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The numerical simulations of the dynamics of high Reynolds number ($$Re>100{,}000$$) swirling flows in pipes with varying geometries of engineering applications continues to be a challenging computational problem, specifically when vortex-breakdown zones or wall-separation regions naturally evolve in the flows. To tackle this challenge, the present paper describes a simulation scheme of the evolution of inviscid, axisymmetric and incompressible swirling flows in expanding or contracting pipes. The integration in time of the circulation together with azimuthal vorticity uses an explicit, first-order accurate finite-difference scheme with a second-order accurate upwind difference formulation in the axial and radial directions. The Poisson solver for advancing in time the spatial distribution of the stream function as a function of azimuthal vorticity uses a second-order accurate over-relaxation difference scheme. No additional numerical steps are needed for computing the natural evolution of flows including the dynamics to states with slow-speed recirculation zones along the pipe centerline or attached to pipe wall. This numerical method shows convergence of the computed results with mesh refinement for various swirl levels and pipe geometry variations. The computed results of time-asymptotic states also present agreement with available theoretical predictions of steady vortex flows in diverging or contracting pipes. In addition, comparison with available experimental data demonstrates that the present algorithm accurately predicts instability processes and long-term mean-flow dynamics of vortex flows in pipes at high Re. The inviscid-flow simulation results support the theoretical predictions and clarify the nature of high-Re flow evolution.

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.

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.

The dynamics of inviscid-limit, incompressible and axisymmetric swirling flows in finite-length, diverging or contracting, long circular pipes is studied through global analysis techniques and numerical simulations. The inlet flow is described by the profiles of the circumferential and axial velocity together with a fixed azimuthal vorticity while the outlet flow is characterized by a state with zero radial velocity. A mathematical model that is based on the Squire–Long equation (SLE) is formulated to identify steady-state solutions of the problem with special conditions to describe states with separation zones. The problem is then reduced to the columnar (axially-independent) SLE, with centreline and wall conditions for the solution of the outlet flow streamfunction. The solution of the columnar SLE problem gives rise to the existence of four types of solutions. The SLE problem is then solved numerically using a special procedure to capture states with vortex-breakdown or wall-separation zones. Numerical simulations based on the unsteady vorticity circulation equations are also conducted and show correlation between time-asymptotic states and steady states according to the SLE and the columnar SLE problems. The simulations also shed light on the stability of the various steady states. The uniqueness of steady-state solutions in a certain range of swirl is proven analytically and demonstrated numerically. The computed results provide the bifurcation diagrams of steady states in terms of the incoming swirl ratio and size of pipe divergence or contraction. Critical swirls for the first appearance of the various types of states are identified. The results show that pipe divergence promotes the appearance of vortex-breakdown states at lower levels of the incoming swirl while pipe contraction delays the appearance of vortex breakdown to higher levels of swirl and promotes the formation of wall-separation states.

Three-dimensional direct numerical simulations of a solid-body rotation superposed on a uniform axial flow entering a rotating constant-area pipe of finite length are presented. Steady in time profiles of the radial, axial, and circumferential velocities are imposed at the pipe inlet. Convective boundary conditions are imposed at the pipe outlet. The Wang and Rusak (Phys. Fluids 8:1007–1016, 1996. doi: 10.1063/1.86882 ) axisymmetric instability mechanism is retrieved at certain operational conditions in terms of incoming flow swirl levels and the Reynolds number. However, at other operational conditions there exists a dominant, three-dimensional spiral type of instability mode that is consistent with the linear stability theory of Wang et al. (J. Fluid Mech. 797: 284–321, 2016). The growth of this mode leads to a spiral type of flow roll-up that subsequently nonlinearly saturates on a large amplitude rotating spiral wave. The energy transfer mechanism between the bulk of the flow and the perturbations is studied by the Reynolds-Orr equation. The production or loss of the perturbation kinetic energy is combined of three components: the viscous loss, the convective loss at the pipe outlet, and the gain of energy at the outlet through the work done by the pressure perturbation. The energy transfer in the nonlinear stage is shown to be a natural extension of the linear stage with a nonlinear saturated process.

The interaction between flow inertia and elasticity in high-Reynolds-number, axisymmetric and near-critical swirling flows of an incompressible and viscoelastic fluid in an open finite-length straight circular pipe is studied at the limit of low elasticity. The stresses of the viscoelastic fluid are described by the generalized Giesekus constitutive model. This model helps to focus the analysis on low fluid elastic effects with shear thinning of the viscosity. The application of the Giesekus model to columnar streamwise vortices is first investigated. Then, a nonlinear small-disturbance analysis is developed from the governing equations of motion. It reveals the complicated interactions between flow inertia, swirl and fluid rheology. An effective Reynolds number that links between steady states of swirling flows of a viscoelastic fluid and those of a Newtonian fluid is revealed. The effects of the fluid viscosity, relaxation time, retardation time and mobility parameter on the flow development in the pipe and on the critical swirl for the appearance of vortex breakdown are explored. It is found that in vortex flows with either an axial jet or an axial wake profile, increasing the shear thinning by decreasing the ratio of the viscoelastic characteristic times from one (with fixed values of the Weissenberg number and the mobility parameter) increases the critical swirl ratio for breakdown. Increasing the fluid elasticity by increasing the Weissenberg number from zero (with a fixed ratio of the viscoelastic characteristic times and a fixed value of the mobility parameter) or increasing the fluid mobility parameter from zero (with fixed values of the Weissenberg number and the ratio of viscoelastic times) causes a similar effect. The results may explain the trend of changes in the appearance of breakdown zones as a function of swirl level that were observed in the experiments by Stokes et al. (J. Fluid Mech., vol. 429, 2001, pp. 67–115), where Boger fluids were used. This work extends for the first time the theory of vortex breakdown to include effects of non-Newtonian fluids.

The three-dimensional, inviscid and viscous flow instability modes that appear on a solid-body rotation flow in a finite-length straight, circular pipe are analysed. This study is a direct extension of the Wang & Rusak (Phys. Fluids, vol. 8 (4), 1996a, pp. 1007–1016) analysis of axisymmetric instabilities on inviscid swirling flows in a pipe. The linear stability equations are the same as those derived by Kelvin (Phil. Mag., vol. 10, 1880, pp. 155–168). However, we study a general mode of perturbation that satisfies the inlet, outlet and wall conditions of a flow in a finite-length pipe with a fixed in time and in space vortex generator ahead of it. This mode is different from the classical normal mode of perturbations. The eigenvalue problem for the growth rate and the shape of the perturbations for any azimuthal wavenumber $m$ consists of a linear system of partial differential equations in terms of the axial and radial coordinates ($x,r$). The stability problem is solved numerically for all azimuthal wavenumbers $m$. The computed growth rates and the related shapes of the various perturbation modes that appear in sequence as a function of the base flow swirl ratio (${\it\omega}$) and pipe length ($L$) are presented. In the inviscid flow case, the $m=1$ modes are the first to become unstable as the swirl ratio is increased and dominate the perturbation’s growth in a certain range of swirl levels. The $m=1$ instability modes compete with the axisymmetric ($m=0$) instability modes as the swirl ratio is further increased. In the viscous flow case, the viscous damping effects reduce the modes’ growth rates. The neutral stability line is presented in a Reynolds number ($Re$) versus swirl ratio (${\it\omega}$) diagram and can be used to predict the first appearance of axisymmetric or spiral instabilities as a function of $Re$ and $L$. We use the Reynolds–Orr equation to analyse the various production terms of the perturbation’s kinetic energy and establish the elimination of the flow axial homogeneity at high swirl levels as the underlying physical mechanism that leads to flow exchange of stability and to the appearance of both spiral and axisymmetric instabilities. The viscous effects in the bulk have only a passive influence on the modes’ shapes and growth rates. These effects decrease with the increase of $Re$. We show that the inviscid flow stability results are the inviscid-limit stability results of high-$Re$ rotating flows.

A nonlinear stability analysis of the viscous circular Couette flow to axisymmetric finite-amplitude perturbations under axial periodic boundary conditions is developed. The analysis is based on investigating the properties of a reduced Arnol’d energy-Casimir function $\mathscr{A}_{rd}$ of Wang (Phys. Fluids, vol. 2, 2009, 084104). A weighted kinetic energy of the perturbation, which has a form of ${\rm\Delta}\mathscr{A}_{rd}$, the difference between the reduced Arnol’d function and its base flow value, is used as a Lyapunov function. We show that all the inviscid flow effects as well as all the viscous-dependent terms that are related to the flow boundaries vanish. The evolution of ${\rm\Delta}\mathscr{A}_{rd}$ depends only on the viscous effects of the perturbation’s dynamics inside the flow domain. The requirement for the temporal decay of ${\rm\Delta}\mathscr{A}_{rd}$ leads to two novel sufficient conditions for the nonlinear stability of the circular Couette flow in response to axisymmetric perturbations. The linearized version of these conditions for infinitesimally small perturbations recovers the recent linear stability results by Kloosterziel (J. Fluid Mech., vol. 652, 2010, pp. 171–193). By examining the nonlinear stability conditions, we establish a definite operational region of the viscous circular Couette flow that is independent of the fluid viscosity. In this region of operation, the flow is nonlinearly stable in response to perturbations of any size, provided that the initial total circulation function is above a minimum level determined by the operational conditions of the base flow. Comparisons with historical studies show that our results shed light on the experimental measurements of Wendt (Ing.-Arch., vol. 4, 1933, pp. 577–595) and extend the classical nonlinear stability results of Serrin (Arch. Rat. Mech. Anal., vol. 3, 1959, pp. 1–13) and Joseph & Hung (Arch. Rat. Mech. Anal., vol. 44, 1971, pp. 1–22). When the flow is nonlinearly stable and evolves axisymmetrically for all time, then it always decays asymptotically in time to the circular Couette flow determined uniquely by the set-up of the rotating cylinders. Finally, we derive upper-bound estimates on the decay rate of finite-amplitude perturbations for the solid-body rotation flow between two coaxial rotating cylinders and for the circular Couette flow. We demonstrate via numerical simulations that the theoretical upper bound is relevant to the dynamics of various axisymmetric perturbations tested, where it is strictly obeyed. This present study provides new physical insights into a classical flow problem that was studied for many decades.

A global analysis of steady states of low Mach number inviscid premixed reacting swirling flows in a straight circular finite-length open pipe is developed. We focus on modelling the basic interaction between the swirl and heat release of the reaction. For analytic simplicity, a one-step first-order Arrhenious reaction kinetics is considered in the limit of high activation energy and infinite Peclet number. Assuming a complete reaction with chemical equilibrium upstream and downstream of the reaction zone, a nonlinear partial differential equation is derived for the solution of the flow stream function downstream of the reaction zone in terms of the specific total enthalpy, specific entropy and circulation functions prescribed at the inlet. Several types of solutions of the nonlinear ordinary differential equation for the columnar flow case describe the outlet states of the flow in a long pipe. These solutions are used to form the bifurcation diagram of steady reacting flows with swirl as the inlet swirl level is increased at a fixed heat release from the reaction. The approach is applied to two profiles of inlet flows, the solid-body rotation and the Lamb–Oseen vortex, both with constant profiles of the axial velocity, temperature and mixture reactant mass fraction. The computed results provide theoretical predictions of the critical inlet swirl levels for the appearance of vortex breakdown states and for the size of the breakdown zone as a function of the inlet flow swirl level, Mach number and heat release of the reaction. For the inlet solid-body rotation, flow is decelerated to breakdown as the inlet swirl is increased above the critical swirl level, and there is a delay in the appearance of breakdown with the increase of the heat release of the reaction. For the inlet Lamb–Oseen vortex at low values of heat release, the critical swirl for breakdown is decreased with the increase of heat release while, at high values of heat release, the appearance of breakdown is delayed to higher incoming flow swirl levels with the increase of heat release. The analysis sheds light on the global dynamics of low Mach number reacting flows with swirl and vortex breakdown and on the interaction between vortex breakdown and heat release that affects the shape of the reaction zone in the domain.

The influence of various inlet swirling flow profiles on the manifold of steady axisymmetric states of flows, their domain of attraction, and on flow dynamics in a straight, long circular pipe is investigated.

A model problem of low-Mach number lean premixed reacting swirling flows with wall-separation zones in a contracting open chamber is studied. A theoretical feasibility for a technology of swirl-assisted combustion where the reaction zone is supported by a wall-separation zone is established.

A global analysis of steady states of inviscid compressible subsonic swirling flows in a finite-length straight circular pipe is developed. A nonlinear partial differential equation for the solution of the flow stream function is derived in terms of the inlet flow specific total enthalpy, specific entropy and circulation functions. The equation reflects the complicated thermo–physical interactions in the flows. Several types of solutions of the resulting nonlinear ordinary differential equation for the columnar case together with a flow force condition describe the outlet state of the flow in the pipe. These solutions are used to form the bifurcation diagram of steady compressible flows with swirl as the inlet swirl level is increased at a fixed inlet Mach number. The approach is applied to two profiles of inlet flows, solid-body rotation and the Lamb–Oseen vortex, both with a uniform axial velocity and temperature. The computed results provide for each inlet flow profile theoretical predictions of the critical swirl levels for the appearance of vortex breakdown states as a function of the inlet Mach number, suggesting that the results are robust for a variety of inlet swirling flows. The analysis sheds light on the dynamics of compressible flows with swirl and vortex breakdown, and shows the delay in the appearance of breakdown with increase of the inlet axial flow Mach number in the subsonic range of operation. The present theory is limited to axisymmetric dynamics of swirling flows in pipes where the wall boundary layer is thin and attached and does not interact with the flow in the bulk.

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.

The 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.

The physical properties of a recently proposed feedback-stabilization method of a vortex flow in a finite-length straight pipe are studied for the case of a solid-body rotation flow. In the natural case, when the swirl ratio is beyond a certain critical level, linearly unstable modes appear in sequence as the swirl level is increased. Based on an asymptotic long-wave (long-pipe) approach, the global feedback control method is shown to enforce the decay in time of the perturbation’s kinetic energy and thereby quench all of the instability modes for a swirl range above the critical swirl level. The effectiveness of an extended version of this feedback flow control approach is further analysed through a detailed mode analysis of the full linear control problem for a solid-body rotation flow in a finite-length pipe that is not necessarily long. We first rigourously prove the asymptotic decay in time of all modes with real growth rates. We then compute the growth rate and shape of all modes according to the full linearized control problem for swirl levels up to 50 % above the critical level. We demonstrate that the flow is stabilized in the whole swirl range and can be even further stabilized for higher swirl levels. However, the control effectiveness is sensitive to the choice of the feedback control gain. A potentially best range of the gain is identified. An inadequate level of gain, either insufficient or excessive, could lead to a marginal control or failure of the control method at high swirl levels. The robustness of the proposed control law to stabilize both initial waves and continuous inlet flow perturbations and the elimination of the vortex breakdown process are demonstrated through numerical computations.

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (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.

The dynamics of a perturbed incompressible, inviscid, axisymmetric, near-critical swirling flow in a long, finite-length, straight, circular pipe is studied through a weakly nonlinear analysis. The flow is subjected to non-periodic inlet and outlet conditions. The long-wave approach involves a rescaling of the axial distance and time. It results in a separation of the perturbation’s structure into a critical standing wave in the radial direction and an evolving wave in the axial direction, that is described by a nonlinear model problem. The approach is first validated by establishing the bifurcation of non-columnar states from the critical swirl and the linear stability modes of these states. Examples of the flow dynamics at various near-critical swirl levels in response to different initial perturbations demonstrate the important role of the nonlinear steepening terms in perturbation dynamics. The computed dynamics shows quantitative agreement with results from numerical simulations that are based on the axisymmetric Euler equations for various swirl levels and as long as perturbations are small, thereby verifying the accuracy of each computation and capturing the essence of flow dynamics. Results demonstrate the various stages of the flow dynamics, specifically during the transition to vortex breakdown states. They reveal the evolution of faster-than-exponential and shape-changing modes as perturbations grow into the vortex breakdown process. These explosive modes provide the sudden and abrupt nature of the vortex breakdown phenomenon. Further analysis of the model problem shows the important role of the nonlinear evolution of perturbations and its relevance to the transfer of the perturbation’s kinetic energy between the boundaries and flow bulk, the evolution of perturbations in practical concentrated vortex flows, and the design of control methods of vortex flows. A robust feedback control method to stabilize a solid-body rotation flow in a pipe at a wide range of swirl levels above critical is developed. The applicability of this method to stabilizing medium and small core-size vortices is also discussed.

The rate of change of the perturbation's kinetic energy E of a perturbed inviscid, incompressible, axisymmetric, columnar and near-critical swirling flow in a finite-length, straight, circular pipe with periodic and non-periodic inlet–outlet conditions is studied using the Reynolds–Orr equation. The perturbation's mode shape and growth rate are computed from the linear-stability eigenvalue problem using a novel asymptotic solution in the case of a flow in a long pipe. This solution technique is general and can be applied to any vortex flow profile, in a range of swirl levels around the critical level, and for various boundary conditions. The solutions are used to analytically estimate the production (or loss) of E at the pipe boundaries and inside the domain and to shed new light on the Wang–Rusak mechanism of exchange of global stability around the critical swirl, that is leading to the vortex breakdown process. It is shown that the production of E inside the domain is modulated by the base flow strain-rate tensor. For the special case of a solid-body rotating flow, this term vanishes and the stability is determined only by the asymmetric transfer of E at the boundaries. For a general base flow, the dominant perturbation's mode shape develops deviations in response to the non-periodic inlet–outlet conditions. These deviations couple with the base flow strain-rate tensor to generate production or loss of E in the bulk. Together with the asymmetric transfer of E at the boundaries, they form a critical balance of production of E and determine the flow stability around the critical state. This behaviour is demonstrated for the Lamb–Oseen and Q vortex models. This analysis reveals a more complicated, as well more realistic, interaction between the perturbed flow in the domain and at the boundaries that dominates vortex flow dynamics.

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.

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.

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 effect of small pipe divergence on an inviscid, incompressible, near critical axisymmetric swirling flow is investigated. The singular behavior of a regular expansion solution, in terms of the pipe divergence parameter, around the critical swirl of a flow in a straight pipe is demonstrated. This singularity infers that large-amplitude disturbances may be induced by the small pipe divergence when incoming flows have a swirl level near the critical swirl. In order to gain insight to the behavior of flows in this swirl range, a small-disturbance analysis is developed. It is found that a small but finite pipe divergence breaks the transcritical bifurcation of solutions of a flow in a straight pipe into two equilibrium solution branches. These branches fold at limit swirl levels near the critical swirl resulting in a finite gap of swirl that separates the two branches. This suggests that no near-columnar axisymmetric state can exist within this range of incoming swirl around the critical level; the flow must develop large disturbances in this swirl range. Beyond this range, two steady states may exist under the same inlet/outlet conditions. However, when the pipe divergence is increased, this special behavior uniformly changes into a branch of solutions with no fold. A weakly nonlinear approach to study the effect of slight pipe divergence on standing waves in a long pipe is also derived. The behavior of the asymptotic solutions match the bifurcation diagrams from previous theoretical and numerical studies and extends their results. The relevance of the results to axisymmetric vortex breakdown in a diverging pipe is discussed.

The effect of slight viscosity on a near-critical axisymmetric incompressible swirling flow in a straight pipe is studied. We demonstrate the singular behavior of a regular-expansion solution in terms of the slight viscosity around the critical swirl. This singularity infers that large-amplitude disturbances may be induced by the small viscosity when the incoming flow to the pipe has a swirl level around the critical swirl. It also provides a theoretical understanding of Hall’s boundary layer separation analogy to the vortex breakdown phenomenon. In order to understand the nature of flows in this swirl range, we develop a small-disturbance analysis. It shows that a small but finite viscosity breaks the transcritical bifurcation of solutions of the Euler equations at the critical swirl into two branches of solutions of the Navier–Stokes equations. These branches fold at limit points near the critical swirl with a finite gap between the two branches. This means that no near-columnar equilibrium state can ex...

The linear stability of an inviscid, axisymmetric and rotating columnar flow in a finite length pipe is studied. A well posed model of the unsteady motion of swirling flows with compatible boundary conditions that may reflect the physical situation is formulated. A linearized set of equations for the development of infinitesimal axially‐symmetric disturbances imposed on a base rotating columnar flow is derived. Then, a general mode of axisymmetric disturbances, that is not limited to the axial‐Fourier mode, is introduced and an eigenvalue problem is obtained. Benjamin’s critical state concept is extended to the case of a rotating flow in a finite length pipe. It is found that a neutral mode of disturbance exists at the critical state. In the case of a solid body rotating flow with a uniform axial velocity component, analytical solution of the eigenvalue problem is found. It is demonstrated that the flow changes its stability characteristics as the swirl changes around the critical level. When the flow is ...

This article extends the perturbation method introduced by Wang (2008, “A Novel Method for Analyzing the Global Stability of Inviscid Columnar Swirling Flow in a Finite Pipe,” Phys. Fluids, 20(7), p. 074101) to determine the global stability of a swirling flow in a straight circular pipe with specified inlet and outlet boundary conditions. To accurately compute the flow stability characteristics, a general procedure to treat the complexity arising from high-order terms is developed. It extends the previous fourth-order method to an eighth-order method. The technique is first applied to the benchmark case of a solid-body rotation flow with a uniform axial speed. It is demonstrated that the eighth-order method is sufficient to construct the growth rate curve between the first and second critical swirls of this flow. Note that this range of swirl is crucial for the study of the vortex breakdown phenomenon since the base flow is unstable and starts the initial stage of transition to a breakdown state. The method is then applied to the Lamb–Oseen vortex to construct the growth rate curve between the first and second critical swirls of this flow. Calculated results are compared with the growth rate curve computed from direct numerical simulations and an overall agreement between the two computations is found. This demonstrates that the Wang and Rusak (1996, “On the Stability of an Axisymmetric Rotating Flow in a Pipe,” Phys. Fluids, 8(4), pp. 1007–1076) instability mechanism captures quantitatively the initial growth of disturbance, which eventually evolves into a breakdown state.

The statistical characteristics of turbulent two-phase pipe flow have been evaluated. In particular, the autocorrelation functions and the power spectral density functions of the axial turbulence fluctuations in the liquid phase were determined. The high frequency content of the power spectrum in bubbly two-phase pipe flow was found to be significantly larger than in single-phase pipe flow and, in agreement with previous studies of homogeneous two-phase flows (Lance et al., 1983), diminished asymptotically with a characteristic −8/3 slope at high frequency. The power spectrum and the autocorrelation functions in two-phase pipe flow, although distinctively different from those in single-phase pipe flow, were insensitive to the local void fraction and the mean liquid velocity when plotted against wave number and spatial separation, respectively. Finally, the dissipation scale, determined from the shape of the autocorrelation function, indicated that the turbulent dissipation rate in two-phase pipe flow was significantly greater than that in single-phase pipe flow.

The nonlinear stability of columnar swirling flows in a finite constant-area pipe with periodic boundary conditions imposed at the inlet and outlet of the pipe is investigated by the Arnold energy-Casimir method. Szeri and Holmes [“Nonlinear stability of axisymmetric swirling flows,” Philos. Trans. R. Soc. London, Ser. A 326, 327 (1988)] applied the Arnold energy-Casimir method to the swirling flow and developed a general form of Arnold function, which forms the basis for the current investigation. It has been shown that for the base flow which has a uniform axial velocity profile, a reduced form of the Arnold function can be used to obtain nonlinear stability. This essentially extends Rayleigh’s linear stability theory to the nonlinear theory which is applicable to finite-amplitude disturbances, and leads to several nonlinear stability criteria which can readily be applied to swirling flows with various configurations. The nonlinear stability criteria are then applied to the Lamb–Oseen vortex and the Tay...

We developed a general strategy to study the stability problem of the inviscid columnar swirling flow in a finite pipe based on the perturbation method of linear operators. By virtue of the columnar base state, we were able to derive all the necessary formulas used in perturbation method in a manner of separation of variables. We then conduct a necessary benchmark case study based on the solid body rotation flow. We then applied the general method to the Lamb–Oseen vortex and the q-vortex and found their approximated growth rate functions. The perturbation method proved effective and robust in application to these vortex flows. We also extended the method by using the analytic continuation to find the unstable modes with complex growth rates. This analytic continuation method reflects the global nature of the perturbation method. Using the new method, we investigated the energy transfer mechanism of the Lamb–Oseen vortex in a finite pipe and revealed a new energy transfer mechanism related to the flow sta...