## Turbulent wake flows: lie group analysis and conservation laws

##### Date

2016

##### Authors

Hutchinson, Ashleigh Jane

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##### Abstract

We investigate the two-dimensional turbulent wake and derive the governing equations
for the mean velocity components using both the eddy viscosity and the Prandtl
mixing length closure models to complete the system of equations. Prandtl’s mixing
length model is a special case of the eddy viscosity closure model. We consider an
eddy viscosity as a function of the distance along the wake, the perpendicular distance
from the axis of the wake and the mean velocity gradient perpendicular to the
axis of thewake. We calculate the conservation laws for the system of equations using
both closure models. Three main types of wakes arise from this study: the classical
wake, the wake of a self-propelled body and a new wake is discovered which we call
the combination wake. For the classical wake, we first consider the case where the
eddy viscosity depends solely on the distance along the wake. We then relax this condition
to include the dependence of the eddy viscosity on the perpendicular distance
from the axis of the wake. The Lie point symmetry associated with the elementary
conserved vector is used to generate the invariant solution. The profiles of the mean
velocity show that the role of the eddy viscosity is to increase the effective width of
the wake and decrease the magnitude of the maximum mean velocity deficit. An infinite
wake boundary is predicted fromthis model. We then consider the application
of Prandtl’s mixing length closure model to the classical wake. Previous applications
of Prandtl’s mixing length model to turbulent wake flows, which neglected the kinematic
viscosity of the fluid, have underestimated the width of the boundary layer. In
this model, a finite wake boundary is predicted. We propose a revised Prandtl mixing
length model by including the kinematic viscosity of the fluid. We show that this
model predicts a boundary that lies outside the one predicted by Prandtl. We also
prove that the results for the two models converge for very large Reynolds number
wake flows. We also investigate the turbulentwake of a self-propelled body. The eddy
viscosity closure model is used to complete the system of equations. The Lie point
symmetry associated with the conserved vector is derived in order to generate the
invariant solution. We consider the cases where the eddy viscosity depends only on
the distance along the wake in the formof a power law and when a modified version
of Prandtl’s hypothesis is satisfied. We examine the effect of neglecting the kinematic
viscosity. We then discuss the issues that arisewhenwe consider the eddy viscosity to
also depend on the perpendicular distance from the axis of the wake. Mean velocity
profiles reveal that the eddy viscosity increases the boundary layer thickness of the
wake and decreases the magnitude of the maximum mean velocity. An infinite wake
boundary is predicted for this model. Lastly, we revisit the discovery of the combination
wake. We show that for an eddy viscosity depending on only the distance along
the axis of the wake, a mathematical relationship exists between the classical wake,
the wake of a self-propelled body and the combination wake. We explain how the
solutions for the combination wake and the wake of a self-propelled body can be
generated directly from the solution to the classical wake.

##### Description

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. March 2016.