Inviscid axisymmetric absolute instability of swirling jets

2008 ◽  
Vol 613 ◽  
pp. 1-33 ◽  
Author(s):  
J. J. HEALEY
Keyword(s):  
Shear Layer ◽  
Axial Velocity ◽  
Finite Thickness ◽  
Outer Cylinder ◽  
Velocity Profiles ◽  
Dispersion Relations ◽  
Absolute Instability ◽  
Pinch Point ◽  
Swirling Jets ◽  
Codimension Two

The propagation characteristics of inviscid axisymmetric linearized disturbances to swirling jets are investigated for two families of model velocity profiles. Briggs' method is applied to dispersion relations to determine when the basic swirling jets are absolutely or convectively unstable. The method is also applied to the neutral inertial waves used by Benjamin to characterize the subcritical or supercritical nature of the flow. Although these waves are neutral, Briggs' method nonetheless indicates a qualitative change in behaviour at Benjamin's criticality condition. The first model jet has uniform axial velocity, rigid-body rotation and issues into still fluid. A known difficulty in the application of Briggs' method to the analytical dispersion relation for inviscid waves in this flow is resolved. The difficulty is that the pinch point can cross into the left half of the complex-wavenumber plane, where waves grow exponentially with radius and fail to satisfy homogeneous boundary conditions. In this paper the physical consequences of this behaviour are explained. It is shown that if the still fluid is of infinite extent in the radial direction, then the jet is convectively unstable to axisymmetric waves, but if the jet is confined by an outer cylinder concentric with the jet axis, then it becomes absolutely unstable to axisymmetric waves provided that the swirl (ratio of azimuthal to axial velocity) is large enough. This destabilizing effect of confinement occurs however large the radius of the outer cylinder. A second family of model swirling jets with smooth profiles and a finite thickness shear layer at the jet edge is also studied. The inviscid stability equations are solved numerically in this case. The results from the analytical dispersion relations are reproduced as the shear layer thickness tends to zero. However, increasing this thickness acts to destabilize the absolute instability of axisymmetric waves, apparently due to the centrifugal instability present in the shear layer. It is suggested that the transition from convective to absolute instability could be associated with the onset of an unsteady vortex breakdown. The swirl required to produce this transition can be either greater, or less, than the swirl required to produce the transition from supercritical to subcritical flow, depending on the details of the basic velocity profiles. A codimension-two point in parameter space can exist where the unsteady bifurcation associated with the convective–absolute transition coincides with the steady bifurcation associated with the supercritical–subcritical transition. Such codimension-two points can control a rich variety of nonlinear dynamical behaviour.

scholarly journals Stability analysis and breakup length calculations for steady planar liquid jets

2010 ◽  
Vol 668 ◽  
pp. 384-411 ◽  
Author(s):  
M. R. TURNER ◽  
J. J. HEALEY ◽  
S. S. SAZHIN ◽  
R. PIAZZESI
Keyword(s):  
Stability Analysis ◽  
Velocity Profile ◽  
Shear Layer ◽  
Fluid Velocity ◽  
Finite Thickness ◽  
Liquid Jet ◽  
Absolute Instability ◽  
Breakup Length ◽  
The Stability ◽  
Thickness Shear

This study uses spatio-temporal stability analysis to investigate the convective and absolute instability properties of a steady unconfined planar liquid jet. The approach uses a piecewise linear velocity profile with a finite-thickness shear layer at the edge of the jet. This study investigates how properties such as the thickness of the shear layer and the value of the fluid velocity at the interface within the shear layer affect the stability properties of the jet. It is found that the presence of a finite-thickness shear layer can lead to an absolute instability for a range of density ratios, not seen when a simpler plug flow velocity profile is considered. It is also found that the inclusion of surface tension has a stabilizing effect on the convective instability but a destabilizing effect on the absolute instability. The stability results are used to obtain estimates for the breakup length of a planar liquid jet as the jet velocity varies. It is found that reducing the shear layer thickness within the jet causes the breakup length to decrease, while increasing the fluid velocity at the fluid interface within the shear layer causes the breakup length to increase. Combining these two effects into a profile, which evolves realistically with velocity, gives results in which the breakup length increases for small velocities and decreases for larger velocities. This behaviour agrees qualitatively with existing experiments on the breakup length of axisymmetric jets.

The effect of confinement on the stability of viscous planar jets and wakes

2010 ◽  
Vol 656 ◽  
pp. 309-336 ◽  
Author(s):  
S. J. REES ◽  
M. P. JUNIPER
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Layer Thickness ◽  
Finite Thickness ◽  
Temporal Analysis ◽  
Velocity Profiles ◽  
Shear Layers ◽  
Channel Width ◽  
Planar Jet ◽  
Spatio Temporal

This theoretical study examines confined viscous planar jet/wake flows with continuous velocity profiles. These flows are characterized by the shear, confinement, Reynolds number and shear-layer thickness. The primary aim of this paper is to determine the effect of confinement on viscous jets and wakes and to compare these results with corresponding inviscid results. The secondary aim is to consider the effect of viscosity and shear-layer thickness. A spatio-temporal analysis is performed in order to determine absolute/convective instability criteria. This analysis is carried out numerically by solving the Orr–Sommerfeld equation using a Chebyshev collocation method. Results are produced over a large range of parameter space, including both co-flow and counter-flow domains and confinements corresponding to 0.1 < h2/h1 < 10, where the subscripts 1 and 2 refer to the inner and outer streams, respectively. The Reynolds number, which is defined using the channel width, takes values between 10 and 1000. Different velocity profiles are used so that the shear layers occupy between 1/2 and 1/24 of the channel width. Results indicate that confinement has a destabilizing effect on both inviscid and viscous flows. Viscosity is found always to be stabilizing, although its effect can safely be neglected above Re = 1000. Thick shear layers are found to have a stabilizing effect on the flow, but infinitely thin shear layers are not the most unstable; having shear layers of a small, but finite, thickness gives rise to the strongest instability.

ABSOLUTE INSTABILITY OF A SUPERCRITICAL SHEAR LAYER

2016 ◽  
Vol 26 (8) ◽  
pp. 815-826 ◽  
Author(s):  
Qing-fei Fu ◽  
Li-Jun Yang ◽  
Chao-Jie Mo
Keyword(s):  
Shear Layer ◽  
Absolute Instability

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.

scholarly journals Viscous effects in the absolute–convective instability of the Batchelor vortex

2002 ◽  
Vol 459 ◽  
pp. 371-396 ◽  
Author(s):  
C. OLENDRARU ◽  
A. SELLIER
Keyword(s):  
Reynolds Number ◽  
Convective Instability ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Viscous Effects ◽  
Swirling Jets ◽  
Bending Mode ◽  
Transitional Mode ◽  
The Absolute

The effects of viscosity on the instability properties of the Batchelor vortex are investigated. The characteristics of spatially amplified branches are first documented in the convectively unstable regime for different values of the swirl parameter q and the co-flow parameter a at several Reynolds numbers Re. The absolute–convective instability transition curves, determined by the Briggs–Bers zero-group velocity criterion, are delineated in the (a, q)-parameter plane as a function of Re. The azimuthal wavenumber m of the critical transitional mode is found to depend on the magnitude of the swirl q and on the jet (a > −0.5) or wake (a < −0.5) nature of the axial flow. At large Reynolds numbers, the inviscid results of Olendraru et al. (1999) are recovered. As the Reynolds number decreases, the pocket of absolute instability in the (a, q)-plane is found to shrink gradually. At Re = 667; the critical transitional modes for swirling jets are m = −2 or m = −3 and absolute instability prevails at moderate swirl values even in the absence of counterflow. For higher swirl levels, the bending mode m = −1 becomes critical. The results are in good overall agreement with those obtained by Delbende et al. (1998) at the same Reynolds number. However, a bending (m = +1) viscous mode is found to partake in the outer absolute–convective instability transition for jets at very low positive levels of swirl. This asymmetric branch is the spatial counterpart of the temporal viscous mode isolated by Khorrami (1991) and Mayer & Powell (1992). At Re = 100, the critical transitional mode for swirling jets is m = −2 at moderate and high swirl values and, in order to trigger an absolute instability, a slight counterflow is always required. A bending (m = +1) viscous mode again becomes critical at very low swirl values. For wakes (a < −0.5) the critical transitional mode is always found to be the bending mode m = −1, whatever the Reynolds number. However, above q = 1.5, near-neutral centre modes are found to define a tongue of weak absolute instability in the (a, q)-plane. Such modes had been analytically predicted by Stewartson & Brown (1985) in a strictly temporal inviscid framework.

Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles

2009 ◽  
Vol 21 (2) ◽  
pp. 023602 ◽  
Author(s):  
A. C. Verkaik ◽  
B. W. A. M. M. Beulen ◽  
A. C. B. Bogaerds ◽  
M. C. M. Rutten ◽  
F. N. van de Vosse
Keyword(s):  
Axial Velocity ◽  
Computational Analysis ◽  
Velocity Profiles ◽  
Volume Flow

scholarly journals Optimal Taylor–Couette flow: direct numerical simulations

2013 ◽  
Vol 719 ◽  
pp. 14-46 ◽  
Author(s):  
Rodolfo Ostilla ◽  
Richard J. A. M. Stevens ◽  
Siegfried Grossmann ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Angular Velocity ◽  
Couette Flow ◽  
Scaling Laws ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Experimental Result ◽  
Counter Rotation ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractWe numerically simulate turbulent Taylor–Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh–Bénard flow. Reynolds numbers of $R{e}_{i} = 8\times 1{0}^{3} $ and $R{e}_{o} = \pm 4\times 1{0}^{3} $ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers $Ta$ up to $1{0}^{8} $. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente Turbulent Taylor–Couette (${T}^{3} C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a= - {\omega }_{o} / {\omega }_{i} $ of about ${a}_{\mathit{opt}} = 0. 33$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of ${a}_{\mathit{opt}} = 0. 15$ for $Ta= 2. 5\times 1{0}^{7} $. An explanation for this shift is elucidated, consistent with the experimental result that ${a}_{\mathit{opt}} $ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities.

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