1. Introduction
The ability of vortex-induced vibration (VIV) to increase the strength of vortices shed through the vibration (e.g., see [
1,
2,
3]) could promote the turbulence generated by a flexible protruding surface. Vortex shedding (typically a Kármán vortex) happens when flowing past a bluff body. Under conditions where the structure is rigid enough (high stiffness), to resist the different pressure distribution on the structure caused by the unsymmetrical vortices, it experiences little to negligible motion (oscillation or vibration as their meaning are interchangeable in this context). It is the same for rigid protruding surfaces, as some portion of the oncoming fluid loses kinetic energy as it meets the rigid protruding surface. Given a flexible body with low Young’s modulus properties and a geometry with a low second moment of area, the bending stiffness will inherently be lower. Therefore, a smaller fluid force is needed to drive the vibration for the VIV to happen.
The main component that determines the flexibility or elasticity of a flexible cylinder is the structural stiffness, as shown in
Figure 1. Stiffness relies on material properties and geometry. Therefore, it is the combination of the material’s Young’s modulus,
E, and the element’s second moment of inertia,
I.
Under general situations, two types of vortex shedding can affect the oscillation mode of the VIV, one of which is perhaps more commonly known than the other—the Kármán vortex shedding. The Kármán vortex street consists of alternately shed vortices due to the asymmetric vortex shedding behavior. It is the asymmetric vortex shedding behavior that alters the pressure distribution on the surface of each side of the cylinder, thus leading to the vibration of the cylinder in transverse direction to the flow. This type of vibration is also widely known as cross-flow motion. The second type of vortex shedding is the symmetric vortex shedding. As the name suggests, it consists of a pair of symmetrical vortices when shed in one cycle. The vibration mode of this shedding is in-line or parallel to the flow, hence the name of in-line VIV. Under certain conditions, in-line VIV accompanied by symmetric vortex shedding may occur at lower flow velocities than the Kármán vortex shedding [
4]. Another type of shedding that would happen in addition to either the Kármán vortex shedding or symmetric vortex shedding is the tip vortex shedding. It happens whenever the free end of the cylinder is exposed to the flow. These tip vortices are generally shed at a frequency one-third of the Kármán vortex shedding and may cause large-amplitude vibrations.
The back and forth motions, which are lift and drag, are due to vortex shedding at which the vortex shedding frequency,
fs, is close to the cylinder’s natural frequency,
fn, at the velocity. The Strouhal number,
St, is the frequency of the excitation force in the lift direction whereas the frequency of the excitation force in the drag direction is normally two times that in the lift direction. The lift force can be explained as a group of vortices shed to one side of the cylinder once per cycle and then the other side. The lift force, which appears when the vortex shedding starts to occur, causes the cross-flow motion (perpendicular to the fluid flow direction). Similarly, the drag force appears as a result of vortex shedding but with all vortices shed downstream. In-line motion (in the same direction as the fluid flow) of the cylinder is caused by drag force. Since all the vortices are shed downstream of the cylinder in the drag direction, the drag force associated with vortex shedding occurs at twice the frequency of the lift force, as explained by [
5].
Higher amplitudes of vibration in the cross-flow direction have been documented by various experimental results in [
5,
6]. When the amplitude reaches its maximum, lock-in is said to happen. Lock-in occurs when the natural frequency of the structure,
fn, is in proximity to the vortex shedding,
fv, or in mathematical expression,
= 1. Both frequencies synchronize and large-amplitude vortex-induced structural vibration can occur. The vortices in lock-in conditions can pack a colossal amount of energy. Besides the increase in vortex strength, consequences of lock-in are also an increase in correlation length, in-line drag force and lock-in bandwidth; all of which increase maximum amplitude. The lock-in bandwidth was found to increase with increasing response amplitude [
7]. It is easy to see that lower-mass-ratio cylinders require less energy to vibrate because they are lighter compared to higher-mass-ratio cylinders. Thus, they can achieve the maximum amplitude (lock-in) more effortlessly in a wider range of reduced velocity, as opposed to the higher-mass-ratio cylinders.
The wake structure differs according to the critical aspect ratio. The
AR of the cylinder is defined by its length over diameter ratio,
AR =
L/
D. From various studies by several authors, the value of the critical aspect ratio seems to be sensitive to experimental conditions, especially the relative thickness of the boundary layer [
8,
9]. The critical aspect ratio varied from 1–7 in many studies, which is a wide range. When a cylinder at a very low
AR is immersed in an atmospheric boundary layer, the vortex formation length, the width of the near wake and the value of the Strouhal number at mid-height are reduced, comparable to the case of the small relative thickness of the boundary layer [
10,
11]. Liu et al. [
12] supported that a two-dimensional region exists when the
AR is greater than the critical aspect ratio but the area of the two-dimensional region decreases when the
Re increases. It should, however, be noted that the range of the critical aspect ratio is only loosely defined based on the observation of the wake structure. It can be seen that the critical aspect ratio is sensitive and scales with the boundary layer thickness and
Re because the
AR has to be large enough to sustain the two-dimensional region. A low
AR or an
AR lower than the critical aspect ratio that is put in a higher
Re and a thicker boundary layer signifies that the region of free end effects will engulf the whole span of the cylinder, suppressing the two-dimensional region.
Tip vortices are known to have more turbulent energy than the regular Kármán vortices. The turbulence intensity has also been reported to increase considerably at the free end of the cylinder. In the recent work of [
13], they demonstrated that the streamwise turbulence intensity (
u′/U) and wall-normal turbulence intensity (
w′/U) are indeed higher at the free end and within the recirculation region on the wake centerline (
y/
D = 0). On the other hand, Park and Lee [
14] also showed in their results that the turbulent kinetic energy is at its highest level at the free end for the cylinder of
AR = 6 at
ReD = 7500. Rostamy et al. [
13] noticed that the elevated turbulence intensity extends into the downwash region. One noticeable distinct pattern that separates the cylinder of
AR = 3 from the rest is that the location of the highest wall-normal turbulence intensity moves downward and further away to a distance of
x/
D = 2. The location of the highest wall-normal turbulence intensity appears to remain at
x/
D < 2 as the
AR of the cylinder increases.
Similarly, the Reynolds shear stress (-
) was also reported to be at a higher value in the experimental work from [
13]. They noticed a region of positive shear stress and a region of negative shear stress at an evaluated value near the wake region. However, the region of positive shear stress is absent below the free end, leaving only the region of negative shear stress for a cylinder of
AR = 3. On the contrary, the positive shear stress dominates the region just below the free end for
AR = 9 but slowly decreases in size and level as the
AR decreases until it disappears at
AR = 3. In their results, the Reynolds shear stress is found to be at its highest for a cylinder of
AR = 9 with a peak at 0.018. However, the Reynolds shear stress decreases as
AR decreases. Additionally, according to
Figure 2, ref. [
8] also showed in their experimental results that the vorticity contour is immensely elevated near the free end. Thus, the tip vortices can greatly improve the quality of mixing. However, due to the downwash phenomenon that is always present at the free end, the tip vortices are always brought downwards to the ground plane. Therefore, its influence could only impinge on the fluid in the very near wake. The mean velocity, turbulence intensity and Reynolds shear stress distributions are similar for cylinders above the critical aspect ratio but dissimilar for cylinders below the critical aspect ratio [
15].
The Reynolds normal (
) and shear (
) stress of a flexible cylinder have also been reported by [
16] to increase significantly compared to that of a rigid cylinder. They first investigated the Reynolds shear stress of a rigid cylinder and compared it with the results from [
17]. The periodic parts of Reynolds normal and shear stress of both studies are a near match though that of the experiment conducted by [
17] was much higher at
Re = 140,000. The peak total Reynolds stress is, however, much higher for the higher
Re experiments. Govardhan and Williamson [
16] therefore concluded that the periodic part of Reynolds stress, which is generated from the repeatable large-scale coherent structures, holds true over the range of
Re = 3900 to 140,000. As the other component of total Reynolds stress—the random part of Reynolds stress—gains strength from the increasing strength of the Kelvin–Helmholtz instability of the separating shear layer as
Re increases, the total Reynolds stress value increases. They also compared the Reynolds normal and shear stress value of a vibrating cylinder to that of a rigid cylinder and found that the periodic part of Reynolds stress (
)
max increases by up to 485%, (
)
max increases by 100% and (
)
max increases by 125% in the lower branch.
It was shown in previous work [
6,
18,
19] that the flexible cylinders bring significant changes to the turbulence in the near wake—the
x deflection of the flexible cylinders increases the turbulence wake region, which effectively increases the region of turbulent activities—and the aspects of the structural motion such as oscillating amplitude, oscillating frequency and the oscillating motion can enhance the turbulence energy production as demonstrated by the production term (in the kinetic energy budget) and the Reynolds stresses. Despite the remarkable findings, they are unable to confirm which parameter(s) of the structural motion could affect the enhancement of turbulence energy in the near wake. It is therefore, this work mainly investigates the influence of different properties of the flexible cylinders to bridge the gap. As an effort to encourage an organized oscillating motion, materials such as aluminum and carbon steel with moderately high stiffness and low damping coefficient are employed in this study. The metal group flexible cylinders have a Reynolds number of 2500 due to the limitation of the highest possible freestream velocity the water tunnel could provide. Apart from that,
AR = 12 and 14 of polymer-based EVA flexible cylinders at Reynolds numbers of 4000, 6000 and 8000 are also investigated in this work.