Lagrangian Structures and Mixing in the Wake of a Streamwise Oscillating Cylinder

1 Lagrangian analysis is capable of revealing the underlying structure and complex phenomena in unsteady flows. We present Particle-Image Velocimetry measurements of the wake of a cylinder undergoing streamwise vortex-induced vibrations and calculate the Finite-Time Lyapunov Exponents (FTLE) in backward-and forward-time. The FTLE fields are compared to the phase-averaged vorticity fields for the four di↵er-ent wake modes observed while the cylinder experiences streamwise vortex-induced vibrations. The backward-time FTLE fields characterise the formation of vortices, with the roll up of spiral-shaped ridges coinciding with the roll up of the shear layers to form the vortices. Ridges in the forward-time fields tend to lie perpendicular to the flow direction and separate nearby vortices. The shedding of vortices coincides with a 'peel o↵' process in the forward-time FTLE fields, in which a ridge connected to the cylinder splits into two strips, one of which moves downstream. Particular attention is given to the 'wake breathing' process, in which the streamwise motion of the cylinder causes both shear layers to roll up simultaneously and two vortices of opposite sign to be shed into the wake. In this case, the ridges in forward-time FTLE fields are shown to define 'vortex cells', in which the new vortices form, and the FTLE fields allow the wake to be decomposed into three distinct regions. Finally, the mixing associated with each wake mode is examined, and it is shown that cross-wake mixing is significantly enhanced when the vibration amplitude is large and the vortices are shed alternately. However, while the symmetric shedding induces large amplitude vibrations, no increase in mixing is observed relative to the von Kármán vortex street observed behind near-stationary bodies.


I. INTRODUCTION
Flow past a cylinder is a classical case of unsteady fluid-structure interaction, with the shear layers on either sides of the cylinder rolling up periodically to form vortices that are then shed downstream. The shedding process results in very interesting Lagrangian dynamics, which have been examined in detail using an unsteady potential flow model, 4 demonstrating the presence of chaos in the near wake. 5The presence of chaotic advection significantly increases mixing efficiency, 6,7 which has been examined with respect to chemical reactions 5 and the mixing in ocean flows around islands. 8 The latter problem was investigated by Crimaldi and Kawakami, 9,10 who performed numerical simulations of steady flow past a stationary cylinder to investigate the mixing of passive tracers, and how this may affect the fertilisation of the eggs of marine species near islands.
There has been relatively little experimental work investigating the Lagrangian dynamics in cylinder wakes.Bourgeois et al. 11 examined the three-dimensional Lagrangian structures in the wake of a square cylinder attached to a plate using the Finite-Time Lyapunov Exponent (FTLE) fields, which represent the local rate of fluid stretching and are often used to approximate the Lagrangian Coherent Structures (LCSs). 12Using the FTLE fields, they were able to characterise the influence of the plate on the three-dimensional shedding process.Similarly, Kourentis and Konstantinidis 13 used a combination of proper-orthogonal decomposition and the FTLE fields to identify the effect of flow pulsations on the structures in the near wake of a circular cylinder and used the resulting approximations of the LCSs to derive measures of the width and decay of the wake pattern.
Celik and Beskok 14 examined the flow of passive tracers past a cylinder in a channel and showed that the mixing in the cylinder wake is significantly enhanced if the cylinder is forced to oscillate transverse to the flow direction.][17] The unsteady wake dynamics are capable of causing the cylinder to oscillate in the absence of external forcing.The vortices are shed at a fixed frequency, f St , given by the Strouhal number, St = f St D/U 0 ; this can lead to Vortex-Induced Vibrations (VIVs) if the frequency of the fluid forces induced by the shedding process coincides with the natural frequency of the cylinder, f n , or its actual vibration frequency, f x , which may be slightly different due to added mass effects.These vibrations can reach large amplitudes and cause significant fatigue damage. 18The condition for VIV is given by the "true" reduced velocity, U r St/ f * , where U r = U 0 / f n D is the conventional reduced velocity and f * = f x / f n is the frequency ratio.The "true" reduced velocity (henceforth referred to simply as the reduced velocity) is equal to the ratio of the predicted shedding frequency to the vibration frequency, f St / f x .VIV typically occurs in the transverse (lift) direction at U r St/ f * ≈ 1; however, as the unsteady drag force typically occurs at twice the shedding frequency, VIV occurs at U r St/ f * ≈ 0.5 in the streamwise (drag) direction.
The vibration amplitudes tend to be smaller in the streamwise direction compared to the transverse; 19,20 however, streamwise VIV is also relevant to the study of stationary cylinders experiencing wave-loading or in pulsatile flow, as occurs in blood flow, tidal flows and various industrial contexts.Provided that the amplitude of the pulsations is low, the case of a cylinder freely oscillating in the streamwise direction in steady flow is dynamically equivalent to a fixed cylinder in pulsating flow. 21,22or streamwise VIV of a circular cylinder, four wake modes have been observed, which are shown in Figure 1, and the response regime is characterised by two branches, separated by a region of low amplitude vibration at resonance, i.e., U r St/ f * ≈ 0.5.When the vibration amplitude, A, of the cylinder is low the A-II mode occurs, [23][24][25] which is essentially the von Kármán vortex street seen behind stationary cylinders (Figure 1).The first response branch occurs over the range U r St/ f * ≈ 0.25-0.45and is characterised by the presence of the S-I mode, in which two vortices of opposite sign are shed simultaneously from either sides of the cylinder 23,24,26,27 (Figure 1).This symmetric shedding is apparently unique to bodies experiencing streamwise VIV or periodic wave loading.Naudascher 26 described the process in detail (based on the work of Aguirre 23 ); the streamwise motion of the cylinder causes the shear layers on either sides of the cylinder to roll up simultaneously, in a process he referred to as "wake breathing."The symmetric arrangement is unstable and tends to rearrange downstream into an alternate pattern. 28,29Aguirre showed that the wake reverts to the A-II mode at the peak of the first branch, which was also observed by the present authors. 24,30he vibration amplitude is low at resonance, U r St/ f * ≈ 0.5.The reason for this counterintuitive reduction in amplitude is not clear; previous authors have argued that it arises from a change in the wake mode, 23 a change in the phase of vortex-shedding with respect to the cylinder motion 22 or a reduction in the amplitude of the fluid forces. 31As the reduced velocity is increased further, the vibration amplitude recovers to form the second response branch in the range U r St/ f * ≈ 0.55-0.65,which is characterised by the SA mode. 27This is a special case of the A-II mode, in which the vortices form very close to the cylinder, have a greater circulation, and remain close to the wake centreline as they convect downstream 24 (Figure 1).The A-IV mode has been observed in studies in which a cylinder is forced to oscillate 17,25 or is placed in pulsating flow. 32This mode is characterised by the shedding of four vortices per wake cycle, with a pair of counter-rotating vortices shed in each cylinder oscillation cycle (Figure 1), and has recently been shown to also occur in the wake of freely oscillating cylinders in the second response branch, 33 using the same experimental data presented in this paper.
Wake modes are typically identified in flow-visualisation studies using dye 1,25,34 or laserinduced fluorescence. 17When the velocity fields in the wake are measured, the wake mode can also be identified from the vorticity fields, 15,24,27,35 although in practice it can be difficult to distinguish vortices in the near wake due to the high shear surrounding the cylinder.7][38] While these Eulerian measures can be extremely useful, they suffer some drawbacks: 37,38 they typically require a user-defined threshold and therefore the identified boundaries of vortices are subjective, and they do not contain any information on the history of the flow.An alternative approach that overcomes these issues is to identify the wake mode and the Lagrangian structures from the FTLE fields computed in forward-and backward-time, which correspond to the repelling and attracting structures in the flow, respectively.This approach is often used to approximate the LCSs and is capable of revealing the complex processes governing flows, 12,39,40 identifying regions of strong or weak mixing, 38,41 as well as producing strikingly beautiful visualisations. 42his paper aims to provide new insight into the dynamics and Lagrangian structures associated with the various wake modes observed in streamwise VIV and to investigate their effect on fluid mixing.The remainder of the paper is structured as follows; Section II describes the experimental system and the Lagrangian calculations; Section III summarises the response regime of the streamwise oscillating cylinder, describes the FTLE fields observed for wake mode and discusses their relationship to the vortex dynamics; the wake modes are examined in terms of their mixing characteristics in Section IV; and finally some concluding remarks are made in Section V.

II. EXPERIMENTAL DETAILS
The experiments were performed in a closed-loop water tunnel, which has been described previously by Konstantinidis et al. 21The test-section had internal dimensions of 72 mm × 72 mm and was positioned downstream of a 9:1 contraction and honeycomb screens, such that the turbulence entering the test-section was less than 2%.A cooling system was used to ensure that the temperature was held constant at 22 ± 0.5 • C. A cylinder was supported within the test-section using fishing wires, such that it was free to vibrate only in the streamwise direction, as shown in Figure 2. The tension in each wire was adjusted, and tests were performed to ensure that any non-streamwise motion (i.e., pitching) that the cylinder experienced was negligible.The cylinder had a diameter of 7.1 mm and was 71 mm in length.It was made of solid perspex and had a mass ratio (i.e., the ratio of the cylinder mass to the displaced fluid mass) of 1.17.A series of tap tests were performed in air and water; the natural frequency in water was f n = 23.7 Hz, and the structural damping ratio (measured in air) was 0.0037.The damping ratio increased to 0.02 when the tests were performed in water, i.e., when the effects of hydrodynamic damping were included.
Particle-Image Velocimetry (PIV) was used to measure the velocity fields surrounding the cylinder.The plane at the mid-span of the cylinder, normal to the cylinder axis, was illuminated using an Nd:Yag laser, as shown in Figure 2, and pairs of images were acquired at 200 Hz using a CMOS camera.Silver-coated hollow glass spheres with an average diameter of 10 µm were used to seed to flow.A three-pass cross-correlation scheme with 50% window overlap was used to estimate the velocity fields, which had a final resolution of 18 vectors per cylinder diameter in both the transverse and streamwise directions.The measurement domain spanned the range x/D = −1.4 to 4.2 and y/D = −1.65 to 1.55, in the streamwise and transverse directions, respectively, with the origin fixed at the mean cylinder position.For each reduced velocity examined, 1000 velocity fields were measured at a sampling rate of 200 Hz.This corresponded to approximately 120 cylinder oscillations per measurement set.The variation in the frequency of traverse velocity fluctuations in the wake, f w , prior to the onset of VIV was used to calculate the Strouhal number, St = 0.2.
The cylinder position was determined directly from the PIV images using a template matching algorithm.The method was found to be accurate to within 0.4 pixels (0.2% of the cylinder diameter). 24The resulting displacement signals were band-pass filtered between 10 Hz and 40 Hz to remove the effects of noise and any low-frequency variations which were not associated with VIV.The mean peak height of the displacement signal was used to measure the vibration amplitude and the dominant vibration frequency was found from the peak of the power-spectral-density of the signal.This was assumed to be accurate to within 0.78 Hz (0.33% of f n ), which was the resolution of the spectra.
The cylinder displacement signals and velocity fields were measured at 16 freestream velocities in the range U 0 = 0.1-0.76m/s, which corresponded to a Reynolds number range of Re = U 0 D/ν (where ν is the kinematic viscosity of water) of 740-5400, and a reduced velocity range of U r St/ f * = 0.126-0.714.A settling time of at least 10 min was allowed between adjusting the flow rate and acquiring the PIV measurements.After this period, the PIV system was elevated and a series of measurements were acquired eight diameters upstream of the cylinder in order to measure the freestream velocity; the system was then lowered to its original position and measurements were acquired from the cylinder wake.
In order to clearly identify the vortex-shedding process and the dominant wake mode, the velocity measurements were phase-averaged.As vortex-shedding does not always occur at f x , it was not possible to phase-average with respect to the cylinder displacement, as is common in transverse VIV studies.Instead, the traverse velocity signal measured at various points in the wake were used as a reference signal, depending on which wake mode was dominant.The vorticity fields were calculated from phase-averaged velocity fields using the least-squares method.

A. Lagrangian analysis
The fluid transport within the wake was studied by numerically advecting passive tracers (with zero size and mass) through the velocity fields using a fourth-order Runge-Kutta scheme and calculating the flow map which defines the dynamic position of tracers that had an initial distribution x 0 at time t 0 .The velocity fields were seeded with tracers with an initial resolution twice that of the velocity vectors (i.e., 36 vectors per cylinder diameter) and the flow map was calculated for both forward-and backward-time.The latter is performed by reversing the flow (i.e., changing the sign of velocity vectors) and allowing the tracers to travel backwards.Lagrangian Coherent Structures (LCSs) are material lines or surfaces which separate qualitatively distinct regions of the flow and are capable of revealing the underlying patterns in complex flows. 12,39,40Ridges in the Finite-Time Lyapunov Exponent (FTLE) fields are often used to identify the LCSs; the fields are defined as where λ max is the largest eigenvalue of This is mathematically equivalent to the rate of separation of particles, 12 such that the maximum rate of separation of two particles at point x with time is proportional to e (σ f (x)(t−t 0 )) .When the FTLE field is calculated in forward-time (i.e., t > t 0 ), the fields are denoted σ + f (where the superscript "+" indicates a forward-time calculation), and regions of high σ + f correspond to points where fluid particles are rapidly moving away from and tend to coincide with repelling structures within the flow.Similarly in backward-time calculations, regions of high σ − f (where the "−" superscript indicates that the property is calculated in backward-time) correspond to points that fluid moves towards over the period (t to t 0 ) and therefore tend to represent attracting structures.
The FTLE fields in forward-and backward-time were calculated from the phase-averaged velocity fields by allowing the tracers to advect over five wake cycles, i.e., the integration time was |t − t 0 | = 5/ f w .The FTLE fields were qualitatively similar to those computed over shorter intervals, but the relatively long integration time of 5/ f w allowed the complex patterns in the near wake to be fully captured.In most cases outside the recirculation zone, the fluid tracers left the PIV domain before this time, and the FTLE field at this point was calculated based on the time at which the local tracers left the domain.

A. Amplitude response
The amplitude response of the cylinder is shown in Figure 3(a) as a function of reduced velocity.The response regime is dominated by two branches, separated by a low amplitude region at U r St/ f * ≈ 0.5, in agreement with previous studies. 23,24,33,34The symbol colours denote the dominant wake mode observed in the PIV fields at each reduced velocity; the A-II mode occurs when the amplitude is very low; both the A-II and S-I modes occur in the first response branch, with unsteady switching between the modes occurring near the peak of the branch; 29 and the SA and A-IV modes are dominant than the second response branch.
Figure 3(b) shows the dominant frequency of transverse velocity fluctuations at various points in the wake.The velocity signals indicate that lock-in occurs throughout both response branches (U r St/ f * ≈ 0.2-0.65)with the velocity fluctuations at some point in the wake locking-in to either f x /2 and f x .This phenomenon is not identical throughout this range.In the first branch symmetric shedding causes the velocity fluctuations close to the cylinder (x/D = 1) to occur at the cylinder vibration frequency, but downstream the vortices rearrange into an alternate pattern causing the velocity fluctuations to occur at the Strouhal frequency. 29Similarly, when the A-IV modes occurs in the second branch the velocity fluctuations at (x/D, y/D) = (3, 0.5) contain peaks at both f x /2 and f x ; the super-harmonic frequency component at f x is likely to be caused by the  doubling in the number of vortices shed per wake cycle.However, the velocity signals measured at (x/D, y/D) = (2, 0) all have a dominant frequency of f x /2 when this mode is present.
The phase-averaged vorticity fields and FTLE fields for each mode were analysed in detail at a sample reduced velocity, as listed in Table I.Based on the data in Figure 3, different reference signals were used in the phase-averaging process depending on which mode was dominant (also indicated in Table I).The results for each mode are discussed in Subsections III B-III E. (A/D = 0.013) and the wake is similar to the von Kármán vortex street observed behind stationary cylinders.The shear layers on either sides of the cylinder extend far into the wake before rolling up to form vortices at x/D ≈ 2. This can be seen in Figures 4(a)-4(d), where the shear layers at y/D ± 0.5 begin to form concentrated circular patches of vorticity in this region.It can also been seen in the backward-time FTLE fields (Figures 4(e)-4(h)), where thin ridges of σ − f extend downstream from the cylinder (in the same positions occupied by the shear layers), before rolling up to form spirals at the same point at which the vortices become apparent in the vorticity fields.An example of this roll up/vortex formation can be seen in the upper shear layer in Figures 4(c) and 4(d) at x/D ≈ 1.5-2.5,where a circular patch of negative (clockwise rotating) vorticity has formed, and the σ − f ridge has rolled up to form a clockwise spiral at the same point (Figures 4(g) and 4(h)).It is apparent that the vortices seen in the ω z fields coincide with rolled up spiral-shaped σ − f ridges, and the vortex formation process is analogous to the roll up phenomenon in the backward-time FTLE fields.

B. A-II mode
The vortex-shedding process is not as immediately clear in the forward-time FTLE (Figures 4(i)-4(l)).There are two ridges of high σ + f extending from the cylinder in roughly the same position as the shear layers, which also roll up in the wake.However, unlike the ridges in the backward-time FTLE fields, the σ + f ridges appear to roll up to surround the newly formed vortices, e.g., at (x/D, y/D) = (2, −0.5) in Figure 4(j).This σ + f ridge has rolled up in an anti-clockwise manner to surround the clockwise rotating vortex as it forms, i.e., the opposite roll up process which we observe in the backward-time fields.
Another interesting feature of the forward-time FTLE fields can be seen in the ridges along the shear layers.While the ridges remain attached to the cylinder throughout the shedding cycle, a strip is "peeled off" each ridge.This can be seen in Figure 4(i), where a single σ + f ridge is attached to the bottom of the cylinder ( y/D = 0 to −0.5), extending to about x/D = 2.2; as the shedding cycle progresses (Figures 4(j) and 4(k)) this layer appears to split in two, with a weaker strip close to the wake centre line remaining attached to the cylinder; by the time the shedding cycle is almost complete (Figure 4(l)), the σ + f ridge has completely split into two separate strips, with a stronger ridge (further from the wake centreline) being cut off from the cylinder and beginning to move downstream.
Finally, two σ + f ridges extend upstream from the front of the cylinder.These features were also observed in previous studies of flow around cylinders 42 and convecting vortex rings, 43 and represent the boundary between regions which are quickly swept downstream and fluid which interacts with the viscous boundary layer surrounding the cylinder.These upstream ridges were observed in the σ + f fields computed for each mode.

C. SA mode
The SA mode is a special case of the A-II mode, in which the vortices are formed very close to the cylinder and remain close to the wake centreline as they convect downstream.These features can be seen in Figure 5  The forward-time fields also reveal some interesting features.The peel off process is again observed; a curved arc of high σ + f surrounds much of the cylinder and near wake in Figure 5(i), and as the shedding progress continues a strip is peeled off and moves downstream (e.g., at x/D ≈ 1.75 in Figure 5(k)).A shed σ + f ridge is visible in each of the forward-time FTLE fields in the range x/D ≈ 1.5-2.5.They tend to lie perpendicular to the σ − f ridges and between the positive and negative vortices in the ω z fields (Figures 5(a)-5(d)).This, along with the fields presented for the A-II mode, suggests that the forward-time FTLE fields separate the regions in which the vortices form from the surrounding flow.For example, in Figures 5(i)-5(l) the σ + f ridges enclose the vortex formation region at the base of the cylinder throughout the shedding cycle, separating it from the ambient flow and the downstream wake.

D. A-IV mode
Figure 6 shows the phase-averaged vorticity and FTLE fields computed at U r St/ f * = 0.58, while the A-IV mode was dominant in the wake.The vorticity patterns are slightly disorganised compared to the other wake modes shown, which is a result of the increased unsteadiness in the vortex-shedding process.The increased number of vortices shed per cycle also means that peak vorticity in each vortex is lower, and they are therefore less clearly distinguishable.
However, the shedding of vortex pairs is still visible; positive and negative vortices can be seen convecting downstream as a pair in Figures 6(b attached to the cylinder appears to enclose the entire near wake region and is connected to both the upper and lower surfaces of the cylinder (e.g., in Figure 6(l)).The shed strip of the σ + f ridge appears to separate the like-signed vortices which are shed sequentially, for example, the weak ridge at x/D ≈ 1.75 in Figure 6(l) lies between the positive vortex at x/D ≈ 2.25 and the newly formed positive vortex at x/D ≈ 1.25 in Figure 6(d).Therefore, while the A-IV mode occurs, the forward-time FTLE ridges separate the vortex pairs rather than the individual shed vortices, as in other modes.

E. S-I mode
The symmetric shedding mode is unique to streamwise oscillations/excitation 26 and can occur for a wide range of streamwise excitation amplitudes and frequencies. 17The symmetric nature of the wake is clear in both the phase-averaged vorticity and FTLE fields in Figure 7.The symmetric arrangement of vortices is unstable and rearranges into an alternate pattern downstream; 28,29 the phase-averaging process cancels out these alternate patterns in the velocity fields and no structures are apparent for x/D 3 in any of the fields in Figure 7.
The vortices formed have a clear circular shape (Figures 7(a)-7(d)) and coincide with "voids" in the forward-time FTLE fields (Figures 7(i)-7(l)), with σ + f ridges rolling up to enclose the regions in which the vortices lies.These regions also coincide with rolled up spirals in the σ − f fields (Figures 7(e)-7(h)), as seen in the other wake modes.The "wake breathing" process has been studied by a number of researchers 17,24,26,28,29,44 since the early flow visualisation study of Aguirre, 23 and the high spatial resolution of the FTLE fields allows this phenomenon to be examined in much greater detail.In particular, we can examine the near-wake processes associated with vortex-formation and shedding which control the overall structures of the fields shown in Figure 7. Figure 8 shows the phase-averaged vorticity fields in the near wake throughout the cylinder oscillation cycle, along with the ridges in the backwardand forward-time FTLE fields and the phase-averaged cylinder position.For the sake of the current discussion, the FTLE ridges are here taken to be all the points where σ − f D/U 0 1.25 and σ + f D/U 0 1.25.A video showing the periodic roll-up of the shear layers and the FTLE ridges in the near wake region as the cylinder vibrates is provided in the supplementary material. 49s the cylinder is in its peak downstream position (Figure 8(a)), a counter-clockwise rotating vortex is present at x/D ≈ 0.75, which can be identified by a patch of positive vorticity and a σ − f spiral (light blue region).This vortex lies in an eye-shaped gap between successive σ + f ridges (dark blue).
The cylinder begins to move upstream, which causes the vortex to separate from the cylinder surface, and by the time the cylinder is passing through its mean position (Figure 8 the spiral has developed a fine-scale structure, with the radial separation between ridges in the spiral approaching the spatial resolution of the PIV fields (i.e., 0.055D).
The forward-time FTLE fields also provide insight into the vortex-formation process.When the cylinder is in its peak downstream position (Figure 8(a)) a σ + f ridge lies along the cylinder surface.As the cylinder moves upstream and sheds its vortex (Figures 8(b) and 8(c)), the σ + f ridge undergoes peel off, with a strong ridge being shed into the wake (moving with the shed vortex) and a thinner strip remaining in contact with the cylinder.There are now two roughly parallel σ + f ridges which separate as the cylinder continues to move upstream and the stronger ridge is shed into the wake (Figures 8(c)-8(e)), forming an eye-shaped void (Figures 8(e)-8(h)).Comparison to the vorticity fields in the same figure indicates that these σ + f ridges are forming a "vortex cell," in which the new vortex forms.There is little vorticity at the cylinder surface before the cell forms (Figures 8(a)-8(h)), and the initial peel off process and the separation of the ridges (Figures 8(b) and 8(c)) coincide with the growth of the patch of positive vorticity at the cylinder surface within the vortex cell.
The FTLE fields reveal an organised structure of the flow, as shown in Figure 9.The flow can be roughly divided into three regions; the "ambient flow" outside of the wake, the "turbulent wake" between the trains of vortices on either sides of the wake centreline, and the vortex cells which are occupied by vortices that form in the near wake and dissipate downstream.The vortex cells will move downstream at the convective velocity of the vortices.The role of the FTLE fields in identifying qualitative boundaries within the wake and revealing the underlying flow structure is similar to the "skeleton" of the flow presented by Mathur et al. 40 for freestream turbulence.

IV. MIXING IN THE WAKE
The Lagrangian analysis can also be used to study the mixing dynamics in the wake, following a similar approach to that of Crimaldi and Kawakami 9,10 who studied the wake of a stationary cylinder.The problem of mixing in the wake is relevant to the dispersion of effluent or pollutants in rivers and coastal regions.Turbulent mixing is required to reduce the concentration of effluent in the water to safe levels; the presence of reeds and other vegetation (or purposely inserted structures susceptible to VIVs 45 ) will result in vortex-shedding which may increase the rate of mixing and Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 128.41 dispersion and decrease the region of the river with dangerously high contaminant concentration.
If the vegetation (which can be approximated as a cylinder) or solid structures are subject to wave loading from wind or sea waves, this case will be equivalent to a cylinder undergoing streamwise VIV in steady flow.It is also relevant to micro-scale flows in which bluff-bodies are purposely inserted and pulsating flow induced to promote mixing and heat transfer. 46,47It is therefore of interest to investigate what effect each of the wake modes in streamwise VIV has on the mixing efficiency, and under what conditions streamwise VIV will promote or inhibit mixing.
To examine this, we take the ideal case of two streams entering the PIV domain on either sides of the cylinder, in the region 0.75 ≤ | y/D| ≤ 1. Mixing in flows is governed by the Péclet number, Pe = U 0 D/d (where d is the mass diffusivity), which is the ratio of the rates of advection and diffusion.The diffusivity for most common fluids in water is of the order of 10 −9 m 2 /s, 48 which corresponds to Pe > 7 × 10 5 for the experiments presented here.This is sufficiently high so that mixing in the wake occurs almost entirely by advection and the effects of diffusion can be neglected in the current analysis.
Tracers were seeded throughout the PIV domain with the same resolution as the FTLE fields (i.e., 36 particles per cylinder diameter) and advected backwards in time using a fourth-order Runge-Kutta scheme to identify the location at which they entered the PIV domain, which was expressed as the instantaneous concentration fields, C (x, y,t).Points at which the local tracer entering the domain from above the cylinder (0.75 ≤ y/D ≤ 1) were assigned a value of 1; a value of −1 was assigned if the local tracer entered from below the cylinder (−1 ≤ y/D ≤ −0.75) and all other points were set to zero.The concentration field was calculated for every PIV field and the results were averaged (omitting the first 100 fields, to avoid erroneously including the effects of any fields in which there was insufficient time to advect the tracers back to the point at which they entered the PIV domain).
This approach was similar to the "coarse-grain density" method of estimating mixing, 7 in which the tracers are advected forward in time and their concentration in a number of bins is examined.However, the current approach of advecting the tracers backwards in time had some advantages for open flows.It guarantees that C can be evaluated at every point in the flow; this is particularly relevant near regions of high vorticity, which tended to be sparsely populated if the tracers are advected forward in time, and near the base of the cylinder where there is reduced mass transport in and out of the recirculation region. 5In addition, it allows for higher resolution of the concentration field relative to the spacing of the tracers, as tracers do not have to be averaged over a bin region.
The mean concentration fields, C, for each wake mode are shown in Figure 10, where the red and blue regions correspond to regions where the local tracers originated in the streams above and below the cylinder, respectively.The horizontal dashed lines indicate the transverse positions from which the streams originated.The mean concentration field corresponding to the A-II mode is shown in Figure 10(a) and can be approximated as the C field which would be seen behind a stationary cylinder.The two streams enter the domain on the left and spread away from the wake centreline as they pass the cylinder before gradually returning to their original transverse positions.The vortex-shedding and turbulence cause the mean concentration streams to disperse downstream, although for x/D 3 virtually no material from either streams has moved closer to the centreline from its original position (i.e., C ≈ 0 for all | y/D| ≤ 0.75).Similarly, little dispersion is observed when the S-I mode occurs (Figure 10(b)), with the streams marginally less dispersed at the downstream boundary of the PIV domain compared to the A-II case.
In contrast, very strong cross-wake mixing is seen when the SA mode dominates (Figure 10(c)).Here, the tendency for vortices to form at the cylinder base and remain close to the wake centreline causes the streams in the mean concentration field to cross over to the other side of the wake at x/D ≈ 2.75, where the averaging process results in a mean value close to zero.At the point where the fluid leaves the PIV domain at x/D = 4.2, the streams in the C field have completely switched sides and the width of each stream is significantly larger than that seen for the A-II or S-I modes, indicating greater mixing and dispersion.
Figure 10(d) shows that mixing is also enhanced when the A-IV mode is present (although not to the same extent as seen for the SA mode), with the streams becoming wider downstream and meeting at the wake centreline at x/D ≈ 3-4.
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 128.41  The mixing characteristics of each mode can be directly compared by examining the transverse profiles of the concentration fields at a given location in the wake.To remove the effect of tracers from either sides of the cylinder "averaging out" in the C fields, as occurred in the A-IV and SA fields, only the tracers which originated below the cylinder ( y/D < 0) were averaged to yield C − ; this corresponds to the mean concentration of the blue streams in Figure 10.The profiles were calculated for a range of different initial positions of the streams.
Figure 11(c) shows the profiles of C − at x/D = 3.5 for the streams entering the domain at −0.75 ≤ y/D ≤ −1, for each of the wake modes.The area underneath each curve will not be the same (as it will depend on the variation in the streamwise velocity component and the flux through the sides of the PIV domain); however, the widths of the peaks in the C − profiles provide a measure of the dispersion of the streams caused by the wake dynamics.
The profiles for the A-II and S-I modes are very similar, with the latter profile occurring slightly closer to the wake centreline.The widths of both peaks are approximately the same, indicating that the two modes do not induce significantly different magnitudes of cross-wake mixing.In contrast, the peak in the profile for the A-IV mode is very broad, spanning the range y/D ≈ −1.5 to 0 and centred at y/D ≈ −0.5, indicating a dramatic increase in mixing.As was seen in the C fields, the profile for the SA mode also indicates very strong mixing; the profile is not characterised by a single pronounced peak but is distributed across the PIV domain, with C − 0.04 for most points.There is a wide bulge at y/D ≈ 0.33-1 (on the opposite side of the wake to the stream's origin) and a small secondary bulge near y/D ≈ −1.5, which can also be seen in Figure 10(c).
Figures 11(a), 11(b), and 11(d) show the C − profiles computed for different initial positions of the streams (as indicated by the grey shaded regions).The profiles are slightly different in each case, tending to be more irregular and broad when streams originate close to the centreline and the tracers are more likely to become entrained in the near wake.However, the general trends are clear; when the SA mode occurs, the cross-wake mixing is strong and tracers are dispersed broadly across the wake; the peak in the C − profile is also broad when the A-IV mode occurs; while the S-I and A-II profiles are narrow.
The C fields and C − profiles are representative of the time-averaged cross-wake mixing, but they do not provide information on how the mixing occurs, or how this process is related to the Lagrangian structures seen in the FTLE fields.To investigate these questions, we examine the case of the streams originating close to the centreline (0.25 ≤ | y/D| ≤ 0.5) while the SA mode was dominant at U r St/ f * = 0.58.This case was chosen as it exhibited the strongest cross-wake mixing Regions of high |∇C| correspond to points at which you would expect to find the strongest mixing if the streams were "active," e.g., streams of reactive chemical agents 5 or in various biological processes such as the fertilisation of marine species. 41The fields contains regions where |∇C|D > 8.5, which correspond to points at which the two fluid streams came very close together, i.e., the separation between the two streams is less than the resolution of the PIV fields.
The points where |∇C|D ≥ 8.5 are indicated on the corresponding σ − f fields by the blue points in Figures 12(i)-12(l).The regions of strong mixing tend to occur along ridges in the σ − f fields, i.e., the attracting lines in the flow.A similar correspondence between regions of intense mixing and attracting Lagrangian structures has been observed in previous studies. 5,41t is clear that mixing is significantly enhanced in the second response branch (where the SA and A-IV modes occur) compared to the case of the near-stationary cylinder, but is not significantly affected by the S-I mode.Symmetric shedding generally occurs when the excitation or cylinder vibration frequency is greater than twice the Strouhal frequency (which is equivalent to U r St/ f * < 0.5). 17,23,26Therefore, if the frequency of wave loading (or the natural frequency of bodies added to the flow to passively increase mixing by undergoing VIV) is high, i.e., the body will not increase the rate of mixing.However, if the frequency is below this value, Figures 10 and 11 indicate that the mixing will be significantly enhanced relative to the stationary cylinder case.The findings of this study have implications for the design of cooling systems for electronics, where pins are often installed in micro-channels to promote vortex-shedding and thus significantly enhance heat transfer. 46,47If the flow is forced, Equation (4) provides a criterion by which the forcing frequency can be tuned to optimise mixing.

V. CONCLUSIONS
The velocity fields surrounding a freely oscillating cylinder were measured using particleimage velocimetry and the finite-time Lyapunov exponent fields were calculated in forward-and backward-time.Analysis of the vorticity and FTLE fields measured while the A-II, SA, and A-IV Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.Downloaded to IP: 128.41 wake modes dominated showed that the roll up of the shear layers to form vortices in the wake coincided with a roll up of the ridges in the backward-time FTLE fields.As the vortices were shed and convected downstream the spirals in the backward-time FTLE fields continued to develop, demonstrating the ability of the FTLE fields to visualise and identify key aspects of the vortex dynamics.The shedding of vortices into the wake coincides with a "peel-off" process in the forward-time FTLE fields, whereby strong ridges which were connected to the cylinder split into two, with the newly formed ridge moving downstream with the shed vortex.The forward-time ridges tended to lie between neighbouring vortices and in the case of the SA and A-IV modes, they were found to separate the near wake (where the vortices were formed) from the surrounding wake.
The FTLE fields and vortex-formation process associated with the S-I mode were analysed in detail.The backward-time fields again identified the roll up of the shear layers as the vortices formed, while the forward-time fields marked the boundaries of "vortex cells," in which new vortices formed.The FTLE fields allowed the flow to be divided into three approximate regions: the ambient flow outside the wake, the turbulent wake between the shed vortices, and the vortex cells.
The mixing efficiency of each of the wake modes was examined by measuring the origin of passive tracers in each instantaneous PIV field and identifying the number of tracers which originated from 0.75 ≤ | y/D| ≤ 1 on either sides of the cylinder.The cross-wake mixing was significantly increased in the second response branch when the SA and A-IV modes occurred, but the S-I mode in the first branch was not found to provide an increase relative to the A-II mode observed behind a near-stationary cylinder.Analysis of the concentration fields indicates that the regions of strongest mixing coincide with ridges in the backward-time FTLE fields.

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FIG.2.Sketch showing the test-section, the cylinder and fishing wire supports, and the camera and laser used to acquired to the PIV measurements.

045107- 6 NFIG. 3 .
FIG. 3. Amplitude response (a) and dominant frequency of velocity fluctuations at various points in the wake (b), as a function of reduced velocity.The dashed line in (b) indicates the Strouhal frequency, while the symbol colours in (a) indicate the dominant wake mode: A-II (white); S-I (blue); SA (red); and A-I (green).Symbols with two colours indicate the presence of mode switching.

Figure 4
Figure 4 shows the vorticity and backward-and forward-time FTLE fields computed at U r St/ f * = 0.126, while the A-II mode occurred.The amplitude response at this reduced velocity is very low for U r St/ f * = 0.538; patches of strong vorticity are seen forming at the base of the cylinder (Figures 5(b) and 5(d)), while σ − f ridges roll up at the same points (Figures 5(f) and 5(h)).The σ − f ridges trace beautiful arcs in the wake, showing the positions of the convecting vortices and the shear layers which connect each vortex.The fields are similar to the dye flow visualisation images of Van Dyke. 1

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FIG. 5. Phase-averaged vorticity fields (a)-(d) and the finite-time Lyapunov exponents calculated in backward-(e)-(h) and forward-time (i)-(l), at U r St/ f * = 0.538, while the SA mode was present.The fields in each row were computed at the same phase in the shedding cycle.
Figure6shows the phase-averaged vorticity and FTLE fields computed at U r St/ f * = 0.58, while the A-IV mode was dominant in the wake.The vorticity patterns are slightly disorganised compared to the other wake modes shown, which is a result of the increased unsteadiness in the vortex-shedding process.The increased number of vortices shed per cycle also means that peak vorticity in each vortex is lower, and they are therefore less clearly distinguishable.However, the shedding of vortex pairs is still visible; positive and negative vortices can be seen convecting downstream as a pair in Figures6(b) and 6(d) at x/D = 2-3.The shed vortex pairs are too weak to be apparent in the FTLE fields (Figures 6(e)-6(l)), but the roll up of vortices in the near wake is clear.Two vortices can be seen in the near wake in Figure6(h), with a large σ − f spiral slightly further downstream (x/D ≈ 1) on the lower surface.In the following stage of the shedding cycle (Figure6(e)) the lower σ − f spiral has been shed and is no longer visible (although it is apparent at x/D ≈ 2.25 in the corresponding vorticity field in Figure6(a)), and a new vortex is starting to form in the same location (x/D ≈ 0.75).The backward-time FTLE fields reveal the dynamics of the vortex shedding process in the near wake in greater detail than is apparent from the slightly disorganised vorticity fields alone.The forward-time FTLE fields are shown in Figures 6(i)-6(l).As in the backward-time case, the shed vortices are not visible downstream, although the σ + f fields show strong ridges close to the cylinder.The peel off process is again apparent; however, in this case the σ + f ridge that remains

FIG. 7 .
FIG. 7. Phase-averaged vorticity fields (a)-(d) and the finite-time Lyapunov exponents calculated in backward-(e)-(h) and forward-time (i)-(l), at U r St/ f * = 0.365, while the S-I mode was present.The fields in each row were computed at the same phase in the shedding cycle.
FIG. 8. Phase-averaged vorticity fields (a)-(h) along with the ridges in the backward-and forward-time FTLE fields (light and dark blue, respectively), at U r St/ f * = 0.365.The ridges were defined as regions where σ − f D/U 0 1.25 and σ + f D/U 0 1.25.The corresponding phase-averaged cylinder position is shown in (j).

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FIG. 10.Mean concentration fields calculated for four wake modes; A-II (a), S-I (b), SA (c), and A-IV (d).The dashed lines show the transverse positions at which the red and blue streams entered the PIV domain, 0.75 ≤ | y/D| ≤ 1.

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FIG. 11.Transverse profiles at x/D = 3.5 of the mean concentration of the tracers entering the PIV domain at y/D = −0.25 to −0.5 (a), y/D = −0.5 to −0.75 (b), y/D = −0.75 to −1 (c), and y/D = −1 to −1.25 (d).The shaded regions represent the initial position of the stream as it enters the PIV domain.

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FIG. 12. Concentration fields (a)-(d), the magnitude of the gradient of C (e)-(h) and the Finite-Time Lyapunov Exponents calculated in backward-time (i)-(l), at U r St/ f * = 0.58 while the SA mode was dominant.All fields were computed from the phase-averaged velocity fields, and the fields in each row were computed at the same phase.The blue points in (i)-(l) correspond to locations where the gradient of the concentration field was large, |∇C |D ≥ 8.5, i.e., where the blue and red stream came into close proximity.These points tend to coincide with ridges in the σ − f fields.

TABLE I .
Sample reduced velocities examined and the point in the wake at which the reference signal was extracted for each wake mode.