On the three-dimensional stability of a solid-body rotation flow in a finite-length rotating pipe

2016 ◽  
Vol 797 ◽  
pp. 284-321 ◽  
Author(s):  
Shixiao Wang ◽  
Zvi Rusak ◽  
Rui Gong ◽  
Feng Liu
Keyword(s):  
Finite Length ◽  
Growth Rates ◽  
Three Dimensional ◽  
Inviscid Flow ◽  
Swirl Ratio ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Instability Modes ◽  
Stability Results ◽  
Flow Case

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.

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

AIP Advances ◽  
2017 ◽  
Vol 7 (9) ◽  
pp. 095112 ◽  
Author(s):  
Chunjuan Feng ◽  
Feng Liu ◽  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Solid Body ◽  
Finite Length ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Instability Modes

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

2014 ◽  
Vol 759 ◽  
pp. 321-359 ◽  
Author(s):  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Solid Body ◽  
Finite Length ◽  
Vortex Breakdown ◽  
Swirling Flows ◽  
Flow Dynamics ◽  
Circular Pipe ◽  
Body Rotation ◽  
Solid Body Rotation

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

Rotating free-shear flows. Part 2. Numerical simulations

1995 ◽  
Vol 293 ◽  
pp. 47-80 ◽  
Author(s):  
Olivier Métais ◽  
Carlos Flores ◽  
Shinichiro Yanase ◽  
James J. Riley ◽  
Marcel Lesieur
Keyword(s):  
Solid Body ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Rossby Number ◽  
Two Dimensional ◽  
Body Rotation ◽  
Mean Flow ◽  
Hairpin Vortices ◽  
Solid Body Rotation ◽  
Rotation Rates

The three-dimensional dynamics of the coherent vortices in periodic planar mixing layers and in wakes subjected to solid-body rotation of axis parallel to the basic vorticity are investigated through direct (DNS) and large-eddy simulations (LES). Initially, the flow is forced by a weak random perturbation superposed on the basic shear, the perturbation being either quasi-two-dimensional (forced transition) or three-dimensional (natural transition). For an initial Rossby number Ro(i), based on the vorticity at the inflexion point, of small modulus, the effect of rotation is to always make the flow more two-dimensional, whatever the sense of rotation (cyclonic or anticyclonic). This is in agreement with the Taylor–Proudman theorem. In this case, the longitudinal vortices found in forced transition without rotation are suppressed.It is shown that, in a cyclonic mixing layer, rotation inhibits the growth of three-dimensional perturbations, whatever the value of the Rossby number. This inhibition exists also in the anticyclonic case for |Ro(i)| ≤ 1. At moderate anticyclonic rotation rates (Ro(i) < −1), the flow is strongly destabilized. Maximum destabilization is achieved for |Ro(i) ≈ 2.5, in good agreement with the linear-stability analysis performed by Yanase et al. (1993). The layer is then composed of strong longitudinal alternate absolute vortex tubes which are stretched by the flow and slightly inclined with respect to the streamwise direction. The vorticity thus generated is larger than in the nonrotating case. The Kelvin–Helmholtz vortices have been suppressed. The background velocity profile exhibits a long range of nearly constant shear whose vorticity exactly compensates the solid-body rotation vorticity. This is in agreement with the phenomenological theory proposed by Lesieur, Yanase & Métais (1991). As expected, the stretching is more efficient in the LES than in the DNS.A rotating wake has one side cyclonic and the other anticyclonic. For |Ro(i)| ≤ 1, the effect of rotation is to make the wake more two-dimensional. At moderate rotation rates (|Ro(i)| > 1), the cyclonic side is composed of Kármán vortices without longitudinal hairpin vortices. Karman vortices have disappeared from the anticyclonic side, which behaves like the mixing layer, with intense longitudinal absolute hairpin vortices. Thus, a moderate rotation has produced a dramatic symmetry breaking in the wake topology. Maximum destabilization is still observed for |Ro(i)| ≈ 2.5, as in the linear theory.The paper also analyses the effect of rotation on the energy transfers between the mean flow and the two-dimensional and three-dimensional components of the field.

On triadic resonance as an instability mechanism in precessing cylinder flow

2018 ◽  
Vol 841 ◽  
Author(s):  
Thomas Albrecht ◽  
Hugh M. Blackburn ◽  
Juan M. Lopez ◽  
Richard Manasseh ◽  
Patrice Meunier
Keyword(s):  
Solid Body ◽  
Three Dimensional ◽  
Frame Of Reference ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Instability Mechanism ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Successful Model ◽  
Basic States

Contained rotating flows subject to precessional forcing are well known to exhibit rapid and energetic transitions to disorder. Triadic resonance of inertial modes has been previously proposed as an instability mechanism in such flows, and that idea was developed into a successful model for predicting instability in a cylindrical container when departures from solid-body rotation are sufficiently small. Using direct numerical simulation and dynamic mode decomposition, we analyse instabilities of precessing cylinder flows whose three-dimensional basic states, steady in the gimbal frame of reference, may depart substantially from solid-body rotation. In the gimbal frame, the instability can be interpreted as resulting from a supercritical Hopf bifurcation that results in a limit-cycle flow. In the cylinder frame of reference, the basic state is a rotating wave with azimuthal wavenumber $m=1$, and the instability satisfies triadic-resonance conditions with the instability mode maintaining a fixed orientation with respect to the basic state. Thus, we are able to demonstrate the existence of two alternative but congruent explanations for the instability. Additionally, we show that basic states may depart substantially from solid-body rotation even with modest cylinder tilt angles, and growth rates for instabilities may be sufficiently large that nonlinear saturation to disordered states can occur within approximately ten cylinder revolutions, in agreement with experimental observations.

Three-dimensional instabilities and inertial waves in a rapidly rotating split-cylinder flow

2016 ◽  
Vol 800 ◽  
pp. 666-687 ◽  
Author(s):  
Juan M. Lopez ◽  
Paloma Gutierrez-Castillo
Keyword(s):  
Solid Body ◽  
Differential Rotation ◽  
Three Dimensional ◽  
Side Wall ◽  
Wall Layer ◽  
Bulk Flow ◽  
Body Rotation ◽  
Rotating Waves ◽  
Solid Body Rotation ◽  
Inertial Wave

The nonlinear dynamics of the flow in a differentially rotating split cylinder is investigated numerically. The differential rotation, with the top half of the cylinder rotating faster than the bottom half, establishes a basic state consisting of a bulk flow that is essentially in solid-body rotation at the mean rotation rate of the cylinder and boundary layers where the bulk flow adjusts to the differential rotation of the cylinder halves, which drives a strong meridional flow. There are Ekman-like layers on the top and bottom end walls, and a Stewartson-like side wall layer with a strong downward axial flow component. The complicated bottom corner region, where the downward flow in the side wall layer decelerates and negotiates the corner, is the epicentre of a variety of instabilities associated with the local shear and curvature of the flow, both of which are very non-uniform. Families of both high and low azimuthal wavenumber rotating waves bifurcate from the basic state in Eckhaus bands, but the most prominent states found near onset are quasiperiodic states corresponding to mixed modes of the high and low azimuthal wavenumber rotating waves. The frequencies associated with most of these unsteady three-dimensional states are such that spiral inertial wave beams are emitted from the bottom corner region into the bulk, along cones at angles that are well predicted by the inertial wave dispersion relation, driving the bulk flow away from solid-body rotation.

On the active feedback control of a swirling flow in a finite-length pipe

2013 ◽  
Vol 737 ◽  
pp. 280-307 ◽  
Author(s):  
Shixiao Wang ◽  
Zvi Rusak ◽  
Steve Taylor ◽  
Rui Gong
Keyword(s):  
Feedback Control ◽  
Control Problem ◽  
Solid Body ◽  
Finite Length ◽  
Critical Level ◽  
Control Method ◽  
Feedback Stabilization ◽  
Mode Analysis ◽  
Body Rotation ◽  
Solid Body Rotation

AbstractThe 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.

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

2018 ◽  
Vol 846 ◽  
pp. 1114-1152 ◽  
Author(s):  
Chunjuan Feng ◽  
Feng Liu ◽  
Zvi Rusak ◽  
Shixiao Wang
Keyword(s):  
Three Dimensional ◽  
Linear Stability Theory ◽  
Neutral Stability ◽  
Instability Mode ◽  
Body Rotation ◽  
Operational Conditions ◽  
Solid Body Rotation ◽  
Flow Evolution ◽  
Spiral Type ◽  
Stability Line

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.

Sign in / Sign up

Export Citation Format

Share Document