Spectrum of an operator chracterising the stability of the pipe flow

• Thesis title in Czech: Spektrum operátoru charakterizujícího stabilitu proudění v trubici
• Thesis title in English: Spectrum of an operator chracterising the stability of the pipe flow
• Key words: stabilita proudění; spektrum diferenciálního operátoru; numerické řešení
• English key words: flow stability; spectrum of differential operator; numerical solution
• Academic year of topic announcement: 2016/2017
• Thesis type: Bachelor's thesis
• Thesis language: angličtina
• Department: Mathematical Institute of Charles University (32-MUUK)
• Supervisor: doc. Mgr. Vít Průša, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 27.10.2016
• Date of assignment: 27.10.2016
• Confirmed by Study dept. on: 10.01.2017
• Date and time of defence: 21.06.2017 00:00
• Date of electronic submission: 17.05.2017
• Date of submission of printed version: 17.05.2017
• Date of proceeded defence: 21.06.2017
• Opponents: prof. RNDr. Josef Málek, CSc., DSc.

Guidelines

The aim of the bachelor thesis is to investigate the relation between the spectrum of the Stokes operator and the spectrum of the Oseen type operator for the pipe flow problem. The exact characterisation of the spectrum of the latter operator is of utmost importance in the theory of stability of fluid flows. Consequently, the first step is to get familiar with the linearised stability theory, see Schmid, Henningson (2001). Second, get familiar with the analytical results concerning the Stokes operator, namely with the exact analytical formulae for the eigenvalues, see Rummler (1997). Third, get familiar with the numerical solution of the eigenvalue problem for the Oseen type operator, see Meseguer, Trefethen (2003). Fourth, compare the spectrum of the Oseen type operator with the spectrum of the Stokes operator. Finally, formulate conjectures concerning the possibility of the approximation of the spectrum of latter operator by the spectrum of the former operator.

References

Schmid, P. J. and D. S. Henningson (2001). Stability and transition in shear flows. Number 142 in Applied Mathematical Sciences. New York: Springer.
Reynolds, O. (1883). An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Proc. R. Soc. Lond. 25, 84–99.
Avila, K., D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof (2011). The onset of turbulence in pipe flow. Science 333 (6039), 192–196.
Rummler, B. (1997). The eigenfunctions of the Stokes operator in special domains. Z. angew. Math. Mech. 77 (8), 619–627.
Meseguer, A. and L. N. Trefethen (2003). Linearized pipe flow to Reynolds number 10 7 . J. Comp. Phys. 186, 178–197.
Yudovich, V. I. (2003). Eleven great problems of mathematical hydrodynamics. Mosc. Math. J. 3 (2), 711–737, 746. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.
Priymak, V. G. (2010). Eigenvalue problem for the Navier–Stokes operator in cylindrical coordinates. Mathematical Models and Computer Simulations 2 (3), 317–333.
Kato, T. (1995). Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin. Reprint of the 1980 edition.
Kerswell, R. (2005). Recent progress in understanding the transition to turbulence in pipe. Nonlinearity 18 (6), 17–44.

Preliminary scope of work

Reynolds v roce 1883 provedl experiment, který podnítil zkoumání stability proudění vůči poruchám a zkoumání přechodu od laminárního k turbulentnímu proudění. Ačkoliv se v daném oboru od roku 1883 mnohé událo, Reynolds původní otázka ohledně stability proudění ve válcové trubici zůstala -- do jisté míry -- nezodpovězena.

Obtížnost problému proudění v trubici spočívá ve skutečnosti, že proudění v trubici je pravděpodobně pro všechny hodnoty Reynoldsova čísla stabilní vůči infinitesimálním poruchám. (V jazyce matematiky to odpovídá vyjádření ohledně polohy spektra jistého lineárního operátoru v komplexní rovině.) Důsledkem je, že popis nestability proudění v trubici je poměrně komplikovaný a je mimo dosah linearizované teorie zaměřené na chování infinitesimálních poruch, viz například Avila (2011). Nové poznatky ovšem nepřinesly odpověď na starou otázku, ta zůstala nerozřešena. Kerswell (2005) uvádí, že "panuje shoda, že proudění v trubici je lineárně stabilní [...] ačkoliv formální důkaz tohoto faktu nám stále uniká." (Všeobecná shoda je založena na velkém množství numerických experimentů.)

Bakalářská práce bude zaměřena na numerické experimenty, které mohou poskytnout vodítko při formulaci hypotéz, které by případně mohly vést k vytouženému důkazu. Cílem je numericky porovnat spektrum zmíněného (složitého) operátoru se spektrem blízkého (jednoduchého) operátoru.

Preliminary scope of work in English

The seminal contribution by Reynolds (1883) started the investigation of the stability of fluid flows with respect to disturbances and the investigation of laminar/turbulent flow transition. Despite many advances in the field, the original problem of the stability of the pipe flow that has been articulated by Reynolds in 1883 still remains, in certain sense, unsolved.

The difficulty of the problem stems from the following fact. Unlike in many systems in fluid mechanics, the pipe flow seems to be stable with respect to infinitesimal disturbances for all values of Reynolds number. (The mathematical reformulation thereof is a statement concerning the location of the spectrum of a corresponding linear operator in the complex plane.) Consequently, the description of the instability mechanism is rather complicated, and is beyond the grasp of infinitesimal (linearised) theory, see for example Avila (2011). But the new advances have not contributed to the solution of the old problem, the old problem is still here. As put by Kerswell (2005), "the consensus now is that the [pipe] flow is linearly stable [...] although a formal proof remains elusive". (The consensus is based on abundant numerical evidence.)

The thesis will be focused on numerical experiments that can be useful in formulating conjectures that would eventually lead to the so much desired formal proof. The idea is to compare the spectrum of the (nasty) operator with the spectrum of a nice "nearby" operator.