Three-dimensionality in the wake of a rapidly rotating cylinder in uniform flow

2013 ◽  
Vol 730 ◽  
pp. 379-391 ◽  
Author(s):  
A. Rao ◽  
J. S. Leontini ◽  
M. C. Thompson ◽  
K. Hourigan
Keyword(s):  
Steady State ◽  
Rotation Rate ◽  
Free Stream ◽  
Three Dimensional ◽  
Rotating Cylinder ◽  
Marginal Stability ◽  
Cylinder Surface ◽  
Rotation Rates ◽  
Centrifugal Instability ◽  
The Stability

AbstractThe flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

Experimental evidence of new three-dimensional modes in the wake of a rotating cylinder

2013 ◽  
Vol 734 ◽  
pp. 567-594 ◽  
Author(s):  
A. Radi ◽  
M. C. Thompson ◽  
A. Rao ◽  
K. Hourigan ◽  
J. Sheridan
Keyword(s):  
Rotation Rate ◽  
Numerical Study ◽  
Water Channel ◽  
Three Dimensional ◽  
Cross Flow ◽  
Reynolds Numbers ◽  
Rotating Cylinder ◽  
Travelling Wave ◽  
Cylinder Surface ◽  
Rotation Rates

AbstractA recent numerical study by Rao et al. (J. Fluid Mech., vol. 717, 2013, pp. 1–29) predicted the existence of several previously unobserved linearly unstable three-dimensional modes in the wake of a spinning cylinder in cross-flow. While linear stability analysis suggests that some of these modes exist for relatively limited ranges of Reynolds numbers and rotation rates, this may not be true for fully developed nonlinear wakes. In the current paper, we present the results of water channel experiments on a rotating cylinder in cross-flow, for Reynolds numbers $200\leqslant \mathit{Re}\leqslant 275$ and non-dimensional rotation rates $0\leqslant \alpha \leqslant 2. 5$. Using particle image velocimetry and digitally post-processed hydrogen bubble flow visualizations, we confirm the existence of the predicted modes for the first time experimentally. For instance, for $\mathit{Re}= 275$ and a rotation rate of $\alpha = 1. 7$, we observe a subharmonic mode, mode C, with a spanwise wavelength of ${\lambda }_{z} / d\approx 1. 1$. On increasing the rotation rate, two modes with a wavelength of ${\lambda }_{z} / d\approx 2$ become unstable in rapid succession, termed modes D and E. Mode D grows on a shedding wake, whereas mode E consists of streamwise vortices on an otherwise steady wake. For $\alpha \gt 2. 2$, a short-wavelength mode F appears localized close to the cylinder surface with ${\lambda }_{z} / d\approx 0. 5$, which is presumably a manifestation of centrifugal instability. Unlike the other modes, mode F is a travelling wave with a spanwise frequency of ${\mathit{St}}_{3D} \approx 0. 1$. In addition to these new modes, observations on the one-sided shedding process, known as the ‘second shedding’, are reported for $\alpha = 5. 1$. Despite suggestions from the literature, this process seems to be intrinsically three-dimensional. In summary, our experiments confirm the linear predictions by Rao et al., with very good agreement of wavelengths, symmetries and the phase velocity for the travelling mode. Apart from this, these experiments examine the nonlinear saturated state of these modes and explore how the existence of multiple unstable modes can affect the selected final state. Finally, our results establish that several distinct three-dimensional instabilities exist in a relatively confined area on the $\mathit{Re}$–$\alpha $ parameter map, which could account for their non-detection previously.

Flow past a rotating cylinder

2003 ◽  
Vol 476 ◽  
pp. 303-334 ◽  
Author(s):  
SANJAY MITTAL ◽  
BHASKAR KUMAR
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Stability Analysis ◽  
Vortex Shedding ◽  
Rotation Rate ◽  
Rotating Cylinder ◽  
Two Dimensional ◽  
Stabilized Finite Element ◽  
Rotation Rates ◽  
Flow Past

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

scholarly journals In-line flow-induced vibrations of a rotating cylinder

2015 ◽  
Vol 781 ◽  
pp. 127-165 ◽  
Author(s):  
Rémi Bourguet ◽  
David Lo Jacono
Keyword(s):  
Rotation Rate ◽  
Vibration Amplitude ◽  
Three Dimensional ◽  
Low Frequency ◽  
Rotating Cylinder ◽  
The Body ◽  
Cylinder Surface ◽  
Force Frequency ◽  
Direction Parallel ◽  
Flow Induced Vibrations

The flow-induced vibrations of an elastically mounted circular cylinder, free to oscillate in the direction parallel to the current and subjected to a forced rotation about its axis, are investigated by means of two- and three-dimensional numerical simulations, at a Reynolds number equal to 100 based on the cylinder diameter and inflow velocity. The cylinder is found to oscillate up to a rotation rate (ratio between the cylinder surface and inflow velocities) close to 2 (first vibration region), then the body and the flow are steady until a rotation rate close to 2.7 where a second vibration region begins. Each vibration region is characterized by a specific regime of response. In the first region, the vibration amplitude follows a bell-shaped evolution as a function of the reduced velocity (inverse of the oscillator natural frequency). The maximum vibration amplitudes, even though considerably augmented by the rotation relative to the non-rotating body case, remain lower than 0.1 cylinder diameters. Due to their trends as functions of the reduced velocity and to the fact that they develop under a condition of wake-body synchronization or lock-in, the responses of the rotating cylinder in this region are comparable to the vortex-induced vibrations previously described in the absence of rotation. The symmetry breaking due to the rotation is shown to directly impact the structure displacement and fluid force frequency contents. In the second region, the vibration amplitude tends to increase unboundedly with the reduced velocity. It may become very large, higher than 2.5 diameters in the parameter space under study. Such structural oscillations resemble the galloping responses reported for non-axisymmetric bodies. They are accompanied by a dramatic amplification of the fluid forces compared to the non-vibrating cylinder case. It is shown that body oscillation and flow unsteadiness remain synchronized and that a variety of wake topologies may be encountered in this vibration region. The low-frequency, large-amplitude responses are associated with novel asymmetric multi-vortex patterns, combining a pair and a triplet or a quartet of vortices per cycle. The flow is found to undergo three-dimensional transition in the second vibration region, with a limited influence on the system behaviour. It appears that the transition occurs for a substantially lower rotation rate than for a rigidly mounted cylinder.

scholarly journals The three-dimensional transition in the flow around a rotating cylinder

2008 ◽  
Vol 607 ◽  
pp. 1-11 ◽  
Author(s):  
R. EL AKOURY ◽  
M. BRAZA ◽  
R. PERRIN ◽  
G. HARRAN ◽  
Y. HOARAU
Keyword(s):  
Rotation Rate ◽  
Critical Reynolds Number ◽  
Rotation Speed ◽  
Three Dimensional ◽  
Rotating Cylinder ◽  
Constant Angular Velocity ◽  
Three Dimensional Flow ◽  
Rotation Rates ◽  
Proper Orthogonal ◽  
Rate Suppression

The flow around a circular cylinder rotating with a constant angular velocity, placed in a uniform stream, is investigated by means of two- and three-dimensional direct numerical simulations. The successive changes in the flow pattern are studied as a function of the rotation rate. Suppression of vortex shedding occurs as the rotation rate increases (>2). A second kind of instabilty appears for higher rotation speed where a series of counter-clockwise vortices is shed in the upper shear layer. Three-dimensional computations are carried out to analyse the three-dimensional transition under the effect of rotation for low rotation rates. The rotation attenuates the secondary instability and increases the critical Reynolds number for the appearance of this instability. The linear and nonlinear parts of the three-dimensional transition have been quantified by means of the amplitude evolution versus time, using the Landau global oscillator model. Proper orthogonal decomposition of the three-dimensional fields allowed identification of the most energetic modes and three-dimensional flow reconstruction involving a reduced number of modes.

Three-dimensionality in the wake of a rotating cylinder in a uniform flow

2013 ◽  
Vol 717 ◽  
pp. 1-29 ◽  
Author(s):  
A. Rao ◽  
J. Leontini ◽  
M. C. Thompson ◽  
K. Hourigan
Keyword(s):  
Unsteady Flow ◽  
Steady Flow ◽  
Parameter Space ◽  
Vortex Shedding ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Rotating Cylinder ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Rotation Rates

AbstractThe wake of a rotating circular cylinder in a free stream is investigated for Reynolds numbers $\mathit{Re}\leqslant 400$ and non-dimensional rotation rates of $\alpha \leqslant 2. 5$. Two aspects are considered. The first is the transition from a steady flow to unsteady flow characterized by periodic vortex shedding. The two-dimensional computations show that the onset of unsteady flow is delayed to higher Reynolds numbers as the rotation rate is increased, and vortex shedding is suppressed for $\alpha \geqslant 2. 1$ for all Reynolds numbers in the parameter space investigated. The second aspect investigated is the transition from two-dimensional to three-dimensional flow using linear stability analysis. It is shown that at low rotation rates of $\alpha \leqslant 1$, the three-dimensional transition scenario is similar to that of the non-rotating cylinder. However, at higher rotation rates, the three-dimensional scenario becomes increasingly complex, with three new modes identified that bifurcate from the unsteady flow, and two modes that bifurcate from the steady flow. Curves of marginal stability for all of the modes are presented in a parameter space map, the defining characteristics for each mode presented, and the physical mechanisms of instability are discussed.

On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

1999 ◽  
Vol 390 ◽  
pp. 127-150 ◽  
Author(s):  
V. A. VLADIMIROV ◽  
H. K. MOFFATT ◽  
K. I. ILIN
Keyword(s):  
Steady State ◽  
Variational Principles ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Steady States ◽  
Mhd Equations ◽  
Quadratic Functional ◽  
The First Variation ◽  
The Stability ◽  
And Variational Principles

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

The influence of a small upstream wire on transition in a rotating cylinder wake

2015 ◽  
Vol 769 ◽  
Author(s):  
Anirudh Rao ◽  
Alexander Radi ◽  
Justin S. Leontini ◽  
Mark C. Thompson ◽  
John Sheridan ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Small Diameter ◽  
Reynolds Numbers ◽  
Rotating Cylinder ◽  
Hydrogen Bubble ◽  
Instability Mode ◽  
Cylinder Wake ◽  
Rotation Rates ◽  
Mode Transitions

Recent experimental research on rotating cylinder wakes has found that a previously numerically predicted subharmonic instability mode, mode C, occurs for considerably lower rotation rates than predicted through stability analysis, yet other mode transitions occur closer to the predicted onset. One difference between the theoretical and experimental set-ups is the use of a small-diameter hydrogen bubble visualisation wire placed upstream of the rotating cylinder. The current paper tests the hypothesis that a wire, of only $1/100$th of the cylinder diameter, placed five diameters upstream of the cylinder, sufficiently perturbs the flow to substantially affect certain wake transitions, including the onset of mode C. This is achieved using stability analysis of a flow that includes the upstream wire. The results indeed show that the wire of a tiny diameter induces a non-negligible asymmetry in the flow, triggering the subharmonic mode at substantially lower rotation rates. Furthermore, at higher rotation rates, the onset of two other three-dimensional modes are delayed to higher Reynolds numbers. These results make the point that even seemingly minute perturbations caused by minimally intrusive methods may result in substantially altered experimental flow behaviour.

Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Ankur Gupta ◽  
Saikat Chakraborty
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Three Dimensional ◽  
Diffusion Reaction ◽  
Dynamic Simulations ◽  
Transverse Mixing ◽  
Autocatalytic Reactions ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Parametric Analyses

Interaction between transport and reaction generates a variety of complex spatio-temporal patterns in chemical reactors. These patterned states, which are typically initiated by autocatalytic effects and sustained by differences in diffusion/local mixing rates, often cause undesired effects in the reactor. In this work, we analyze the dynamic evolution of mixing-limited spatial pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors using two-dimensional (2-D) convection-diffusion-reaction (CDR) models that are obtained through rigorous spatial averaging of the three-dimensional (3-D) CDR model using Liapunov-Schmidt technique of bifurcation theory. We use the spatially-averaged 2-D CDR model (and its "regularized" form) to perform steady-state bifurcation analysis that captures the region of multiple solutions, and we analyze the stability of these multiple steady states to transverse perturbations using linear stability analysis. Parametric analyses of the steady-state bifurcation diagrams and stability boundaries show that when transverse mixing is significantly slower than the rate of autocatalytic reaction, mixing-limited patterns emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Our dynamic simulations show the emergence of three different types of spatial patterns namely, Band, Anti-phase and Target, depending on the nature of transverse perturbation. The temporal evolution of these patterns consists of rapid intensification of the concentration-segregation process (especially when transverse mixing is much slower than reaction) followed by slow diffusion-mediated return to symmetry that occurs at time scales much larger than the reactor residence time. Our parametric analysis of the dynamics reveals that while larger Péclet numbers (both axial and transverse) increase the stability and decay time of the patterned states, larger Damköhler numbers lead to faster ignition resulting in the opposite effect.

scholarly journals Effects of the Control Parameters on the Stability of a Laminar Boundary Layer on a Porous Flat Plate

2021 ◽  
Vol 26 (4) ◽  
pp. 113-127
Author(s):  
T.F. Lihonou ◽  
A.V. Monwanou ◽  
C.H. Miwadinou ◽  
J.B. Chabi Orou
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Temporal Stability ◽  
Three Dimensional ◽  
Darcy Number ◽  
Basic Flow ◽  
Marginal Stability ◽  
Stability Diagrams ◽  
Plane Plate ◽  
The Stability

Abstract This work is devoted to the analysis of the linear temporal stability of a laminar dynamic boundary layer on a horizontal porous plane plate. The basic flow is assumed to be laminar and two-dimensional. The basic flow velocity profiles are obtained by numerically solving the Blasius equation using the Runge-Kutta method. The perturbations of these basic solutions are expressed in the form of three-dimensional Tollmien-Schlichting waves. The formulation of the stability problem leads to the Orr-Sommerfeld equation modified by the permeability parameter (Darcy number) and the small Reynolds number. This equation is given in a general form which can be applied to the Chebyshev domain and the boundary layer domain and solved numerically using the Chebyshev spectral collocation method. The marginal stability diagrams, the critical Reynolds numbers and the eigenvalue spectra are obtained for different values of the parameters which have modified the stability equation. Numerical solutions indicate the importance of the effect of these parameters on the flow stability characteristics.

The dependence of global and local metrics of super-rotation on planetary rotation rate

2020 ◽  
Author(s):  
Neil Lewis ◽  
Greg Colyer ◽  
Peter Read
Keyword(s):  
Angular Momentum ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Axial Component ◽  
Rotation Rates ◽  
The Earth ◽  
Planetary Rotation ◽  
Rate Limit ◽  
Global And Local ◽  
Rotation Strength

&lt;div&gt; &lt;div&gt;Super-rotation is a phenomenon in atmospheric dynamics where the axial angular momentum of an atmosphere in some way exceeds that of the underlying planet. In this presentation, we will discuss the dependency of both globally-integrated, and local metrics of super-rotation on planetary rotation rate, revealed through analysis of idealised General Circulation Model experiments. The model used here is based on the Held-Suarez benchmark for a dry, 'Earth-like' atmosphere, and results from both axisymmetric and three-dimensional experiments will be presented. Previous work has shown that the three-dimensional configuration used here will transition to a state of equatorial super-rotation if the rotation rate is reduced sufficiently from the Earth's. This motivates the question: How does super-rotation strength depend on rotation rate?&lt;/div&gt; &lt;br&gt;&lt;div&gt;We will use the term 'global super-rotation' to refer to an atmosphere with excess of globally-integrated axial angular momentum relative to that achieved by solid body co-rotation with the underlying planet, and 'local super-rotation' to refer to the existence of some region within the atmosphere where axial angular momentum exceeds that of the underlying planet at the equator. In an inviscid, axisymmetric atmosphere, the axial component of specific angular momentum is materially conserved. Consequently, in such a system local super-rotation is forbidden, although global super-rotation may still be achieved if a meridional circulation is able to transport fluid equilibrated with the equatorial surface poleward. If the restriction of axisymmetry is lifted, then local super-rotation may exist if non-axisymmetric disturbances that act to transport angular momentum up-gradient are present. The atmospheres of Venus, the Earth, Mars, and Titan may be considered to be globally super-rotating, however only Venus and Titan exhibit permanent local super-rotation at the equator.&lt;/div&gt; &lt;br&gt;&lt;div&gt;The results from axisymmetric experiments reveal that at high rotation rate (e.g., greater than 1/4 of the Earth's), the degree of global super-rotation scales inversely with the square of the rotation rate. In the low rotation rate limit, the degree of global super-rotation saturates, and becomes independent of rotation rate. We will show that the high, and low rotation rate dependencies can be predicted by a single analytic scaling for global super-rotation. Our three-dimensional experiments exhibit the same scaling behaviour for global super-rotation as observed in the axisymmetric experiments. The degree of global super-rotation achieved by the three-dimensional experiments is less than that of the axisymmetric experiments at high rotation rates, and greater at lower rotation rates, but in both limits the deviation from the axisymmetric 'base circulation' is small. In the low-rotation rate limit, local super-rotation is accelerated at the equator, which is consistent with the three-dimensional experiments obtaining a higher degree of global super-rotation than their axisymmetric counterparts. Estimates for global super-rotation strength on the Earth and Mars agree closely with the results of our three-dimensional numerical experiments, but Venus and Titan achieve substantially stronger global, and local super-rotation than found here. It appears that low rotation rate alone cannot induce substantial excess global super-rotation, relative to the axisymmetric base circulation we identify.&lt;/div&gt; &lt;/div&gt;

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