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Unsteady two-dimensional jet with flexible flaps at the channel exit
- Prashant Das, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 845 / 25 June 2018
- Published online by Cambridge University Press:
- 26 April 2018, pp. 462-498
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The present work studies the effect of passive exit flexibility on a two-dimensional starting jet. The exit flexibility is introduced by attaching two flexible (deformable) flaps at the jet exit of a high aspect ratio rectangular duct with the flaps initially being parallel to the channel walls. A controlled piston motion is used to generate the starting jet, which is composed of a rapid acceleration to a constant velocity ($U_{p}$) that is maintained for a given duration of time, after which it is brought to rest impulsively. The parameters which are varied include the flexural rigidity ($EI$) of the flaps, flap length ($L_{f}$) and piston speed ($U_{p}$), with measurements of the flap kinematics and flow field in each case. The flaps initially bulge due to the acceleration of the piston from rest, with this bulge growing in size and moving downstream as the flow develops, culminating in a large opening at the flap exit. Subsequently, the flaps return to their initial parallel position and remain there as long as the piston is in motion. Once the piston stops, the flaps collapse inwards due to fluid deceleration causing additional flow out of the flap region in the form of a jet that adds to the net amount of fluid pushed by the piston. We find that the flap kinematics is affected by the flap $EI$ and $L_{f}$ besides $U_{p}$. We define a non-dimensional flexural rigidity $EI^{\ast }=EI_{eq}/(1/2\unicode[STIX]{x1D70C}U_{p}^{2}L_{f}^{2}d)$, where $EI_{eq}$ is an equivalent flexural rigidity which takes the self-weight of the flaps into account ($d=\text{channel width}$; $\unicode[STIX]{x1D70C}=$ fluid density). We find that across different $EI_{eq}$, $L_{f}$, and piston speeds, the maximum opening of the flap tip and the time taken to reach this maximum opening in terms of $L/L_{f}$ (where $L=\text{fluid slug length}$) fall on a single curve for all the cases studied, when plotted with $EI^{\ast }$. Particle image velocimetry measurements show that the motion of the flaps results in the formation of additional pairs of vortices when compared to the single vortex pair formed in the absence of flaps. The total final circulation coming out of the flap region remains nearly the same as that of the rigid exit case. However, the final fluid impulse is always found to be higher in the flap cases, with the fluid impulse in most flap cases being approximately two times the fluid impulse of the rigid exit case. This increase in impulse is shown to be linked to the fact that the centroids of vorticity get spread out more in the lateral direction due to the opening of the flaps. The increased impulse and the higher time rate of change of impulse, which is linked with force, suggest that introduction of flexible flaps can help improve thrust performance when looked at from a propulsion point of view.
The kinematic genesis of vortex formation due to finite rotation of a plate in still fluid
- M. Jimreeves David, Manikandan Mathur, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 839 / 25 March 2018
- Published online by Cambridge University Press:
- 02 February 2018, pp. 489-524
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We present a combined experimental and numerical study of an idealized model of the propulsive stroke of the turning manoeuvre in fish. Specifically, we use the framework of Lagrangian coherent structures (LCSs) to describe the kinematics of the flow that results from a thin plate performing a large angle rotation about its tip in still fluid. Temporally and spatially well-resolved velocity fields are obtained using a two-dimensional, incompressible finite-volume solver, and are validated by comparisons with experimentally measured velocity fields and alternate numerical simulations. We then implement the recently proposed variational theory of LCSs to extract the hyperbolic and elliptic LCSs in the numerically generated velocity fields. Detailed LCS analysis is performed for a plate motion profile described by $\dot{\unicode[STIX]{x1D703}}(t)=\unicode[STIX]{x1D6FA}_{max}\sin ^{2}(\unicode[STIX]{x1D714}t)$ during $0\leqslant t\leqslant t_{o}$ and zero otherwise. The stopping time $t_{o}$ is given by $t_{o}=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}=10~\text{s}$, the value of $\unicode[STIX]{x1D6FA}_{max}$ chosen to give a stopping angle of $\unicode[STIX]{x1D703}_{max}=90^{\circ }$, resulting in a Reynolds number $Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$, where $c$ is the plate chord length and $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ the kinematic viscosity of water. The flow comprises a starting and a stopping vortex, resulting in a pair of oppositely signed vortices of unequal strengths that move away from the plate in a direction closely aligned with the final plate orientation at $t/t_{o}\approx 2$. The hyperbolic LCSs are shown to encompass the fluid material that is advected away from the plate for $t>t_{o}$, henceforth referred to as the advected bulk. The starting and stopping vortices, identified using elliptic LCSs and hence more objective than Eulerian vortex detection methods, constitute only around two thirds of the advected bulk area. The advected bulk is traced back to $t=0$ to identify five distinct lobes of fluid that eventually form the advected bulk, and hence map the long-term fate of various regions in the fluid at $t=0$. The five different lobes of fluid are then shown to be delineated by repelling LCS boundaries at $t=0$. The linear momentum of the advected bulk region is shown to account for approximately half of the total impulse experienced by the plate in the direction of its final orientation, thus establishing its dynamical significance. We provide direct experimental evidence for the kinematic relevance of hyperbolic and elliptic LCSs using novel dye visualization experiments, and also show that attracting hyperbolic LCSs provide objective characterization of the spiral structures often observed in vortical flows. We conclude by showing that qualitatively similar LCSs persist for several other plate motion profiles and stopping angles as well.
Thrust generation from pitching foils with flexible trailing edge flaps
- M. Jimreeves David, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 828 / 10 October 2017
- Published online by Cambridge University Press:
- 31 August 2017, pp. 70-103
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In the present experimental study, we investigate thrust production from a pitching flexible foil in a uniform flow. The flexible foils studied comprise a rigid foil in the front (chord length $c_{R}$) that is pitched sinusoidally at a frequency $f$, with a flexible flap of length $c_{F}$ and flexural rigidity $EI$ attached to its trailing edge. We investigate thrust generation for a range of flexural rigidities ($EI$) and flap length to total chord ratio ($c_{F}/c$), with the mean thrust ($\overline{C_{T}}$) and the efficiency of thrust generation ($\unicode[STIX]{x1D702}$) being directly measured in each case. The thrust in the rigid foil cases, as expected, is found to be primarily due to the normal force on the rigid foil ($\overline{C_{TN}}$) with the chordwise or axial thrust contribution ($\overline{C_{TA}}$) being small and negative. In contrast, in the flexible foil cases, the axial contribution to thrust becomes important. We find that using a non-dimensional flexural rigidity parameter ($R^{\ast }$) defined as $R^{\ast }=EI/(0.5\unicode[STIX]{x1D70C}U^{2}c_{F}^{3})$ appears to combine the independent effects of variations in $EI$ and $c_{F}/c$ at a given value of the reduced frequency ($k=\unicode[STIX]{x03C0}fc/U$) for the range of $c_{F}/c$ values studied here ($U$ is free-stream velocity; $\unicode[STIX]{x1D70C}$ is fluid density). At $k\approx 6$, the peak mean thrust coefficient is found to be about 100 % higher than the rigid foil thrust, and occurs at $R^{\ast }$ value of approximately 8, while the peak efficiency is found to be approximately 300 % higher than the rigid foil efficiency and occurs at a distinctly different $R^{\ast }$ value of close to 0.01. Corresponding to these two optimal flexural rigidity parameter values, we find two distinct flap deflection shapes; the peak thrust corresponding to a mode 1 type simple bending of the flap with no inflection points, while the peak efficiency corresponds to a distinctly different deflection profile having an inflection point along the flap. The peak thrust condition is found to be close to the ‘resonance’ condition for the first mode natural frequency of the flexible flap in still water. In both these optimal cases, we find that it is the axial contribution to thrust that dominates ($\overline{C_{TA}}\gg \overline{C_{TN}}$), in contrast to the rigid foil case. Particle image velocimetry (PIV) measurements for the flexible cases show significant differences in the strength and arrangement of the wake vortices in these two cases.
Interaction of a vortex ring with a single bubble: bubble and vorticity dynamics
- Narsing K. Jha, R. N. Govardhan
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- Journal:
- Journal of Fluid Mechanics / Volume 773 / 25 June 2015
- Published online by Cambridge University Press:
- 29 May 2015, pp. 460-497
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The interaction of a single bubble with a single vortex ring in water has been studied experimentally. Measurements of both the bubble dynamics and vorticity dynamics have been done to help understand the two-way coupled problem. The circulation strength of the vortex ring (${\it\Gamma}$) has been systematically varied, while keeping the bubble diameter ($D_{b}$) constant, with the bubble volume to vortex core volume ratio ($V_{R}$) also kept fixed at about 0.1. The other important parameter in the problem is a Weber number based on the vortex ring strength $(\mathit{We}=0.87{\it\rho}({\it\Gamma}/2{\rm\pi}a)^{2}/({\it\sigma}/D_{b});a=\text{vortex core radius},{\it\sigma}=\text{surface tension})$, which is varied over a large range, $\mathit{We}=3{-}406$. The interaction between the bubble and ring for each of the $\mathit{We}$ cases broadly falls into four stages. Stage I is before capture of the bubble by the ring where the bubble is drawn into the low-pressure vortex core, while in stage II the bubble is stretched in the azimuthal direction within the ring and gradually broken up into a number of smaller bubbles. Following this, in stage III the bubble break-up is complete and the resulting smaller bubbles slowly move around the core, and finally in stage IV the bubbles escape. Apart from the effect of the ring on the bubble, the bubble is also shown to significantly affect the vortex ring, especially at low $\mathit{We}$ ($\mathit{We}\sim 3$). In these low-$\mathit{We}$ cases, the convection speed drops significantly compared to the base case without a bubble, while the core appears to fragment with a resultant large decrease in enstrophy by about 50 %. In the higher-$\mathit{We}$ cases ($\mathit{We}>100$), there are some differences in convection speed and enstrophy, but the effects are relatively small. The most dramatic effects of the bubble on the ring are found for thicker core rings at low $\mathit{We}$ ($\mathit{We}\sim 3$) with the vortex ring almost stopping after interacting with the bubble, and the core fragmenting into two parts. The present idealized experiments exhibit many phenomena also seen in bubbly turbulent flows such as reduction in enstrophy, suppression of structures, enhancement of energy at small scales and reduction in energy at large scales. These similarities suggest that results from the present experiments can be helpful in better understanding interactions of bubbles with eddies in turbulent flows.
Effect of hinged leaflets on vortex pair generation
- Prashant Das, R. N. Govardhan, J. H. Arakeri
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- Journal:
- Journal of Fluid Mechanics / Volume 730 / 10 September 2013
- Published online by Cambridge University Press:
- 02 August 2013, pp. 626-658
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We experimentally study the effect of having hinged leaflets at the jet exit on the formation of a two-dimensional counter-rotating vortex pair. A piston–cylinder mechanism is used to generate a starting jet from a high-aspect-ratio channel into a quiescent medium. For a rigid exit, with no leaflets at the channel exit, the measurements at a central plane show that the trailing jet in the present case is never detached from the vortex pair, and keeps feeding into the latter, unlike in the axisymmetric case. Passive flexibility is introduced in the form of rigid leaflets or flaps that are hinged at the exit of the channel, with the flaps initially parallel to the channel walls. The experimental arrangement closely approximates the limiting case of a free-to-rotate rigid flap with negligible structural stiffness, damping and flap inertia, as these limiting structural properties permit the largest flap openings. Using this arrangement, we start the flow and measure the flap kinematics and the vorticity fields for different flap lengths and piston velocity programs. The typical motion of the flaps involves a rapid opening and a subsequent more gradual return to its initial position, both of which occur when the piston is still moving. The initial opening of the flaps can be attributed to an excess pressure that develops in the channel when the flow starts, due to the acceleration that has to be imparted to the fluid slug between the flaps. In the case with flaps, two additional pairs of vortices are formed because of the motion of the flaps, leading to the ejection of a total of up to three vortex pairs from the hinged exit. The flaps’ length (${L}_{f} $) is found to significantly affect flap motions when plotted using the conventional time scale $L/ d$, where $L$ is the piston stroke and $d$ is the channel width. However, with a newly defined time scale based on the flap length ($L/ {L}_{f} $), we find a good collapse of all the measured flap motions irrespective of flap length and piston velocity for an impulsively started piston motion. The maximum opening angle in all these impulsive velocity program cases, irrespective of the flap length, is found to be close to 15°. Even though the flap kinematics collapses well with $L/ {L}_{f} $, there are differences in the distribution of the ejected vorticity even for the same $L/ {L}_{f} $. Such a redistribution of vorticity can lead to important changes in the overall properties of the flow, and it gives us a better understanding of the importance of exit flexibility in such flows.
Defining the ‘modified Griffin plot’ in vortex-induced vibration: revealing the effect of Reynolds number using controlled damping
- R. N. GOVARDHAN, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 561 / 25 August 2006
- Published online by Cambridge University Press:
- 09 August 2006, pp. 147-180
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In the present work, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We employ a technique to accurately control the structural damping, enabling the system to take on both negative and positive damping. This permits a systematic study of the effects of system mass and damping on the peak vibration response. Previous experiments over the last 30 years indicate a large scatter in peak-amplitude data ($A^*$) versus the product of mass–damping ($\alpha$), in the so-called ‘Griffin plot’.
A principal result in the present work is the discovery that the data collapse very well if one takes into account the effect of Reynolds number ($\mbox{\textit{Re}}$), as an extra parameter in a modified Griffin plot. Peak amplitudes corresponding to zero damping ($A^*_{{\alpha}{=}0}$), for a compilation of experiments over a wide range of $\mbox{\textit{Re}}\,{=}\,500-33000$, are very well represented by the functional form $A^*_{\alpha{=}0} \,{=}\, f(\mbox{\textit{Re}}) \,{=}\, \log(0.41\,\mbox{\textit{Re}}^{0.36}$). For a given $\mbox{\textit{Re}}$, the amplitude $A^*$ appears to be proportional to a function of mass–damping, $A^*\propto g(\alpha)$, which is a similar function over all $\mbox{\textit{Re}}$. A good best-fit for a wide range of mass–damping and Reynolds number is thus given by the following simple expression, where $A^*\,{=}\, g(\alpha)\,f(\mbox{\textit{Re}})$: \[ A^* \,{=}\,(1 - 1.12\,\alpha + 0.30\,\alpha^2)\,\log (0.41\,\mbox{\textit{Re}}^{0.36}). \] In essence, by using a renormalized parameter, which we define as the ‘modified amplitude’, $A^*_M\,{=}\,A^*/A^*_{\alpha{=}0}$, the previously scattered data collapse very well onto a single curve, $g(\alpha)$, on what we refer to as the ‘modified Griffin plot’. There has also been much debate over the last three decades concerning the validity of using the product of mass and damping (such as $\alpha$) in these problems. Our results indicate that the combined mass–damping parameter ($\alpha$) does indeed collapse peak-amplitude data well, at a given $\mbox{\textit{Re}}$, independent of the precise mass and damping values, for mass ratios down to $m^*\,{=}\,1$.
Vortex-induced vibrations of a sphere
- R. N. GOVARDHAN, C. H. K. WILLIAMSON
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- Journal:
- Journal of Fluid Mechanics / Volume 531 / 25 May 2005
- Published online by Cambridge University Press:
- 18 May 2005, pp. 11-47
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There are many studies on the vortex-induced vibrations of a cylindrical body, but almost none concerned with such vibrations for a sphere, despite the fact that tethered bodies are a common configuration. In this paper, we study the dynamics of an elastically mounted or tethered sphere in a steady flow, employing displacement, force and vorticity measurements. Within a particular range of flow speeds, where the oscillation frequency ($f$) is of the order of the static-body vortex shedding frequency ($f_{vo})$, there exist two modes of periodic large-amplitude oscillation, defined as modes I and II, separated by a transition regime exhibiting non-periodic vibration. The dominant wake structure for both modes is a chain of streamwise vortex loops on alternating sides of the wake. Further downstream, the heads of the vortex loops pinch off to form a sequence of vortex rings. We employ an analogy with the lift on an aircraft that is associated with its trailing vortex pair (of strength $\Gamma^*$ and spacing $b^*$), and thereby compute the rate of change of impulse for the streamwise vortex pair, yielding the vortex force coefficient ($\cvortex$): \[ \cvortex = \frac{8}{\pi} {U^*_{v}}b^*( - \Gamma^*). \] This calculation yields predicted forces in reasonable agreement with direct measurements on the sphere. This is significant because it indicates that the principal vorticity dynamics giving rise to vortex-induced vibration for a sphere are the motions of these streamwise vortex pairs. The Griffin plot, showing peak amplitudes as a function of the mass–damping ($m^*\zeta$), exhibits a good collapse of data, indicating a maximum response of around 0.9 diameters. Following recent studies of cylinder vortex-induced vibration, we deduce the existence of a critical mass ratio, $m^*_{crit} {\approx} 0.6$, below which large-amplitude vibrations are predicted to persist to infinite normalized velocities. An unexpected large-amplitude and highly periodic mode (mode III) is found at distinctly higher flow velocities where the frequency of vibration ($f$) is far below the frequency of vortex shedding for a static body. We find that the low-frequency streamwise vortex pairs are able to impart lift (or transverse force) to the body, yielding a positive energy transfer per cycle.