Numerical Simulation of Wind-Induced Transverse Vibration of a 2D Square Cylinder

Journal of Applied Science and Engineering, Vol. 20, No. 3, pp. 319-326 (2017) DOI: 10.6180/jase.2017.20.3.06

Numerical Simulation of Wind-induced Transverse Vibration of a 2D Square Cylinder

Deqian Zheng1,2*, Ming Gu1, Aishe Zhang3, Yanjie Xie2, Beibei Huang2 and Haochen Hu2

1State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, P.R. China 2School of Civil Engineering and Architecture, Henan University of Technology, Zhengzhou 450001, P.R. China 3School of Civil Engineering, Shandong Jianzhu University, Ji’nan 250101, P.R. China

Abstract Wind-induced transverse vibration of a two-dimensional square cylinder was numerically simulated, based on commercial code Fluent. The Reynolds number, defined by inflow velocity and the depth of the cylinder, was set to be 22000. The fluid-structure coupled system was solved by employing partitioned coupling scheme. The fluid field was simulated using SST k-w turbulence model, and the structural motion was calculated by Newmark method. The solution procedure was programmed by user define function (UDF). Flow around the cylinder at stationary state was firstly simulated to obtain initial flow field condition for the coupled system. Wind-induced transverse vibration of the cylinder was then simulated at different reduced wind velocities. Wind-induced galloping, the beat-phenomena and vortex-excited resonance of the cylinder in the transverse direction, were all captured, with the increase of the reduced wind velocities. The simulated data were also compared with those of previous studies. The comparison results showed that the present method is applicable in solving wind-induced vibration problems. Finally, parametric analysis of Scruton number’s effect on the across-wind vibrations of the cylinder was investigated. The results indicated that wind-induced vibration of the cylinder was remarkably affected by the Scruton number. The transverse vibrations were obviously divided into galloping and vortex-induced vibration at large Scruton number, while the switch between the two vibration types was not so remarkable when the Scruton number was low.

Key Words: Numerical Simulation, Aeroelasticity, Square Cylinder, Partitioned Coupling Scheme, Wind-induced Vibration

1. Introduction blems. Numerical methods for FSI simulations are usu- ally classified as monolithic and partitioned coupling Wind load is one of the major dynamic loads on st- schemes [1]. Although monolithic models enable en- ructures. The fluid-structure interaction (FSI) problems ergy-conserving coupling, their governing equations for cannot be neglected when wind-induced vibrations are the coupled system are not easily constructed for com- large enough to affect the flow fields around the struc- plex engineering problems. In the partitioned method, ture. Comparing with wind tunnel experiments, numeri- the fluid and structural subsystems are solved stagger- cal simulation would be an efficient solution for FSI pro- ingly. It is widely accepted that the solution of a large class of transient fluid-structure interaction problems *Corresponding author. E-mail: [email protected] characterized by complex fluid and/or structural models 320 Deqian Zheng et al. is computed most efficiently using a partitioned analysis (1a) procedure, based on two independent fluid and structural codes or software modules [2]. The configuration of structures with sharp-edged cross section has a possibility for the onset of galloping (1b) oscillation as well as vortex-induced oscillation in the transverse direction [3]. Based on the partitioned cou- where r is the fluid density; n is the kinematic viscosity pling scheme, there have been so far a great number of coefficient; p is the fluid dynamic pressure; u is the numerical studies focusing on the mechanism of such fluid velocity vector; ug is the grid velocity vector; n is unstable phenomena for square section structures [3-6]. the normal vector; S is the boundary of the control vol-

For simulation of FSI problems, the structural motion in ume V;and(u - ug) is the time-dependent convection the fluid domain should be taken into account, resulting velocity. To ensure the conservation principle, the grid in problems on how to compromise the description dif- conservation law should be satisfied in each control ference between the structure field and the flow field. volume V: One common solution is to modify the flow field governing equations, i.e., the Navier-Stokes (N-S) equations, (2) adopting the Arbitrary Lagrangian-Eulerian (ALE) method [7]. The grid velocities are taken into consideration The direct calculation of the grid velocities ug can be in the modified N-S equations and the flow field grids avoided by replacing them with the mass fluxes through are updated simultaneously according to the changes of the control volume faces (‘mesh fluxes’) which result the structure motion. Transverse galloping and vortex- from the motion of control volume faces during the time induced vibrations of 2D square cylinders were simu- step [8]. The convective term in Eq. (1) containing the lated by employing the ALE method [3-5]. Another so- grid velocity ug can be discretized for a hexahedral con- lution is to solve the time-depended flow field directly, trol volume as follows: and the mesh updating of the flow field is implemented by re-meshing the flow domain at each time-step. Vor- (3) tex-induced vibration for a square cylinder at high Rey- nolds number was numerically simulated by this method where F is the generalized scalar with F =1forthe [6]. continuity equation, and F ={u, v, w}forthemomen- In this paper, numerical simulation platform for FSI tum equation; c ={W, E, S, N, B, T} stands for the six problems was constructed by employing partitioned cou- faces of the control volume. pling scheme based on commercial code Fluent. Wind- After the above transformation, the solution to Eq. induced transverse vibration of a two-dimensional square (1) can be achieved based on the relative fluxes, which cylinder with Reynolds number of 22000 was numeri- are the differences between the fluxes through a control cally simulated. The simulated results were verified in volume face caused by fluid motion and the ‘mesh fluxes’. comparison with results of previous studies. Analysis of Based on this formulation the grid velocity ug is no lon- Scruton number’s effect on the across-wind vibrations of ger required for the internal flow region but it has to be the cylinder was also investigated. known explicitly at moving impermeable wall [8]. For the structural dynamic analysis, the equation of 2. Governing Equations motion can be expressed as

(4) Based on the ALE description [7], the integral form of the modified continuity equation and N-S equations in where M is the mass matrix, C the damping matrix re- each moving boundary control volume V, can be written presenting the structural damping, and K the stiffness & as matrix; üs, u s and us describe the acceleration, velocity, Numerical Simulation of Wind-induced Transverse Vibration of a 2D Square Cylinder 321 and displacement of the structure motion, respectively. applied to the mesh updating module, based on the dy- F(t) characterizes the load acting on the structure caused namic mesh technique of Fluent. Sketches of composi- by the fluid. tion of present simulation platform are shown in Figure Generally, requirements for the computational grids 1a. The conventional partitioned algorithm equipped with for the fluid and structure subsystem will lead to mis- subcycles for the fluid field calculation, shown in Figure match of grid points on the fluid/structure interface boun- 1b, was adopted in the present study to save CPU time in dary. In order to ensure the consistency of the fluid/struc- FSI simulation. The main solution procedure was pro- ture coupled system, the equilibrium and compatible grammed by using visual C++ language. conditions on the fluid/structure coupled boundary should be satisfied as 4. Numerical Simulation of Wind-induced Vibration of a 2D Cylinder (5) us = uF (6) Wind-induced vibration of a 2D square cylinder, de- picted in Figure 2a, was numerical simulated by using where sS denotes the structure stress tensor, sF is the the above method. The depth D of the cylinder is 0.1 m. fluid viscous stress tensor; and uF is the ALE displace- The Reynolds number, defined by inflow velocity U0 and ment field of the fluid. the depth D of the cylinder, was 22000. The turbulence intensity was 2%. 3. Numerical Solutions The 2D square cylinder was built within domain size of 35D(x) ´ 20D(y), shown in Figure 2a, in which the Partitioned coupling scheme was employed as a so- boundary conditions are also given. The domain was me- lution approach to the FSI problem in this study. The shed with non-uniformly structured grids with minimum present numerical simulation platform consists of four grid space of 0.0025D. The numbers of mesh are about modules, i.e. the computational fluid dynamics (CFD) 10800. Flow around the cylinder at stationary state was module, the computational structural dynamics (CSD) firstly simulated by using unsteady SST k-w turbulence module, the data transfer module, and the mesh updating model, to obtain fully developed flow field as the initial module. CFD condition for the FSI calculation. For the initial For the CFD module, the flow field was simulated by CSD condition, velocity and displacement of the struc- using SST k-w turbulence model, based on commercial ture model were both set to be zero. code Fluent. SIMPLEC was adopted as pressure-veloc- In the present study, the wind-induced vibration of ity coupling method. The convective terms of momen- the square cylinder was confined in the transverse direc- tum equations were discretized using QUICK scheme. tion. The cylinder can be regarded as a mass-damping For the CSD module, Newmark-b method was program- system with one degree of freedom, shown in Figure 2a. med adopting user define function (UDF) of Fluent to The structural parameters are listed in Table 1, where m solve the structural motion. Spring analogy method was is the mass of the cylinder; H is the height with value of

Figure 1. Sketches of component (a) and solution procedures (b) of present numerical simulation platform. 322 Deqian Zheng et al.

1.0; zs is the structural damping ratio; Sc is the Scruton (7) number [9]; U* is the reduced wind velocity; and fn is the natural frequency of the structure model. For each case shown in Table 1, mean inflow wind velocity U0 was set (8) to be a fixed value, thus the reduced wind velocity U* varies with the natural frequency of the structure model.

where Fx and Fy denote the along-wind and across-wind 5. Results and Discussions aerodynamic forces, respectively. The simulated aerodynamic force coefficients of the cylinder at stationary 5.1 Verification of the Simulation Results state are compared with previous wind tunnel experi- Time histories of the aerodynamic force coefficients ment [10] and numerical simulation data [3,11]. The of the square cylinder at stationary state are shown in comparison results are shown in Table 2, where Cd,mean

Figure 3a, in which the drag and lift coefficients are is the mean drag coefficient, Cl,rms is the r.m.s lift coeffi- non-dimensionalized by cient and St is the Strouhal number. It can be seen from

Figure 2. Computational model & boundary conditions (a), and CFD mesh (b).

Table 1. Case details for study on wind-induced vibration of 2D square cylinder 2 M*=m/rD H z s Sc =2M* ´ 2pz s U*=U0 /fn D Case 1 120 0.0067 10 5.0, 6.0, 7.0, 7.5, 7.7, 8.0, 8.3, 9.0, 10.0, 15.0 Case 2 120 0.0133 20 5.0, 6.0, 7.0, 7.5, 7.7, 8.0, 8.3, 9.0, 10.0, 15.0, 20.0, 30.0 Case 3 240 0.0067 20 5.0, 6.0, 7.0, 7.5, 7.7, 8.0, 8.3, 9.0, 10.0, 15.0, 20.0, 30.0

Figure 3. Time histories of the aerodynamic forces (a) and vorticity contour (b) of flow around 2D square cylinder at stationary state. Numerical Simulation of Wind-induced Transverse Vibration of a 2D Square Cylinder 323

Table 2 that present simulation results are in good agree- vious studies, in spite of some numerical differences ment with those of previous studies. which may be caused by different inflow turbulence in- The present simulated transverse displacement re- tensity. The above comparison results indicate that the sponses of the square cylinder are compared with those present method is preliminarily proved to be effective in of previous numerical simulation [3]. The comparison simulation of wind-induced vibration problems. results are shown in Figure 4, where inflow turbulence intensity is zero and 10% for cases ‘smooth flow’ and 5.2 Variation of the Simulated Responses at ‘turbulent flow’, respectively. It can be seen from Figure Different Reduced Velocities 4 that present simulation results are consistent with pre- Figures 5-7 show the simulated time histories of dis-

Table 2. Comparison results of aerodynamic characteristics of 2D square cylinder at stationary state

Re Cd,mean Cl,rms St = fst D/U0 Present simulation 22000 2.13 1.40 0.134 Wind tunnel experiment [10] 22000 2.05~2.33 - 0.132 ± 0.004 Numerical simulation [3] 22000 2.13 1.20 0.125 Numerical simulation [11] 20000 2.18 1.62 0.134

Figure 4. Displacement comparison of present simulation results with previous study data.

Figure 5. Aerodynamic force coefficients and displacement responses of Case 1 (M* = 120, Sc = 10). 324 Deqian Zheng et al. placement responses and aerodynamic force coefficients (2) When the reduced wind velocity is beyond 10.0, the of the square cylinder at different U*. Some flow laws displacement amplitude gets significantly large for and cylinder motion phenomena can be obtained from all cases. Inconsistent oscillations are found between those figures. displacement responses and corresponding lift coef- (1) When the reduced wind velocity is less than 10.0, ficients of the cylinder, when U* = 15.0 for Case 1 transverse vibrations of the cylinder are mainly in- and U* = 30.0 for Cases 2 and 3 (Figures 6 and 7). duced by vortex shedding. The displacement ampli- Moreover, the mean value of the lift coefficient is not tude is small and the vibrations present beat-phenom- zero but oscillating with corresponding displacement ena at low reduced wind velocity. With the increase of time history with the same phase, causing galloping U*, the displacement amplitude gradually gets large of the cylinder in the transverse direction. Similar for Case 1, while increases first and then decreases for phenomena can be found in previous simulation stu- Cases 2 and 3. The peak amplitude was found at U*= dies [3,4]. 8.3 for Case 2 and U* = 7.7 for Case 3, respectively, when vortex excited resonance occurs. Phases of dis- 5.3 The Scruton Number Effects placement responses and lift coefficients are reverse at Figure 8 shows comparison results of transverse dis- the beginning of FSI calculation when U* = 9.0 for placement amplitude of the square cylinder at different

Case 1, while become the same direction with the in- Sc number. It can be seen from Figure 8a that variations crease of the computing time. This means that U*= of the displacement amplitude with U* are not the same 9.0 is the phase switch for Case 1. For Case 2 & Case at different Scruton number, although aerodynamic re- 3, the phase switch occurs when the reduced wind ve- sponses of the square cylinder are vortex-induced vibra- locity is around 9.0 and 8.0, respectively. tion at low U* and transverse galloping when U* is large.

Figure 6. Aerodynamic force coefficients and displacement responses of Case 2 (M* = 120, Sc = 20). Numerical Simulation of Wind-induced Transverse Vibration of a 2D Square Cylinder 325

Figure 7. Aerodynamic force coefficients and displacement responses of Case 3 (M* = 240, Sc = 20).

Figure 8. Comparisons of displacement amplitude at different Scruton numbers for the present study.

At small Scruton number (Case 1), the unstable oscilla- ratio zs is slightly larger than that of Case 3. It indicates tion starts from the resonance velocity, while the unsta- that even though the mass and structural damping ratios ble area is separated to galloping and vortex-induced vi- of the cylinder are different, the wind-induced vibration brations at large Scruton numbers (Case 2 & Case 3). pattern and variation trend of the displacement ampli- Figure 8b shows that the tendency of displacement am- tude are similar, provided by the same Scruton number. plitude varying with U* between Case 2 and Case 3 is The comparison indicates that wind-induced vibration of nearly the same. The displacement amplitude of Case 2 the cylinder was remarkably affected by the Scruton with small mass ratio M* and large structural damping number. 326 Deqian Zheng et al.

6. Conclusions linear Compressible Fluid-structure Interaction Prob- lems,” International Journal for Numerical Methods The partitioned coupling scheme was employed in in Engineering, Vol.84, No. 1, pp. 73-107 (2010). doi: this paper to simulate the fluid-structure interaction pro- 10.1002/nme.2883 blems. The two-dimensional square cylinder oscillations [3] Tamura, T. and Ono, Y., “LES Analysis on Aeroelastic in the wind field were computed by using the present Instability of Prisms in Turbulent Flow,” Journal of method. Some conclusions can be obtained. Wind Engineering and Industrial Aerodynamics, Vol. (1) Comparisons of the present numerical data with previ- 91, No. 12-15, pp. 1827-1846 (2003). doi: 10.1016/ ous numerical simulation and experimental results indi- j.jweia.2003.09.032 cate the validity of the proposed method in this study. [4] Deng, J., Ren, A. L. and Zou, J. F., “Numerical Study (2) Wind-induced galloping, the beat phenomena, phase swi- of Transverse Galloping and Vortex-induced Vibra- tch, and the vortex-excited resonance of the square cylin- tions of Square Cylinder,” Journal of Zhejiang Univer- der in the transverse direction, are all captured by present sity (Engineering Science), Vol. 39, No. 4, pp. 595- study. Present method is verified to be applicable in nu- 599 (2005). (in Chinese) merically solving fluid-structure interaction problems. [5] Zhao, M., Cheng, L. and Zhou, T. M., “Numerical (3) Parametric analysis shows that wind-induced vibra- Simulation of Vortex-induced Vibration of a Square tions of the square cylinder strongly dependent on Cylinder at a Low Reynolds Number,” Physics of Flu- the Scruton number. The transverse vibrations are ids, Vol. 25, No. 2 (2013). doi: 10.1063/1.4792351 obviously divided into galloping and vortex-induced [6] Fang, P. Z. and Gu, M., “Numerical Simulation of Vor- vibration at large Scruton number, while the switch tex-induced Vibration for a Square Cylinder at High between the two vibration types is not so remarkable Reynolds Number,” Journal of Tongji University (Natural when the Scruton number is low. Science), Vol. 36, No. 2, pp. 161-165 (2008). (in Chinese) The object used in the present study is a 2-D cylin- [7] Hirt, C. W., Amsden, A. A. and Cook, J. L., “An Arbi- der, which is a simple simulation process. The fluid- trary Lagrangian-Eulerian Computing Method for All structure interaction for real high-rise buildings will be Flow Speeds,” Journal of Computational Physics, Vol. considered in our future work. 135, pp. 203–216 (1997). doi: 10.1006/jcph.1997.5727 [8] Glück, M., “Computation of Wind-induced Vibrations Acknowledgments of Flexible Shells and Membranous Structures,” Jour- nal of Fluids and Structures, Vol. 17, No. 5, pp. 739- The authors gratefully acknowledge the supports 765 (2003). doi: 10.1016/S0889-9746(03)00006-9 from the National Natural Science Foundation of China [9] Scruton, C., “On the Wind Excited Oscillations of (51408196, 91215302, 90715040), and Plan for Scien- Stacks, Towers and Masts,” Proceedings of the Inter- tific Innovation Talent of Henan University of Techno- national Conference on Wind Effects on Buildings and logy (2014CXRC03). Structures, Her Majesty’s Stationary Office, Tedding- ton, Vol. 298 (1963). References [10] Lyn, D. A., “Phase Averaged Turbulence Measure- ments in the Separated Shear Layer Region of Flow [1] Braun, A. L. and Awruch, A. M., “Aerodynamic and Around a Square Cylinder,” Proc 23rd Cong Int Ass Aeroelastic Analyses on the CAARC Standard Tall Hydraulic Research, Ottawa Canada, pp. 85-92 (1989). Building Model Using Numerical Simulation,” Com- [11] Bouris, D. and Bergeles, G., “2D LES of Vortex Shedding puters & Structures, Vol. 87, No. 9-10, pp. 564-581 from a Square Cylinder,” Journal of Wind Engineering (2009). doi: 10.1016/j.compstruc.2009.02.002 and Industrial Aerodynamics, Vol. 80, pp. 31-46 (1999). [2] Farhat, C., Rallu, A., Wang, K., et al., “Robust and Provably Second-order Explicit-explicit and Implicit- Manuscript Received: Oct. 28, 2016 explicit Staggered Time-integrators for Highly Non- Accepted: Feb. 15, 2017