Resonance in vortex-induced in-line vibration at low Reynolds numbers

Figure 1 shows the flow-structure configuration considered in the present study. The elastically-mounted cylinder is modelled using the conventional mass-spring-damper system. The cylinder is constrained so that it can oscillate only in-line with a uniform free stream. The motion of the cylinder is governed by Newton’s second law, which can be expressed per unit span as

$$m\,\ddot{x}_{c}+c\,\dot{x}_{c}+k\,x_{c}=F_{x}(t),$$

(1)

where $x_{c}$, $\dot{x}_{c}$, and $\ddot{x}_{c}$ respectively are the displacement, velocity and acceleration of the cylinder, $m$ is the mass of the cylinder, $c$ is the structural damping, $k$ is the spring stiffness, and $F_{x}(t)$ is the time-dependent sectional fluid force acting on the cylinder.

The condition for the onset of vortex-induced in-line vibration can be broadly expressed as $U^{*}_{a}\approx 1/(2S)$, whereas the corresponding value for cross-stream vibration is $U^{*}_{a}\approx 1/S$, where $U_{a}^{*}=U_{\infty}/f_{n,a}D$ is the reduced velocity and $S=f_{v0}D/U_{\infty}$ is the Strouhal number; here $U_{\infty}$ is the velocity of the free stream, $D$ is the diameter of the cylinder, $f_{v0}$ is the frequency of vortex shedding for a stationary cylinder, and $f_{n,a}$ is the natural frequency of the structure in still fluid, i.e. including the ‘added mass’. Traditionally, response data from experimental studies obtained as the flow velocity is varied over the attainable range of experimental facilities are presented as a function of the reduced velocity based on the structural frequency in still fluid. Other non-dimensional parameters governing the structural response are: the ratio of the cylinder mass to the fluid mass displaced by the cylinder, denoted as the mass ratio $m^{*}$, the ratio of the structural damping to the critical damping at which the mechanical system can exhibit oscillatory response to external forcing, denoted as the damping ratio $\zeta$, as well as the Reynolds number, $Re$, which determines the flow regime. The definitions of the mass ratio and the damping ratio are also not uniform in the literature but depend on whether the added fluid mass is taken into account. In this work, we have selected a set of non-dimensional parameters listed in table 1. It should be noted that we define the reduced velocity using the natural frequency of the system in vacuum, $f_{n}=(1/2\upi)\sqrt{k/m}$, in common with most previous numerical studies of vortex-induced vibration.

Typically, the response amplitude of cylinder vibration is magnified in distinct ranges of the reduced velocity, which mimic the classical resonance of a single degree-of-freedom oscillator to external harmonic forcing. These distinct regions of high-amplitude response have been given various names such as ‘instability regions’ (King, 1977), ‘excitation regions’ (Naudascher, 1987), or ‘response branches’ (Williamson & Govardhan, 2004). It has been established as early as the 1970’s that there exist two distinct excitation regions of free in-line vibration: the first one appears at $U_{a}^{*}\lesssim 2.5$ and has been associated with symmetrical shedding of vortices simultaneously from both sides of the cylinder, whereas the second one appears at $U_{a}^{*}\gtrsim 2.5$ and has been associated with alternating shedding of vortices from each side of the cylinder (Wootton et al., 1972; King, 1977; Aguirre, 1977). The value of $U_{a}^{*}\approx 2.5$ corresponds to $1/(2S)$ assuming a Strouhal number of 0.20, at which reduced velocity the frequency of vortex shedding from a stationary cylinder becomes equal to half the natural frequency of the structure in still fluid, i.e. $f_{v0}\approx\frac{1}{2}f_{n,a}$. A factor of 2 arises in the denominator from the fact that two vortices shed from alternate sides of the cylinder each one induces a periodic oscillation of the fluid force in the streamwise direction. It should be remembered that the Strouhal number is a function of the Reynolds number, defined as $Re=\rho U_{\infty}D/\mu$, where $\rho$ is the density and $\mu$ is the dynamic viscosity of the fluid. Therefore, the above value of $U_{a}^{*}\approx 2.5$ should generally be replaced by $U_{a}^{*}\approx 1/(2S)$ in dealing with response data at different Reynolds numbers.

In experimental studies with elastically mounted rigid cylinders, the structural frequency depends on the oscillating mass and the stiffness of the supporting springs, which provide elastic restoring forces. In an early study, Aguirre (1977) conducted more than a hundred tests in a water channel to investigate vortex-induced in-line vibration. He concluded that the structural mass and the stiffness affected the cylinder response independently and in different ways: for a given value of $f/f_{v0}$, the density ratio (i.e. the mass ratio here) did not affect the response amplitude when normalized with the cylinder diameter, nor did the stiffness affect the normalized frequency of vibration $f/f_{n,a}$, where $f$ is the actual vibration frequency. Beyond that early study, the effects of the structural mass and stiffness, which are embodied in the mass ratio and the reduced velocity values, have not been systematically addressed in more recent studies. Yet, Okajima et al. (2004) found that the response amplitude of in-line oscillation decreases in both excitation regions with increasing the reduced mass-damping, or Scruton number - a non-dimensional parameter that is proportional to the product $m^{*}\zeta$ and is often employed to compile peak amplitude data as a function of a single parameter. The later findings possibly illustrate the influence of structural damping alone since the mass ratio was constant in those tests.

The existence of two distinct response branches of free in-line vibration and corresponding modes of vortex shedding were confirmed in more recent experimental studies at Reynolds numbers in the range approximately from $10^{3}$ to $3.5\times 10^{4}$ (Okajima et al., 2004; Cagney & Balabani, 2013b). The drop in response amplitude in-between the two branches, i.e. at $U_{a}^{*}\approx 2.5$, has been attributed to the phasing of alternating vortex shedding, which provides a positive-damping, or negative-excitation force with respect to the oscillation of the cylinder (Konstantinidis et al., 2005; Konstantinidis, 2014). More recently, a mixed mode of combined symmetric and alternating vortex shedding was also reported to exist in-between the two branches (Gurian et al., 2019).

For self-excited vibrations to be possible, energy must be transferred from the fluid to the structural motion in an average cycle so as to sustain the oscillations, i.e. $E\geqslant 0$ where $E=\oint{F_{x}\mathrm{d}x_{c}}$; ${F_{x}}$ is the instantaneous fluid force driving the body motion and $x_{c}$ is the displacement of the body. A rationale is to determine the energy transfer from forced vibrations of the body in order to predict whether free vibrations can occur for the corresponding case where the cylinder is elastically constrained. This is typically based on the approximation that both the motion of the cylinder $x_{c}(t)$ and the driving fluid force per unit length $F_{x}(t)$ can be expressed as single-harmonic functions of time $t$, e.g.

	$\displaystyle x_{c}(t)$	$\displaystyle=$	$\displaystyle X_{0}+A\cos{(2\upi ft)},$		(2)
	$\displaystyle F_{x}(t)$	$\displaystyle=$	$\displaystyle F_{x0}+F_{x1}\cos{(2\upi ft+\phi_{x})},$		(3)

where $X_{0}$ is the mean streamwise displacement of the cylinder, $A$ is the amplitude and $f$ is the frequency of body oscillation, $F_{x0}$ and $F_{x1}$ respectively are the magnitudes of the mean and unsteady in-line fluid forces, and $\phi_{x}$ is the phase lag between the displacement and the driving force at the oscillation frequency. Using the above harmonic approximations, it can be readily shown that

$$E=\upi AF_{x1}\sin\phi_{x}.$$

(4)

Hence, free vibration is possible only if $\sin\phi_{x}\geqslant 0$, or equivalently the phase lag be within the range $0^{\circ}\leqslant\phi_{x}<180^{\circ}$.

Two independent studies employing forced harmonic in-line vibrations at fixed amplitudes of oscillation in the range from 0.1 to 0.28 diameters peak-to-peak, have shown that energy is transferred from the fluid to the cylinder motion in two excitation regions separated by approximately $U^{*}_{r}\approx 2.5$ for $Re$ values higher than $10^{3}$ (Tanida et al., 1973; Nishihara et al., 2005). Here, $U^{*}_{r}=U_{\infty}/fD$ is the reduced velocity based on the actual frequency of forced oscillation. The use of $U^{*}_{r}$ is also critical in correlating forced-vibration studies with the response from free-vibration studies, in which case the vibration frequency is not necessarily equal to the structural frequency (Williamson & Govardhan, 2004; Konstantinidis, 2014). Overall, predictions using forced harmonic vibration agree well with the excitation regions found in free vibration at relatively high Reynolds numbers, including the wake modes responsible for free vibration (Tanida et al., 1973; Nishihara et al., 2005). On the contrary, Tanida et al. found that energy transfer was always negative for all reduced velocities at $Re=80$. They stated that results obtained for $Re=80$ were representative over the range $40\leqslant Re\leqslant 150$, which indicated that free vibration may not be possible in that range Reynolds numbers.

More recently, detailed results from two-dimensional numerical simulations for a cylinder placed in an oscillating free stream and the equivalent case of a cylinder oscillating in-line with a steady free stream both showed that $E<0$ for all reduced velocities at fixed Reynolds numbers of $Re=150$ (Konstantinidis & Bouris, 2017) and $Re=100$ (Kim & Choi, 2019). The lowest amplitudes of forced oscillation in those studies were 0.1 and 0.05 diameters, respectively. The findings from both studies have also indicated that free in-line vibration may not be feasible for Reynolds numbers in the laminar regime, which is consistent with the earlier experimental study of Tanida et al. (1973). Nevertheless, it is plausible that energy transfer may become positive at lower amplitudes of oscillation than those employed in previous studies, which has not received attention to date since free in-line vibration has scarcely been studied at low Reynolds numbers; the authors are aware of only few numerical studies where in-line vibration of circular cylinder rotating at prescribed rates was investigated at $Re=100$ (Bourguet & Lo Jacono, 2015; Lo Jacono et al., 2018). These studies showed that an elastically-mounted rotating cylinder can be excited into large-amplitude galloping-type vibrations as the reduced velocity increases. However, the response amplitudes were negligible in the case of a non-rotating cylinder compared to the rotating cases and the former results were not discussed.

Apart from addressing the question whether in-line vortex-induced vibration is possible at low Reynolds numbers, a more fundamental issue is to clarify what are the flow physics causing variations in the amplitude and frequency of response when self-excited vibration does occur, not only at low Reynolds numbers. This issue impacts our understanding of vortex-induced vibration as well as its modelling and prediction using semi-empirical codes in industrial applications. To address that issue, it is essential to formulate a theoretical framework with the aid of which results can be interpreted. In this study, we maintain that the in-line free vibration offers a convenient test case because it allows different dynamical effects, which are associated with fluid inertia, fluid damping, and fluid excitation from the unsteady wake, to be segregated.

A long-standing approach is to represent the in-line force per unit length $F_{x}(t)$ based on the equation proposed by Morison et al. (1950). For a cylinder of circular cross section oscillating in-line with a steady free stream, the equation can be written as

$$F_{x}(t)=\frac{1}{2}\rho DC_{dh}\left|U_{\infty}-\dot{x}_{c}\right|\left(U_{\infty}-\dot{x}_{c}\right)-\frac{1}{4}\upi\rho D^{2}C_{mh}\ddot{x}_{c},$$

(5)

where $\rho$ is the density of the fluid. The coefficients $C_{dh}$ and $C_{mh}$ are often referred to as drag and added mass (or inertia) coefficients, respectively, and their values are empirically determined from measurements or simulations. Inherent to this approach is the harmonic approximation since the coefficients $C_{dh}$ and $C_{mh}$ are often determined from tests where the cylinder is forced to vibrate harmonically. Even when the cylinder motion is self-excited, it is still necessary to characterize the vibration in terms of the least number of appropriate non-dimensional parameters for compiling fluid forcing data; this usually boils down to the use of two parameters, i.e. the normalized amplitude and normalized frequency of oscillation, which can fully characterize only single-harmonic oscillations.

Another similar approach is to decompose the fluctuating part of the in-line force into harmonic components in-phase with the displacement (or alternately acceleration) and in-phase with the velocity of the oscillating cylinder, in addition to a steady term for the mean drag. By using harmonic approximations, the steady-state response can be predicted as we present in appendix A. This approach is analogous to using Morison et al.’s equation and linearising the drag term as $\left|U_{\infty}-\dot{x}_{c}\right|\left(U_{\infty}-\dot{x}_{c}\right)\approx U_{\infty}^{2}-2U_{\infty}\dot{x}_{c}$. However, Morison et al.’s equation comprises two force coefficients whereas the harmonic approximation comprises three force coefficients. The lack of an independent term for the mean drag may explain, at least partially, why Morison et al.’s equation reconstructs the in-line force unsatisfactorily in the case of vibrations in-line with a steady free stream. However, it should be noted that the addition of a third term for steady drag in Morison’s equation did not considerably improve the empirical-fit results (see Konstantinidis & Bouris, 2017).

Sarpkaya (2001) discussed some limitations of Morison et al.’s equation to represent the in-line force on a cylinder placed perpendicular to zero-mean oscillatory flow. Recently, Konstantinidis & Bouris (2017) demonstrated the inability of the thus reconstructed force acting on a cylinder in non-zero-mean oscillatory flows to capture fluctuations due to vortex shedding in the drag-dominated regime, where the vortex shedding and the cylinder motion (the wave motion in that study) are not synchronized. When the primary mode of sub-harmonic synchronization occurs, Morison’s equation provides a fairly accurate fit to the in-line force but subtle differences still exist, which may have detrimental effects when the model equation is used for predicting the free response. A disadvantage of previous approaches based on Morison et al.’s equation as well as on the harmonic representation, is that the values of the force coefficients show some dependency on the best-fitting method, e.g. Fourier averaging vs. least-squares method (see Konstantinidis & Bouris, 2017). Another disadvantage, more important, is that force coefficients are empirically determined and as a consequence it is difficult to decipher the flow physics from the variation of the force coefficients, which is our primary goal in this work.

Despite the empirical use of Morison et al.’s equation, its inventors stated that it originates from the summation of a quasi-steady drag force and the added-mass force resulting from ‘wave theory’ (Morison et al., 1950). A comprehensive discussion of the theory can be found in Lighthill (1986). For a body accelerating rectilinearly within a fluid medium, there is an ideal ‘potential’ force acting on the body, which can be expressed as $F_{x,\mathrm{potential}}=-C_{a}m_{d}\ddot{x}_{c}$, where $C_{a}$ is the added mass coefficient, $m_{d}$ is the mass of fluid displaced by the body, and $\ddot{x}_{c}$ is the acceleration of the body. Thus, the body behaves as if it has a total mass of $m+m_{a}$, where $m_{a}=C_{a}m_{d}$ is the added mass of fluid. According to the theory of inviscid flow, in which case the velocity field can be defined by the flow potential, the added mass coefficient $C_{a}$ of any body is assumed to depend exclusively on the shape of the body. For a circular cylinder, $C_{a}=1$. Free-decay oscillation tests in quiescent fluid have shown that $C_{a}$ is quite close to the ideal value of unity. However, the applicability of the ideal $C_{a}$ value in general flows, including cylinders oscillating normal to a free stream, has been criticized (Sarpkaya, 1979, 2001, 2004). On the other hand, Khalak & Williamson (1996) argued that removing the ideal added-mass force from the total force will leave a viscous force that may still comprise a component in-phase with acceleration, i.e. the decomposition does not have to separate all of the acceleration-depended forces as done in empirical approaches. Ever since the separation of ‘potential’ (inviscid) and ‘vortex’ (viscous) components has been widely employed to shed light into the vortex dynamics around oscillating bodies transversely to a free stream (see, e.g., Govardhan & Williamson, 2000; Carberry et al., 2005; Morse & Williamson, 2009; Zhao et al., 2014, 2018; Soti et al., 2018).

For body oscillations in-line with a free stream, one may also split the streamwise force as

$$F_{x}(t)=F_{x,\mathrm{potential}}(t)+F_{x,\mathrm{vortex}}(t),$$

(6)

in order to explore the link between fluid forcing and vortex dynamics. This was previously done for the case of a fixed cylinder placed normal to a free stream with small-amplitude sinusoidal oscillations superimposed on a mean velocity (Konstantinidis & Liang, 2011). This case is kinematically equivalent to the forced vibration of the cylinder in-line with a steady free stream. In that study, large-eddy simulations corresponding to $Re=2150$ showed that alternating vortex shedding provides positive energy transfer for $U^{*}_{r}>2.5$, in very good agreement with previous experimental studies discussed earlier. It was also observed that the streamwise vortex force diminished in magnitude while the instantaneous phase of the vortex force with respect to the imposed oscillation drifted continuously near the middle of the wake resonance (synchronization) region. This was considered to be inconsistent with the flow physics in the following sense: within the synchronization region the vortex shedding and the oscillation are strongly phase-locked and the corresponding wake fluctuations are resonantly intensified. Therefore, the magnitude of the vortex force would have been expected to increase and its instantaneous phase to remain fairly constant in this region. The irregular phase dynamics observed in that study indicates that the vortex force remaining from subtracting the ideal inertial force from the total force may not fully represent the effect of the unsteady vortex motions on the fluid forcing.

In this paper, we develop a new theoretical model for representing the streamwise force on a cylinder oscillating in-line with a free stream. The model stems from some recent observations. In particular, it was recently shown that Morison et al.’s equation based on the sum of a quasi-steady viscous drag force and an inviscid inertial force represents the in-line force with comparable accuracy as does the equation with best-fitted coefficients over a wide range of parameters from the inertia to drag-dominated regimes (Konstantinidis & Bouris, 2017). However, neither method could capture fluctuations at the vortex shedding frequency in the drag-dominated regime, as noted earlier. Thus, the idea here is to introduce an independent force term $F_{dw}$, i.e. to express the total force as

$$F_{x}(t)=\frac{1}{2}\rho DC_{d}\left|U_{\infty}-\dot{x}_{c}\right|\left(U_{\infty}-\dot{x}_{c}\right)-\frac{1}{4}\upi\rho D^{2}C_{a}\ddot{x}_{c}+F_{dw}(t),$$

(7)

where the first term represents the quasi-steady drag, the second term represents the inviscid added-mass force, and the third term represents the unsteady force due to periodic vortex formation in the wake. In this new approach, there are two viscous contributions: the quasi-steady drag, which is an ‘instantaneous’ reaction force, and the wake drag, which represents the ‘memory’ effect in a time-dependent flow. These contributions may be thought of as originating from the vorticity in the thin boundary and free shear layers, and from the vorticity in the near-wake region, respectively. Both are affected by the rate of diffusion of the vorticity, which is finite. However, at sufficiently high Reynolds numbers for which separation occurs, the diffusion within the thin vortex layers occurs fast enough, almost ‘instantaneously’. Then, the force required to supply the rate of increase of the kinetic energy of the rotational motion in these regions may be taken to be proportional to the square of the relative velocity $U_{\infty}-\dot{x}_{c}$, which gives rise to a quasi-steady drag (Lighthill, 1986). On the other hand, the diffusion of vorticity at the back of the cylinder is a very complex process involving its cross-annihilation as oppositely-signed vortices roll-up close together in the formation region (see, e.g., Konstantinidis & Bouris, 2016); as a consequence the resulting fluid force acting on the body depends on the history of the vortex motions in the near wake.

The splitting of the viscous drag to quasi-steady and wake components is consistent with the contribution of vorticity in distinguishable flow regions around a cylinder to the fluid forces as shown in the work of Fiabane et al. (2011). They revealed these separable contributions by displaying force-density distributions based on the volume-integral expression proposed by Wu et al. (2007). Fiabane et al. were able to separate an ‘external-flow’ region containing the thin vortex structures in the attached and free shear layers, which contributed 90% of the mean drag, and a ‘back-flow’ region between these two vortex layers behind the cylinder, which contributed almost all the drag fluctuations. Moreover, they found that the intensification of the vortex roll-up closer to the cylinder with increasing Reynolds number in the range $Re=50-400$ resulted an increase of the drag fluctuations imposed by the back-flow region. Their findings suggest that – when the cylinder is oscillating – the ‘external flow’ is at the origin of the quasi-steady drag whereas the contribution from the ‘back flow’ is captured by the wake drag. Interestingly, Wu et al. (2007) also considered the flow around a circular cylinder in a steady free stream as a test case in their study; they remarked that while a concentrated vortex after its feeding sheet is cut off makes little direct contribution to the fluid force, it plays an indirect but major role through its induced effect on unsteadiness of the boundary-layer separation and the motion of separated shear layers, which implies that wake vortices influence the phasing of the fluid forces.

Assuming that a single periodic mode of vortex shedding occurs, the wake-induced force can be further modelled as a single-harmonic function of time, i.e.

$$F_{dw}(t)=\frac{1}{2}\rho U_{\infty}^{2}DC_{dw}\cos{(2\upi f_{dw}t+\phi_{dw})},$$

(8)

where the coefficient $C_{dw}$ represents the magnitude of the unsteady wake drag, and $\phi_{dw}$ the phase between the wake drag and the displacement of the cylinder. The frequency $f_{dw}$ excited by the unsteady wake depends on the vortex-shedding mode, e.g.  $f_{dw}=2f_{vs}$ for the alternating mode, whereas $f_{dw}=f_{vs}$ for the symmetrical mode. Equation (8) serves as a reduced-order model with the aid of which the fluid dynamics of vortex-induced vibration can be analysed more thoroughly than possible heretofore by employing previous semi-empirical approaches from the literature.

In this study, we conducted numerical simulations of the flow-structure interaction of a circular cylinder elastically constrained so as to oscillate only in-line with a free stream. The main objectives are: (a) use the numerical data from simulations to obtain the variations of the model parameters $C_{dw}$ and $\phi_{dw}$ as functions of the problem parameters, and (b) use the expressions derived to calculate the model parameters in conjunction with the equation of cylinder motion to develop a theoretical framework for interpreting the phenomenology of vortex-induced in-line vibration. Simulations were restricted to the two-dimensional laminar regime at low Reynolds numbers to keep computer time within reason so as to examine the influence of the reduced velocity and the mass ratio over wide ranges and with a good resolution. The numerically produced sets of data for flow fields and induced fluid forces and their interaction with the resulting free motion of the cylinder allowed us to address the issues raised in the foregoing paragraphs and hopefully make a contribution to the understanding of the complex flow physics.

The flow is assumed incompressible and two-dimensional while physical properties of the fluid are constant. The fluid motion is governed by the momentum (Navier–Stokes) and continuity equations, which can be written in non-dimensional form using the pressure-velocity formulation as

	$\displaystyle\frac{\partial u_{x}}{\partial t}+u_{x}\frac{\partial u_{x}}{\partial x}+u_{y}\frac{\partial u_{x}}{\partial y}$	$\displaystyle=$	$\displaystyle-\frac{\partial p}{\partial x}+\frac{1}{Re}\left(\frac{\partial^{2}u_{x}}{\partial x^{2}}+\frac{\partial^{2}u_{x}}{\partial y^{2}}\right)-\ddot{x}^{*}_{c},$		(9)
	$\displaystyle\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{\partial x}+u_{y}\frac{\partial u_{y}}{\partial y}$	$\displaystyle=$	$\displaystyle-\frac{\partial p}{\partial y}+\frac{1}{Re}\left(\frac{\partial^{2}u_{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}\right),$		(10)
	$\displaystyle\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}$	$\displaystyle=$	$\displaystyle 0.$		(11)

Coordinates are normalized with $D$, fluid velocities with $U_{\infty}$, time with $D/U_{\infty}$, and pressure with $\rho U_{\infty}^{2}$. The acceleration of the cylinder $\ddot{x}^{*}_{c}$ appears on the right-hand side of equation (9) because the Navier–Stokes equations are applied in a non-inertial frame of reference that moves with the vibrating cylinder. Instead of explicitly enforcing the continuity equation, the pressure field was computed by solving the following Poisson equation at each time step,

$$\nabla^{2}p=2\left(\frac{\partial u_{x}}{\partial x}\frac{\partial u_{y}}{\partial y}-\frac{\partial u_{x}}{\partial y}\frac{\partial u_{y}}{\partial x}\right)-\frac{\partial\mathcal{D}}{\partial t},$$

(12)

where $\mathcal{D}=\partial u_{x}/\partial x+\partial u_{y}/\partial y$ is the dilation. Although the dilation is zero by default in incompressible flows (Eq. 11), the term $\partial\mathcal{D}/\partial t$ is kept in Eq. (12) to avoid the propagation of numerical inaccuracies (Harlow & Welch, 1965).

On the cylinder surface, the no-slip boundary condition gives

$$u_{x}=0,\qquad u_{y}=0,$$

$(13a,b)$

and the following condition for the normal pressure gradient at the wall

$$\frac{\partial p}{\partial n}=\frac{1}{Re}\nabla^{2}u_{n}-\ddot{x}^{*}_{c,n},$$

(14)

where $n$ refers to the component normal to the cylinder surface pointing to the fluid side. At the far field, a potential flow field is assumed so that

$$u_{x}=u_{x,pot}-\dot{x}_{c}^{*},\qquad u_{y}=u_{y,pot},$$

$(15a,b)$

where $u_{x,pot}$ and $u_{y,pot}$ is the velocity field from the known potential of irrotational flow. The corresponding condition for the far-field pressure is

$$\frac{\partial p}{\partial n}=0.$$

(16)

The initial field corresponds to the potential flow around a circular cylinder. The dimensionless form of the equations illustrates that the fluid motion depends solely on the Reynolds number given the boundary and initial conditions.

The displacement of a cylinder elastically constrained so that it can oscillate only in-line with a uniform free stream is governed by Newton’s law of motion, which can be written in non-dimensional form as

$$\ddot{x}_{c}^{*}+\frac{4\upi\zeta}{U^{*}}\dot{x}_{c}^{*}+\left(\frac{2\upi}{U^{*}}\right)^{2}x_{c}^{*}=\frac{2C_{x}(t)}{\upi m^{*}},$$

(17)

where $x_{c}^{*}$, $\dot{x}_{c}^{*}$, and $\ddot{x}_{c}^{*}$ respectively are the non-dimensional displacement, velocity, and acceleration of the cylinder, normalized using $D$ and $U_{\infty}$ as length and velocity scales; $C_{x}(t)$ is the sectional fluid force on the cylinder normalized by $0.5\rho U^{2}_{\infty}D$. Here, the fluid forcing is provided by the ambient flow through balancing the normal and shear stresses on the cylinder surface. The cylinder is initially at rest. The dimensionless form of equation (17) illustrates that the cylinder motion depends on the reduced velocity $U^{*}$, the mass ratio $m^{*}$, and the damping ratio $\zeta$. The full set of independent dimensionless parameters of the problem comprises $Re,\,U^{*},\,m^{*}$, and $\zeta$.

Because of the two-dimensional approach, simulations were limited up to $Re=250$. Our main simulations are at $Re=180$, a value which is slightly lower than the threshold for which mode-A spanwise instability occurs in the wake of a stationary cylinder, i.e. $Re_{c}\approx 190$ (see Williamson, 1996; Barkley & Henderson, 1996), and may provide some indication of the corresponding threshold of three-dimensional transition for oscillating cylinders. Yet, the in-line oscillation of the cylinder causes wake synchronization, which has been shown to suppress the mode-A spanwise instability into two-dimensional laminar flow with strong Kármán vortices at $Re=220$ (Kim et al., 2009). Therefore, the flow may well be expected to remain strictly two dimensional for our main simulations at $Re=180$. For simulations at the highest value of $Re=250$, it is plausible that some three-dimensional instability might exist. Three dimensionality usually appears first in the form of weak coherent structures of streamwise vorticity with specific wavelength riding on the primary spanwise vorticity that remains in-phase along the length of freely-vibrating cylinders (see Lo Jacono et al., 2018; Bourguet, 2020). Under such flow conditions, the spanwise vorticity component is much higher than the other two components by one order and the direct effect of the three-dimensionality of the vortical structures on the force is weak as noted by Wu et al. (2007). Thus, the plausible existence of three-dimensional vortex structures in the cylinder wake may be reasonably expected to not directly influence the magnitude and phase of the streamwise and transverse fluid forces, which are primarily determined by the two-dimensional wake instability, nor influence the cylinder response at the highest Reynolds number of 250 at which we conducted simulations to check the trends of results in the laminar regime.

An in-house code based on the finite difference method was used to solve the equations of fluid motion (Baranyi, 2008). The flow domain is enclosed between two concentric circles: the inner circle is the boundary fitted to the cylinder surface while the outer circle represents the far field boundary. The polar physical domain is mapped into a rectangular computational domain using linear mapping functions. The computational mesh of the ‘physical domain’ is fine in the vicinity of the cylinder and coarse in the far field while the corresponding mesh of the transformed domain is equidistant. Space derivatives are approximated using a fourth order finite-difference scheme except for the convective terms for which a third-order modified upwind difference scheme is employed. The pressure Poisson equation is solved using the successive over-relaxation (SOR) method and the continuity equation is implicitly satisfied at each time step. The Navier-Stokes equations are integrated explicitly using the first-order Euler method and the fourth-order Runge–Kutta scheme is employed to integrate the equation of cylinder motion in time. At each time step the fluid forces acting on the cylinder are calculated by integrating the pressure and shear stresses around the cylinder surface, which are obtained from the flow solver. The streamwise force is supplied to the right-hand side of Eq. (17) which is integrated to advance the cylinder motion. At the next time step, the cylinder acceleration is updated and the equations of fluid motion are integrated to complete the fluid-solid coupling. For all simulations reported here, the cylinder is initially at rest and the initial field around the cylinder satisfies the potential flow.

In this section, we present results from preliminary simulations to check the dependence of main output parameters on a) the size of the computational domain in terms of the radius ratio $R_{2}/R_{1}$, b) the grid resolution $\xi_{max}\times\eta_{max}$, and c) the dimensionless time step $\Delta t$. Here $\xi_{max}$ and $\eta_{max}$ are the number of grid points in peripheral and radial directions, respectively. During these computations, the Reynolds number, the reduced velocity, the mass ratio, and the structural damping ratio were fixed at $Re=180$, $U^{*}=2.55$, $m^{*}=10$, and $\zeta=0$, respectively. The main output parameters of interest are the amplitude $A^{*}$ and frequency $f^{*}$ of cylinder response (normalized with $D$ and $U_{\infty}$) as well as the standard deviations of the in-line and transverse fluid forces (normalized with $0.5\rho U_{\infty}^{2}D$), which are respectively denoted $C^{\prime}_{x}$ and $C^{\prime}_{y}$.

First, three different values of the radius ratio $R_{2}/R_{1}$ of the inner and outer circles defining the computational domain were tested and the corresponding results are shown in table 2. The number of circumferential nodes $\eta_{max}$ of the physical domain was adjusted in each case in order to keep the grid equidistant in the transformed domain. For these computations, the dimensionless time step was fixed at $\Delta t=10^{-4}U^{*}\cong 0.0002$. Table 2 shows that the most sensitive quantity on the domain size is $C^{\prime}_{x}$ displaying a relative difference of 1.7% between the small and large domains, whereas the corresponding differences for $A^{*}$, $f^{*}$ and $C^{\prime}_{y}$ are below 0.7%. The relative differences for all quantities of interest become less than 0.6% between the medium and large domains. Thus, the medium-sized domain with a radius ratio of $R_{2}/R_{1}=160$ was chosen for the rest of the computations.

Next, we tested three grids with different resolutions $\xi_{max}\times\eta_{max}$ where the number of peripheral and radial grid points was increased so that the grid remains equidistant in the transformed plane. For these computations, the radius ratio and dimensionless time step values were fixed at $R_{2}/R_{1}=160$ and $\Delta t=10^{-4}U^{*}\cong 0.0002$, respectively. The results of these tests are shown in table 3. Again, $C^{\prime}_{x}$ is the most sensitive quantity displaying a relative difference of 0.7% between coarse and fine grids. All quantities of interest display relative differences less than 0.3% between medium and fine grids. Thus, the medium-resolution grid was chosen for further computations.

Finally, we tested the dependence on the dimensionless time step, $\Delta t$ for three different values corresponding to $\Delta t\cong 2\times 10^{-4}U^{*},10^{-4}U^{*}$ and $5\times 10^{-5}U^{*}$. For these computations, the radius ratio and grid resolution values were fixed at $R_{2}/R_{1}=160$ and $360\times 292$, respectively. The results are shown in table 4. In this tests, the quantity that is most sensitive to the time step is $A^{*}$, which shows a relative difference of 0.4% between the largest and smallest time steps whereas the corresponding relative differences for $f^{*}$, $C^{\prime}_{x}$ and $C^{\prime}_{y}$ are all below 0.15%. The relative differences of all quantities of interest are below 0.1% between the intermediate and the smallest time steps. Thus, a dimensionless time step of $\Delta t=10^{-4}U^{*}$ was chosen for the main computations.

The computational code used in the present study was previously employed in several studies of flows about stationary and oscillating cylinders and results have been extensively compared against data from the literature (see Baranyi, 2008; Dorogi & Baranyi, 2018, 2019). For instance, Baranyi (2008) found good agreement with the study of Al-Mdallal et al. (2007) in terms of the time history of the lift coefficient and Lissajous patterns of lift vs. cylinder displacement at comparable situations for the case of a cylinder forced to oscillate in the streamwise direction. In addition, the extended code handling flow-structure interaction has been validated for the case of a cylinder undergoing vortex-induced vibration with two degrees of freedom with equal natural frequencies in the streamwise and transverse directions against results from Prasanth & Mittal (2008) for $m^{*}=10$, $\zeta=0$, and Reynolds numbers in the range from 60 to 240 (for details see Dorogi & Baranyi, 2018). Furthermore, Dorogi & Baranyi (2019) showed that results obtained with the present code compare well with published results in Navrose & Mittal (2017) for purely transverse free vibration, as well as in Prasanth et al. (2011) and Bao et al. (2012) for free vibration with two degrees of freedom with equal or unequal, respectively, natural frequencies in the streamwise and transverse directions, for similar conditions in each case.

In addition to the validation tests presented in previous studies, we compare in figure 2 results obtained with the present code against the study of Bourguet & Lo Jacono (2015) for purely in-line free vibration. Here, we employed a finer step in the reduced velocity to resolve the maximum in $A^{*}$ as well as the minimum in $C^{\prime}_{x}$. There is excellent agreement of results for $A^{*}$ and $C^{\prime}_{x}$ but there are some minor deviations for $f^{*}$ and $C^{\prime}_{y}$ of 0.2% and 1.2%, respectively, which might be attributable to different numerical methods employed in those studies (finite difference vs. spectral element). Overall, previous and present validation tests show that the numerical code employed in the present study provides accurate solutions.

To start with, we consider effect of the Reynolds number on the in-line response of a cylinder with a mass ratio of $m^{*}=5$. The structural damping was set to zero so as to allow for the highest possible amplitude response to take place. Figure 3 (top plot) shows that there exists a single excitation region in which the response amplitude $A^{*}$ displays a marked peak for all Reynolds numbers considered. The $U^{*}$ value at which peak amplitudes occur decreases with $Re$, which can be attributable to the corresponding increase of the Strouhal number since peak amplitudes occur at approximately $U^{*}_{a}\approx 1/(2S)$. The peak amplitude over the entire $U^{*}$ range, denoted as $A^{*}_{\mathrm{max}}$, increases from 0.002 at $Re=100$ to 0.024 at $Re=250$, i.e. an increase in $Re$ by a factor of 2.5 results in an increase in $A^{*}_{\mathrm{max}}$ by a factor of 12. This is a remarkable increase of the order of magnitude, which contrasts the constancy of peak amplitudes of purely transverse free vibration in the corresponding range of Reynolds numbers (a compilation of $A^{*}_{\mathrm{max}}$ data as a function of $Re$ from several studies can be found in Govardhan & Williamson, 2006).

For all simulations conducted in this study, the vortex shedding remained synchronized at half the frequency of the cylinder oscillation. As shown in the bottom plot in figure 3 the normalized response frequency $f^{*}$ is approximately twice the corresponding Strouhal number at each Reynolds number. However, $f^{*}$ displays a trough within the excitation region that becomes more pronounced as the Reynolds number increases; the trough can be hardly discerned for $Re=100$. The variations of $A^{*}$ and $f^{*}$ appear to be strongly correlated. The characteristics of free response remain similar for all Reynolds numbers considered here. However, a sudden drop in $A^{*}$ appears just after the peak amplitude for $Re=250$, a feature which is not present for lower Reynolds numbers. This might indicate the onset of branching behaviour similar to that observed in vortex-induced vibration purely transverse to the free stream (see Leontini et al., 2006).

In comparison to previous experimental studies, which typically correspond to Reynolds numbers above $10^{3}$, we did not observe another excitation region associated with symmetrical vortex shedding. In contrast, we observed only the alternating mode of vortex shedding in the present simulations corresponding to the laminar wake regime. Figure 4 shows vorticity distributions in the wake at $U^{*}$ values corresponding to peak amplitudes for different Reynolds numbers. In all cases, the vorticity distributions display the familiar von Kármán vortex street similar to the wake of a stationary cylinder. As the Reynolds number is increased, the contours of individual vortices become more concentrated and peak vorticity values within them increase. This might be partly attributable to the increase in the amplitude of cylinder oscillation with Reynolds number, which accrues the generation of vorticity on the cylinder surface (Konstantinidis & Bouris, 2016). In addition, the streamwise spacing between the centres of subsequent vortices decreases due to the increase of the normalized frequency of cylinder oscillation $f^{*}$ with Reynolds number. The absence of the other excitation region associated with the symmetrical vortex shedding may be attributable to the fact that, as has been shown in several previous studies where the cylinder is forced to oscillate in the streamwise direction at correspondingly low Reynolds numbers, the onset of this mode occurs at relatively high amplitudes above 0.1 diameters (Al-Mdallal et al., 2007; Marzouk & Nayfeh, 2009; Kim & Choi, 2019). Since streamwise amplitudes of free vibration are much lower than that threshold, it is not surprising that the mode of symmetrical shedding and the corresponding excitation region were not observed in the present study.

Next, we concentrate on the effect of mass ratio on the in-line response at a fixed Reynolds number of $Re=180$, at which the flow is expected to remain laminar and strictly two dimensional. The structural damping was set to zero $(\zeta=0)$ to allow the highest possible amplitudes.

We begin with linearising the quasi-steady drag term in equation (7) and substituting the harmonic approximations in equations (2) and (8) to obtain the linearised force, which we can express explicitly as a function of time $t$ as

	$\displaystyle F_{x}(t)=\frac{1}{2}\rho U_{\infty}^{2}D\left[C_{d}+2\left(\frac{\omega A}{U_{\infty}}\right)C_{d}\sin{(\omega t)}+C_{dw}\cos{(\omega t+\phi_{dw})}\right]$
	$\displaystyle+\frac{1}{4}\upi\rho D^{2}C_{a}\omega^{2}A\cos{(\omega t)},$		(23)

where the angular frequency $\omega=2\upi f$ has been introduced for brevity and $f_{dw}$ has been replaced by $f$ due to the synchronization of the wake and the cylinder vibration.

The first term on the right-hand side of equation (23) represents the steady part of the in-line force whereas the remaining terms represent the unsteady part, which comprises the combination of cosine and sine functions at the frequency of cylinder oscillation. Equating the steady parts in equations (3) and (23) yields

$$C_{d}=\overline{C}_{x},$$

(24)

i.e. $C_{d}$ is equal to the mean drag coefficient. Equating the cosine and sine terms in equations (3) and (23) yields the following relationships:

	$\displaystyle C_{dw}\sin{\phi_{dw}}$	$\displaystyle=$	$\displaystyle C_{x1}\sin\phi_{x}+4\upi f^{*}A^{*}C_{d},$		(25)
	$\displaystyle C_{dw}\cos{\phi_{dw}}$	$\displaystyle=$	$\displaystyle C_{x1}\cos\phi_{x}-2\upi^{3}f^{*2}A^{*}C_{a}.$		(26)

The set of equations (24-26) establishes relationships between fluid forcing components as functions of $A^{*}$ and $f^{*}$ alone, i.e. the structural parameters do not appear. Therefore, the same values of the fluid forcing components also hold when the cylinder is forced to oscillate at the same operating points in the $A^{*}:f^{*}$ parameter space as the free vibration. We use the above set of relationships in order to determine $C_{dw}$ and $\phi_{dw}$ appearing in the new theoretical model from the cylinder response $(A^{*},\,f^{*})$ and the fluid forcing $(C_{x1},\,\phi_{x})$ data obtained from the simulations. It should be remembered here that $C_{a}=1$ is the known ideal added-mass coefficient of unity.

Figure 11 shows the variation of $C_{dw}$ and $\phi_{dw}$ as functions of the parameter $U^{*}_{a}f^{*}$. We have employed this parameter because peak amplitudes occur at $U^{*}_{a}f^{*}\approx 1$. As can be seen in the top plot, $C_{dw}$ also displays a peak at the same point with a maximum value of 0.067 for all $m^{*}$. The mutual amplification of the response amplitude and the wake drag are representative of resonance. Hence, we refer to the condition $U^{*}_{a}f^{*}=1$ as the ‘resonance point’. Away from resonance $C_{dw}$ tends asymptotically to 0.036, which corresponds to the amplitude of the fluctuating drag coefficient for a non-vibrating cylinder at $Re=180$.

In the bottom plot in figure 11 can be seen that $\phi_{dw}$ follows an S-type increase as a function of $U^{*}_{a}f^{*}$ from approximately $0^{\circ}$ to $180^{\circ}$. At $U^{*}_{a}f^{*}=1$, $\phi_{dw}$ passes through $90^{\circ}$ for all $m^{*}$. Overall, the variation of $\phi_{dw}$ is very similar to that of $\phi_{y}$ shown earlier; in fact there is a direct relationship between them, which is illustrated in figure 12. Since the variation of $\phi_{y}$ is invariably linked to the vortex dynamics, in particular to the timing of vortex shedding as discussed earlier, it follows that $\phi_{dw}$ appropriately captures related changes as the reduced velocity is varied.

The new hydrodynamic model can be combined with the equation of cylinder motion in order to illustrate the interaction between the fluid and the structural dynamics. The steady-state solution can be obtained following a similar procedure as described in appendix A, which results in the following two relationships:

$$C_{dw}\sin{\phi_{dw}}=4\upi f^{*}A^{*}\left(C_{d}+\frac{\upi^{2}m^{*}\zeta}{U^{*}}\right),\\$$

(27)

$$C_{dw}\cos{\phi_{dw}}=2\upi^{3}\frac{m^{*}A^{*}}{U^{*2}}\left[1-\left(\frac{f}{f_{n,a}}\right)^{2}\right].$$

(28)

When the structural damping is zero $(\zeta=0)$, the solution of equation (27) apparently does not depend on $m^{*}$. It should be noted however that the feasible operating points in the $A^{*}:f^{*}$ space are limited to those that satisfy equation (27), which correspond to the contour of zero energy transfer $(\sin\phi_{x}=0)$. In addition, equation (28) becomes indeterminate at the resonance point where $f=f_{n,a}$ and $\phi_{dw}=90^{\circ}$ simultaneously. It should be noted that there is no a priori guarantee that the ‘resonance point’ is attainable since the solution pursued here is open form. Nevertheless, as we can see from the simulations the resonance point is attained and corresponds to the maximum amplitude, in which case we can write equation (27) as

$$A^{*}_{\mathrm{max}}=\frac{C_{dw}}{4\upi f^{*}C_{d}}.$$

(29)

This result shows that the steady-state solution at the resonance point does not depend on $m^{*}$; rather the operating point is solely determined by the fluid dynamics.

When the structural damping is finite positive $(\zeta>0)$, equation (29) may still be used to estimate approximately the peak amplitude for very low values of the mass and the damping such that $\upi^{2}m^{*}\zeta/U^{*}\ll C_{d}$, on the condition that the resonance point is attained. The early experiments of Aguirre (1977) support the above inference; he observed little variation of the peak amplitude in the second excitation region, i.e. the one associated with alternating vortex shedding, which occurred at $U_{a}^{*}\approx 1.2/(2S)$ for $m^{*}=1.2$ and 4.3 (see figure 43 of his thesis). The mass–damping was reported in terms of a stability parameter $k^{\prime}_{s0}$ to be approximately 0.5, which was estimated from free-decay tests in still water, i.e. it includes contributions from hydrodynamic damping from the cylinder as well as its mountings. From his data, we estimate here that the corresponding $k_{s}$ value in air was significantly lower than the one measured in water so that we can assume $\upi^{2}m^{*}\zeta/U^{*}<0.05$; this is small compared to the mean drag coefficient $C_{d}$. Thus, we argue that the present analysis is consistent with the experimental facts at Reynolds numbers higher than considered in this study.

At the resonance point ($\phi_{dw}=90^{\circ}$), equations (25) and (26) reduce to

	$\displaystyle C_{dw}$	$\displaystyle=$	$\displaystyle 4\upi f^{*}A^{*}C_{d},$		(30)
	$\displaystyle C_{x1}$	$\displaystyle=$	$\displaystyle 2\upi^{3}f^{*2}A^{*}C_{a}.$		(31)

The above set of equations establishes relationships between the wake, mean, and total drag coefficients at resonance point, which involve variations in $A^{*}$ and $f^{*}$. Considering that all three force coefficients, $C_{dw}$, $C_{d}$, and $C_{x1}$, may be ‘uniquely’ specified in the parameter space $A^{*}:f^{*}$, this could suggest a single operating point inside this space at which a steady-state harmonic solution is feasible, which implies that the cylinder peak response is drawn towards this operating point irrespectively of the mass ratio. Nonetheless, it should be cautioned that bimodal dynamics can also appear at particular regions of the parameter space $A^{*}:f^{*}$ (Cagney & Balabani, 2013a, b; Gurian et al., 2019). Assuming that a unique solution exists at the resonance point, equation (30) illustrates that the component of the wake drag in-phase with the cylinder velocity counterbalances the quasi-steady drag, whereas equation (31) illustrates that the only contribution of the fluid force in-phase with the cylinder acceleration is the inviscid added-mass force. This is commensurate to the situation where an elastically mounted cylinder oscillates freely within a fluid medium with no net viscous forces; in such a situation, it would be anticipated that the frequency of cylinder oscillation is equal to the natural frequency of the structure in an inviscid fluid, i.e. $f=f_{n,a}$. This is as if the fluid was phenomenologically interacting with the cylinder motion only through its ideal inertia. It should be emphasized that in the case of a viscous fluid, the zero net viscous force results from the cancelling out of the wake drag and the quasi-steady drag, which is brought about by a gradual change in the phasing of the vortex shedding, thereby in $\phi_{dw}$, as the reduced velocity is varied.

The above changes can be viewed from another perspective, which is more insightful. As $U^{*}$ increases, the phasing of vortex shedding gradually shifts to follow the changes in the frequency of oscillation with the reduced velocity. Since these variations occur in a continuous manner for the low-$Re$ cases considered here, a point is reached where the timing of vortex shedding induces a wake drag exactly in-phase with the cylinder velocity, $\phi_{dw}=90^{\circ}$. At this point, the net viscous force must become null (Eq. 30), whereas only the added mass contributes to the force in-phase with the cylinder acceleration (Eq. 31). It is important to note again that these changes are brought in entirely by the fluid dynamics.

At first glance, it seems counter-intuitive that the cylinder experiences almost no net force at the coincidence point. However, this can be explained by the cancelling out of the fluid-force components. It should be remembered that when the structural damping is zero $(\zeta=0)$ the linearised solution of the equation of cylinder motion (Eq. 22) shows that the component of the streamwise force at the main harmonic is null $(C_{x1}=0)$ at the coincidence point $(f^{*}U^{*}=1)$. In this case, equations (25) and (26) reduce to

	$\displaystyle C_{dw}\sin{\phi_{dw}}$	$\displaystyle=$	$\displaystyle 4\upi f^{*}A^{*}C_{d},$		(32)
	$\displaystyle-C_{dw}\cos{\phi_{dw}}$	$\displaystyle=$	$\displaystyle 2\upi^{3}f^{*2}A^{*}C_{a}.$		(33)

From these equations, we can see that the wake drag in-phase with the velocity of the cylinder balances the quasi-steady drag while the wake drag in-phase with acceleration balances the inviscid inertia. This is physically possible as these force contributions originate from different flow structures. Furthermore, the cylinder must be oscillating for this case to be realized; the terms on the right-hand side of equations (32) and (33) are null for a non-oscillating cylinder. Although the above scenario is idealised as some damping will be present in a real situation, extending the analysis for a system with very low structural damping shows that the cylinder will experience a very low net in-line force at the coincidence point, if this point can still be attained. It should be noted that we approached very close to the coincidence point by employing a very fine step around this region, but exact coincidence was unattainable. A similar observation was made by Shiels et al. (2001) who showed that the net transverse force $C_{y}(t)$ exerted on a cylinder undergoing free vibration in the transverse direction becomes null at limiting values of the structural parameters, $m=c=k=0$. Here, we show that a null net streamwise force $C_{x}(t)$ on a cylinder undergoing free in-line vibration is physically possible at the coincidence point for finite $m$ and $k$ values.

The variation of the peak amplitude as a function of the Reynolds number can be estimated using the following assumptions. The mean drag coefficient $C_{d}$ and the inverse of the Strouhal number $1/S$ for stationary bluff bodies display similar variations as functions of $Re$ so that the product $SC_{d}$ remains almost constant (Alam & Zhou, 2008). For vibrating cylinders, there is also evidence that the corresponding product $(f^{*}/2)C_{d}$ remains constant (Griffin, 1978). Alam & Zhou (2008) showed that $SC_{d}\approx 0.25$ for stationary bluff bodies of various cross-sectional shapes throughout the sub-critical regime. However, $SC_{d}$ appears to increase slightly with $Re$ in the laminar regime as seen in their data compilation (see their figure 5). We have confirmed this dependency on $Re$ in the laminar regime for the stationary as well as the freely-vibrating circular cylinder as shown in table 7. Nevertheless, the variation of $SC_{d}$ and $(f^{*}/2)C_{d}$ with $Re$ is small, and in order to keep some generality in the result, we assume that $f^{*}C_{d}$ remains approximately constant. In this case, equation (29) shows that $A^{*}_{\mathrm{max}}\propto C_{dw}$. In addition, we assume $C_{dw}$ is magnified by some constant factor at resonance so that $A^{*}_{\mathrm{max}}\propto C_{dw0}$, where $C_{dw0}$ is the fluctuating drag coefficient for a stationary cylinder. Then, peak amplitudes at different $Re$ can be estimated from data at one particular Reynolds number $Re_{0}$, i.e.

$$A^{*}_{\mathrm{max}}(Re)=\left(\frac{A^{*}_{\mathrm{max}}}{C_{dw0}}\right)_{Re_{0}}C_{dw0}(Re).$$

(34)

Figure 13 shows predictions of $A^{*}_{\mathrm{max}}$ using $C_{dw0}$ data at different $Re$ (taken from Qu et al., 2013) and the known response at $Re=180$. It can be seen that $A^{*}_{\mathrm{max}}$ predictions fit well with the amplitude from the simulations at $Re=100$ while they also extrapolate to the result at $Re=250$. The enhancement of $A^{*}_{\mathrm{max}}$ with $Re$ is close to quadratic, which is quite marked compared to the nearly constant amplitude of purely transverse vortex-induced vibration in the laminar regime (see Govardhan & Williamson, 2006).