Numerical investigation of the vortex-induced vibration of an elastically mounted circular cylinder having low mass ratio using the

RANS code

*Niaz B Khan1), Zainah B Ibrahim2), Muhammad Faisal Javed3)

1),2) Department of Civil, Faculty of Engineering, University of Malaya,

Kuala Lumpur, WP Kuala Lumpur Malaysia. 3) Department of Civil Engineering, COMSATS Institute of Information Technology

Abbottabad Campus, Abbottabad 22010, Pakistan 1) n_bkhan@yahoo.com

ABSTRACT

The study present numerical investigation of vortex-induced vibration of elastically mounted rigid cylinder, free to oscillate in cross flow direction, having low mass damping ratio. Numerical simulations are performed using 2-Dimensional incompressible Reynolds-Averaged Navier–Stokes (RANS) equations. The study is carried out for range of reduced velocity= 2 to 16 which corresponds to Reynolds number 1700 to 14000. The mass-ratio is 2.4 and mass-damping ratio=0.0013. Previously, similar studies have been performed numerically but having deficiency in achieving the maximum cylinder response. In current study, comparatively higher value of maximum amplitude of cylinder is computed. However, delayed in transition from ‘upper branch’ to ‘lower branch’ is observed. The results extracted from current simulations are compared with previous experimental and numerical studies.

1. INTRODUCTION The fluid-structure interaction problem associated between cylinder and fluid has gained tremendous attention of the researchers due to its importance in the wide range of application that include, marine industry, nuclear reactor, skyscrapers, bridges, wind turbines, chimneys etc. The recent trend toward the exploration of deep-sea oil production also increased the need to accurately predict the behaviour of fluid past bluff bodies in offshore structure industry. When flow past a circular cylinder, vortices shedding would occur behind the cylinder alternatively at top and bottom side which will result in oscillating force. The regular shedding of vortices in the wake is known as ‘Karman Vortex Street’. The vortex shedding may result in unwanted structural vibration

1) PhD student 2) Associate Professor

especially when the frequency of vortex shedding is equal to or near to natural frequency of structure. This phenomenon is known as vortex induced vibration. The dramatic collapse of the Tacoma Narrows Bridge in 1940 is the famous example which occurs due to vortex induced vibration phenomenon (VIV). After the collapse of Tacoma Narrow Bridge, it is realized by researchers to assess the influence of wind forces in design consideration. The VIV phenomenon which may result in large oscillation cannot be underestimated in design of longer and slender structures. Also, the pseudo-static design analysis is not enough to tackle the disastrous aerodynamic vibration. In 1995, the failure in thermowell in Japan nuclear reactors (Monju) was also due to VIV phenomenon. In 2002, one of the tower of the thriller ride, VertiGo, at Cedar point in Sandusky, Ohio also collapsed due to vortex induced vibration effects. These major incidents again draw the attention of researchers to minimize the unwanted VIV phenomenon investigation during construction of towers and other structures. Wind turbines are also comes in fluid-structure interaction field application. Due to these wide ranges of application, large number of fundamental research has been performed to investigate the VIV phenomenon. Williamson‘s group has significant contribution in this area and provided solid base for basic understanding of VIV phenomenon (Jauvtis & Williamson, 2004; Khalak & Williamson, 1996; A. Khalak & C. Williamson, 1997; Williamson & Govardhan, 2004; Williamson & Roshko, 1988). In case of flow around elastically mounted rigid cylinder, two type of response is observed as explained by (A. Khalak & C. H. Williamson, 1997) i.e. small amplitude with two branches and large-amplitude with three branches . According to authors, higher amplitude is observed for the low mass damping ratio and the three branches are termed as “initial”, “lower” and “upper” branch. Smaller amplitude with two branches is observed at high mass damping ratio and the two branches are termed as “initial” and “lower” branch. Lower amplitude with two branches was observed in study of (Feng, 1968), in which air was used as working fluid. In study of (A. Khalak & C. H. Williamson, 1997), authors used water as working fluid with extremely low damping (low mass damping ratio) and higher amplitude equal to 1D of cylinder was observed in experimental study. It is also observed that with low mass damping ratio, a wider response range is observed. This behaviors was also noted by (O. M. Griffin & Ramberg, 1975) in small mass system. Wake behind an oscillating structure is investigated by number of researchers in literature but most of the early research was limited to small amplitude with two branches mode. (O. Griffin, Skop, & Koopmann, 1973; O. M. Griffin, 1972) computed the phase angle, lift coefficient, energy transfer and mean velocity fluctuation behind the cylinder. Smoke visualization approach is used by (Brika & Laneville, 1993) to observe the wake structure of smaller amplitude having initial and lower branches. (Govardhan & Williamson, 2000; Khalak & Williamson, 1999) investigated the flow structure in the wake for higher amplitude and smaller amplitude case. Authors also measured the fluid forces on cylinder, non-dimensional amplitude and other coefficients for the range of reduce velocity Ur=2-14. Two single vortices shedding per cycle were observed at initial branch and two pairs of vortices shedding at upper and lower branches.Recently with continuous improvement

in computer technology, the trend is diverted toward the numerical testing to investigate the VIV phenomenon for circular cylinder. The complex nature of the flow around a cylinder makes it an excellent case to assess the ability of computational packages and extend the study to the real environmental flow conditions. Despite the rapid advancement in computational field, there are still lot of limitations in performing CFD analysis as explained in the reviews of (Breuer, 1998) (Sarpkaya, 2004) and (Bearman, 2011). The objective of the current study is to numerically investigate VIV phenomenon for the range of reduced velocity Ur= 2 to 16, which correspond to Reynolds number Re = 1700 to 14000, using RANS SST-kw model. Results extracted from the simulations are compared with experimental study of (Khalak & Williamson, 1996) and numerical study of (Li, Li, & Liu, 2014) and (Pan, Cui, & Miao, 2007).

2. NUMERICAL MODEL The unsteady incompressible RANS equation for flow around cylinder can be written as: 𝜕𝑢𝑖

𝜕𝑥𝑖= 0

(1)

𝜕

𝜕𝑡(𝜌𝑢𝑖) +

𝜕

𝜕𝑥𝑗(𝜌𝑢𝑖𝑢𝑗) = −

𝜕𝜌

𝜕𝑥𝑖+

𝜕

𝜕𝑥𝑗(2𝜇𝑆𝑖𝑗 − 𝜌𝑢𝑖

′𝑢𝑗′)

(2)

where ui and ρ represent the time-average values of velocity and pressure, respectively; μ represents molecular viscosity; and Sij and 𝑢𝑖𝑗 are the mean stress tensor and

Reynolds stress tensor, respectively. In current study shear-stress-transport (SST-kw model) is used. This model involves the two transport equations, i.e. one for turbulent kinetic energy and another for specific dissipation rate. The details about the model is explained by Menter (Menter, 1994). The unsteady segregated algorithm is adopted in the calculation. Pressure–velocity coupled equations are solved with the SIMPLE algorithm [explained in the ANSYS manual (ANSYS 2016. Ansys fluent manual)], and the implicit second-order scheme is used for unsteady terms. The second-order scheme is used for k-ω transport equations and for convection terms in the momentum equations. According to (Guilmineau & Queutey, 2004), the cross-flow oscillation of circular cylinder can be represented by following equation.

𝑑2𝑌

𝑑𝜏2+4𝜋𝜁

𝑈𝑟

𝑑𝑌

𝑑𝜏+4𝜋2

𝑈𝑟2 𝑌 =

2𝐶𝑦

𝜋𝑚∗

(3)

where Y = Ay/D represents the displacement in the cross-flow direction normalized by the cylinder diameter; Ur, ζ, m*, and Cy are the reduced velocity, structural damping ratio, mass ratio, and life coefficient, respectively.

3. COMPUTATIONAL MESH AND BOUNDARY CONDITIONS Figure 1 shows the computational domain used in the current study. Domain size of 20D X 40D is used in the study. Inlet is located at 15D toward the left hand side, while outlet is at distance of 25D right hand side of the cylinder. An average static reference pressure of 0 Pa is applied at the outlet boundary. Top and bottom sides, which are assigned with symmetry condition, are at a distance of 10D from the cylinder surface. No-slip boundary condition is assigned on the surface of the cylinder. Hybrid mesh is used in the current study as shown in Figure 1(b and c). In order to avoid the negative volume error during mesh deformation, prism layer having quad elements are made around the cylinder. Since the simulation involve the oscillation of cylinder, therefore tri elements are made in surrounding of prism layers to facilitate the remeshing process. Other regions are made of quad elements to reduce the cost. During all the case studies, first node is kept at a distance which ensure the y+ value equal to or less than unity. Furthermore, cluster of elements are made around the cylinder in all the cases, while mesh at the far regions are coarse. Grids sensitivity tests are performed by varying number of elements on the cylinder surface. Finally, 240 nodes at surface of cylinder and total of 45230 elements are chosen, based on the mesh independence test. In order to maintain Courant–Friedrichs–Lewy (CFL) condition value equal to or less than unity for RANS SST k-w model, a non-dimensional timestep is set to 0.0001 in all case studies. Figure 2 shows the schematic representation of flow around a circular which is free to oscillate in cross flow direction. The elastic system has ‘c’ as a damping constant and ‘k’ as a spring constant in y-direction.

(a)

(b) (c)

Figure 1 (a) Computational domain size and (b,c) mesh details

Figure 2 Cylinder free to oscillate in cross-flow direction

4. RESULTS AND DISCUSSIONS Numerical simulations are performed for flow around elastically mounted rigid cylinder free to oscillate in cross flow direction as show in Figure 22. As depicted in the Figure 22, the cylinder oscillation is constrained in the y-direction by a spring damper system. Analysis are performed for range of Reynolds number Re=1700 to 14000 which correspond to Ur=2 to 16. The change in reduced velocity is obtained by changing the uniform inlet velocity. All the physical parameters used in the current study are same as used by (Khalak & Williamson, 1996), (Li, Li, & Liu, 2014) and (Pan, Cui, & Miao, 2007). Figure 13 shows the time history of cylinder response (Ay/D) and corresponding instantaneous force coefficients at different reduced velocity. Figure 24 presents the cylinder response as a function of reduced velocity together with the experimental and numerical results. Cylinder response (Ay/D) is extracted from the time history of displacement. All the three branches are observed in the study as depicted from Figure 24. Initial branch is observed in the range of reduced velocity Ur=2-3, followed by upper branch in the range of reduced velocity Ur 5-9. It is clearly depicted that at region of upper branch, high value of displacement is observed. In current study, maximum non dimensional cylinder response value Ay/D=0.86 is observed at reduced velocity Ur=7.5 which is comparatively better than the numerical results of (Li, Li, & Liu, 2014) and (Pan, Cui, & Miao, 2007). (Pan, Cui, & Miao, 2007) and (Li, Li, & Liu, 2014) found the maximum cylinder response Ay/D=0.70 and Ay/D=0.747, respectively, which is quite smaller than experimental value Ay/D=1. The higher value of cylinder response also validated the occurrence of lock-in phenomenon in upper branch. After upper branch, the transition to the lower branch is observed at reduced velocity Ur=11 where cylinder response reduced to Ay/D=0.2-0.1 which agreed well with the other experimental and numerical results. From Figure 13, it is also observed that at higher value of cylinder response, the mean drag values are higher as shown is Figure 13(d and e) where at out of the ‘lock-in’ region, the smaller mean value of drag is observed.

(a) Ur=2 (b) Ur=2

-1

-0.5

0

0.5

1

30300 40300 50300 60300 70300 80300 90300 100300

Dis

pla

cem

en

t, A

y/D

Time

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

30300 40300 50300 60300 70300 80300 90300 100300

Forc

e C

oe

ffic

ien

t

Time

(c) Ur=5 (d) Ur=5

(d) Ur 7.5 (e) Ur 7.5

(e) Ur 11 (f) Ur 11

Figure 1 Time history of cylinder response (Ay/D) and force coefficients (drag force-red

color and lift force-black color) at different reduced velocity

-1

-0.5

0

0.5

1

1600 11600 21600 31600

Dis

pla

cem

en

t, A

y/D

Time (s)

-2

-1

0

1

2

3

4

5

16000 21000 26000 31000

Forc

e c

oe

ffic

ien

t

Time (s)

-1

-0.5

0

0.5

1

1000 6000 11000 16000 21000

Dis

pla

cem

en

t, A

y/D

Time (s)

-2

-1

0

1

2

3

4

5

1000 6000 11000 16000 21000

Forc

e c

oe

ffic

ien

t

Time (s)

-1

-0.5

0

0.5

1

5000 6000 7000 8000 9000 10000 11000 12000

Dis

pla

cem

en

t, A

y/D

Time (s)

-1.5

-1

-0.5

0

0.5

1

1.5

2

5000 6000 7000 8000 9000 10000 11000 12000

Forc

e c

oe

ffic

ien

ts

Time (s)

Figure 2 Cylinder response (Ay/D) at range of reduced velocity (Ur)

5. CONCLUSION The current work presents the numerical investigation of vortex-induced vibration of an elastically mounted rigid cylinder, having low mass ratio, using RANS SST k-w model. The numerical results are compared with experimental and numerical studies available in literature. All the three branches of VIV phenomenon is observed in the study. Smaller value of cylinder response and drag forces is found at the initial branch while the maximum value of cylinder response is observed at upper branch. The maximum cylinder amplitude observed in the current study is comparatively better than the results available in the literature. However, this value of higher cylinder response occurs in the later phase of the upper branch. The lock-in phenomenon having higher value of cylinder amplitude and drag forces are observed at reduced velocity 4-9 which is more wide range compared to other studies. Author also come up with the conclusion that there is need to investigate the impact of blockage ratio and prism mesh distribution on achieving the good quality results.

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16

Dis

pla

cem

en

t (A

y/D

)

Reduced Velocity, Ur

Williamson

Current SST - kw

Pan 207

welie

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