The stagnation point von Kármán coefficient

Details The stagnation point von Kármán coefficient The stagnation point von Kármán coefficient

2009-02-03

arxiv.org/abs/0902.0579v3

Phys. Rev. E 80, 046306 (2009)

Physics

Fluid Dynamics

14 pages, 20 figures. This version of the article has been accepted for publication in PRE. Note though that the PRE paper for

Scientific paper

10.1103/PhysRevE.80.046306

On the basis of various DNS of turbulent channel flows the following picture is proposed. (i) At a height y from the y = 0 wall, the Taylor microscale \lambda is proportional to the average distance l_s between stagnation points of the fluctuating velocity field, i.e. \lambda(y) = B_1 l_s(y) with B_1 constant, for \delta_\nu << y \lesssim \delta. (ii) The number density n_s of stagnation points varies with height according to n_s = C_s y_+^{-1} / \delta_\nu^3 where C_s is constant in the range \delta_\nu << y \lesssim \delta. (iii) In that same range, the kinetic energy dissipation rate per unit mass, \epsilon = 2/3 E_+ u_\tau^3 / (\kappa_s y) where E_+ is the total kinetic energy per unit mass normalised by u_\tau^2 and \kappa_s = B_1^2 / C_s is the stagnation point von K\'arm\'an coefficient. (iv) In the limit of exceedingly large Re_\tau, large enough for the production to balance dissipation locally and for - ~ u_\tau^2 in the range \delta_\nu << y << \delta, dU_+/dy ~ 2/3 E_+/(\kappa_s y) in that same range. (v) The von K\'arm\'an coefficient \kappa is a meaningful and well-defined coefficient and the log-law holds only if E_+ is independent of y_+ and Re_\tau in that range, in which case \kappa ~ \kappa_s. The universality of \kappa_s = B_1^2 / C_s depends on the universality of the stagnation point structure of the turbulence via B_1 and C_s, which are conceivably not universal. (vi) DNS data of turbulent channel flows which include the highest currently available values of Re_\tau suggest E_+ ~ 2/3 B_4 y_+^{-2/15} and dU_+/dy_+ ~ B_4/(\kappa_s) y_+^{-1 - 2/15} with B_4 independent of y in \delta_\nu << y << \delta if the significant departure from - ~ u_\tau^2 is taken into account.

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