Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers

Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers S.P. Singh a,n, G. Biswas b,1 a b

Simulation and Modelling Laboratory, CSIR-CMERI, Durgapur, WB 713 209, India Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, UP 208 016, India

a r t i c l e i n f o

abstract

Article history: Received 31 March 2012 Received in revised form 8 March 2013 Accepted 25 March 2013

Vortex induced vibration (VIV) of an elastically mounted square cylinder of low nondimensional mass is simulated at subcritical Reynolds numbers (Re), i.e., Re≤50. The cylinder is allowed to vibrate in the transverse direction to the incoming flow. Four cases are considered for understanding the behavior of VIV of the square cylinder at subcritical Re. In the first case, the non-dimensional frequency varies as 3.1875/Re. In the second case, the non-dimensional frequency is kept constant at 0.1333. In both the cases, Re is varied. In third and fourth cases, studies are conducted for Re¼ 25, 30 and 35, and Re¼ 80, respectively. In the third and fourth cases, the non-dimensional velocity, Un, is varied. It is found that maximum transverse displacement is approximately 0.15D when the nondimensional frequency, Fn, varies with Re. The maximum transverse displacement is 0.25D when the non-dimensional frequency is constant. For the first case, VIV starts at Re as low as 23.9 and it ceases at Re∼33:5. In all these cases, it is observed that the phase difference between the lift coefficient and the transverse displacement depends upon non-dimensional mass, Re, and non-dimensional velocity. In all the cases, the lock-in phenomenon is observed. In the fourth case of supercritical Re, hysteresis is also observed and it is seen that its extent depends upon non-dimensional mass. Stabilized finite-element space–time formulations (SUPG and PSPG) are utilized to solve the two-dimensional incompressible Navier–Stokes equations together with the equations of motion of the body. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Unsteady flows Hysteresis Vortex shedding Finite elements

1. Introduction The study of the unsteady wake of bluff bodies of various cross sections is very useful in understanding the flow past the complicated structures. The square and circular cylinders are the most studied cross sections. Both the cylinders show similar type of instabilities but the flow separation points for the square cylinder are the corners (endpoints) of the leading edges or the rear edges depending on the values of various parameters. For a circular cylinder, the separation points always vary depending on the pertinent flow parameters, such as, Re, roughness of the cylinder surface, etc. The shedding frequency and the drag and lift coefficients differ significantly for the square and the circular cylinders. If a cylinder with either type of cross-section is elastically mounted with a damper, it can undergo VIV due to the unsteady lift and drag forces

n

Corresponding author. Tel.: +91 343 6510452; fax: +91 343 2548204. E-mail addresses: [email protected], [email protected] (S.P. Singh), [email protected] (G. Biswas). 1 On deputation at CSIR-CMERI (Central Mechanical Engineering Research Institute), Durgapur, WB 713 209, India.

0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011

Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

2

acting upon either type of bluff body. VIV of structures occurs in heat exchanger tubes, riser tubes, bridges, chimney stacks, and marine and land vehicles. Buffoni (2003) has experimentally studied the phenomena of the vortex shedding in subcritical conditions for a circular cylinder. It was discovered that vortex shedding could be triggered under subcritical Re by imparting low-amplitude transverse vibrations to a cylinder through the incoming flow. For supercritical Re (Re 4 49), the Strouhal number variation with Reynolds number can be expressed by the equation of Roshko (1954): as St ¼0.21−4.5/Re. It is observed by Buffoni (2003) that the relationship between the Strouhal number and Re is St ¼ ð1=406Þ  ðγ 2 −2γ þ 2ÞRe for vibrations at Re o 49. Here γ ¼ Re0 =Re and Re0 is equal to 49. Cossu and Morino (2000) have done global stability analysis of such an aero-elastic system and found that the critical Re for the cylinder, when the ratio of the density of the solid to the fluid is smaller than 70, is less than half of the critical Reynolds number for the stationary case. Mittal and Singh (2005) investigated VIV of an elastically mounted circular cylinder of low non-dimensional mass at subcritical Re. The cylinder is allowed to move in both the directions, i.e., transverse as well as in-line directions. They have found that vortex shedding and self-excited vibrations of the cylinder are possible for Re as low as 20 for certain natural frequencies of the spring-mass system. They have plotted the variation of the Strouhal number with Re. The Strouhal number variation is considered for various cases including fixed non-dimensional velocity, mass, forced vibrations as well as stationary cylinders. Reddy (2007) has carried out numerical simulations of VIV of a square cylinder with the blockage ratio of approximately 0.1. The topics of vortex induced vibration and the related topics are studied widely by Sumner (2010), Huera-Huarte and Gharib (2011a, 2011b, 2011c), Carmo et al. (2011), Bearman (2011), Etienne and Pelletier (2012), Nguyen et al. (2012) and Vandiver (2012). In the present paper, results of VIV for a square cylinder at subcritical Re, i.e., Re o 50 (Sohankar et al., 1998) are presented for low non-dimensional mass mn ¼10 and/or 5, with the value of structural damping coefficient assigned to zero. The square cylinder is allowed to move in the transverse direction to the incoming flow. Four sets of computations were carried out to observe the VIV of a square cylinder and compare the results with their counterpart for a circular cylinder flow. In the first set, Fn varies as 3.1875/Re, while it is constant at 0.1333 for the second case. In the third set of the simulations, VIV is examined at Re¼25, 30 and 35, by varying the non-dimensional velocity Un. In the fourth set of the simulations, VIV is studied at Re¼80, which is in the supercritical range. In addition to these observations, the dependence of phase difference between the lift coefficient and the transverse displacement on various governing parameters such as, Re, the nondimensional velocity, mass are observed and reported in the present work. The paper is organized in the following way. The problem and mesh description are presented followed by the boundary conditions. Then the results for the stationary square cylinder are reported for the purpose of validation. This section is followed by the VIV results, which consist of the results for the above-mentioned cases.

2. The governing equations 2.1. The incompressible flow equations Let Ωt ⊂Rnsd and (0,T) be the spatial and temporal domains, respectively, where nsd is the number of space dimensions and let Γ t denote the boundary of Ωt . The spatial and temporal coordinates are denoted by x and t, respectively. The Navier– Stokes equations governing incompressible fluid flow are
 ∂u þ u  ∇u−f −∇  r ¼ 0 ρ ∂t ∇u¼0

on Ωt for ð0; TÞ;

on Ωt for ð0; TÞ:

ð1Þ

ð2Þ

Here ρ, u, f and r are the density, velocity, body force and the stress tensor, respectively. The stress tensor is written as the sum of its isotropic and deviatoric parts r ¼ −pI þ T;

T ¼ 2μεðuÞ;

εðuÞ ¼ ðð∇uÞ þ ð∇uÞT Þ;

ð3Þ

where p and μ are the pressure and dynamic viscosity, respectively. Both Dirichlet and Neumann-type boundary conditions are accounted for, represented as u ¼ g on ðΓ t Þg ;

n  r ¼ h on ðΓ t Þh ;

ð4Þ

where ðΓ t Þg and ðΓ t Þh are complementary subsets of the boundary Γ t and n is its outward unit normal vector, respectively. The initial condition on the velocity is specified on Ωt at t¼ 0 uðx; 0Þ ¼ u0

on Ω0 ;

ð5Þ

where u0 is divergence free. Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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2.2. The equations of motion for a rigid body A solid body immersed in the fluid experiences unsteady forces and in certain cases may exhibit rigid body motion. The motion of the body, in the two directions along the Cartesian axes, is governed by the following equations: CD X€ þ 2πF n ζ X_ þ ðπF n Þ2 X ¼ πmn

for ð0; TÞ;

ð6Þ

CL Y€ þ 2πF n ζ Y_ þ ðπF n Þ2 Y ¼ πmn

for ð0; TÞ:

ð7Þ

Here, Fn is the reduced natural frequency of the oscillator, ζ is the structural damping coefficient, mn is the non-dimensional mass of the body, while CL and CD are the instantaneous lift and drag coefficients for the body, respectively. The free-stream flow is assumed to be along the x-axis. X€ , X_ and X denote the normalized in-line acceleration, velocity and displacement of the body, respectively, while Y€ , Y_ and Y represent the same quantities associated with the cross-flow motion. In the present study, in which the rigid body is a circular cylinder, the displacement and velocity are normalized by the characteristics dimension, D, of the cylinder and the free-stream speed, U ∞ , respectively. The reduced natural frequency of the system, Fn is defined as f n D=U ∞ where fn is the actual frequency of the oscillator. Another governing parameter is the reduced velocity, Un. It is defined as U n ¼ U ∞ =f n D ¼ 1=F n . The non-dimensional mass of the cylinder is defined as mn ¼ m=ρ∞ D2 where m is the actual mass of the oscillator per unit length and ρ∞ is the density of the fluid. The force coefficients are computed by carrying out an integration, that involves the pressure and viscous stresses, around the circumference of the cylinder Z 1 ðrnÞ  nx dΓ; ð8Þ CD ¼ 1 2 2ρ∞ U ∞ D Γ cyl CL ¼ 1

1

Z

2 2ρ∞ U ∞ D Γ cyl

ðrnÞ  ny dΓ:

ð9Þ

Here nx and ny are the Cartesian components of the unit vector n that is normal to the cylinder boundary Γ cyl . The finite-element formulation that can handle moving boundaries and interfaces is employed and the function spaces are constructed for the space–time method. The variational formulation includes certain stabilization terms added to the basic Galerkin formulation to enhance its numerical stability. Details on the formulation, including the definitions of the various coefficients can be found in the papers by Tezduyar et al. (1992a,b,c). The equations of motion for the oscillator given by Eqs. (6)–(7) are also cast in the space–time formulation in the same manner as described in the work by Tezduyar et al. (1992b) and Mittal (1992). 3. The problem description and validation of method Fig. 1 shows the schematic diagram of the problem along with the boundary conditions. An elastically mounted square cylinder with low non-dimensional mass (mn ¼5 or 10) is placed within a rectangular domain. The length of each side of the square cylinder is D. The cylinder is allowed to vibrate in the transverse direction. The length of the domain is 35.5D, while the blockage ratio of the domain is 1/20, i.e.,0.05. Distance between the upstream boundary and the center of the stationary cylinder is 10D while that between the downstream boundary and the cylinder is 25.5D. The structural damping coefficient is set to zero so that the vortex-induced vibrations are maximized.

Fig. 1. Vortex induced vibration of a square cylinder: schematic diagram of an elastically mounted square cylinder allowed to vibrate in the transverse direction.

Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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At the upstream boundary, normal and in-line velocities are assigned free-stream values. The viscous stress vector at the downstream velocity is set to zero. For the top and bottom boundaries, the normal component of the velocity vector and the component of stress vector along the boundaries are set to zero. 3.1. The finite-element mesh and validation Fig. 2 shows the finite-element mesh with 31 610 nodes and 31 200 quadrilateral four-noded elements. The cylinder is at the center of a square region of 5D by 5D and the mesh in this square region moves with the square cylinder like a rigid body. The outer boundary of the domain is fixed and hence the only elements deformed at each time step are those between the square region and the outermost fixed boundary. This mesh-moving scheme is free of projection errors as mentioned by Tezduyar et al. (1992b) and Mittal (1992). The variation of the drag coefficient with Re for the stationary square cylinder is shown in Fig. 3. The present results are compared with experimental results of Shimizu and Tanida (1997) and computational results of Okajima et al. (1997). The present results are in accordance with these two sets of results. Sohankar et al. (1998) have reported the critical Reynolds number for a square cylinder at 50. Lankadasu and Vengadesan (2008) have concluded that the critical number is 46 71 for a stationary square cylinder for the blockage ratio of 0.1. Saha et al. (2003) have simulated flow past a stationary square cylinder for Re ¼50–500. They have reported the values of the Strouhal number. For the present computations for the stationary cylinder, the Strouhal numbers for Re ¼ 60, 100 and 110 are 0.113, 0.146 and 0.154, respectively. These values are in excellent agreement with those reported by Saha et al. (2003). This mesh is utilized for all the VIV simulations of the present study. 4. Vortex-induced vibrations In the present work, the vortex-induced vibration of a square cylinder is simulated. The cylinder is allowed to move in the transverse direction to the incoming flow for subcritical and supercritical Re, i.e., Re o 50 and Re 450. The structural damping coefficient, ζ, is set to zero to maximize the amplitude of the vibrations. The non-dimensional mass of the cylinder is chosen as 5 for all sets of simulations, while an additional value of mn ¼10 is employed for the last two sets of simulations. The effect of the non-dimensional mass on VIV is investigated for two different values. Fig. 4(a)–(c) illustrates the variations of maximum drag coefficient (C D;max ), maximum lift coefficient (C L;max ), amplitude (normalized by the cylinder width) and the Strouhal number, with Re. It is observed that vibration starts at Re∼23:9 and ceases at Re ¼33.5. The maximum of C L;max occurs at Re¼25 while that of YD is attained at Re¼26. Thus we observe that the maximum YD occurs at slightly higher Re than that for the lift. The maximum value of YD is noted to be approximately 0.15D.

H

Lu

Ld

Fig. 2. Vortex induced vibration of a square cylinder: the finite-element mesh with 31 600 nodes and 31 200 quadrilateral four-noded elements. H¼ 20D. Lu ¼10D, Lu ¼ 25.5D where D is the side of the square cylinder.

Fig. 3. Vortex induced vibration of a square cylinder: variation of the maximum drag coefficient with Re.

Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

5

YD, CL,max

Lock-in is observed in all the cases (not shown in the figure though). Instantaneous vorticity fields are shown for Re ¼24, 27, 30 and 33 in Fig. 5. These fields are shown for instants corresponding to the maximum value of the lift coefficient. The vortex shedding mode is either 2S or C(2S). In the 2S mode, two single vortices are shed per shedding cycle, while vortices of the 2S mode coalesce in the C(2S) mode. Various vortex shedding modes, including 2S and C(2S), are discussed by Williamson and Roshko (1988). In the second set of simulations, the non-dimensional frequency, Fn (1/Un), is maintained at a constant value of 0.1333 in order to observe the persistence of VIV at various Re and variations of parameters pertaining to oscillations. The nondimensional mass of the oscillator is 5. The variations of the maximum lift coefficient, normalized amplitude, maximum drag coefficient with Re are shown in Fig. 6(a) and (b). It is seen that the lift coefficient reaches a local maximum at Re ¼ 35 and a minimum at Re¼ 70. In Fig. 6(c), the variation of the Strouhal number with Re is shown. It is compared with the Strouhal number variation for the circular cylinder due to the prediction of Mittal and Singh (2005) and the experimental results of Buffoni (2003). The trends match qualitatively. Fig. 6(d) shows the comparison of the normalized amplitude for the square and circular cylinders for constant and varying non-dimensional frequency. The amplitude variation is due to Mittal and Singh (2005). However, the non-dimensional mass used by Mittal and Singh (2005) is 4.73, while it is 5 for the present study involving a square cylinder. For mn ¼5, the comparison is expected to be better. It is found that the amplitude of the displacement of the square cylinder is around 0.15D and 0.25D for varying and constant non-dimensional frequencies, respectively. These values for the circular cylinder are higher than those for the square cylinder by approximately 50%, as reported by Mittal and Singh (2005). Mittal and Singh (2005) also observed that the maximum value of the displacement amplitudes is slightly higher for the circular cylinder with the non-dimensional mass of 100. Fig. 7 shows the variation of the lift coefficient with time for Re¼60, 70 and 180. Three local maxima/minima peaks are discerned for Re ¼70 while the variation is sinusoidal for other two values of Re. Fig. 8 shows the plots of the lift coefficient and displacement vis-a-vis time for Re¼60, 80 and 160, respectively. The phase difference (ϕ) between the lift coefficient and displacement is found to be 01 for Re¼60 and 1801 for Re¼80 and 160, respectively. The square cylinder undergoes VIV at subcritical Re when the non-dimensional velocity (Un) is varied. In the present work, VIV is examined for Re¼25, 30 and 35 for two values of non-dimensional mass, i.e., mn ¼5 and 10. Fig. 9(a)–(c) shows the variation of Strouhal number, the maximum lift coefficient and normalized amplitude with the non-dimensional

0.3

CL,max YD

0.2

0.11

St

0.1 0 22

24

26

28

30

32

34

2.3

CD,max

0.105

St

Re

0.1

CD,max 2.1 1.9 22

0.095 22 24

26

28

30

32

34

24

26

28

30

32

34

Re

Re Fig. 4. Vortex induced vibration of a square cylinder: (a) shows the variation of the maximum lift coefficient and normalized transverse amplitude with Re; (b) shows the variation of the maximum drag coefficient with Re and in (c) variation of the Strouhal number with Re is shown.

Fig. 5. Vortex induced vibration of a square cylinder: instantaneous vorticity field for Re as shown in each subfigure. Fn varies as 3.1875/Re. Flow field is taken at the instant when the lift coefficient is maximum.

Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

Present study,U*=1/7.5 Mittal and Singh (2005) Buffoni, Expt. (2003) Present study

CL,max YD

0.4 0.3 0.1 0 10

50

90

130

170

Re 2.6

CD,max

(U*=3.1875/Re)

0.18

0.2

St

YD, CL,max

6

0.12

CD,max

2.2

0.15

1.8 1.4 10

50

90

130

0.09 10

170

50

90

Re

130

170

Re

0.5

Present study,U*=1/7.5 Present study, U* varying Mittal and Singh (2005), U* constant Mittal and Singh (2005), U* varying Reddy (2007)

0.4

YD

0.3 0.2 0.1 0 25

40

55

70

85

100

115

130

145

160

Re Fig. 6. Vortex induced vibration of a square cylinder: (a) variation of the maximum lift coefficient and normalized transverse amplitude with Re; (b) variation of the maximum drag coefficient with Re; (c) comparison of the various variations of Strouhal number with Re and (d) comparison of the variations of the normalized transverse amplitude with Re.

velocity for the three values of Re, respectively. These plots reveal that the square cylinder with the smaller mass starts vibrating at a relatively lower value of the non-dimensional velocity. The amplitude, the lift coefficient and the Strouhal number are higher for the higher mass of the cylinder at the same the non-dimensional velocity. Fig. 9(d)–(g) shows the variation of the lift coefficient and normalized displacement with time for Re ¼ 35, mn ¼5 and 10 and Un ¼8 and 10. It is observed that the phase differences are 231 and 131 for mn ¼10 and 5, respectively, while the values of Re is 35 and that of Un is 10. For Re¼35 and Un ¼8, the phase differences are 01 for both the values of mn ¼10 and 5. At the same time, it is observed that the variation of the maximum lift coefficient (C L;max ) attains its minimum value at Un ¼9.5 for Re¼35 and mn ¼10. Similarly, the same variation for Re¼35 and mn ¼5 shows its minimum value at Un ¼ 9.5. Thus we observe that the phase difference is observed when the variation of C L;max attains its minima. These results match well with the results of Reddy (2007). Fig. 10 shows how the pertinent output parameters of the oscillator behave for Re ¼80, which is in the supercritical range (Re 4 50) for the non-dimensional mass (mn) of 5 and 10. The variations are plotted against the non-dimensional velocity, Un. Fig. 10(a) shows the plot for the pertinent response parameters against F n =St. It may be noted that Fn is inversely proportional to Un, and hence higher values of Un represent lower values of Fn. Hence the graph starts from the right hand side of the x-axis. Moreover, F n =St is equivalent to f v =f n , i.e., ratio of the actual frequency of the oscillator and the vortex shedding frequency. It may be noted that the maximum YD is achieved at F n =St∼1 for mn ¼10, while the corresponding value for mn ¼ 5 is approximately 1.2. Williamson and Govardhan (2008) have discussed about the phenomenon of lock-in or synchronization. Traditionally synchronization means that as Un is increased, a speed is reached at which the vortex shedding frequency fv becomes close to the natural frequency of the structure (fn), and the two frequencies synchronize. For bodies in water, they observed for mn ¼2.4, the body oscillates at a distinctly higher frequency (f v =f n ¼ 1:4). Fig. 10(b)–(d) Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

Re=180

0.4

CL

7

0 -0.4 400

425

450

475

500

0.4

CL

Re = 70 0

.005 −.005 400

-0.4 400

600

425

450

475

500

0.4

CL

Re = 60 0 -0.4 400

425

450

475

500

Time Fig. 7. Vortex induced vibration of a square cylinder: variation of the lift coefficient with time for Re ¼ 180, 70 and 60. For Re¼ 70, variation is magnified in the inset.

0.6

φ = 180°

Re = 160, CL

CL, y/D

y/D

0

-0.6 400

425

CL, y/D

0.35

φ = 180°

475

500

Re = 80, CL y/D

0

-0.35 400

425

0.6

CL, y/D

450

450

φ = 0°

475

500

Re = 60, CL y/D

0

-0.6 400

425

450

475

500

Time Fig. 8. Vortex induced vibration of a square cylinder: variation of the normalized transverse displacement with time for Re¼160, 80 and 60. The phase difference between the two signals is indicated in each of the subfigures.

shows the variation of Strouhal number, the maximum lift, drag coefficient and the normalized amplitude for mn ¼5 and 10. The phenomenon of hysteresis is observed for all the variables but shape of the hysteresis loop does vary as the nondimensional mass (mn) varies. For lower values of mn, this extent is reduced. Such type of reduction in the hysteresis loop is observed by Prashanth and Mittal (2008) for VIV of a circular cylinder when the blockage of the domain is varied. It can be noted that the hysteresis loop for C L;max for mn ¼10 (Fig. 10(c)) shows a somewhat different trend at Un ¼5.77. The value of C L;max is different when Un is increased and reduced. In general, only one peak is observed in hysteresis when Un (or Re as the Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

8

Re = 25, m*=10 m*=5 Re = 30, m*=10 m*=5 Re = 35, m*=10 m*=5 Fn

St

0.13

0.1

Re = 35, m* = 10, U = 10, CL YD

φ =23° CL,YD

0.15

0

-0.1 900

0.11

Time

1000

0.09 6

7

8

9

10

11 0.1

CL,max

0.6

CL,YD

U*

Re = 25, m*=10 m*=5 Re = 30, m*=10 m*=5 Re = 35, m*=10 m*=5

0.4

-0.1 900

0 8

9

10

11

CL,YD

0.3

7

φ=13°

0

0.2

6

Re = 35, m* = 5, U = 10, CL Yd

Time

1000

Re = 35, m* = 10, U = 8, Cl Yd °

φ =0

0

U* -0.3 900

Re = 25, m*=10 Re = 25, m*=5 Re = 30, m*=10 Re = 30, m*=5 Re = 35, m*=10 Re = 35, m*=5

0.3

0.3

CL,YD

YD

0.2

0.1

0 6

7

8

9

U*

10

11

Time

1000

Re = 35, m* = 5, U = 8, Cl Yd

φ=0°

0

-0.3 900

1000

Time

Fig. 9. Vortex induced vibration of a square cylinder: in (a), variation of the of Strouhal number with Un; (b) shows the variation of the maximum lift coefficient with the non-dimensional velocity, Un and (c) shows the variation of the normalized transverse amplitude with the non-dimensional velocity, Un. For (d)–(g), variation of the transverse displacement and the lift coefficient with time for the parameters indicated. The phase difference, ϕ, is indicated in each case.

case may be) is reduced. However, such behavior is observed bySen et al. (2011) for the VIV of a square cylinder for Re¼ 60– 150. Fig. 10(e) shows the variation of the lift coefficient and displacement for Re ¼ 80, mn ¼10 and Un ¼ 6 and 8. The phase differences observed for Un ¼6 and 8 are 01 and 1141, respectively. In Fig. 10(f), a plot for CL versus y/D is shown for Re¼80, mn ¼5, and Un ¼6 and 8. The phase difference is 551 and 1801. Thus we see that the phase difference and its magnitude depend upon both mn and Un. Fig. 10(g)–(k) shows various instantaneous vorticity plots at the time instant when the lift coefficient is maximum. The vortex shedding mode for all cases is 2S. 5. Conclusions and observations Vortex induced vibrations are simulated for a square cylinder for subcritical Re for the non-dimensional masses (mn) ¼5 and 10. Four cases are considered. In the first case, the non-dimensional frequency (Fn) is varied as 3.1875/Re. In the second case, the non-dimensional frequency, Fn, is constant at 0.1333. In both the cases, Re is varied. In the third and fourth cases, studies are undertaken for Re¼25, 30 and 35 and Re¼80, respectively. In these two latter cases, the non-dimensional velocity is varied. The following conclusions are drawn based upon the observation of the four cases. Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

.5

9

C L ,y/D for m*=10,

U*=6 ,φ=0ο

0.18

m* = 10, YD m* = 5, YD C L ,y/D for m*=10,U*=8,

YD

0.12

φ=114°

−.5 200

0.06

Time

.5

0 0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

225

C L ,y/D for m*=5,U*=6

φ=55°

2.4

Fn/St

St

0.17

m* = 10, St m* = 5, St

0.15

−.5

CL ,y/D for m*=5,U*=8, ° φ=180 200

0.13 3

5

7

9

11

CL,max CD,max

m* = 10, C L,max m* = 5, C L,max m* = 10, C D,max m* = 5, C D,max

2.4 1.8

225

m*=5,U*=5.5

U* 3

Time

m*=5,U*=6

1.0

1.2 0.6

0.3

0 3

5

6.0

5.2

7

9

m*=5,U*=8

11

U*

YD

0.2

m*=10,U*=5.4

m* = 10, Y D m* = 5, Y D

0.15

U* Decreasing

0.1 0.05

m*=10,U*=5.77

0 3

5

7

9

11

U* Fig. 10. Vortex induced vibration of a square cylinder: (a) shows the variation of the normalized transverse amplitude with the ratio of the reduced frequency and Strouhal number, F n = St, for the non-dimensional mass (mn) ¼ 5 and 10; (b) shows the variation of Strouhal number with Un for mn ¼ 5 and 10; (c) shows the variation of the maximum lift and drag coefficient with the non-dimensional velocity (Un) and (d) shows the variation of the normalized transverse amplitude with the non-dimensional velocity (Un). In (e) and (f), variations of the lift coefficient and the transverse displacement with time for the indicated parameters. From (g)–(j), instantaneous vorticity field at the instant when the lift coefficient is at its peak value.

In the first case, it is observed that the vortex induced vibrations and vortex shedding are observed at a low Reynolds number (Re), which is 23.9. In this case, VIV stops at Re∼34. It can be noted that VIV starts and ends in subcritical range. For the second case when the non-dimensional frequency (Fn) is constant, the maximum transverse amplitudes are observed up to 0.25D for Re¼ 35, which is in the subcritical range. It is observed that a phase change takes place by 1801 after the lift coefficient attains a minima at Re ¼70. In the third case, it is observed that the oscillator with the lower mass starts vibrating at a relatively lower value of Un. The amplitude, the lift coefficient, the normalized displacement and the Strouhal number are higher for the higher mass of the cylinder at the same Un. The phase difference is observed for the two values of nondimensional mass as Un varies. In the fourth set of simulation, it is observed that the maximum YD occurs at F n = St¼1 for mn ¼ 10 but it occurs at F n = St¼1.2 for the lower value of mn ¼5. The phase difference and its magnitude depend on both the non-dimensional mass and velocity. Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i

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S.P. Singh, G. Biswas / Journal of Fluids and Structures ] (]]]]) ]]]–]]]

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Please cite this article as: Singh, S.P., Biswas, G., Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers. Journal of Fluids and Structures (2013), http://dx.doi.org/10.1016/j.jfluidstructs.2013.03.011i