scholarly journals Nonlinear optimal perturbations in a Couette flow: bursting and transition

2013 ◽  
Vol 716 ◽  
pp. 251-279 ◽  
Author(s):  
S. Cherubini ◽  
P. De Palma
Keyword(s):  
Couette Flow ◽  
Short Path ◽  
Basic Structure ◽  
Initial Energy ◽  
Transition To Turbulence ◽  
Streamwise Vortices ◽  
Target Time ◽  
Bursting Events ◽  
Optimal Disturbances ◽  
Transition Scenario

AbstractThis paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed of inclined streamwise vortices along localized regions of low and high momentum. This basic structure closely recalls that found in boundary-layer flow (Cherubini et al., J. Fluid Mech., vol. 689, 2011, pp. 221–253), indicating that this structure may be considered the most ‘energetic’ one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, direct numerical simulations have shown that optimal disturbances obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence, skipping the phases of streak formation and secondary instability which are typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the nonlinear optimal disturbances: (i) the large initial value of the streamwise velocity component; (ii) the streamwise dependence of the disturbance; (iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the edge state (Schneider et al., Phys. Rev. E, vol. 78, 2008, 037301), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van Veen & Kawahara (Phys. Rev. Lett., vol. 107, 2011, p. 114501), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.

scholarly journals Minimal-energy perturbations rapidly approaching the edge state in Couette flow

2015 ◽  
Vol 764 ◽  
pp. 572-598 ◽  
Author(s):  
S. Cherubini ◽  
P. De Palma
Keyword(s):  
State Space ◽  
Couette Flow ◽  
Edge State ◽  
The State ◽  
Transition To Turbulence ◽  
Minimal Distance ◽  
Minimal Energy ◽  
Edge Of Chaos ◽  
Optimal Disturbances ◽  
Laminar State

AbstractTransition to turbulence in shear flows is often subcritical, thus the dynamics of the flow strongly depends on the shape and amplitude of the perturbation of the laminar state. In the state space, initial perturbations which directly relaminarize are separated from those that go through a chaotic trajectory by a hypersurface having a very small number of unstable dimensions, known as the edge of chaos. Even for the simple case of plane Couette flow in a small domain, the edge of chaos is characterized by a fractal, folded structure. Thus, the problem of determining the threshold energy to trigger subcritical transition consists in finding the states on this complex hypersurface with minimal distance (in the energy norm) from the laminar state. In this work we have investigated the minimal-energy regions of the edge of chaos, by developing a minimization method looking for the minimal-energy perturbations capable of approaching the edge state (within a prescribed tolerance) in a finite target time $T$. For sufficiently small target times, the value of the minimal energy has been found to vary with $T$ following a power law, whose best fit is given by $E_{min}\propto T^{-1.75}$. For large values of $T$, the minimal energy achieves a constant value which corresponds to the energy of the minimal seed, namely the perturbation of minimal energy asymptotically approaching the edge state (Rabin et al., J. Fluid Mech., vol. 738, 2012, R1). For $T\geqslant 40$, all of the symmetries of the edge state are broken and the minimal perturbation appears to be localized in space with a basic structure composed of scattered patches of streamwise velocity with inclined streamwise vortices on their flanks. Finally, we have found that minimal perturbations originate in a small low-energy zone of the state space and follow very fast similar trajectories towards the edge state. Such trajectories are very different from those of linear optimal disturbances, which need much higher initial amplitudes to approach the edge state. The time evolution of these minimal perturbations represents the most efficient path to subcritical transition for Couette flow.

Dynamical systems and the transition to turbulence in linearly stable shear flows

2007 ◽  
Vol 366 (1868) ◽  
pp. 1297-1315 ◽  
Author(s):  
Bruno Eckhardt ◽  
Holger Faisst ◽  
Armin Schmiegel ◽  
Tobias M Schneider
Keyword(s):  
Couette Flow ◽  
Coherent Structures ◽  
Pipe Flow ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Transition To Turbulence ◽  
Plane Couette Flow ◽  
Numerical Studies ◽  
Lifetime Studies ◽  
Transition Scenario

Plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures, which transiently show up in the temporal evolution of the turbulent flow. Here we summarize the evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow and on the coherent structures in pipe flow.

scholarly journals Large-scale structures in stratified turbulent Couette flow and optimal disturbances

2019 ◽  
pp. 1-31
Author(s):  
Grigory Vladimirovich Zasko ◽  
Andrey Vasilyevich Glazunov ◽  
Evgeny Valeryevich Mortikov ◽  
Yuri Mikhailovich Nechepurenko
Keyword(s):  
Couette Flow ◽  
Large Scale ◽  
Large Scale Structures ◽  
Scale Structures ◽  
Optimal Disturbances

Optimal energy growth in stably stratified turbulent Couette flow

2021 ◽  
Author(s):  
Grigory Zasko ◽  
Andrey Glazunov ◽  
Evgeny Mortikov ◽  
Yuri Nechepurenko ◽  
Pavel Perezhogin
Keyword(s):  
Boundary Layer ◽  
Turbulent Flow ◽  
Couette Flow ◽  
Large Scale ◽  
Thin Layers ◽  
Plane Couette Flow ◽  
Turbulent Layer ◽  
Wide Range ◽  
The Mean ◽  
Optimal Disturbances

<p>In this report, we will try to explain the emergence of large-scale organized structures in stably stratified turbulent flows using optimal disturbances of the mean turbulent flow. These structures have been recently obtained in numerical simulations of turbulent stably stratified flows [1] (Ekman layer, LES) and [2] (plane Couette flow, DNS and LES) and indirectly confirmed by field measurements in the stable boundary layer of the atmosphere [1, 2]. In instantaneous temperature fields they manifest themselves as irregular inclined thin layers with large gradients (fronts), spaced from each other by distances comparable to the height of the entire turbulent layer, and separated by regions with weak stratification.</p><p>Optimal disturbances of a stably stratified turbulent plane Couette flow are investigated in a wide range of Reynolds and Richardson numbers. These disturbances were computed based on a simplified linearized system of equations in which turbulent Reynolds stresses and heat fluxes were approximated by isotropic viscosity and diffusion with coefficients obtained from DNS results. It was shown [3] that the spatial scales and configurations of the inclined structures extracted from DNS data coincide with the ones obtained from optimal disturbances of the mean turbulent flow.</p><p>Critical value of the stability parameter is found starting from which the optimal disturbances resemble inclined structures. The physical mechanisms that determine the evolution, energetics and spatial configuration of these optimal disturbances are discussed. The effects due to the presence of stable stratification are highlighted.</p><p>Numerical experiments with optimal disturbances were supported by the RSF (grant No. 17-71-20149). Direct numerical simulation of stratified turbulent Couette flow was supported by the RFBR (grant No. 20-05-00776).</p><p>References:</p><p>[1] P.P. Sullivan, J.C. Weil, E.G. Patton, H.J. Jonker, D.V. Mironov. Turbulent winds and temperature fronts in large-eddy simulations of the stable atmospheric boundary layer // J. Atmos. Sci., 2016, V. 73, P. 1815-1840.</p><p>[2] A.V. Glazunov, E.V. Mortikov, K.V. Barskov, E.V. Kadantsev, S.S. Zilitinkevich. Layered structure of stably stratified turbulent shear flows // Izv. Atmos. Ocean. Phys., 2019, V. 55, P. 312–323.</p><p>[3] G.V. Zasko, A.V. Glazunov, E.V. Mortikov, Yu.M. Nechepurenko. Large-scale structures in stratified turbulent Couette flow and optimal disturbances // Russ. J. Num. Anal. Math. Model., 2010, V. 35, P. 35–53.</p>

scholarly journals A novel subcritical transition to turbulence in Taylor–Couette flow with counter-rotating cylinders

2020 ◽  
Vol 892 ◽  
Author(s):  
Christopher J. Crowley ◽  
Michael C. Krygier ◽  
Daniel Borrero-Echeverry ◽  
Roman O. Grigoriev ◽  
Michael F. Schatz
Keyword(s):  
Couette Flow ◽  
Transition To Turbulence ◽  
Rotating Cylinders ◽  
Taylor Couette ◽  
Image Position ◽  
Taylor Couette Flow

On the secondary instabilities of transient growth in Couette flow

2017 ◽  
Vol 813 ◽  
pp. 528-557 ◽  
Author(s):  
Michael Karp ◽  
Jacob Cohen
Keyword(s):  
Couette Flow ◽  
Initial Energy ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Transient Growth ◽  
Vortical Structures ◽  
Inflection Points ◽  
Theoretical Predictions ◽  
Secondary Instabilities ◽  
Good Agreement

The secondary instability of linear transient growth (TG) in Couette flow is explored theoretically, utilizing an analytical representation of the TG based on four modes and their nonlinear interactions. The evolution of the secondary disturbance is derived using the multiple time scales method. The theoretical predictions are compared with direct numerical simulations and very good agreement with respect to the growth of the disturbance energy and associated vortical structures is observed, up to the final stage just before the breakdown to turbulence. The theoretical model enables us to perform a full parametric study, including TG symmetry type, various wavenumbers, initial energy, TG nonlinearity and Reynolds number, to find all possible routes to transition and the optimal parameters for each type of the secondary disturbance. It is found that the most dangerous secondary disturbances are associated with spanwise wavenumbers which generate the strongest inflection points, i.e. those having maximal shear, rather than with those maximizing the energy gain during the TG phase.

Direct numerical simulations of roughness-induced transition in supersonic boundary layers

2012 ◽  
Vol 693 ◽  
pp. 28-56 ◽  
Author(s):  
Suman Muppidi ◽  
Krishnan Mahesh
Keyword(s):  
Numerical Simulations ◽  
Shear Layer ◽  
Boundary Layers ◽  
Plate Boundary ◽  
Direct Numerical Simulations ◽  
Transition To Turbulence ◽  
Streamwise Vortices ◽  
Mean Flow ◽  
The Mean ◽  
Near Wall

AbstractDirect numerical simulations are used to study the laminar to turbulent transition of a Mach 2.9 supersonic flat plate boundary layer flow due to distributed surface roughness. Roughness causes the near-wall fluid to slow down and generates a strong shear layer over the roughness elements. Examination of the mean wall pressure indicates that the roughness surface exerts an upward impulse on the fluid, generating counter-rotating pairs of streamwise vortices underneath the shear layer. These vortices transport near-wall low-momentum fluid away from the wall. Along the roughness region, the vortices grow stronger, longer and closer to each other, and result in periodic shedding. The vortices rise towards the shear layer as they advect downstream, and the resulting interaction causes the shear layer to break up, followed quickly by a transition to turbulence. The mean flow in the turbulent region shows a good agreement with available data for fully developed turbulent boundary layers. Simulations under varying conditions show that, where the shear is not as strong and the streamwise vortices are not as coherent, the flow remains laminar.

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