A Fourier-Chebyshev Petrov-Galerkin spectral method is described for high-accuracy computation of linearized dynamics for ow in an in nite circular pipe. Our code is unusual in being based on solenoidal velocity variables and in being written in MATLAB. Systematic studies are presented of the dependence of eigenvalues, transient growth factors, and other quantities on the axial and azimuthal wave numbers and the Reynolds number R for R ranging from 10 2 to the idealized (physically unrealizable) value 10 7. Implications for transition to turbulence are considered in the light of recent theoretical results of S. J. Chapman.
Direct numerical simulation of transitional pipe flow is carried out in a long computational domain in order to characterize the dynamics within the saddle region of phase space that separates laminar flow from turbulent intermittency. For Reynolds numbers ranging from Re=1800 to 2800, a shoot and bisection method is used to compute critical trajectories. The chaotic saddle or edge state approached by these trajectories is studied in detail. For Re< or =2000 the edge state and the corresponding intermittent puff are shown to share similar averaged global properties. For Re> or =2200, the puff length grows unboundedly whereas the edge state varies only little with Re. In this regime, transition is shown to proceed in two steps: first the energy grows to produce a localized turbulent patch, which then, during the second stage, spreads out to fill the pipe.
This work is devoted to the study of transient growth of perturbations in the Taylor-Couette problem due to linear mechanisms. The study is carried out for a particular small gap case and is mostly focused on the linearly stable regime of counter-rotation. The exploration covers a wide range of inner and outer angular speeds as well as axial and azimuthal modes. Significant transient growth is found in the regime of stable counter-rotation. The numerical results are in agreement with former analyses based on energy methods. Similarities with transient growth mechanisms in plane Couette flow and in Hagen-Poiseuille flow are discussed. This is reflected in the modulation of the basic circular Couette flow by the presence of azimuthal streaks as a result of the nonmodal growth of initial axisymmetric perturbations. This study might shed some light on the subcritical transition to turbulence which is found experimentally in Taylor-Couette flow when the cylinders rotate in opposite directions.
Alternating laminar and turbulent helical bands appearing in shear flows between counterrotating cylinders are accurately computed and the near-wall instability phenomena responsible for their generation identified for the first time. The computations show that this intermittent regime can only exist within large domains and that its spiral coherence is not dictated by endwall boundary conditions. A supercritical transition route, consisting of a progressive helical alignment of localised turbulent spots, is carefully studied. Subcritical routes disconnected from secondary laminar flows have also been identified. 47.20.Lz, 47.27.Cn A comprehensive understanding of turbulent phenomena necessarily requires a previous explanation of the mechanisms that mediate between laminar and fully disordered fluid motion. One of the most challenging shear flow problems is the understanding of laminar-turbulent coexistence phenomena or intermittency, i.e., spatio-temporal coexistence between laminar and turbulent regions in a fluid flow. Canonical shear flows such as plane Couette flow between inertially countersliding parallel plates or pipe flow in a very long straight pipe of circular cross section exhibit localised turbulence as a prelude to fully developed turbulent flow [1][2][3][4][5][6]. Open shear flows share many common drawbacks when studying the long term behaviour of turbulent or intermittent regimes, since localised turbulent spots are often advected downstream and leave the domain. Computation of these flows usually assumes streamwise periodicity, overlooking the real boundary conditions at the entrance and exit of the domains and potentially leading to artificial interaction of the leading and trailing edges of localised turbulent spots. A naturally streamwise-periodic problem such as the Taylor-Couette system between independently rotating concentric cylinders solves these difficulties. Furthermore, while transition in open shear flows is typically subcritical, i.e., bypassing linear stability, Taylor-Couette flow exhibits a huge variety of secondary supercritical steady, time periodic, or almost periodic laminar flows before an eventual transition to chaotic regimes [7]. This enables to study transition in a supercritical setting, along with degeneration into subcriticality. We refer the reader to standard monographs and references therein [8,9].Laminar-turbulent coexistence in Taylor-Couette flow was originally reported by Coles and Van Atta in the 1960s [10,11]. They observed interlaced laminar-turbulent helical patterns (see Fig. 1a) so called spiral turbulence or barber pole turbulence, according to Feynman [12]. This pattern has been studied experimentally by many authors later in the 1980s [7,13] and during the current decade [1,14]. Spiral turbulence, henceforth termed as SPT, may exhibit hysteretic subcritical behaviour, being sustained even in situations where linear theory predicts stability of the base laminar flow. Linear non-modal analysis has shown a strong correlation between the hysteretic ...
A numerical exploration of the linear stability of a fluid confined between two coaxial cylinders rotating independently and with an imposed axial pressure gradient (spiral Poiseuille flow) is presented. The investigation covers a wide range of experimental parameters, being focused on co-rotation situations. The exploration is made for a wide gap case in order to compare the numerical results with previous experimental data available. The competition between shear and centrifugal instability mechanisms affects the topological features of the neutral stability curves and the critical surface is observed to exhibit zeroth-order discontinuities. These curves may exhibit disconnected branches which lower the critical values of instability considerably. The same phenomenon has been reported in similar fluid flows where shear and centrifugal instability mechanisms compete. The stability analysis of the rigid-body rotation case is studied in detail and the asymptotic critical values are found to be qualitatively similar to those obtained in rotating Hagen-Poiseuille and spiral Couette flows. The results are in good agreement with the previous experimental explorations.
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