Columnar State Research Articles

Linear global stability analysis of a transient swirling flow leading to the formation of a helical precessing vortex breakdown in a straight diffuser

Vortex breakdown occurs in highly swirling flows in various engineering applications, such as hydraulic turbines working in either high-load or part-load conditions, leading to pressure pulsations and performance degradation. In the present study, the formation of a helical vortex breakdown having rotational and plunging components (axial oscillation) in a straight diffuser is investigated with transient simulation from its columnar state to a helical vortex breakdown state. It is similar to the helical precessing vortex rope occurring in hydraulic turbines operating at part-load condition. In addition, considering vortex breakdown as a global instability, a linear global stability analysis of the time-averaged turbulent flow field is conducted for different flow states. The main objective is to understand the role of plunging mode in vortex rope formation in straight draft tubes and identify its differences with the one observed in hydraulic turbines with elbow draft tubes operating at part-load condition. Results showed that in contrast to the elbow draft tubes where the plunging mode appears earlier than the rotating mode, the transient simulation showed that the plunging mode appears after several vortex rope rotations in the straight draft tube. This is confirmed by the stability analysis showing that the flow is unstable to the asymmetrical disturbances with a frequency close to the rotating frequency not the plunging ones. Finally, the stability analysis showed a higher critical flow rate than the one obtained from the contour of local swirl number and time history of pressure and velocity.

Columnar Liquid Crystalline Glasses by Combining Configurational Flexibility with Moderate Deviation from Planarity: Extended Triaryltriazines.

Triply phenanthryl- and tetrahelicenyl-substituted triazine-hexaalkyl esters with short alkyl chains show glass transitions conveniently above room temperature within the hexagonal columnar liquid crystalline state, resulting in a solid columnar order at room temperature. As the hexagonal columnar mesophase is easily aligned with the director perpendicular to a solid substrate, such glassy columnar liquid matrices are aimed at for the orientation of guest emitters, to obtain anisotropic emission. A condition for face-on alignment on substrates are attainable melting and clearing temperatures, which is achieved with the moderately nonplanar tetrahelicenyl derivatives in spite of their short alkyl periphery. An unusual phase transition between two columnar mesophases of same hexagonal symmetry, but very different long-distance regularity of the column lattice, is found in one phenanthryl homolog.

Direct 129Xe-NMR spectroscopy evidence of a mesogenic dendrimer with free void space

A mesogenic dendrimer with three-fold symmetry was determined, using 129 Xe-NMR spectroscopy, to contain free void space in the columnar or solid state.

Delayed Fluorescence by Triplet-Triplet Annihilation from Columnar Liquid Crystal Films.

Delayed fluorescence (DF) by triplet–triplet annihilation (TTA) is observed in solutions of a benzoperylene-imidoester mesogen that shows a hexagonal columnar mesophase at room temperature in the neat state. A similar benzoperylene-imide with a slightly smaller HOMO–LUMO gap, that also is hexagonal columnar liquid crystalline at room temperature, does not show DF in solution, and mixtures of the two mesogens show no DF in solution either, because of collisional quenching of the excited triplet states on the imidoester by the imide. In contrast, DF by TTA from the imide but not from the imidoester is observed in condensed films of such mixtures, even though neat films of either single material are not displaying DF. In contrast to the DF from the monomeric imidoester in solution, DF of the imide occurs from dimeric aggregates in the blend films, assisted by the imidoester. Thus, the close contact of intimately stacked molecules of the two different species in the columnar mesophase leads to a unique mesophase-assisted aggregate DF. This constitutes the first observation of DF by TTA from the columnar liquid crystalline state. If the imide is dispersed in films of polybromostyrene, which provides an external heavy-atom effect facilitating triplet formation, DF is also observed. Organic light-emitting diodes (OLEDs) devices incorporating these liquid crystal molecules demonstrated high external quantum efficiency (EQE). On the basis of the literature and to the best of our knowledge, the EQE reported is the highest among nondoped solution-processed OLED devices using a columnar liquid crystal molecule as the emitting layer.

Oil Production Optimization by Means of a Combined “Plugging, Profile Control, and Flooding” Treatment: Analysis of Results Obtained Using Computer Tomography and Nuclear Magnetic Resonance

Due to long-term water injection, often oilfields enter the so-called medium and high water cut stage, and it is difficult to achieve good oil recovery and water reduction through standard methods (single profile control and flooding measures). Therefore, in this study, a novel method based on “plugging, profile control, and flooding” being implemented at the same time is proposed. To assess the performances of this approach, physical simulations, computer tomography, and nuclear magnetic resonance are used. The results show that the combination of a gel plugging agent, a polymer microsphere flooding agent, and a high-efficiency oil displacement agent leads to better results in terms of oil recovery with respect to the situation in which these approaches are used separately (the oil recovery is increased by 15.37%). Computer tomography scan results show that with the combined approach, a larger sweep volume and higher oil washing efficiency are obtained. The remaining oil in the cluster form can be recovered in the middle and low permeability layer, increasing the proportion of the columnar and blind end states of the oil. The nuclear magnetic resonance test results show that the combined “plugging, profile control, and flooding” treatment can also be used to control more effectively the dominant channels of the high permeability layer and further expand the recovery degree of the remaining oil in the pores of different sizes in the middle and low permeability layers. However, for the low permeability layer (permeability difference of 20), the benefits in terms of oil recovery are limited.

Air curtain dust-collecting technology: An experimental study on the performance of a large-scale dust-collecting system

The escape of dust particles from dust-producing sources in welding, wood-, and crop-processing sites is hazardous for human health and can violate strict technological requirements. This study proposed artificial unloading of dust-producing links in tobacco plants and designed an air curtain dust-collecting system with large size of 2m × 2m × 2m. The dust was first isolated by air curtain jets produced by the air curtain generator and then driven away by the suction airflow on the top of the device to achieve the effective dust collection. The flow field characteristics in the dust-collecting system and smoke fine particles' capturing behavior were qualitatively analyzed via a tracer smoke. Moreover, the effects of the air curtain's jet velocity, angle, and the exhaust-to-pressure ratio (EPR) on the system's dust-collecting performance were experimentally evaluated. The airflow's entrainment and smoke exhaust capacity were improved with increased air supply and air suction quantities. At a total air supply and air suction quantities of 3706 and 4322 m3/h, respectively, the tracer smoke exhibited a pronounced columnar collecting state. Excessive and insufficient jet velocities were found inappropriate for the developed large-scale dust-collecting system. The optimal jet velocity range was determined as 12–15 m/s, with a corresponding air supply quantity range of 3110.4–3708 m3/h. Within a reasonable EPR range of 0.9–1.0, the efficient and cost-effective dust-collecting performance was achieved. As the jet angle increased, the system's dust-collecting efficiency dropped steadily, and a zero jet angle was optimal. The optimal parameters determined through the orthogonal experiment were as follows: jet velocity of 15 m/s, EPR of 1.0, and jet angle of 0°. Under the optimal parameter setting, the dust-collecting efficiency reached a maximum of 96.88%.

Supramolecularly Engineered J-Aggregates Based on Perylene Bisimide Dyes.

The discovery of the self-assembly of cyanine dyes into J-aggregates had a major impact on the development of dye chemistry due to the emergence of new useful properties in the aggregated state. The unique optical features of these J-aggregates are narrowed, bathochromically shifted absorption bands with almost resonant fluorescence with an increased radiative rate that results from the coherently coupled molecular transition dipoles arranged in a slip-stacked fashion. Because of their desirable properties, J-aggregates gained popularity in the field of functional materials and enabled the efficient photosensitization of silver halide grains in color photography. However, despite a good theoretical understanding of structure-property relationships by the molecular exciton model, further examples of J-aggregates remained scarce for a long time as supramolecular designs to guide the formation of dye aggregates into the required slip-stacked arrangement were lacking.Drawing inspiration from the bacteriochlorophyll c self-organization found in the chlorosomal light-harvesting antennas of green sulfur bacteria, we envisioned the use of nature's supramolecular blueprint to develop J-aggregates of perylene bisimides (PBIs). This class of materials is applied in high-performance color pigments and as n-type organic semiconductors in transistors and solar cells. Combining outstanding photochemical and thermal stability, high tinctorial strength and excellent fluorescence, PBIs are therefore an ideal model system for the preparation of J-aggregates with a wide range of potential applications.In this Account, we elucidate how a combination of steric constraints and hydrogen bonding receptor sites can guide the self-assembly of PBI dyes into slip-stacked packing motifs with J-type exciton coupling. We will discuss the supramolecular polymerization of multiple hydrogen-bonded PBI strands in organic and aqueous media and how minor structural modifications in monomeric PBI molecules can be used to obtain near-infrared absorbing J-aggregates, organogels, or thermoresponsive hydrogels. Pushing the boundaries of self-assembly into the bulk, engineering of the substituents' steric requirements by a dendron-wedge approach afforded adjustable numbers of helical strands of PBI J-aggregates in the columnar liquid-crystalline state and the preparation of lamellar phases. To fully explore their potential, we have studied PBI J-aggregates in collaborative work with spectroscopists, physicists, and theoreticians. In this way, exciton migration over distances of up to 180 nm was shown, and insights into the influence of static disorder on the transport of excitation energy in PBI J-aggregates were derived. Furthermore, the application of PBI J-aggregates as functional materials was demonstrated in photonic microcavities, thin-film transistors, and organic solar cells.

Symmetry Breaking in Large Columnar Frozen Wave Patterns in Weightlessness

The large columnar patterns resulting from the frozen wave instability in zero gravity conditions are investigated numerically for finite containers of immiscible liquids. The mode selection process is affected by forcing amplitude, frequency, container size and the vibroequilibria effect. The final columnar states are often approximately left-right symmetric and a quantitative measure of their asymmetry is used to distinguish three different growth regimes depending on the applied vibration. Solutions at low frequencies are affected by the antiphase forcing at the lateral walls and by early collisions of the longwave modes with the upper and lower boundaries. At moderate frequencies, the development of the pattern is characterized by centrally located defects and nearly symmetric modes. At higher frequencies, there are off-center defects and asymmetric modes. The critical value of the symmetry-breaking transition separating these final two types of behavior is determined as a function of the container aspect ratio.

Effect of Crowding Agent Polyethylene Glycol on Lyotropic Chromonic Liquid Crystal Phases of Disodium Cromoglycate

Adding crowding agents such as polyethylene glycol (PEG) to lyotropic chromonic liquid crystals (LCLCs) formed by water dispersions of materials such as disodium cromoglicate (DSCG) leads to a phase separation of the isotropic phase and the ordered phase. This behavior resembles nanoscale condensation of DNAs but occurs at the microscale. The structure of condensed chromonic regions in crowded dispersions is not yet fully understood, in particular, it is not clear whether the condensed domains are in the nematic (N) or the columnar (C) state. In this study, we report on small angle X-ray scattering (SAXS) and wide-angle X-ray scattering (WAXS) measurements of mixtures of aqueous solutions of DSCG with PEG and compare results to measurements of aqueous solutions of pure DSCG. X-ray measurements demonstrate that addition of PEG to DSCG in the N phase triggers appearance of the C phase that coexists with the isotropic (I) phase. Within the coexisting region, the lateral distance between the columns of the chromonic aggregates decreases as the temperature is increased.

Deposition of Cr films by hot target magnetron sputtering on biased substrates

The Cr films were deposited by hot target magnetron sputtering on grounded and biased substrates. The deposition modes with low and high deposition rates (different by 4 times) were selected. The sublimation of hot Cr target decreased the re-sputtering effect of the growing films under ion bombardment and reduced the losses of deposition rate from 28.7 to 13.4% (for −40 V bias and 1.9 kW discharge power). The calculations of energy flux density on the substrate revealed that the major contribution has a target radiation (56.2…85.5%) and kinetic energy of deposited particles and ions (4.8…34.3%) become a significant only with substrate biasing. The XRD measurements showed cubic structured (110) Cr films with compressive stresses (0.48…0.90 GPa), which is non-dependent from bias voltage for high-rate deposition modes. The Cr films mainly had a columnar structure and columnar width increased from 50 to 450 nm with rise of discharge power (W). The substrate biasing and high deposition rate resulted in the formation of irregular microstructure of the Cr films along their thickness from porous to columnar state. The indentation tests showed the strong dependence of film hardness and toughness (H/E) from W and different behavior of the mechanical properties of the Cr films for the deposition on biased substrates with low and high deposition rates.

Swirling flow states of compressible single-phase supercritical fluids in a rotating finite-length straight circular pipe

Steady states of inviscid, compressible and axisymmetric swirling flows of a single-phase, inert, thermodynamically supercritical fluid in a rotating, finite-length, straight, long circular pipe are studied. The fluid thermodynamic behaviour is modelled by the van der Waals equation of state. A nonlinear partial differential equation for the solution of the flow streamfunction is derived from the fluid equations of motion in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. This equation reflects the complicated, nonlinear thermo-physical interactions in the flows, specifically when the inlet state temperature and density profiles vary around the critical thermodynamic point, flow compressibility is significant and the inlet swirl ratio is high. Several types of solutions of the resulting nonlinear ordinary differential equation for the axially independent case describe the flow outlet state when the pipe is sufficiently long. The approach is applied to an inlet flow described by a solid-body rotation with uniform profiles of the axial velocity and temperature. The solutions are used to form the bifurcation diagrams of steady compressible flows of real fluids as the inlet swirl level and the centreline inlet density are increased at a fixed inlet Mach number and temperature. Focus is on heavy-molecule fluids with low values of $R/C_{v}$. Computed results provide theoretical predictions of the critical swirl levels for the exchange of stability of the columnar state and for the appearance of non-columnar states and of vortex breakdown states as a function of inlet centreline density. The difference in the dynamical behaviour between that of a calorically perfect gas and of a real gas is explored. The analysis sheds new fundamental light on the complex dynamics of high-Reynolds-number, compressible, subsonic swirling flows of real gases.

Liquid crystalline oligomers derived from cholesterol: synthesis and columnar mesomorphism

ABSTRACTThe novel columnar liquid crystals based on symmetric hairpin-shaped cholesterol tetramers bridged by rigid or hydrogen-bonding Schiff-base spacers were prepared by reacting cholesterol dimer with phenylenediamines or dihydrazides in yields of 70–85%. Structural characterisation of these new compounds had been achieved by infrared, nuclear magnetic resonance and mass spectrometry. Their mesomorphic properties were investigated by polarising optical microscopy, differential scanning calorimetry and X-ray diffraction. They exhibited good hexagonal columnar mesophase with three molecules in a disc of the columnar hexagonal state. This study implied that the symmetric cholesterol tetramers with rigid cores or hydrogen-bonding cores prefer to columnar mesophase.

S-Triazine-Based Functional Discotic Liquid Crystals: Synthesis, Mesomorphism and Photoluminescence.

A new series of C3 -symmetric, π-conjugated molecules was designed, synthesized and characterized. The materials were derived from electron-accepting s-triazine, appended covalently to electron-donating styrylbenzene arms, and were readily prepared in excellent yield with high purity by means of three-fold condensation of triphosphonate with n-alkoxybenzaldehydes under Horner-Wadsworth-Emmons reaction conditions. Examination of the phase transitional properties by several complementary techniques evidenced self-assembly into a hexagonal columnar phase, occurring over wide and reasonable thermal ranges. The photophysical properties were studied both in solution and in the fluid/frozen columnar states by UV/Vis absorption and photoluminescence spectroscopy. The emission spectra obtained as a function of the temperature rule out the breaking-up of larger columns and a non-radiative, thermally activated process. A study carried out on thin films of the glassy columnar state, which accounts for conserved fluorescence, defect-free orientation, and freezing ionic species, with the help of atomic force microscopy (AFM) images, suggested a homogeneous granular morphology comprising fibrillar structures. Dissimilarities in the surface morphology and birefringence of thin films of the solid and frozen columnar states were clearly shown by Raman spectroscopy. An electrochemical investigation revealed a LUMO energy of -4.0 eV. Thus, the discotic motifs presented herein meet certain criteria of organic materials, which are essential for developing electronic devices.

Stabilization of the Columnar Mesophase of Perylenediimide by Racemic Triple Tails.

Whereas perylene tetracarboxdiimides derived from amino-n-alkanes if at all only show monotropic (thermodynamically unstable) mesogenic self-assembly, the hexagonal columnar liquid crystalline state can be stabilized over a broad temperature range with doubly branched, doubly racemic alkyl residues. An improved tendency to homeotropic surface orientation is observed, and the orientation of the liquid crystalline domains is maintained upon cycling through the crystalline state at room temperature.

Nonlinear Control of Axisymmetric Swirling Flows in a Long Finite-Length Pipe

Feedback stabilization of inviscid and high Reynolds number, axisymmetric, swirling flows in a long finite-length circular pipe using active variations of pipe geometry as a function of the evolving inlet radial velocity is studied. The complicated dynamics of the natural flow requires that any theoretical model that attempts to control vortex stability must include the essential nonlinear dynamics of the perturbation modes. In addition, the control methodology must establish a stable desired state with a wide basin of attraction. The present approach is built on a weakly nonlinear model problem for the analysis of perturbation dynamics on near-critical swirling flows in a slightly area-varying, long, circular pipe with unsteady changes of wall geometry. In the natural case with no control, flows with incoming swirl ratio above a critical level are unstable and rapidly evolve to either vortex breakdown states or accelerated flow states. Following an integration of the model equation, a perturbation kinetic-energy identity is derived, and an active feedback control methodology to suppress perturbations from a desired columnar state is proposed. The stabilization of both inviscid and high-Re flows is demonstrated for a wide range of swirl ratios above the critical swirl for vortex breakdown and for large-amplitude initial perturbations. The control gain for the fastest decay of perturbations is found to be a function of the swirl level. Large gain values are required at near-critical swirl ratios while lower gains provide a successful control at swirl levels away from critical. This feedback control technique cuts the feed-forward mechanism between the inlet radial velocity and the growth of perturbation's kinetic energy in the bulk and thereby enforces the decay of perturbations and eliminates the natural explosive evolution of the vortex breakdown process. The application of this proposed robust active feedback control method establishes a branch of columnar states with a wide basin of attraction for swirl ratios up to at least 50% above the critical swirl. This study provides guidelines for future flow control simulations and experiments. However, the present methodology is limited to the control of high-Reynolds number (nearly inviscid), axisymmetric, weakly nonparallel flows in long pipes.

An active feedback flow control theory of the axisymmetric vortex breakdown process

An active feedback flow control theory of the axisymmetric vortex breakdown process in incompressible swirling flows in a finite-length straight circular pipe is developed. Flow injection distributed along the pipe wall is used as the controller. The flow is subjected to non-periodic inlet and outlet conditions where the inlet profiles of the axial velocity, circumferential velocity and azimuthal vorticity are prescribed, along with no radial velocity at the outlet. A long-wave asymptotic analysis at near-critical swirl ratios, which involves a rescaling of the axial distance and time, results in a model problem for the dynamics and the nonlinear control of both inviscid and high-Reynolds-number ($\mathit{Re}$) flows. The approach provides the bifurcation diagram of steady states and the stability characteristics of these states. In addition, an energy analysis of the controlled flow dynamics suggests a feedback control law that relates the flow injection to the evolving maximum radial velocity at the inlet. Computed examples of the flow dynamics based on the full Euler and Navier–Stokes formulations at various swirl levels demonstrate the evolution to near-steady breakdown states when swirl is above a critical level that depends on $\mathit{Re}$. Moreover, applying the proposed feedback control law during flow evolution shows for the first time the successful and robust elimination of the breakdown states and flow stabilization on an almost columnar state for a wide range of swirl (up to at least 30 %) above critical. The feedback control cuts the natural feed-forward mechanism of the breakdown process. Specifically, in the case of high-$\mathit{Re}$ flows, the control approach establishes a branch of columnar states for all swirl levels studied, where in the natural flow dynamics no such states exist. The present theory is limited to the control of axisymmetric flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

Magnon dispersion and single hole motion in 2D frustrated antiferromagnets with four-sublattice structures

We study a two dimensional spin-12J1–J2 antiferromagnet in a square lattice using the linearized spin wave theory recognizing the 4-sublattice nature of the underlying magnetic lattice. Multiple magnon modes with optical and acoustic branches about the stable Neel ordered and double acoustic branches about the columnar reference states are obtained for small and large values of λ(=J2/J1) respectively. An additional uniaxial anisotropy, for large λ, can lead to distinct spin gaps in such systems, as also witnessed experimentally. The single hole spectral behavior in a 2D t–J1–J2 model, for small frustration, is then calculated within the non-crossing approximation. Our results match fairly well with exact diagonalization results from a 4×4 cluster. Hole spectral features and their evolution with λ resulting in “water-fall”-like smooth spectral weight transfer are discussed. Hole energy bands are identified and the corresponding energy-shift and reduction in width with spin-frustration are indicated.

Transitions to valence-bond solid order in a honeycomb lattice antiferromagnet

We use Quantum Monte-Carlo methods to study the ground state phase diagram of a S=1/2 honeycomb lattice magnet in which a nearest-neighbor antiferromagnetic exchange J (favoring N\'eel order) competes with two different multi-spin interaction terms: a six-spin interaction Q_3 that favors columnar valence-bond solid (VBS) order, and a four-spin interaction Q_2 that favors staggered VBS order. For Q_3 ~ Q_2 >> J, we establish that the competition between the two different VBS orders stabilizes N\'eel order in a large swathe of the phase diagram even when J is the smallest energy-scale in the Hamiltonian. When Q_3 >> (Q_2,J) (Q_2 >> (Q_3,J)), this model exhibits at zero temperature phase transition from the N\'eel state to a columnar (staggered) VBS state. We establish that the N\'eel-columnar VBS transition is continuous for all values of Q_2, and that critical properties along the entire phase boundary are well-characterized by critical exponents and amplitudes of the non-compact CP^1 (NCCP^1) theory of deconfined criticality, similar to what is observed on a square lattice. However, a surprising three-fold anisotropy of the phase of the VBS order parameter at criticality, whose presence was recently noted at the Q_2=0 deconfined critical point, is seen to persist all along this phase boundary. We use a classical analogy to explore this by studying the critical point of a three-dimensional XY model with a four-fold anisotropy field which is known to be weakly irrelevant at the three-dimensional XY critical point. In this case, we again find that the critical anisotropy appears to saturate to a nonzero value over the range of sizes accessible to our simulations.

Semiclassical evidence of columnar order in the fully frustrated transverse-field Ising model on the square lattice

We investigate the zero-temperature phase diagram of the fully frustrated transverse-field Ising model on the square lattice both in the classical limit and in the presence of quantum fluctuations. At the classical level (the limit of infinite spin $S$), we find that upon decreasing the transverse field $\ensuremath{\Gamma}$ this model exhibits a phase transition from the fully polarized state into an eightfold degenerate translational symmetry breaking state. This phase can be identified to correspond to plaquette order in the dimer language and remains the lowest-energy state in the entire range of fields below the critical one, ${\ensuremath{\Gamma}}_{c}$. The eightfold degenerate solution that corresponds to columnar order in the dimer language is a saddle point of the classical energy. It is degenerate with the plaquette solution at $\ensuremath{\Gamma}=0$ and is only slightly higher in energy in the whole interval $0<\ensuremath{\Gamma}<{\ensuremath{\Gamma}}_{c}$. The effect of quantum fluctuations is investigated in the context of a large S expansion both for the plaquette and columnar structures. For this purpose we employ an approximate method allowing to estimate from above the fluctuation-induced correction to the energy of a configuration which at the classical level is a saddle point of the energy, not a local minimum, and find that the harmonic quantum fluctuations show a clear tendency to overcome the energy difference between the two states. For relatively high fields, the transition from the plaquette to the columnar state takes place at values of $S$ large enough to be in the domain of validity of the harmonic approximation.

Wall-separation and vortex-breakdown zones in a solid-body rotation flow in a rotating finite-length straight circular pipe

AbstractThe incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ${\it\omega}_{1}$. We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ${\it\omega}$. We prove that the base columnar flow is a unique steady-state solution when ${\it\omega}$ is below ${\it\omega}_{1}$. This state is asymptotically stable and a global attractor of the flow dynamics. However, when ${\it\omega}>{\it\omega}_{1}$, we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ${\it\omega}>{\it\omega}_{1}$, the base columnar flow is a min–max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min–max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007–1016), followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.