Journal of Marine Science and Technology https://doi.org/10.1007/s00773-017-0503-5

ORIGINAL ARTICLE

Numerical simulations of steady flow over a circular cylinder near a boundary Linwei Shen1 · Yan Wei1 Received: 8 April 2017 / Accepted: 15 November 2017 © JASNAOE 2017

Abstract A well-developed numerical scheme is used to simulate the steady current over a circular cylinder in proximity with a rigid plane at low Reynolds numbers. The numerical results, including the hydrodynamic forces acting on the cylinder and the flow fields, are obtained and discussed. The reduction of the vorticity generated at the cylinder surface of the gap side is attributed to the decrease of the amplitude of harmonic force components and eventually causes the complete suppression of the vortex shedding process. Particularly, a distinct flow scenario characterized by the agglomeration of two adjacent vortices takes place in the wake when the cylinder is positioned close to the free-slip wall just before the complete cessation of the vortex shedding. This finding might be served to explain the disappearance of regularly arranged vortices observed when the cylinder was towed above the stationary wall in the experiment. In contrast, the flow with a no-slip plane is also considered. The effective gap between the cylinder and the plane is reduced due to the presence of the boundary layer near the plane, and earlier cessation of vortex shedding is observed in comparison with that in the situations of free-slip plane. Keywords  Immersed boundary method · Vortex shedding · Hydrodynamic forces · Strouhal number · Plane wall

1 Introduction The interactions between a steady current flow and a circular cylinder in proximity to a boundary have a lot of engineering applications, such as pipelines’ design in offshore industry. A lot of effort has been made both experimentally and numerically in an attempt to understand the flow around the cylinder near the boundary, but there is still a considerable uncertainty about the flow [1, 2]. There are a number of published papers on the study of fluid flow around a circular cylinder close to a solid plane wall. In his work, Taneda [3] carried out the experiment by towing a circular cylinder close to a wall at a Reynolds number of 170. He observed a single row of vortices at the gap ratio G∕D = 0.1 , where G is the gap height between the cylinder and the wall and D the cylinder diameter, while two rows of vortices were found for G∕D = 0.6 . Bearman * Yan Wei [email protected] Linwei Shen [email protected] 1



Ocean College, Zhejiang University, Zhoushan Campus, Zhoushan 316021, People’s Republic of China

and Zdravkovich [4] conducted the experiments in a wind tunnel at two Reynolds numbers of 2.5 × 104 and 4.8 × 104 . Their spectral analysis of hot-wire signals showed that the regular vortex shedding was suppressed for G∕D < 0.3 and the Strouhal number around 0.2 was found for all gap ratios larger than 0.3. The experiment was extended by Taniguchi and Miyakoshi [5] to study the effects of the wall boundary layer thickness on the hydrodynamic forces acting on the cylinder at the Reynolds number of 9.4 × 104 . They found that the fluctuating forces were sharply decreased at some critical gap ratios, which were related with the boundary layer thickness at the plane wall, and suggested that suppression of vortex shedding at small G/D was due to the cancellation of vorticity generated on the wall and that shed from the cylinder. Lei et al. [6] continued the experimental investigation at Reynolds numbers ranging from 1.30 × 104 to 1.45 × 104 . They proposed the root-mean-square (RMS) lift coefficient to identify the onset or suppression of the vortex shedding, and their experiments showed that the vortex shedding was suppressed at a gap ratio of about 0.2–0.3, depending on the thickness of the boundary layer. In their laboratory investigation, Price et al. [1] made use of flow visualization, particle image velocimetry (PIV) and hotfilm anemometry to study the same problem at Reynolds

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numbers between 1200 and 4960. Four distinct regions of the gap ratios were defined in terms of the characteristics of the fluid flow around the cylinder. Apart from the laboratory study, the above-mentioned problem was also examined by numerical simulations. For example, the two-dimensional (2D) Navier–Stokes (N–S) equations were solved in a curvilinear coordinate system by Lei et al. [7] at the Reynolds numbers ranging from 80 to 1000. Kazeminezhad et al. [8] studied the effects of the bed proximity and boundary layer thickness on the forces and vortex shedding frequency by solving the Reynolds-averaged N–S equations in 2D. In summary, the flow patterns and the hydrodynamic loading on the cylinder are affected by the presence of a rigid boundary. However, the mechanism behind the suppression of the Karman vortex shedding and the effect of the boundary on the flow in the wake needs to be further explored. Moreover, the possible occurrence of flow irregularity behind the cylinder when it was towed very close to the stationary plane wall in laboratory experiment [9] has not been addressed. In this paper, numerical simulations are carried out to study the flow passing over a stationary circular cylinder in proximity to a rigid wall. The Reynolds numbers of 150, 170 and 200 are considered so that the 2D flow assumption is reasonable. The Reynolds number is defined as Re = U∞ D∕𝜈, where U∞ is the upstream velocity, D the cylinder diameter, and 𝜈 the kinematic viscosity coefficient. In fact, the numerical results at the low Reynolds numbers might share some common features exhibited at high Reynolds numbers and, therefore, may be served to give insight into the nature of the flow phenomena. A careful study is firstly conducted for the case of the rigid wall imposed with a free-slip boundary condition. More complicated cases of a solid wall with a no-slip boundary condition are then presented for further investigations. The immersed boundary (IB) method [10] is Fig. 1  Computational domain for steady flow around a stationary circular cylinder placed close to a rigid wall. A free-slip boundary condition 𝜕u∕𝜕z = w = 0 or no-slip boundary condition u = w = 0 is imposed on bottom wall

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used to deal with the no-slip boundary condition imposed on the cylinder. In Sect. 2, a brief description is given about the mathematical model and the numerical method as well. The numerical results are presented and discussed in Sect. 3. Concluding remarks are given in Sect. 4.

2 Mathematical model and numerical method A solid circular cylinder of diameter D is considered and its center is located at a distance of h from a nearby plane wall, sketched in Fig. 1. Cartesian coordinates x and z are also introduced in the figure. The flow of an incompressible viscous fluid is governed by the N–S equations:

𝜕u 𝜕u 𝜕u 1 𝜕p +u +w =− + v∇2 u, 𝜕t 𝜕x 𝜕z 𝜌 𝜕x

(1)

𝜕w 𝜕w 𝜕w 1 𝜕p +u +w =− + v∇2 w + g 𝜕t 𝜕x 𝜕z 𝜌 𝜕z

(2)

and the continuity equation:

𝜕u 𝜕w + = 0, 𝜕x 𝜕z

(3)

where u and w are the velocity components along axes x and z, respectively, p is the pressure, t the time, 𝜌 the water density, and g the gravitational acceleration. In the simulations, the water is initially at rest. The boundary conditions are given in Fig. 1 for this particular case. In the framework of the IB method, the solid body of circular cylinder is replaced with a proper force term being imposed on the body surface. Consequently, the N–S equations become:

Journal of Marine Science and Technology Table 1  Flow variables comparison for uniform flow over an isolated circular cylinder at Re = 150

Source

Method

C̄ D

CDAmp

CLrms

CLAmp

St

Mesh system 1 (Δmin = 0.01D) Mesh system 2 (Δmin = 0.02D) Williamson [16] Hammache and Gharib [17] Zhang et al. [18] Persillon and Braza [19] Posdziech and Grundmann [20]

Num. Num. Exp. Exp. Num. Num. Num.

1.357 1.353 – – 1.402 1.268 1.313

0.025 0.026 – – – – –

0.364 0.353 – – 0.403 0.169 –

0.547 0.518 – – – – –

0.186 0.186 0.184 0.186 0.191 0.181 0.183

𝜕u 𝜕u 𝜕u 1 𝜕p +u +w = + 𝜈∇2 u + fx , 𝜕t 𝜕x 𝜕z 𝜌 𝜕x

(4)

𝜕w 𝜕w 𝜕w 1 𝜕p +u +w =− + 𝜈∇2 w + g + fz , 𝜕t 𝜕x 𝜕z 𝜌 𝜕z

(5)

A SIMPLEC-type two-step computational scheme is used to solve the N–S equations. The detailed scheme, including the determination of the force term, was presented in Shen and Chan [11].

where fx and fz are the force term components, which are zero everywhere except on the body surface.

Fig. 2  a Time series of drag and lift coefficients acting on a circular cylinder close to a freeslip rigid boundary at h∕D = 1.5 under Re = 150 , and b their associated spectra

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Fig. 3  Harmonic components of fluctuating forces acting on a circular cylinder close to a freeslip boundary at various h∕D

3 Numerical results and discussion The numerical simulations are carried out by the IB method, which has been well developed and validated. For example, the same numerical scheme was successfully applied to the simulations of fluid–structure interactions [11]–15]. In this paper, one more validation of a steady flow around an isolated circular cylinder at Re = 150 is carried out. The numerical results are compared with published data in Table 1 and a reasonable agreement is achieved. The lift 2 and drag coefficients are defined as CL = Fz ∕0.5𝜌U∞ D and 2  , where CD = Fx ∕0.5𝜌U∞ D Fx and Fz are the hydrodynamic forces acting on the object in x and z directions, respectively, and the Strouhal number is St = fL D∕U∞ , where fL is the vortex shedding frequency obtained from the fluctuating lift. It is noted that two sets of the mesh system, with different

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minimum grid sizes near the cylinder, are able to produce stable results. Figure  1 illustrates the computational domain of 38D × 20D . A relatively large domain is used in the computation so as to minimize the impact of the top and outflow boundary conditions. Additional grid convergence tests are also carried out for various Reynolds numbers considered in the paper. A uniform grid size of Δx = Δz = 0.01D is employed in the vicinity of the solid boundaries where the no-slip boundary conditions are imposed. The time step is determined by satisfying the criterion of Courant–Friedrichs–Lewy (CFL) number of 0.1.

3.1 Circular cylinder near a free‑slip rigid wall Flow around the cylinder close to a free-slip wall is relatively simple due to the exclusion of the boundary layer effects.

Journal of Marine Science and Technology Fig. 4  Time-averaged hydrodynamic forces as a function of h∕D . a C̄ D and b C̄ L

a

b

Fig. 5  Variation of Strouhal number as a function of h∕D

However, it has some practical analogy to the object moving in fluid at rest near the rigid boundary and two equalsized side-by-side cylinders as well. Moreover, this result may help explain the phenomena observed in the laboratory, such as the vortex shedding suppression and the irregularity of vortices in the wake when the object was towed very close to the stationary wall. 3.1.1 Hydrodynamic forces on cylinder Figure 2 shows the hydrodynamic forces acting on the cylinder and their associated frequency spectra at h∕D = 1.5

and Re = 150 . The dominant frequency of fL is found in the fluctuating lift coefficient CL (t) and its harmonic component amplitude is denoted by CL(1) . In contrast, two peaks are obtained in the spectrum for the fluctuating drag coefficient, and CD(1) and CD(2) represent the amplitudes of two harmonic components corresponding to the frequency fD(1) and fD(2) , respectively. It is noted that fD(2) = 2fD(1) = 2fL. Figure 3 shows the variation of CL(1) , CD(1) and CD(2) as a function of h∕D for the Reynolds number of 150, 170 and 200. The value of CL(1) at h∕D = 4 and Re = 150 is 2.4% more than that for an isolated cylinder. The values of CL(1) , under Re = 150 , 170 and 200, continue to increase gradually as

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Fig. 6  Vorticity field and streamlines for a circular cylinder close to a free-slip rigid boundary at a h∕D = 0.75 , b h∕D = 0.67 and c h∕D = 0.63 under Re = 150 . The difference of the stream functions representing two neighboring streamlines is Δ𝜓 = 2.5 × 10−5

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h∕D is decreased and reach their maximums near h∕D = 1.1 . These values fall sharply when h∕D is smaller than 0.7. The values of CD(1) are negligibly small at h∕D = 4 and increase as the cylinder is positioned closer to the wall. These values start to drop rapidly after the maximums reached around h∕D = 0.7 . On the other hand, CD(2) does not vary significantly in the range of 1 ⩽ h∕D ⩽ 4 under these Reynolds numbers. These values of CD(2) decrease when h∕D is smaller than 1.3 and fall rapidly after the small kink at h∕D = 0.7. It is well known that in the uniform steady flow over an isolated circular cylinder at the low Reynolds number such

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as 150, 170 and 200, the only frequency detected from the drag force is twice the vortex shedding frequency that from the lift force. However, the appearance of the harmonic component of CD(1) , particularly when h∕D is smaller than 2, indicates the difference of the vortices shedding alternately from the upper and the lower side of the cylinder into the wake. Apparently, this stems from the presence of the rigid boundary. Besides, the rapid reduction of the amplitude of these harmonic components is a clear signal of the attenuation of vortex shedding process when h∕D is smaller than 0.7. This

Journal of Marine Science and Technology Fig. 6  (continued)

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is attributed to the reduction of the vorticity contained in the boundary layer at the lower side of the cylinder. Figure 4 displays the time-averaged force coefficients at various values of h∕D . The drag coefficient C̄ D increases gradually when the cylinder moves closer to the wall, and turns to decline quickly after the maximum values near h∕D  =  0.8–0.9 for different Reynolds numbers. On the other hand, the lift coefficient C̄ L increases monotonously as h∕D is decreased. The Strouhal number, shown in Fig. 5, slightly increases as h∕D is decreased until it reaches the maximum at h∕D = 1.2 . The value declines rapidly when h∕D is smaller than 0.8. 3.1.2 Fundamental flow fields The instantaneous vorticity fields, together with their associated streamline topologies at h∕D = 0.75 , 0.67 and 0.63 under Re = 150 are exhibited in Fig. 6. The effect of the nearby free-slip plane boundary on the attenuation of the vortex shedding is clearly demonstrated. A process of vortices shedding from both sides of the cylinder is observed at h∕D = 0.75 , with a clear dominance of vorticity shed from the free stream side in the wake. A single street of vortices with positive vorticity is observed at h∕D = 0.67 . There is no regular vortex shedding at h∕D = 0.63. Figure 7 shows the circulation of vorticity concentration as a function of h∕D when the concentration center, which is identified by the peak vorticity, passes the section of (x − x0 )∕D = 5 . For example, the vorticity concentration

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labeled by ‘A’ in Fig. 6a has the non-dimensional circulation Γ∗L ≈ −0.73 , where Γ∗L = ΓL ∕𝜋U∞ D and ΓL = ∬ 𝜔y dxdz , the subscript ‘L’ represents the vorticity shed from the lower side of the cylinder. It is found that Γ∗U and Γ∗L have very close absolute values in the range of h∕D from 4 to 1.5, but a sharp change of both the values, i.e. an increase in Γ∗U and | | a decrease in |Γ∗L | , occurs as h∕D is smaller than 1. This | | result indicates that the existence of the free-slip boundary may start to suppress the vortex shedding from the lower side of the cylinder approximately at h∕D = 1 , but the reduc| | tion of |Γ∗L | is compensated to a degree by the increment of | | Γ∗U as a result of the increased shear stress at the upper side | | of the body surface. In fact, the variation of Γ∗U + |Γ∗L | illus| | trated in Fig. 7c reaches the maximum at h∕D = 0.78–0.8 under Re = 200. 3.1.3 Secondary flow fields It is interesting to see a distinct scenario in the wake just before the complete cessation of the vortex shedding from the cylinder when it is very close to the free-slip wall. Figure 8 demonstrates the snapshots of the vorticity field at h∕D = 0.64 and Re = 200 . The corresponding streamline topologies are displaced in Fig. 9. At the dimensionless time t∕T = 282.90 , a vortex with negative vorticity, labeled by ‘F’, is observed behind the cylinder. Meanwhile, the boundary layer with positive vorticity separates from the upper side of the cylinder and forms a new vortex, labeled by ‘B’. Particularly, the flow in the wake features the agglomeration

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Fig. 7  Circulation of vorticity concentration, which is shed from the cylinder surface, passing the section of (x − x0 )∕D = 5 , where x0 is the x-axis coordinate of the cylinder center. The subscripts ‘U’ and ‘L’ represent the vorticity shed from the upper and lower side of the cylinder, respectively

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a

b

c

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of two adjacent vortices into a large-scale vortex, evidently demonstrated in Fig. 9a. The vortex ‘B’ grows in strength and size at t∕T = 286.70 . Simultaneously, a new smaller vortex with negative vorticity, tagged by ‘D’, is observed. This vortex increases in strength as a result of its merging with the negative vorticity ‘F’ into a larger-sized one, labeled by ‘D’ in Fig. 8d. During the period from t∕T = 292.42 to 296.94, a new vortex labeled by ‘E’ forms from the curlup of the boundary layer with positive vorticity which is

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separated from the upper side of the cylinder. Again, an agglomeration of two adjacent positive vortices occurs in the wake. It is noted that two neighboring vortices, namely vortex ‘A’ and ‘B’ in Fig. 8b, have distinguishably different vorticity concentration and position when they pass the same section, at x = 0.3 m for instance. This might be the result of the combination of two negative vortices ‘D’ and ‘F’, and also the cause of the agglomeration of these two positive vortices.

Journal of Marine Science and Technology Fig. 8  Snapshots of instantaneous vorticity field for a circular cylinder close to a free-slip rigid boundary under Re = 200 at h∕D = 0.64 at dimensionless time a t∕T = 282.90 , b t∕T = 286.70 , c t∕T = 289.08 , d t∕T = 290.99 , e t∕T = 292.42 , f t∕T = 294.08 , and g t∕T = 296.94

The similar process of the wake evolution is also observed at h∕D = 0.63 and 0.65 under Re = 200 , and at h∕D = 0.65 and 0.66 under Re = 170 . This may explain, to some extent, the disappearance of the regular arrangement of vortices behind the towing cylinder just before the complete suppression of vortex shedding, which was observed in the laboratory visualization in Zdravkovich [9]. It is noted that the boundary layer effects from the wall was excluded in the experiments, in the similar fashion with these particular cases.

The frequency spectra of the fluctuating forces at h∕D = 0.64 and Re = 200 are shown in Fig. 10. Apart from the fundamental frequency fL , corresponding to the vortex shedding, an additional one, half the value of fL , is found in both the lift and the drag coefficients. The appearance of the new smaller frequency is believed to have some correlation with the agglomeration of two adjacent vortices described above.

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Fig. 8  (continued)

3.2 Circular cylinder near a no‑slip rigid wall The no-slip boundary condition imposed on the bottom will give rise to the development of the boundary layer there and its separation as well. Consequently, the flow patterns in the wake of the cylinder will be different from that presented above. Figure 11 shows the variation of CL(1) , CD(1) and CD(2) as a function of h∕D under Re = 150 and 200. The harmonic component amplitude CL(1) , which dominates the fluctuating lift coefficient, increases slightly as h∕D is decreased from 4, and then drops smoothly as h∕D is reduced further from 2.5 to 2 for the Reynolds number of 150 and 200, respectively. The component CD(1) appears at h∕D = 4 and reaches the maximum at h∕D = 1.5 for Re = 150 and at 1.4 for Re = 200 . The component CD(2) remains almost unchanged in the range of h∕D from 4 to 2, and then decreases as the cylinder is placed closer to the wall. It is easy to understand that the boundary layer developed on the bottom reduces the effective gap between the cylinder and the bottom. Consequently, it becomes earlier

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for the presence of the plane wall to take effect when the cylinder approaches closer to the boundary. Meanwhile, that the fluctuating force amplitudes, lift force in particular, are smaller than the counterparts with the free-slip boundary might be partly ascribed to the reduction of the vorticity formed on the cylinder surface as well as the cancellation of the vorticity generated on the wall and that shed from the lower side of the object. Figure 12 shows the time-averaged hydrodynamic force coefficients at various h∕D under Re = 150 and 200. C̄ D increases gradually as h∕D is reduced from 4 to 2 and then decreases as the cylinder is positioned closer to the boundary. In comparison, C̄ L increases at first and then turns to drop as h∕D is reduced to 1.8 and 1.6 for Re = 150 and 200, respectively. Eventually, C̄ L increases as h∕D is smaller than 1.1 for Re = 150 and 1 for 200. This result is largely due to the fact that the cylinder seems to be immersed in the boundary layer when the cylinder is positioned very close to the bottom. This feature is consistent with experimental results reported or edited by Zdravkovich [21], Lei et al. [6] and

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Fig. 9  Instantaneous streamlines for a circular cylinder close to a free-slip rigid boundary under Re = 200 at h∕D = 0.64 at dimensionless time a t∕T = 282.90 , b t∕T = 286.70 , c t∕T = 289.08 , d t∕T = 290.99 , e t∕T = 292.42 , f t∕T = 294.08 , and g t∕T = 296.94 . The difference of the stream functions representing two neighboring streamlines is Δ𝜓 = 2.5 × 10−5

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Sumer and Fredsoe [22], in terms of the trend of the curve. The variation of the Strouhal number is shown in Fig. 13. Figure 14 exhibits the vorticity field and its associated streamline topologies under Re = 150 . Two rows of vorticity, positive from the free-stream side and negative from the other side, are formed in the wake at h∕D = 2.5 . The positive

vorticity generated at the bottom is mixed into the wake at a regular frequency of vortex shedding from the cylinder. At h∕D = 1.5 , a large amount of positive vorticity generated at the bottom is brought into the wake and the negative vorticity concentration shedding from the cylinder becomes elongated in shape, evidently shown in Fig. 14b. A single row of

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Fig. 9  (continued)

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positive vorticity concentrations is observed at h∕D = 1 and no periodic vortex shedding is found at h∕D = 0.9 . Clearly, the cylinder is immersed in the boundary layer of the bottom at smaller values of h∕D.

4 Conclusions The well-developed numerical scheme is applied to the simulations of steady current flow over a circular cylinder placed close to a rigid plane at the low Reynolds numbers in a systematic manner. The numerical results, including the hydrodynamic forces acting on the cylinder, instantaneous vorticity fields and streamline topologies near the cylinder, are obtained and discussed.

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The flow around the cylinder which is positioned in proximity with a free-slip boundary is studied first for its simplicity. The spectral analysis of the hydrodynamic forces shows that the component associated with the vortex shedding frequency is dominant in the fluctuating lift force at various values of h∕D . The appearance of the harmonic component of CD(1) , particularly when h∕D < 2 , indicates the flow difference induced by the vortices shedding alternately from the two sides of the cylinder into the wake in the presence of the boundary. Besides, the rapid reduction of the amplitude of these harmonic components is a clear signal of the attenuation of vortex shedding process when h∕D is smaller than 0.7. This is attributed to the reduction of the vorticity contained in the boundary layer at the lower side of the cylinder, which is confirmed by the quantitative investigation of the vorticity concentration in the wake.

Journal of Marine Science and Technology Fig. 10  Frequency spectrum for fluctuating a CD and b CL under Re = 200 at h∕D = 0.64 from free-slip boundary

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Fig. 11  Harmonic components of CL and CD acting on a cylinder close to no-slip boundary at various values of h∕D

Re=150 Re=200

0.06 0.04 0.02 0 0.5

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2 h/D

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a time-averaged CD

Fig. 12  Time-averaged hydrodynamic forces as a function of h∕D when the cylinder is close to no-slip rigid boundary. a C̄ D ; and b C̄ L

Journal of Marine Science and Technology

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Fig. 13  Variation of Strouhal number as a function of h∕D when the cylinder is close to no-slip rigid boundary

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The gradual attenuation of the vortex shedding process is also revealed by the flow fields when the cylinder is located very close to the free-slip plane. Moreover, a distinct scenario in the wake takes place when the cylinder is positioned just before the complete cessation of the vortex shedding. For example, at h∕D = 0.63–0.65 and Re = 200 , the flow is characterized by the agglomeration of two adjacent vortices which have distinguishably different amount of vorticity concentrations. Consequently, an additional harmonic component corresponding to half the vortex shedding frequency presents in the force frequency spectrum. This finding might be served to explain the disappearance of regularly arranged vortices in the wake observed in the laboratory

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experiment when the cylinder was towed above the stationary wall immediately before the complete suppression of the vortex shedding. In contrast, the flow is more complicated in the presence of the no-slip plane bottom. The boundary layer formed on the bottom reduces the effective gap between the cylinder and the boundary, and its separated vortices may reduce the amount of the vorticity generated at the lower side of the cylinder in opposite sign shed into the wake through cancellation. These features may be blamed for the smaller amplitude of the fluctuating force components and the earlier cessation of vortex shedding, in comparison with those under the free-slip boundary condition.

Journal of Marine Science and Technology Fig. 14  Vorticity field and streamlines for a circular cylinder close to a no-slip rigid boundary under Re = 150 at a h∕D = 2.5 , b h∕D = 1.5 , c h∕D = 1 , and d h∕D = 0.9 . The difference of the stream functions representing two neighboring streamlines is Δ𝜓 = 2.5 × 10−5

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In particular, that the cylinder seems to be immersed in the bottom boundary layer when it is positioned very close to the boundary may explain, to some degree, the variation

of the time-averaged lift coefficients as a function of h∕D , a remarkable feature consistent with experiments in terms of the trend of the curves.

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Journal of Marine Science and Technology

c

Fig. 14  (continued)

z(m)

-0.3

-0.35

-0.4

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.35

0.4

0.45

0.5

x(m)

d

z(m)

-0.3

-0.35

-0.4

0.2

0.25

0.3 x(m)

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