The stagnation point von K\'arm\'an coefficient

Abstract

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.