## Doctoral Dissertations

Winter 2017

Dissertation

#### Program or Major

Mechanical Engineering

#### Degree Name

Doctor of Philosophy

Joseph C Klewicki

Joseph C Klewicki

Gregory P Chini

#### Abstract

Wall-bounded turbulent flows are ubiquitous in numerous technological applications, and thus much effort has been devoted to investigate their properties. Scaling analyses involving the application multiple-scale approaches are effectively used to explore parameters (Reynolds, Prandtl numbers) dependent scaling behaviors of these flows. The objective of this dissertation research is to firstly extend the analysis of self-similar behaviors on the inertial domain as admitted by the mean dynamics in wall-bounded turbulent flows (WBTF). It then mathematically and physically characterizes the existence of a leading order balance structure in both the kinetic energy and passive scalar transport budgets, and subsequently uses this leading balance structure for scaling purposes.

Recent evidence indicates that, at sufficiently high Reynolds number, a number of the statistical measures of wall-turbulence exhibit self-similar behaviors on an interior inertial domain. Experimental measurements in the Flow Physics Facility at the University of New Hampshire have been acquired, and well-resolved streamwise velocity measurements up to high Reynolds number are used to investigate three measures of self-similarity in turbulent boundary layers, and compare their behaviors with those revealed through analysis of the mean momentum equation. The measures include the Kullback-Leibler divergence (KLD), the logarithmic decrease of even statistical moments, and the so-called diagnostic plot. The findings indicate that the approximately constant KLD profiles and the approximately logarithmic moment profiles follow the same scaling but reside interior to the bounds of the self-similar inertial domain associated with the mean dynamics. Conversely, the bounds of the self-similar region on the diagnostic plot correspond closely to the theoretically estimated bounds.

Multiple-scale analysis involving the consideration of the relative magnitude of terms in the governing equation is applied to kinetic energy budgets for fully developed turbulent flow in pipes and channels, and in the zero-pressure gradient turbulent boundary layer. These analyses are based on available high-quality numerical simulation data. The mean kinetic energy budget is analytically verified to exhibit the same four-layer structure as the mean momentum equation, while the turbulence budget only shows either a two- or three-layer structure depending on channel/pipe versus boundary layer flow. A distinct four-layer structure is observed in position and size for the total kinetic energy budget. Here the width of the third layer, which is located in the inertia domain of the mean dynamics, is mathematically reasoned to scale with $\delta^+-\sqrt{\delta^+}$ at finite Reynolds number.

Like the velocity field, the passive scalar field equation in WBTF can also be quantified in terms of its leading balance structure. Both the mean scalar and scalar variance equations with constant heat generation for fully-developed turbulent channel are explored. A similar four-layer structure is found using the same methodology. Both the Reynolds number and Prandtl number dependent scaling of the layer thickness is empirically quantified with available DNS data and verified through rigorous scaling analysis. The analysis also indicates that the mean scalar equation can be cast into an invariant form that properly reflects the local dominant physical mechanism, which uncovers the governing effect of a small and constant parameter on an underlying scaling layer hierarchy. There exists a linear region in the distribution of the inner-normalized widths of this layer hierarchy. Like the momentum equation, analysis indicates that this region coincides with where the mean scalar profile exhibits a logarithmic increase and leads to a distinct expression for the scalar log law. The scalar variance equation manifests itself like the total kinetic energy budget with a distinctive four-layer structure, in which the third layer size has a special scaling under the effects of both Reynolds number and Prandtl Number. The underlying causes of the difference between the K{\'a}rm{\'a}n constant and the scalar K{\'a}rm{\'a}n constant, i.e., $k_\theta >k$, are also investigated and clarified.

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