Surface Roughness Effects on Flows Past Two Circular Cylinders in Tandem Arrangement at Co-Shedding Regime

Abstract

This paper contributes by investigating surface roughness effects on temporal history of aerodynamic loads and vortex shedding frequency of two circular cylinders in tandem arrangement. The pair of cylinders is immovable; of equal outer diameter, D; and its geometry is defined by the dimensionless center-to-center pitch ratio, L/D. Thus, a distance of L/D = 4.5 is chosen to characterize the co-shedding regime, where the two shear layers of opposite signals, originated from each cylinder surface, interact generating counter-rotating vortical structures. A subcritical Reynolds number of Re = 6.5 × 10⁴ is chosen for the test cases, which allows some comparisons with experimental results without roughness effects available in the literature. Two relative roughness heights are adopted, nominally ε/D = 0.001 and 0.007, aiming to capture the sensitivity of the applied numerical approach. Recent numerical results published in the literature have reported that the present two-dimensional model of surface roughness effects is able to capture both drag reduction and full cessation of vortex shedding for an immovable cylinder near a moving ground. That roughness model was successfully blended with a Lagrangian vortex method using sub-grid turbulence modeling. Overall, the effects of relative roughness heights on flows past two cylinders reveal changing of behavior of the vorticity dynamics, in which drag reduction, intermittence of vortex shedding, and wake destruction are identified under certain roughness effects. This kind of study is very useful for engineering conservative designs. The work is also motivated by scarcity of results previous discussing flows past cylinders in cross flow with surface roughness effects.

1. Introduction

Flows around cylindrical structures are present in different practical engineering problems (e.g., chemical reaction-towers, chimneys, groups of buildings, power transmission lines, stacks, supports of off-shore platform, tube banks in heat exchangers, etc.). In this scenery, the circular cylinder is the most known geometry between the bluff bodies for experimental and numerical investigations involving fluid mechanics problems. In the last five decades, such investigations have undoubtedly contributed for advances in science and technology. To the best of our knowledge, the well-known aerodynamics of a single circular cylinder can (definitively) be modified because of different interference effects, such as freestream turbulence, surface texture, flow-induced vibrations, buoyance forces, and the presence of other solid boundaries.

Zdravkovich [1] defined two basic interference effects for flows past two circular cylinders in cross flow (center-to-center, longitudinal, and transverse distances): (i) wake interference, when the downstream cylinder is near to or completely immerse in the viscous wake of the upstream cylinder and (ii) proximity interference, when both cylinders are close to each other; however, neither is immersed in the viscous wake of the other. A no-interference region was also defined by him such that the interference effect is negligible, and the aerodynamics of each cylinder tends to that of single cylinder. Summer [2] reported that flows around two circular cylinders in cross flow can generate a pair of shear layers of opposite signals, one from each cylinder, and, consequently, the presence of two von Kármán vortex streets. Furthermore, these two mechanisms can interact between them changing generation processes of vortical structures.

Of particular interest for this paper, the center-to-center spacing, L, and the Reynolds number value determine if two circular cylinders in tandem arrangement will behave as one extended bluff body or as two bodies, each one generating a von Kármán vortex street [3,4,5]. For small values of L, the downstream cylinder can experiment until negative drag force. On the other hand, the downstream cylinder presents a jump to positive drag as L increases, and consequently, there is a critical center-to-center spacing where a signal reverse is clearly identified. These phenomena are complex, and for subcritical Reynolds numbers, they were classified into three flow regimes and viscous wake patterns [4,5]: (i) extended-body regime, (ii) reattachment regime, and (iii) co-shedding regime. It is important to comment that the subcritical regime for Reynolds numbers is characterized when the boundary layer remains laminar, and the transition to turbulence occurs after the flow separation region, i.e., in the near wake region.

The extended-body regime is commonly identified for dimensionless center-to-center pitch ratio in the range 1.0 < L/D < 1.5, in which the two cylinders are closest between them and the two shear layers of opposite signals from the upstream cylinder can totally involve the downstream cylinder surface, producing a flow rather stagnant in the gap space between two bodies. The reattachment regime normally manifests for 1.5 < L/D < 3.5, where the two shear layers separated from the upstream cylinder are able to reattach to the downstream cylinder surface. Newly, the flow in the gap space between two bodies still remains stagnant. For both regimes, the dominant interference effect is the proximity between two bodies forming one common vortex street from the downstream cylinder. The third flow regime, named co-shedding regime, commonly manifests for L/D > 3.5, already at large spacing. The remarkable characteristic is that two shear layers from the upstream cylinder govern the large-scale vortex formation mode for both cylinders. The von Kármán-type vortices take place in the gap space between two bodies and will determine the vortex formation regime from the downstream cylinder. The value L/D ≈ 3.5, named as critical spacing, is important to establish the transition from regimes of extended-body and reattachment to co-shedding regime [5]. The literature also reports that a bistable state appears around the critical spacing, where the flow intermittently changes from reattachment regime to co-shedding regime [3].

The objective of the present paper is to investigate surface roughness effects interfering with flows at subcritical Reynolds number of Re = 6.5 × 10⁴ past a pair of immovable circular cylinders, of equal diameter, in tandem arrangement. The dimensionless center-to-center spacing between the two bodies is kept constant at L/D = 4.5, thus characterizing a co-shedding regime. There is a lack of results available in the literature discussing surface roughness effects of a pair of cylinders in cross flow, which justifies the present work. The methodology employs a Lagrangian vortex method with model of surface roughness effects; the key idea is to inject momentum into the boundary layer by simulating additional inertial effects because of the roughness. Previous works reported that the present methodology is able to capture intermittence or complete cessation of vortex shedding for flows past a circular cylinder with or without wall confinement [6,7,8,9,10,11]. Experimental results available in the literature without surface roughness effects [12,13] have been effectively used to support new physical interpretations when surface roughness effects are numerically investigated using the present methodology. With the same philosophy, the present numerical results obtained with the model of surface roughness effects are also compared with other experimental results without surface roughness effects [3]. Likewise, experimental results without roughness effects support new physical interpretations when the model of surface roughness effects is activated, and changing of behavior of the vorticity dynamics is clearly identified. The most interesting numerical results indicate drag reduction accomplished by intermittence of vortex shedding and wake destruction. The behavior of the base pressure coefficient obtained for both cylinders is physically consistent with the mentioned drag reduction caused by surface roughness effects.

2. The Lagrangian Vortex Method with Model of Surface Roughness Effects

Figure 1 sketches the flow past two circular cylinders, immovable and of same outer diameter, D. The fluid domain, Ω, is completely defined by surface S = S₁∪S₂∪S_∞, in which S₁ represents the upstream cylinder surface, S₂ the downstream cylinder surface, and S_∞ the fluid surface far from the cylinders. The cylinders boundary is assumed as a hydraulically smooth wall for inclusion of surface roughness effects [7]. The inlet flow is defined by U, and the global coordinate system is identified by x and y. The different in tandem arrangements can be defined by varying L (center-to-center spacing or pitch). The quantities and equations below were non-dimensionalized by adopting the representative length D, the representative velocity U, and the time scale U/D.

The flow that develops from the cylindrical surfaces is assumed to be two-dimensional, incompressible, and unsteady. The flow includes turbulence manifestations using Large-Eddy Simulation (LES) modeling such that the large eddies are disconnected from the smaller ones [7,8,9,10,11]. The rheological behavior of the fluid is Newtonian, with constant thermodynamics properties. The problem is originally governed by the continuity and the Navier–Stokes equations.

The Reynolds number, physically interpreted as inertial forces in opposition to the viscous forces, is defined as:

$$\mathrm{R}\mathrm{e}=\frac{\mathrm{U}\mathrm{D}}{\mathrm{\nu }}$$

(1)

where ν is the fluid kinematics viscosity coefficient.

The formation of two shear layers of opposite signal from bluff body has its origin linked to boundary layer separation [8,14]. That mechanism generates pairs of counter-rotating vortical structures alternatively shedding, one by one, from the body surface. The frequency of vortex shedding, f, of every pair of vortical structures is measured by the dimensionless Strouhal number in accordance with the following definition:

$$\mathrm{S}\mathrm{t}=\frac{\mathrm{f}\mathrm{D}}{\mathrm{U}}$$

(2)

where the frequency, f, is numerically captured from the time history of the lift force, and the latter oscillates around zero once for each pair of vortical structures shedding from bluff body [8,10].

In Lagrangian manner, all vorticity generated from solid boundaries is discretized and represented by a collection of vortex blobs; the present approach utilizes Lamb vortex elements [15]. In two dimensions, every vortex blob is completely identified by position vector, x; strength of vortex, Γ; viscous core (or Lamb vortex core), σ₀; and velocity vector, u [16]. Every vortex blob is individually followed during a typical numerical simulation. Therefore, the flow field is solved only for fluid regions where the vorticity field is not null. As an advantage, the boundary condition far from the solid boundaries is automatically satisfied.

The Panels Method [17] is utilized to discretize the solid boundaries of the problem, and thus, the boundary conditions necessary on solid boundaries can be numerically imposed and satisfied. Here, each boundary solid is discretized and represented by a finite number of flat panels. Every panel has a centered point, or pivotal point, where the boundary conditions must be imposed during each time step of the numerical simulation. Figure 2 illustrates the proposed boundary layer model, which introduces nascent vortex blobs into the fluid domain. Only the first panel of a typical discretization is represented in Figure 2a–c, where two extreme points can be identified, one pivotal point and one shedding point, the latter to shed nascent vortex blobs. In this way, only the extreme points used to discretize solid boundaries intercept the real boundary of a body.

The impermeability condition and the mass conservation are imposed on every pivotal point through the distribution of source elements with constant density, σ, over the length of every panel [17]. The no-slip condition and the global circulation conservation are imposed on every pivotal point through the distribution of nascent vortex blobs of strength Γ and viscous core σ₀. As consequence, two linear systems of algebraic equations are obtained and iteratively solved via method of least squares.

Figure 2a sketches as a nascent vortex blob sheds from a panel during one time step, $∆\mathrm{t}$. The boundary layer modeling illustrated in Figure 2a does not include surface roughness effects. For this case, the Lamb vortex core is estimated as [7]:

$${\sigma }_0=1.41421\sqrt{\frac{∆\mathrm{t}}{\mathrm{R}\mathrm{e}}}$$

(3)

The model of surface roughness effects aims to change the viscous core, σ₀, of every nascent vortex blob by introducing extra inertial effects in the vicinity of the shedding point. Firstly, the local process to compute surface roughness effects starts considering that the center of the viscous core, σ₀, coincides with the shedding point, and thus, the vortex strength, Γ, without roughness effects is computed, as shown in Figure 2a. Secondly, the turbulent activities near the shedding point are added into the boundary layer, where a semicircle of radius $‖\mathbf{b}‖=2\epsilon -{\sigma }_0$, and centered on the shedding point is required, as shown in Figure 2b. In this way, ε ≡ ε/D represents the effect of a selected relative roughness height. Bimbato et al. [7] proposed a set of NR = 21 “rough points” on each semicircle to compute the average velocity differences necessaries to evaluate the second-order velocity structure function of the filtered field, $\overline{\mathbf{u}}$. And these average velocity differences must be estimated between the center of each semicircle and the NR points over the same semicircle in accordance with [7,18]:

$${\overline{\mathrm{F}}}_{2_i}\left(\mathrm{t}\right)=\frac{1}{\mathrm{N}\mathrm{R}}{\displaystyle \sum _{w=1}^{\mathrm{N}\mathrm{R}}{‖{\overline{\mathbf{u}}}_{{\mathrm{t}}_{\mathrm{i}}}\left({\mathbf{x}}_{\mathrm{i}},\mathrm{t}\right)-{\overline{\mathbf{u}}}_{{\mathrm{t}}_{\mathrm{w}}}\left({\mathbf{x}}_{\mathrm{i}}+\mathbf{b},\mathrm{t}\right)‖}_{\mathrm{w}}^2}\left(1+\epsilon \right)$$

(4)

Thirdly, the local eddy viscosity coefficient must be computed on every shedding point, placed very close the pivotal point, attending the following equation [7,18]:

$${\upsilon }_{{\mathrm{t}}_i}\left(\mathrm{t}\right)=0.105{\mathrm{C}}_{\mathrm{k}}^{-3/2}{\sigma }_{0_{\mathrm{k}}}\sqrt{{\overline{\mathrm{F}}}_{2_i}\left(\mathrm{t}\right)}$$

(5)

where C_k = 1.4 is the Kolmogorov constant.

After that, the new viscous core, including roughness effects, is recalculated as [7]:

$${\sigma }_{0{\mathrm{c}}_{\mathrm{k}}}\left(\mathrm{t}\right)=1.41421\sqrt{\frac{∆\mathrm{t}}{\mathrm{R}\mathrm{e}}\left(1+\frac{{\mathrm{\nu }}_{{\mathrm{t}}_{\mathrm{i}}}\left(\mathrm{t}\right)}{\mathrm{\nu }}\right)}$$

(6)

Lastly, the local process to compute roughness effects is completed by solving the two linear systems of algebraic equations, one for sources distributions over panels and the other for nascent vortex blobs. Figure 2c illustrates the gain of vorticity attributed to the vortex strength, Γ. This process also represents a new source term of momentum for the problem formulation.

The vorticity dynamics is simulated through the vorticity transport equation (i.e., an obtained version of the Navier–Stokes equations after the curl operator to be applied on them), which assumes the following scalar version in two dimensions [7,19]:

$$\frac{\partial \overline{\omega }}{\partial \mathrm{t}}+\left(\overline{\mathbf{u}}\cdot \nabla \right)\overline{\omega }=\frac{1}{\mathrm{R}{\mathrm{e}}_{\mathrm{c}}}{\nabla }^2\overline{\omega }$$

(7)

where $\mathrm{R}{\mathrm{e}}_{\mathrm{c}}$ is interpreted as the “local Reynolds number” because of inclusion of the local eddy viscosity coefficient and defined as:

$$\mathrm{R}{\mathrm{e}}_{{\mathrm{c}}_{\mathrm{i}}}\left(\mathrm{t}\right)=\frac{\mathrm{U}\mathrm{D}}{\mathrm{\nu }+{\mathrm{\nu }}_{{\mathrm{t}}_{\mathrm{i}}}\left(\mathrm{t}\right)}$$

(8)

In Equation (8), the local eddy viscosity coefficient must be computed for every vortex blob that instantaneously constructs the two viscous wakes for the fluid domain Ω. The Equation (7) is also solved for the same vortex blobs during each time step.

According to Chorin [20], an algorithm that splits in two parts the advective-diffusive operator of Equation (7) must be employed such that:

$$\frac{\mathrm{D}\overline{\omega }}{\mathrm{D}\mathrm{t}}=\frac{\partial \overline{\omega }}{\partial \mathrm{t}}+\left(\overline{\mathbf{u}}\cdot \nabla \right)\overline{\omega }=0$$

(9)

$$\frac{\partial \overline{\omega }}{\partial \mathrm{t}}=\frac{1}{\mathrm{R}{\mathrm{e}}_{\mathrm{c}}}{\nabla }^2\overline{\omega }$$

(10)

Equation (9) governs the vortex blobs advection; in Lagrangian manner, that equation implies that every vortex blob is advected as a set of fluid material particles. On the other hand, Equation (10) includes viscous effects and turbulence modeling via LES theory.

In a sequential way, the velocity and pressure fields must be evaluated before solving Equations (9) and (10).

The instantaneous velocity is only calculated at positions where the vortex blobs are there. Thus, three contributions are necessary to evaluate the velocity vector of the filtered field, i.e., (i) the inlet flow or freestream (Figure 1); (ii) the surfaces S₁ and S₂ (Figure 1), which are represented by panels with sources distribution of constant density [17]; and (iii) the vortex-vortex interaction, obtained from the Biot–Savart Law [16,19].

The inlet flow contribution is directly obtained from mainstream velocity (Figure 1).

The solid boundaries of the problem contribute to the velocity filed computation through the following expression [17]:

$$\overline{\mathrm{u}{\mathrm{b}}_{\mathrm{k}}^{\mathrm{n}}}\left({\mathbf{x}}_{\mathrm{k}},\mathbf{t}\right)={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}\mathrm{P}}{\sigma }_{\mathrm{i}}{\mathrm{c}}_{\mathrm{k}\mathrm{i}}^{\mathrm{n}}\left[{\mathbf{x}}_{\mathrm{k}}\left(\mathrm{t}\right)-{\mathbf{x}}_{\mathrm{i}}\right]},\mathrm{n}=1,2;\mathrm{k}=1,\mathrm{N}\mathrm{V}$$

(11)

where ${\sigma }_{\mathrm{i}}$ is associated with the source density distributed over length of the i-th panel, ${\mathrm{c}}_{\mathrm{k}\mathrm{i}}^{\mathrm{n}}\left[{\mathbf{x}}_{\mathrm{k}}\left(\mathrm{t}\right)-{\mathbf{x}}_{\mathrm{i}}\right]$ defines the n-th component of the velocity that the i-th source panel (of unitary source density) induces at position occupied by k-th vortex blob, NP is the number of panels used to discrete the two solid boundaries, and NV is the total number of vortex blobs that instantaneously construct the two viscous wakes of the problem.

The velocity induced by the vortex blobs over themselves is computed in accordance with the Biot–Savart law [15] such that:

$$\overline{\mathrm{u}{\mathrm{v}}_{\mathrm{k}}^{\mathrm{n}}}\left({\mathbf{x}}_{\mathrm{k}},\mathrm{t}\right)={\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{N}\mathrm{V}}{\mathrm{\Gamma }}_{\mathrm{j}}{\mathrm{c}}_{\mathrm{k}\mathrm{j}}^{\mathrm{n}}\left[{\mathbf{x}}_{\mathrm{k}}\left(\mathrm{t}\right)-{\mathbf{x}}_{\mathrm{j}}\left(\mathrm{t}\right)\right]},\mathrm{n}=1,2;\mathrm{k}=1,\mathrm{N}\mathrm{V}$$

(12)

Where ${\mathrm{\Gamma }}_{\mathrm{j}}$ is associated with the strength of the j-th vortex blob, and ${\mathrm{c}}_{\mathrm{k}\mathrm{j}}^{\mathrm{n}}\left[{\mathbf{x}}_{\mathrm{k}}\left(\mathrm{t}\right)-{\mathbf{x}}_{\mathrm{j}}\left(\mathrm{t}\right)\right]$ defines the n-th component of the velocity that the j-th vortex blob (of unitary strength) induces at position occupied by k-th vortex blob.

The drag and lift coefficients are calculated starting with the following definition of stagnation pressure:

$$\overline{\mathrm{Y}}=\frac{\overline{\mathrm{p}}}{\rho }+\frac{{\overline{\mathrm{u}}}^2}{2};\overline{\mathrm{u}}=\left|\overline{\mathbf{u}}\right|$$

(13)

In Equation (13), the static pressure is given by $\overline{\mathrm{p}}$, ρ is the fluid density, and the velocity vector induced at computation points of interest is defined by $\overline{\mathbf{u}}$. The aerodynamic loads computation is associated with a Poisson equation for the pressure, which is mathematically established through the following integral formulation [21]:

$$\mathrm{H}{\overline{\mathrm{Y}}}_{\mathrm{i}}-{\displaystyle \underset{\mathrm{S}}{\int }\overline{\mathrm{Y}}\nabla {\mathsf{\Xi }}_{\mathrm{i}}\cdot {\mathbf{e}}_{\mathrm{n}}\mathrm{d}\mathrm{S}}={\displaystyle \underset{\Omega }{\iint }\nabla {\mathsf{\Xi }}_{\mathrm{i}}\cdot \left(\overline{\mathbf{u}}\times \overline{\mathsf{\omega }}\right)\mathrm{d}\Omega }-\frac{1}{\mathrm{R}\mathrm{e}}{\displaystyle \underset{\mathrm{S}}{\int }\left(\nabla {\mathsf{\Xi }}_{\mathrm{i}}\times \overline{\mathsf{\omega }}\right)\cdot {\mathbf{e}}_{\mathrm{n}}\mathrm{d}\mathrm{S}}$$

(14)

In Equation (14), the filtered pressure field can be obtained at calculation points of interest, with $\mathrm{H}=1.0$ and $\mathrm{H}=1/2$ for computational points of the fluid domain and over solid boundaries, respectively. $\Xi$ is the fundamental solution of Laplace equation, and ${\mathbf{e}}_{\mathrm{n}}$ defines the normal unit vector pointing from a solid boundary to fluid domain.

In the present numerical approach, Equation (14) has as unknown value: the static pressure distribution over the pivotal point of every panel. The solution of Equation (14) enables the drag and lift coefficients evaluation, such as, respectively [6,7,8,9,10,11]:

$${\mathrm{C}}_{\mathrm{D}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}\mathrm{P}}2\left({\overline{\mathrm{p}}}_{\mathrm{i}}-{\mathrm{p}}_{\infty }\right)∆{\mathrm{S}}_{\mathrm{i}}\mathrm{s}\mathrm{i}\mathrm{n}{\mathrm{\beta }}_{\mathrm{i}}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}\mathrm{P}}{\mathrm{C}}_{{\mathrm{P}}_{\mathrm{i}}}∆{\mathrm{S}}_{\mathrm{i}}\mathrm{s}\mathrm{i}\mathrm{n}{\mathrm{\beta }}_{\mathrm{i}}}$$

(15)

$${\mathrm{C}}_{\mathrm{L}}=-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}\mathrm{P}}2\left({\overline{\mathrm{p}}}_{\mathrm{i}}-{\mathrm{p}}_{\infty }\right)∆{\mathrm{S}}_{\mathrm{i}}\mathrm{c}\mathrm{o}\mathrm{s}{\mathrm{\beta }}_{\mathrm{i}}}=-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}\mathrm{P}}{\mathrm{C}}_{{\mathrm{P}}_{\mathrm{i}}}∆{\mathrm{S}}_{\mathrm{i}}\mathrm{c}\mathrm{o}\mathrm{s}{\mathrm{\beta }}_{\mathrm{i}}}$$

(16)

In Equations (15) and (16), ${\mathrm{p}}_{\infty }$ represents the reference pressure far from the solid boundaries, $∆{\mathrm{S}}_{\mathrm{i}}$ refers to length of the i-th panel, and ${\mathrm{\beta }}_{\mathrm{i}}$ is the anti-clockwise orientation angle of the same panel with reference to the x-axis, as shown in Figure 1.

It is important to comment that the distributed and integrated aerodynamic loads are instantaneously evaluated always that the boundary conditions are satisfied over pivotal points.

Then, the problem of the vortex blobs advection, Equation (9), can be solved by integrating the path equation of every vortex blob, in which an explicit Euler scheme [22] is utilized such that:

$$\frac{\mathrm{d}{\mathbf{x}}_{\mathrm{k}}}{\mathrm{d}\mathrm{t}}={\overline{\mathbf{u}}}_{{\mathrm{t}}_{\mathrm{k}}}\left({\mathbf{x}}_{\mathrm{k}},\mathrm{t}\right),\mathrm{k}=1,\mathrm{N}\mathrm{V}$$

(17)

In Equation (17), dt is interpreted as time stepping, i.e., $∆\mathrm{t}$ such that its value is evaluated from an estimate of the velocity and advective length of the flow. As a test of this idea, Figure 3 sketches a vortex blobs pair of equal strength Γ and distance AB apart, which starts to experience an advection velocity and subsequent advective motion in accordance with Equations (12) and (17), respectively.

In this way, the vortex blob A, initially placed at (0.0−1.0), will induce velocity at vortex blob B, the latter initially located at (0.0,1.0), as shown in Figure 3. Simultaneously, an advective velocity is induced at vortex blob A by vortex blob B. As the time runs, the vortex blobs will each experience new advection velocities because of the other acting in the direction normal to the line AB joining them. Moreover, Equation (17) governs the advective motion of the vortex blobs pair in a circular path about the midpoint of AB (point M). The vortex blobs move in clockwise direction over several time steps ∆t. In

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

Distance AB (Figure 3)	ΓA	σ0	∆t	Steps	Error (%)
2.0D	1.0	0.001D	0.05	795	0.4406
2.0D	1.0	0.001D	0.1	400	0.8780
2.0D	1.0	0.001D	0.2	203	1.7478

, the column identified by “Steps” indicates the number of finite steps necessary for the vortex blob A to approximately attain the position (0.0,1.0). In the same Table 1, the last column estimates the relative error between the true circular drift path of constant distance given by AB = 2.0D and that numerically calculated by Equation (17) exactly at the instant that the vortex blob A approximately reaches the position (0.0,1.0). The vortex blob B behavior is analogous when it simultaneously and approximately attains the position (0.0−1.0), as identified in Figure 3. As a result, the time step ∆t = 0.05 is chosen for the explicit Euler scheme of all test cases in Section 3.

It is remarkable that the Lagrangian manner to solve Equation (9) dispenses the explicit treatment of advective derivatives [23].

The problem of vorticity diffusion, Equation (10), is numerically solved by random walk method, in which the vorticity distribution is given by [20,24]:

$$\overline{\omega }\left(\mathbf{x},\mathrm{t}\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[\mathrm{G}\left(\mathrm{x},\mathrm{y},\mathrm{t}\right)-\mathrm{G}\left(\mathrm{x}-\mathrm{y},\mathrm{t}\right)\right]\mathrm{f}\left(\mathrm{y}\right)\mathrm{d}\mathrm{y}}$$

(18)

where:

$$\mathrm{G}\left(\mathrm{x},\mathrm{y},\mathrm{t}\right)=\frac{1}{\sqrt{\frac{4\pi \mathrm{t}}{\mathrm{R}{\mathrm{e}}_{\mathrm{c}}}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-{\left(\mathrm{x}-\mathrm{y}\right)}^2}{\frac{4\mathrm{t}}{\mathrm{R}{\mathrm{e}}_{\mathrm{c}}}}\right)$$

(19)

The function G(x,y,t) is the Green function related with the heat equation, the latter is solved by convolution of the initial data associated with the Green function [24].

As practical result, after the increment $∆t$ the new position occupied by k-th vortex, the blob will become [7,8,9,10,11,25]:

$${\varsigma }_{\mathrm{k}}\left(\mathrm{t}\right)=\sqrt{\frac{4∆\mathrm{t}}{\mathrm{R}{\mathrm{e}}_{\mathrm{c}}}\mathrm{l}\mathrm{n}\left(\frac{1}{\mathrm{P}}\right)}\left[\mathrm{c}\mathrm{o}\mathrm{s}{\left(2\pi \mathrm{Q}\right)}_x+\mathrm{s}\mathrm{i}\mathrm{n}{\left(2\pi \mathrm{Q}\right)}_{\mathrm{y}}\right]$$

(20)

$\mathrm{P}$ and $\mathrm{Q}$ are random numbers assumed in the range: $0<\mathrm{P}<1$ and $0<\mathrm{Q}<1$. Consequently, the solution of Equation (20) needs two random numbers generated for every vortex blob during each time increment.

The random walk method applicability can be demonstrated through the radial diffusion of a vortical structure of strength Γ_v = 1.0, centered on the origin (0.0,0.0) in Figure 4a. That problem presents an exact solution for the vorticity distribution in space and time, calculated as [19,25]:

$$\omega \left(\mathrm{r},\mathrm{t}\right)=\frac{{\mathrm{\Gamma }}_{\mathrm{v}}}{4\mathrm{\pi }\mathrm{\nu }\mathrm{t}}{\mathrm{e}}^{\left(-{\mathrm{r}}^2/4\mathrm{\nu }\mathrm{t}\right)}$$

(21)

The exact solution is exemplified in Figure 4b for the case t = 1.0, Γ_v = 1.0, and Re = Γ_v/ν = 1.0. A numerical solution is provided for comparison, where the vortical structure Γ_v is replaced by Z = 5000 vortex blobs each of strength Γ_v/Z. All vortex blobs are initially placed on the origin (0.0,0.0) at t = 0, and after twenty time steps ∆t = 0.05, they will diffuse, as presented in Figure 4a, in accordance with Equation (20) without turbulence modeling. The vorticity distribution is numerically computed as [25]:

$$\omega \left(\mathrm{r}\right)=\frac{{\mathrm{z}}_{\mathrm{j}}}{\mathrm{Z}\mathrm{\pi }\left({\mathrm{r}}_{\mathrm{j}+1}^2-{\mathrm{r}}_{\mathrm{j}}^2\right)}$$

(22)

where, in Figure 4a, twenty annular bins of same thickness ∆r = 0.4 and centered on the origin (0.0,0.0) were used to capture all z_j vortex blobs into the correspondent annular area lying between 0.0 ≤ r_j ≤ 7.6 and 0.4 ≤ r_j+1 ≤ 8.0, and a very good numerical prediction is obtained in comparison with the exact solution, Figure 4b. In Equation (21), the exact solution for the vorticity distribution adopted the r.m.s. radius $\sqrt{\left[\frac{1}{2}\left({\mathrm{r}}_{\mathrm{j}}^2+{\mathrm{r}}_{\mathrm{j}+1}^2\right)\right]}$ for every strip j of a annular bin. Thus, the time step ∆t = 0.05 is also chosen for random diffusion of all test cases in Section 3.

In Equation (20), the local Reynolds number, already defined in Equation (8), includes turbulence modeling, whose interpretation is associated with average velocity differences existing among the vortex blob under analysis and other vortex blobs around it [7,8,9,10,11]. The objective is to compute the local eddy viscosity coefficient (Equation (5)) supported by the following calculation of the second-order velocity structure function of the filtered field [18]:

$${\overline{\mathrm{F}}}_{2_{\mathrm{k}}}=\frac{1}{\mathrm{N}}{\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{N}}{‖{\overline{\mathbf{u}}}_{{\mathrm{t}}_{\mathrm{k}}}\left({\mathbf{x}}_{\mathrm{k}}\right)-{\overline{\mathbf{u}}}_{{\mathrm{t}}_{\mathrm{j}}}\left({\mathbf{x}}_{\mathrm{k}}+{\mathbf{r}}_{\mathrm{j}}\right)‖}_{\mathrm{j}}^2{\left(\frac{{\sigma }_{0_{\mathrm{k}}}}{{\mathbf{r}}_{\mathrm{j}}}\right)}^{2/3}}$$

(23)

where ${\overline{\mathbf{u}}}_{\mathrm{t}}$ is the filtered velocity field induced on points of interest ($\left|\overline{\mathbf{u}}{}_{\mathrm{t}}\right|=\overline{\mathrm{u}\mathrm{i}}+\overline{\mathrm{u}\mathrm{b}}+\overline{\mathrm{u}\mathrm{v}}$, respectively, the contributions of the freestream, source panels, and cluster of vortex blobs), and N is the number of vortex blobs around vortex blob (point where the local eddy viscosity coefficient needs to be evaluated) [7,8,9,10,11].

The Equation (4) was originally inspired from Equation (23) to develop the model of surface roughness effects [7]. The great computational advantage of using the second-order velocity structure function of the filtered field over the model of Smagorinsky [26] is that the concept of velocity fluctuations (differences of velocity) is used instead of rate of deformation (derivatives). This concept is easily incorporated to the Lagrangian vortex method [7,8,9,10,11].

Finally, the algorithm described above was implemented through an in-house code using an available academic license of OpenMP with Fortran in the following sequence of eight steps:

(i)

simultaneous solution of the linear systems of algebraic equations to generate source densities and vortex blob strengths (the model of roughness effects can be activated);

(ii)

velocity vector calculation at every vortex blob;

(iii)

static pressure evaluation on pivotal points followed by drag and lift coefficients computation for two bodies;

(iv)

solution of the vortex blobs advection;

(v)

solution of the vorticity diffusion including turbulence modeling using LES theory;

(vi)

reflection of stray vortex blobs from each body profile interior;

(vii)

velocity vector calculation at every pivotal point for new generation of source densities and vortex blobs strength (this step restores the boundary conditions on every pivotal point); and

(viii)

advance by ∆t.

3. Results and Discussion

The main objective of this section is to present numerical results of the model of surface roughness effects of two circular cylinders, immovable, of same diameter, in which the problem geometry is established by the dimensionless center-to-center pitch ratio of L/D = 4.5 and thus characterized in tandem arrangement at co-shedding regime. The test case of smoothed upstream cylinder at L/D→∞ is also presented for some comparisons. The applied technique in this work was specially validated for bluff body aerodynamics in the presence of wall confinement [8,9,10,11]. The chosen turbulence modeling was earlier validated by Bimbato et al. [7]. In the same way, the accuracy of the main numerical parameters utilized here was also investigated and discussed [6,7,8,9,10,11].

3.1. Simulation Setup

The key parameters of each test case for two cylinders in tandem at L/D = 4.5 are Reynolds number value of Re = 6.5 × 10⁴ (subcritical regime); two cylinders surface discretized by NP = 600 source flat panels of the same length (300 panels for each cylinder); time increment of ∆t = 0.05; displacement of δ = σ₀ = 0.001D to move every nascent vortex blob to δ-layer normal to the discretized cylinder surface (Figure 2a) [9]; and two relative roughness heights of ε/D = 0.001 and 0.007 (ε/D = 0.000 when the roughness effects are not simulated). Each test case was carried out up to dimensionless time t = 80 (1600 time increments), where were computed mean values of pressure coefficient distribution, drag and lift coefficients, and Strouhal number.

As described in Section 2, the present work only computes form (or pressure) drag force. Alcântara Pereira et al. [10] observed that the drag force on a smooth cylinder is almost totally dominated by form component contributing about approximately 98% of the total drag; on the other hand, the skin friction (or viscous) component contributes around 1–2% for the total drag [27,28]. Other important commentary is that surface roughness effects are able to produce differences on drag force measurement near 60% for flows with same Reynolds number because of some influencing factors, including turbulence intensity and surface texture [29].

3.2. Smoothed Upstream Circular Cylinder without the Presence of the Downstream Circular Cylinder

This study investigates the flow past upstream circular cylinder without the presence of the downstream circular cylinder, nominally for dimensionless center-to-center pitch ratio L/D→∞. The surface roughness model is not activated (ε/D = 0.000), because numerical results with roughness effects for this cylinder configuration already was published earlier [7,9,10].

Table 2. Comparisons among results of drag force and Strouhal number (Re = 1.0 × 10⁵, ε/D = 0.000 and L/D→∞).

Case	CD	CL	St
Experimental [12]	1.2000	−	0.1900
Numerical [20]	1.0700	−	−
Numerical [30]	1.1000	−	−
Numerical [31]	1.2200	−	0.2200
Present simulation	1.1873	−0.0152	0.2000

compares, when possible, the present numerical results of integrated aerodynamic loads and Strouhal number with experimental results of Blevins [12] and with numerical results of Chorin [20], Ogami and Ayano [30], and Mustto et al. [31]; the results of the latter were obtained using different versions of the Lagrangian vortex method at high Reynolds number flow of Re = 1.0 × 1 0⁵. The present time-averaged numerical values were computed between t = 17.25 and t = 48.85.

As can be seen in Figure 5a, a periodic behavior of the drag coefficient is established around a mean value about C_D ≈ 1.1873, while the experimental value reported by Blevins [12] is about C_D ≈ 1.2000, with 10% uncertainty (Table 2). That periodic behavior is attained after the numerical transient; consequently, a periodic steady state regime is established from dimensionless time t = 12, approximately. Figure 5b presents the spectral analysis of the lift coefficient curve (Figure 5a) using Fast Fourier Transform, where the Strouhal number (dimensionless frequency) is about St ≈ 0.2000, while the experimental value is St ≈ 0.1900 [12], with 10% uncertainty (Table 2).

Figure 6 compares the computed values for the time-averaged pressure coefficient distribution with the experimental values reported by Blevins [12], where θ is the angular position of every pivotal point starting from anterior stagnation point of the body in clockwise direction (Figure 1). The pressure coefficient is numerically computed on all pivotal points; and Figure 6 clearly shows the presence of a stagnation point for the cylinder, where the pressure coefficient is about C_p ≈ 1.0. The anterior stagnation point of the cylinder is slightly placed below of the first pivotal point, because of body discretization criteria using flat panels [17]. The discretization of the solid boundary always starts coinciding with the stagnation point of the real solid boundary; however, the pivotal point of the first panel is placed at center of the panel, see also Figure 2a.

It also is important to comment that, in Figure 6, the dimensionless frequency of the lift curve is about twice the frequency of the drag curve; the lift coefficient oscillates once for each pair of vortical structures shedding from the cylinder surface, while the drag force oscillates once for each of upper and lower shedding [14,32]. The value of the mean lift coefficient, C_L ≈ −0.0152 (Table 1), is not zero because of numerical approximations.

Figure 7 shows magnitude of the instantaneous velocity field of the upstream cylinder without the presence of the downstream cylinder (L/D→∞) at dimensionless time t = 40. The instantaneous velocity field is computed for each vortex blob, including the following contributions: freestream, U; source panels, Equation (11); and vortex-vortex interaction, Equation (12). As already mentioned, the velocity field is non-dimensionalized using the freestream, U.

The gain of experience acquired earlier [7,8,9,10,11] has pointed that all chosen numerical parameters guarantee a statistical equilibrium, and the saturation state of the numerical simulations is attained [8]. Thus, the flow past two circular cylinders in tandem arrangement can be numerically investigated (next section).

3.3. Upstream Circular Cylinder with the Presence of the Downstream Circular Cylinder

In this study, the surface roughness effects are included aiming to capture drag reduction accomplished by intermittence of vortex shedding for flows past two circular cylinders in tandem arrangement at co-shedding regime (L/D = 4.5).

Table 3. Comparisons among results of drag force and Strouhal number with effects of the smaller relative roughness height (Re = 6.5 × 10⁴ and L/D = 4.5).

Case	Upstream Cylinder			Downstream Cylinder
	ε/D	CD	St	ε/D	CD	St
Experimental [3]	−	1.2612	0.1867	−	0.2766	0.1867
Present Simulation	0.000	1.0790	0.2000	0.000	0.4897	0.2000
Present Simulation	0.000	1.1035	0.2000	0.001	0.4409	0.2000
Present Simulation	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

and

Table 4. Comparisons among results of drag force and Strouhal number with effects of the higher relative roughness height (Re = 6.5 × 10⁴ and L/D = 4.5).

Case	Upstream Cylinder			Downstream Cylinder
	ε/D	CD	St	ε/D	CD	St
Experimental [3]	−	1.2612	0.1867	−	0.2766	0.1867
Present Simulation	0.000	1.0790	0.2000	0.000	0.4897	0.2000
Present Simulation	0.000	1.0321	0.2000	0.007	0.3798	0.2000
Present Simulation	0.007	0.9904	0.2000	0.000	0.4860	0.2000
Present Simulation	0.007	1.0246	0.1875	0.007	0.4194	0.1875

compare time-averaged values of C_D and St with experimental data [3], with 92% confidence, approximately. Alam et al. [3] conducted their experiment in a low-speed, closed-circuit wind tunnel. The test section dimensions were 0.6-m height, 0.4-m width, and 5.4-m length. According to the authors, the level of turbulence in the test section was 0.19%. The two cylinders were made of brass with the same diameter of 49 mm. Other important information was that the geometric blockage ratio and aspect ratio at the working section were 8.1 and 8.2%, respectively; and results still reported that neither measured value was corrected because of the wind-tunnel blockage effects.

Figure 8 registers the temporal history of drag and lift coefficients for two cylinders without model of surface roughness effects (ε/D = 0.000). The observed force oscillations for upstream cylinder present similar pattern with the classical behavior of the curves of single cylinder, as seen in Figure 5a, in which the Strouhal number is around of St = 0.2000 (Table 3). The drag force newly oscillates once for each vortical structure shedding from the upstream cylinder, while the lift force completes every period of oscillation for one pair counter-rotating of vortical structures shedding from the same surface (Figure 8a). The interaction of the viscous wake generated from upstream cylinder with the downstream cylinder surface governs the vortex shedding mechanism of the latter cylinder; however, the temporal evolution of the drag and lift forces is influenced by upstream viscous wake effects (Figure 8b); that behavior is coherent with the expected co-shedding regime.

As consequence of news numerical results with roughness effects, there are different behaviours to be discussed and compared with the numerical results for both cylinders without roughness effects. For instance, the interaction of wake from upstream smoothed surface with downstream roughened surface (ε/D = 0.001) is able to reduce the drag force acting on downstream cylinder around 9.97% (Table 3). Other result is that the smaller relative roughness height (ε/D = 0.001), when simulated for both cylinders, provokes higher drag reduction for the upstream cylinder (around of 16.42%). That latter effect slightly increases the drag force on downstream cylinder about 3.33% (Table 3). The oscillations with the time of the drag and lift coefficients for both the cylinders with ε/D = 0.001 are presented in Figure 9.

Other interesting result is that when the smaller relative roughness height (ε/D = 0.001) is only simulated for the upstream cylinder, the drag force decreases around of 14.32% for upstream cylinder and increases about 31.71% for downstream cylinder. The interaction of wake from upstream roughened surface with downstream smoothed surface is able to increase the drag force acting on downstream cylinder (Table 3).

The Strouhal number behavior is always governed by the vortex shedding mechanism of the upstream cylinder when different combinations of smaller relative roughness height (ε/D = 0.001) are investigated. Additionally, a Strouhal number reduction around of 6.25% is identified for both the cylinders when the upstream cylinder includes roughness effects into the co-shedding regime (Table 3).

On the other hand, Table 4 indicates that the higher relative roughness height (ε/D = 0.007), when numerically activated for any cylinder, always reduces the drag force for that body. The Strouhal number only reduces by about 6.25% and for both cylinders when the roughness effects are simulated for both bodies, which implies an intermittence of vortex shedding. For that latter behavior, the oscillations with the time of the drag and lift coefficients can be observed in Figure 10, in which a drag reduction around of 14.36% occurs on downstream cylinder, and the upstream cylinder presents a drag reduction about 5.04% (Table 4).

The higher drag reduction was identified about 22.44% for only the downstream cylinder simulating roughness effects (ε/D = 0.007). In this test case, the upstream cylinder presents a drag reduction around of 4.35%. Here, the near wake dynamics because of the smoothed upstream surface also governs the vortex shedding mechanism of the roughened downstream cylinder, where the Strouhal number for both bodies does not decrease, and a possible intermittence of vortex shedding is not identified (Table 4).

Figure 11a,b can be explored to justify the drag force behaviours, summarized in Table 3 and Table 4, in which the greater base pressure of a rough cylinder is linked with higher drag reduction.

Still, in Figure 11a,b, (ε/D)_u and (ε/D)_d represent relative roughness heights numerically simulated for upstream and downstream cylinders, respectively. Figure 11a also indicates the presence of a stagnation point for the upstream cylinder, where the pressure coefficient is about C_p ≈ 1.0. In Figure 11b, however, the experimental pressure coefficient behavior [3] needs to be better investigated for a fair comparison with the present numerical results.

Figure 12, Figure 13 and Figure 14 exemplify magnitudes of non-dimensionalized, instantaneous velocity fields of three test cases at dimensionless time t = 40. The co-shedding regime is identified, where anti-symmetrical disturbances from the upstream cylinder are felt by the downstream cylinder. In Figure 12, the counter-rotating vortical structures generated from the upstream surface impact with the downstream cylinder surface and impose the vortex shedding process for the downstream cylinder. The formation mode of a large-scale vortex is characterized behind the downstream cylinder.

In Figure 13, a low velocity zone can be visualized behind the downstream cylinder, where both cylinders simulate effects of smaller relative roughness height (ε/D = 0.001). That behavior can be associated with wake destructive behavior provoked by roughness effects.

The wake destruction mechanism is more remarkable when both cylinders simulate roughness effects for ε/D = 0.007 (Figure 14). The numerical results of drag reduction about 14.36% and Strouhal number St = 0.1875 (Table 4) for downstream cylinder are physically consistent with the new vorticity dynamics imposed by model of surface roughness effects.

Most situations of flows past bodies with bluff cross-section are commonly associated with high Reynolds numbers, where the control of vortex shedding and its suppression are very important for engineering applications. In general, the vortex shedding mechanism [32] reduces the lifetime of cylindrical structures in cross flow. The use of wake pattern controllers, particularly the surface roughness, may decrease vortex shedding deficiencies. Therefore, the present numerical technique looks promising in presenting potentialities to control the wake destructive behavior and vortex shedding suppression of bluff bodies in cross flow.

4. Conclusions

This work has utilized a purely Lagrangian version of the Vortex Method with model of surface roughness effects to discuss different flow patterns around two circular cylinders in tandem arrangement at subcritical Reynolds number of Re = 6.5 × 10⁴. The numerical analyse of distributed and integrated aerodynamic loads and of vortex shedding frequency for two cylinders at co-shedding regime (named at center-to-center spacing of L/D = 4.5) has identified different behaviours than that obtained without any roughness effects. The relative roughness heights of ε/D = 0.001 and 0.007 have been chosen, aiming to describe the potentialities of the present in-house vortex code to produce significant changes on aerodynamics of two bluff bodies when the two shear layers from the upstream cylinder are able to reproduce the large-scale vortex formation mode. Put in other words, the von Kármán-type vortices take place in the gap space between two bodies and importantly interfere in the vortex-forming regime from the downstream cylinder. The main conclusions are summarized below, in which the case numerically simulated for both cylinders without roughness effects (ε/D = 0.000) is used for important comparisons:

(i)

The relative roughness height of ε/D = 0.001 reduces the Strouhal number of the upstream cylinder about 6.25% when it simulate roughness effects. The mentioned reduction also occurs for the downstream cylinder without or with simulation of surface roughness effects (Table 3). That flow pattern ratifies the expected physics, where the vortical structures from upstream cylinder travel impinge on the downstream cylinder and also govern the vortex-forming regime from the downstream cylinder, as illustrated in Figure 13.

(ii)

For ε/D = 0.001, the higher drag reduction on upstream cylinder surface (around of 16.42%) is identified when both the cylinders simulate surface roughness effects. That flow pattern increases the drag force on downstream cylinder about 3.33% (Table 3).

(iii)

For ε/D = 0.001, the higher drag reduction on downstream cylinder surface (around of 9.97%) manifests for downstream cylinder simulating roughness effects and smoothed upstream cylinder (ε/D = 0.000). That flow pattern increases the drag force on upstream cylinder around of 2.27% (Table 3).

(iv)

Still, for ε/D = 0.001, interestingly, the drag force on downstream cylinder increases about 31.71% when only the upstream cylinder simulates roughness effects. That flow pattern reduces the drag force acting on upstream cylinder by around 14.32% (Table 3).

(v)

On the other hand, the relative roughness height of ε/D = 0.007 only reduces the Strouhal number about 6.25% when both cylinders simulate roughness effects (Table 4). That flow pattern indicates that counter-rotating vortical structures from upstream cylinder also travel, impinging on the downstream cylinder and governing the vortex-forming regime from the latter cylinder. As illustration, Figure 14 shows that the higher relative roughness height, simultaneously simulated for both cylinders, provides the most representative wake destructive behavior behind two bodies. A drag reduction on upstream cylinder surface around of 5.04% and about 14.36% on downstream cylinder surface are also identified when both the cylinders include model of surface roughness effects (Table 4).

(vi)

For ε/D = 0.007, the higher drag reduction on upstream cylinder surface (around of 8.21%) is identified when the downstream cylinder does not simulate surface roughness effects and upstream cylinder includes roughness effects. That flow pattern slows down the drag force on the downstream cylinder a little, by about 0.76% (Table 4).

(vii)

Moreover, for ε/D = 0.007, the higher drag reduction on downstream cylinder surface (around of 22.44%) occurs for upstream cylinder without roughness effects and roughened downstream cylinder. That flow pattern decreases the drag force on downstream cylinder around of 4.35% (Table 4).

As a general conclusion, the relative roughness height of ε/D = 0.001, when compared with the configuration of a hydraulically smooth wall for both cylinders, can increase or decrease the drag force under certain roughness effects. The relative roughness height of ε/D = 0.007, however, always provokes drag reduction when compared with the numerical configuration of a hydraulically smooth wall for both cylinders. It is important to mention that the experimental result from Alam et al. [3] without roughness effects, particularly, indicates a good approximation with the numerical result of drag force acting on an upstream cylinder without a model of surface roughness effects (Table 3).

The present numerical results for behavior of base pressure coefficient support interpretations about drag force variations on cylinders’ surfaces. In Figure 11a, the experimental result [3] of pressure coefficient for an upstream cylinder presents very good agreement with the captured behavior of the numerical simulation for two cylinders without roughness effects (ε/D = 0.000). Further investigation is required, however, to fully clarify the behavior of the experimental results [3] of drag force and pressure coefficient distribution for a downstream cylinder to be carefully compared with numerical results (Table 3 and Table 4 and Figure 11b).

In general, experimental results without roughness effects are extremely important to support new investigations of sensitivity of the present vortex code with a roughness model because there is a lack of experimental and numerical results published in the literature with roughness effects for the investigated problem in this work.

The new flow patterns identified in the co-shedding regime can be summarized as: viscous wake generated from roughened surface acting on smoothed surface, viscous wake generated from smoothed surface acting on roughened surface, viscous wake generated from roughened surface acting on roughened surface, and also all new combinations of wake-wake interaction. This study about roughness effects opens doors and can motivate scientists around the world to investigate changing of behavior of the vorticity dynamics and its relationship with drag reduction and intermittence of vortex shedding for flows past two circular cylinders in cross flow. For motivation and inspiration, the relative roughness height that increases the destructive behavior of the von Kármán-type vortices can be visualized by comparing Figure 7 and Figure 12, Figure 13 and Figure 14, which present magnitude of instantaneous velocity fields.

The numerical results reported here can be useful for engineers developing conservative designs, besides enabling science to evolve. The results can also be compared with future studies in the literature. It can be reported that the present methodology seems to capture the critical behavior of flows past two circular cylinders in tandem arrangement with roughness effects. The numerical results ratify that a model of two-dimensional roughness is able to capture drag reduction in a good physical sense [6,7,8,9,10,11]. Within the last decade, our research group has made an effort to develop the present two-dimensional Lagrangian vortex method with a model of surface roughness effects.

In closing, the analysis of flows past cascades of turbomachines is planned as a future study [33,34]. This kind of flow is motivated by the challenge of the interaction between more than two bodies.