Nonlinear evolution of two fast-particle-driven modes near the linear stability threshold

Three-dimensional and time-dependent simulations of viscoelastic Taylor–Couette flow of dilute polymer solutions are performed using a fully implicit parallel spectral time-splitting algorithm to discover flow patterns with various spatio-temporal symmetries, namely rotating standing waves (RSWs), disordered oscillations (DOs) and solitary vortex structures referred to as oscillatory strips (OSs) and diwhirls (DWs). A detailed account of the impact of flow transitions on molecular conformation and viscoelastic stress, velocity profiles, hydrodynamic drag force and energy spectra of time-dependent flow states is presented. Overall, predicted pattern selection and flow features compare very favourably with experimental observations. For elasticity number E, that signifies the ratio of elastic to viscous forces, >0.1, and when the shear rate (cylinder rotation speed) is increased above the linear stability threshold, the circular Couette flow (CCF) becomes unstable to RSWs which are characterized by a checkerboard-like pattern in the space–time plot of radial velocity, implying symmetry between inflow/outflow (I/O) regions. As the shear rate is further increased, perturbations that break the I/O symmetry are amplified leading to DOs and/or flame-like patterns with spectral mechanical energy transfer reminiscent of elastically induced low-Reynolds-number turbulence. However, when the shear rate is decreased from those at which such chaotic states are observed, the radially inward acting polymer body force created by flow-induced molecular stretching causes the development of narrow inflow regions surrounded by much broader weak outflow domains. This promotes the formation of solitary vortex structures, which can be stationary and axisymmetric (DWs) or time-dependent (OSs). The dynamics of the formation of these structures by merging and coalescence of vortex pairs and the implication of such events on instantaneous hydrodynamic force are studied. For O(1) values of E, OSs and DWs appear approximately at constant values of the We, defined as the ratio of polymer relaxation time to the inverse shear rate in the gap. As shear rate is decreased further, DWs decay to CCF although at We values less than the linear stability threshold. The flow transitions are hysteretic with respect to We, as evidenced by a plot of drag force versus We.

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Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer

Journal of Fluid Mechanics ◽

10.1017/s0022112010003903 ◽

2010 ◽

Vol 665 ◽

pp. 209-237 ◽

Cited By ~ 17

Author(s):

J. GULA ◽

V. ZEITLIN ◽

F. BOUCHUT

Keyword(s):

Shallow Water ◽

Linear Stability ◽

Nonlinear Evolution ◽

Lower Layer ◽

Short Wave ◽

Finite Volume Scheme ◽

Coastal Currents ◽

Mean Flow ◽

The Mean ◽

Rotating Shallow Water

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linear stability problem for coastal currents and study the nonlinear evolution of the instabilities with the help of high-resolution direct numerical simulations. We show how nonlinear saturation of the ageostrophic instabilities leads to reorganization of the mean flow and emergence of coherent vortices. We follow the same lines as in Part 1 and, first, perform a complete linear stability analysis of the baroclinic coastal currents for various depths and density ratios. We then study the nonlinear evolution of the unstable modes with the help of the recent efficient two-layer generalization of the one-layer well-balanced finite-volume scheme for rotating shallow water equations, which allows the treatment of outcropping and loss of hyperbolicity associated with shear, Kelvin–Helmholtz type, instabilities. The previous single-layer results are recovered in the limit of large depth ratios. For depth ratios of order one, new baroclinic long-wave instabilities come into play due to the resonances among Rossby and frontal- or coastal-trapped waves. These instabilities saturate by forming coherent baroclinic vortices, and lead to a complete reorganization of the initial current. As in Part 1, Kelvin fronts play an important role in this process. For even smaller depth ratios, short-wave shear instabilities with large growth rates rapidly develop. We show that at the nonlinear stage they produce short-wave meanders with enhanced dissipation. However, they do not change, globally, the structure of the mean flow which undergoes secondary large-scale instabilities leading to coherent vortex formation and cutoff.

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Linear stability and nonlinear evolution of a polar vortex cap on a rotating sphere

European Journal of Mechanics - B/Fluids ◽

10.1016/j.euromechflu.2020.09.006 ◽

2021 ◽

Vol 85 ◽

pp. 102-109

Author(s):

Sun-Chul Kim ◽

Sung-Ik Sohn

Keyword(s):

Linear Stability ◽

Nonlinear Evolution ◽

Polar Vortex ◽

Rotating Sphere

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Destructive interactions between two counter-rotating quasi-geostrophic vortices

Journal of Fluid Mechanics ◽

10.1017/s0022112009990954 ◽

2009 ◽

Vol 639 ◽

pp. 195-211 ◽

Cited By ~ 9

Author(s):

JEAN N. REINAUD ◽

DAVID G. DRITSCHEL

Keyword(s):

Linear Stability ◽

Tilt Angle ◽

Nonlinear Evolution ◽

Volume Ratio ◽

Separation Distance ◽

Margin Of Stability ◽

Critical Gap

This paper illustrates the linear stability and the nonlinear evolution of two opposite-signed quasi-geostrophic vortices. We investigate the influence of the volume ratio between the two vortices as well as the influence of their vertical offset. Instability is always found for sufficiently close vortices. A convenient measure of the separation distance between the two vortices at their margin of stability is the horizontal gap between their two outermost edges. When the vortex volume ratio is very close to unity, the critical gap at the margin of stability tends to increase with the vertical offset. However, for volume ratios greater than 1.1, it decreases with the vertical offset. This is due to differences in the magnitude of the tilt angle of the vortices. The nonlinear evolution of unstable equilibria tends to be destructive, often breaking one vortex or both vortices into smaller vortices.

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Electrohydrodynamic linear stability analysis of dielectric liquids subjected to unipolar injection in a rectangular enclosure with rigid sidewalls

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.537 ◽

2014 ◽

Vol 758 ◽

pp. 586-602 ◽

Cited By ~ 15

Author(s):

A. T. Pérez ◽

P. A. Vázquez ◽

Jian Wu ◽

P. Traoré

Keyword(s):

Aspect Ratio ◽

Linear Stability ◽

Element Formulation ◽

Nonlinear Behaviour ◽

Algebraic Equations ◽

Stability Threshold ◽

Rectangular Enclosure ◽

Linear Partial Differential Equations ◽

Symmetry Properties ◽

Unipolar Injection

AbstractWe investigate the linear stability threshold of a dielectric liquid subjected to unipolar injection in a two-dimensional rectangular enclosure with rigid boundaries. A finite element formulation transforms the set of linear partial differential equations that governs the system into a set of algebraic equations. The resulting system poses an eigenvalue problem. We calculate the linear stability threshold, as well as the velocity field and charge density distribution, as a function of the aspect ratio of the domain. The stability parameter as a function of the aspect ratio describes paths of symmetry-breaking bifurcation. The symmetry properties of the different linear modes determine whether these paths cross each other or not. The resulting structure has important consequences in the nonlinear behaviour of the system after the bifurcation points.

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Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.134 ◽

2015 ◽

Vol 770 ◽

pp. 319-349 ◽

Cited By ~ 17

Author(s):

Mengqi Zhang ◽

Fulvio Martinelli ◽

Jian Wu ◽

Peter J. Schmid ◽

Maurizio Quadrio

Keyword(s):

Stability Analysis ◽

Electric Field ◽

Linear Stability ◽

Poiseuille Flow ◽

Cross Flow ◽

Transient Growth ◽

Unstable Flow ◽

Electrohydrodynamic Flow ◽

Stability Threshold ◽

Energy Growth

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

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Linear Vibrations of Statically Indeterminate Rotors on Angular Misaligned Hydrodynamic Bearings

Volume 4: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; Education; IGTI Scholar Award; General ◽

10.1115/99-gt-333 ◽

1999 ◽

Cited By ~ 2

Author(s):

Demetrio C. Zachariadis

Keyword(s):

Finite Elements ◽

Linear Stability ◽

Design Stage ◽

Angular Misalignment ◽

Hydrodynamic Bearings ◽

Early Design ◽

Rotor Systems ◽

Early Design Stage ◽

Stability Threshold ◽

Linear Vibrations

The influence of journal’s static and dynamic angular misalignment on the synchronous unbalance response and linear stability threshold of statically indeterminate rotors is analysed. Both short and finite bearing models are considered in order to calculate hydrodynamic reactions, and rotating parts are modeled using beam finite elements. The results provide descriptions of the effects of the consideration of the 32 coefficient bearing model on linear vibrations analyses, together with guidelines for early design stage identification of rotor systems sensitive to angular misalignments.

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Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2018.15 ◽

2018 ◽

Vol 840 ◽

pp. 5-24 ◽

Cited By ~ 6

Author(s):

Junho Park ◽

Paul Billant ◽

Jong-Jin Baik ◽

Jaemyeong Mango Seo

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Couette Flow ◽

Linear Stability Analysis ◽

Reynolds Numbers ◽

Axisymmetric Mode ◽

Stability Threshold ◽

Taylor Couette ◽

The Stability ◽

Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

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