Hydrodynamic Stability and transition to turbulence
Alvaro MESEGUER
Hydrodynamic Stability of Hagen-Poiseuille (pipe) flow.
Transition to turbulence in a straight pipe of circular cross-section
still remains an unexplained phenomenon in fundamental fluid
dynamics. Since the work of Osborne Reynolds in 1883, many engineers,
physicists and also mathematicians have tried to provide an answer to
the following questions:
Since the parabolic Hagen-Poiseuille flow is linearly stable,
what is the minimum amplitude of a perturbation required to trigger transition?
Is pipe turbulence just a transient phenomenon?.
Or:
Are there any other attractors apart from the basic solution?
What role do the recently found finite amplitude families of
Travelling Wave solutions (found independently by Holger Faisst, Bruno
Eckhardt, Hakan Wedin and Rich Kerswell) play in the transition process?
Even I do not provide any answer to the questions above,
the studies below (and their references therein) may shed some light
on this problem:
Since G.I. Taylor published his famous paper in 1923, the
Taylor-Couette problem (study of the behaviour of an incompressible
viscous fluid contained between two independently rotating concentric
cylinders) has been one of the most studied problems of fluid
dynamics. We could see Taylor-Couette problem as the Hydrogen Atom
of fluid dynamics, where all the mathematical formulation
provided by Dynamical Systems / Bifurcation Theory applies nicely. The
ammount of literature regarding this topic is huge, from experimental
studies to numerical simulations. A lot of explorations have been
carried out and many secondary supercritical laminar flows (steady,
time periodic or almost periodic) have been identified numerically and
experimentally.
Similarly to what happens in the pipe problem described above, the
circular Couette flow may exhibit subcritical transition to turbulence
in the absence of linear instabilities or local bifurcations. This
phenomenon was originally reported by Coles and Van Atta in the mid
1960's. The two works below study in detail the transient growth of
perturbations in this problem and its possible implications in the
aforementioned subcritical transition observed by Coles:
Meseguer, A. (2002) `On
nonnormal effects in the Taylor-Couette problem', Theoretical and
Computational Fluid Dynamics, 16, 71-77, doi:
10.1007/s00162-002-0067-8. .
Meseguer, A. (2002)
`Energy transient growth in the Taylor Couette problem,'
Physics of Fluids, 14 (5).
Spectral solenoidal Petrov-Galerkin schemes for incompressible Navier-Stokes
equations.
There are many methodologies of approximation of the Navier-Stokes
equations. The most used by engineers are finite differences/elements
because they are easy to implement and parallelize. Spectral methods
are well known for their unbeatable accuracy but also for their
difficult formulation. Things become a bit more complicated when
working with polar coordinates, where Navier-Stokes equations lead to
an apparent singularity at the origin. Besides, the incompressibility
condition and the pressure term increase the complexity of the
spectral formulation. In these two reports we provide a spectral
method to solve Navier-Stokes equations in polar coordinates, with
suitable regularity conditions at the polar axis, incompressibility
identically satisfied, and pressure term elegantly eliminated from the
formulation:
Meseguer, A., Trefethen, L. N. `A Spectral Petrov-Galerkin
formulation for pipe flow II: Nonlinear transitional stages ',
Oxford University Computing Laboratory, Tech. Rep. 01/19
(2001).
Meseguer, A. and Trefethen, L. N., `A spectral Petrov-Galerkin
formulation for pipe flow I: Linear stability and transient growth',
Oxford University Computing Laboratory, Tech. Rep. 00/18
(2000).
Stability of Spiral Poiseuille and Spiral Couette flows.
Spiral-Couette and Spiral-Poiseuille are a generalization of the
Taylor-Couette problem described above. Apart from the rotation
induced by the cylinders, the fluid is also enforced to advect
downstream by means of either a uniform axial pressure gradient
(Spiral Poiseuille Flow) or an inertial sliding of one of the two
cylinders in the axial direction (Spiral Couette Flow). In both
problems, centrifugal and shear mechanisms of instability compete. In
these works we provided a comprehensive linear stability analysis of
both problems:
Meseguer, A. & Marques, F. (2002)
`On the competition between centrifugal and shear instability in
spiral Poiseuille flow,'
Journal of Fluid Mechanics, 455, 129-148.
Meseguer, A. & Marques, F. (2000)
`Axial Effects in the Taylor-Couette Problem: Spiral-Couette and
Spiral-Poiseuille flows,' in Physics of Rotating Fluids.
Ed. C. Egbers and G. Pfister. Lecture Notes in Physics,
549, 118-136. Springer-Verlag.
Meseguer A.& Marques F. (2000)
`On the competition between centrifugal and shear instability in
spiral Couette flow,'
Journal of Fluid Mechanics402, 33-56.