EFFECTS OF AXIS RATIO ON THE VORTEX-INDUCED VIBRATION AND ENERGY HARVESTING OF RHOMBUS CYLINDER

Transcription

1 Proceedings of the SME 2014 Power Conference POWER2014 July 28-31, 2014, Baltimore, Maryland, US POWER EFFECTS OF XIS RTIO ON THE VORTEX-INDUCED VIBRTION ND ENERGY HRVESTING OF RHOMBUS CYLINDER Li Zhang Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education of China(Chongqing University) Chongqing, China Heng Li Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education of China(Chongqing University) Chongqing, China Lin Ding Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education of China(Chongqing University) Chongqing, China BSTRCT The vortex-induced vibrations of a rhombus cylinder are investigated using two-dimensional unsteady Reynolds- veraged Navier-Stokes simulations at high Reynolds numbers ranging from 10,000 to 120,000. The rhombus cylinder is constrained to illate in the transverse direction, which is perpendicular to the flow velocity direction. Three rhombus cylinders with different axis ratio (R=0.5, 1.0, 1.5) are considered for comparison. The simulation results indicate that the vibration response and the wake modes are dependent on the axis ratio of the rhombus cylinder. The amplitude ratios are functions of the Reynolds numbers. nd as the R increases, higher peak amplitudes can be made over a significant wide band of Re. On the other hand, a narrow lock-in area is observed for R=0.5 and R=1.5 when 30,000<Re<50,000, but the frequency ratio of R=1.0 monotonically increases at a nearly constant slope in the whole Re range. The vortex shedding mode is always 2S mode in the whole Re range for R=0.5. However, the wake patterns become diverse with the increasing of Re for R=1.0 and 1.5. In addition, the mechanical power output of each illating rhombus cylinder is calculated to evaluate the efficiency of energy transfer in this paper. The theoretical mechanical power P between water and a transversely illating cylinder is achieved. On the base of analysis and comparison, the rhombus cylinder with R=1.0 is more suitable for harvesting energy from fluid. INTRODUCTION Flow-induced motion (FIM) is a canonical problem of fluid-structure interaction. Elastically mounted, rigid, bluff bodies that are long in one direction perpendicular to a free stream are susceptible to this phenomenon. The classic research on FIM is that of a circular cylinder, elastically mounted, immersed in a free stream. Much research has been made in this field for the past decades. Generally speaking, vortex-induced vibration is affected by a large number of system parameters, which include the mass ratio, structure stiffness, system damping, surface roughness of the cylinder, etc (1-5). The illating amplitude is an extremely important measurement, which can quantify the displacement response of vortex-induced vibration (VIV). Moreover, the frequency is another crucial measurement to describe the dynamic characteristic of the illation system. s for a circular cylinder subjected to flow-induced motion, several different frequencies are defined to fully describe the phenomenon, such as the natural frequency of the cylinder, the illating frequency and the vortex shedding frequency. With the increasing of flow velocity, the illating frequency will remain around the natural frequency over a range of flow velocity, this is called a lock-in area and the cylinder will illate at a relatively high amplitude. n experimental study of a circular cylinder with different mass ratios were conducted by Govardhan and Williamson (7), the results show that the illation frequency is a function that depends on the reduced velocity. On the other hand, in order to gain a better understanding of VIV, Williamson and Roshko (9) studied the relationship between the vibration amplitude and the vortex shedding mode. Digital Particle Image Velocimetry (DPIV) was adopted in the experiments to identify the vortex shedding patterns behind the illating cylinder. Based on a large number of tests results, the Williamson-Roshko wake mode maps (6, 9) were established. 1 Copyright 2014 by SME

2 Their findings suggest that the vortex shedding mode is closely related to the vibration of the cylinder. Different wake modes can be observed for a rigid cylinder with low mass and low damping undergoing vortex-induced vibration, including the 2S, 2P, P+S and 2P+2S, etc(7-11). Here, the S represents a single vortex and the P represents a pair of vortices. Need to add that, more complicated wake modes could be created in some circumstances, but these complex modes are also made up of some basic modes. Flow-induced motion is the product of the interaction between fluid and the cylinder. This kind of interaction depends upon two factors: the density and velocity of the fluid, the stiffness and mass distribution of the structure. It s significant to note that the influence of the geometry of the illating cylinder is one that has received little attention with regard to VIV (12). ctually, geometrical features can have a significant impact on the exciting force of flow-induced motion and the study on this effect can offer some insight into the excitation mechanism. fter studying the flow-induced motion of twodimensional bluff body, Pakinson and Simith (13) pointed out that the shape and size of a bluff body are the most important parameters affecting the exciting force. Deniz and Staubli (14, 15) performed controlled illation experiments of rectangular- and octagonal- section cylinders. The visualization analysis of the flow field was adopted to show the process of vortex shedding and collision. The research highlighted that the length of the after-body (the body downstream of the separation points of the shear layers) can have a substantial impact on the illation response of the cylinder. It can be inferred from the existing literature that data on the flow over rhombus cylinders is limited, especially at high Reynolds numbers. Meanwhile, research on the energy transfer of rhombus cylinders has not been reported yet. Hence, the focus of this numerical study is to investigate the influence of geometry on vortex-induced vibration of rhombus cylinder and to identify the passive geometric features that can be used to generate large amplitude illations suitable to drive a generator. In the present work, two-dimensional unsteady turbulent flow over a rhombus cylinder with different axis ratio (R) was investigated numerically. The rhombus cylinder is constrained to illate in the transverse direction. Numerical simulations were performed for high Reynolds numbers ranging from 10,000 to 120,000 in three different axis ratios, R=0.5, 1.0 and 1.5. The influences of the R on the amplitude and frequency ratio, wake structure, and theoretical mechanical power are obtained. PHYSICL SYSTEM The physical model considered in this paper consists of an illatory system as depicted in Fig.1. The elements of the illatory system are a rigid rhombus cylinder, K is the stiffness of two supporting linear springs, and the system damping C due to friction. The rhombus cylinders are constrained to illate in the y-direction, which is perpendicular to the flow direction (x). Three rhombus cylinders with different axis ratio (R=0.5, 1.0, 1.5) are considered for comparison, where R=L/D and L is the length of diagonal of the rhombus cylinder parallel to the flow direction, D is the length of diagonal vertical to the flow direction. The Reynolds number is defined as Re =UD/ν. Fig. 1 Schematic of the physical model In this paper, the system parameters of the models in the 2- D URNS simulation are listed in Table 1 and Table 2. Table 1 Nomenclature Peak amplitude peak Root-mean-square amplitude D L Re K n n, water Length of diagonal vertical to the flow direction Length of diagonal parallel to the flow direction Reynolds number Spring constant T 1/ f Natural period in water U C system Flow velocity Structural damping fn, water K / ( m ma ) / 2 System natural frequency in water f Oscillating frequency of cylinder m d m a a d Displaced fluid mass C m dded mass m Oscillating system mass * m m / md C a Mass ratio Kinematic molecular viscosity Density of the fluid dded mass coef. 2 Copyright 2014 by SME

3 Table 2 Physical model parameters Item R=0.5 R=1.0 R=1.5 Diameter D [m] Length l [m] Oscillating system mass m [kg] Spring const. K [N/m] Damping Natural freq. in water C system f n, water moving wall boundary condition is applied for the cylinder when the cylinder is in FIM. In order to see the effect of the axis ratio (R) on the vortex-induced vibration of rhombus cylinder, numerical simulations were done for high Reynolds numbers ranging from 10,000 to 120,000 in three different axis ratios, R=0.5, 1.0 and 1.5. GOVERNING EQUTION In this section, the mathematical modeling for the fluid dynamics is provided. In this study, the flow is assumed to be two-dimensional and unsteady, and the fluid is incompressible. The flow is modeled using the Unsteady Reynolds-veraged Navier-Stokes (URNS) equations together with the one-equation Spalart llmaras (S ) turbulence model. The basic URNS equations are: Ui 0 (1) xi Ui 1 p ( UiU j ) (2 Sij u iu j ) (2) t x x x j i j where, is the molecular kinematic viscosity and mean strain-rate tensor. 1 U U j ( i Sij ) 2 x x j i S ij is the U is the mean flow velocity vector. i ij uu is known as i j the Reynolds-stress tensor. The Boussinesq eddy-viscosity approximation is employed to solve the URNS equations for the mean-flow properties of the turbulence flow. The dynamics of the rhombus cylinder is modeled by the classical linear illator model. m y cy Ky f () t (4) Here, (3) m is the total illating mass of cylinder, c is the damping of the system, and K is the linear spring constant. COMPUTTIONL DOMIN ND GRID GENERTION The computational domain is 50D 50D for the single rhombus cylinder. s shown in Fig.2, the entire domain includes five boundaries: inflow, outflow, top, bottom, and the cylinder. The cylinder is located in the center of the computational domain. The inflow velocity is considered as uniform and constant velocity. t the out flow boundary, a zero gradient condition is specified for velocity. The bottom and top conditions are defined the same as the inflow boundary. Fig. 2 Computational domain. Sample of the grid points for R=1.5. grid sensitivity study was conducted on three different grid levels for the rhombus cylinder with different R. The vibration displacement and lift coefficient were calculated for comparison. For the present study, all three grids obtain similar results. In view of the computer resources and computing time, grid number was chosen for all simulations. Fig.2 displays a sample of the grid points for R=1.5. RESULTS ND DISCUSSION Simulations were conducted to study the effect of axis ratio (R=0.5, 1.0, 1.5) on the fluid flow around a rhombus cylinder. In particular, the correlations between the geometry and the vibration response, the energy output have been analyzed. The traditional measure of flow-induced motion response has been the peak amplitude of illation. This measure admits only harmonic body illations, meaning that the peak 3 Copyright 2014 by SME

4 amplitude and frequency of the illating cylinder are required to fully describe the motion. It s not a problem during fully synchronized VIV. However, the vibration properties are not clear when the flow is not completely synchronized to the body motion. In order to get a better description of the amplitude response of rhombus cylinder, the peak amplitude ( ) and the root-mean-square (RMS) amplitude ( this paper. The peak amplitude ratio ( peak peak ) are presented in /D) for the numerical study are plotted in Fig.3a. Fig.3b presents a measure of the RMS amplitude ratio ( /D), and provides a more meaningful measure of magnitude for asymmetric illations. The peak amplitude is below 0.1D in this region. Further increasing Re sees the onset of the upper branch for 40,000<Re<60,000. The peak amplitude risen steeply from 0.08D to 0.68D when the Reynolds number reaches 40,000, then stays around 0.7D. With the increasing of Reynolds number, the following region is identified as the lower branch in which the peak amplitude gradually dropped to about 0.4D and stays stable when Re increases from 70,000 to 120,000. b) R=1.0: No obvious branches are observed in the amplitude response of this rhombus cylinder. When Re increases from 10,000 to 70,000, the peak amplitude rises sharply from 0.05D to 0.97D. Then it stays around 1D and the maximum amplitude 1.07D is obtained when Re=120,000. c) R=1.5: No obvious branches are observed in the amplitude response of this rhombus cylinder. s can be seen from Fig. 3a, the peak amplitude increases continuously with Re in the whole Re range. The maximum amplitude 1.52D is achieved when Re=120,000. However, the RMS amplitude has a different changing trend, this difference indicates that the illating loses its periodicity. Fig. 3b shows that the RMS amplitude stays around 0.5D when Re>60,000. Fast Fourier Transform (FFT) is used to process the simulation records for each run and for each cylinder. The major harmonic frequency of illation for each cylinder is non-dimensionalized by the corresponding system natural frequency in water, f. The frequency ratio is plotted versus n, water Reynolds number in Fig.4. Fig. 3 Peak amplitude ratio ( amplitude ratio ( peak /D). /D). RMS Fig.3 shows the peak amplitude and the root-mean-square amplitude of illation for all three geometries as functions of Reynolds number: a) R=0.5: It should be noted that the peak amplitude ratio and the RMS amplitude ratio follow the same trends as the Re increases. number of regions have been identified for this cylinder in two different amplitude ratio curves. The first region is the initial branch in VIV for Re<30,000. f / f ) Fig. 4 Frequency ratio (, n water a) R=0.5: s shown in Fig.4, The frequency ratio quickly rises to 1.1 when Re reaches 30,000. With the increases of Re, a narrow lock-in area is observed for 30,000<Re<50,000, which corresponding to the upper branch. Following the lock-in area, the curve shows a 4 Copyright 2014 by SME

5 nearly constant slope when Re>50,000. Eventually, the maximum frequency ratio 2.84 is achieved at Re=120,000. b) R=1.0: The frequency ratio of this cylinder monotonically increases at a nearly constant slope in the whole Re range. It's important to point out that the frequency ratio curve of R=1.0 keeps above the other two curves when Re>50,000. The maximum value reaches up to 4.1 at Re=120,000. c) R=1.5: s the Re increases to 30,000, the frequency ratio reaches fter Re=30,000, frequency ratio stabilizes around 1.1 for a narrow Re range, with the illation frequency of the cylinder being very close to the system natural frequency. When Re>60,000, the frequency ratio increases sharply to relatively high value. In fact, amplitude and frequency are integral properties of the fluid-structure dynamics. The differences in vortex shedding patterns are caused by the interaction of vortices being shed from the sharp corners (including the leading edge and the trailing edge) of the cylinder. To get a better understanding of the relationship between the illation response and the vortex shedding process, the vortex structures around the cylinder are presented. The simulation results of three typical Reynolds numbers are presented. The vortex structure for each rhombus cylinder at Re=30,000, Re=60,000, Re=90,000, and Re=120,000 are presented in Figs.5-8, respectively. The most striking feature of rhombus cylinders is the sharp trailing edge can have a significant impact on the wake response, and the vortex pattern becomes more complex when the R increases. It's important to note that the vortex shedding mode is always 2S mode in the whole Re range for R=0.5. However, the wake patterns become diverse with the increasing of Re for R=1.0 and 1.5. s for R=0.5, the size of the trailing edge is relatively small and the trailing edge has slight effect on the interaction between the two shear layers. The two separating shear layers could interact with each other freely, the vortex formed on one side of the cylinder will be chopped by the shear layer from another side of the cylinder. In this process, 2S mode is formed. However, the impact of the sharp trailing edge can no longer be ignored for R=1.0 and R=1.5. The vortex formation is influenced by the interaction between the shear layer and the sharp trailing edge. s can be observed in the images of vorticity, the sharp trailing edge delays the interaction of the separation shear layers and snips the vortices formed from the shear layers. For the 2P mode, both forming vortices are snipped by the sharp trailing edge. During the development of P+S, P+S+S and P+S+P mode, a similar effect of trailing edge can be observed in the vortex shedding process. a) Re=30,000 In Fig.5, images of vorticity clearly describe the vortex structure of a rhombus cylinder. For R=0.5, the 2S mode of vortex shedding is observed for the rhombus cylinder. Two single vortices shed from the cylinder per cycle of illation, one by the top shear layer and another one by the bottom shear layer. For R=1.0, the wake is in a P+S mode, consisting of a pair of vortices on one side, and a single vortex on the other side per illation cycle. 2P mode is observed for R=1.5, consisting of two pairs of vortices per illation cycle. (c) Fig. 5 Vortex structures for different rhombus cylinders at Re=30,000. R=0.5, R=1.0, (c) R=1.5. b) Re= 60,000 The wake is still in 2S mode for R=0.5. But for R=1.0, two different modes are observed. For a period of illation at a relatively low amplitude, the mode is P+S. However, this mode is not stable. Fig.6 shows that a P+S+S mode is observed. During the downstroke of the first cycle, the top shear layer fo a pair of vortices shed from the trailing edge at first, then another single vortex shed from the top shear layer. This change in wake mode corresponding to a peak amplitude which can be seen in the displacement curve. For R=1.5, the wake modes are 2P and P+S+P. One more S shed from the shear layer per illation cycle, which lead to development from 2P to P+S+P. This development in the wake pattern also generates a sudden change in the displacement of the cylinder. 5 Copyright 2014 by SME

6 calculated on the base of a mathematical model, which was developed by Bernitsas et al. (16). In this model, the theoretical power output P is proportional to the fourth power of major harmonic frequency ( amplitude ( ). f ) and the square of root-mean-square (c) Fig. 6 Vortex structures for different rhombus cylinders at Re=60,000. R=0.5, R=1.0, (c) R=1.5. It should be mentioned that the wake mode (R=1.0 and 1.5) becomes very unstable when Re>60,000. It will transform between several different modes. The unstable wake pattern creates chaos in the wake of a cylinder. c) Re= 90,000 s shown in Fig.7, when Re=90,000, the wake mode of the three rhombus cylinders is approximately the same with that of Re=60,000. But the entrainment between the shedding vortices are much more stronger. d) Re= 120,000 Fig.8 shows that, even when Re reaches 120,000, the wake mode is still typically 2S for R=0.5. In comparison, the wake modes of other two rhombus cylinders are more complex. s for R=1.0 and R=1.5, three different modes can be seen in the wake, including 2P, P+S and P+S+S. The entrainment and merging between the shedding vortices are so strong that it s more difficult to identify the different modes. In order to evaluate the efficacy of energy transfer, the mechanical power output of each illating rhombus cylinder is obtained in this section. The mechanical energy transfer P between water and a transversely illating cylinder is (c) Fig. 7 Vortex structures for different rhombus cylinders at Re=90,000. R=0.5, R=1.0, (c) R=1.5. Fig.9 shows the mechanical power output dependence with the R and Reynolds number. The mechanical power output will grow with increasing Re for each rhombus cylinder. When Re<70,000, the power is all below 0.5-watt. But when Re>70,000, significant increasing of P is observed. Especially for cases of R=1.0 and R=1.5. The output P increases sharply to a relatively high value. This is due to the rapidly increases in illation frequency of these two cylinders, which can be seen in Fig. 4. It should be noted that the output of R=1.0 is always higher than the other two cylinders when Re>60,000 and the Re range of high output is much wider. The maximum P reaches 5.02-watt at Re=120,000, around 50% greater than the maximum output of R=1.5. On the base of analysis and comparison, the rhombus cylinder with R=1.0 is 6 Copyright 2014 by SME

7 more suitable for harvesting energy from fluid. In other words, it has a greater potential to drive a generator among these three rhombus cylinders. (c) Fig.8. Vortex structures for different rhombus cylinders at Re=120,000. R=0.5, R=1.0, (c) R=1.5. Fig. 9 Mechanical power output P (in watts). CONCLUSIONS In this study, two-dimensional unsteady turbulent flow over a rhombus cylinder with different axis ratio (R) was investigated numerically. The rhombus cylinder is constrained to illate in the transverse direction. Numerical simulations were performed for high Reynolds numbers ranging from 10,000 to 120,000 in three different axis ratios, R=0.5, 1.0, and 1.5. The influences of the R on the amplitude and frequency ratio, wake structure, and theoretical mechanical power are obtained: 1) Both the peak amplitude ratio and root-mean-square amplitude ratio are functions of the Reynolds numbers. s the R increases, higher peak amplitudes can be achieved over a significant wide band of Re. For R=0.5, three branches have been observed, including the initial branch, the upper branch and the lower branch. However, the branches can not be clearly observed in the amplitude response of the other two cylinders. 2) The frequency-ratio results show that a narrow lock-in area is observed for R=0.5 and R=1.5 when 30,000<Re<50,000, but the frequency ratio of R=1.0 monotonically increases at a nearly constant slope in the whole Re range. It should be noted that the frequency ratio curve of R=1.0 keeps above the other two curves when Re>50,000 and the maximum value reaches up to 4.1 finally. 3) The most striking feature of rhombus cylinders is the sharp trailing edge can have a significant impact on the wake response, and the vortex pattern becomes more complex when the R increases. The vortex shedding mode is always 2S mode in the whole Re range for R=0.5. However, the wake patterns become diverse with the increasing of Re for R=1.0 and 1.5. When Re>60,000, it will transform between several different modes. 4) The mechanical power output will grow with increasing Re for each rhombus cylinder. For R=1.0, the output is always higher than the other two cylinders when Re>60,000 and the Re range of high output is also much wider. Thus, it can be concluded that the cylinder with R=1.0 has a greater potential to drive a generator among these three rhombus cylinders. 5) The results of this paper are only limited to three different axis ratios, the case of R=1.0 may not be the most appropriate rhombus cylinder for energy harvesting. Hence, much more work must be done to fully analyze the vibration response of rhombus with plenty of different axis ratios. Moreover, passive control can also be used to improve the disorder and chaos in the wake of a rhombus cylinder, such as rounding or chamfering the corners of a rhombus cylinder. CKNOWLEDGMENTS This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No ). 7 Copyright 2014 by SME

8 REFERENCES 1 Govardhan R, Williamson C H K. Critical mass in vortexinduced vibration of a cylinder[j]. European Journal of Mechanics B-Fluids, 2004, 23(1): Vikestad K, Vandiver J K, Larsen C M. dded mass and illation frequency for a circular cylinder subjected to vortex-induced vibrations and external disturbance[j]. Journal of Fluids and Structures, 2000, 14(7): Skop R, Luo G. n inverse-direct method for predicting the vortex-induced vibrations of cylinders in uniform and nonuniform flows[j]. Journal of Fluids and Structures, 2001, 15(6): Guilmineau E, Queutey P. Numerical simulation of vortexinduced vibration of a circular cylinder with low massdamping in a turbulent flow[j]. Journal of Fluids and Structures, 2004, 19(4): Jauvtis N, Williamson C H K. The effect of two degrees of freedom on vortex-induced vibration at low mass and damping[j]. Journal of Fluid Mechanics, 2004, 509: Khalak, Williamson C H K. Motions, forces and mode transitions in vortex-induced vibrations at low massdamping[j]. Journal of Fluids and Structures, 1999, 13(7-8): Williamson C H K, Govardhan R. Vortex-induced vibrations[j]. nnual Review of Fluid Mechanics, 2004, 36: Govardhan R, Williamson C H K. Modes of vortex formation and frequency response of a freely vibrating cylinder[j]. Journal of Fluid Mechanics, 2000, 420: Williamson C H K, Roshko. Vortex formation in the wake of an illating cylinder[j]. Journal of Fluids and Structures, 1988, 2(4): Williamson C H K. Sinusoidal flow relative to circularcylinders[j]. Journal of Fluid Mechanics, 1985, 155(1): Jauvtis N, Williamson C H K. Vortex-induced vibration of a cylinder with two degrees of freedom[j]. Journal of Fluids and Structures, 2003, 17(7): Leontini J S, Thompson M C. Vortex-induced vibrations of a diamond cross-section: Sensitivity to corner sharpness[j]. Journal of Fluids and Structures, Parkinson G V, Smith J D. The square prism as an aeroelastic non-linear illator[j]. The Quarterly Journal of Mechanics and pplied Mathematics, 1964, 17(2): Deniz S, Staubli T. Oscillating rectangular and octagonal profiles: modelling of fluid forces[j]. Journal of fluids and structures, 1998, 12(7): Deniz S, Staubli T. Oscillating rectangular and octagonal profiles: Interaction of leading-and trailing-edge vortex formation[j]. Journal of Fluids and Structures, 1997, 11(1): Bernitsas M M, Raghavan K, Ben-Simon Y, et al. VIVCE (vortex induced vibration aquatic clean energy): new concept in generation of clean and renewable energy from fluid flow[j]. Journal of Offshore Mechanics and rctic Engineering, 2008, 130(4): Copyright 2014 by SME