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:
• Mellibovsky, F., Meseguer A. (2007) `Pipe flow transition thresholds following localized impulsive perturbations'. Physics of Fluids, (2007).
• Mellibovsky, F., Meseguer, A. (2006) `The role of streawise perturbations in pipe flow transition'. Physics of Fluids 18, 074104, (2006).
• Meseguer, A (2003). `Streak breakdown instability in pipe Poiseuille flow'. Physics of Fluids (2003).
• Meseguer, A. & Trefethen, L. N. (2003) `Linearized pipe flow to Reynolds number 10.000.000' Journal of Computational Physics (2003)

Subcritical transition in Taylor-Couette flow .

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 Mechanics 402, 33-56.

Miscellaneous.

• Cavity flow.

  References

• Meseguer A., Marques F. & Sanchez, J. (1996) `Feigenbaum's Universality in a Low-Dimensional Fluid Model,' International Journal of Bifurcations and Chaos 6, 1587-1594.

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