Date of Award

Fall 2023

Project Type


Program or Major

Applied Mathematics

Degree Name

Doctor of Philosophy

First Advisor

John Gibson

Second Advisor

Gregory Chini

Third Advisor

Ben Chandran


This thesis focuses on two wall-bounded shear flows, namely, plane Poiseuille flow, where a fluid is pushed between two infinite parallel plates and radially heated Taylor-Couette flow, where a fluid is sheared between two independently rotating cylinders maintained at different temperatures. An overarching theme of this thesis is understanding and exploiting the interplay between symmetries and dynamics.

Chapter 1 presents an overview of the field of transition to turbulence and introduces the dynamical systems viewpoint of turbulence. In particular, we discuss subcritical transition to turbulence, where flows are observed to be turbulent below critical thresholds predicted by linear stability analysis. We also present an outline of the thesis in this chapter.

Chapter 2 is dedicated to the classification of the invariant subspaces of plane Poiseuille flow. The governing Navier-Stokes equations and boundary conditions are equivariant (i.e., symmetries and dynamics commute) under the Lie group $G_{\text{PPF}}$, generated by reflections in the wall--normal and spanwise coordinates, along with continuous translations in the streamwise and spanwise directions. We show that for many combinations of reflections and continuous translations, a periodicity is implied. When a periodic domain of the implied periodicity is considered, translations often reduce to half--box shifts. Instead of considering continuous translations on an infinite domain, we henceforth consider discrete translations on a periodic domain. When half--box shifts in the streamwise and spanwise directions are considered as generators along with reflections in the wall--normal and spanwise coordinates, a subgroup of $G_{\text{PPF}}$ is generated, denoted by $G_h$, with the subscript (`h') standing for half--box shifts. Not all subgroups of $G_h$ are truly independent. When two velocity fields are related to each other by a coordinate transformation, their isotropy groups (groups of transformations that leave the fields unchanged) are shown to be related by conjugation. This ``conjugacy under coordinate transformations'' is shown to be an equivalence relation that partitions the set of subgroups of $\Gppfh$ into different equivalence classes. Different equivalence classes support qualitatively distinct dynamics in accordance with symmetries. We classify all subgroups of $G_h$ into different equivalence classes using a symbolic code in Julia.

Chapter 3 explores dynamics in the invariant subspaces of $G_h$ classified in chapter 2. We report fifteen new traveling wave solutions in seven different invariant subspaces of plane Poiseuille flow. Furthermore, we also show numerical continuations with respect to Reynolds number, a dimensionless number that captures relative strengths of inertial and viscous forces. Some symmetry--breaking bifurcations are located by numerical continuations. To the best of our knowledge, these solutions are found for the first time and add to the current repository of known finite--amplitude solutions.

Chapter 4 is concerned about the ``edge of chaos'' of plane Poiseuille flow, the invariant manifold in the state space that separates initial conditions that have long transient lifetimes from those that laminarize. Traveling wave and relative periodic orbit solutions are found as edge states in different invariant spaces. We study spanwise localization of traveling waves with increasing spanwise widths. We locate codimension--two bifurcations for one of the traveling waves, considering spanwise width and Reynolds number as control parameters. We also discuss some possible unfoldings near codimension--two bifurcation points.

Chapter 5 concentrates on the effect of outer cylinder rotation on the radially heated Taylor-Couette flow.We perform linear stability analysis to determine the Taylor number for the onset of instability. We consider two radius ratios corresponding to wide and thin gaps with several rotation rate ratios. The rotation of the outer cylinder is found to have a general stabilizing effect on the stability threshold as compared to pure inner--cylinder rotation, with a few exceptions. The radial heating sets up an axial flow which breaks the reflection symmetry of isothermal Taylor--Couette flow in the axial coordinate. This symmetry breaking separates linear stability thresholds and we find fastest growing modes with both positive and negative azimuthal numbers for different parameters. Another important finding of this chapter is the discovery of unstable modes in the Rayleigh-stable regime. Furthermore, instability islands, also known as closed disconnected neutral curves (CDNCs) are observed for both wide and thin gaps which can separate from or merge into open neutral stability curves. Alternatively, CDNCs can also morph into open neutral stability curves as the rotation rate ratio is changed. CDNCs are observed to be sensitive to changes in control parameters and their appearance/disappearance is shown to induce discontinuous jumps in the critical Taylor number. For both wide and thin gaps, the fastest--growing modes found in the pure co-rotation case are shown to have their origins in the instability islands at smaller values of rotation rate ratios.

Chapter 6 presents summary, conclusions and possible directions for future work.