Abstract
The main objective of
the research is to study the evolution of the key structures in turbulent
flows: streaks and hairpins vortices, using experimental, theoretical and
numerical methods. Two canonical flows are considered, plane Poiseuille flow
(PPF) and flow in a pipe.
Streaks are generated
experimentally in a laminar PPF using injection through a streamwise slot. Flow
visualization and velocity measurements (representing mean, instantaneous and
phase-locked data) are carried out using hot wire and PIV techniques. The
measured streaks grow in accordance with the linear mechanism of transient
growth, and the distance of maximum growth (Xmax) is
proportional to Reynolds number (Re). New theoretical considerations
show that many aspects of the transient growth mechanism can be predicted using
just the two least stable modes of laminar PPF. As is observed in experiments,
when streaks undergo secondary instability, they become wavy and localized
disturbances are formed at the crest of these waves, giving rise to hairpin
vortices. The linear stability analysis of the inflectional streak profile
correctly predicts the wavelength and the frequency observed in the
experiments. However, it under-predicts the growth rate of the hairpin
vortices, suggesting a different mechanism for the growth of the hairpins, even
though the secondary instability correctly predicts their initiation.
Experiments show that the amplitude (in this case the injection velocity of the
disturbance, v0) required to produce hairpins varies as
Re-3/2.
The hairpins once produced continue to sustain in the PPF.
A similar process of
streak development (as in PPF) and their secondary instability to produce
hairpin vortices are observed experimentally in pipe flow, by artificial
injection of continuous disturbances. In pipe flow the scaling is, v0
~ Re-1 at the generation point. However, the streaks and
hairpins decay along the downstream distance (X); a variation in the
scaling of v0 Vs
Re along
X for the initiation
of hairpins, confirms this. Measurements of energy in the cross-sectional plane
of the pipe, and maps of disturbance velocity at various
X-locations
show the transient growth and decay of energy. Owing to the increase in the
area of the disturbance influence along the X-direction, the energy can
transiently increase even when the total disturbance magnitude is decreasing.
Finally, localized
hairpin vortices are treated analytically by the temporal evolution of their
associated integral characteristic, the fluid impulse. Closed form expressions
are derived for the evolution of nonlinear, localized disturbance in an
irrotational shear flow, and compared with numerical results.
