Empirical scaling laws for wall-bounded turbulence deduced from direct numerical simulations

Statistical quantities in channel (Poiseuille), Couette, and pipe flow, including Reynolds stresses and their budgets, are studied for their dependence on the normalized distance from the wall y(+) and the friction Reynolds number Re-tau. Any quantity Q can be normalized in wall units, based on the friction velocity u(tau). and viscosity nu, and it is accepted that the physics of fully developed turbulence in ducts leads to standard results of the type Q(+) = f (y(+), Re-tau), in which f (. . .) means "only a function of," and f is different in different flow types. Good agreement between experiments and simulations is expected. We are interested in stronger properties for Q, generalized from those long recognized for the velocity, but not based on first principles. These include the law of the wall Q(+) = f (y(+)); the logarithmic law for velocity; and for the Reynolds-number dependence, the possibility that at a given y+ it is proportional to the inverse of Re-tau, that is, Q(+)(y(+), Re-tau) = f(infinity)(y(+)) + f(Re)(y(+))/Re-tau. This has been proposed before, also on an empirical basis, and recent work by Luchini [Phys. Rev. Lett. 118, 224501 (2017)] for Q U is of note. The question of whether the profiles are the same in all three flows, in other words, that there is a single function f(infinity), is still somewhat open. We arrive at different conclusions for different Q quantities. The inverse-Re, dependence is successful in some cases. Its failure for some of the Reynolds stresses can be interpreted physically by invoking "inactive motion," following Townsend [The Structure of Turbulent Shear Flow, 2nd ed. (Cambridge University Press, Cambridge, 1976)] and Bradshaw [J. Fluid Mech. 30, 241 (1967)], but that is difficult to capture with any quantitative theory or turbulence model. The case of the boundary layer is studied, and it is argued that a direct generalization of Re, is questionable, which would prevent a sound extension from the internal flows.