Localised and bifurcating structures in planar shear flows

• Main Author: Wang, Zhe
• Other Authors: Claude Guet
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2020
• Subjects: Science::Physics
• Online Access: https://hdl.handle.net/10356/136642
• Institution: Nanyang Technological University

Turbulent spots in planar shear flows are always accompanied by the presence of (i) the quadrupolar flows outside the spot; and (ii) the counter-rotating streamwise vortices inside the spot. In the present thesis, those flow structures, i.e. quadrupolar flows and streamwise vortices have been treated analytically : By exploiting the geometric scale separation inscribed in all planar shear flows, equations governing the large-scale flow surrounding turbulent spots has been derived from the Navier-Stokes equations and solved analytically. Based exclusively on the obtained analytical solutions, the origin of the quadrupolar angular dependence is identified as (i) the presence of the flow shear in the streamwise direction; and (ii) the breaking of the spatial homogeneity in the spanwise direction. The quadrupolar flow decays algebraically with power-law exponent -3 in the far-field and characterised by an exponentially localised reversed flow centred at the turbulent spot. Moreover, the topological origin of the quadrupolar flow is unveiled. By linearising the Navier-Stokes equations about a pair of counter-rotating rigid vortices and upon a Fourier transform in three spatial directions, a three-dimensional dynamical system is obtained. The proposed model corresponds to the Z_2-symmetric pitchfork/Hopf problem, whose linear instability is the centrifugal instability. As long as the rotational symmetry is preserved, the system can have at most a pair of reflectional symmetric invariant two-tori, signifying the quasiperiodicity. With broken rotational symmetry, periodic orbits can experience a Neimark-Sacker bifurcation leading to a torus, a phase locking, a period-doubling cascade, a heteroclinic bifurcation, a coexistence between the quasiperiodic and chaotic behaviours. All of these phenomena occur in a narrow region of parameter space, complicating the assignment of the single route to chaos.