Zvi Rusak

The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown

This paper provides a new study of the axisymmetric vortex breakdown phenomenon. Our approach is based on a thorough investigation of the axisymmetric unsteady Euler equations which describe the dynamics of a swirling flow in a finite-length constant-area pipe. We study the stability characteristics as well as the time-asymptotic behaviour of the flow as it relates to the steady-state solutions. The results are established through a rigorous mathematical analysis and provide a solid theoretical understanding of the dynamics of an axisymmetric swirling flow. The stability and steady-state analyses suggest a consistent explanation of the mechanism leading to the axisymmetric vortex breakdown phenomenon in high-Reynolds-number swirling flows in a pipe. It is an evolution from an initial columnar swirling flow to another relatively stable equilibrium state which represents a flow around a separation zone. This evolution is the result of the loss of stability of the base columnar state when the swirl ratio of the incoming flow is near or above the critical level.

On the stability of an axisymmetric rotating flow in a pipe

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 ...

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. 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) and the time-asymptotic behaviour of the flow as described by steady-state solutions of the problem presented in Wang & Rusak. 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

The dynamics of a laminar flow in a symmetric channel with a sudden expansion

Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio D / d ( d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Re c ( D / d ) where asymmetric states appear in addition to the basic symmetric states when Re [ges ] Re c ( D / d ). The bifurcation of asymmetric states at Re c has a pitchfork nature and the asymmetric perturbation grows like √ Re − Re c ( D / d ). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Re c ( D / d ) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re Re c ( D / d ) the symmetric states have an asymptotically stable mode of disturbance. However, when Re > Re c ( D / d ), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re > Re c ( D / d ) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Re c ( D / d ), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.

The effect of slight viscosity on a near-critical swirling flow in a pipe

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...

On the stability of non‐columnar swirling flows

The linear stability of an inviscid, axisymmetric and non‐columnar swirling flow in a finite length pipe is studied. A novel linearized set of equations for the development of infinitesimal axially‐symmetric disturbances imposed on a base non‐columnar rotating flow is derived. Then, a general mode of an axisymmetric disturbance, that is not limited to the axially‐periodic mode, is introduced and an eigenvalue problem is obtained. A neutral mode of disturbance exists at the critical state. The asymptotic behavior of the branches of non‐columnar solutions that bifurcate at the critical state is given. Using asymptotic techniques, it is shown that the critical state is a point of exchange of stability for these branches of solutions. This result, together with a previous result of Wang and Rusak [Phys. Fluids 8, 1007 (1996)] on the stability of columnar vortex flows, completes the investigation on the stability of all branches of solutions near the critical state. Results reveal the important relation betwee...

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

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

The stability of noncolumnar swirling flows in diverging streamtubes

A linear stability analysis of a family of steady, noncolumnar and axisymmetric, swirling flows that may develop in a finite-length slightly diverging pipe is presented. These flow states are described by the asymptotic analysis of Rusak et al. (1998). There exists a limit level of the incoming flow swirl ratio ωcσ1 which is the corrected critical swirl as a result of the pipe divergence. When the swirl ratio is in a certain range below ωcσ1, two steady states can exist for the same inlet, outlet, and wall conditions: One which describes a near-columnar vortex state and another which describes a swirling flow with a large-amplitude disturbance. When the swirl level is above ωcσ1, no near-columnar, steady, and axisymmetric state exists. The stability of this family of flows is examined by studying the linearized dynamics of an unsteady and axially symmetric perturbation which also satisfies the boundary conditions. The stability analysis shows that ωcσ1 is a point of exchange of stability for the family of...

The interaction of near-critical swirling flows in a pipe with inlet azimuthal vorticity perturbations

The interaction of a near-critical axisymmetric incompressible swirling flow in a straight pipe with small inlet azimuthal vorticity perturbations is studied. Certain flow conditions that may reflect the physical situation are prescribed along the pipe inlet and outlet. It is first demonstrated that under these conditions a regular-expansion solution in terms of the small azimuthal vorticity perturbations has a singular behavior around the critical swirl. This singularity infers that large-amplitude disturbances may be induced by the small perturbations when the incoming flow to the pipe has a swirl level around the critical swirl. In order to understand the nature of flows in this swirl range, a small-disturbance analysis is developed. It shows that under the prescribed inlet/outlet conditions, a small but finite inlet azimuthal vorticity perturbation breaks the transcritical bifurcation of solutions of the Euler equations at the critical swirl into two branches of perturbed solutions. When the azimuthal...