Axisymmetric Vortex Breakdown Research Articles

Global modes of variable-viscosity two-phase swirling flows and their triadic resonance

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$ . They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

Axisymmetric vortex breakdown: a barrier to mixing

In this paper, we investigate mixing in flows dominated by bubble-type vortex breakdown. Three-dimensional Navier–Stokes equations in cylindrical polar coordinates are solved to simulate flow in a cylindrical container. We have found that in steady regime of the flow, the vortex breakdown bubble is axisymmetric and apparent non-axisymmetric features observed in experiments are artifacts of imperfections in experimental set-ups. We also find that the heteroclinic manifold joining hyperbolic points of the vortex breakdown bubble is stable in the absence of any perturbation and no chaotic advection was found within vortex breakdown bubble. This makes the vortex breakdown bubble impermeable to outer fluid and hence, the vortex breakdown bubble inhibits mixing. We conclude that symmetry is a barrier to mixing.

Instability modes on a solid-body-rotation flow in a finite-length pipe

Numerical solutions of the incompressible Navier-Stokes equations are obtained to study the time evolution of both axisymmetric and three-dimensional perturbations to a base solid-body-rotation flow in a finite-length pipe with non-periodic boundary conditions imposed at the pipe inlet and outlet. It is found that for a given Reynolds number there exists a critical swirl number beyond which the initial perturbations grow, in contrast to the solid-body rotation flow in an infinitely-long pipe or a finite-length pipe with periodic inlet and exit boundary conditions for which the classical Kelvin analysis and Rayleigh stability criterion affirm neutrally stable for all levels of swirl. This paper uncovers for the first time the detailed evolution of the perturbations in both the axisymmetric and three-dimensional situations. The computations reveal a linear growth stage of the perturbations with a constant growth rate after a brief initial period of decay of the imposed initial perturbations. The fastest growing axisymmetric and three-dimensional instability modes and the associated growth rates are identified numerically for the first time. The computations show that the critical swirl number increases and the growth rate of instability decreases at the same swirl number with decreasing Reynolds number. The growth rate of the axisymmetric mode at high Reynolds number agrees well with previous stability theory for inviscid flow. More importantly, three-dimensional simulations uncover that the most unstable mode is the spiral type m = 1 mode, which appears at a lower critical swirl number than that for the onset of the axisymmetric mode. This spiral mode grows faster than the unstable axisymmetric mode at the same swirl. Moreover, the computations reveal that after the linear growing stage of the perturbation the flow continues to evolve nonlinearly to a saturated axisymmetric vortex breakdown state.

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.

Transition in vortex breakdown modes in a coaxial isothermal unconfined swirling jet

This paper reports first observations of transition in recirculation pattern from an open-bubble type axisymmetric vortex breakdown to partially open bubble mode through an intermediate, critical regime of conical sheet formation in an unconfined, co-axial isothermal swirling flow. This time-mean transition is studied for two distinct flow modes which are characterized based on the modified Rossby number (Rom), i.e., Rom ≤ 1 and Rom > 1. Flow modes with Rom ≤ 1 are observed to first undergo cone-type breakdown and then to partially open bubble state as the geometric swirl number (SG) is increased by ∼20% and ∼40%, respectively, from the baseline open-bubble state. However, the flow modes with Rom > 1 fail to undergo such sequential transition. This distinct behavior is explained based on the physical significance associated with Rom and the swirl momentum factor (ξ). In essence, ξ represents the ratio of angular momentum distributed across the flow structure to that distributed from central axis to the edge of the vortex core. It is observed that ξ increases by ∼100% in the critical swirl number band where conical breakdown occurs as compared to its magnitude in the SG regime where open bubble state is seen. This results from the fact that flow modes with Rom ≤ 1 are dominated by radial pressure gradient due to swirl/rotational effect when compared to radial pressure deficit arising from entrainment (due to the presence of co-stream). Consequently, the imparted swirl tends to penetrate easily towards the central axis causing it to spread laterally and finally undergo conical sheet breakdown. However, the flow modes with Rom > 1 are dominated by pressure deficit due to entrainment effect. This blocks the radial inward penetration of imparted angular momentum thus preventing the lateral spread of these flow modes. As such these structures fail to undergo cone mode of vortex breakdown which is substantiated by a mere 30%–40% rise in ξ in the critical swirl number range.

Control of axisymmetric vortex breakdown in a constricted pipe: Nonlinear steady states and weakly nonlinear asymptotic expansions

The present paper investigates numerically and theoretically the axisymmetric vortex breakdown occurring in a constricted pipe of infinite extension, i.e., the transition from a smooth columnar state to a breakdown state exhibiting a recirculation bubble. Velocity distributions are prescribed at the pipe inlet under the form of Batchelor vortices with uniform axial velocity and variable levels of confinement. A numerical continuation technique is developed to follow the branches of nonlinear steady solutions when varying the swirl parameter. In the most general case, vortex breakdown occurs abruptly owing to a subcritical, global instability of the non-parallel, viscous columnar solution, and results in the coexistence of multiple stable solutions over a finite range of swirl. For highly confined vortices, a second scenario prevails, where the flow transitions smoothly from the columnar to the breakdown state without any instability. The effect of a low-flow rate jet positioned at the pipe wall is then characterized in the perspective of control. Its effectiveness is evaluated in light of several practically meaningful criteria, namely, the ability of the control to optimize either the stability domain or the topology of the columnar state and its ability to alleviate hysteresis. For each criterion, an optimal jet position is determined from nonlinear simulations, the results being in good agreement with that issuing from an asymptotic expansion of the Navier–Stokes equations. Finally, we illustrate the importance of physically motivated control strategies by demonstrating how the wall jet technique can be outdone by an appropriate manipulation of the axial velocity profile prescribed at the pipe inlet.

Experimental and Computational Study of Nonreacting Vortex Breakdown in a Swirl-Stabilized Combustor

Non-reacting flow experiments are conducted in a swirl-stabilized combustor with several configurations of a triple annular research swirler (TARS) fuel injector. Particle Imaging Velocimetry (PIV) is used to measure mean axial and radial distribution of the velocity field from which swirl ratio and position of vortex breakdown are calculated for each injector configuration. The numerical simulat ions are based on the ReynoldsAveraged-Navier-Stokes (RANS) model equations of non-reacting flows, and provide insight into the characteristics of the axisymmetric vortex breakdown phenomenon in a circular, finite-length pipe with a sudden expansion downstream of a concentric, circular inlet pipe. Results show that flow states with vortex breakdown can be simulated numerically. Good agreement is found between the results of the simul ations and available theoretical predictions for the first appearance of breakdown downstream of the pipe expansion plane as well as its first appearance in the inlet pipe s ection as upstream swirl level is increased. We also show good agreement between the measured experimental data, simulations, and theoretical predictions. It is demonstrated that th e theoretical criteria and numerical simulations can be used to predict the appearance a nd location of vortex breakdown for several different nozzles of varying swirl numbers relevant to the stability of lean, premixed combustion.

The bifurcation structure of viscous steady axisymmetric vortex breakdown with open lateral boundaries

The effect of small viscosity on the behavior of the incompressible axisymmetric flow with open lateral and outlet boundaries near the critical swirling number has been studied by numerical simulations and asymptotic analysis. This work extends the theoretical studies of Wang and Rusak and numerical results of Beran and Culik to the case of flow with open lateral and outlet boundaries. In the inviscid limit the columnar flow state constitutes a solution that is known to become unstable at a particular swirl parameter. An asymptotic expansion shows that for small perturbations about this inviscid state an exchange of stability gives rise to a double saddle node bifurcation. The solution of the Euler equations breaks into two branches of the Navier–Stokes equations with a gap between the branches in which no near-columnar flow can exist. Around this region, two steady-state solutions exist for the same boundary conditions, one close to the columnar state and the other corresponding to either an accelerated or a decelerated state. This bifurcation structure is verified by numerical simulations, where the Navier–Stokes solutions are computed using branch continuation techniques based on the recursive projection method. For relatively small Reynolds numbers the numerically computed bifurcation curve does not exhibit any characteristic fold, and thus no hysteresis behavior. In this case, only a single equilibrium solution is found to exist, which changes monotonically from the quasicolumnar state to the breakdown solution. For large Reynolds numbers, however, the numerically determined bifurcation diagram confirms the fold structure characterized by the disappearance of the nearly columnar state via a saddle node bifurcation. Using the minimum axial velocity on the axis as a measure of the flow state we show that the agreement between theory and numerics is asymptotically good.

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization, regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.

On the failure of the quasicylindrical approximation and the connection to vortex breakdown in turbulent swirling flow

This paper analyzes the properties of viscous swirling flow in a pipe. The analysis is based on the time-averaged quasicylindrical Navier-Stokes equations and is applicable to steady, unsteady, and turbulent swirling flow. A method is developed to determine the critical level of swirl (vortex breakdown) for an arbitrary vortex. The method can also be used for an estimation of the radial velocity profile if the other components are given or measured along a single radial line. The quasicylindrical equations are rearranged to yield a single ordinary differential equation for the radial distribution of the radial velocity component. The equation is singular for certain levels of swirl. It is shown that the lowest swirl level at which the equation is singular corresponds exactly to the sufficient condition for axisymmetric vortex breakdown as derived by Wang and Rusak [J. Fluid Mech. 340, 177 (1997)] and Rusak et al. [AIAA J. 36, 1848 (1998)]. In narrow regions around the critical levels of swirl, the solution violates the quasicylindrical assumptions and the flow must undergo a drastic change of structure. The critical swirl level is determined by the sign change of the smallest eigenvalue of the discrete linear operator which relates the radial velocities to effects of viscosity and turbulence. It is shown that neither viscosity nor turbulence directly alters the critical level of swirl.

Transcritical Transitions In SwirlingQuasi-columnar Flows

Complete mathematical similarity between nonlinear transcritical transitions in swirled flows and transonic phenomena in compressible gas dynamics is established. It is shown that transition of base supercritical columnar swirling flow to another subcritical columnar swirling flow is governed by an equation identical to that of the shock waves formation in gases. It is proven that under some assumptions transition follows the scenario suggested by Benjamin [1]. The presented study is based on the nonlinear wave theory of swirled flows developed by Leibovich [7]. The results are established through a rigorous mathematical analysis and provide a solid theoretical understanding of the wave dynamics of axisymmetric swirling flows. The unsteady wave analysis suggests a consistent explanation of the mechanism leading to the axisymmetric vortex breakdown phenomenon in high-Reynolds swirling flows.

Numerical Study of Axisymmetric Vortex Breakdown Flows in an Open Cylindrical Container with Rotating Bottom Wall.

Although extended studies have been made on the vortex breakdown inside a cylindrical container by means of numerical simulation, majority of the studies conducted on this flow treated the case of flows inside a closed container with bottom and top disks. The occurrence of vortex breakdown in a cylindrical container is considered to be influenced by the swirling velocity component and the upstream vorticity distribution. Therefore, in the present study, by changing the top disk non-slip boundary condition to the free-slip boundary condition, the influence of the free surface on the emergence of vortex breakdown is investigated in detail. Numerical results are compared with the existing experimental study and novel flow behavior is noticed for container with small aspect ratios.

Core dynamics of a strained vortex: instability and transition

We study the instability of a laminar vortex column (in an external orthogonal strain field) to an axisymmetric core size perturbation, and the resulting transition to fine-scale turbulence. The perturbation, which evolves as a standing wave oscillation (i.e. core dynamics, CD), is inviscidly amplified by the external strain. Analysis of a weakly strained Rankine vortex explains the physical mechanism of instability: resonant interaction between the perturbation – the azimuthal wavenumber m = 0 wave – and m = ±2 waves. The CD instability (CDI) – a type of elliptic instability – experiences the fastest growth when the CD oscillation frequency equals vortex column's fluid angular velocity, such matching occurring only at specific discrete values of the axial wavenumber k. At this resonant frequency, the net effect of the swirl-induced tilting of perturbation vorticity and the CD-induced tilting of base flow vorticity is such that perturbation vorticity is continually aligned with the stretching direction of the external strain. Such strain–vorticity locking occurs for all m; hence all waves are unstable, the instability oscillation frequency being dependent on m. In a viscous Gaussian-like vortex, CDI has low-strain, low-Re and high-k cutoffs – consequences of the competing effects of inviscid amplification and viscous damping. Direct numerical simulation reveals two physical-space mechanisms of transition: (i) formation of a thin annular vortex sheath surrounding a low-enstrophy ‘bubble’ (similar to axisymmetric vortex breakdown) and the sheath's subsequent roll-up into smaller ‘vortexlets’; and (ii) folding and reconnection of core vortex filaments giving rise to additional fine-grained random vorticity within the bubble – both mechanisms caused by CD-induced intense axial flow within the vortex column. The resulting finer tubular vortices (similar to ‘worms’) have in turn their own CD, and thus this transition scenario suggests a physical-space cascade process in developed turbulence (as well as a concomitant anti-cascade process during the bubble's collapse phase). Additionally, we show that bending waves, in spite of their faster growth, effect surprisingly much slower transfer of energy into fine scales than CDI does, and hence are less effective than CDI in vortex transition and in turbulence cascade.

Chaotic advection in three-dimensional stationary vortex-breakdown bubbles: šil'nikov's chaos and the devil's staircase

We study the motion of non-diffusive, passive particles within steady, three-dimensional vortex breakdown bubbles in a closed cylindrical container with a rotating bottom. The velocity fields are obtained by solving numerically the three-dimensional Navier–Stokes equations. We clarify the relationship between the manifold structure of axisymmetric (ideal) vortex breakdown bubbles and those of the three-dimensional real-life (laboratory) flow fields, which exhibit chaotic particle paths. We show that the upstream and downstream fixed hyperbolic points in the former are transformed into spiral-out and spiral-in saddles, respectively, in the latter. Material elements passing repeatedly through the two saddle foci undergo intense stretching and folding, leading to the growth of infinitely many Smale horseshoes and sensitive dependence on initial conditions via the mechanism discovered by šil'nikov (1965). Chaotic šil'nikov orbits spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then downward near the surface, wrapping around the toroidal region in the interior of the bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent with what we would generically anticipate for a mildly perturbed, volume-preserving, three-dimensional dynamical system (MacKay 1994; Mezić & Wiggins 1994a). Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. We calculate residence times of upstream-originating non-diffusive particles and show that when mapped to initial release locations the resulting maps exhibit fractal properties. We argue that there exists a Cantor set of initial conditions that leads to arbitrarily long residence times within the breakdown region. We also show that the emptying of the bubble does not take place in a continuous manner but rather in a sequence of discrete bursting events during which clusters of particles exit the bubble at once. A remarkable finding in this regard is that the rate at which an initial population of particles exits the breakdown region is described by the devil's staircase distribution, a fractal curve that has been already shown to describe a number of other chaotic physical systems.

Observations of Swirling Flows Behind Circular Disks

Thee owpatternsofaswirlingjetsubjecttotheine uencesofvariouscentralblockagesarestudiedexperimentally via the smoke-wire e ow visualization method. Details of the observed e ow structures are described and discussed. Thee owstructuresdisplay multiplemodes,depending on theblockageratio and swirlnumber. When theblockage ratio is less than about 0.1, swirling wake, vortex breakdown, and three-dimensional vortex shedding are observed for different ranges of swirl number. When the blockage ratio is greater than about 0.1, the bluff-body effect becomes signie cant. In this case, a pair of counter-rotating vortex rings and a central swirling jet characterize the e owe eld. Axisymmetric vortex breakdown and three-dimensional vortex shedding may appear on the central swirling jetwhen theswirlnumberorblockageratio islargerthan critical values.Somecharacteristicquantities of thee owe eld aremeasuredandcorrelated,includingthefreeseparationlines,recirculationlengths,andfrequencies of the vortex shedding.

Evolution of an initially columnar vortex terminating normal to a no-slip wall

The early evolution of an initially columnar vortex normal to a solid wall was examined. The vortex was generated by a pair of flaps in a water tank. Detrimental effects from the wall during the vortex generation were avoided by producing the vortex normal to a free surface and subsequently bringing a horizontal plate into contact with the surface. Digital particle image velocimetry (DPIV) measurements of the velocity and vorticity, together with laser induced fluorescence (LIF) visualizations, in a meridional plane revealed a toroidal structure with the appearance of an axisymmetric vortex breakdown bubble. Agreement was found between the measurements and numerical simulations of the axisymmetric Navier–Stokes equations. The results show that the flow in the effusive corner region is dominated by a Bodewadt-type spatially oscillatory boundary layer within the core region and a potential-like vortex boundary layer at large radii. The toroidal structure results from the interaction between these two boundary layers, leading to the roll up of a dominant shear layer within the Bodewadt structure, and does not develop from the columnar vortex itself.

Instability of a free swirling jet driven by a half–line vortex

By studying similarity solutions of the Navier–Stokes equations, which represent swirling jets of a viscous incompressible fluid, we develop a new stability approach, and elucidate the nature of perturbations that cause hysteresis and break axisymmetry. As an example, we consider a jet in an infinite fluid driven by a half–line vortex: a model of a tornado and of a leading–edge vortex above the delta wing of an aircraft. The approach reduces the problem of spatial stability of these strongly non–parallel flows to an ordinary differential system and thereby eases the analysis. We show how non–uniqueness of the solutions appears through cusp and fold catastrophes as the swirl Reynolds number, Res , increases, and find that the fold instability is due to disturbances at the outer boundary of the flow. Also, we study the breaking of axisymmetry due to steady three–dimensional disturbances, and reveal that a helical instability occurs due to disturbances at the inner boundary of the flow. Both the fold and helical instabilities occur for moderate values of Res . Finally, we deduce an amplitude equation, similar to the Ginzburg–Landau equation, to describe the weakly nonlinear spatio–temporal growth of disturbances when Res is slightly above its critical value for linear stability. Thus, our results reveal new features of axisymmetric and helical vortex breakdown in jets.

Numerical Study on the Effect of Free Surface and Solid Wall on the Axisymmetric Vortex Breakdown in a Cylindrical Container at Aspect Ratio of h=2.0.

Numerical Study on the Effect of Free Surface and Solid Wall on the Axisymmetric Vortex Breakdown in a Cylindrical Container at Aspect Ratio of h=2.0.

Numerical Study of Steady Axisymmetric Vortex Breakdown in a Cylindrical Container. 1st Report, Limiting Reynolds Number of the Occurrence of Vortex Breakdown and the Location of Stagnation Point.

Numerical study is performed on the steady axsymmetric vortex breakdown within a cylindrical container with a rotating top lid. The cylindrical container is filled with incompressible viscous fluid and the swirling flow is driven in the container by the top lid rotating at a constant anguler velocity. This flow problem is governed by a combination of two non-dimensional physical parameters, i.e., the Reynolds number Re and the container aspect ratio h. In the past experimental studies, the occurrence of vortex breakdown is known to be confined in a narrow range of h and Re and the region diagram of vortex breakdown was constructed. In the present numerical investigation, the region diagram is reconstructed for the first time by the numerical computation. The axisymmetry of the flow is assumed and the resulting system of governing equations are solved by the Ψ-Ω method. The comparison of the present region diagram obtained for the steady solutions and that obtained experimentally by Escudier exhibited good agreement. This cross check of limiting Re values for the occurrence of vortex breakdown confirms the reliability of both computational method and the experimental methodology. Based on the present numerical solutions, the locations of stagnation points for various aspect ratio h are calculated as functions of Re and these values are also compared with available experimental data. The outcome of present comparison confirms the belief that the steady vortex breakdown within a cylindrical container is actually axisymmetric. Further argument will be discussed in the proceeding studies.

Prediction of Vortex Breakdown in Leading-Edge Vortices Above Slender Delta Wings

This paper applies a recent theory on the axisymmetric vortex breakdown process to experimental flow profiles of leading edge vortices above slender delta wings at high angles of attack. Using several simplifying assumptions about the nature of the vortex generation process and its swirl ratio dependence on angle of attack, the necessary conditions for the breakdown of the vortices are calculated. The computations show good agreement with the available experimental data and, thus, provide an efficient method to accurately predict the burst locations as function of angle of attack along a delta wing. Specifically, this can be predicted out of a single set of measurements at a lower angle of attack where no sign of breakdown yet exists. In addition, the calculation of the swirl ratio based on the ratio of the maximum circumferential speed over the maximum axial speed, at the breakdown conditions reveals that it is almost independent of both the angle of attack and location along the wing chord, having an average value of 0.58. This swirl ratio may, therefore, serve as a simple universal criterion for the appearance of breakdown along leading edge vortices above slender wings with sharp edges. Nomenclature