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Vibration of two elastically mounted cylinders of different diameters in oscillatory flow
Applied Ocean Research 69 (2017) 173–190
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Vibration of two elastically mounted cylinders of different diameters in oscillatory flow Toni Pearcey, Ming Zhao ∗ , Yang Xiang, Mingming Liu School of Computing Engineering and Mathematics, Western Sydney University, Penrith, NSW 2751, Australia
a r t i c l e
i n f o
Article history: Received 30 April 2017 Received in revised form 1 November 2017 Accepted 1 November 2017 Keywords: Flow induced vibration Numerical method Oscillatory flow Circular cylinder
a b s t r a c t Vibration of two elastically mounted cylinders in an oscillatory flow at a Keulegan-Carpenter number of 10 is simulated numerically. The two cylinders are rigidly connected with each other and are allowed to vibrate in the cross-flow direction only. The aim of this paper is to identify the effects of the orientation of the cylinders and the gap between the cylinders on the vibration. The two-dimensional ReynoldsAveraged Navier-Stokes equations are solved to predict the flow and the cylinder vibration is predicted using the equation of motion. When the two cylinders are in a tandem arrangement, a combined single pair flow regime and attached pair flow regime are observed as reduced velocity exceeds 10 and this combined regime and the single pair regime occurs intermittently. Periodic vibration is found when the two cylinders are in a staggered arrangement with a 45◦ flow attack angle. When the two cylinders are in a side-by-side arrangement, a new single vortex regime is observed. This single vortex remains attached to the cylinder surface and rotates around the cylinder. The intermittent switch between this single vortex regime and the single pair regime are observed. © 2017 Published by Elsevier Ltd.
1. Introduction With increasing demand for energy resources from ocean including offshore oil and gas, renewable energy from wind, tidal currents and waves, more and more offshore structures are constructed. Offshore structures for extracting energy from ocean have to survive severe storms without damaging their functionality. Many cylindrical structures are used in offshore engineering such as subsea pipelines, risers, mooring cables, etc. Oscillatory flow is often used to model the water motion due to waves when the impact of the waves on small scale cylindrical structures is studied. Many studies have been performed to understand the hydrodynamics and flow patterns around circular cylinders in oscillatory flows. It has been found that the hydrodynamic forces on cylindrical structures are mainly affected by the Keulegan-Carpenter (KC) number and the Reynolds number. The KC number is defined as KC = (Um T)/D, where Um and T are the velocity amplitude and period of the oscillatory flow, respectively, and D is the diameter of the cylinder. The Reynolds number is defined as Re = Um D/, where is the kinematic viscosity of the fluid. The ratio of the KC number to the Reynolds number is called the viscous parameter, ˇ [15].
∗ Corresponding author. E-mail address: [email protected] (M. Zhao). https://doi.org/10.1016/j.apor.2017.11.003 0141-1187/© 2017 Published by Elsevier Ltd.
[21] conducted an experimental study of oscillatory flow past a circular cylinder for KC numbers ranging from 1 to 40 and classified the vortex flow into different flow regimes: Paring of attached vortices (non-vortex shedding regime) when KC < 7, single pair regime when 7 < KC < 15, double pair regime when 15 < KC < 24, three-pair regime when 24 < KC < 32 and four–pair regime when 32 < KC < 40. Obasaju et al. [13] conducted a detailed study of the relationship between the vortex shedding regime and the hydrodynamic forces on a circular cylinder in an oscillatory flow. It was found that the spanwise correlation of the flow is good when KC is at the center of a regime and poor when KC is at the boundary between two regimes. Numerical studies have been successfully conducted to investigate oscillatory flow past a circular cylinder. Some studies are mainly focused on the inception of the three-dimensionality of flow at low Reynolds numbers and low KC numbers [3,1,17,4]. Recently, research has been performed to study flow induced vibration (FIV) of circular cylinders in oscillatory flows. In addition to the Reynolds number and the KC number, FIV of a cylinder in oscillatory flow is also dependent on the mass ratio and the reduced velocity. The mass ratio is defined as m* = m/md , where m is the cylinder mass and md is the displaced fluid mass, and the reduced velocity is defined as Vr = Um /(fn D), where fn is the structural natural frequency measured in vacuum in this study and many numerical studies. In many experimental studies of FIV in water, the natural frequency measured in still water (defined as fnw in this study) is used to define
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Nomenclature Amax CL d D fnw Fy fn fw G KC kT m m∗ md p Re Sij Sy T Um u ui uˆ i u i u j Vr xi Y Y˙ Y¨ ␣ ˇ ω ωf ωT
Maximum displacement Lift coefficient Small cylinder diameter Large cylinder diameter Natural structural frequency measured in water Lift force Structural natural frequency Oscillatory flow frequency Gap between two cylinders Keulegan-Carpenter number Turbulent kinetic energy Cylinder mass Mass ratio Displaced fluid mass Pressure Reynolds number Strain-rate tensor Displacement of the mesh Oscillatory flow period Velocity amplitude Sinusoidal velocity Velocity component in the xi -direction Mesh velocity Reynolds stress tensor Reduced velocity The ith Cartesian coordinate Displacement of the cylinders Velocity of the cylinders Acceleration of the cylinders Flow attack angle Viscous parameter Kinematic viscosity Fluid density Vorticity Angular frequency of the flow Specific dissipation of turbulence
the reduced velocity. Experimental studies of the vibration of an elastically mounted circular cylinder in oscillatory flow show that the vibration of the cylinder locks in with a frequency which is a multiple of the oscillatory flow frequency [18,9]. Two pipelines of different diameters are often bundled together in offshore engineering to form a so-called piggyback pipeline. The piggyback pipeline configuration ensures the stability of the small-diameter pipeline. Many studies have been performed to investigate two cylinders of different diameters in fluid flow [11,10,7,20,8]. The attachment of the shear layer from the gap to the back surface of the larger cylinder was observed when two cylinders of different diameters are in a side-by-side arrangement in a steady flow [20,19]. Rahmanian et al. [14] studied vortex inducedvibration of two cylinders of different diameters at Re = 8000 and found that the vibration of two cylinders is significantly different from that of a single cylinder. FIV of two cylinders of different diameters in an oscillatory flow, which is relevant to piggyback pipelines in waves in offshore engineering, has never been studied. In this study, FIV of two cylinders of different diameters as sketched in Fig. 1 (a) is studied numerically. The cylinders are allowed to vibrate only in the cross-flow direction. The diameters of the large and small cylinders are represented by D and d, respectively. The gap between the two cylinders and the flow attack angle is defined as G and ␣, respectively. The
main objective of this study is to understand the effect of the position of the small cylinder relative to the large cylinder on the FIV. Simulations are conducted for a constant diameter ratio of d/D = 0.2, a constant mass ratio of 2, a constant Reynolds number (based on D) of 2 × 104 , a constant KC number of 10, three gap ratios of G/D = 0, 0.1 and 0.2 and reduced velocities in a range of 1–20 with an interval of 1. A two-dimensional numerical model based on the RANS equations is used considering the large parametric space used in this study. A constant KC number of 10 is used, as the flow at this KC number presents the typical one pair flow regimes which occurs when 7 < KC < 15. This regime is the KC number range that piggyback pipelines generally experience in ocean engineering. 2. Numerical method The two-dimensional incompressible Reynolds-Averaged Navier Stokes (RANS) equations are used as the governing equations for simulating the flow. To account for the moving boundaries of the two cylinders, the Arbitrary Langrangian-Eulerian (ALE) scheme is applied to solve the RANS equations. The ALE scheme can avoid large deformations of the computational mesh because it allows the mesh to move with a velocity different from the fluid velocity. The effects of the mesh movement are considered by modifying the convection terms of the RANS equations. A Cartesian coordinate system is defined with its origin located at the center of the large cylinder and its x-coordinate pointing in the flow direction. The RANS equations in the ALE method are written as
∂ui 1 ∂p ∂ui ∂ + − 2Sij − u i u j = 0 + uj − uˆ j ∂t ∂xj ∂xi ∂xi
(1)
∂ui =0 ∂xi
(2)
where x1 = x and x2 = y are the Cartesian coordinates, ui is the fluid velocity in the xi direction, uˆ j is the mesh velocity, is the fluid density, p is the pressure, Sij is the strain-rate tensor and u i u j is the Reynolds stress tensor. The shear stress transport (SST) k-ω turbulence model developed by Menter [12] is used to close the RANS equations. The equation of motion is used to calculate the displacement of the cylinder. For a one-degree-of-freedom problem, the equation of motion is: mY¨ + c Y˙ + kY = Fy
(3)
where m is the total mass of the two cylinders, c is the damping coefficient of the system and k is the spring constant, Y, Y˙ and Y¨ are the displacement, velocity and acceleration of the cylinders, respectively, Fy is the total hydrodynamic force of both cylinders in the cross-flow direction. The forces are found by integrating the shear stress and pressure over the cylinder surfaces. The RANS equations are solved by the Petrov-Galerkin Finite Element Method (PG-FEM) developed by Zhao et al. [24]. Initially the fluid velocity and pressure are zero in the whole fluid domain and the cylinders are at their static balance position, i.e. Y and Y˙ are zero. The boundary conditions are described as follows. At the left and right boundaries, the velocity is given as a sinusoidal flow as u = Um sin(ωf t), where ωf = 2/T is the angular frequency of the flow and the pressure is given based on the undisturbed flow condition as p = − xωf Um cos(ωf t). The turbulent quantities at the left and right boundaries of the computational domain are the specific dissipation turbulence ωT = 1 s−1 and turbulent kinetic energy kT = 0.001ωT2 2. On the two side boundaries that are parallel to the flow direction, the velocity perpendicular to the boundary is zero and the gradient of all other quantities in the normal direction of the boundary are zero. On the cylinder surfaces, no-slip boundary condition is used, i.e. the velocity of the fluid is the same as the
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Fig. 1. (a) A sketch for two elastically mounted cylinder in an oscillatory flow, the two cylinders are rigidly connected with each other; (b) Computational mesh near the cylinders at ˛ = 45◦ and G/D = 0.
velocity of the cylinders. The turbulent energy is zero on the cylinder surfaces and the specific dissipation rate of the turbulence is specified on the layer of mesh nodes next to the cylinder surface as ωT = 61 /, where 1 is the distance from a node to the boundary surface. If the cylinders move, the mesh nodes move according to the displacement of the cylinders. The governing equation for calculating the displacements of the computational mesh nodes is [23]
∇ · ( ∇ Sy ) = 0
(4)
where Sy is the displacement of the mesh and is a parameter that controls the mesh deformation. To ensure that the mesh near the cylinders, where refined finite elements are used, has small deformation, is calculated by = 1/A with A being the area of a finite element. The boundary condition of Eq. (4) is that Sy is the same as the displacement of the cylinders on the cylinder surfaces and zero on the rest of the boundaries. The procedure of the calculation in each computational step is summarized as follows.
(1) The simulation starts from solving the RANS equations. Then the hydrodynamics forces are calculated using the solution of the RANS equations. (2) The velocity and the displacement of the cylinders in the next step is calculated by solving the equations of motion (Eq. (3)) using the fourth-order Runge-Kutta method. (3) After the new displacement of the cylinders is known, the displacement and the velocity of the mesh are calculated by solving Eq. (4) using the Galekin Finite Element Method. (4) The above three steps are repeated until the last time step.
3. Numerical results A square computational domain is used in the simulation. The size of the computational domain is 110D in both the cross flowdirection and the flow directions and the large cylinder is located at the center of the domain. The blockage of the computational domain (defined as the ratio of D + d to the fluid domain width) is 0.011. Anagnostopoulos and Minear [2]conducted a study of the blockage effects on the forces on cylinders in an oscillatory flow 0.1 ≤ KC ≤ 6. It was found that the blockage effect is negligible if the blockage is less than 0.2 and the blockage effects for various KC numbers are consistent. The blockage ratio used in this study is only 0.0545 times the one suggested by Anagnostopoulos and Minear [2] to ensure that the motion of the vortices that are shed from the cylinders are not affected by the side boundaries of the computational domain. The whole computational domain is divided into 38480 quadrilateral finite elements. Fig. 1 (b) shows the computational mesh near the cylinder surfaces for G/D = 0 and ˛ = 45◦ . The surfaces of the large and small cylinders are discretised into 212 and 80 elements, respectively. The minimum mesh size in the radial direction next to the cylinder surfaces is 0.0007D. In all the simulations, the Reynolds number, the KC number and the damping ratio are 20000, 10 and 0, respectively. Simulations are carried out for three gap ratios of G/D = 0, 0.1 and 0.2, three arrangements of ␣=0◦ , 45◦ and 90◦ (referred to as tandem, staggered and side-by-side arrangements, respectively) and reduced velocities ranging from 1 to 20. A mesh dependency study is conducted to ensure the meshes used are fine enough to get converged numerical results. In the mesh dependency study, simulations for G/D = 0.1, ␣ = 45◦ and two reduced velocities of 8 and 20 are conducted at a denser mesh with a radial minimum mesh size of 0.0003D on the cylinder surfaces. The numbers of the elements along the large and small cylinders in the dense mesh are 256 and 96, respectively. ˛ = 45◦ is chosen because
Fig. 2. Comparison between the displacements calculated from normal mesh and dense mesh for G/D = 0.1, ˛ = 45◦ and two reduced velocities of Vr = 8 and 20.
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Fig. 3. Time histories of the displacement of the cylinder for ˛ = 0◦ . The spacing between the horizontal gridlines is Y/D = 1.
the vibration at this attack angle is periodic, making it easy to compare the results from different meshes. Vr = 8 is the reduced velocity where the vibration amplitude reaches its maximum and Vr = 20 is the largest reduced velocity simulated in this study. Fig. 2 is a comparison between the displacements of the cylinders calculated from the normal and dense meshes. In the figure, the normal mesh is the mesh used in all the simulations. The difference between the vibra-
tion amplitudes calculated from both meshes is less than 0.5% for both reduced velocities. No experimental data of FIV of two cylinders in an oscillatory flow were found in literature for validating the numerical model. However, the validation of the present numerical model on the FIV of a cylinder in other similar cases has been performed in a number of previous studies. Using the same numerical model, Zhao and
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Fig. 4. Variation of the maximum displacement with the reduced velocity for ˛ = 0◦ , 45◦ and 90◦ .
Cheng [23] achieved excellent agreement between the numerical results of FIV of a cylinder in steady flow and the experimental data by Jauvtis and Williamson [5]. The validity of the model on FIV of a cylinder in oscillatory flow are demonstrated in Zhao et al. [25]. No further numerical validation will be performed in this paper.
3.1. Tandem arrangement (˛ = 0◦ ) Fig. 3 shows the time histories of the displacement of the cylinders for ␣=0◦ and reduced velocities from 2 to 20 with an interval of 2, together with the results in the case of a single cylinder. The reduced velocity for two cylinders of different diameters is defined based on the diameter of the large cylinder as Vr = Um /(fn D). The static balance positions of the cylinders for different reduced velocities are marked as horizontal gridlines and the reduced velocities are labelled at the right hand side of diagrams in Fig. 3. In Fig. 3 (a), the vibration of a single cylinder changes between a low frequency, high amplitude stage and a high frequency, low amplitude stage intermittently as Vr = 12. As Vr ≥ 16, the vibration becomes stable and periodic. The mean position of the cylinder is found to be either positive or negative, because vortices are shed from only one side of the cylinder. If the two cylinders of different diameters are in a tandem arrangement in the fluid flow, the periodicity of the vibration was broken when the reduced velocity exceeds a value of Vr = 10. It is interesting to see in Fig. 3(b)–(d) that the mean position of the two
cylinders changes between positive and negative values intermittently when Vr > 14. It has been demonstrated that vortices only shed from one side (top or bottom) of the cylinder if KC = 10, forming the so-called transverse vortex street [21]. The transverse vortex street leads to a mean lift force pointing to one side of the cylinder. The changing of the mean position of the cylinders from negative to positive or the other way around is due to the switching of the vortex shedding side of the cylinder. When the mean position of the cylinder is changing its sign, the vibration displacement becomes high and recovers gradually. For example, during t = 40 to 50 for Vr = 20, when the mean position of the cylinder is changing from positive to negative, the displacement of the cylinder is more than doubled and gradually changes to its normal condition. Fig. 4(a) shows the variation of the maximum displacement of the two tandem cylinders with the reduced velocity. To make it convenient to compare the maximum displacements for different ␣, the maximum displacements for ˛ = 45◦ and 90◦ are presented in Fig. 4(b) and (c) also and will be discussed in the next two sections. The maximum displacement is defined as Amax = max(Ymax , − Ymin ), where Ymax and Ymin are the maximum and minimum displacements, respectively, when the vibrations have stabilised after 30% of the calculated time, which is at least 150 periods of the oscillatory flow. The variation trend of the maximum vibration displacement of the two cylinders with the reduced velocity appears similar to that of a single cylinder until Vr = 10. The vibration amplitude of a single cylinder decreases with increasing Vr as 11 < Vr < 15,
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Fig. 5. Contour of the real part of the wavelet transform of the vibration displacement for ˛ = 0◦ . The corresponding time histories of the vibration are added on the top of each diagram.
and slowly increases as Vr > 15, while that of two tandem cylinders increases significantly as Vr > 11 with a maximum amplitude of triple that of the single cylinder. The maximum vibration displacement continuously increases with increasing Vr , mainly because of high amplitude vibration during the time when the vortex shedding switches between the top and bottom side of the cylinders. It can be seen in Fig. 3 that each time after the mean position of the cylinder changes its sign, the magnitude of the deflection of the cylinder is large and reduces gradually until the vortex shedding side changes again. From Fig. 3(b)–(d) it can be clearly seen that neither the vibration nor the amplitude of the vibration remains constant during the vibration at large reduced velocities. To see how the vibration frequency varies with time, the Morlet wavelet analysis of the vibration displacement is conducted and Fig. 5 shows the contours of the real part of the wavelet transform of the vibration displacement for two typical cases where the vibration is stable and two cases where the vortex shedding is switching intermit-
tently between different patterns. The time history of the vibration displacement is added on the top of each wavelet diagram in Fig. 5. When ˛ = 0◦ , the vibration of the cylinders mainly has two frequency components of f/fw = 1 and 2, where fw is the oscillatory flow frequency, and the contribution of these two frequency components vary with time. Whenever the amplitude of f/fw = 1 increases, the amplitude of f/fw = 2 decreases, and vice versa. The vibration for G/D = 0.1 and Vr = 12 is periodic and dominated by a frequency of f/fw = 0.5, indicating that the vibration repeats every two cycles of the flow. As the reduced velocity is increased to 14, the vibration becomes very irregular and mainly contains three frequencies of f/fw = 0.5, 1 and 2. However, only one or two frequencies coexist at a time. The vibration of Vr = 14 actually irregularly changes between the three frequency modes. To see how the reduced velocity and the gap ratio affect the frequency of the vibration for all of the simulated cases, the Fast Fourier Transform (FFT) spectra of the displacements are shown in Fig. 6. To see the correlation between the vibration and the
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Fig. 6. FFT spectra of the displacement for ˛ = 0◦ . The displacement in each spectrum is normalized by the amplitude corresponding to the highest peak of the spectrum.
lift coefficient, the FFT spectra of the total lift coefficient on both cylinders are shown in Fig. 7. The total lift coefficient is defined as 2 /2). The ratios of the natural frequency of the cylinCL = Fy /(DUm ders measured in fluid to the oscillatory flow frequency (fnw /fw ) are marked as square symbols on the horizontal axes in Fig. 6 and 7. While the natural frequency measured in vacuum (structural natural frequency) fn is used as an input parameter in the numerical simulations, the natural frequency measured in water fnw is calm∗ /(CA + m∗ ), where culated using the relationship of fnw /fn = CA = 1 is the added mass coefficient. The added mass coefficient CA was found to be 1 with an error less than 5% in the experiments by Khalak and Williamson [6] As the reduced velocity is 1, the flow pattern should be the same as that of two stationary cylinders since the vibration amplitude is almost zero (Fig. 4), i.e. a pair of vortices are shed from the cylinders in one oscillatory period. Because the flow direction reverses twice within one oscillatory flow period, the vibration frequency is twice the oscillatory frequency and so is the lift force [16]. The vibration frequency is found to follow the natural frequency in water when the reduced velocity is between 2 and 4 for all the gap ratios. As the reduced velocity exceeds 5, the vibration frequency of a single cylinder locks onto fw as 9 ≤ Vr ≤ 14 and 2fw for higher reduced velocities. In a tandem arrangement, the vibration is found to be dominated by 0.5fw at high reduced velocities. It should be noted that the 0.5fw does not always dominate the vibration throughout the simulated vibration time. In fact, the vibration frequency changes between 0.5fw , fw and 2fw , intermittently. The frequency of 0.5fw dominates in a FFT spectrum because the amplitude when the vibration frequency is 0.5fw is higher than those when the vibration frequency is fw and 2fw , as shown in the wavelet analysis results in Fig. 5 (c). For large reduced velocities,
the vibration of the cylinders in tandem is mainly in a multiple frequency mode, and each frequency is multiple of 0.5fw . Same as the vibration displacement, the lift coefficient of two cylinders in tandem also has multiple frequencies that are multiples of 0.5fw . It appears that the lift coefficient is always dominated by the frequency of 2fw . The predominant frequencies of the displacement and the lift coefficient are different from each other at large reduced velocities. For example, the vibration frequency is close to either 0.5fw or fw as Vr ≥ 10 and G/D = 0.1, while the dominant frequency of the lift coefficient remains to be 2fw . The dominant frequencies of the vibration and the lift coefficient are different from each other mainly because the non-dominant frequency of the lift coefficient has locked onto the natural frequency and resulted in the excitation of the vibration. From Fig. 5 and 6 it can be seen that although the vibration histories are seemingly very chaotic and irregular, distinct peak frequencies can be identified from the spectra of the vibration displacement and lift coefficient. It has been found that the vibration frequency of a single cylinder in an oscillatory flow is a multiple of the oscillatory flow frequency [18,9]. Based on the frequency of the vibration, Zhao [22] classified the vibrations into single-, double-, triple- frequencies modes, etc. It is well known that for a stationary cylinder in an oscillatory flow with KC = 10, the vortex shedding flow pattern is a single pair flow pattern and the frequency of the lift coefficient is twice the oscillatory frequency [13,21]. If the frequency of the lift coefficient is close to the natural frequency, the vibration of the cylinder increases (Vr = 4 to 6 in Fig. 4 (a)) and the vibration frequency will be the same as the frequency of the lift coefficient. As Vr = 7 and 8, although both the frequencies of the displacement and the lift coefficient are 2fw , the vibration amplitude is reduced because the difference between the frequency of the lift coefficient and the nat-
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Fig. 7. FFT spectra of the lift coefficient for ˛ = 0◦ . The lift coefficient in each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
Fig. 8. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 5 within one oscillatory flow period.
ural frequency has increased significantly. With further increasing reduced velocity, the vibration frequency becomes fw because fw is close to the natural frequency fnw . The vibration frequency is closely related to the vortex shedding flow pattern, which is presented by the contours of the nondi-
mensional vorticity defined as ω = (∂v/∂x − ∂u/∂y)/(Um /D). Fig. 8–10 shows the contours of the nondimensional vorticity as G/D = 0.1 and Vr = 5, 11 and 12, respectively. The vortices in the figures are given labels to facilitate the discussion. The vortex shedding for Vr = 5 is a typical one pair regime. After the flow reverses at t/Tw = 114, vor-
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Fig. 9. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 11 within one oscillatory flow period.
Fig. 10. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 12 within one oscillatory flow period.
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Fig. 11. Time histories of the displacement of the cylinder for ˛ = 45◦ . The spacing between the horizontal gridlines is Y/D = 1.
tex A generated from the bottom side of the cylinder moves along the cylinder surface to the top side and is shed from the cylinder at t/Tw = 114.5. Another vortex B will be shed from the cylinder in the second half period. Because only two vortices are generated and shed from the cylinder and they are all shed from the top side of the cylinder in one cycle of vibration, the vortex shedding regime is called one pair regime or transverse vortex shedding regime. The frequencies of the cylinder displacement and the lift coefficient are the same in the one pair regime as shown in Fig. 8(f). When Vr = 11, vortex A is generated and shed from the top side of the cylinder, while vortex B is generated from the bottom side but is shed from the top side of the cylinder in the first half oscillatory flow period. The vortex shedding in the first half oscillatory flow period (during t/Tw = 103 and 103.5) is similar to its corresponding pattern for Vr = 5, i.e. one vortex A is shed from the cylinder. In the second half flow period, two vortices C and D are generated and separated from the bottom and top sides of the cylinder, respectively, forming a symmetric vortex flow pattern as shown in Fig. 9(d). In the next
half cycle, vortices E and F in Fig. 9(e) will repeat what vortices A and B did in Fig. 9(a), respectively. Comparing Fig. 9(b) with (d) it can be seen that the vortex flow patterns in two halves of a flow period are very different from each other, resulting in a vibration frequency the same as the oscillatory flow frequency. The single pair regime and the regime of ‘paring of attached vortices’ are found to switch from one to another after every half flow period. The vortex shedding pattern in Fig. 9(b) is very similar to that of the single pair regime. The vortex shedding pattern in Fig. 9(d), where a pair of vortices (C and D in Fig. 9) are formed before flow reverses, is very similar to the regime of ‘Pairing of attached vortices’ observed by [21]. After the flow direction reverses, the pair of vortices is split up by the newly generated pair of smaller vortices (E and F in Fig. 9). As Vr = 12, the vibration frequency is 0.5 times the oscillatory flow frequency because the vortex shedding flow repeats every two flow periods. The vortex flow pattern from t/Tw = 127.5 and 128.5 are very similar to that of the single pair regime. Transverse vortex flow patterns can be clearly identified in Fig. 10(c) and (e). The vor-
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Fig. 12. FFT spectra of the displacement and the lift coefficient of the cylinder for ˛ = 45◦ . Each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
Fig. 13. Contours of the nondimensional vorticity for ˛ = 45◦ , G/D = 0 and Vr = 9 within one oscillatory flow period.
tex flow pattern from t/Tw = 128.5 to t/Tw = 129.5 are in the regime of ‘pairing of attached vortices’ (see Fig. 10(h) and (j)). The lift coefficient for Vr = 12 is also found to repeat after every two flow periods. The vortex shedding in Fig. 10 is also a combination of single pair regime and the regime of ‘pairing of attached vortices’. The difference between the flow patterns in Fig. 9 and 10 is that the flow repeats every half period in Fig. 9 and every whole period
in Fig. 10. To differentiate from each other, the flow regimes in Fig. 9 and 10 are defined as combined flow regime 1 and combined flow regime 2, respectively. The single pair flow regime has a vibration frequency of 2fw , the combined flow regime 1 has a vibration frequency of fw and the combined flow regime 2 has a vibration frequency of 0.5fw . In many cases the vortex shedding flow switches from one regime to another alternatively, and which regime the
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Fig. 14. Contours of the nondimensional vorticity for ˛ = 45◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period.
flow is in can be determined based on the instantaneous vibration frequency. For example as Vr = 14 and G/D = 0.1 (see Fig. 5(d)), the flow is in the single pair regime during t/Tw = 75 to 90 (f/fw = 2), combined regime 1 during t/Tw = 50 to 60 (two frequencies coexist but f/fw = 1 dominates) and combined regime 2 during t/Tw = 130 to 145 (f/fw = 0.5 dominates). If alternative switching of the flow between flow regimes occurs, the FFT spectrum shown in Fig. 6 has multiple distinct frequencies as the vibrations change frequencies and amplitudes as the flow switches regimes. 3.2. Staggered arrangement (˛ = 45◦ ) Fig. 11 shows the time histories of the vibration displacement of the cylinders for three gap ratios and reduced velocities from 2 to 20 for ␣=45◦ . The vibrations for all the three simulated gaps are periodic at ␣=45◦ , indicating the vibration and the vortex shedding flow remains unchanged. The periodicity of the vibration can also be evidenced by the FFT spectra of the vibration displacement and the lift coefficient shown in Fig. 12. Each of the FFT spectra in Fig. 12 has very sharp spikes with frequencies that are multiples of the oscillatory frequency. Due to the asymmetric configuration of the system in both the x- and y-directions, the vibration histories in two halves of one oscillatory flow period are always different from each other as Vr ≥ 7, resulting in a dominant vibration frequency same as fw . There is a secondary frequency component of 2fw in the vibration because the vortex shedding pattern is in the one pair regime. Fig. 13 and 14 show the contours of the nondimensional vorticity for Vr = 9 and G/D = 0 and 0.2, respectively. It is clear that two vortices (labelled as vortices A and B) are shed from the cylinder in one period of the oscillatory flow for both gap ratios. When G/D = 0 and Vr = 9, a positive vortex A is generated from the top of the cylinder, moves to the bottom of the cylinder and is shed from the bottom side of the cylinder as shown in Fig. 13(a) to (c). In Fig. 13(b) to (e) another vortex B is generated from the top side of the cylinder, moves to the bottom side of the cylinder and is shed from the bottom side of the cylinder. Vortex A is much stronger than Vortex B because of two reasons. Firstly, vortex A is strengthened because after flow reverses (Fig. 13(c)), the vortex A, that has been shed from the cylinder is convected back by the flow and combines with the newly generated vortex A. Secondly, vortex B is generated from the top side of the cylinder (Fig. 13(b)) does not grow much due to the influence of the small cylinder and it is weakened by the small
cylinder when it moves along the right surface of the large cylinder to the bottom side. The difference in the lift coefficients produced by vortices A and B can be clearly seen in Fig. 13(f). The vortex shedding for Vr = 9 and G/D = 0 is similar to that for G/D = 0.2 with the same reduced velocity. Compared with G/D = 0, the effect of the small cylinder for G/D = 0.2 on the flow is weaker, the strengths of Vortex A and B are more balanced and the difference between the lift coefficients generated by vortices A and B is smaller. The variation of the maximum vibration displacement Amax /D for ␣=45◦ is similar to that for ␣=0◦ as shown in Fig. 4. Two maximum values occur at Vr ≈ 4 and 9, respectively. The vibration amplitude reaches its maximum at Vr ≈ 4 and 9 because 2fw and fw are close to the natural frequency of the cylinder in water, respectively as shown in Fig. 12. The maximum displacement is increased significantly at Vr = 9 and G/D = 0 because of the very strong positive vortex A as shown in Fig. 12. By observing Fig. 11, it appears that the asymmetry of the configuration leads to periodic and regular vibration, because the instability of changing between two flow patterns is only likely to occur in a symmetric configuration.
3.3. Side-by-side arrangement (˛ = 90◦ ) Fig. 15(a)–(c) shows the time histories of the vibration displacement for ␣=90◦ . It appears that the switching between different vibration modes occurs very frequently in the asymmetric configuration of ␣=90◦ , especially as G/D = 0.2. The switching between different vibration frequencies can be clearer seen in Fig. 15(d), where the contours of the real part of the wavelet transform of the vibration displacement for Vr = 13 are shown. The vibration is found to switching from double-frequency to single-frequency modes alternatively. Fig. 16 shows the FFT spectra of the displacement and the lift coefficient for ␣=90◦ and two gap ratios of G/D = 0 and 0.2. Same as the ones for ␣=0◦ and 45◦ , the nondimensional frequency of the displacement f/fw decreases with increasing reduced velocity until Vr = 5. As G/D = 0, the predominant frequency of the displacement is f/fw = 1 in the reduced velocity range between 8 and 15 and f/fw = 2 when Vr ≥ 16. As G/D = 0.2, it appears that the frequencies of f/fw = 1 and 2 equally dominate the vibration as Vr ≥ 16. This is because the vibration switches between high frequency vibration and low frequency vibration intermittently as shown in the time histories
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Fig. 15. Time histories of the vibration displacement for ˛ = 90◦ . The spacing between the horizontal gridlines is Y/D = 1.
of the displacement in Fig. 15. The switching of the vibration from one mode to another occurs very frequently when G/D = 0.2. Fig. 17 shows the contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period. It can be seen that the flow is dominated by a negative vortex A, in the whole vibration cycle. In Fig. 17(a), the flow starts to reverse and the cylinders have almost moved to their lowest position. This allows the negative vortex A to be convected to the right hand side from the top of the cylinder. When the flow reverses at t/Tw = 178.5, the cylinders are almost at their highest position and vortex A moves to
the left hand side from the bottom of the cylinders. The large vortex A keeps attached to the cylinders throughout the whole cycle of vibration. In addition of very weak vortices B and C, vortex shedding of large vortices from the cylinder is not observed. Because the dominant frequency of the lift coefficient is the same as the oscillatory flow frequency (Fig. 17(f)) and close to the natural frequency fnw (Fig. 16(c)), high amplitude vibration is achieved when Vr = 9. Fig. 18 shows the contours of the nondimensional vorticity for ␣=90◦ , G/D = 0.2 and Vr = 16 within one oscillatory flow period. The vibration has two frequencies of f/fw = 1 and 2 as shown in Fig. 18
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Fig. 16. FFT spectra of the displacement and the lift coefficient of the cylinder for ˛ = 90◦ . Each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
(f) and is slightly dominated by f/fw = 1. The vortex shedding in Fig. 18 is a typical one pair vortex shedding regime. In one period of the oscillatory flow, two vortices A and B are shed from the top of the two cylinders. This one pair vortex shedding pattern makes the frequency of the lift coefficient twice the oscillatory flow frequency. Because 2fw is very different from the natural frequency fnw as Vr = 16 (see Fig. 16(c)), the vibration amplitude in Fig. 18 is much smaller than that in Fig. 17. The switching between the vortex patterns shown in Fig. 17 and 18 are observed and occur very frequently when G/D = 0.2 and reduced velocity is greater than 11. Fig. 19 is an example of changes of the vortex flow pattern when ␣=90◦ , G/D = 0.2 and Vr = 13. During the oscillatory flow period of t/Tw = 157–158, the vortex shedding is in a one pair regime and vortices A and B are shed from the bottom side of the two cylinders. In the period starting from t/Tw = 189, the flow is in a no-shedding regime similar to the one shown in Fig. 18. The vortex shedding in the period starting from t/Tw = 246 is also in a one-pair pattern. Different from those in Fig. 19(a) and (b), the vortices in Fig. 19(e) and (f) are shed from the top side of the cylinder, instead of the bottom side. The vibration amplitude becomes small whenever the vortex shedding is in a one pair regime and large whenever the vortex shedding is in a no-shedding regime. 4. Discussion The vibration is periodic for all the simulated reduced velocities for ␣=45◦ . The configuration of ␣=45◦ is asymmetric in both the x- and y-directions, making it difficult to produce a bi-stable
flow. When ␣=0◦ , the periodicity of the vibration is broken when Vr ≥ 14. Irregular vibrations are also found when ␣=90◦ and at large reduced velocities. Periodic vibration occurs when the natural frequency of the system measured in water is close to the oscillatory flow frequency fw or a multiple number of fw . If a cylinder is stationary, the vortex shedding is in a transverse vortex street at KC = 10, i.e. a pair of vortices are only shed from one side of the cylinder in every period of the vibration. The vibration becomes irregular at large reduced velocities mainly because the vortex shedding side of the cylinder changes between the two sides of the cylinder very randomly and the flow is disturbed significantly during the changes. Before the vortex shedding side of the cylinder changes completely, a no-shedding regime is found and may last a number of oscillatory flow periods. The vibration frequency changes at large reduced velocities for both ␣ = 0◦ and 90◦ mainly because the vortex shedding mode changes. More vortices are shed in one period of vibration, higher the vibration frequency. The effects of Vr on the maximum displacement, Amax /D can be found in Fig. 4. A maximum Amax /D occurs in the reduced velocity range of 4 ≤ Vr ≤ 6 and a minimum Amax /D occurs as 6 ≤ Vr ≤ 8 for all the simulated gap ratios and flow attack angles. The variation of Amax /D with Vr for different flow attack angles are different from each other as Vr ≥ 8. At the largest simulated reduced velocity of Vr = 20, Amax /D continues increases with increasing Vr when ␣ = 0◦ , Amax /D continues increases with increasing Vr but with smaller rate when ␣ = 45◦ and Amax /D overall decreases with increasing Vr as ␣ = 90◦ . The variation of Amax /D with Vr is very similar to that of a single cylinder as ␣ = 0◦ and 45◦ .
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Fig. 17. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period.
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Fig. 18. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 16 within one oscillatory flow period.
5. Conclusion The effects of the flow approaching angle on the vibration of two cylinders of different diameters in oscillatory flow are investigated through two-dimensional numerical simulations for tandem, side-by-side and staggered arrangements of the cylinders and three gap ratios of 0, 0.1 and 0.2. The KC number remains constant at KC = 10, which belongs to the single pair vortex shedding regime for a stationary cylinder. The main conclusions are summarized as follows. The striking phenomenon found for the tandem arrangement is that the vortex shedding changes between three regimes intermittently as the reduced velocity exceeds 10. These three modes are: (1) a pure single pair regime and two combined regimes 1 and 2. A combined regime is a combination of the single pair regime and the regime of pairing of attached vortices. In combined regime 1, the vibration frequency is fw and the two regimes repeat every half flow period. In combined regime 2, the vibration frequency is 0.5fw and the two regimes repeat every two periods. If the flow is in a pure single pair regime, the vibration frequency is twice the oscillatory flow frequency. The vibration displacement is found to be increases significantly immediately after the flow switches from one regime to another. All the frequencies of the three regimes (0.5fw , fw and 2fw ) can be seen in the FFT spectra of the displacement.
The vibration for a staggered arrangement with ˛ = 45◦ is perfectly periodic and stable for all the calculated reduced velocities. The vortex shedding for ˛ = 45◦ is always in the single pair regime and the vibration has two frequencies of fw or 2fw . The vibration with G/D = 0.2 is more dominated by 2fw than that with G/D = 0. The switching between different flow regimes is also observed when ˛ = 90◦ . In addition to the single pair flow regime, an interesting single vortex regime is found where a single vortex always attaches to the cylinders and rotates around the cylinders. This single vortex regime occurs at intermediate reduced velocities and the lift coefficient frequency is the same as the oscillatory flow frequency at this flow regime. The vibration may change between the single vortex regime and one pair regime alternatively in some cases.
Acknowledgements The authors would like to acknowledge the support from the Natural Science Foundation of China (Grant 51628091) and the Australian Research Council (Grant LP150100249). The calculations were carried out on the computational facilities of Intersect Australia Ltd. in NSW, Australia.
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Fig. 19. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 13 within one oscillatory flow period.
References [1] H. An, L. Cheng, M. Zhao, Direct numerical simulation of oscillatory flow around a circular cylinder at low Keulegan?Carpenter number, J. Fluid Mech. 666 (2011) 77–103. [2] P. Anagnostopoulos, R. Minear, Blockage effect of oscillatory flow past a fixed cylinder, Appl. Ocean Res. 26 (2004) 147–153. [3] H. Dütsch, F. Durst, S. Becker, H. Lienhart, Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers, J. Fluid Mech. 360 (1998) 249–271. [4] G. Iliadis, P. Anagnostopoulos, Numerical visualization of oscillatory flow around a circular cylinder at Re = 200 and KC = 20–An aperiodic flow case, Commun. Numer. Methods Eng. 14 (1998) 181–194. [5] N. Jauvtis, C.H.K. Williamson, The effect of two degrees of freedom on vortex-induced vibration at low mass and damping, J. Fluid Mech. (2004) 23–62. [6] A. Khalak, C.H.K. Williamson, Motions: forces and mode transitions in vortex-induced vibrations at low mass-damping, J. Fluids Struct. 13 (1999) 813–851. [7] N.W.M. Ko, P.T.Y. Wong, Flows past two circular cylinders of different diameters, J. Wind Eng. Ind. Aerodyn. 41 (1992) 563–564. [8] N.W.M. Ko, P.T.Y. Wong, R.C.K. Leung, Interaction of flow structures within Bistable flow behind two circular cylinders of different diameters, Exp. Therm. Fluid Sci. 12 (1996) 33–44. [9] A. Kozakiewicz, B.M. Sumer, J. Fredsøe, E.A. Hansen, Vortex regimes around a freely vibrating cylinder in oscillatory flow, Int. J. Offshore Polar Eng. 7 (1997) 94–103.
[10] K.M. Lam, P.T.Y. Wong, N.W.M. Ko, Interaction of flows behind two circular cylinders of different diameters in side-by-side arrangement, Exp. Therm. Fluid Sci. 7 (1993) 189–201. [11] M. Mahbub Alam, Y. Zhou, Strouhal numbers: forces and flow structures around two tandem cylinders of different diameters, J. Fluids Struct. 24 (2008) 505–526. [12] R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (1993) 1598–1605. [13] E.D. Obasaju, P.W. Bearman, J.M.R. Chraham, A study of forces: circulation and vortex patterns around a circular cylinder in oscillating flow, J. Fluid Mech. 196 (1988) 467–494. [14] M. Rahmanian, M. Zhao, L. Cheng, T. Zhou, Two-degree-of-freedom vortex-induced vibration of two mechanically coupled cylinders of different diameters in steady current, J. Fluids Struct. 35 (2012) 133–159. [15] T. Sarpkaya, Force on a circular cylinder in viscous oscillatory flow at low Keulegan—Carpenter numbers, J. Fluid Mech. 165 (1986) 61–71. [16] T. Sarpkaya, M. Storm, In-line force on a cylinder translating in oscillatory flow, Appl. Ocean Res. 7 (1985) 188–196. [17] P. Scandura, V. Armenio, E. Foti, Numerical investigation of the oscillatory flow around a circular cylinder close to a wall at moderate Keulegan-Carpenter and low Reynolds numbers, J. Fluid Mech. 627 (2009) 259–290. [18] B.M. Sumer, J. Fredsøe, Transverse vibrations of an elastically mounted cylinder exposed to an oscillating flow, J. Offshore Mech. Arct. Eng. 110 (1988) 387–394. [19] J. Thapa, M. Zhao, L. Cheng, T. Zhou, Three-dimensional flow around two circular cylinders of different diameters in a close proximity, Phys. Fluids 27 (2015) 085106. [20] T. Tsutsui, T. Igarashi, K. Kamemoto, Interactive flow around two circular cylinders of different diameters at close proximity: experiment and
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numerical analysis by vortex method, J. Wind Eng. Ind. Aerodyn. (1997) 279–291 (69–71). [21] C.H.K. Williamson, Sinusoidal flow relative to circular cylinders, J. Fluid Mech. 155 (1985) 141–174. [22] M. Zhao, Numerical investigation of two-degree-of-freedom vortex-induced vibration of a circular cylinder in oscillatory flow, J. Fluids Struct. 39 (2013) 41–59. [23] M. Zhao, L. Cheng, Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary, J. Fluids Struct. 27 (2011) 1097–1110.
[24] M. Zhao, L. Cheng, B. Teng, G. Dong, Hydrodynamic forces on dual cylinders of different diameters in steady currents, J. Fluids Struct. 23 (2007) 59–83. [25] M. Zhao, L. Cheng, T. Zhou, Numerical investigation of vortex-induced vibration (VIV) of a circular cylinder in oscillatory flow, Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering − OMAE, 2011 (2011) 597–603.
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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
Vibration of two elastically mounted cylinders of different diameters in oscillatory flow Toni Pearcey, Ming Zhao ∗ , Yang Xiang, Mingming Liu School of Computing Engineering and Mathematics, Western Sydney University, Penrith, NSW 2751, Australia
a r t i c l e
i n f o
Article history: Received 30 April 2017 Received in revised form 1 November 2017 Accepted 1 November 2017 Keywords: Flow induced vibration Numerical method Oscillatory flow Circular cylinder
a b s t r a c t Vibration of two elastically mounted cylinders in an oscillatory flow at a Keulegan-Carpenter number of 10 is simulated numerically. The two cylinders are rigidly connected with each other and are allowed to vibrate in the cross-flow direction only. The aim of this paper is to identify the effects of the orientation of the cylinders and the gap between the cylinders on the vibration. The two-dimensional ReynoldsAveraged Navier-Stokes equations are solved to predict the flow and the cylinder vibration is predicted using the equation of motion. When the two cylinders are in a tandem arrangement, a combined single pair flow regime and attached pair flow regime are observed as reduced velocity exceeds 10 and this combined regime and the single pair regime occurs intermittently. Periodic vibration is found when the two cylinders are in a staggered arrangement with a 45◦ flow attack angle. When the two cylinders are in a side-by-side arrangement, a new single vortex regime is observed. This single vortex remains attached to the cylinder surface and rotates around the cylinder. The intermittent switch between this single vortex regime and the single pair regime are observed. © 2017 Published by Elsevier Ltd.
1. Introduction With increasing demand for energy resources from ocean including offshore oil and gas, renewable energy from wind, tidal currents and waves, more and more offshore structures are constructed. Offshore structures for extracting energy from ocean have to survive severe storms without damaging their functionality. Many cylindrical structures are used in offshore engineering such as subsea pipelines, risers, mooring cables, etc. Oscillatory flow is often used to model the water motion due to waves when the impact of the waves on small scale cylindrical structures is studied. Many studies have been performed to understand the hydrodynamics and flow patterns around circular cylinders in oscillatory flows. It has been found that the hydrodynamic forces on cylindrical structures are mainly affected by the Keulegan-Carpenter (KC) number and the Reynolds number. The KC number is defined as KC = (Um T)/D, where Um and T are the velocity amplitude and period of the oscillatory flow, respectively, and D is the diameter of the cylinder. The Reynolds number is defined as Re = Um D/, where is the kinematic viscosity of the fluid. The ratio of the KC number to the Reynolds number is called the viscous parameter, ˇ [15].
∗ Corresponding author. E-mail address: [email protected] (M. Zhao). https://doi.org/10.1016/j.apor.2017.11.003 0141-1187/© 2017 Published by Elsevier Ltd.
[21] conducted an experimental study of oscillatory flow past a circular cylinder for KC numbers ranging from 1 to 40 and classified the vortex flow into different flow regimes: Paring of attached vortices (non-vortex shedding regime) when KC < 7, single pair regime when 7 < KC < 15, double pair regime when 15 < KC < 24, three-pair regime when 24 < KC < 32 and four–pair regime when 32 < KC < 40. Obasaju et al. [13] conducted a detailed study of the relationship between the vortex shedding regime and the hydrodynamic forces on a circular cylinder in an oscillatory flow. It was found that the spanwise correlation of the flow is good when KC is at the center of a regime and poor when KC is at the boundary between two regimes. Numerical studies have been successfully conducted to investigate oscillatory flow past a circular cylinder. Some studies are mainly focused on the inception of the three-dimensionality of flow at low Reynolds numbers and low KC numbers [3,1,17,4]. Recently, research has been performed to study flow induced vibration (FIV) of circular cylinders in oscillatory flows. In addition to the Reynolds number and the KC number, FIV of a cylinder in oscillatory flow is also dependent on the mass ratio and the reduced velocity. The mass ratio is defined as m* = m/md , where m is the cylinder mass and md is the displaced fluid mass, and the reduced velocity is defined as Vr = Um /(fn D), where fn is the structural natural frequency measured in vacuum in this study and many numerical studies. In many experimental studies of FIV in water, the natural frequency measured in still water (defined as fnw in this study) is used to define
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Nomenclature Amax CL d D fnw Fy fn fw G KC kT m m∗ md p Re Sij Sy T Um u ui uˆ i u i u j Vr xi Y Y˙ Y¨ ␣ ˇ ω ωf ωT
Maximum displacement Lift coefficient Small cylinder diameter Large cylinder diameter Natural structural frequency measured in water Lift force Structural natural frequency Oscillatory flow frequency Gap between two cylinders Keulegan-Carpenter number Turbulent kinetic energy Cylinder mass Mass ratio Displaced fluid mass Pressure Reynolds number Strain-rate tensor Displacement of the mesh Oscillatory flow period Velocity amplitude Sinusoidal velocity Velocity component in the xi -direction Mesh velocity Reynolds stress tensor Reduced velocity The ith Cartesian coordinate Displacement of the cylinders Velocity of the cylinders Acceleration of the cylinders Flow attack angle Viscous parameter Kinematic viscosity Fluid density Vorticity Angular frequency of the flow Specific dissipation of turbulence
the reduced velocity. Experimental studies of the vibration of an elastically mounted circular cylinder in oscillatory flow show that the vibration of the cylinder locks in with a frequency which is a multiple of the oscillatory flow frequency [18,9]. Two pipelines of different diameters are often bundled together in offshore engineering to form a so-called piggyback pipeline. The piggyback pipeline configuration ensures the stability of the small-diameter pipeline. Many studies have been performed to investigate two cylinders of different diameters in fluid flow [11,10,7,20,8]. The attachment of the shear layer from the gap to the back surface of the larger cylinder was observed when two cylinders of different diameters are in a side-by-side arrangement in a steady flow [20,19]. Rahmanian et al. [14] studied vortex inducedvibration of two cylinders of different diameters at Re = 8000 and found that the vibration of two cylinders is significantly different from that of a single cylinder. FIV of two cylinders of different diameters in an oscillatory flow, which is relevant to piggyback pipelines in waves in offshore engineering, has never been studied. In this study, FIV of two cylinders of different diameters as sketched in Fig. 1 (a) is studied numerically. The cylinders are allowed to vibrate only in the cross-flow direction. The diameters of the large and small cylinders are represented by D and d, respectively. The gap between the two cylinders and the flow attack angle is defined as G and ␣, respectively. The
main objective of this study is to understand the effect of the position of the small cylinder relative to the large cylinder on the FIV. Simulations are conducted for a constant diameter ratio of d/D = 0.2, a constant mass ratio of 2, a constant Reynolds number (based on D) of 2 × 104 , a constant KC number of 10, three gap ratios of G/D = 0, 0.1 and 0.2 and reduced velocities in a range of 1–20 with an interval of 1. A two-dimensional numerical model based on the RANS equations is used considering the large parametric space used in this study. A constant KC number of 10 is used, as the flow at this KC number presents the typical one pair flow regimes which occurs when 7 < KC < 15. This regime is the KC number range that piggyback pipelines generally experience in ocean engineering. 2. Numerical method The two-dimensional incompressible Reynolds-Averaged Navier Stokes (RANS) equations are used as the governing equations for simulating the flow. To account for the moving boundaries of the two cylinders, the Arbitrary Langrangian-Eulerian (ALE) scheme is applied to solve the RANS equations. The ALE scheme can avoid large deformations of the computational mesh because it allows the mesh to move with a velocity different from the fluid velocity. The effects of the mesh movement are considered by modifying the convection terms of the RANS equations. A Cartesian coordinate system is defined with its origin located at the center of the large cylinder and its x-coordinate pointing in the flow direction. The RANS equations in the ALE method are written as
∂ui 1 ∂p ∂ui ∂ + − 2Sij − u i u j = 0 + uj − uˆ j ∂t ∂xj ∂xi ∂xi
(1)
∂ui =0 ∂xi
(2)
where x1 = x and x2 = y are the Cartesian coordinates, ui is the fluid velocity in the xi direction, uˆ j is the mesh velocity, is the fluid density, p is the pressure, Sij is the strain-rate tensor and u i u j is the Reynolds stress tensor. The shear stress transport (SST) k-ω turbulence model developed by Menter [12] is used to close the RANS equations. The equation of motion is used to calculate the displacement of the cylinder. For a one-degree-of-freedom problem, the equation of motion is: mY¨ + c Y˙ + kY = Fy
(3)
where m is the total mass of the two cylinders, c is the damping coefficient of the system and k is the spring constant, Y, Y˙ and Y¨ are the displacement, velocity and acceleration of the cylinders, respectively, Fy is the total hydrodynamic force of both cylinders in the cross-flow direction. The forces are found by integrating the shear stress and pressure over the cylinder surfaces. The RANS equations are solved by the Petrov-Galerkin Finite Element Method (PG-FEM) developed by Zhao et al. [24]. Initially the fluid velocity and pressure are zero in the whole fluid domain and the cylinders are at their static balance position, i.e. Y and Y˙ are zero. The boundary conditions are described as follows. At the left and right boundaries, the velocity is given as a sinusoidal flow as u = Um sin(ωf t), where ωf = 2/T is the angular frequency of the flow and the pressure is given based on the undisturbed flow condition as p = − xωf Um cos(ωf t). The turbulent quantities at the left and right boundaries of the computational domain are the specific dissipation turbulence ωT = 1 s−1 and turbulent kinetic energy kT = 0.001ωT2 2. On the two side boundaries that are parallel to the flow direction, the velocity perpendicular to the boundary is zero and the gradient of all other quantities in the normal direction of the boundary are zero. On the cylinder surfaces, no-slip boundary condition is used, i.e. the velocity of the fluid is the same as the
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Fig. 1. (a) A sketch for two elastically mounted cylinder in an oscillatory flow, the two cylinders are rigidly connected with each other; (b) Computational mesh near the cylinders at ˛ = 45◦ and G/D = 0.
velocity of the cylinders. The turbulent energy is zero on the cylinder surfaces and the specific dissipation rate of the turbulence is specified on the layer of mesh nodes next to the cylinder surface as ωT = 61 /, where 1 is the distance from a node to the boundary surface. If the cylinders move, the mesh nodes move according to the displacement of the cylinders. The governing equation for calculating the displacements of the computational mesh nodes is [23]
∇ · ( ∇ Sy ) = 0
(4)
where Sy is the displacement of the mesh and is a parameter that controls the mesh deformation. To ensure that the mesh near the cylinders, where refined finite elements are used, has small deformation, is calculated by = 1/A with A being the area of a finite element. The boundary condition of Eq. (4) is that Sy is the same as the displacement of the cylinders on the cylinder surfaces and zero on the rest of the boundaries. The procedure of the calculation in each computational step is summarized as follows.
(1) The simulation starts from solving the RANS equations. Then the hydrodynamics forces are calculated using the solution of the RANS equations. (2) The velocity and the displacement of the cylinders in the next step is calculated by solving the equations of motion (Eq. (3)) using the fourth-order Runge-Kutta method. (3) After the new displacement of the cylinders is known, the displacement and the velocity of the mesh are calculated by solving Eq. (4) using the Galekin Finite Element Method. (4) The above three steps are repeated until the last time step.
3. Numerical results A square computational domain is used in the simulation. The size of the computational domain is 110D in both the cross flowdirection and the flow directions and the large cylinder is located at the center of the domain. The blockage of the computational domain (defined as the ratio of D + d to the fluid domain width) is 0.011. Anagnostopoulos and Minear [2]conducted a study of the blockage effects on the forces on cylinders in an oscillatory flow 0.1 ≤ KC ≤ 6. It was found that the blockage effect is negligible if the blockage is less than 0.2 and the blockage effects for various KC numbers are consistent. The blockage ratio used in this study is only 0.0545 times the one suggested by Anagnostopoulos and Minear [2] to ensure that the motion of the vortices that are shed from the cylinders are not affected by the side boundaries of the computational domain. The whole computational domain is divided into 38480 quadrilateral finite elements. Fig. 1 (b) shows the computational mesh near the cylinder surfaces for G/D = 0 and ˛ = 45◦ . The surfaces of the large and small cylinders are discretised into 212 and 80 elements, respectively. The minimum mesh size in the radial direction next to the cylinder surfaces is 0.0007D. In all the simulations, the Reynolds number, the KC number and the damping ratio are 20000, 10 and 0, respectively. Simulations are carried out for three gap ratios of G/D = 0, 0.1 and 0.2, three arrangements of ␣=0◦ , 45◦ and 90◦ (referred to as tandem, staggered and side-by-side arrangements, respectively) and reduced velocities ranging from 1 to 20. A mesh dependency study is conducted to ensure the meshes used are fine enough to get converged numerical results. In the mesh dependency study, simulations for G/D = 0.1, ␣ = 45◦ and two reduced velocities of 8 and 20 are conducted at a denser mesh with a radial minimum mesh size of 0.0003D on the cylinder surfaces. The numbers of the elements along the large and small cylinders in the dense mesh are 256 and 96, respectively. ˛ = 45◦ is chosen because
Fig. 2. Comparison between the displacements calculated from normal mesh and dense mesh for G/D = 0.1, ˛ = 45◦ and two reduced velocities of Vr = 8 and 20.
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Fig. 3. Time histories of the displacement of the cylinder for ˛ = 0◦ . The spacing between the horizontal gridlines is Y/D = 1.
the vibration at this attack angle is periodic, making it easy to compare the results from different meshes. Vr = 8 is the reduced velocity where the vibration amplitude reaches its maximum and Vr = 20 is the largest reduced velocity simulated in this study. Fig. 2 is a comparison between the displacements of the cylinders calculated from the normal and dense meshes. In the figure, the normal mesh is the mesh used in all the simulations. The difference between the vibra-
tion amplitudes calculated from both meshes is less than 0.5% for both reduced velocities. No experimental data of FIV of two cylinders in an oscillatory flow were found in literature for validating the numerical model. However, the validation of the present numerical model on the FIV of a cylinder in other similar cases has been performed in a number of previous studies. Using the same numerical model, Zhao and
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Fig. 4. Variation of the maximum displacement with the reduced velocity for ˛ = 0◦ , 45◦ and 90◦ .
Cheng [23] achieved excellent agreement between the numerical results of FIV of a cylinder in steady flow and the experimental data by Jauvtis and Williamson [5]. The validity of the model on FIV of a cylinder in oscillatory flow are demonstrated in Zhao et al. [25]. No further numerical validation will be performed in this paper.
3.1. Tandem arrangement (˛ = 0◦ ) Fig. 3 shows the time histories of the displacement of the cylinders for ␣=0◦ and reduced velocities from 2 to 20 with an interval of 2, together with the results in the case of a single cylinder. The reduced velocity for two cylinders of different diameters is defined based on the diameter of the large cylinder as Vr = Um /(fn D). The static balance positions of the cylinders for different reduced velocities are marked as horizontal gridlines and the reduced velocities are labelled at the right hand side of diagrams in Fig. 3. In Fig. 3 (a), the vibration of a single cylinder changes between a low frequency, high amplitude stage and a high frequency, low amplitude stage intermittently as Vr = 12. As Vr ≥ 16, the vibration becomes stable and periodic. The mean position of the cylinder is found to be either positive or negative, because vortices are shed from only one side of the cylinder. If the two cylinders of different diameters are in a tandem arrangement in the fluid flow, the periodicity of the vibration was broken when the reduced velocity exceeds a value of Vr = 10. It is interesting to see in Fig. 3(b)–(d) that the mean position of the two
cylinders changes between positive and negative values intermittently when Vr > 14. It has been demonstrated that vortices only shed from one side (top or bottom) of the cylinder if KC = 10, forming the so-called transverse vortex street [21]. The transverse vortex street leads to a mean lift force pointing to one side of the cylinder. The changing of the mean position of the cylinders from negative to positive or the other way around is due to the switching of the vortex shedding side of the cylinder. When the mean position of the cylinder is changing its sign, the vibration displacement becomes high and recovers gradually. For example, during t = 40 to 50 for Vr = 20, when the mean position of the cylinder is changing from positive to negative, the displacement of the cylinder is more than doubled and gradually changes to its normal condition. Fig. 4(a) shows the variation of the maximum displacement of the two tandem cylinders with the reduced velocity. To make it convenient to compare the maximum displacements for different ␣, the maximum displacements for ˛ = 45◦ and 90◦ are presented in Fig. 4(b) and (c) also and will be discussed in the next two sections. The maximum displacement is defined as Amax = max(Ymax , − Ymin ), where Ymax and Ymin are the maximum and minimum displacements, respectively, when the vibrations have stabilised after 30% of the calculated time, which is at least 150 periods of the oscillatory flow. The variation trend of the maximum vibration displacement of the two cylinders with the reduced velocity appears similar to that of a single cylinder until Vr = 10. The vibration amplitude of a single cylinder decreases with increasing Vr as 11 < Vr < 15,
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Fig. 5. Contour of the real part of the wavelet transform of the vibration displacement for ˛ = 0◦ . The corresponding time histories of the vibration are added on the top of each diagram.
and slowly increases as Vr > 15, while that of two tandem cylinders increases significantly as Vr > 11 with a maximum amplitude of triple that of the single cylinder. The maximum vibration displacement continuously increases with increasing Vr , mainly because of high amplitude vibration during the time when the vortex shedding switches between the top and bottom side of the cylinders. It can be seen in Fig. 3 that each time after the mean position of the cylinder changes its sign, the magnitude of the deflection of the cylinder is large and reduces gradually until the vortex shedding side changes again. From Fig. 3(b)–(d) it can be clearly seen that neither the vibration nor the amplitude of the vibration remains constant during the vibration at large reduced velocities. To see how the vibration frequency varies with time, the Morlet wavelet analysis of the vibration displacement is conducted and Fig. 5 shows the contours of the real part of the wavelet transform of the vibration displacement for two typical cases where the vibration is stable and two cases where the vortex shedding is switching intermit-
tently between different patterns. The time history of the vibration displacement is added on the top of each wavelet diagram in Fig. 5. When ˛ = 0◦ , the vibration of the cylinders mainly has two frequency components of f/fw = 1 and 2, where fw is the oscillatory flow frequency, and the contribution of these two frequency components vary with time. Whenever the amplitude of f/fw = 1 increases, the amplitude of f/fw = 2 decreases, and vice versa. The vibration for G/D = 0.1 and Vr = 12 is periodic and dominated by a frequency of f/fw = 0.5, indicating that the vibration repeats every two cycles of the flow. As the reduced velocity is increased to 14, the vibration becomes very irregular and mainly contains three frequencies of f/fw = 0.5, 1 and 2. However, only one or two frequencies coexist at a time. The vibration of Vr = 14 actually irregularly changes between the three frequency modes. To see how the reduced velocity and the gap ratio affect the frequency of the vibration for all of the simulated cases, the Fast Fourier Transform (FFT) spectra of the displacements are shown in Fig. 6. To see the correlation between the vibration and the
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Fig. 6. FFT spectra of the displacement for ˛ = 0◦ . The displacement in each spectrum is normalized by the amplitude corresponding to the highest peak of the spectrum.
lift coefficient, the FFT spectra of the total lift coefficient on both cylinders are shown in Fig. 7. The total lift coefficient is defined as 2 /2). The ratios of the natural frequency of the cylinCL = Fy /(DUm ders measured in fluid to the oscillatory flow frequency (fnw /fw ) are marked as square symbols on the horizontal axes in Fig. 6 and 7. While the natural frequency measured in vacuum (structural natural frequency) fn is used as an input parameter in the numerical simulations, the natural frequency measured in water fnw is calm∗ /(CA + m∗ ), where culated using the relationship of fnw /fn = CA = 1 is the added mass coefficient. The added mass coefficient CA was found to be 1 with an error less than 5% in the experiments by Khalak and Williamson [6] As the reduced velocity is 1, the flow pattern should be the same as that of two stationary cylinders since the vibration amplitude is almost zero (Fig. 4), i.e. a pair of vortices are shed from the cylinders in one oscillatory period. Because the flow direction reverses twice within one oscillatory flow period, the vibration frequency is twice the oscillatory frequency and so is the lift force [16]. The vibration frequency is found to follow the natural frequency in water when the reduced velocity is between 2 and 4 for all the gap ratios. As the reduced velocity exceeds 5, the vibration frequency of a single cylinder locks onto fw as 9 ≤ Vr ≤ 14 and 2fw for higher reduced velocities. In a tandem arrangement, the vibration is found to be dominated by 0.5fw at high reduced velocities. It should be noted that the 0.5fw does not always dominate the vibration throughout the simulated vibration time. In fact, the vibration frequency changes between 0.5fw , fw and 2fw , intermittently. The frequency of 0.5fw dominates in a FFT spectrum because the amplitude when the vibration frequency is 0.5fw is higher than those when the vibration frequency is fw and 2fw , as shown in the wavelet analysis results in Fig. 5 (c). For large reduced velocities,
the vibration of the cylinders in tandem is mainly in a multiple frequency mode, and each frequency is multiple of 0.5fw . Same as the vibration displacement, the lift coefficient of two cylinders in tandem also has multiple frequencies that are multiples of 0.5fw . It appears that the lift coefficient is always dominated by the frequency of 2fw . The predominant frequencies of the displacement and the lift coefficient are different from each other at large reduced velocities. For example, the vibration frequency is close to either 0.5fw or fw as Vr ≥ 10 and G/D = 0.1, while the dominant frequency of the lift coefficient remains to be 2fw . The dominant frequencies of the vibration and the lift coefficient are different from each other mainly because the non-dominant frequency of the lift coefficient has locked onto the natural frequency and resulted in the excitation of the vibration. From Fig. 5 and 6 it can be seen that although the vibration histories are seemingly very chaotic and irregular, distinct peak frequencies can be identified from the spectra of the vibration displacement and lift coefficient. It has been found that the vibration frequency of a single cylinder in an oscillatory flow is a multiple of the oscillatory flow frequency [18,9]. Based on the frequency of the vibration, Zhao [22] classified the vibrations into single-, double-, triple- frequencies modes, etc. It is well known that for a stationary cylinder in an oscillatory flow with KC = 10, the vortex shedding flow pattern is a single pair flow pattern and the frequency of the lift coefficient is twice the oscillatory frequency [13,21]. If the frequency of the lift coefficient is close to the natural frequency, the vibration of the cylinder increases (Vr = 4 to 6 in Fig. 4 (a)) and the vibration frequency will be the same as the frequency of the lift coefficient. As Vr = 7 and 8, although both the frequencies of the displacement and the lift coefficient are 2fw , the vibration amplitude is reduced because the difference between the frequency of the lift coefficient and the nat-
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Fig. 7. FFT spectra of the lift coefficient for ˛ = 0◦ . The lift coefficient in each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
Fig. 8. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 5 within one oscillatory flow period.
ural frequency has increased significantly. With further increasing reduced velocity, the vibration frequency becomes fw because fw is close to the natural frequency fnw . The vibration frequency is closely related to the vortex shedding flow pattern, which is presented by the contours of the nondi-
mensional vorticity defined as ω = (∂v/∂x − ∂u/∂y)/(Um /D). Fig. 8–10 shows the contours of the nondimensional vorticity as G/D = 0.1 and Vr = 5, 11 and 12, respectively. The vortices in the figures are given labels to facilitate the discussion. The vortex shedding for Vr = 5 is a typical one pair regime. After the flow reverses at t/Tw = 114, vor-
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Fig. 9. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 11 within one oscillatory flow period.
Fig. 10. Contours of the nondimensional vorticity for ˛ = 0◦ , G/D = 0.1 and Vr = 12 within one oscillatory flow period.
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Fig. 11. Time histories of the displacement of the cylinder for ˛ = 45◦ . The spacing between the horizontal gridlines is Y/D = 1.
tex A generated from the bottom side of the cylinder moves along the cylinder surface to the top side and is shed from the cylinder at t/Tw = 114.5. Another vortex B will be shed from the cylinder in the second half period. Because only two vortices are generated and shed from the cylinder and they are all shed from the top side of the cylinder in one cycle of vibration, the vortex shedding regime is called one pair regime or transverse vortex shedding regime. The frequencies of the cylinder displacement and the lift coefficient are the same in the one pair regime as shown in Fig. 8(f). When Vr = 11, vortex A is generated and shed from the top side of the cylinder, while vortex B is generated from the bottom side but is shed from the top side of the cylinder in the first half oscillatory flow period. The vortex shedding in the first half oscillatory flow period (during t/Tw = 103 and 103.5) is similar to its corresponding pattern for Vr = 5, i.e. one vortex A is shed from the cylinder. In the second half flow period, two vortices C and D are generated and separated from the bottom and top sides of the cylinder, respectively, forming a symmetric vortex flow pattern as shown in Fig. 9(d). In the next
half cycle, vortices E and F in Fig. 9(e) will repeat what vortices A and B did in Fig. 9(a), respectively. Comparing Fig. 9(b) with (d) it can be seen that the vortex flow patterns in two halves of a flow period are very different from each other, resulting in a vibration frequency the same as the oscillatory flow frequency. The single pair regime and the regime of ‘paring of attached vortices’ are found to switch from one to another after every half flow period. The vortex shedding pattern in Fig. 9(b) is very similar to that of the single pair regime. The vortex shedding pattern in Fig. 9(d), where a pair of vortices (C and D in Fig. 9) are formed before flow reverses, is very similar to the regime of ‘Pairing of attached vortices’ observed by [21]. After the flow direction reverses, the pair of vortices is split up by the newly generated pair of smaller vortices (E and F in Fig. 9). As Vr = 12, the vibration frequency is 0.5 times the oscillatory flow frequency because the vortex shedding flow repeats every two flow periods. The vortex flow pattern from t/Tw = 127.5 and 128.5 are very similar to that of the single pair regime. Transverse vortex flow patterns can be clearly identified in Fig. 10(c) and (e). The vor-
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Fig. 12. FFT spectra of the displacement and the lift coefficient of the cylinder for ˛ = 45◦ . Each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
Fig. 13. Contours of the nondimensional vorticity for ˛ = 45◦ , G/D = 0 and Vr = 9 within one oscillatory flow period.
tex flow pattern from t/Tw = 128.5 to t/Tw = 129.5 are in the regime of ‘pairing of attached vortices’ (see Fig. 10(h) and (j)). The lift coefficient for Vr = 12 is also found to repeat after every two flow periods. The vortex shedding in Fig. 10 is also a combination of single pair regime and the regime of ‘pairing of attached vortices’. The difference between the flow patterns in Fig. 9 and 10 is that the flow repeats every half period in Fig. 9 and every whole period
in Fig. 10. To differentiate from each other, the flow regimes in Fig. 9 and 10 are defined as combined flow regime 1 and combined flow regime 2, respectively. The single pair flow regime has a vibration frequency of 2fw , the combined flow regime 1 has a vibration frequency of fw and the combined flow regime 2 has a vibration frequency of 0.5fw . In many cases the vortex shedding flow switches from one regime to another alternatively, and which regime the
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Fig. 14. Contours of the nondimensional vorticity for ˛ = 45◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period.
flow is in can be determined based on the instantaneous vibration frequency. For example as Vr = 14 and G/D = 0.1 (see Fig. 5(d)), the flow is in the single pair regime during t/Tw = 75 to 90 (f/fw = 2), combined regime 1 during t/Tw = 50 to 60 (two frequencies coexist but f/fw = 1 dominates) and combined regime 2 during t/Tw = 130 to 145 (f/fw = 0.5 dominates). If alternative switching of the flow between flow regimes occurs, the FFT spectrum shown in Fig. 6 has multiple distinct frequencies as the vibrations change frequencies and amplitudes as the flow switches regimes. 3.2. Staggered arrangement (˛ = 45◦ ) Fig. 11 shows the time histories of the vibration displacement of the cylinders for three gap ratios and reduced velocities from 2 to 20 for ␣=45◦ . The vibrations for all the three simulated gaps are periodic at ␣=45◦ , indicating the vibration and the vortex shedding flow remains unchanged. The periodicity of the vibration can also be evidenced by the FFT spectra of the vibration displacement and the lift coefficient shown in Fig. 12. Each of the FFT spectra in Fig. 12 has very sharp spikes with frequencies that are multiples of the oscillatory frequency. Due to the asymmetric configuration of the system in both the x- and y-directions, the vibration histories in two halves of one oscillatory flow period are always different from each other as Vr ≥ 7, resulting in a dominant vibration frequency same as fw . There is a secondary frequency component of 2fw in the vibration because the vortex shedding pattern is in the one pair regime. Fig. 13 and 14 show the contours of the nondimensional vorticity for Vr = 9 and G/D = 0 and 0.2, respectively. It is clear that two vortices (labelled as vortices A and B) are shed from the cylinder in one period of the oscillatory flow for both gap ratios. When G/D = 0 and Vr = 9, a positive vortex A is generated from the top of the cylinder, moves to the bottom of the cylinder and is shed from the bottom side of the cylinder as shown in Fig. 13(a) to (c). In Fig. 13(b) to (e) another vortex B is generated from the top side of the cylinder, moves to the bottom side of the cylinder and is shed from the bottom side of the cylinder. Vortex A is much stronger than Vortex B because of two reasons. Firstly, vortex A is strengthened because after flow reverses (Fig. 13(c)), the vortex A, that has been shed from the cylinder is convected back by the flow and combines with the newly generated vortex A. Secondly, vortex B is generated from the top side of the cylinder (Fig. 13(b)) does not grow much due to the influence of the small cylinder and it is weakened by the small
cylinder when it moves along the right surface of the large cylinder to the bottom side. The difference in the lift coefficients produced by vortices A and B can be clearly seen in Fig. 13(f). The vortex shedding for Vr = 9 and G/D = 0 is similar to that for G/D = 0.2 with the same reduced velocity. Compared with G/D = 0, the effect of the small cylinder for G/D = 0.2 on the flow is weaker, the strengths of Vortex A and B are more balanced and the difference between the lift coefficients generated by vortices A and B is smaller. The variation of the maximum vibration displacement Amax /D for ␣=45◦ is similar to that for ␣=0◦ as shown in Fig. 4. Two maximum values occur at Vr ≈ 4 and 9, respectively. The vibration amplitude reaches its maximum at Vr ≈ 4 and 9 because 2fw and fw are close to the natural frequency of the cylinder in water, respectively as shown in Fig. 12. The maximum displacement is increased significantly at Vr = 9 and G/D = 0 because of the very strong positive vortex A as shown in Fig. 12. By observing Fig. 11, it appears that the asymmetry of the configuration leads to periodic and regular vibration, because the instability of changing between two flow patterns is only likely to occur in a symmetric configuration.
3.3. Side-by-side arrangement (˛ = 90◦ ) Fig. 15(a)–(c) shows the time histories of the vibration displacement for ␣=90◦ . It appears that the switching between different vibration modes occurs very frequently in the asymmetric configuration of ␣=90◦ , especially as G/D = 0.2. The switching between different vibration frequencies can be clearer seen in Fig. 15(d), where the contours of the real part of the wavelet transform of the vibration displacement for Vr = 13 are shown. The vibration is found to switching from double-frequency to single-frequency modes alternatively. Fig. 16 shows the FFT spectra of the displacement and the lift coefficient for ␣=90◦ and two gap ratios of G/D = 0 and 0.2. Same as the ones for ␣=0◦ and 45◦ , the nondimensional frequency of the displacement f/fw decreases with increasing reduced velocity until Vr = 5. As G/D = 0, the predominant frequency of the displacement is f/fw = 1 in the reduced velocity range between 8 and 15 and f/fw = 2 when Vr ≥ 16. As G/D = 0.2, it appears that the frequencies of f/fw = 1 and 2 equally dominate the vibration as Vr ≥ 16. This is because the vibration switches between high frequency vibration and low frequency vibration intermittently as shown in the time histories
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Fig. 15. Time histories of the vibration displacement for ˛ = 90◦ . The spacing between the horizontal gridlines is Y/D = 1.
of the displacement in Fig. 15. The switching of the vibration from one mode to another occurs very frequently when G/D = 0.2. Fig. 17 shows the contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period. It can be seen that the flow is dominated by a negative vortex A, in the whole vibration cycle. In Fig. 17(a), the flow starts to reverse and the cylinders have almost moved to their lowest position. This allows the negative vortex A to be convected to the right hand side from the top of the cylinder. When the flow reverses at t/Tw = 178.5, the cylinders are almost at their highest position and vortex A moves to
the left hand side from the bottom of the cylinders. The large vortex A keeps attached to the cylinders throughout the whole cycle of vibration. In addition of very weak vortices B and C, vortex shedding of large vortices from the cylinder is not observed. Because the dominant frequency of the lift coefficient is the same as the oscillatory flow frequency (Fig. 17(f)) and close to the natural frequency fnw (Fig. 16(c)), high amplitude vibration is achieved when Vr = 9. Fig. 18 shows the contours of the nondimensional vorticity for ␣=90◦ , G/D = 0.2 and Vr = 16 within one oscillatory flow period. The vibration has two frequencies of f/fw = 1 and 2 as shown in Fig. 18
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Fig. 16. FFT spectra of the displacement and the lift coefficient of the cylinder for ˛ = 90◦ . Each spectrum is normalized by the amplitude corresponding to the highest peak value of the spectrum.
(f) and is slightly dominated by f/fw = 1. The vortex shedding in Fig. 18 is a typical one pair vortex shedding regime. In one period of the oscillatory flow, two vortices A and B are shed from the top of the two cylinders. This one pair vortex shedding pattern makes the frequency of the lift coefficient twice the oscillatory flow frequency. Because 2fw is very different from the natural frequency fnw as Vr = 16 (see Fig. 16(c)), the vibration amplitude in Fig. 18 is much smaller than that in Fig. 17. The switching between the vortex patterns shown in Fig. 17 and 18 are observed and occur very frequently when G/D = 0.2 and reduced velocity is greater than 11. Fig. 19 is an example of changes of the vortex flow pattern when ␣=90◦ , G/D = 0.2 and Vr = 13. During the oscillatory flow period of t/Tw = 157–158, the vortex shedding is in a one pair regime and vortices A and B are shed from the bottom side of the two cylinders. In the period starting from t/Tw = 189, the flow is in a no-shedding regime similar to the one shown in Fig. 18. The vortex shedding in the period starting from t/Tw = 246 is also in a one-pair pattern. Different from those in Fig. 19(a) and (b), the vortices in Fig. 19(e) and (f) are shed from the top side of the cylinder, instead of the bottom side. The vibration amplitude becomes small whenever the vortex shedding is in a one pair regime and large whenever the vortex shedding is in a no-shedding regime. 4. Discussion The vibration is periodic for all the simulated reduced velocities for ␣=45◦ . The configuration of ␣=45◦ is asymmetric in both the x- and y-directions, making it difficult to produce a bi-stable
flow. When ␣=0◦ , the periodicity of the vibration is broken when Vr ≥ 14. Irregular vibrations are also found when ␣=90◦ and at large reduced velocities. Periodic vibration occurs when the natural frequency of the system measured in water is close to the oscillatory flow frequency fw or a multiple number of fw . If a cylinder is stationary, the vortex shedding is in a transverse vortex street at KC = 10, i.e. a pair of vortices are only shed from one side of the cylinder in every period of the vibration. The vibration becomes irregular at large reduced velocities mainly because the vortex shedding side of the cylinder changes between the two sides of the cylinder very randomly and the flow is disturbed significantly during the changes. Before the vortex shedding side of the cylinder changes completely, a no-shedding regime is found and may last a number of oscillatory flow periods. The vibration frequency changes at large reduced velocities for both ␣ = 0◦ and 90◦ mainly because the vortex shedding mode changes. More vortices are shed in one period of vibration, higher the vibration frequency. The effects of Vr on the maximum displacement, Amax /D can be found in Fig. 4. A maximum Amax /D occurs in the reduced velocity range of 4 ≤ Vr ≤ 6 and a minimum Amax /D occurs as 6 ≤ Vr ≤ 8 for all the simulated gap ratios and flow attack angles. The variation of Amax /D with Vr for different flow attack angles are different from each other as Vr ≥ 8. At the largest simulated reduced velocity of Vr = 20, Amax /D continues increases with increasing Vr when ␣ = 0◦ , Amax /D continues increases with increasing Vr but with smaller rate when ␣ = 45◦ and Amax /D overall decreases with increasing Vr as ␣ = 90◦ . The variation of Amax /D with Vr is very similar to that of a single cylinder as ␣ = 0◦ and 45◦ .
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Fig. 17. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 9 within one oscillatory flow period.
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Fig. 18. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 16 within one oscillatory flow period.
5. Conclusion The effects of the flow approaching angle on the vibration of two cylinders of different diameters in oscillatory flow are investigated through two-dimensional numerical simulations for tandem, side-by-side and staggered arrangements of the cylinders and three gap ratios of 0, 0.1 and 0.2. The KC number remains constant at KC = 10, which belongs to the single pair vortex shedding regime for a stationary cylinder. The main conclusions are summarized as follows. The striking phenomenon found for the tandem arrangement is that the vortex shedding changes between three regimes intermittently as the reduced velocity exceeds 10. These three modes are: (1) a pure single pair regime and two combined regimes 1 and 2. A combined regime is a combination of the single pair regime and the regime of pairing of attached vortices. In combined regime 1, the vibration frequency is fw and the two regimes repeat every half flow period. In combined regime 2, the vibration frequency is 0.5fw and the two regimes repeat every two periods. If the flow is in a pure single pair regime, the vibration frequency is twice the oscillatory flow frequency. The vibration displacement is found to be increases significantly immediately after the flow switches from one regime to another. All the frequencies of the three regimes (0.5fw , fw and 2fw ) can be seen in the FFT spectra of the displacement.
The vibration for a staggered arrangement with ˛ = 45◦ is perfectly periodic and stable for all the calculated reduced velocities. The vortex shedding for ˛ = 45◦ is always in the single pair regime and the vibration has two frequencies of fw or 2fw . The vibration with G/D = 0.2 is more dominated by 2fw than that with G/D = 0. The switching between different flow regimes is also observed when ˛ = 90◦ . In addition to the single pair flow regime, an interesting single vortex regime is found where a single vortex always attaches to the cylinders and rotates around the cylinders. This single vortex regime occurs at intermediate reduced velocities and the lift coefficient frequency is the same as the oscillatory flow frequency at this flow regime. The vibration may change between the single vortex regime and one pair regime alternatively in some cases.
Acknowledgements The authors would like to acknowledge the support from the Natural Science Foundation of China (Grant 51628091) and the Australian Research Council (Grant LP150100249). The calculations were carried out on the computational facilities of Intersect Australia Ltd. in NSW, Australia.
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Fig. 19. Contours of the nondimensional vorticity for ˛ = 90◦ , G/D = 0.2 and Vr = 13 within one oscillatory flow period.
References [1] H. An, L. Cheng, M. Zhao, Direct numerical simulation of oscillatory flow around a circular cylinder at low Keulegan?Carpenter number, J. Fluid Mech. 666 (2011) 77–103. [2] P. Anagnostopoulos, R. Minear, Blockage effect of oscillatory flow past a fixed cylinder, Appl. Ocean Res. 26 (2004) 147–153. [3] H. Dütsch, F. Durst, S. Becker, H. Lienhart, Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers, J. Fluid Mech. 360 (1998) 249–271. [4] G. Iliadis, P. Anagnostopoulos, Numerical visualization of oscillatory flow around a circular cylinder at Re = 200 and KC = 20–An aperiodic flow case, Commun. Numer. Methods Eng. 14 (1998) 181–194. [5] N. Jauvtis, C.H.K. Williamson, The effect of two degrees of freedom on vortex-induced vibration at low mass and damping, J. Fluid Mech. (2004) 23–62. [6] A. Khalak, C.H.K. Williamson, Motions: forces and mode transitions in vortex-induced vibrations at low mass-damping, J. Fluids Struct. 13 (1999) 813–851. [7] N.W.M. Ko, P.T.Y. Wong, Flows past two circular cylinders of different diameters, J. Wind Eng. Ind. Aerodyn. 41 (1992) 563–564. [8] N.W.M. Ko, P.T.Y. Wong, R.C.K. Leung, Interaction of flow structures within Bistable flow behind two circular cylinders of different diameters, Exp. Therm. Fluid Sci. 12 (1996) 33–44. [9] A. Kozakiewicz, B.M. Sumer, J. Fredsøe, E.A. Hansen, Vortex regimes around a freely vibrating cylinder in oscillatory flow, Int. J. Offshore Polar Eng. 7 (1997) 94–103.
[10] K.M. Lam, P.T.Y. Wong, N.W.M. Ko, Interaction of flows behind two circular cylinders of different diameters in side-by-side arrangement, Exp. Therm. Fluid Sci. 7 (1993) 189–201. [11] M. Mahbub Alam, Y. Zhou, Strouhal numbers: forces and flow structures around two tandem cylinders of different diameters, J. Fluids Struct. 24 (2008) 505–526. [12] R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32 (1993) 1598–1605. [13] E.D. Obasaju, P.W. Bearman, J.M.R. Chraham, A study of forces: circulation and vortex patterns around a circular cylinder in oscillating flow, J. Fluid Mech. 196 (1988) 467–494. [14] M. Rahmanian, M. Zhao, L. Cheng, T. Zhou, Two-degree-of-freedom vortex-induced vibration of two mechanically coupled cylinders of different diameters in steady current, J. Fluids Struct. 35 (2012) 133–159. [15] T. Sarpkaya, Force on a circular cylinder in viscous oscillatory flow at low Keulegan—Carpenter numbers, J. Fluid Mech. 165 (1986) 61–71. [16] T. Sarpkaya, M. Storm, In-line force on a cylinder translating in oscillatory flow, Appl. Ocean Res. 7 (1985) 188–196. [17] P. Scandura, V. Armenio, E. Foti, Numerical investigation of the oscillatory flow around a circular cylinder close to a wall at moderate Keulegan-Carpenter and low Reynolds numbers, J. Fluid Mech. 627 (2009) 259–290. [18] B.M. Sumer, J. Fredsøe, Transverse vibrations of an elastically mounted cylinder exposed to an oscillating flow, J. Offshore Mech. Arct. Eng. 110 (1988) 387–394. [19] J. Thapa, M. Zhao, L. Cheng, T. Zhou, Three-dimensional flow around two circular cylinders of different diameters in a close proximity, Phys. Fluids 27 (2015) 085106. [20] T. Tsutsui, T. Igarashi, K. Kamemoto, Interactive flow around two circular cylinders of different diameters at close proximity: experiment and
190
T. Pearcey et al. / Applied Ocean Research 69 (2017) 173–190
numerical analysis by vortex method, J. Wind Eng. Ind. Aerodyn. (1997) 279–291 (69–71). [21] C.H.K. Williamson, Sinusoidal flow relative to circular cylinders, J. Fluid Mech. 155 (1985) 141–174. [22] M. Zhao, Numerical investigation of two-degree-of-freedom vortex-induced vibration of a circular cylinder in oscillatory flow, J. Fluids Struct. 39 (2013) 41–59. [23] M. Zhao, L. Cheng, Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary, J. Fluids Struct. 27 (2011) 1097–1110.
[24] M. Zhao, L. Cheng, B. Teng, G. Dong, Hydrodynamic forces on dual cylinders of different diameters in steady currents, J. Fluids Struct. 23 (2007) 59–83. [25] M. Zhao, L. Cheng, T. Zhou, Numerical investigation of vortex-induced vibration (VIV) of a circular cylinder in oscillatory flow, Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering − OMAE, 2011 (2011) 597–603.