 ## Doctoral Dissertations

Summer 2019

Dissertation

#### Program or Major

Mechanical Engineering

#### Degree Name

Doctor of Philosophy

Christopher M White

Joseph C Klewicki

Gregory P Chini

#### Abstract

\setlength{\baselineskip}{1.5\baselineskip} {Recent studies reveal that at large friction Reynolds number $\delta^+ = u_{\tau}\delta/\nu$ the inertially dominated region of the turbulent boundary layer is composed of large-scale zones of nearly uniform momentum segregated by narrow fissures of concentrated vorticity. Here $u_{\tau} = \sqrt{\tau_{\omega}/\rho}$, where $\tau_{\omega}$ is the shear stress at the wall and $\rho$ the fluid density, $\delta$ is the boundary layer thickness and $\nu$ is the kinematic viscosity of the fluid. The aim of this thesis is better understand the role of this binary structure with respect to the wall-normal transport of momentum and heat in turbulent boundary layers at high $\delta^+$. This is addressed by assuming that the dynamically important processes governing turbulent transport are owed to the interactions between the vorticity $\omega$ field, which quantifies the level of fluid rotation, and the wall-normal velocity $v$. Effectively, it is assumed that turbulent transport is a consequence of the wall-normal motions of concentrated zones of vorticity (or heat). The basis of this assumption is evidenced by the following relation

\begin{equation}\label{eq:re_stress_abs}

-\frac{\partial \overline{u v}}{\partial y} \cong \overline{v \omega_{z}}-\overline{w \omega_{y}},

\end{equation}

where $u$, $v$ and $w$ denote the streamwise, wall-normal and spanwise fluctuating velocity respectively, the subscript on $\omega$ denotes the component of the fluctuating vorticity, and an overbar denotes a correlation. The left-hand side of the equation is the Reynolds stress gradient (responsible for turbulent transport) and the right-hand side of the equation are the velocity-vorticity correlations.

The present research is divided into an experimental and numerical study of the $\overline{v \omega_{z}}$ correlation. In the experimental study, the contributions of the $v$ and $\omega_z$ motions to the vorticity transport ($\overline{v\omega_z}$) mechanisms are evaluated at large friction

Reynolds numbers $\delta^+$. Here the primary contributions to $v$ and $\omega_z$ are estimated by identifying the peak wavelengths of their streamwise spectra. The magnitudes of these peaks are of the same order, and are shown to exhibit a weak $\delta^+$ dependence. The peak wavelengths of $v$, however, exhibits a strong wall-distance ($y$) dependence, while the peak wavelengths of $\omega_z$ shows only a weak $y$ dependence, and remains almost $O(\sqrt{\delta^+})$ in size throughout the inertial domain.

In the numerical study, a simple model that exploits the binary structure of the turbulent boundary layer, i.e., uniform momentum zones (UMZ) separated by vortical fissures (VFs), is developed. First, a master wall-normal profile of streamwise velocity is constructed by placing a discrete number of fissures across the boundary layer. The number of fissures and their wall-normal locations follow scalings informed by analysis of the mean momentum equation. The fissures are then randomly displaced in the wall-normal direction, exchanging momentum as they move, to create an instantaneous velocity profile. This process is repeated to generate ensembles of streamwise velocity profiles from which statistical moments are computed. The modelled statistical profiles are shown to agree remarkably well with those acquired from direct numerical simulations of turbulent channel flow at large . In particular, the model robustly reproduces the empirically observed sub-Gaussian behaviour for the skewness and

kurtosis profiles over a large range of input parameters. Encouraged by the success of this simple model with respect to momentum transport, a similar model is developed with respect to the wall-normal transport of a passive scalar (i.e, temperature). Similarly, this model robustly reproduces the statistical moments of the scalar field and the fluctuating streamwise velocity-temperature correlation.}

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