Friction in a Turbulent Boundary Layer

  • Felix V. Dolzhansky
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 103)


We return now to the equations of motion of a viscous fluid described in Chap. 19, and consider briefly an approach to the description of turbulent flows in the framework of an incompressible fluid. The basis of a statistical approach to the theory of turbulence is the passage from considering a single turbulent flow (an implementation) to considering a statistical ensemble of possible implementations for fixed external conditions. In other words, velocity, temperature and other characteristics of a turbulent flow are now to be considered as random fields. Because of the uncertainty in the probability distribution in the space of realizations, we approach the very delicate issue of calculating the average values in a way that is common in turbulence theory. Under the average value 〈f(r, t)〉 of a random field f(r, t) we mean the average over the set of possible implementations (or, in other words, “the ensemble average”), which in practical applications is replaced by the average over time, based on the ergodic hypothesis. In this case the quantity f(r, t) itself can be written as f(r, t) = 〈f(r, t)〉 + f′(r, t), where f′(r, t) are fluctuations, pulsations, deviations from the mean-field, 〈f′(r, t)〉 = 0. Taking into account the above definitions and using the medium incompressibility (ρ = ρ 0 = const) the averaged Navier–Stokes equations can be rewritten.


Planetary Boundary Layer Potential Vorticity Turbulent Viscosity Free Atmosphere Ekman Layer 
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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix V. Dolzhansky
    • 1
  1. 1.

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