Velocity derivatives in a high Reynolds number turbulent boundary layer. Part I: Dissipation and Energy Balance

Abstract

An experiment was performed using SPIV in the LMFL boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy, $\varepsilon = 2 \nu \langle s_{ij}s_{ij} \rangle$ where $s_{ij}$ is the fluctuating strain-rate and $\langle~\rangle$ denotes ensemble averages. Also measured were all the moments of the full average deformation rate tensor, as well as all of the first, second and third fluctuating velocity moments except those involving pressure. The Reynolds number was $Re_\theta = 7500$ or $Re_\tau = 2300$. The results are presented in three separate papers. This first paper (Part I) presents the measured average dissipation, $\varepsilon$ and the derivative moments comprising it. It compares the results to the earlier measurements of \cite{balint91,honkan97} at lower Reynolds numbers and a new results from a plane channel flow DNS at comparable Reynolds number. It then uses the results to extend and evaluate the theoretical predictions of \cite{george97b,wosnik00} for all quantities in the overlap region. Of special interest is the prediction that $\varepsilon^+ \propto {y^+}^{-1}$ for streamwise homogeneous flows and a nearly indistinguishable power law, $\varepsilon \propto {y^+}^{\gamma-1}$, for boundary layers. In spite of the modest Reynolds number, the predictions seem to be correct. It also predicts and confirms that the transport moment contribution to the energy balance in the overlap region, $\partial \langle - pv /\rho - q^2 v/2 \rangle/ \partial y$ behaves similarly. An immediate consequence is that the usual eddy viscosity model for these terms cannot be correct. The second paper, Part II, examines in detail the statistical character of the velocity derivatives. The details of the SPIV methodology is in Part III, since it will primarily be of interest to experimentalists.