Past Event: Oden Institute Seminar

Nikuradse meets Kolmog'orov, or: How to derive the diagram from the spectrum

Gustavo Gioia, Professor, Mechanical Science and Engineering University of Illinois - Urbana

Abstract

A diagram published in 1933 remains among the weightier contributions to experimental turbulence ever. In that diagram, Nikuradse plotted six log-log curves evincing the dependence on the Reynolds number (Re) of the friction coefficient ($f$) of the turbulent flow in six pipes of fixed roughness. (The roughness of a pipe is the ratio $r/D$, where $r$ is the size of the roughness elements that line the interior of the pipe and $D$ the diameter of the pipe.) Nikuradse's diagram is rich in distinctive features, including a pronounced "hump" where $f$ attains a maximum shortly after the transition to turbulence; a "smooth regime" governed by Blasius's empirical scaling, $f \sim {\rm Re}^{-1/4}$; shallow "bellies" where $f$ attains local minima at intermediate values of Re; and a "rough regime" governed by Strickler's empirical scaling, $f \sim (r/D)^{1/3}$. For seventy years now, our understanding of Nikuradse's diagram has been aided by little beyond a pictorial narrative of roughness elements being progressively exposed to the turbulence as Re increases. In this seminar we identify the eddies that effect most of the momentum transfer between the viscous layer and the turbulent flow, and derive an expression for $f$ in terms of the characteristic velocity of those eddies, $u_s$. Then we use Kolmog\'orov's spectrum for the inertial range to determine $u_s$ and show that the resulting expression for $f$ gives a gradual transition between the scalings of Blasius and Strickler, but fails to give the hump or the bellies of Nikuradse's diagram. To obtain an expression for $f$ that also gives the bellies, we include an exponential spectrum for the dissipation range. Last, to obtain an expression for $f$ that also gives the hump, we include von K\'arm\'an's spectrum for the energy-containing range. This final expression for $f$ is in minute qualitative agreement with Nikuradse's diagram; it affords a way of interpreting successive portions of the diagram as manifestations of the varying habits of momentum transfer; and it reveals the existence of close ties between two milestones of experimental and theoretical turbulence. This research is joint work with Pinaki Chakraborty. NOTE: I will try to explain all that is needed to understand this seminar even if you are an undergraduate student who has not been exposed to turbulence, do not be intimidated!