Numerical Investigation of Vortex Induced Vibration of Two Cylinders in Side by Side Arrangement Sourav Kumar Kar a, 1,, Harshit Mishra a, 2, Rishitosh Ranjan b, 3 Undergraduate Student a, Assitant Proffessor b, School of Mechanical Engineering KIIT University, Bhubneswar-751024. Abstract In this work, Vortex Induced Vibrations (VIV) and Vortex Shedding characteristics of elastically mounted two cylinders in side by side arrangement has been investigated. The cylinders have 2DoF (Two Degrees of Freedom) and a mass ratio (ratio of mass of cylinder and mass of fluid displaced by cylinder), m*=3. The study was conducted for various frequency ratios (ratio of natural frequency in x and y direction), 1 fr 2, and different reduced velocity, U* values ranging from 4 to 10 with incompressible laminar flow at Reynolds Number 160 and transverse gap ratio (T/D) 3. The Reynolds number is based on the diameter of cylinder (D) and free stream velocity (U). The numerical simulation was performed using finite volume method (FVM) by solving 2D Navier Strokes Equation with SIMPLE pressure velocity coupling scheme using Block Structured Quadrilateral Grid with pressure based solver in ANSYS Fluent. The oscillating motion produced by interacting vortices on the two cylinders were studied by calculating the vibrating frequency, amplitude of vibration and the trajectory of both the cylinders. Keywords: VIV, Laminar, 2DOF, Trajectory Plot I. INTRODUCTION Flow past circular cylinders is a frequently studied topic both in industry and academia due to its large number of applications. By analyzing the fluid flow around such bluff bodies we can predict the aerodynamic force (Drag and Lift) acting on the bodies. These forces cause vibrations in the bodies. These vibrations are called vortex Induced Vibrations (VIV). The main reasons for these vibrations are Vortex shedding and fluid elastic excitation. Among these two vortex shedding is the most important. It is calculated by studying the boundary separation phenomenon. As the fluid moves it collides against the cylinder. Thus a pressure is created in the vicinity of the bluff body. Thus we can imagine a pressure gradient with pressure increasing from free stream pressure to the high pressure just in front of the cylinder. But this pressure is not high enough to push the boundary layer all the way to the back of the cylinder. Thus flow separation occurs along with the formation of free shear layers. The separated flow leads to the formation of a wake region behind the cylinder where the flow velocity is less than the free stream velocity. The free shear layers rolls up and forms vortices which are shed alternatively from either sides of the bluff body. These vortices impart oscillating forces to the body. Resonance occurs if the vortex shedding frequency matches the natural frequency of the body. Nowadays newer material are being developed which are of low weight and high strength. These materials are more prone to vibrations due to the light weight. Thus the importance of research in the field of flow induced vibration is increasing. Some of the applications are heat exchangers, overhead wires, boilers and super heater structures, offshore structures, vibration and failure of suction line strainers, chimney stacks and cooling towers. In vortex induced vibrations the body vibrates both in in-line and in transverse directions. The forces causing the vibration have unsteadiness both in Lift and Drag and the in-line vibrations must also be considered along with the transverse vibrations. Also the in-line fluctuations may have significant effect on the wake structure at the back of the body. Nature prefers a figure eight type motion which can only be achieved by considering the in-line vibrations as only 1D trace cannot produce such a figure. The drag force acting on the cylinder has steady and oscillating component. The oscillating part also excites the body in in-line direction with significant amplitude. Vibration analysis of elastically mounted circular cylinder in flow, is a fundamental case of VIV. It has been studied for many decades so that the forces acting on the body can be calculated and finally the motion of the body can be predicted. Feng [1]carried out studies for VIV in 1DOF in cross flow direction for high m* (mass ratios). Williamson and Roshko [6] found that as ISSN: 2231-5381 http://www.ijettjournal.org Page 291
the vortex shedding frequency is forced to interact with the cylinder vibration therefore the vortex shedding frequency becomes equal to the vibration frequency. Yang et al. [7]-[8] experimentally studied the flow around a cylinder near a rigid boundary and found that the amplitude and frequency of VIV have certain relationships with the Gap/Diameter ratio and U* (reduced velocity). Mahir and Rockwell [4] studied the forces out of phase and in phase vibrations of cylinders in side by side arrangement. Huera-Huarte and Gharib [2]-[3] studied the side by side arrangement of two flexible cylinders and found that the interference between the cylinders weakened as the center gap to diameter ratio increased above the value of 3.5. Williamson [5] conducted a qualitative study of Vortex Street created by two circular cylinders placed in side by side arrangement at different spacing, using flow visualization. They found that two parallel vortex streets can be formed in such an arrangement as an in-phase or an out of phase system. In an out of phase system two separate vortex streets are formed whereas an in phase vortex street can lead to the formation of a single large wake with binary vortices. When the T/D ratio decreases below a certain threshold value, the flow pattern becomes asymmetric in the downstream region of the cylinders and a large wake is formed. II. PROBLEM STATEMENT AND GOVERNING EQUATIONS WITH BOUNDARY CONDITIONS A. Problem Statement The flow field consists of a 2D incompressible fluid flowing over two elastically mounted cylinders of circular cross section in side by side arrangement, with two degrees of freedom. The transverse gap ratio (T/D) is 3. The fluid flow velocity is U. The diameter of the cylinder is D. The domain in the upstream direction is 15D and in the downstream direction is 25D from the Y- axis. Both the lateral walls are at a distance of 8D from the X-axis in the cross stream direction. Fluid enters the domain with X-direction velocity equal to U and Y- direction velocity equal to zero. B. Governing Equation In this study the flow was investigated solving the Navier strokes equation which can be written in its nondimensional form as follows: (1) (2) Where, Reynolds Number, Re= UD/, = dynamic viscosity, D= reference length, U= reference speed; U=(u x, u y ), this is the velocity field, t= time, p= modified static pressure, = density. In this study we considered a rigid cylinder mounted elastically; thus it was allowed to oscillate in X and Y direction. The oscillation is described by the equations of a linear mass spring damper system when forced by a fluid load. The non-dimensional form of this equation can be written as follows: (3) (4) Where, M * = non-dimensional mass, C * =nondimensional damping coefficient (0 in this case), K * = non-dimensional stiffness coefficient, F * = nondimensional force imparted on fluid. Where, L= axial length of cylinder, C L = lift coefficient, the variables x * c and y * * * * c, c and c, c and * c are the non-dimensional displacement, velocity and acceleration of the body in x and y directions respectively, t * = non-dimensional time. Fig. 1. Computational domain with Boundary condition To make the non-dimensional structure form and flow equations consistent and to facilitate easy coupling between structure and flow solvers, the non-dimensional scheme was used. To make it easily comparable with the previously published data the results of this study are shown as functions of mass ratio m * and reduced velocity U *. ISSN: 2231-5381 http://www.ijettjournal.org Page 292
Where, natural frequency of the structure in vacuum,. C. Boundary Condition At the inlet velocity, inlet boundary condition was used as. The outlet was taken as a pressure outlet boundary condition in ANSYS Fluent with gauge pressure equal to zero. Here the default pressure outlet boundary is similar to the fully developed flow conditions and the velocity gradient in axial direction is zero but gradient exists in the lateral direction, i.e.. The lateral boundaries are assumed to be slip boundaries i.e..the surface of the circular cylinder is implemented with no-slip condition for velocity i.e.. D. Parameters In this study, simulations are performed for an oscillating isolated cylinder. The cylinder has a diameter D. A uniform steady fluid flows around the cylinder with velocity u x =1.6 at Reynolds number 160. The cylinder is free to vibrate in transverse and in in-line directions. The mass ratio (ratio of mass of cylinder to mass of fluid displaced by the cylinder) of the body is m*=3. The damping coefficients of the structure are set to zero, to have high amplitude of vibrations. The reduced velocity, U*, is increased from 4 to 10. Five frequency ratios, f r, (= fnx/fny, =1.0, 1.25, 1.5, 1.75 and 2.0) are studied for each reduced velocity. III. NUMERICAL METHOD The entire domain was divided into many elements by meshing. The finite Volume Method was used in the present simulations and it was run in transient mode. The laminar model is used. The SIMPLE scheme is used for pressure velocity coupling. In spatial discretization, least squares cell based option is used in gradient. Second order upwind and first order implicit options were used in momentum and Transient formulation respectively. A grid independence test was conducted to find out the grid size that would be optimum for the present simulation. Grids with 160 nodes on the circumference of circular (?) was selected to ensure a reasonable computation time. Similarly, time independence test was done for solution stability and a reasonable computation time. 10 Fig. 2. Extended domain mesh and Enlarged view IV. RESULT AND DISCUSSION U* 4 5 6 7 8 9 Fig. 3. Trajectory of centroid of cylinder with frequency ratio 1 and at different reduced velocity. 10 U* 4 5 6 7 8 9 ISSN: 2231-5381 http://www.ijettjournal.org Page 293
at the reduced velocity of 4 for frequency ratio 1 and 1.5. At the highly reduced velocity values, the ratio curve for frequency ratio 1 and 2 somewhat overlaps. Defection of both the cylinders from their respective mean positions increases in the downstream direction with increase in their reduced velocities, but the defection decreases with increase in frequency ratios. Fig. 4 Trajectory of centroid of cylinder with frequency ratio 1 and at different reduced velocity. 10 U* 4 5 6 7 8 9 Fig. 5. Trajectory of centroid of cylinder with frequency ratio 1 and at different reduced velocity. In the plots of the trajectory of centre of gravity of the cylinders, Fig.3-5, it is observed that as the reduced velocity (U*) increases, the regular Lissajous figure (resembles figure eight) becomes more and more irregular. The rms (root mean square) amplitude curves in Fig.6, shows that the deflection of in the stream direction is less than 10% of the diameter of the cylinders and the deflection in transverse direction is around 50% of the diameter of the cylinder. The amplitude decreases by an order of ten from higher reduced velocity to lower reduced velocity. As the frequency ratio decreases from 2 to 1, the eight figure becomes thinner and deformed. Further it is observed in Fig.8 that the vibrating frequency is lower than the natural frequency for all the cases for both the cylinders. The ratio increases as the reduced velocity increases in both the cylinders. Whereas in Fig.9, it is found that the vibration frequency of the oscillating cylinder is less than the shedding frequency of the rigidly mounted cylinder for almost all the cases except Fig.6. Plots of ratio of rms of amplitude in stream direction to the diameter of cylinder against reduced velocity of upper and lower cylinders. ISSN: 2231-5381 http://www.ijettjournal.org Page 294
Fig.7. Plots of ratio of rms of amplitude in transverse direction to the diameter of cylinder against reduced velocity of upper and lower cylinders Fig. 9. Plots of ratio of vibrating frequency of oscillating cylinder and shedding frequency of stationary cylinder against reduced velocity of upper and lower cylinders. Fig. 8. Plots of ratio of vibrating frequency and natural frequency of oscillating cylinder against reduced velocity of upper and lower cylinders. Fig. 10. Plots of ratio of deflection of cylinder in stream wise direction to the diameter of cylinder against reduced velocity of upper and lower cylinders. V. CONCLUSION Numerical computation has been successfully performed for 2D uniform flow around a pair of cylinders in side-by-side arrangement. The effects on vibration with change in the frequency ratio and reduced velocity on cylinders were studied by plotting amplitude response, frequency response and trajectory of centroid. Some partial conclusions are as follows: 1) With increase in reduced velocity, amplitude of vibration falls down. 2) At low reduced velocity, the trajectories of the cylinders follow the famous Lissajous figure. 3) With decrease in frequency ratio, the Lissajous figure gets distorted. ISSN: 2231-5381 http://www.ijettjournal.org Page 295
4) With increase in reduced velocity and frequency ratio, deflection of cylinder in downstream direction increases. REFERENCES [1] Feng, C.C., 1968. In: The Measurements of Vortex-Induced Effects on Flow Past a Stationary and Oscillation and Galloping (Master's thesis)university BC, Vancouver, Canada [2] Huera-Huarte,F.J.,Gharib,M.,2011a.Flow-induced vibrations of a side-by-side arrangement of two flexible circular cylinders. Journal of fluids and Structures. [3] Huera-Huarte,F.J.,Gharib,M.,2011b.Vortex and wake-induced vibrations of a tandem arrangement of two flexible circular cylinders with far wake interference. Journal of fluids and structures. [4] Mahir, N.,Rockwell,D.,1996.Vortex formation from a forced system of two cylinders part ii: side-by-side arrangement. Journal of fluids and structures. [5] Williamson, C., 1985. Evolution of a single wake behind a pair of bluff bodies. Journal of Fluid Mechanics. [6] Williamson, C.H.K., Roshko,A.,1988.Vortex formation in the wake of an oscillating cylinder, Journal of Fluids and Structures. [7] Yang,B.,Gao,F.P.,Jeng,D.S.,Ying- Xiangwu,Y.X.,2008.Experimental study of vortex-induced vibrations of a pipeline near an erodible sandy seabed. Ocean Engineering. [8] Yang,B.,Gao,F.P.,Jeng,D.S.,Wu,Y.X.,2009.Experimental study of vortex-induced vibrations of a cylinder near a rigid plane boundary in steady flow. Acta Mechanica Sinica ISSN: 2231-5381 http://www.ijettjournal.org Page 296