Issue 
Eur. Phys. J. Appl. Phys.
Volume 81, Number 2, February 2018



Article Number  21101  
Number of page(s)  14  
Section  Physics and Mechanics of Fluids, Microfluidics  
DOI  https://doi.org/10.1051/epjap/2018170385  
Published online  08 June 2018 
https://doi.org/10.1051/epjap/2018170385
Regular Article
Numerical investigation of the vibration effect of a flexible membrane on the flow behaviour around a circular cylinder
^{1}
Unit of Computational Fluid Dynamics and Transfer Phenomena, National Engineering School of Sfax, University of Sfax,
B.P. 1171,
3038
Sfax, Tunisia
^{2}
Sfax Preparatory Engineering Institute, University of Sfax,
B.P. 1172,
3018
Sfax, Tunisia
^{3}
Faculty of Sciences of Sfax, Department of Physics, University of Sfax,
B.P. 1171,
3000
Sfax, Tunisia
^{*} email: nabaouia.maktouf@gmail.com
Received:
21
November
2017
Accepted:
5
March
2018
Published online: 8 June 2018
Active control of the flow behind a bluff body is obtained by integrating a vibrating membrane. A numerical study has been conducted to investigate the effect of the vibration of a flexible membrane, stuck to the rear side of a circular cylinder, on the global flow parameters such as the Strouhal number, the drag and lift coefficients. The shape of the membrane is evolving as a vibrating chord using a dynamic mesh. The governing equations of 2D and laminar flow have been solved using ANSYS Fluent 16.0 as a solver and the Gambit as a modeler. The motion of the membrane is managed by two parameters: frequency f and amplitude A. The effect of the flexible membrane motion is studied for the range of conditions as 0.1 Hz ≤ f ≤ 6 Hz and 5 × 10^{−4} m ≤ A ≤ 10^{−3} m at a fixed Reynolds number, Re = 150. Three different sizes of the flexible membrane have been studied. Results show that a beat phenomenon affects the drag coefficient. The amplitude does not affect significantly the Strouhal number as well as drag and lift coefficients. By increasing the size of the flexible membrane, we show a lift enhancement by a growth rate equal to 39.15% comparing to the uncontrolled case.
© EDP Sciences, 2018
1 Introduction
Several aerodynamic studies have striven to control passively or actively the flow around bluff bodies in order to improve vehicles performance, reducing the noise and fuel consumption. The circular cylinder, despite its simple geometry, constitutes the reference of all the fluid dynamic phenomena as well as it can be observed in many engineering applications. Up to now, many techniques have been studied to control the flow with the aim to suppress the von Karman sheet or to influence the aerodynamic forces. These techniques varied from passive control which affects the surface of the bluff body, such as the addition of splitter plates [1,2], flexible filament [3], or changing its property like adding dimples [4] or grooves [5], to active control using synthetic jets [6–9], plasma actuators [10–12], blowing suction [13,14], flapping foil [15,16], electromagnetic forces [17,18] acoustic actuator [19,20].
It seems that nowadays, the flexible objects represent an attractive research topic in the fluid control field, from free oscillation to the forced one many phenomena have been found. For example, inspired by the fish swimming, the free motion of a flexible plate attached to the rear side of a circular cylinder was investigated by Wu et al. [3]. Their numerical results conducted for Reynolds number equal to 150 show a suppression of the vortex shedding, the vortex induced vibration as well as the fluctuation of the lift force.
Despite the free motion of the flat plate is considered as a passive control way since it needs no energy input, it leads to many problems. In fact, the free motion leads to a nonuniform fluid distribution over the flexible plate, so that the determination of the Strouhal number of vortex shedding and the frequency of plate deflection, are hard to estimate using the natural plate frequencies [21]. Thereafter, a series of numerical and experimental studies on forced oscillation objects have been performed. The flexibility of the rear part of the circular cylinder was one of the active control technique. Wu et al. [22] have produced a traveling transverse wave called fluid roller bearing (FRB) due to the flexibility of the rear part of the cylinder. Their twodimensional numerical simulation was conducted for a Reynolds number equal to 500, 2000, and 5000 shows a reduction on the drag average by 85% and a total suppression of the vortex shedding for Reynolds Re ≥ 2000. Chen et al. [23] have reproduced the FRB due to a partial flexible part of the cylinder located at [90–165°] and [225–270°]. Their numerical and experimental study shows a decrease in the mean drag coefficient, narrow in the wake width, and a suppression of the vortex shedding. Bergmann et al. [24] observed a decrease of the relative mean drag value from 30 to 75% when they applied a specific sinusoidal control to a well selected upstream part of a circular cylinder instead of its whole surface.
Nowadays, studies focus to search the optimal size and place of the flexible surface on the bluff body. The flexible membrane is one of the most recent techniques. This idea has been inspired from the bio flight and applied especially in the micro air vehicles and widely applied to the control of aircraft [25]. Kang et al. [26], studied numerically the effect of a local oscillating membrane attached to an airfoil NACA0012 with an attack angle of 6° at Re = 5000. A parametric study of frequency, amplitude, and flexibility of the membrane shows a lift enhancement of 69.86% for membrane's excitation frequency f = 1.22 Hz compared to the uncontrolled case.
The present research work was carried out to contribute to the existing knowledge of active control using a flexible membrane attached to the rear side of a circular cylinder. The motion of the flexible membrane is imposed in order to study the response of the flow on forced vibration and not the vortex induced vibration phenomena. The numerical study particularly focuses on the effect of frequency, amplitude, and size of the membrane on the flow pattern. Numerical simulations are performed out for frequency f and amplitude A ranging respectively from 0.1 Hz to 6.0 Hz and 5 × 10^{−4} m to 10^{−3} m at a fixed Reynolds number Re = 150 with three different sizes of the membrane.
2 Problem statement and governing equations
Active control of a twodimensional laminar flow of air around a circular cylinder using a flexible membrane attached to its rear side is simulated numerically. The computational domain and coordinate system along are shown in Figure 1.
Three different sizes of flexible membrane have been studied for different lengths ℓ (, , ) corresponding to an angle of aperture (α = 8°, α = 16°, α = 32°) respectively. The motion of the membrane, attached to the backside of the circular cylinder, is governed by the following equation so that it generates a longitudinal stationary wave: (1) where D = 15 mm, ℓ, A and f are respectively the diameter of the circular cylinder, the length of the membrane, the amplitude and the frequency of the flexible membrane.
Noslip boundary conditions for the velocity have been imposed along the solid walls. The cylinder is exposed to a free stream with uniform axial velocity () corresponding to a Reynolds number . The outlet is set at pressure outlet with zerogauge pressure. Under these conditions, the following conservation equations that include continuity and momentum equations are expressed as follows:
Momentum (3) where, , and μ = 1.7894 kg.m^{−1}.s^{−1} are respectively the local pressure, the velocity vector, the fluid density and the dynamic viscosity of air.
The drag and lift coefficients are the two important factors reflecting the flow around the circular cylinder. They are calculated at each time step and defined as: (4) (5) where F_{D} and F_{L} are the drag and lift forces, exerted by the fluid on the cylinder, respectively. These forces are calculated by integrating the viscous shear forces and pressure over the surface of the cylinder.
Fig. 1 Computational domain. 
3 Numerical computation
Equations (2) and (3) combined with equation (1) are discretized by the control volume approach using computational fluid dynamics software Fluent 16.0. The secondorder upwind difference scheme was employed in the discretization of momentum equations. The wellknown SIMPLE algorithm was adopted in order to link the pressure with the velocity for an incompressible flow. The convection and diffusion terms in the flow conservation equations are discretized using the least Squares CellBased. During the numerical computations, the residuals were carefully monitored and the solutions were considered to be converged when the residuals for all governing equations were lower than 10^{−6}. From the postprocessing of these data, we compute values of drag and lift coefficients as well as the Strouhal number St characterizing the frequency of the vortex shedding behind the cylinder.
3.1 Grid independence study
The grid structure was generated using the GAMBIT 2.3.16. We have highly refined the grid in the vicinity of the cylinder in order to well evaluate the strong variation of the dynamic variables, especially near the flexible membrane. The generated mesh was then exported to Fluent software to solve the governing equations.
The grid independence study was performed to answer that all the flow details are captured, employing the least number of elements that can yield accurate computational results. Four different mesh sizes, containing respectively G_{1} = 25756 nodes, G_{2} = 54345 nodes, G_{3} = 64784 nodes and G_{4} = 75981 nodes, were generated in order to study the effect of the grid spacing on the flow characteristic. Figure 2 illustrates the mesh independence test by plotting the temporal evolution of the drag and of the lift coefficients obtained at Re_{ }= 150 and A = 0 m. It is clear that the results of meshes G_{3} and G_{4} agree fairly well with each other verifying that the mesh independence was achieved from mesh G_{3} and further refinement is not necessary. Finer grid spacing may give a more accurate solution; however, refinement of the accuracy of computation will be small after the number of grid points is increased beyond a certain level. At the same time, computational cost becomes larger. In the present computation, the G_{3} grid system was adopted based on the tradeoff between the accuracy and the cost of the computation.
Figure 3 shows the grid generation around the flexible membrane for one period of membrane motion at A = 10^{−3}m and f = 4.5 Hz. This grid is adapted to the dynamic mesh model given by Fluent. In fact, the connectivity between nodes is considered as a network of springs, and a spring force is applied for each displacement at a boundary node. According to the model of Hooke's law, the spring force applied for each displacement at a boundary node, is given by the following expression: (6) where , are the displacements of node neighbor j and i.
n_{i} The number of nodes connected to i
where the spring factor is between 0 and 1.
It should be noted that this moving mesh is used in each iteration.
Concerning the time integration, the second order implicit scheme is used. The current CFL condition Δt ≪0.1 s is verified by Δt = 0.005 s. We have also simulated the flow with Δt = 0.0005 s and no significant change in the numerical results were observed. A parallel processing has been investigated to save the computation time using a subdivision of the grid size into 4 processors.
Fig. 2 (a) Instantaneous drag coefficient for different mesh sizes (Re = 150 and A = 0 m). (b) Instantaneous lift coefficient for different mesh sizes (Re = 150 and A = 0 m). 
Fig. 3 Dynamic mesh of flexible membrane for f = 4.5 Hz and A = 10^{−3} m_{.} 
3.2 Validation of the present numerical model
To validate the present computational procedure, a preliminary calculation was carried out for Re=150 and an amplitude of the membrane A = 0 m. We note that for this amplitude, the membrane appears to be at rest. The Strouhal number St, the mean drag coefficient ⟨C_{d}⟩ and the amplitude of the C_{l} oscillation (C_{lmax}) are compared in Table 1 with those of Golani et al. [27], Farhoud et al. [28] and, Wu et al. [3]. Results obtained with our computational model have a maximum deviation of the order of 3.5%, 6.8% and 4.5% on the Strouhal number, the mean drag coefficient and the amplitude of the C_{l} oscillation respectively. In Figure 4 we compare the present temporal traces of the lift coefficient with those of Golani and Dhiman [27]. Our results are in good agreement with those found in the literature. We can conclude that our numerical model can predict correctly the flow across a circular cylinder by introducing a flexible membrane attached to the backside of the bluff body.
Validation of Strouhal number and lift coefficient results with published values at Re = 150.
Fig. 4 Lift coefficient for controlled and uncontrolled case. 
4 Results and discussion
In this section, we represent the results of the parametric study at Re = 150. The subsections are presented as follow: in Section 4.2, we investigate the effects of the frequency on velocity fields and on the aerodynamic forces, in Section 4.3 we show the main results of varying the amplitude of the flexible membrane, in Section 4.4 we present the effect of size of the flexible membrane on the global flow parameters.
4.1 Qualitative study of uncontrolled wake behavior
The vortex velocity fields around uncontrolled circular cylinder are presented in Figure 5 for Re = 150. The fluid at the inlet moves slowly towards the circular cylinder and slides over its walls, thereby creating a boundary layer. The detachment of the boundary layer begins from the separation points. As the fluid moves slowly, the length of the wake increases from the shear layer and a symmetric mode appears showing two recirculation zones (Fig. 5(a) and (b)). After some iterations, an antisymmetric mode appears due to the sliding motion (Fig. 5(c)). The interaction of the eddies forms von Karman sheets as shown all over the other snapshots in (Fig. 5(e)–(h)). The rotating vortices are in opposite senses. This periodic motion induces a sinusoidal variation of average lift and drag coefficients (Figs. 2–4).
The effect of introducing a flexible membrane on the rear surface of the circular cylinder is discussed in the next section in term of wake topology and aerodynamic forces.
Fig. 5 instantaneous vorticity contours of uncontrolled cylinder for Re = 150. 
4.2 Frequency effect on wake behavior behind controlled cylinder
In this section, the effect of frequency oscillation of the flexible membrane is investigated while fixing other parameters such as amplitude and size of the membrane. The size of the membrane studied in this part is corresponding to an angle of aperture α = 8°. The flexible membrane attached to the rear surface of the cylinder is vibrating with a fixed amplitude equal to A = 10^{−3}m.
4.2.1 Qualitative study
A comparison of the effect of frequency control on the instantaneous vorticity fields around the circular cylinder versus the uncontrolled case is presented in Figure 6 into five snapshots. The instantaneous fields for the controlled cylinder are captured for a flexible membrane vibrating with a frequency f = 4.5 Hz, and amplitude A = 10^{−3}m.
From Figure 6, the flexible membrane can be seen physically in different ways. In fact, at given instant, the flexible membrane takes the form of circular arc groove or dimple [5]. Due to the timedependent motion of the membrane described in Section 2 by equation (1), the flexible membrane can be considered as multitudes of dimples. Previously Kimura and Tsutahara [5] have studied the effect of grooves, with different depths of dimples, practiced on the surface of a circular cylinder. Their study focused on velocity fields, separation points, and experimental visualizations. The aerodynamic forces have not been presented in that study due to the timeconsuming problem. In our numerical simulation, a parallel computing into 4 processors has been used to save the CPU time. The motion of the flexible membrane managed by equation (1) is also similar to a string fixed at both ends generating standing waves with a fundamental mode of oscillation. The particles of fluid stuck to the flexible membrane are considered as particles in a moving frame. Their velocities are different comparing to the rest fluid particles.
As the flexible membrane oscillates in the (ox) direction, the near fluid follows this motion creating a longitudinal wave as shown in Figure 7(a) and (b).
Fig. 6 Instantaneous vorticity fields for uncontrolled (I) and controlled (II) case with f = 4.5 Hz. 
Fig. 7 Instantaneous vorticity fields for forced flexible membrane motion showing longitudinal waves. 
4.2.2 Quantitative study
In this section, the oscillating frequency effect on the aerodynamic coefficients is studied with a constant amplitude A = 0.001 for a range of frequency varied from 0.1 Hz to 6 Hz. The size of the membrane is ℓ = 10^{−3}m. In order to quantify the variation of the aerodynamic coefficients for the cylinder with local oscillation with respect to rigid one, we have defined the growth rate for drag and lift coefficient as follow: where:
is the growth rate of the mean drag coefficient, and is the growth rate of the maximum of lift coefficient.
and are respectively the mean value of uncontrolled and controlled drag coefficient.
and are respectively the maximum value of uncontrolled and controlled lift coefficient.
The variation of the growth rate of the mean drag and the growth rate of the maximum of lift in terms of oscillating frequency are summarized in Table 2 and presented in Figure 8.
It is seen from Table 1 and Figure 8 that the growth rate of the maximum lift coefficient increases from 2.5 to 4.91% for f varying between 0.1 Hz and 0.8 Hz, and then decreases to 3.84% for f = 2 Hz. The maximum of is reached for a frequency of oscillation of the membrane f = 3.8 Hz, decreases for f = 4.5 Hz beyond which it increases again as f increases.
The curve of in terms of frequency has approximately an antisymmetric behavior comparing to the curve of . In fact, the growth rate of the mean drag coefficient drops progressively with increasing f from 0.5 Hz until reaching its minimal value 1.4% for f = 3.8 Hz and then increases to reach its maximum for f = 5.5 Hz. Therefore, the extreme values of the aerodynamic forces are obtained when the frequency of excitation of the flexible membrane coincides with the natural frequency of the drag coefficient vortex for the uncontrolled cylinder. This frequency is the double of the shedding frequency given by the Strouhal number since the drag coefficient varies twice as faster as the lift coefficient as shown in Figure 2(a) and (b).
In order to understand the behavior of drag and lift coefficient in response to the motion of the membrane, time signal traces have been represented in Figure 9 for selected frequencies {2 Hz, 4.5 Hz, 6 Hz} where a noticeable variation of curves in terms of flow time has been detected.
It is noticeable from Figure 9 that the lift coefficient conserves the same curve behavior for all the values of frequency control. However, the drag coefficient seems to oscillate between distinct states. It is important to mention that the curve of drag coefficient for f = 3.8, which not has been presented here, looks stable and similar to its trace in the uncontrolled case since this frequency of control coincides with the frequency of drag for the uncontrolled case. From these observations, it can be seen that a beat phenomenon is affecting the drag coefficient for a circular cylinder with a local flexible membrane. A spectrum analysis of drag coefficient presented in Figure 10 demonstrates the existence of two different frequencies.
The fast Fourier transformation, as shown in Figure 10, of the drag coefficient clearly demonstrates the presence of two different shedding frequencies with different magnitudes. The primary frequency is approximated to the frequency of excitation of the flexible membrane, while the second frequency is equal to the frequency of drag for an uncontrolled circular cylinder for Reynolds number equal to 150. From the temporal evolution and the spectral decomposition given respectively by Figures 9 and 10, an empirical relation of the drag coefficient was established. where ⟨C_{d}⟩ is the mean value of the drag coefficient, a_{1}, a_{2} amplitudes, f_{1}, f_{2} frequencies and ϕ_{1}, ϕ_{2} the phases. The values of these terms are given in Table 3.
From Table 3 we observe that the mean drag value conserves the value of 1.52 for all the excitation's frequencies. The term of a_{1}sin(2πf_{1}t + ϕ_{1}) conserves also the same value of (a_{1} = 0.022) and (f_{1}=3.86 Hz) for all the cases of control, however, the term a_{2}sin(2πf_{2}t + ϕ_{2}) behave differently. In fact, the frequency f_{2} locks on the frequency of excitation of the membrane for all the observed cases (f_{2} = f). The amplitude a_{2} increases and it becomes superior to a_{1} for f > f_{1}. For f = 4.5 Hz, a_{2} becomes equal to a_{1} and the beat phenomenon is clearly observed (Fig. 9). Therefore, we can analyze the total variation of the drag coefficient for the cylinder with a local vibrating membrane as the superposition of two drag forces; one is related to the forced term a_{2}sin(2πf_{2}t + ϕ_{2}) and the other to the unforced term a_{1}sin(2πf_{1}t + ϕ_{1}). As the frequency of control is inferior to the drag natural frequency f = 3.8 Hz, the unforced term is predominant (the amplitude a_{1} = 0.022 > a_{2} = 0.005 for f = 2 Hz). By increasing the frequency of oscillation of the vibrating membrane, the forced term becomes predominant. Consequently, the fluid excited by the vibrating membrane can be observed like a coupled oscillator forced by two elastic forces with two frequencies f_{1} and f_{2}. Thus, the fluid behaves as a nonlinear oscillator as shown in Figures 8 and 9.
Growth rate of drag and lift coefficient for active control.
Fig. 8 Influence of frequency of control on the growth rate of the aerodynamic forces. 
Fig. 9 Influence of the frequency on the temporal variation of the drag and lift coefficients. 
Fig. 10 FFT of the temporal evolution of the drag coefficient for different frequencies of flexible membrane. 
Values of mathematical parameters of the drag coefficient for different controlled frequencies.
4.3 Amplitude effect of the flexible membrane
In this section, we study the effect of the oscillation amplitude of the flexible membrane, sized of ℓ = 10^{−3}m, on the aerodynamic forces for a fixed excitation frequency f = 4.5 Hz.
Figure 11 shows the drag and lift variations for a range of excitation amplitudes A equal to (ℓ, 0.5 ℓ, 0.6 ℓ, 0.7 ℓ) where ℓ = 10^{−3}m.
The drag coefficient increases linearly from 1.3 to 12% when the amplitude of oscillation of the flexible membrane increases from 0.5ℓ to ℓ. The beat phenomenon persists even for the lowest value of amplitude vibration A = 0.7× ℓ. The lift coefficient is insensitive to the oscillation amplitude of the membrane. Some research shows a variation of the drag curve for the turbulent case called as drag crisis [29] for the uncontrolled flow around a circular cylinder. However, in laminar flow and at Re = 100, a similar behavior for lift coefficient has been observed by Kim et al. [30] by putting two blowing suctions in the upper and lower side of the cylinder parallel to the lift direction. From previous works and our study, we can predict that a perturbation near a region of the wake would affect the force parallel to its direction. In fact, the control energizes the particles of fluid near the flexible membrane in our case. Placed at the rear surface of the controlled cylinder, the motion of the vibrating membrane has created a pressure gradient in the supposed to be a low pressure region before the control. This pressure gradient is responsible for the creation of the forced drag oscillating with the frequency f_{2}.
Fig. 11 Effect of amplitude on the instantaneous drag and lift coefficients. 
4.4 Size effect of the flexible membrane
In this section, we have investigated the effect of the size of the membrane on aerodynamic forces. Three different sizes of the flexible membrane have been studied (ℓ_{1}, ℓ_{2} = 2 ℓ _{1}, ℓ_{3} = 3 ℓ _{1}) where ℓ_{1} = 10^{−3}m, corresponding respectively to aperture angle α = (8^{°}, 16^{°}, 32^{°}).
Figure 12(a) and (b) show the variation of the maximum value of lift coefficient with respect to the frequency of excitation of flexible membrane sized (ℓ_{2} = 2 ℓ _{1}) and (ℓ_{3} = 3 ℓ _{1}) respectively. The study of membrane sized ℓ_{1} have been investigated in Section 4.1. For a vibrating membrane sized of ℓ_{2} = 2 ℓ _{1}, the lift coefficient is increased more than 34%, relatively to the uncontrolled case, for a range of frequency f = (0.1 Hz, 4.5 Hz.). The maximum of lift coefficient occurs for f = 4.5 Hz. For a vibrating membrane sized of ℓ_{3} = 3 ℓ _{1}, the lift coefficient increased by 39.15%, relatively to the uncontrolled case, for f = 3.8 Hz. Comparing the given results with those found in Section 4.1 by a membrane sized ℓ_{1}, we notice that an enhancement of the lift coefficient is related to the size of the membrane. The maximum growth rate of the lift coefficient is increased from 7% to 39.15% when the size of the membrane increases from ℓ_{1} to ℓ_{3} = 3 ℓ _{1}.
Figure 13 shows the influence of the size of the membrane on the drag trace for a constant frequency f = 4.5 Hz. For the given frequency, we notice that beat phenomena, previously observed for a membrane sized ℓ_{1}, persists even by increasing the size of the flexible membrane.
We notice from Figure 13 that the amplitude of the drag coefficient has increased linearly with respect to the size of the flexible membrane. A growth of approximately 10% in the amplitude of the drag coefficient is observed when increasing the size of the membrane from ℓ_{1} to ℓ_{3} = 3 ℓ _{1}_{.} We conclude that the beat phenomenon could affect the drag coefficient even for the smallest perturbation (membrane sized ℓ_{1}).
Fig. 12 Effect of frequency control on the maximum lift variation (a) vibrating membrane sized ℓ_{2} = 2 × ℓ _{1} (b) vibrating membrane sized ℓ_{3} = 3 × ℓ _{1}. 
Fig. 13 beat phenomenon at f = 4.5 Hz for different vibrating membrane size: (ℓ_{1}, ℓ_{2} = 2 ℓ _{1}, ℓ_{3} = 3 ℓ _{1}). 
5 Conclusion
In this paper, we have studied the effect of a flexible membrane attached to the rear side of a circular cylinder as an active control at a fixed Reynolds number equal to 150. To this end, we investigated numerically the effect of amplitude, frequency, and size of the flexible membrane on the aerodynamic forces and in the vortex fields. Three different sizes have been compared (ℓ _{1}, ℓ _{2} = 2 ℓ _{1}, ℓ _{3} = 3 ℓ _{1}) where ℓ_{1} = 10^{−3}m, corresponding to an angle of aperture α = (8^{∘}, 16^{∘}, 32^{∘}) respectively. The flexible membrane is forced to move as a vibrating chord leading to the appearance of longitudinal waves. The equation of the motion of the membrane is implemented into ANSYS Fluent as a userdefined function. A dynamic mesh was chosen to update the grid size deformed due to the motion of the membrane. The computational domain is considered as a deforming zone. A parallel processing has been investigated to save the computation time using a subdivision of the grid size into 4 processors. A primary study was conducted for a flexible membrane sized ℓ_{1}. The parametric study shows that the drag curves has changed significantly compared to the uncontrolled case for a range of frequency f from 0.1 to 6 Hz and an oscillation amplitude of the membrane A = ℓ _{1} = 10^{−3}m. A spectral decomposition shows two frequencies present in each case of control, which are respectively the natural vortex frequency related to the Strouhal number and the excitation frequency of the membrane. The beat phenomenon is clearly observed in the drag curve variation for f around 4.5 Hz. The effect of the oscillation amplitude of the membrane on lift and drag coefficient is studied while fixing other parameters (frequency f = 4.5 Hz, size ℓ_{1}). Results show that for a range of amplitude varying from 0.5 ℓ _{1} to ℓ_{1}, an increase of 12% and 1.3% are observed, respectively in the maximum and in the mean drag variation of a flexible membrane sized ℓ_{1}. The influence of the size of the flexible membrane have been investigated for a constant amplitude A = ℓ _{1} = 10^{−3}m. An enhancement of 39.15% in the maximum variation of lift coefficient has been observed for a flexible membrane sized ℓ_{3} = 3 ℓ _{1} = 3 × 10^{−3}m comparing to the uncontrolled case. The beat phenomenon clearly observed for the drag coefficient for the frequency f = 4.5 Hz persists when increasing the size of the membrane and a growth of approximately 10% in the amplitude of the drag coefficient has been observed.
We conclude that the frequency and size are the main parameters of the control. The longitudinal motion of the flexible membrane affects the aerodynamic force parallel to its direction. Further study will be carried out considering membrane location, and modes of vibration of the flexible membrane.
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Cite this article as: Nabaouia Maktouf, Ali Ben Moussa, Saïd Turki, Numerical investigation of the vibration effect of a flexible membrane on the flow behaviour around a circular cylinder, Eur. Phys. J. Appl. Phys. 81, 21101 (2018)
All Tables
Validation of Strouhal number and lift coefficient results with published values at Re = 150.
Values of mathematical parameters of the drag coefficient for different controlled frequencies.
All Figures
Fig. 1 Computational domain. 

In the text 
Fig. 2 (a) Instantaneous drag coefficient for different mesh sizes (Re = 150 and A = 0 m). (b) Instantaneous lift coefficient for different mesh sizes (Re = 150 and A = 0 m). 

In the text 
Fig. 3 Dynamic mesh of flexible membrane for f = 4.5 Hz and A = 10^{−3} m_{.} 

In the text 
Fig. 4 Lift coefficient for controlled and uncontrolled case. 

In the text 
Fig. 5 instantaneous vorticity contours of uncontrolled cylinder for Re = 150. 

In the text 
Fig. 6 Instantaneous vorticity fields for uncontrolled (I) and controlled (II) case with f = 4.5 Hz. 

In the text 
Fig. 7 Instantaneous vorticity fields for forced flexible membrane motion showing longitudinal waves. 

In the text 
Fig. 8 Influence of frequency of control on the growth rate of the aerodynamic forces. 

In the text 
Fig. 9 Influence of the frequency on the temporal variation of the drag and lift coefficients. 

In the text 
Fig. 10 FFT of the temporal evolution of the drag coefficient for different frequencies of flexible membrane. 

In the text 
Fig. 11 Effect of amplitude on the instantaneous drag and lift coefficients. 

In the text 
Fig. 12 Effect of frequency control on the maximum lift variation (a) vibrating membrane sized ℓ_{2} = 2 × ℓ _{1} (b) vibrating membrane sized ℓ_{3} = 3 × ℓ _{1}. 

In the text 
Fig. 13 beat phenomenon at f = 4.5 Hz for different vibrating membrane size: (ℓ_{1}, ℓ_{2} = 2 ℓ _{1}, ℓ_{3} = 3 ℓ _{1}). 

In the text 
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