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.

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.

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.

Simulation of swept-wing vortices using nonlinear parabolized stability equations

2000 ◽  
Vol 405 ◽  
pp. 325-349 ◽  
Author(s):  
TIM S. HAYNES ◽  
HELEN L. REED
Keyword(s):  
Pressure Gradient ◽  
Three Dimensional ◽  
Experimental Results ◽  
Linear Stability Theory ◽  
Neutral Stability ◽  
Boundary Layer Flows ◽  
Unstable Flow ◽  
Parabolized Stability Equations ◽  
Unstable Boundary Layer ◽  
Dimensional Boundary

The nonlinear development of stationary crossflow vortices over a 45° swept NLF(2)-0415 airfoil is studied. Previous investigations indicate that the linear stability theory (LST) is unable to accurately describe the unstable flow over crossflow-dominated configurations. In recent years the development of nonlinear parabolized stability equations (NPSE) has opened new pathways toward understanding unstable boundary-layer flows. This is because the elegant inclusion of nonlinear and non-parallel effects in the NPSE allows accurate stability analyses to be performed without the difficulties and overhead associated with direct numerical simulations (DNS). NPSE results are presented here and compared with experimental results obtained at the Arizona State University Unsteady Wind Tunnel. The comparison shows that the saturation of crossflow disturbances is responsible for the discrepancy between LST and experimental results for cases with strong favourable pressure gradient. However, for cases with a weak favourable pressure gradient the stationary crossflow disturbances are damped and nonlinearity is unimportant. The results presented here clearly show that for the latter case curvature and non-parallel effects are responsible for the previously observed discrepancies between LST and experiment. The comparison of NPSE and experimental results shows excellent agreement for both configurations.Through this work, a detailed quantitative comparison and validation of NPSE with a careful experiment has now been provided for three-dimensional boundary layers. Moreover, the results validate the experiments of Reibert et al. (1996), and Radeztsky et al. (1993, 1994) suggesting that their databases can be used by future researchers to verify theories and numerical schemes. The results show the inadequacy of linear theories for modelling these flows for significant crossflow amplitude and demonstrate the effects of weak curvature to be more significant than slight changes in basic state, especially near neutral-stability locations.

STABILIZATION ANALYSIS OF A MULTIPLE LOOK-AHEAD MODEL WITH DRIVER REACTION DELAYS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250048 ◽  
Author(s):  
JIANZHONG CHEN ◽  
ZHONGKE SHI ◽  
YANMEI HU
Keyword(s):  
Reaction Time ◽  
Time Delay ◽  
Kdv Equation ◽  
Mkdv Equation ◽  
Linear Stability Theory ◽  
Neutral Stability ◽  
Look Ahead ◽  
Unfavorable Effect ◽  
The Stability ◽  
Stability Line

A multiple look-ahead model is extended to take into account the reaction-time delay of drivers. The stability condition of this model is obtained by using the linear stability theory. Through nonlinear analysis, the Korteweg–de Vries (KdV) equation near the neutral stability line and the modified KdV (mKdV) equation near the critical point are derived. Both the analytical and simulation results demonstrate that the stabilization of traffic flow is weakened with increasing the reaction-time delay of drivers, and multiple look-ahead consideration could partially remedy this unfavorable effect.

Oval Flow Mode Between Two Corotating Disks With Stationary Shroud

2014 ◽  
Vol 137 (3) ◽  
Author(s):  
Ching Min Hsu ◽  
Jia-Kun Chen ◽  
Min Kai Hsieh ◽  
Rong Fung Huang
Keyword(s):  
Flow Structure ◽  
Solid Body ◽  
Circumferential Velocity ◽  
Body Rotation ◽  
Flow Core ◽  
Core Flow ◽  
Node Point ◽  
Solid Body Rotation ◽  
Characteristic Flow ◽  
Buffer Region

The characteristic flow behavior, time-averaged velocity distributions, phase-resolved ensemble-averaged velocity profiles, and turbulence properties of the flow in the interdisk midplane between shrouded two corotating disks at the interdisk spacing to disk radius aspect ratio 0.2 and rotation Reynolds number 3.01 × 105 were experimentally studied by flow visualization method and particle image velocimetry (PIV). An oval core flow structure rotating at a frequency 60% of the disks rotating frequency was observed. Based on the analysis of relative velocities, the flow in the region outside the oval core flow structure consisted of two large vortex rings, which move circumferentially with the rotation motion of the oval flow core. Four characteristic flow regions—solid-body-rotation-like region, buffer region, vortex region, and shroud-influenced region—were identified in the flow field. The solid-body-rotation-like region, which was featured by its linear distribution of circumferential velocity and negligibly small radial velocity, was located within the inscribing radius of the oval flow core. The vortex region was located outside the circumscribing radius of the oval flow core. The buffer region existed between the solid-body-rotation-like region and the vortex region. In the buffer region, there existed a “node” point that the propagating circumferential velocity waves diminished. The circumferential random fluctuation intensity presented minimum values at the node point and high values in the solid-body-rotation-like region and shroud-influenced region due to the shear effect induced by the wall.

scholarly journals Azimuthal velocity profiles in Rayleigh-stable Taylor–Couette flow and implied axial angular momentum transport

2015 ◽  
Vol 774 ◽  
pp. 342-362 ◽  
Author(s):  
Freja Nordsiek ◽  
Sander G. Huisman ◽  
Roeland C. A. van der Veen ◽  
Chao Sun ◽  
Detlef Lohse ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Azimuthal Velocity ◽  
Velocity Profiles ◽  
Body Rotation ◽  
Solid Body Rotation ◽  
Angular Momentum Transport ◽  
Taylor Couette ◽  
Taylor Couette Flow

We present azimuthal velocity profiles measured in a Taylor–Couette apparatus, which has been used as a model of stellar and planetary accretion disks. The apparatus has a cylinder radius ratio of ${\it\eta}=0.716$, an aspect ratio of ${\it\Gamma}=11.74$, and the plates closing the cylinders in the axial direction are attached to the outer cylinder. We investigate angular momentum transport and Ekman pumping in the Rayleigh-stable regime. This regime is linearly stable and is characterized by radially increasing specific angular momentum. We present several Rayleigh-stable profiles for shear Reynolds numbers $\mathit{Re}_{S}\sim O(10^{5})$, for both ${\it\Omega}_{i}>{\it\Omega}_{o}>0$ (quasi-Keplerian regime) and ${\it\Omega}_{o}>{\it\Omega}_{i}>0$ (sub-rotating regime), where ${\it\Omega}_{i,o}$ is the inner/outer cylinder rotation rate. None of the velocity profiles match the non-vortical laminar Taylor–Couette profile. The deviation from that profile increases as solid-body rotation is approached at fixed $\mathit{Re}_{S}$. Flow super-rotation, an angular velocity greater than those of both cylinders, is observed in the sub-rotating regime. The velocity profiles give lower bounds for the torques required to rotate the inner cylinder that are larger than the torques for the case of laminar Taylor–Couette flow. The quasi-Keplerian profiles are composed of a well-mixed inner region, having approximately constant angular momentum, connected to an outer region in solid-body rotation with the outer cylinder and attached axial boundaries. These regions suggest that the angular momentum is transported axially to the axial boundaries. Therefore, Taylor–Couette flow with closing plates attached to the outer cylinder is an imperfect model for accretion disk flows, especially with regard to their stability.

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