeneous turbulence
Summary
Turbulence is a ubiquitous phenomenon encountered in very diverse natural systems, from the large-scale atmosphere [1] and oceans [2] all the way down to quantum fluids [3], as well as in engineered systems, such as pipelines, heat exchangers, wind turbines, etc. We propose a generalization of the spectral theory that includes fluctuations in the streamwise velocity due to an essentially fluctuating wall shear stress These velocity fluctuations are characterized by an interplay between the Kolmogorov-Obukhov energy spectrum and the 1/k spectrum in the buffer and inertial layers. It starts with the Prandtl-von Kármán law in the inertial region but extends the mean velocity across the boundary and viscous layers by means of detached and attached eddies The latter is a mathematical formulation of Townsend’s theory [17] that connects all the eddies, and the former is similar. V, we use the attached eddy hypothesis and the SCT [13,14] to derive the form of the Townsend-Perry and the generalized TownsendPerry constants This allows us to derive the streamwise fluctuations in the wall shear stress and remove the assumption made in Refs. VIII, we conclude with a discussion on the proposed spectral theory and the role that Townsend’s attached eddies play in it
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