Axisymmetric Breakdown Research Articles

Transient energy growth of a swirling jet with vortex breakdown

We investigate the existence of short-time, local transient growth in the helical modes of a rapidly swirling, high-speed jet that has transitioned into an axisymmetric bubble breakdown state. The time-averaged flow consisting of the bubble and its wake downstream constitute the base state, which we show to exhibit strong transient amplification owing to the non-modal behaviour of the continuous eigenspectrum. A pseudospectrum analysis mathematically identifies the so-called potential modes within this continuous spectrum and the resultant non-orthogonality between these modes and the existing discrete stable modes is shown to be the main contributor to such growth. As the swirling flow develops post the collapsed bubble, the potential spectrum moves further toward the unstable half-plane, which along with the concurrent weakening of exponential growth from the discrete unstable modes, increases the dynamic importance of transient growth inside the wake region. The transient amplifications calculated at several locations inside the bubble and wake confirm this, where strong growths inside the wake far outstrip the corresponding modal growths (if available) at shorter times, but especially at the higher helical orders and smaller streamwise wavenumbers. The corresponding optimal perturbations at initial times consist of streamwise streaks of azimuthal velocity, which if concentrated inside the core vortical region, unfold via the classical Orr mechanism to yield structures resembling core (or viscous) Kelvin waves of the corresponding Lamb–Oseen vortex. However, in contrast to that in Lamb–Oseen vortex flow, where critical-layer waves are associated with higher transient gains, here, such core Kelvin modes with the more compact spiral structure at the vortex core are seen to yield the maximum transient amplifications.

Dynamics of a perturbed solid-body rotation flow in a finite-length straight rotating pipe

Direct numerical simulations are used to study the three-dimensional, incompressible and viscous flow dynamics of a base solid-body rotation flow with a uniform axial velocity entering a rotating, finite-length, straight circular pipe. Steady in time profiles of the axial, radial and circumferential velocities are prescribed along the pipe inlet. The convective boundary conditions for each velocity flux component is set at the pipe outlet. The simulation results describe the neutral stability line in response to either axisymmetric or three-dimensional perturbations in a diagram of Reynolds number ($Re$, based on inlet axial velocity and pipe radius) versus the incoming flow swirl ratio ($\unicode[STIX]{x1D714}$). This line is in good agreement with the neutral stability line recently predicted by the linear stability theory of Wang et al. (J. Fluid Mech., vol. 797, 2016, pp. 284–321). The computed time history of the velocity components at a certain point in the flow is used to describe three-dimensional phase portraits of the flow global dynamics and its long-term behaviour. They show three types of flow evolution scenarios. First, the Wang & Rusak (Phys. Fluids, vol. 8 (4), 1996, pp. 1007–1016) axisymmetric instability mechanism and evolution to a stable axisymmetric breakdown state is recovered at certain operational conditions in terms of $Re$ and $\unicode[STIX]{x1D714}$. However, at other operational conditions with same $\unicode[STIX]{x1D714}$ but with a higher $Re$, a second scenario is found. The axisymmetric breakdown state continues to evolve and a spiral instability mode appears on it and grows to a rotating spiral breakdown state. Moreover, at higher levels of $\unicode[STIX]{x1D714}$ a third scenario is found where there exists a dominant three-dimensional spiral type of instability mode that agrees with the linear stability theory of Wang et al. (J. Fluid Mech., vol. 797, 2016, pp. 284–321). The growth of this mode leads directly to a spiral type of flow roll-up and nonlinearly saturates on a rotating spiral type of vortex breakdown. The Reynolds–Orr equation is used to reveal the mechanism that drives all the instabilities as well as the nonlinear global flow evolution. At high swirl ratios, the confined kinetic energy in the swirling flow can be triggered to be released through various physical agents, such as the asymmetric inlet–outlet conditions, that eliminate axial homogeneity along the pipe and induce flow instabilities and evolution to breakdown states. It is also shown that local instability analysis or its extension using the assumption of a weakly non-parallel flow to conduct convective instability–absolute instability analyses is definitely not related to any of the instability modes found in the present study. Moreover, a stability study based on the strongly non-parallel flow character, including axial inhomogeneity due to a finite-domain boundary conditions, must be conducted to reveal instabilities in such flows.

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

A study of the geometry and parameter dependence of vortex breakdown

The types of vortex breakdown observed in the torsionally driven cylinder (TDC) flow and in the flow through an open-ended pipe are compared. The connection between the various breakdown types is specifically addressed, and the differences in manifestation of breakdown are attributed to the different Reynolds number regimes involved. Here, in both cases, the Reynolds number is based on quantities associated with the vortex core immediately upstream of breakdown, rather than the more geometry-specific Reynolds number defined in the previous work. Thus, the relationship between the TDC flow and the flows observed in other, more open geometries, is clarified. The predominantly asymmetric breakdown observed in open high Reynolds number flows is replaced by a closed bubble form with decreasing Reynolds number in the TDC. Three-dimensional numerical simulations support this interpretation, showing that the 3D spiral type of breakdown is replaced by a TDC-type axisymmetric breakdown in an open pipe as the Reynolds number is reduced. The stability of the three-dimensional solutions indicates that spiral breakdown modes stabilise at lower Reynolds number, leading to an axisymmetric breakdown state as a stable evolved flow solution.

Formation of Turbulent Vortex Breakdown: Intermittency, Criticality, and Global Instability

This study provides quantitative insight into the formation of vortex breakdown and the onset of global instability in a turbulent swirling jet. A water jet is guided through a rotating honeycomb that imparts the rotational motion, passed through a contraction, and discharged into a large water tank. The flow states evolving at increasing swirl are mapped out via time-resolved particle image velocimetry. The experimental results scale properly with the swirl number based on the axial momentum flux when the commonly used boundary-layer approximations are omitted. The instantaneous velocity field reveals that vortex breakdown occurs intermittently at a wide range of swirl numbers before it appears in the mean flow. At this intermittent state, the evolving breakdown bubble oscillates heavily between two streamwise locations where the vortex core is subcritical. Upon further increasing the swirl, the breakdown oscillations decay and a region of reversed flow appears in the mean flowfield. The formation of this socalled axisymmetric breakdown state is accompanied by a supercritical-to-subcritical transition of the inflowing vortex core. The reversed flow region is found to grow linearly with increasing swirl until the flow undergoes a supercritical Hopf bifurcation to a global single-helical mode, and vortex breakdown adopts a spiral shape. The global mode shape is extracted from the particle image velocimetry snapshots by means of proper orthogonal decomposition and Fourier analysis. The present experiment reveals that, at gradually increasing swirl, the jet first transitions to an axisymmetric breakdown state that remains globally stable until a critical swirl number is exceeded. This sequence of flow states agrees well with the transient formation of vortex breakdown observed in laminar flows.

A weakly nonlinear mechanism for mode selection in swirling jets

AbstractGlobal linear and nonlinear bifurcation analysis is used to revisit the spiral vortex breakdown of nominally axisymmetric swirling jets. For the parameters considered herein, stability analyses single out two unstable linear modes of azimuthal wavenumber $m= \ensuremath{-} 1$ and $m= \ensuremath{-} 2$, bifurcating from the axisymmetric breakdown solution. These modes are interpreted in terms of spiral perturbations wrapped around and behind the axisymmetric bubble, rotating in time in the same direction as the swirling flow but winding in space in the opposite direction. Issues are addressed regarding the role of these modes with respect to the existence, mode selection and internal structure of vortex breakdown, as assessed from the three-dimensional direct numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, pp. 331–378). The normal form describing the leading-order nonlinear interaction between modes is computed and analysed. It admits two stable solutions corresponding to pure single and double helices. At large swirl, the axisymmetric solution bifurcates to the double helix which remains the only stable solution. At low and moderate swirl, it bifurcates first to the single helix, and subsequently to the double helix through a series of subcritical bifurcations yielding hysteresis over a finite range of Reynolds numbers, the estimated bifurcation threshold being in good agreement with that observed in the direct numerical simulations. Evidence is provided that this selection is not to be ascribed to classical mean flow corrections induced by the existence of the unstable modes, but to a non-trivial competition between harmonics. Because the frequencies of the leading modes approach a strong $2$:$1$ resonance, an alternative normal form allowing interactions between the $m= \ensuremath{-} 2$ mode and the first harmonics of the $m= \ensuremath{-} 1$ mode is computed and analysed. It admits two stable solutions, the double helix already identified in the non-resonant case, and a single helix differing from that observed in the non-resonant case only by the presence of a slaved, phase-locked harmonic deformation. On behalf of the finite departure from the $2$:$1$ resonance, the amplitude of the slaved harmonic is however low, and the effect of the resonance on the bifurcation structure is merely limited to a reduction of the hysteresis range.

Dye visualization near a three-dimensional stagnation point: application to the vortex breakdown bubble

An analytical model, based on the Fokker–Planck equation, is constructed of the dye visualization expected near a three-dimensional stagnation point in a swirling fluid flow. The model is found to predict dye traces that oscillate in density and position in the meridional plane in which swirling flows are typically visualized. Predictions based on the model are made for the steady vortex breakdown bubble in a torsionally driven cylinder and compared with computational fluid dynamics predictions and experimental observations. Previous experimental observations using tracer visualization techniques have suggested that even for low-Reynolds-number flows, the steady vortex breakdown bubble in a torsionally driven cylinder is not axisymmetric and has an inflow/outflow asymmetry at its tail. Recent numerical and theoretical studies show that the asymmetry of the vortex breakdown bubble, and consequently its open nature, can be explained by the very small imperfections that are present in any experimental rig. Distinct from this, here it is shown that even for a perfectly axisymmetric flow and breakdown bubble, the combined effect of dye diffusion and the inevitable small errors in the dye injection position lead to the false perception of an open bubble structure with folds near the lower stagnation point. Furthermore, the asymmetries in the predicted flow structures can be remarkably similar to those observed in flow observations and computational predictions with geometric asymmetries of the rig. Thus, when interpreting dye-visualization patterns in steady flow, even if axisymmetric flow can be achieved, it is important to take into account the relative diffusivity of the dye and the accuracy of its injection.

Numerical simulation of tangentially injected turbulent swirling flow in a divergent tube

AbstractThis paper studies the properties of turbulent swirling decaying flow induced by tangential inlets in a divergent pipe using the realizable k–ε turbulence model and discusses the effects of the injector pressure and injection position. The results of transient solutions show that both the recirculation zone near the wall in upstream of the injectors and the vortex breakdown in downstream of the injectors increase in size during the whole period. A nearly axisymmetric conical breakdown is formed and its internal structure consists of two asymmetric spiral‐like vortices rotating in opposite directions. The stagnation point shifts slowly toward the pipe outlet over time. The maxima of the three velocity components, which are located near the wall, decrease gradually with streamwise direction. It can also be inferred that Mach number approaches 1.0 near the injector outlets. The velocities increase with the increasing injector pressure. However, its increasing trend is not significant. With the increase of the injection position, vortex breakdown moves in downstream direction and the pitch along the streamwise direction increases. Copyright © 2008 John Wiley & Sons, Ltd.

A coaxial vortex ring model for vortex breakdown

A simple–yet plausible–model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about all of the characteristics that have been observed in what is now a substantial body of published research on the phenomenon of vortex breakdown. Moreover, it is demonstrated how the very nature of the fluid dynamics in axisymmetric breakdown flows can be predicted and controlled by the choice of the initial ring configurations and their vortex strengths. The dynamic intricacies produced by the two ring + swirl model are illustrated with several numerical experiments.

On the stability of a compressible axisymmetric rotating flow in a pipe

The linear stability of a compressible inviscid axisymmetric and rotating columnar flow of a prefect gas in a finite-length straight circular pipe is investigated. This work extends a previous analysis to include the influence of Mach number on the flow dynamics. A well-posed model of the unsteady motion of a swirling flow, with inlet and outlet conditions that may reflect the physical situation, is formulated. The linearized equations of motion for the evolution of infinitesimal axially symmetric disturbances are derived. A general mode of disturbance, that is not limited to the axial-Fourier mode, is introduced and an eigenvalue problem is developed. It is found that a neutral mode of disturbance exists at ‘the critical swirl ratio for a compressible vortex flow’. The flow changes its stability characteristics as the swirl ratio increases across this critical level. When the swirl ratio is below the critical level (supercritical flow), an asymptotically stable mode is found and, when it is above the critical level (subcritical flow), an unstable mode of disturbance develops. This result cannot be predicted by any of the previous stability criteria. When the characteristic Mach number of the base flow tends to zero, the results are the same as found for incompressible swirling flows in pipes. The growth rate ratio is positive for all Mach numbers, but decreases as Mach number is increased. This ratio vanishes at the limit Mach number at which the critical swirl tends to infinity. The present results also demonstrate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed to higher swirl ratios with the increase of the incoming flow Mach number.

Experimental investigation of a swirling flow in a cubic container

A flow induced by a rotating disk mounted at the top of a cubic container totally filled with a liquid is studied experimentally. The flow pattern is visualized for Reynolds numbers in the interval 1500–6000, and the velocity variation along the axis of the container, which coincides with the axis of revolution of the disk, is observed by means of Doppler laser anemometry (DLA). As Re grows, the velocity axial component starts fluctuating because of the vortex core precession. The breakdown of the vortex helical structure becomes pronounced at Re>4000 without the formation of the return flow region (vortex breakdown bubble) at the axis. With the Reynolds number and the container height-to-disk radius ratio being the same, the axial flow patterns in standard cylindrical [1] and cubic containers differ radically. In the latter vessel at low Re, the steady flow regime and axisymmetric bubble breakdown of the vortex structure near the axis are absent.

Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation

Vortex breakdown of nominally axisymmetric, swirling incompressible flows with jet- and wake-like axial velocity distributions issuing into a semi-infinite domain is studied by means of direct numerical simulations. By selecting a two-parametric velocity profile for which the steady axisymmetric breakdown is well-studied (Grabowski & Berger 1976), issues are addressed regarding the role of three-dimensionality and unsteadiness with respect to the existence, mode selection, and internal structure of vortex breakdown, in terms of the two governing parameters and the Reynolds number. Low Reynolds numbers are found to yield flow fields lacking breakdown bubbles or helical breakdown modes even for high swirl. In contrast, highly swirling flows at large Reynolds numbers exhibit bubble, helical or double-helical breakdown modes, where the axisymmetric mode is promoted by a jet-like axial velocity profile, while a wake-like profile renders the flow helically unstable and ultimately yields non-axisymmetric breakdown modes. It is shown that a transition from super- to subcritical flow, as defined by Benjamin (1962), accurately predicts the parameter combination yielding breakdown, if applied locally to flows with supercritical inflow profiles. Thus the basic form of breakdown is axisymmetric, and a transition to helical breakdown modes is shown to be caused by a sufficiently large pocket of absolute instability in the wake of the bubble, giving rise to a self-excited global mode. Two distinct eigenfunctions corresponding to azimuthal wavenumbers have been found to yield a helical or double-helical breakdown mode, respectively. Here the minus sign represents the fact that the winding sense of the spiral is opposite to that of the flow.

The effect of compressibility on the critical swirl of vortex flows in a pipe

The effect of compressibility on the critical swirl level for breakdown of subsonic vortex flows in a straight circular pipe of finite length is studied. This work extends the critical-state concept of Benjamin (1962) to include the influence of Mach number on the flow behaviour. The analysis is based on a linearized version of the equations for the motion of a steady, axisymmetric, inviscid and compressible swirling flow of a perfect gas. The relationship between the velocity, density, temperature and pressure perturbations to a base columnar flow state are derived. An eigenvalue problem is formulated to determine the first critical level of swirl at which a special mode of a non-columnar small disturbance may appear on the base flow. It is found that when the characteristic Mach number of the base flow tends to zero the eigenvalue problem and the critical swirl are the same as defined by Wang & Rusak (1996a, 1997a) in their study of incompressible swirling flows in pipes. As the characteristic Mach number is increased, the critical swirl level increases and the flow perturbation expands in the radial direction. As the Mach number is increased toward a certain limit value related to the core size of the vortex, the critical swirl reaches very large values and becomes singular. The present results indicate that the axisymmetric breakdown of high-Reynolds-number compressible vortex flows may be delayed with the increase of the flow Mach number.

Axisymmetric Breakdown of a Q-Vortex in a Pipe

Necessary and sufficient conditions for the axisymmetric breakdown of a Q-vortex in a pipe are presented. These unique calculations are guided by a recent rigorous theoretical approach on this subject (Wang, S., and Rusak, Z.). The fundamental characteristics that lead to vortex instability and breakdown in high-Reynolds-number swirling flows are computed from the solutions of a single, nonlinear, reduced-order, ordinary differential equation, representing a columnar flow problem. The breakdown criteria for the Q-vortex for various core radii and jet/wake axial flow profiles are described. Results show good agreement with available experimental data of axisymmetric breakdown in swirling flows in a pipe. The correlation between the present results and other criteria for vortex breakdown is also discussed

The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown

The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown is studied through numerical computations. These unique simulations are guided by a recent rigorous theory on this subject presented by Wang & Rusak (1997a). Using the unsteady and axisymmetric Euler equations, the nonlinear dynamics of both small- and large-amplitude disturbances in a swirling flow are described and the transition to axisymmetric breakdown is demonstrated. The simulations clarify the relation between our linear stability analyses of swirling flows (Wang & Rusak 1996a, b) and the time-asymptotic behaviour of the flow as described by steady-state solutions of the problem presented in Wang & Rusak (1997a). The numerical calculations support the theoretical predictions and shed light on the mechanism leading to the breakdown process in swirling flows. It has also been demonstrated that the fundamental characteristics which lead to vortex instability and breakdown in high-Reynolds-number flows may be calculated from considerations of a single, reduced-order, nonlinear ordinary differential equation, representing a columnar flow problem. Necessary and sufficient criteria for the onset of vortex breakdown in a Burgers vortex are presented.

Axisymmetric breakdown of a Q-vortex in a pipe

Axisymmetric breakdown of a Q-vortex in a pipe

Asymptotic theory of vortex breakdown

Axisymmetric breakdown of a vortex in a steady flow of incompressible fluid is investigated by means of an asymptotic analys is of the Navier-Stokes equations for large Reynolds number (R→∞). A criterion for vortex breakdownis formulated. The region of slow motion resulting from vortex breakdown is shown to be bounded by an ellipsoid with length of order R−1/4 and thickness of the order R−3/8.

Asymptotic model of the axisymmetric breakdown of a vortex filament in an incompressible fluid

The axisymmetric flow of an incompressible fluid is considered. An exact solution of the Euler equations corresponding to the breakdown of a straight vortex filament of intensity Γ0 into a vortex filament of lesser intensity and a conical vortex surface is obtained. It is shown that beyond the breakdown point in the region bounded by the conical vortex surface reverse flows occur. An investigation of the problem with allowance for viscous effects at large Reynolds numbers makes it possible to establish a relation between the free parameters entering into the solution of the Euler equations. The results obtained are useful for investigating the problem of the breakdown of a swirled jet, whose solution has recently been receiving much attention [1, 2].

Vortex Breakdown Phenomena in a Circular Pipe : 5th Report, Axisymmetric Breakdown in Rotating Conical Pipes

It is known that a swirling flow in a diverging pipe breaks down more easily than a flow in a converging one. In this report, experiments are made using a divergent pipe, a convergent pipe and a straight pipe, in order to clarify the relation between the shape of pipes and the occurrence of the breakdown. A swirling flow field, which is induced in a rotating conical-pipe, is measured by a laser-Doppler-velocimeter. When each pipe rotates faster than a certain critical value, a stationary internal wave occurs. The vortex breakdown phenomenon is ascribed to the internal waves in swirling flows, as pointed out in preceding reports. In the diverging pipe, the amplitude of the wave is smaller than that in the converging pipe; but the average flow near the axis is fairly retarded, so that the flow reverses at the trough of the superposed wave component, and a bubble type breakdown appears. A mathematical model is presented to give a qualitative explanation for the effect of a sectional area variation.

Vortex breakdown phenomena in a circular pipe. 5th report Axisymmetric breakdown in rotating conical pipes.

It is known that a swirling flow in a diverging pipe breaks down more easily than a flow in a converging one. In this report, experiments are made using diverging, converging and straight pipes, in order to clarify the relation between the shape of pipes and the occurrence of the breakdown. Swirling flow fields, that are realized by rotation of pipes, are measured by a Laser-Doppler-Velocimeter. When each pipe rotates faster than a certain critical value, a stationary internal wave occurs. The vortex breakdown phenomenon is due to the internal waves in swirling flows, as pointed out in preceding reports. In the diverging pipe, the amplitude of the wave is rather smaller than that in the converging pipe ; but the mean axial flow near the axial is fairly retarded, so that the flow reverses at the trough of the superposed wave component, and bubble type breakdown appears. A mathematical model is also presented to give a qualitative explanation of the effect of a sectional area variation.