Analysis of Forced Transverse Vibration of a Rough Circular Cylinder Subjected to Wake Interference—Numerical Investigation Using the Lagrangian Discrete Vortex Method

Abstract

This paper numerically investigates the flow past two circular cylinders of equal diameter arranged in tandem with respect to the incident flow. The upstream cylinder is fixed, and the downstream cylinder is located within the wake interference region for a streamwise center-to-center spacing of L = 5D (D is the cylinder diameter). The downstream cylinder is forced to vibrate transversely in the wake of the upstream cylinder to investigate a regime of wake-induced vibration ($WIV$) at a Reynolds number of $Re=65,000$. The non-dimensional vibration amplitude is fixed at $A/D=0.15$, and the reduced velocity is set to $V_R=5$. The literature has reported that $WIV$ is a phenomenon resulting from the interaction between the incoming wake and the downstream flexible structure, in which the downstream cylinder vibrates significantly over a wide velocity range, and the cross-flow fluid force is not in phase with the body’s motion. The phenomenon of $WIV$ appears combined with a resonant regime, in which the downstream cylinder vibrates at the resonant velocity similar to the vortex-induced vibration ($VIV$) of a single cylinder. The results show that the individual resonant regime is captured for both surfaces without roughness effects. The main contribution of this paper is to demonstrate that the roughness effect variation of the downstream cylinder surface desynchronizes the $WIV$ regime and simultaneously promotes synchronization through the emergence of harmonic frequencies, indicating competition between $VIV$ and $WIV$.

1. Introduction

External flow around bluff bodies at high Reynolds numbers remains a central problem in applied fluid mechanics due to its relevance to different engineering structures and to the strong sensitivity of boundary-layer separation, vortex shedding, and wake formation to flow and surface conditions. Among these geometries, the circular cylinder constitutes the classical configuration for the study of aerodynamic interference, instabilities, and fluid–structure interaction [1,2]. When two cylinders interact, especially in a tandem arrangement, the problem becomes more complex than that of an single body, since the downstream cylinder responds not only to its local flow field but also to the incident wake generated by the upstream cylinder. In this configuration, the spacing between the bodies and the Reynolds number govern the transition among the extended-body, reattachment, and co-shedding regimes as well as the intensity of wake interference [2,3].

Within this context, wake-induced vibrations ($WIV$s) constitute a class of instabilities of particular importance for the downstream cylinder. For two parallel circular cylinders, these wake-induced responses may be classified as wake-induced vortex vibration, wake galloping, and wake-induced flutter [4]. In the same work, the authors showed that wake galloping is predominantly associated with the transverse oscillation of the downstream cylinder, whereas wake-induced flutter involves simultaneous components in the longitudinal and transverse directions with an elliptical trajectory. Consistently, Dao et al. [3] showed that the type of $WIV$ strongly depends on the wake-interference pattern and on the Reynolds-number regime with wake galloping occurring at closer spacings and coupled flutter arising over distinct reduced-velocity ranges and aerodynamic-interference conditions.

Recent studies have further refined the physical interpretation of wake-induced vibration in tandem-cylinder systems. Lin et al. [5] mapped the properties of $WIV$ on a circular cylinder and showed that the downstream response may evolve into distinct dynamical states depending on the organization of the incoming wake. In a related experimental investigation, Dulac et al. [6] demonstrated that the oscillatory characteristics of the upstream cylinder may significantly alter the downstream $WIV$ response, reinforcing the importance of prescribed wake kinematics. Assi [7] identified the retro lock-in mechanism in closely spaced tandem cylinders, showing that the motion of the downstream body may feed back into the upstream shedding process. Alam et al. [8] further organized the responses $WIV$ into different branches associated with desynchronization and galloping-type behavior, while Li et al. [9] proposed an activation–modulation framework for the response of the downstream cylinder in tandem configurations. Taken together, these studies show that $WIV$ should not be viewed as a single uniform mechanism but rather as a family of wake-governed responses that depend strongly on the spacing, wake structure, synchronization state, and body–wake feedback.

In addition to wake interference itself, surface roughness constitutes a first-order parameter because it alters the boundary-layer transition, the mean drag, the Strouhal number, and the wake structure [4,10,11,12]. Du et al. [4] showed that increasing roughness shifts the transition to the critical and supercritical regimes to lower Reynolds numbers. In a specific study on circular-cylinder wakes with large relative roughness heights, Sun et al. [10] observed a reduction in the Strouhal number, thickening of the shear layers, and modification of the wake formation length as roughness increased. In vibration problems, Ghazali et al. [11] reported a reduction in response amplitude with increasing roughness for a bluff body subjected to transverse $VIVs$ (vortex-induced vibrations), whereas Gao et al. [12], for a cylinder vibrating in the vicinity of a wall, showed that roughness alters hydrodynamic forces, vibration trajectories, and aspects of the lock-in regions. Taken together, these results indicate that roughness does not act merely as a secondary geometric detail but rather as a parameter capable of reorganizing the dynamics of both the flow and the structural response.

More recent studies have reinforced this interpretation by showing that roughness may substantially modify vibration response even for single cylinders. Han et al. [13] reported that surface roughness changes the $VIV$ response branches and may introduce more complex oscillatory behavior at moderate Reynolds numbers. Chen et al. [14] numerically showed that at subcritical Reynolds numbers, roughness alters the vibration amplitude, hydrodynamic coefficients, and wake-vortex characteristics. Similarly, Jiang et al. [15] demonstrated that for a two-degree-of-freedom circular cylinder in oscillatory flow, increasing roughness modifies both the vibration response and the associated wake patterns. These results reinforce the idea that roughness may act as an effective dynamic parameter in fluid–structure interaction problems rather than merely as a passive geometric perturbation.

When moving from an isolated cylinder to two-cylinder arrangements, the influence of roughness becomes even more sensitive because it also affects the very nature of wake interference. Du et al. [4] showed that the roughness of the downstream cylinder has a more pronounced effect on $WIV$s than the roughness of the upstream cylinder, and when the downstream cylinder is rough, the response may cease to be divergent and instead occur within a limited reduced-velocity range, characterizing velocity-restrained vibrations. Dao et al. [3] further showed that interference between two parallel cylinders may take different forms, including reattachment of the upstream-cylinder shear layer onto the downstream cylinder, gap-flow formation, and vortex impingement, with these mechanisms being strongly dependent on Reynolds number. In a different, yet mechanistically relevant, configuration, Yang et al. [16] showed that for two cylinders with roughness strips, the gap-flow formation depends on the spacing and Reynolds number, and this flow may become the main factor governing vibration under certain geometric and kinematic combinations.

This combined effect of roughness and interference has also been examined in more recent tandem configurations. Hu et al. [17], in an experimental study on rough risers arranged in tandem, showed that the interference effects and surface condition should be analyzed jointly, since the downstream-body response is sensitive to the coupling between the incoming-wake structure and roughened-body dynamics. Although that configuration differs from the present one, the study supports the broader interpretation that roughness may alter not only the local hydrodynamic loads but also the interference-mediated vibration regime itself.

From a numerical standpoint, the analysis of these phenomena requires a formulation capable of adequately representing detached vortical structures, wake interference, and surface effects without losing the dominant physics of the problem. Carvalho et al. [1] showed that the Discrete Vortex Method ($DVM$), in a purely Lagrangian formulation, constitutes a particularly suitable strategy for bluff-body aerodynamic problems with heat transfer, since it discretizes the vorticity-carrying regions into vortex particles that move with the fluid, restricting the computations to the regions of interest and avoiding the need for the explicit treatment of far-field boundaries. The same authors also showed that this formulation can incorporate viscous diffusion, the panel discretization of solid surfaces, turbulence modeling, and roughness effects. In particular, a two-dimensional roughness model, coupled with LES theory, was employed to modify the boundary-layer flow and the separation point through an effective momentum injection into the near-wall region [1]. Previous applications of this methodology have already shown that it is capable of capturing relevant physical changes associated with roughness, such as modifications in vortex-shedding frequency, the intermittency or cessation of vortex shedding, and significant changes in aerodynamic loads [1,2].

In the specific context most closely related to this paper, Moraes et al. [18] employed a purely Lagrangian $DVM$ to numerically investigate the two-dimensional flow around two identical circular cylinders aligned with the incoming flow, considering a fixed upstream cylinder and a downstream cylinder subjected to forced transverse vibration at high Reynolds numbers. In that study, in addition to the analysis of two fixed cylinders in tandem, the fluid-dynamic characteristics of the downstream oscillating cylinder were investigated for $g/D$ = 4.5 (g is the gap spacing center-to-tenter between two cylinders), reduced amplitudes $A/D$ = 0.15 and $A/D$ = 0.50, and the reduced-velocity range 5 < $V_R$ < 50, leading to the conclusion that in this arrangement, the $WIV$ mechanism does not constitute a purely resonant phenomenon [18]. This research therefore represents a direct basis for this paper, as it established the fluid-dynamic and structural problem of the tandem arrangement with no roughness effect.

In addition, recent evidence indicates that the application of vortex-based Lagrangian formulations to vibration problems involving rough cylinders remains limited. Chiaradia et al. [19] employed the Lagrangian vortex method to investigate the $VIV$ response and drag of a rough circular cylinder, showing that roughness can substantially modify both the aerodynamic loading and the oscillatory response. This study is especially relevant from a methodological standpoint because it confirms the suitability of purely Lagrangian vortex formulations for rough-cylinder problems while also highlighting that their extension to tandem arrangements with direct wake interference and forced transverse oscillation is still scarce in the recent literature.

Despite these advances, it still remains insufficiently understood how the distribution of the surface roughness on the downstream cylinder modifies, in a tandem arrangement aligned with the flow, the coexistence, alternation, and transition among mechanisms associated with lock-in and wake-induced response. A significant part of the literature focuses on freely vibrating or elastically mounted cylinders [4,11,12] staggered arrangements with gap flow. Dao et al. [3] and Yang et al. [16] studied cylinders with roughness strips or helical protrusions, or they analyzed fixed cylinders in the co-shedding regime. Even in the numerical antecedent closest to the present problem, the study of Moraes and Alcântara Pereira [2] was conducted for smooth cylinders without incorporating surface roughness on the downstream cylinder. In contrast, the combination of the forced oscillation of the downstream cylinder, direct interference from the wake of the upstream cylinder, and distributed roughness as a dynamic control parameter still lacks specific investigation, particularly under high-Reynolds-number conditions in which wake reorganization and fluid–structure synchronization become especially relevant.

Accordingly, this paper numerically investigates the two-dimensional, incompressible, and unsteady flow around two identical circular cylinders aligned with the incoming flow with the upstream cylinder stationary and the downstream cylinder subjected to forced transverse oscillation. The numerical formulation is based on a purely Lagrangian description through the $DVM$, combined with large-eddy simulation ($LES$) subgrid-scale turbulence modeling and the two-dimensional roughness model, in which roughness acts through local modification of the boundary layer and flow separation [1]. The main contribution of this paper is to extend the problem formulated by Moraes and Alcântara Pereira [2] to the case in which the downstream cylinder presents distributed surface roughness, thereby allowing an integrated analysis of how the combination of forced vibration and roughness reorganizes wake interference and the synchronization between lift force and body motion. In particular, this paper focuses on three representative relative roughness levels, $\epsilon /D$ = 0, $\epsilon /D$ = 0.0045 and $\epsilon /D$ = 0.01, which were selected because they produce qualitatively distinct responses, thus making it possible to highlight the transition among regimes associated with lock-in, desynchronization, and wake-induced behavior. Therefore, this paper expands the existing literature by addressing a problem in which roughness effects are not analyzed in isolation but rather in direct coupling with forced vibration and with the incident wake of a second cylinder, which constitutes the novel and central element of this paper’s contribution [1,2].

The two-dimensional roughness model used in this paper was developed and validated by [20]. The roughness model was able to capture supercritical flow patterns from a typical subcritical Reynolds number flow. The numerical results successively indicated that two-dimensional roughness modeling is able to capture the drag crisis in a good physical sense at a high Reynolds number of $Re=1.0\times {10}^5$. The second-order velocity structure function model of the filtered vorticity field presented by Lesieur and Métais [21] utilizes the concept of velocity fluctuations (velocity differences) instead of the rate of deformation (derivatives), which is a very appropriate association for Lagrangian discrete vortex methods for developing a roughness model, while the latter is based on the physics involved in turbulent boundary layer flows. The sensitivity of the present two-dimensional roughness model has been demonstrated by our research group by capturing the main features of turbulent boundary layer flows past bluff bodies, such as drag crises, separation point displacements, changes in the lift force direction, stagnation point displacements and interference from vortex formation regimes from a bluff body forced to oscillate [2,19,20].

In general, the two-dimensional roughness model can improve the prediction of Strouhal numbers more than other studies without roughness models. The aerodynamic characteristics of a bluff body, as well as the control of intermittence and complete interruption of von Kármán-type vortex shedding, can be discussed in a very good physical sense. The idea behind our simulations is the development of a numerical technique that produces results for aerodynamic loads and vortex-shedding frequencies that are very useful for conservative designs for engineering applications.

2. Mathematical Formulation of the Problem

2.1. Definition of the Physical Problem

Consider a two-dimensional viscous and incompressible flow with constant velocity $U_{\infty }$ around two circular cylinders arranged in tandem with spacing ratio L. The configuration is illustrated in Figure 1, in which both cylinders have the same diameter, D, the upstream cylinder is fixed, and the downstream cylinder is forced to vibrate in the cross-stream direction. The fluid domain $\mathrm{\Omega }$ is defined by the surface $S=S_1\cup S_2\cup S_{\infty }$, where $S_1$ corresponds to the surface of the upstream cylinder, $S_2$ corresponds to the surface of the downstream cylinder, and $S_{\infty }$ represents the far-field fluid boundary.

The surface of the upstream cylinder is described as

$$F_1(x,y)=x^2+y^2-{\left({\displaystyle \frac{D}{2}}\right)}^2=0,$$

(1)

whereas the instantaneous surface of the downstream cylinder is given by

$$F_2(x,y,t)={(x-L)}^2+{[y-y_b\left(t\right)]}^2-{\left({\displaystyle \frac{D}{2}}\right)}^2=0$$

(2)

where $y_b\left(t\right)$ represents the cross-stream displacement imposed on the moving body. The uniform incident flow is defined by ${\mathit{U}}_{\infty }=(U_{\infty },0)$.

In the present problem, the upstream cylinder acts as the generator of the incident wake, whereas the downstream cylinder constitutes the moving body on which the combined effects of forced vibration and surface roughness are investigated.

2.2. Modeling of the Forced Oscillation

The oscillation of the downstream cylinder is prescribed in the cross-stream direction relative to the incident flow by a harmonic law. Since, in the present problem, the vibration is cross-stream, the instantaneous position of a material point on the moving surface, $y_b\left(t\right)$, is described by

$$y_b\left(t\right)=Acos\left(2\pi f_{0}t\right),$$

(3)

where A is the forced oscillation amplitude and $f_0$ is the excitation frequency. The instantaneous velocity associated with this motion, ${\dot{y}}_b\left(t\right)$, is given by

$${\dot{y}}_b\left(t\right)=-2\pi f_{0}Asin\left(2\pi f_{0}t\right).$$

(4)

Thus, the velocity vector of the downstream-cylinder surface can be written according to

$$v_j={\dot{y}}_b\left(t\right).$$

(5)

2.3. Governing Equations and Boundary Conditions

The governing differential equations are written in terms of the filtered equations for two-dimensional incompressible flow, and they are presented in the attached material in the same form. The continuity equation is given by

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial x_i}}=0,$$

(6)

whereas the filtered momentum equation is written as

$${\displaystyle \frac{\partial {\overline{u}}_i}{\partial t}}+{\displaystyle \frac{\partial }{\partial x_j}}\left({\overline{u}}_i{\overline{u}}_j\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+2{\displaystyle \frac{\partial }{\partial x_j}}\left[(\nu +{\nu }_t){\overline{S}}_{ij}\right],$$

(7)

where ${\overline{u}}_i$ and $\overline{p}$ are, respectively, the filtered velocity and pressure fields, $\rho$ is the fluid density, $\nu$ is the molecular kinematic viscosity, ${\nu }_t$ is the subgrid-scale turbulent viscosity, and ${\overline{S}}_{ij}$ is the filtered strain-rate tensor.

The local turbulent viscosity is modeled in the following manner [21]:

$${\nu }_t(\mathit{x},{\mathrm{\Delta }}^{+},t)=0.105C_K^{-3/2}{\mathrm{\Delta }}^{+}\sqrt{{\overline{F}}_2(\mathit{x},{\mathrm{\Delta }}^{+},t)},$$

(8)

where $C_K=1.4$ is the Kolmogorov constant [22], ${\mathrm{\Delta }}^{+}$ represents the local filtering scale, and ${\overline{F}}_2$ is the second-order velocity structure function of the filtered field.

On the solid surfaces, the impermeability and no-slip conditions are imposed in relative form between the fluid and the surface, respectively:

$$(u_i-v_j)·\widehat{n}=0\mathrm{at}{S}_1\mathrm{and}{S}_2,$$

(9)

$$(u_i-v_j)·\widehat{\mathit{\tau }}=0\mathrm{at}{S}_1\mathrm{and}{S}_2,$$

(10)

where $\widehat{n}$ and $\widehat{\mathit{\tau }}$ are, respectively, the normal and tangential unit vectors pointing outwards to $S_1$ and $S_2$, and $v_j$ is the solid-surface velocity. At $S_{\infty }$, it is assumed that the disturbance induced by the bodies vanishes, so that it is represented by

$$\parallel u_i\parallel \to U_{\infty }\mathrm{at}{x}_i\to S_{\infty }$$

(11)

For the upstream cylinder, since the body is stationary, $v_j=0$. For the downstream cylinder, these velocities are determined by the forced oscillation law defined in the next item.

2.4. Assumptions and Non-Dimensionalization

It is assumed that the fluid is Newtonian, with constant properties, and that the flow is two-dimensional, viscous, incompressible, and unsteady. The equations are written in terms of filtered quantities so that the effects of the smaller turbulent scales are incorporated by a subgrid-scale LES model, according to the formulation of the attached material. The effect of surface roughness is not introduced through an explicit modification of the macroscopic geometry of the cylinder but rather through a two-dimensional model coupled to the Lagrangian methodology and to turbulence modeling, which is applied only to the downstream cylinder. All quantities are non-dimensionalized by adopting the cylinder diameter D as the characteristic length and the incident-flow velocity $U_{\infty }$ as the velocity scale. Thus,

$$x^{*}={\displaystyle \frac{x}{D}},y^{*}={\displaystyle \frac{y}{D}},t^{*}={\displaystyle \frac{t{U}_{\infty }}{D}},u_i^{*}={\displaystyle \frac{u_i}{U_{\infty }}},p^{*}={\displaystyle \frac{p}{\rho U_{\infty }^2}}.$$

(12)

In addition, the following quantities are defined as shown below:

$$A^{*}={\displaystyle \frac{A}{D}}$$

(13)

being the dimensionless amplitude,

$${\epsilon }^{*}={\displaystyle \frac{\epsilon }{D}}$$

(14)

being the relative roughness size,

$$f_0^{*}={\displaystyle \frac{f_{0}D}{U_{\infty }}}$$

(15)

being the dimensionless frequency,

$$St={\displaystyle \frac{fD}{U_{\infty }}},$$

(16)

being the Strouhal number that measures the classical vortex shedding frequency [23].

For simplicity of notation, the non-dimensional superscripts are omitted in the subsequent equations. The Reynolds number of the problem is defined as

$$Re={\displaystyle \frac{U_{\infty }D}{\nu }},$$

(17)

whereas the reduced velocity is defined as [24]

$$V_R={\displaystyle \frac{U_{\infty }}{f_{0}D}}.$$

(18)

3. Numerical Solution

3.1. Numerical Algorithm

The in-house code developed by our research group follows a procedural computation paradigm in which the vorticity field is obtained in a step-wise manner. First, the algorithm is presented, which is followed by details of the numerical implementation.

The numerical solution is obtained through a sequential time-marching procedure based on the Lagrangian discrete vortex method. At the beginning of the computation, (i) the physical and numerical input parameters are defined, including the flow conditions, geometric configuration, time step, number of panels, forced-oscillation parameters, and relative roughness height; (ii) the solid boundaries are discretized using the panel method, in which the surfaces of the upstream and downstream cylinders are represented by flat panels and the boundary conditions are imposed at the pivotal points located at the midpoint of each panel; and (iii) constant-strength sources are generated over the panels to satisfy the impermeability condition and enforce mass conservation. This step accounts for the incident flow, the velocity induced by the existing vortex cloud, and the prescribed velocity of the downstream cylinder.

After the impermeability condition has been imposed, (iv) discrete Lamb vortex particles are generated near the surface of each body to satisfy the no-slip condition. This step accounts for the incident flow, the vorticity field represented by the vortex cloud, the prescribed velocity of the downstream cylinder, and the surface roughness effects. The roughness model is applied only to the downstream cylinder and modifies the generation of nascent vortices through the local viscous core associated with the near-wall turbulent viscosity. Once the source strengths and the circulation of the nascent vortices have been determined, (v) the velocity field is evaluated at each discrete vortex particle by considering the combined contribution of the incident flow, the solid boundaries represented by constant-strength sources, and the vortex cloud.

Then, (vi) the pressure distribution is computed at the control points on the cylinder surfaces, allowing the drag and lift coefficients to be obtained from the integration of the pressure field. The vortex cloud is subsequently advanced in time by means of the viscous splitting algorithm, in which (vii) the advective displacement of each vortex particle is calculated using a first-order Euler scheme, and (viii) the diffusive displacement is modeled through the random walk method combined with the turbulence model. After the advection and diffusion steps, (ix) the position of the downstream cylinder is updated according to the prescribed transverse harmonic motion.

Finally, (x) vortex particles that move inside the solid boundaries are reflected back into the fluid domain. Since the downstream cylinder undergoes forced oscillation, (xi) the boundary coordinates and the corresponding coupling matrices are updated at the end of each time step to account for the new body position. The computation then advances to the next time step, in which (xii) a new distribution of Lamb vortex particles and constant-strength sources is generated, and the same numerical sequence is repeated until the prescribed final simulation time is reached.

3.2. Solid Boundaries Contribution

The two surfaces are represented using the panels method [25] by means of constant strength sources placed on each flat panel. The Neumann boundary condition is imposed to ensure that the flow velocity at each pivotal point, located at the midpoint of a panel, matches the body velocity. Given a source i, the velocity ${\mathit{U}}_{s_i}\left({\mathit{x}}_P\right)=(u_{s_i},v_{s_i})$ induced at a point ${\mathit{x}}_P=(x_p,y_p)$ is calculated in the local frame (denoted by the superscript L) of the source in the following manner:

$$u_{s_i}^L\left({\mathit{x}}_P\right)={\displaystyle \frac{{\sigma }_i}{2\pi }}ln\left({\displaystyle \frac{r_1}{r_2}}\right),$$

(19)

$$v_{s_i}^L\left({\mathit{x}}_P\right)={\displaystyle \frac{{\sigma }_i}{2\pi }}({\theta }_2-{\theta }_1),$$

(20)

where $r_1$ and $r_2$ are the distance between the left (1) and right (2) vertices of the flat source to ${\mathit{x}}_P$. ${\theta }_1$ and ${\theta }_2$ are the angles between the horizontal of the local frame of the panel and the line that connects the left (1) and right (2) of the panel to the point ${\mathit{x}}_P$. The result is then returned to the global inertial frame. The self-induction singularity is avoided using a limit process where $v_{self}={\sigma }_{self}/2$. For $NP$ panels, the total velocity induced by the sources at ${\mathit{x}}_P$, $\mathit{ub}\left({\mathit{x}}_P\right)$, is

$$\mathit{ub}\left({\mathit{x}}_P\right)={\displaystyle \sum _{i=1}^{NP}}{\mathit{U}}_{s_i}\left({\mathit{x}}_P\right).$$

(21)

3.3. Vortex Cloud Contribution

The vorticity field, represented by overlapped discrete vortex particles [26], induces velocity at each point of the flow. The contribution to the total velocity at a point ${\mathit{x}}_P$ due to a single vortex point, ${\mathit{u}}_{v_i}\left({\mathit{x}}_P\right)=(u_{v_i},v_{v_i})$ located at ${\mathit{x}}_i=(x_i,y_i)$, is given by

$$u_{v_i}\left({\mathit{x}}_P\right)={\displaystyle \frac{{\mathrm{\Gamma }}_i}{2\pi }}{\displaystyle \frac{y_P-y_i}{\parallel {\mathit{x}}_P-{\mathit{x}}_i{\parallel }^2}}\left[1-exp\left({\displaystyle \frac{\parallel {\mathit{x}}_P-{\mathit{x}}_i{\parallel }^2}{{\sigma }_i^2}}\right)\right],$$

(22)

$$v_{v_i}\left({\mathit{x}}_P\right)=-{\displaystyle \frac{{\mathrm{\Gamma }}_i}{2\pi }}{\displaystyle \frac{x_P-x_i}{\parallel {\mathit{x}}_P-{\mathit{x}}_i{\parallel }^2}}\left[1-exp\left({\displaystyle \frac{\parallel {\mathit{x}}_P-{\mathit{x}}_i{\parallel }^2}{{\sigma }_i^2}}\right)\right],$$

(23)

where ${\mathrm{\Gamma }}_i$ and ${\sigma }_i$ are, respectively, the strength and viscous core of the i-th discrete vortex particle. For $NV$ vortex particles, the total velocity $\mathit{uv}\left({\mathit{x}}_P\right)$ induced by the vortex cloud using the Biot–Savart law [27], is given by

$$\mathit{uv}\left({\mathit{x}}_P\right)={\displaystyle \sum _{i=1}^{NV}}{\mathit{u}}_{v_i}\left({\mathit{x}}_P\right).$$

(24)

3.4. Boundary Conditions

The boundary conditions are imposed by the solution of two linear systems: one to generate sources (impermeability condition) and another to generate vorticity and ensure the no-slip condition.

3.4.1. Impermeability Condition

The impermeability condition consists of solving the following linear system [25]:

$$\left[COUPS\right]\left\{\sigma \right\}=\left\{RHSS\right\},$$

(25)

the coupling coefficients matrix, $\left[COUPS\right]$, consists of the normal velocity induced by each panel at each pivotal point. $\left\{\sigma \right\}$ is the unknown source density of each panel and $\left\{RHSS\right\}$ is the right-hand side vector containing the normal velocity contributions due to the incident flow, the vortex cloud and the oscillation of the downstream cylinder (the upstream cylinder is fixed during the entire simulation). The right-hand sides vector is calculated at each pivotal point, ${\mathit{x}}_{pi{v}_i}$:

$${RHSS}_i=[-\mathit{U}-\mathit{uv}\left({\mathit{x}}_{pi{v}_i}\right)-{\mathit{v}}_{osc}]·{\widehat{\mathit{n}}}_i,$$

(26)

where $\mathit{U}=(1,0)$, ${\mathit{v}}_{osc}$ is the velocity due to the forced vibration (only for the downstream panels) calculating using Equation (4), and ${\widehat{\mathit{n}}}_i$ is the normal unit vector pointing to the interior of the fluid domain at each pivotal point. The mass conservation is then imposed by adding two new lines on $\left[COUPS\right]$: the first fills columns 1 to $N{P}_{up}$ with the length of each panel, $\mathrm{\Delta }{s}_i$. The remaining of the line is filled with zeroes. The second line has the first $N{P}_{up}$ values zeroed and the following $N{P}_{down}$ columns filled with $\mathrm{\Delta }{s}_i$. The solution for $\left\{\sigma \right\}$ is obtained via the least squares method.

3.4.2. No-Slip Condition and Roughness Effects

The verification of the no-slip condition follows Chorin’s implementation [28], where vorticity is generated to ensure that the tangential velocity of the flow at each pivotal point is equal to the body motion. However, the generation is computed together with the roughness effects. The roughness scheme consists of generating $NR$ “rough points” in a semicircle located at a distance $b=2\epsilon /D-\sigma$ from the collocation point of the downstream cylinder (the upstream cylinder is smooth). Then, at each rough point, the local eddy viscosity coefficient, ${\nu }_t$, is calculated using the LES approach discussed in Section 3.6.2. The eddy viscosity coefficient is then used to change the viscous core of the k-th nascent vortex, ${\sigma }_{0c_k}$, in the following manner [20]:

$${\sigma }_{0c_k}=1.41421\sqrt{{\displaystyle \frac{\mathrm{\Delta }t}{Re}}\left(1+{\displaystyle \frac{{\nu }_{t_k}\left(t\right)}{\nu }}\right)}.$$

(27)

Then, a linear system is solved to impose the no-slip condition and the global conservation of circulation:

$$\left[COUPV\right]\left\{\mathrm{\Gamma }\right\}=\left\{RHSV\right\},$$

(28)

where $\left[COUPV\right]$ is the vortex coupling coefficients matrix, and each component $COUP{V}_{ij}$ represents the tangential velocity at the i-th panel induced by the j-th nascent vortex if it had unitary strength. $\left\{\mathrm{\Gamma }\right\}$ is the unknown vortex strength vector and $\left\{RHSV\right\}$ is the right-hand size vector which considers the tangential velocity induced by the body, the vortex cloud, the incident flow, and the body motion in the following manner:

$$RHS{V}_i=[-\mathit{U}-\mathit{uv}\left({\mathit{x}}_{pi{v}_i}\right)-\mathit{ub}\left({\mathit{x}}_{pi{v}_i}\right)-{\mathit{v}}_{osc}]·{\widehat{\mathit{\tau }}}_i,$$

(29)

where ${\widehat{\mathit{\tau }}}_i$ is the tangential unit vector at each pivotal point, and ${\mathit{v}}_{osc}$ is the velocity due to the forced vibration (only for the downstream panels) calculating using Equation (4).

The matrix $\left[COUPV\right]$ needs two additional lines to impose the conservation of global circulation in a similar fashon to $\left[COUPS\right]$, however, with the value $1.0$ instead of the length of each panel in the corresponding position. The linear system in Equation (28) is then solved by the least squares method.

3.5. The Pressure Field

The pressure field is obtained using an integral formulation [29]:

$$H{\overline{Y}}_i-{\int }_{S_1\cup S_2}\overline{Y}\nabla G_i·\mathit{n}dS=\underset{\mathrm{\Omega }}{\int \int }\nabla G_i·(\overline{\mathit{u}}\times \overline{\mathit{\omega }})d\mathrm{\Omega }-{\displaystyle \frac{1}{Re}}{\int }_{S_1\cup S_2}(\nabla G_i\times \overline{\mathit{\omega }})·\mathit{n}dS,$$

(30)

where H is a switch function where $H=1$ if a point is in the fluid domain $\mathrm{\Omega }$ and $H=1/2$ if the point belongs to the contour $S_1\cup S_2$. $\overline{Y}$ is the stagnation pressure, the unknown variable, $G_i$ is the Green function that is the solution to the Poisson equation, $\overline{\mathit{u}}$ is the velocity at a given point of the surface, and $\mathit{n}$ is the unit vector pointing outward from the surface. Equation (30) is solved by forming a linear system maintaining the left and right hand sides:

$$\left[COUPP\right]\left\{\overline{Y}\right\}=\left\{RHSP\right\},$$

(31)

where $\left[COUPP\right]$ is the coupling coefficients matrix and $\left\{RHSP\right\}$ is the right-hand side’s pressure vector. The solution of the system gives ${\overline{Y}}_i$, which is used to calculate the pressure coefficient, $C_{P_i}$, due to body forces [30]:

$$C_{P_i}={\overline{Y}}_i+1$$

(32)

The drag and lift coefficients are then calculated as, respectively [31],

$$C_D={\displaystyle \sum _{j=1}^M}{C}_{P_j}\mathrm{\Delta }{S}_{j}sin\left({\theta }_{pj}\right),$$

(33)

$$C_L=-{\displaystyle \sum _{j=1}^M}{C}_{P_j}\mathrm{\Delta }{S}_{j}cos\left({\theta }_{pj}\right),$$

(34)

where ${\theta }_{pj}$ is the orientation angle of each panel with respect to the x-axis.

3.6. Viscous Splitting Algorithm

The vorticity transport equation is obtained by taking the curl of the Navier–Stokes Equation (7) supported by the mass conservation Equation (6), yielding [32]

$${\displaystyle \frac{D\overline{\omega }}{Dt}}=\left({\displaystyle \frac{1}{Re}}+{\displaystyle \frac{{\nu }_t}{U_{\infty }D}}\right){\nabla }^2\overline{\omega }.$$

(35)

The vorticity transport equation is solved using the viscous splitting algorithm proposed by Chorin [28]. This algorithm guarantees that for small time-steps and given that the vortex particles overlap, the vorticity transport equation can be separated into an advection and a diffusion problem and then the motions can be summed, converging to the original equation.

3.6.1. The Advective Problem

The advection is represented by the following equation:

$${\displaystyle \frac{D\overline{\omega }}{Dt}}=0.$$

(36)

This means that the vorticity is conserved over the trajectory of a particle, and therefore, the advection can be solved by a numerical method. In this paper, the solution is obtained using the first-order Euler scheme [33]:

$${\mathit{x}}_i^{t+\mathrm{\Delta }t}={\mathit{x}}_i^t+{\mathit{u}}_i\left({\mathit{x}}_{\mathbf{i}}^{\mathbf{t}}\right)\mathrm{\Delta }t,$$

(37)

where ${\mathit{x}}_i^t$ is the position vector of the i-th vortex particle at a given instant (the value i ranges from 1 up to the total number of discrete lamb vortices) t, $\mathrm{\Delta }t$ is the time step and ${\mathit{u}}_i$ is the velocity calculated at the position of each particle:

$${\mathit{u}}_i\left({\mathit{x}}_i^t\right)=\mathit{U}+\mathit{uv}\left({\mathit{x}}_i^t\right)+\mathit{ub}\left({\mathit{x}}_i^t\right).$$

(38)

3.6.2. Diffusion and Turbulence Simulation

The two-dimensional turbulence modeling and diffusion of vorticity are calculated in sequence in the algorithm. Turbulence physically affects the advection, creating more gradients that increase the molecular diffusion. Using the Boussinesq hypothesis, this effect is modeled as an additional term in the diffusive part of the vorticity transport equation. The additional derivatives that would appear when applying the curl operator to the Navier–Stokes equations are neglected in the present model, since its contribution is less representative.

Moraes and Alcântara Pereira [2] improved a two-dimensional LES model based on the Second-Order Velocity Structure Function (SOVSF). This model was accelerated using an acceleration method based on a stencil set around the fluid domain with vortical activity [34]. The turbulence model consists of using an annular region around each discrete vortex and computing the velocity fluctuation due to the $NA$ vortex particles inside the annular region. Firstly, the SOVSF is obtained in the following manner [35]:

$${\overline{F}}_{2_i}={\displaystyle \frac{1}{NA}}{\displaystyle \sum _{j=1}^{NA}}\left[\parallel {\overline{\mathit{u}}}_i-{\overline{\mathit{u}}}_j{\parallel }^2{\left({\displaystyle \frac{{\sigma }_i}{\parallel {\mathit{x}}_j-{\mathit{x}}_i\parallel }}\right)}^{{\displaystyle \frac{2}{3}}}\right],$$

(39)

where ${\overline{F}}_{2_i}$ is the SOVSF for the i-th vortex particle. The term $({\sigma }_i/\parallel {\mathit{x}}_j-{\mathit{x}}_i{\parallel )}^{2/3}$ is a correction made to compensate the non-uniform particle distribution inside each annular region. The eddy viscosity of this particle is then calculated in the following manner:

$${\nu }_{t_i}=0.105C_K^{-3/2}{\sigma }_i\sqrt{{\overline{F}}_{2_i}},$$

(40)

for the roughness effects eddy viscosity coefficient, the SOVSF is calculated using the $NR$ rough points to average the fluctuations [20]:

$${\overline{F}}_{2_i}^r={\displaystyle \frac{1}{NR}}{\displaystyle \sum _{j=1}^{NR}}[\parallel {\mathit{u}}_i-{\mathit{u}}_j{\parallel }^2(1+\epsilon )].$$

(41)

The diffusion is calculated using the random walk method [28] after the advection step shown in Equation (37):

$${\mathit{x}}_i^{t+\mathrm{\Delta }t}={\mathit{x}}_i^t+{\mathit{u}}_i\left({\mathit{x}}_{\mathit{i}}^{\mathit{t}}\right)\mathrm{\Delta }t+\mathrm{\Delta }{\mathit{x}}_{d_i}^t,$$

(42)

and $\mathrm{\Delta }{\mathit{x}}_{d_i}^t$ is the diffusion step:

$$\mathrm{\Delta }{\mathit{x}}_{d_i}^t=\sqrt{{\displaystyle \frac{4\mathrm{\Delta }t}{Re}}(1+{\displaystyle \frac{{\nu }_t}{U_{\infty }D}})ln{\displaystyle \frac{1}{P}}}\left[cos\left(2\pi Q\right)\widehat{\mathit{i}}+sin\left(2\pi Q\right)\widehat{\mathit{j}}\right],$$

(43)

being P and Q pseudo-random numbers between 0 and 1.

4. Results and Discussion

4.1. Convergence of Important Parameters

The convergence of the advection problem including the chosen time stepping is shown in Figure 2. The strategy is to demonstrate that a pair of Lamb discrete vortices of equal strength $\mathrm{\Gamma }$, distant AB apart, start to experience an advection velocity and subsequent advective motion in accordance with Equations (37) and (38), respectively.

As can be seen in Figure 2, the vortex particle A is placed at (0.0;−1.0) and induces velocity at vortex particle B; the latter initially located at (0.0;1.0) also induces velocity at vortex particle A. As the time runs, the vortex particles will each experience new advection velocities because of the other acting in the direction normal to the line AB joining them. The advective motion of every vortex particle develops in the form of a circular path about the midpoint of AB (point M). The vortex particles move in clockwise direction over some time steps $\mathrm{\Delta }t$.

Table 1. Numerical estimates of advective motions for the vortex particle A by using an explicit Euler scheme.

Distance AB	$\Gamma$	$\mathit{\sigma }$	$\Delta \mathit{t}$	Steps	Error (%)
2	1.0	$0.001D$	0.05	795	0.4406
2	1.0	$0.001D$	0.1	400	0.8780
2	1.0	$0.001D$	0.2	203	1.7478

shows the column Steps, in which a number of finite steps, necessary for the vortex particle A to approximately attain the position (0.0,1.0), is reproduced. Additionally, the last column shows the estimated relative error between the true circular drift path of constant distance given by $AB=2.0$. That error is computed at the instant that the vortex particle A approximately reaches the position (0.0;1.0). The vortex particle B behavior is analogous when it simultaneously and approximately attains the position (0.0;−1.0) identified in Figure 2. It can be concluded that the time step $\mathrm{\Delta }t$ = 0.05, used in all numerical simulations of this paper, is appropriated to be used with the explicit Euler scheme.

The convergence of the random walk method, employed to model the diffusion process according to Equation (44), is also evaluated by considering the radial diffusion of a vortical structure with unit circulation, $\left\{\mathrm{\Gamma }\right\}$ = 1.0, which was initially located at the origin (0.0;0.0), as shown in Figure 3. This radial diffusion problem admits an analytical solution for the spatial and temporal evolution of the vorticity distribution, as reported in Lewis [36]:

$$\omega (r,t)={\displaystyle \frac{\mathrm{\Gamma }}{4\pi vt}}{e}^{-\left({\displaystyle \frac{r^2}{4vt}}\right)}.$$

(44)

The analytical solution is presented in Figure 4 for the case where $t=1.0$, ${\mathrm{\Gamma }}_v=1.0$, and $Re={\mathrm{\Gamma }}_v/\nu =1.0$. For comparison purposes, a numerical approximation is also obtained by representing the vortical structure of circulation ${\mathrm{\Gamma }}_v$ with $Z=5000$ vortex particles, each carrying a circulation equal to ${\mathrm{\Gamma }}_v/Z$. At the initial instant, $t=0$, all vortex particles are placed at the origin $(0.0,0.0)$. After twenty time steps of $\mathrm{\Delta }t=0.05$, the particles diffuse according to Equation (44) without the inclusion of turbulence modeling, as illustrated in Figure 3. The corresponding vorticity distribution is then numerically computed according to Equation (45):

$$\omega \left(r\right)={\displaystyle \frac{z_j}{Z\pi \left(r_{j+1}^2-r_j^2\right)}}$$

(45)

In Figure 3, the numerical vorticity field was evaluated using twenty concentric annular bins of uniform thickness $\mathrm{\Delta }r=0.4$, which were centered at the origin $(0.0,0.0)$. These annular regions were used to group the $z_j$ vortex particles according to their radial position, covering the interval from $0.0\le r_j\le 7.6$ to $0.4\le r_{j+1}\le 8.0$. The comparison with the analytical solution shows a very good agreement, indicating that the random walk procedure accurately reproduces the radial diffusion process. In Equation (46), the analytical vorticity distribution is expressed in terms of the r.m.s. radius:

$$\sqrt{{\displaystyle \frac{1}{2}}\left(r_j^2+r_{j+1}^2\right)}$$

(46)

for every strip j of a annular bin. Thus, the time step $\mathrm{\Delta }t=0.05$ is also chosen for the random diffusion of all test cases in Section 3.

Preliminary verifications were carried out to define the number of panels employed in the discretization of each body considered in the investigated problem. This analysis was performed by evaluating the ability of the panel method to reproduce the analytical potential-flow solution around a two-dimensional circular cylinder. In this validation procedure, a cylinder of radius $r_0$ is represented by $M_B$ flat panels with uniform source distribution and subjected to a uniform incident flow. This classical problem in fluid mechanics admits an analytical solution obtained by superposing the incident flow with a dipole located at the center of the cylinder. Using the complex-potential formulation, the velocity components u and v at an arbitrary point $z=x+iy$ in the complex plane can be written as follows:

$$W\left(z\right)=u-iv=1-{\displaystyle \frac{r_0^2}{z^2}}.$$

(47)

The numerical solution obtained from the panel method was then compared with the analytical solution for different discretization levels. This comparison was used to verify the fulfillment of the impermeability condition, associated with the Neumann boundary condition, while maintaining an adequate computational cost in terms of CPU time. Based on the results obtained, the use of $M_B=200$ panels per body was adopted, since from this refinement level onward, the numerical solution did not show a significant dependence on any further increase in the number of panels. Therefore, this discretization was considered sufficient to accurately represent the geometry of the bodies and to ensure the numerical accuracy required for the problem addressed in this paper. Figure 5 presents the comparison between the numerical solutions obtained with different numbers of panels and the analytical potential-flow solution, evidencing the convergence of the panel method as the discretization is refined.

4.2. The Surface Roughness Effects on WIV

Moraes and Alcântara Pereira [2] validated the present vortex code for a configuration of two immovable circular cylinders at a Reynolds number of $Re=65,000$ and streamwise center-to-center spacing of $L/D=4.5D$.

Table 2. Comparison between the present numerical results and the experimental data reported by Alam et al. [37] for two circular cylinders in tandem arrangement.

Case	Upstream Cylinder			Downstream Cylinder
	$\mathit{\epsilon }/\mathit{D}$	${\mathit{C}}_{\mathit{d}}$	$\mathit{St}$	$\mathit{\epsilon }/\mathit{D}$	${\mathit{C}}_{\mathit{d}}$	$\mathit{St}$
Experimental [37]	0.000	1.2612	0.1867	0.000	0.2766	–
	0.000	1.0790	0.2000	0.000	0.4897	0.2000
	0.000	1.1035	0.2000	0.001	0.4409	0.2000
	0.001	0.9245	0.1875	0.000	0.6450	0.1875
Present Simulation	0.001	0.9018	0.1875	0.001	0.5060	0.1875
	0.000	1.0321	0.2000	0.007	0.3798	0.2000
	0.007	0.9904	0.2000	0.000	0.4860	0.2000
	0.007	1.0246	0.1875	0.007	0.4194	0.1875

reproduce some results of time-averaged values given by Moraes and Alcântara Pereira [2], which are compared with experimental data by Alam et al. [37] at about 92% confidence, when possible.

The common simulation parameters are shown in

Table 3. Shared input parameters for the simulation.

Parameter	Value
Viscous core ($\sigma$)	0.0015
Number of panels in each cylinder (M)	200
Time step ($\mathrm{\Delta }t$)	0.05
Number of time steps	1000
Incident flow velocity ($U_{\infty }$)	1.0
Diameter of each cylinder (D)	1.0
Center-center gap between the cylinders ($L/D$)	5.0
Reynolds number (Re)	65,000
Downstream cylinder oscillation frequency ($f_0$)	0.2
Reduced velocity ($V_R$)	5.0
Vibration amplitude ($A/D$)	0.15

. The summary of the results is presented in

Table 4. Summary of the test cases.

Case	$\mathit{\epsilon }/\mathit{D}$	St	${\mathit{C}}_{\mathit{L}}$	Vibration Model
1	0.0000	0.1993	−0.0033	$WIV$
2	0.0045	0.1663 and 0.2329	−0.4391	Opposite $WIV$
3	0.0100	0.1996	−0.4120	$VIV$ (Prevailing) + $WIV$

.

4.3. Case 1: Downstream Circular Cylinder Oscillating Transversely Without Roughness Effect—$\epsilon /D$ = 0.0000

For the oscillating downstream cylinder without a rough surface $\epsilon /D$ = 0.0000, subjected to harmonic transverse vibration with an amplitude ratio $A/D=0.15$ and dimensionless frequency $f_0=0.2$, the time series of the lift coefficient and the distribution of the pressure coefficient along the body surface were analyzed. Figure 6 presents the time evolution of the lift coefficient with the displacement and velocity curves. The results indicate the occurrence of $VIV$, evidenced by the synchronization between the lift coefficient and the cylinder displacement, or by the proximity between the vortex shedding frequency and the imposed oscillation frequency. Additionally, the occurrence of $WIV$ may be associated with the wake-galloping phenomenon, which is identified when the lift coefficient is in phase with the vibration velocity, characterizing a net positive transfer of energy from the fluid to the body.

For the hydraulically smooth surface condition ($\epsilon /D=0.0000$), Figure 6 shows that the lift coefficient remains in phase with the displacement curve between the dimensionless times 0 and 12.5, characterizing the occurrence of lock-in within this interval. Between the dimensionless times 12.5 and approximately 24, neither lock-in nor wake galloping is observed. From that instant onward, the lift coefficient remains in phase with the velocity curve until the end of the simulation, indicating the occurrence of wake galloping between the dimensionless times 24 and 50. Therefore, since this behavior is not maintained throughout the entire time series, the analyzed case cannot be characterized as either pure lock-in or pure wake galloping.

The mean lift coefficient was $C_L=-0.0033$, whereas the Strouhal number, obtained by applying Fast Fourier Transform (FFT) to the lift coefficient time series, was $St=0.1993$. Both parameters were determined over the dimensionless time interval from 30 to 50. These results indicate hydrodynamic behavior associated with a stable regime, which is characterized by smooth and well-defined oscillations over time.

The analysis of the Strouhal number can be complemented by Figure 7, which presents the FFT of the lift coefficient time series in greater detail. In this figure, the presence of a single dominant frequency is observed, evidencing the predominance of a well-defined oscillatory mode in the flow.

As indicated in Figure 7, a single significant peak is observed in the frequency spectrum obtained from the lift coefficient time series. This behavior is consistent with the predominance of the wake-galloping phenomenon after the interaction between the wake of the upstream cylinder and the oscillating downstream cylinder, which persists until the end of the numerical simulation. Thus, no significant alternation between different phenomena is observed throughout the time series.

Figure 8 presents the vortical structure generated by the upstream cylinder interacting with the downstream cylinder to the dimensionless time 15. This instant is associated with point P in the lift coefficient time series shown in Figure 8, through which the downstream cylinder reaches its maximum positive displacement. Therefore, this configuration may be interpreted as an initial stage of the interaction between the wake of the upstream cylinder and the oscillating solid boundary of the downstream cylinder.

It is observed in Figure 8 that a vortical structure originating from the upstream cylinder crosses the lower region of the downstream cylinder, inducing velocities in the nascent vortical structures generated by the latter. The presence of a vorticity sheet connecting this structure to another large vortical structure originating from the upstream cylinder is also observed, and this second structure is about to interact with the solid boundary of the downstream cylinder.

Point $Q_1$, corresponding the dimensionless time 38.6 in the lift coefficient time series presented in Figure 6, is associated with a high oscillation amplitude resulting from the interaction between the vortical structure shed from the upstream cylinder and the oscillating downstream cylinder. Figure 9 presents the visualization of this interaction at the instant corresponding to point $Q_1$.

From Figure 9, it is verified that at point $Q_1$, a vortical structure crosses the upper region of the cylinder, inducing velocities in the flow and forming a zone of intense circulation. As a consequence, a high-amplitude peak is observed in the lift coefficient.

Additionally, the instantaneous pressure coefficient distribution as a function of angle $\theta$ was analyzed at point $Q_1$ in order to investigate this portion of the lift coefficient time series in greater detail. Figure 10 presents the instantaneous pressure coefficient curve corresponding to this point.

As shown in Figure 10, intense negative fluctuations of the pressure coefficient are observed in the upper region of the cylinder, corroborating the interpretation that at this instant, a vortical structure originating from the upstream cylinder crosses this region, inducing velocities and promoting strong local circulation. This interaction substantially modifies the instantaneous pressure field, producing a pronounced suction region and leading to a global minimum of the pressure coefficient on this portion of the surface. As a consequence of this asymmetric pressure distribution, a positive lift force of high magnitude is generated.

The occurrence of these localized pressure fluctuations is therefore a consequence of the resolved unsteady vortical interaction and should not be associated with numerical noise or residual numerical oscillations. These fluctuations are directly related to the physical response of the flow in regions of intense vorticity and the strong local circulation generated by the interaction between the upstream wake and the oscillating downstream cylinder. In the present formulation, this behavior is captured by the Lagrangian vortex representation combined with the LES-based turbulence model, which accounts for the influence of unresolved turbulent scales on the transport and diffusion of vorticity. Thus, the localized variations of the pressure coefficient reflect the unsteady vortical dynamics resolved by the numerical method rather than a spurious numerical effect.

In the cases with nonzero roughness, the same LES-based framework also provides the local turbulent viscosity required by the roughness model near the wall. This coupling allows the roughness effects to modify the generation of nascent vortical structures and, consequently, the pressure redistribution around the downstream cylinder. Therefore, the pressure fluctuations observed in the present analysis are consistent with the physical mechanisms represented by the numerical formulation, especially in regions where strong vortical structures interact with the moving solid boundary.

4.4. Case 2: Downstream Circular Cylinder Oscillating Transversely with Roughness Effect—$\epsilon /D=0.0045$

The time series of the lift coefficient and the pressure coefficient distribution along the body surface were analyzed. Figure 11 presents the time evolution of the lift coefficient together with the displacement and velocity curves.

Based on Figure 11, it is observed that the action of the upstream-cylinder wake on the downstream cylinder interrupts the occurrence of lock-in. In addition, the manifestation of wake galloping, a phenomenon characteristic of $WIV$, is not observed. Its main feature is the synchronization between the lift coefficient and the oscillation velocity with coincidence between the peaks and valleys of CL and those of the velocity curve. Under this condition, there is a net positive transfer of energy from the fluid to the body.

However, this behavior is not observed for the roughness height analyzed. After approximately the dimensionless time $t=14$, when the coincidence between the valleys of the lift coefficient and those of the displacement curve associated with lock-in ceases to occur, a behavior opposite to wake galloping is identified: the peaks of the lift coefficient coincide with the valleys of the velocity curve, while the valleys of $C_L$ coincide with the peaks of that same curve. Therefore, by definition, this regime does not characterize wake galloping but rather a behavior opposite to the mechanism of net positive energy transfer from the fluid to the body.

For the analyzed case, the mean lift coefficient was $C_L=-0.4391$, whereas the Strouhal number, obtained by applying the FFT to the lift coefficient time series, was $St=0.1663$. In addition, the presence of a second relevant frequency was observed in the spectrum with a value of 0.2329. The joint analysis of these frequencies indicates the absence of synchronization in the dimensionless time interval from 30 to 50. This result is consistent with the flow behavior during this period, in which, under the influence of the wake of the upstream cylinder on the downstream cylinder, neither the lock-in phenomenon nor wake galloping is observed. The results obtained from the application of the FFT to the lift coefficient curve are presented in Figure 12.

As shown in Figure 12, two relevant peaks are observed in the frequency spectrum obtained from the lift coefficient time series. However, neither of these frequencies approaches the imposed oscillation frequency, evidencing the absence of synchronization. This result reinforces the interpretation that for the roughness height $\epsilon /D=0.0045$, synchronization associated with either the lock-in phenomenon or wake galloping does not occur. Thus, this case exhibits behavior distinct from the other cases analyzed, even indicating an apparent response opposite to that observed in wake galloping.

The response observed for $\epsilon /D=0.0045$ may be interpreted as a transitional behavior between the hydraulically smooth configuration, in which the wake-induced response becomes clearly established after the interaction with the upstream wake, and the rougher case, $\epsilon /D=0.01$, in which the lock-in mechanism becomes more persistent over a significant portion of the time series. Therefore, this roughness level does not lead to a purely $WIV$ dominated regime nor does it promote a predominantly $VIV$ like response. Instead, it disturbs the phase relation required for wake galloping while not establishing a stable synchronization between the lift coefficient and the imposed displacement of the downstream cylinder.

In this sense, the designation “Opposite WIV” is used to describe a wake-induced response in which the phase relation between the lift coefficient and the oscillation velocity is contrary to that expected in a wake-galloping regime. In classical wake-galloping behavior, the lift force tends to remain in phase with the transverse velocity of the body, favoring a positive aerodynamic energy transfer from the fluid to the structure. For $\epsilon /D=0.0045$, however, the opposite trend is observed: the peaks of the lift coefficient tend to coincide with the valleys of the velocity curve, whereas the valleys of $C_L$ tend to coincide with the velocity peaks. This anti-phase behavior indicates that the incoming wake still governs the aerodynamic loading, but the resulting force does not reinforce the prescribed motion in the manner typically associated with $WIV$.

From a physical standpoint, this behavior suggests that the intermediate roughness modifies the near-wall vorticity generation and the local pressure redistribution around the downstream cylinder, altering the way in which the incident vortical structures interact with the moving surface. The upstream wake continues to induce strong unsteady pressure fluctuations; however, the roughness level $\epsilon /D=0.0045$ promotes an irregular balance between suction and pressure-recovery regions on the upper and lower portions of the cylinder. Consequently, the aerodynamic force loses a stable phase relationship with both the displacement and the velocity of the body. The presence of two relevant spectral peaks further supports this interpretation, indicating that the flow response is governed by competing unsteady mechanisms rather than by a single dominant synchronized regime.

Figure 13 presents the vortical structure generated by the upstream cylinder interacting with the oscillating downstream cylinder with surface roughness $\epsilon /D=0.0045$ to the dimensionless time $t=15$. This instant is associated with point P, of maximum positive displacement, in the lift coefficient time series. Therefore, this configuration may be interpreted as an initial stage of the interaction between the wake of the upstream cylinder and the oscillating solid boundary of the downstream cylinder.

It is observed in Figure 13 that a vortical structure originating from the upstream cylinder crosses the lower region of the downstream cylinder, inducing velocities in the nascent vortical structures generated in this region. Simultaneously, another vortical structure from the upstream cylinder is observed that is about to cross the upper region of the downstream cylinder. This arrangement promotes a balance of velocity induction between the lower and upper regions of the body, favoring the equalization of the pressure field and resulting in a lift coefficient close to zero. This behavior is consistent with that observed at point P in the lift coefficient time series presented in Figure 11. In addition, the interaction between the vortical structures of the upstream and downstream cylinders, connected by vorticity sheets, is evident.

Point $Q_2$, corresponding to the dimensionless time $t=52.15$ in the lift coefficient time series shown in Figure 11, is characterized by a high-amplitude peak associated with the interaction between a vortical structure shed from the upstream cylinder and the oscillating downstream cylinder. Figure 14 presents the visualization of this interaction at the instant corresponding to point $Q_2$.

From Figure 14, it is observed that at point $Q_2$, a vortical structure originating from the upstream cylinder begins to cross the upper region of the downstream cylinder, inducing velocities in the nascent vortices of this region. Another vortical structure from the upstream cylinder that has already crossed the lower region of the downstream cylinder is also observed, although it is farther from the body wall. These velocity inductions generate a zone of intense circulation and, consequently, a reduction in pressure in this region, resulting in a peak in the lift coefficient.

In order to investigate the behavior observed at point $Q_2$ in greater detail, the instantaneous pressure coefficient distribution as a function of angle $\theta$ was analyzed at this instant. This analysis allows the pressure distribution along the surface of the downstream cylinder to be evaluated and the mechanisms associated with the identified $C_L$ peak to be understood more rigorously. Figure 15 presents the instantaneous pressure coefficient curve corresponding to this point, highlighting the regions of suction and pressure recovery around the cylinder, as well as the effects of the interaction between the vortical structures from the upstream cylinder and the nascent structures on the downstream cylinder.

It is observed that at point $Q_2$, corresponding to a peak in the lift coefficient time series, a pronounced pressure drop occurs on the upper surface of the downstream cylinder. In addition, intense pressure fluctuations are observed as a result of the combined action of the wake generated by the upstream cylinder, the prescribed transverse motion of the downstream cylinder, and the surface roughness effects. This coupling favors pressure recovery in the lower region of the body; however, this recovery is not smooth or uniform along the surface. Instead, it occurs in a strongly unsteady manner with significant local variations in the pressure coefficient. Thus, although there is a tendency for pressure to increase in this region, the pressure field remains highly irregular due to the interaction between the incident vortical structures and the rough oscillating surface.

The occurrence of these intensified pressure fluctuations should not be interpreted as numerical noise or residual numerical oscillations. Rather, they are associated with the physical response of the flow to regions of intense vorticity and strong local circulation generated by the wake–body interaction. For the roughness height $\epsilon /D=0.0045$, this behavior becomes more pronounced because the roughness modifies the near-wall vorticity generation and changes the way in which the incoming wake interacts with the downstream-cylinder surface. Within the set of roughness heights investigated in this paper, this case represents a transitional response between the smooth configuration, in which the wake-induced vibration mechanism is more clearly established, and the rougher case, $\epsilon /D=0.01$, in which the lock-in mechanism becomes more persistent. Therefore, the stronger pressure fluctuations observed for $\epsilon /D=0.0045$ are consistent with the transitional character of this response and with the previously identified behavior opposite to the classical $WIV$/wake-galloping regime.

In the present formulation, these localized variations of the pressure coefficient are captured by the Lagrangian vortex representation combined with the LES-based turbulence model and the roughness model applied to the downstream cylinder. The LES framework accounts for the influence of unresolved turbulent scales on the transport and diffusion of vorticity, while the roughness model uses the local turbulent viscosity near the wall to modify the generation of nascent vortical structures. Consequently, the intensified fluctuations observed in the pressure coefficient are a physical consequence of the modeled interaction among the wake interference, forced transverse motion, and surface roughness. This interpretation supports the conclusion that for $\epsilon /D=0.0045$, the aerodynamic loading is governed by competing unsteady mechanisms rather than by a single synchronized $VIV$ or $WIV$-dominated regime.

4.5. Case 3: Downstream Circular Cylinder Oscillating Transversely with Roughness Effect—$\epsilon /D$ = 0.01

Figure 16 presents the time evolution of the lift coefficient together with the displacement and velocity curves. Based on these results, the localized occurrence of lock-in is identified, which is a phenomenon characteristic of $VIV$ that is associated either with the synchronization between the lift coefficient and the displacement of the cylinder or with the proximity between the vortex-shedding frequency and the imposed oscillation frequency.

Based on Figure 16, it is observed that at the beginning of the time series, the lock-in phenomenon occurs up to approximately the dimensionless time $t=10$. Within this interval, the wake of the upstream cylinder does not yet exert a significant influence on the fluid-dynamic loads acting on the downstream cylinder. From $t=10$ onward, the localized manifestation of wake galloping, characteristic of $WIV$, is identified in the approximate interval between $t=10$ and $t=15$. In this range, the peaks of the lift coefficient coincide with the peaks of the velocity curve, characterizing a net positive transfer of energy from the fluid to the body. Thus, for the roughness $\epsilon /D=0.01$, wake galloping occurs in a localized and brief manner, and it is associated with the onset of the influence of the wake of the upstream cylinder.

After $t=15$, the lock-in phenomenon reappears over a longer interval, extending up to approximately $t=47$, even with the passage of vortical structures originating from the upstream cylinder through the region of the oscillating downstream cylinder. Thus, unlike the other cases with surface roughness, this case presents a predominant regime over a significant portion of the lift coefficient time series. Still, it cannot be classified as a pure $VIV$ case, since there was a previous manifestation of $WIV$, even if over a short interval. In addition, after $t=47$, a desynchronization of $C_L$ with respect to the displacement curve is observed, and later, near $t=60$, synchronization is observed again. Therefore, although lock-in predominates during a significant part of the simulation, the presence of intermittencies and episodes of desynchronization prevents the regime from being characterized as strictly $VIV$.

For this case, the mean lift coefficient was $C_L$ = −0.4120, whereas the Strouhal number, obtained by applying Fast Fourier Transform to the lift coefficient time series, was St = 0.1996. Both parameters were calculated over the dimensionless time interval from $t=30$ to $t=50$. The result of the FFT applied to the lift coefficient curve is presented in Figure 17.

As indicated in Figure 17, only one relevant peak is observed in the frequency spectrum obtained from the lift coefficient time series. This dominant frequency is consistent with the predominance of a single regime throughout the CL time series, especially in the interval considered for the application of the FFT, between the dimensionless time $t=30$ and $t=50$.

Figure 18 presents the vortical structure generated by the upstream cylinder interacting with the oscillating downstream cylinder, for the surface roughness $\epsilon /D=0.01$, to the dimensionless time $t=15$. This instant is associated with point P, of maximum positive displacement, in the lift coefficient time series shown in Figure 16. Thus, this configuration may be interpreted as the initial stage of the interaction between the wake generated upstream and the oscillating solid boundary of the downstream cylinder.

It is observed in Figure 18 that a vortical structure originating from the upstream cylinder crosses the upper region of the downstream cylinder, while another structure, also originating from the upstream cylinder, is about to cross its lower region. This arrangement promotes a balance of velocity induction between the upper and lower portions of the body, favoring the equalization of the pressure field and resulting in a slightly negative lift coefficient, which is mainly influenced by the vortical structure approaching the lower region. This behavior is consistent with that observed at point P in the lift coefficient time series presented in Figure 16.

In addition, it is verified that the binary wake downstream of the oscillating cylinder tends to form and move downward. This occurs because at the instant when the vortical structure originating from the upstream cylinder crosses the vicinity of the downstream cylinder, the latter is near its maximum positive displacement, causing most of the incident wake to pass through the lower region of the body. This effect is consistent with the negative mean displacement of the lift coefficient observed in Figure 16. The interaction between the vortical structures of the upstream and downstream cylinders, connected by vorticity sheets, is also evident.

Point $Q_3$, corresponding the dimensionless time $t=22.65$ in the lift coefficient time series shown in Figure 19, is characterized by a high-amplitude valley ($C_L$ < −2), which is associated with the interaction between a vortical structure shed from the upstream cylinder and the oscillating downstream cylinder. Figure 19 presents the visualization of this interaction at the instant corresponding to point $Q_3$.

From Figure 19, it is observed that at point $Q_3$, a vortical structure originating from the upstream cylinder is about to cross the lower region of the downstream cylinder, inducing high velocities in the nascent vortices of this region. This process generates a zone of intense circulation and, consequently, a local pressure drop. As a result, the occurrence of a high-amplitude valley in the lift coefficient is observed.

Additionally, the instantaneous pressure coefficient distribution as a function of angle $\theta$ was analyzed at point $Q_3$ in order to investigate this portion of the lift coefficient time series in greater depth. Figure 20 presents the instantaneous pressure coefficient curve corresponding to this point.

As shown in Figure 20, intense negative fluctuations of the pressure coefficient are observed in the lower region of the cylinder, corroborating the hypothesis that at this instant, a vortical structure originating from the upstream cylinder is about to cross this region, inducing velocities and promoting strong local circulation. This dynamic significantly modifies the pressure field, making it more negative and unstable, which results in a negative lift force of high magnitude.

The use of parallel computing (OpenMP) to compute the vortex–vortex interactions (the Biot–Savart law) has permitted an increase in the number of particles constructing the viscous wakes and, consequently, a reduction of the final CPU time. The computer used to run the simulations has the following specifications: CPU Intel Core i9-13900KF 3.00 GHz with 32 threads and L2 cache of 32 MB. Every test case needed approximately 150 h of wall clock time to complete.

5. Conclusions

A purely Lagrangian $DVM$, coupled with a surface roughness model, was developed to simulate the two-dimensional, incompressible and unsteady flow around two identical circular cylinders arranged in tandem and immersed in an incident flow at a high Reynolds number. The investigated problem corresponds to a wake-induced vibration configuration in which the upstream cylinder remains fixed, while the downstream cylinder is forced to oscillate in the transverse direction for $L/D=5$, $Re=65,000$, $A/D=0.15$ and $V_R=5$. In this context, the main contribution of this paper is to demonstrate, in a systematic manner, how the surface roughness of the downstream cylinder alters the coupling between the imposed vibration and the incident wake generated by the upstream cylinder, thus redefining the dominant fluid–structure interaction regime.

The results show that in all the cases analyzed, the initial response of the system is marked by the occurrence of lock-in before the interference from the upstream wake becomes fully dominant. However, the subsequent evolution of the response is strongly dependent on the roughness height. For the smooth surface, $\epsilon /D=0.000$, the flow evolves from an initial lock-in regime to a regime dominated by wake galloping with a mean lift coefficient close to zero, $C_L$ = −0.0033, and a spectrum characterized by a single dominant frequency, $St=0.1993$, indicating a globally more stable and well-defined response. This result reproduces the classical $WIV$ behavior of the downstream cylinder and establishes the reference condition for interpreting the effects of roughness.

For the intermediate roughness, $\epsilon /D=0.0045$, the most singular behavior was observed among the investigated cases. Although lock-in still manifests itself during the initial stage, the action of the upstream-cylinder wake interrupts this regime without inducing a typical wake-galloping state. Instead, a non-canonical response is established that is incompatible with both synchronized vortex-induced vibration and a classical wake-galloping-type $WIV$ mechanism. This interpretation is supported by the presence of two relevant frequencies in the spectrum, $St=0.1663$ and $St=0.2329$, neither of which is close to the imposed oscillation frequency, as well as by the significantly negative value of the mean lift coefficient, $C_L$ = −0.4391. This result shows, quite importantly, that a moderate roughness does not merely modify the response quantitatively but may also disrupt the typical synchronization of the system and drive it into a distinct fluid-dynamic regime governed by more complex interactions among the incident wake, the nascent vortices, and the instantaneous pressure field around the cylinder.

For the highest roughness analyzed, $\epsilon /D=0.01$, a partial reorganization of the dynamic response was observed. In this case, an alternation between lock-in and wake galloping occurs with a brief manifestation of $WIV$ at the onset of wake influence and a subsequent predominance of lock-in over a longer interval, although it was still with intermittencies and localized episodes of desynchronization. The spectrum again exhibits a single dominant frequency, $St=0.1996$, while the mean lift coefficient remains strongly negative, $C_L$ = −0.4120. These results indicate that a further increase in roughness does not restore the behavior observed for the smooth cylinder; rather, it favors a more coherent regime than that obtained for $\epsilon /D=0.0045$, although it is still not purely associated with $VIV$. In physical terms, this suggests that the higher roughness reorganizes the interaction between the incident wake and the vorticity layer of the oscillating cylinder, promoting a more structured response over a large portion of the simulation.

Taken together, the three cases show that surface roughness does not act merely as a parameter for local boundary-layer modification but rather as a mechanism capable of redefining the dominant fluid–structure interaction regime in aligned cylinders. The smooth condition favors the transition to wake galloping after the arrival of the upstream wake; the intermediate roughness suppresses the classical synchronization and leads to an anomalous response; and the higher roughness re-establishes a predominance of lock-in, although intermittently. Accordingly, the main advance of this paper lies in showing that small variations in relative roughness can shift the system among different synchronization states, including transitions between typical $WIV$ responses, desynchronized states, and mixed regimes.

In addition, the results show that extreme lift events are directly associated with the passage of vortical structures shed by the upstream cylinder close to the surface of the downstream cylinder, leading to localized suction peaks and marked pressure asymmetry. This confirms that the instantaneous dynamics of the incident wake is the key mechanism controlling the modulation of the fluid-dynamic loads acting on the oscillating body. Within this framework, the proposed methodology, based on the coupling between a Lagrangian $DVM$ and a roughness model, proved capable of capturing not only variations in drag and dominant vortex-shedding frequency but also qualitative changes in the response regime itself.

Therefore, this paper provides a novel contribution to the specialized literature on wake-induced vibration by demonstrating that the surface roughness of the downstream cylinder may play a decisive role in the maintenance, suppression, or reorganization of fluid–structure synchronization in tandem arrangements. This finding is particularly relevant because it shows that roughness should be regarded not merely as a passive boundary-layer feature but also as a parameter with potential for the passive control of dynamic response and fluid-dynamic loads in systems subjected to flow-induced vibrations.

As future improvements to refine the vorticity field solution, CUDA technology will be used to implement the present algorithm. That technology is very efficient in reducing the CPU time and, consequently, provides the means for longer-time simulations to be carried out. As a consequence, the present methodology will be extended to include mixed convection heat transfer [1] and the moving ground effect [34] combined with the vortex-induced vibration of a downstream elastically mounted rigid cylinder [38].