- Email: [email protected]
Force reduction of flow around a sinusoidal wavy cylinder
308
2009,21(3):308-315 DOI: 10.1016/S1001-6058(08)60151-1
FORCE REDUCTION OF FLOW AROUND A SINUSOIDAL WAVY CYLINDER* ZOU Lin School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China, E-mail: [email protected] LIN Yu-feng Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China
(Received May 13, 2008, Revised January 8, 2009)
Abstract: A large eddy simulation of cross-flow around a sinusoidal wavy cylinder at Re = 3000 was performed and the load cell measurement was introduced for the validation test. The mean flow field and the near wake flow structures were presented and compared with those for a circular cylinder at the same Reynolds number. The mean drag coefficient for the wavy cylinder is smaller than that for a corresponding circular cylinder due to the formation of a longer wake vortex generated by the wavy cylinder. The fluctuating lift coefficient of the wavy cylinder is also greatly reduced. This kind of wavy surface leads to the formation of 3-D free shear layers which are more stable than purely 2-D free shear layers. Such free shear layers only roll up into mature vortices at further downstream position and significantly modify the near wake structures and the pressure distributions around the wavy cylinder. Moreover, the simulations in laminar flow condition were also performed to investigate the effect of Reynolds number on force reduction control. Key words: wavy cylinder, force reduction, Large Eddy Simulation (LES), Reynolds number effect
1. Introduction Flow past a circular cylinder has been a subject of interest for a long time. The alternate vortex shedding from the cylinder will produce very complicated flow effects such as the flow-induced body vibration[1]. How to control vortex shedding and hence reduce the flow-induced vibration becomes a challenging problem in the fluid dynamics area[2,3]. By modifying the geometry of the cylindrical structures, a
* Project supported by the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200804971025), the Council of the Hong Kong Special Administrative Region, China (Grant No. PolyU 5311/04E) . Biography: ZOU Lin (1970-), Female, Ph. D., Associate Professor Corresponding author: LIN Yu-feng, E-mail: [email protected]
type of cylinder with sinusoidally varying diameter along its spanwise direction, called a wavy cylinder, is introduced (see Fig.1). It is hoped that this kind of cylinder could lead to better control of the vortex shedding and hence the suppression of flow-induced vibration of a cylindrical structure. Ahmed et al.[4] experimentally investigated the surface-pressure distributions of wavy cylinders with four spanwise wavelengths, and found that the separated flow structures near the geometric nodes are distinctly asymmetric for a large fraction of time and the sectional drag coefficients at the geometric nodes are greater than those at the geometric saddles. Ahmed et al.[5] studied the turbulent wake patterns behind a wavy cylinder. The topology of the boundary layer separation lines and the subsequent 3-D development of the turbulent wake structures were obtained. Lam et al.[6] found that drag reduction of up to 20% could be achieved by changing the geometric wavelength and
309
wave amplitude of the wavy cylinder in the range of subcritical Reynolds numbers from 2×104 to 5×104. It was also found that the root-mean-square fluctuating lift coefficients of wavy cylinders are much lower than those of circular cylinders. On the other hand, Zhang et al.[7,8] and Lee et al.[9] experimentally investigated the 3-D near wake structures behind a wavy cylinder. Along the spanwise direction, the well-organized streamwise vortices with alternating positive and negative vortices were observed. Lam et al.[10] also captured the detailed well-organized 3-D vortex structures for different wavy cylinders at a low Reynolds number of 100 by using the numerical simulation method. The current work is mainly aimed to investigate the relationship between sinusoidal wavy surface and flow characteristics of a wavy cylinder at a subcritical Reynolds number of Re = 3000 . The instantaneous 3-D vortex structures which can not be illustrated clearly using experimental methods will be captured using the Large Eddy Simulation (LES) method[11,12]. The experimental measurements will also be preformed for validating the LES results. Some valuable data, such as drag, lift, pressure, velocity and turbulent kinetic energy, etc., will be discussed compared with a corresponding circular cylinder at the same Reynolds number. The laminar flow condition simulations are also considered to discuss the effect of Reynolds numbers.
amplitude a / Dm = 0.15 are adopted in the present simulations considering the previously investigation results[3-10].
3. Experimental setup and flow measurements The experiments were carried out in a closed-loop water tunnel with the test section of 0.3 m×0.6 m×2.1 m (width, height, length). Forces on the wavy cylinder were measured using a threecomponent quartz piezoelectric load cell characterized by high response, resolution and stiffness. The load cell mounted on the top end of the wavy cylinder was bolted tightly between two machine-polished stainless steel blocks, which could measure instantaneous integral fluid forces acting on the length of the cylinder exposed within the water tunnel. Static calibrations of the load cell in the lift and drag directions were carried out using dead weights. 4. Computational models 4.1 Governing equations By using the 3-D LES turbulence model, the large scale eddies are solved directly by the filtered Navier-Stokes equations and the small eddies are modeled using a Sub-Grid Scale (SGS) model. The governing equations employed for the LES are
wui =0 wxi
Fig.1 Geometry of a wavy cylinder
wW wui wui u j w 2ui 1 wp + = +Q ij U wxi wt wx j wx j wx j wx j (i = 1, 2, 3)
2. Parametric definition of body geometry The geometry of a wavy cylinders is described by the equation, Dz = Dm + 2a cos(2Sz / O ) . As is shown in Fig.1, Dz denotes the local diameter of the wavy cylinder. The mean diameter is defined with Dm = ( Dmin + Dmax ) / 2 . Dmin is the minimum diameter of the wavy cylinder along the spanwise direction, while Dmax is the maximum diameter. The amplitude of the curved surface is denoted by a , and O is the wavelength along the spanwise direction. The axial location with maximum local diameter is referred to as “node”, while the minimum diameter is called “saddle”. The “middle” is also defined at the mid-point between the node and the saddle. All geometrical lengths are scaled with Dm . The dimensionless wavelength O / Dm = 1.5 and wave
(1)
(2)
where ui are the filtered velocity components in the Cartesian coordinates xi , p is the pressure, U is the fluid density and Q is the kinematic viscosity of the fluid. The influence of the small scales on the large (resolved) scales takes place through the subgrid scale stress defined by
W ij = ui u j ui u j
(3)
resulting from the filtering operation, which are unknown and must be modeled with a subgrid model. All computations in the present work were carried out with a Smagorinsky constant of Cs = 0.1 . (The detailed description see Zou et al.[12])
310
4.2 Numerical method To solve the 3-D unsteady incompressible Navier-Stokers equation, the finite volume method applied on unstructured grids is employed. The PISO algorithm is used to deal with the pressure-velocity coupling between the momentum and the continuity equations and a second-order central differencing scheme is used for momentum discretization while a second-order implicit scheme is employed to advance the equations in time.
the circular cylinder ( 3O wavelength along the spanwise direction). Figure 2(b) shows that the grid is nonuniform in the x-y plane but uniform along the z direction. The distance from the cylinder surface to the nearest grid points are fixed at y + = 1 . A uniform velocity profile ( u = 1 , v = w = 1 ) is imposed at the inlet, while the convective boundary condition is used at the outlet boundary. A periodic boundary condition is employed at the boundaries in the spanwise direction and a no-slip boundary condition is prescribed at the surface of the wavy cylinder. The lateral surfaces are treated as slip surfaces using symmetry conditions. 5. Validation test of numerical methods The grid independence tests were carried out for both wavy and circular cylinder cases and the grid number of 1.28×106 was adopted at last with considering the time consumption of the present simulations and the accuracy of the results. A dimensionless time step 't U f Dm = 0.02 was chosen for the simulation. The overall drag coefficient is defined by CD = 2 FD UU f2 Dm Lz , while the
Fig.2 Computational model of a wavy cylinder
4.3 Computational domain and boundary conditions Figure 2(a) shows the schematic diagram of the computational domain (in a fixed Cartesian coordinate system ( x, y , z ) ). The origin of the coordinate system is located at the end of the wavy cylinder. The x -axis is aligned with the inlet flow direction (streamwise direction). The z -axis is parallel to the cylinder axis (spanwise direction) and the y -axis is perpendicular to both the x and z axes (crosswise direction). In the x-y plane, the computational boundaries in the streamwise and crosswise directions are set at 24 Dm and 16 Dm , respectively. The upstream boundary is set at 8Dm away from the centerline of wavy cylinder. The height (spanwise direction) of computational domain is set at 4.5 Dm which will be sufficient for the present simulations (refer to the discussions by Zou et al.[12]). At subcritical Reynolds numbers, the periodic repeated vortex structures were observed to be consistent with the periodic repetition of the wavy cylinder spanwise wavelength O [7-9]. As a result, in the present simulation, the spanwise domain of the wavy cylinder is set equal to the same range as that of
overall lift coefficient is CL = 2 FL UU f2 Dm Lz . The total drag force and total lift force are given by FD and FL , respectively. The present results given by LES simulation and load cell measurement of both the wavy and circular cylinders are shown in Table 1 compared with other experimental and numerical results at Re = DmU f Q = 3000 . The mean drag coefficients ( CD ), fluctuating lift coefficients ( C Lc ) and Strouhal number ( St = f s Dm U f ) in the present simulation for circular cylinder agree with those of others[13,14]. Here, f s is the vortex shedding frequency. Moreover, the present experimental results of CD , C Lc and St for the wavy and circular cylinders are all in consistence with results obtained by the preset LES method. 6. Results and discussion 6.1 Force characteristics and Strouhal number As is summarized in Table 1, the values of CD and C Lc of the wavy cylinder are smaller than those of a corresponding circular cylinder at Re=3000. Compared with a circular cylinder, the drag coefficient reduction up to 15.8% by the experimental measurement and 16.5% by the LES simulation are obtained. The evident reduction of the fluctuating lift coefficient is also obtained. The Strouhal number of
311
the wavy cylinder is slightly lower than that of the circular cylinder at the same Reynolds number. It means that the variation in surface geometry has little or no effect on the frequency of vortex shedding for the wavy cylinder at such subcritical Reynolds number. The previous experimental measurement results on the Strouhal numbers also show that the wave shape has little effect on vortex shedding of wavy cylinders at subcritical Reynolds numbers[6].
Table 1 Experimental and LES results on force coefficients and Strouhal numbers of wavy and circular cylinders at Re = 3000 Case
CD
C Lc
St
CY (Exp.) Norberg[13]
0.98-1.03
í
0.210-0.213
CY (LES) Lu et al.[14]
1.07
0.48
í
CY (Exp.) Present
1.01
0.13
0.208
CY (LES) Present
1.09
0.18
0.210
WY (Exp.) Present
0.85
0.01
0.197
WY (LES) Present
0.91
0.01
0.198
6.2 Pressure and velocity vector distributions The mean velocity vectors distributions for wavy and circular cylinders are shown in Fig.3. The near wake width in the nodal plane is different from that in the saddle plane. The vortex shedding phenomenon is associated with flow separation from the boundary layer of bluff bodies. The stability of the free shear layer and the resulting vortex structure are highly dependent on the flow separation point (the points in the surface of bluff bodies where the shear stress vanishes) of the wavy cylinder. In the nodal plane of the wavy cylinder, the flow separation angle (104o) is much larger than that of a circular cylinder (88o), while it is smaller than that of a circular cylinder in the saddle plane (75o). As a result, the near wake velocity vectors expand along both the streamwise direction and crosswise direction and the wake width in the saddle plane is increased giving rise to a wide
wake at further downstream. However, a large value of separation angle in the nodal plane suppresses the shear layer development. The near wake velocity vectors in the nodal plane seem to be extended only in the streamwise direction and noticeably suppressed in the crosswise direction. As a result, it produces a narrower wake downstream. The significant spanwise flow moves from the saddle plane toward the nodal plane and depicts surface streamlines near the separation line similar to that shown in Fig.4. It means that this kind of wavy surface can significantly modify the near wake structures. Thus the near wake 3-D flow structures are formed with periodic variation along the spanwise direction of the wavy cylinder. In general, the 3-D free shear layers developed from the wavy cylinder suppress the vortex roll-up process and give rise to complex vortex formation. It explains why the value of fluctuating lift coefficient is very small.
Fig.3 Mean pressure coefficients and mean velocity vector distributions for wavy and circular cylinders in the x-y plane at Re = 3000
The mean pressure coefficient distributions in different planes of the wavy cylinder are also captured (see Fig.3). The maximum negative mean pressure coefficient near the wake of the wavy cylinder is smaller than that of the circular cylinder and its center is far away behind the wavy cylinder. In general, the
312
distributions of the mean pressure coefficient for the wavy cylinder have the same trend as the mean velocity vector distributions in both nodal and saddle planes. Considering the discussed above, we conclude that a wavy cylinder with smaller pressure difference can lead to the reduction of drag.
smaller than that of the circular cylinder.
Fig.4 Surface streamlines and the topology of the 3-D eparation lines for wavy and circular cylinders at Re = 3000
6.3 Turbulent kinetic energy Figure 5 shows the contour plots of the normalized Turbulent Kinetic Energy (TKE) in the x-y planes ( TKE = (u c2 + vc2 + wc2 ) / 2U f2 ) of the wake behind the wavy and circular cylinders at Re = 3000 . The region of negligible kinetic energy is noticeably larger behind the wavy cylinder than that of a circular cylinder. The maximum TKE and the high-value TKE regime in the nodal plane are larger than those in the saddle plane. However, all the TKE values in different cross-section planes of the wavy cylinder are smaller than those of the circular cylinder. The position of maximum TKE is farther away from the back of the wavy cylinder than that of the circular cylinder. Therefore, the overall TKE of the flow behind the wavy cylinder is lower than that of the circular cylinder at such subcritical Reynolds number. Figure 6 shows that in the x-z plane with y/Dm = 0 , the maximum TKE regime behind the circular cylinder is around the position x/Dm = 2 and the values of TKE are uniform along the spanwise direction. For the wavy cylinder at the downstream location of approximately x/Dm = 3.5 , however, the TKE exhibits local maxima behind the nodes (see Fig.6(a)). This is possibly due to the converging flow of the free shear layer from the saddle to the node (the significant spanwise flow moves from the saddle plane) and additional transport of kinetic energy near the node by the streamwise vortices. In the regimes behind the saddles, the values of TKE are smaller than those behind the nodes. In general, the spanwise variation of TKE shows periodic repetition for the wavy cylinder as contrast to the uniform distribution of the circular cylinder. Furthermore, the maximum of TKE along the spanwise direction for wavy cylinder is
Fig.5 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the x-y plane at
Re = 3000
Figure 7 shows the contours of the normalized TKE in the y-z plane at the downstream location x/Dm = 2 for the circular and wavy cylinders. For a circular cylinder, the TKE contours are nearly parallel to the cylinder axis. The significantly large value of TKE is in the wake center regime due to active momentum exchange, while it decreases along the crosswise direction. The TKE distribution of the flow behind the wavy cylinder shows periodic variation along the spanwise direction. Compared with the circular cylinder, the values of TKE in the near wake behind the wavy cylinder are smaller, especially at the saddle positions. It indicates that for the wavy cylinder, the strength of TKE decreases substantially in the near wake region which is the physical reason for the reduction of mean drag and fluctuating lift acting on the cylinder. In general, the flow around a wavy cylinder shows an evidently periodic characteristic in
313
the flow patterns along the spanwise direction. This kind of wavy surface leads to a significant reduction of TKE and in turn, the reduction of the mean drag force and the suppression of the fluctuating lift.
drag force reduction and the weakening of the lift force vibration.
Fig.7 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the y-z plane at x/Dm = 2 Fig.6 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the x-z plane at y/Dm = 0
and Re = 3000
and Re = 3000
6.4 3-D vortex structures Figure 8 shows the iso-surfaces of streamwise vorticity ( Z x ) of the wavy cylinder. It exhibits the periodic patterns along the spanwise direction with alternating negative and positive vortices and symmetrically distributed with respect to the central axis of the cylinders. The iso-surface of streamwise vorticity behind the wavy cylinder shows distinctively the regimes of zero vorticity (the recovery region) centered adjacent to the saddle planes. While no clear zero vorticity patterns can be found in other ranges of wavelength and no zero vorticity regimes can be observed for the circular cylinder. Zhang et al.[7] investigated a wavy cylinder with O /Dm = 2 at the same Reynolds number using the PIV method also observed this phenomenon. The periodic patterns of near-wake streamwise vorticity appear to be well organized and coherent. It means that the 3-D free shear layer generated from the wavy surface is more difficult to roll up to form vortex structures at the near wake positions. Finally, it rolls up to mature vortex at a further downstream position in the cylinder wake. It also means that a large value of vortex formation length is generated which leads to the significantly
Fig.8 Instantaneous streamwise vorticity iso-surfaces ( Z x = r1 ) for wavy and circular cylinders at Re = 3000
6.5 Reynolds number effect Table 2 shows the force coefficients and Strouhal numbers for wavy and circular cylinders at low Reynolds numbers of Re = 60, 100 and 150, respectively. The periodic vortex structures along the spanwise direction of the wavy cylinder are also observed in Fig.9. In laminar flow condition, the values of CD , C Lc and St for wavy cylinder are close
314
to those of a corresponding circular cylinder[15]. No significant force reduction can be obtained in such low Reynolds number regime. In general, the reduction ratios of CD and C Lc increase with the increase of the Reynolds number. Considering the discussion above in turbulent flow condition (refer to Table 1), we conclude that the force reduction character of wavy cylinder has a strong relationship with the Reynolds number.
Fig.9 Instantaneous vorticity iso-surfaces for wavy and circular cylinders in laminar flow condition Table 2 Force coefficients and Strouhal numbers of wavy and circular cylinders in laminar flow condition
CD
Case
Re=60
100
150
CY (Henderson[15])
1.42
1.34
1.35
CY (Present)
1.43
1.32
1.34
7. Conclusions The cross-flow past a wavy cylinder has been simulated using LES at Re = 3000 in the present study. The present LES results show good agreement with the experimental measurements. Due to the wavy flow separation line along the spanwise direction of the wavy cylinder, the wake width expands in the region behind the saddles of cylinders and shrinks behind the nodes. As a result, the near wake vortex structures exhibit a periodic variation along the spanwise direction. Furthermore, the free shear layer is more difficult to roll up and develop to a mature vortex at a further downstream position by a longer vortex formation length. Hence, the effects of pressure and pressure fluctuation are less strongly felt by the cylinder. It significantly modifies and controls the 3-D vortex structures behind the wavy cylinder and weakens the strength of near wake vortex. All of the above results have explained why such wavy cylinder can produces a significant drag reduction and suppression of the cylinder vibration. Moreover, the simulations in laminar flow condition have also been carried out to examine the Reynolds number effect. In general, we conclude that the force reduction character of wavy cylinder has a strong relationship with the Reynolds number. A large value of Reynolds number may give rise to a high force reduction of the wavy cylinder. Acknowledgement This work was supported by the Research program of the Wuhan University of Technology, China (Grant No. 471-38650324). References [1]
C Lc
WY (present)
1.55
CY (Present)
0.087
0.233
0.381
WY (Present)
0.115
0.212
0.183
1.43
1.31 [2] [3] [4] [5]
CY (Present)
0.137
St
0.164
0.185 [6]
0.158 WY (Present)
0.138
0.171 [7]
WILLIAMSON C. H. K. Vortex dynamics in the cylinder wake[J]. Annual Review of Fluid Mechanics, 1996, 28: 477- 539. DAREKAR R. M., SHERWIN S. J. Flow past a bluff body with a wavy stagnation face[J]. Journal of Fluids and Structures, 2001, 15(3-4): 587- 596. DOBRE A., HANGAN H. and VICKERY B. J. Wake control based on spanwise sinusoidal perturbations[J]. AIAA Journal, 2006, 44(3): 485-492. AHMED A., BAYS-MUCHMORE B. Transverse flow over a wavy cylinder[J]. Physics of Fluids A, 1992, 4(9): 1959-1967. AHMED A., KHAN M. J. and BAYS-MUCHMORE B. Experimental investigation of a three-dimensional bluff-body wake[J]. AIAA Journal, 1993, 31(3): 559563. LAM K., WANG F. H. and LI J. Y. et al. Experimental investigation of the mean and fluctuating forces of wavy (varicose) cylinders in a cross-flow[J]. Journal of Fluids and Structures, 2004, 19(3): 321- 334. ZHANG W., Daichin and LEE S. J. PIV measurements
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of the near-wake behind a sinusoidal cylinder[J]. Experiments in Fluids, 2005, 38(6): 824- 832. [8] ZHANG Wei, Daichin. Experimental study of flow structure behind a circular cylinder with wavy surface by PIV[J]. Journal of Hydrodynamics, Ser. B, 2006, 18(1): 97- 103. [9] LEE Sang-joon, NGUYEN Anh-tuan. Experimental investigation on wake behind a wavy cylinder having sinusoidal cross-sectional area variation[J]. Fluid Dynamics Research, 2007, 39(4): 292-304. [10] LAM K., LIN Y. F. Drag force control of flow over wavy cylinders at low Reynolds number[J]. Journal of Mechanical Science and Technology, 2007, 21(9): 1331-1337. [11] XU Chang-yue, CHEN Li-wei and LU Xi-yun. Largeeddy and detached-eddy simulations of the separated flow around a circular cylinder[J]. Journal of
Hydrodynamics, Ser. B, 2007, 19(5): 559-563. ZOU Lin, LIN Yu-feng and LAM Kit. Large-eddy simulation of flow around cylinder arrays at a subcritical Reynolds number[J]. Journal of Hydrodynamics, 2008, 20(4): 403-413. [13] NORBERG C. Fluctuating lift on a circular cylinder: Review and new measurements[J]. Journal of Fluids and Structures, 2003, 17(1): 57-96. [14] LU X., DALTON C. and ZHANG J. Application of large eddy simulation to flow past a circular cylinder[J]. ASME Journal of Offshore Mechanics and Arctic Engineering, 1997, 119(4): 219-225. [15] HENDERSON R. D. Details of the drag curve near the onset of vortex shedding[J]. Physics of Fluids, 1995, 7(14): 2102-2104. [12]
2009,21(3):308-315 DOI: 10.1016/S1001-6058(08)60151-1
FORCE REDUCTION OF FLOW AROUND A SINUSOIDAL WAVY CYLINDER* ZOU Lin School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China, E-mail: [email protected] LIN Yu-feng Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China
(Received May 13, 2008, Revised January 8, 2009)
Abstract: A large eddy simulation of cross-flow around a sinusoidal wavy cylinder at Re = 3000 was performed and the load cell measurement was introduced for the validation test. The mean flow field and the near wake flow structures were presented and compared with those for a circular cylinder at the same Reynolds number. The mean drag coefficient for the wavy cylinder is smaller than that for a corresponding circular cylinder due to the formation of a longer wake vortex generated by the wavy cylinder. The fluctuating lift coefficient of the wavy cylinder is also greatly reduced. This kind of wavy surface leads to the formation of 3-D free shear layers which are more stable than purely 2-D free shear layers. Such free shear layers only roll up into mature vortices at further downstream position and significantly modify the near wake structures and the pressure distributions around the wavy cylinder. Moreover, the simulations in laminar flow condition were also performed to investigate the effect of Reynolds number on force reduction control. Key words: wavy cylinder, force reduction, Large Eddy Simulation (LES), Reynolds number effect
1. Introduction Flow past a circular cylinder has been a subject of interest for a long time. The alternate vortex shedding from the cylinder will produce very complicated flow effects such as the flow-induced body vibration[1]. How to control vortex shedding and hence reduce the flow-induced vibration becomes a challenging problem in the fluid dynamics area[2,3]. By modifying the geometry of the cylindrical structures, a
* Project supported by the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200804971025), the Council of the Hong Kong Special Administrative Region, China (Grant No. PolyU 5311/04E) . Biography: ZOU Lin (1970-), Female, Ph. D., Associate Professor Corresponding author: LIN Yu-feng, E-mail: [email protected]
type of cylinder with sinusoidally varying diameter along its spanwise direction, called a wavy cylinder, is introduced (see Fig.1). It is hoped that this kind of cylinder could lead to better control of the vortex shedding and hence the suppression of flow-induced vibration of a cylindrical structure. Ahmed et al.[4] experimentally investigated the surface-pressure distributions of wavy cylinders with four spanwise wavelengths, and found that the separated flow structures near the geometric nodes are distinctly asymmetric for a large fraction of time and the sectional drag coefficients at the geometric nodes are greater than those at the geometric saddles. Ahmed et al.[5] studied the turbulent wake patterns behind a wavy cylinder. The topology of the boundary layer separation lines and the subsequent 3-D development of the turbulent wake structures were obtained. Lam et al.[6] found that drag reduction of up to 20% could be achieved by changing the geometric wavelength and
309
wave amplitude of the wavy cylinder in the range of subcritical Reynolds numbers from 2×104 to 5×104. It was also found that the root-mean-square fluctuating lift coefficients of wavy cylinders are much lower than those of circular cylinders. On the other hand, Zhang et al.[7,8] and Lee et al.[9] experimentally investigated the 3-D near wake structures behind a wavy cylinder. Along the spanwise direction, the well-organized streamwise vortices with alternating positive and negative vortices were observed. Lam et al.[10] also captured the detailed well-organized 3-D vortex structures for different wavy cylinders at a low Reynolds number of 100 by using the numerical simulation method. The current work is mainly aimed to investigate the relationship between sinusoidal wavy surface and flow characteristics of a wavy cylinder at a subcritical Reynolds number of Re = 3000 . The instantaneous 3-D vortex structures which can not be illustrated clearly using experimental methods will be captured using the Large Eddy Simulation (LES) method[11,12]. The experimental measurements will also be preformed for validating the LES results. Some valuable data, such as drag, lift, pressure, velocity and turbulent kinetic energy, etc., will be discussed compared with a corresponding circular cylinder at the same Reynolds number. The laminar flow condition simulations are also considered to discuss the effect of Reynolds numbers.
amplitude a / Dm = 0.15 are adopted in the present simulations considering the previously investigation results[3-10].
3. Experimental setup and flow measurements The experiments were carried out in a closed-loop water tunnel with the test section of 0.3 m×0.6 m×2.1 m (width, height, length). Forces on the wavy cylinder were measured using a threecomponent quartz piezoelectric load cell characterized by high response, resolution and stiffness. The load cell mounted on the top end of the wavy cylinder was bolted tightly between two machine-polished stainless steel blocks, which could measure instantaneous integral fluid forces acting on the length of the cylinder exposed within the water tunnel. Static calibrations of the load cell in the lift and drag directions were carried out using dead weights. 4. Computational models 4.1 Governing equations By using the 3-D LES turbulence model, the large scale eddies are solved directly by the filtered Navier-Stokes equations and the small eddies are modeled using a Sub-Grid Scale (SGS) model. The governing equations employed for the LES are
wui =0 wxi
Fig.1 Geometry of a wavy cylinder
wW wui wui u j w 2ui 1 wp + = +Q ij U wxi wt wx j wx j wx j wx j (i = 1, 2, 3)
2. Parametric definition of body geometry The geometry of a wavy cylinders is described by the equation, Dz = Dm + 2a cos(2Sz / O ) . As is shown in Fig.1, Dz denotes the local diameter of the wavy cylinder. The mean diameter is defined with Dm = ( Dmin + Dmax ) / 2 . Dmin is the minimum diameter of the wavy cylinder along the spanwise direction, while Dmax is the maximum diameter. The amplitude of the curved surface is denoted by a , and O is the wavelength along the spanwise direction. The axial location with maximum local diameter is referred to as “node”, while the minimum diameter is called “saddle”. The “middle” is also defined at the mid-point between the node and the saddle. All geometrical lengths are scaled with Dm . The dimensionless wavelength O / Dm = 1.5 and wave
(1)
(2)
where ui are the filtered velocity components in the Cartesian coordinates xi , p is the pressure, U is the fluid density and Q is the kinematic viscosity of the fluid. The influence of the small scales on the large (resolved) scales takes place through the subgrid scale stress defined by
W ij = ui u j ui u j
(3)
resulting from the filtering operation, which are unknown and must be modeled with a subgrid model. All computations in the present work were carried out with a Smagorinsky constant of Cs = 0.1 . (The detailed description see Zou et al.[12])
310
4.2 Numerical method To solve the 3-D unsteady incompressible Navier-Stokers equation, the finite volume method applied on unstructured grids is employed. The PISO algorithm is used to deal with the pressure-velocity coupling between the momentum and the continuity equations and a second-order central differencing scheme is used for momentum discretization while a second-order implicit scheme is employed to advance the equations in time.
the circular cylinder ( 3O wavelength along the spanwise direction). Figure 2(b) shows that the grid is nonuniform in the x-y plane but uniform along the z direction. The distance from the cylinder surface to the nearest grid points are fixed at y + = 1 . A uniform velocity profile ( u = 1 , v = w = 1 ) is imposed at the inlet, while the convective boundary condition is used at the outlet boundary. A periodic boundary condition is employed at the boundaries in the spanwise direction and a no-slip boundary condition is prescribed at the surface of the wavy cylinder. The lateral surfaces are treated as slip surfaces using symmetry conditions. 5. Validation test of numerical methods The grid independence tests were carried out for both wavy and circular cylinder cases and the grid number of 1.28×106 was adopted at last with considering the time consumption of the present simulations and the accuracy of the results. A dimensionless time step 't U f Dm = 0.02 was chosen for the simulation. The overall drag coefficient is defined by CD = 2 FD UU f2 Dm Lz , while the
Fig.2 Computational model of a wavy cylinder
4.3 Computational domain and boundary conditions Figure 2(a) shows the schematic diagram of the computational domain (in a fixed Cartesian coordinate system ( x, y , z ) ). The origin of the coordinate system is located at the end of the wavy cylinder. The x -axis is aligned with the inlet flow direction (streamwise direction). The z -axis is parallel to the cylinder axis (spanwise direction) and the y -axis is perpendicular to both the x and z axes (crosswise direction). In the x-y plane, the computational boundaries in the streamwise and crosswise directions are set at 24 Dm and 16 Dm , respectively. The upstream boundary is set at 8Dm away from the centerline of wavy cylinder. The height (spanwise direction) of computational domain is set at 4.5 Dm which will be sufficient for the present simulations (refer to the discussions by Zou et al.[12]). At subcritical Reynolds numbers, the periodic repeated vortex structures were observed to be consistent with the periodic repetition of the wavy cylinder spanwise wavelength O [7-9]. As a result, in the present simulation, the spanwise domain of the wavy cylinder is set equal to the same range as that of
overall lift coefficient is CL = 2 FL UU f2 Dm Lz . The total drag force and total lift force are given by FD and FL , respectively. The present results given by LES simulation and load cell measurement of both the wavy and circular cylinders are shown in Table 1 compared with other experimental and numerical results at Re = DmU f Q = 3000 . The mean drag coefficients ( CD ), fluctuating lift coefficients ( C Lc ) and Strouhal number ( St = f s Dm U f ) in the present simulation for circular cylinder agree with those of others[13,14]. Here, f s is the vortex shedding frequency. Moreover, the present experimental results of CD , C Lc and St for the wavy and circular cylinders are all in consistence with results obtained by the preset LES method. 6. Results and discussion 6.1 Force characteristics and Strouhal number As is summarized in Table 1, the values of CD and C Lc of the wavy cylinder are smaller than those of a corresponding circular cylinder at Re=3000. Compared with a circular cylinder, the drag coefficient reduction up to 15.8% by the experimental measurement and 16.5% by the LES simulation are obtained. The evident reduction of the fluctuating lift coefficient is also obtained. The Strouhal number of
311
the wavy cylinder is slightly lower than that of the circular cylinder at the same Reynolds number. It means that the variation in surface geometry has little or no effect on the frequency of vortex shedding for the wavy cylinder at such subcritical Reynolds number. The previous experimental measurement results on the Strouhal numbers also show that the wave shape has little effect on vortex shedding of wavy cylinders at subcritical Reynolds numbers[6].
Table 1 Experimental and LES results on force coefficients and Strouhal numbers of wavy and circular cylinders at Re = 3000 Case
CD
C Lc
St
CY (Exp.) Norberg[13]
0.98-1.03
í
0.210-0.213
CY (LES) Lu et al.[14]
1.07
0.48
í
CY (Exp.) Present
1.01
0.13
0.208
CY (LES) Present
1.09
0.18
0.210
WY (Exp.) Present
0.85
0.01
0.197
WY (LES) Present
0.91
0.01
0.198
6.2 Pressure and velocity vector distributions The mean velocity vectors distributions for wavy and circular cylinders are shown in Fig.3. The near wake width in the nodal plane is different from that in the saddle plane. The vortex shedding phenomenon is associated with flow separation from the boundary layer of bluff bodies. The stability of the free shear layer and the resulting vortex structure are highly dependent on the flow separation point (the points in the surface of bluff bodies where the shear stress vanishes) of the wavy cylinder. In the nodal plane of the wavy cylinder, the flow separation angle (104o) is much larger than that of a circular cylinder (88o), while it is smaller than that of a circular cylinder in the saddle plane (75o). As a result, the near wake velocity vectors expand along both the streamwise direction and crosswise direction and the wake width in the saddle plane is increased giving rise to a wide
wake at further downstream. However, a large value of separation angle in the nodal plane suppresses the shear layer development. The near wake velocity vectors in the nodal plane seem to be extended only in the streamwise direction and noticeably suppressed in the crosswise direction. As a result, it produces a narrower wake downstream. The significant spanwise flow moves from the saddle plane toward the nodal plane and depicts surface streamlines near the separation line similar to that shown in Fig.4. It means that this kind of wavy surface can significantly modify the near wake structures. Thus the near wake 3-D flow structures are formed with periodic variation along the spanwise direction of the wavy cylinder. In general, the 3-D free shear layers developed from the wavy cylinder suppress the vortex roll-up process and give rise to complex vortex formation. It explains why the value of fluctuating lift coefficient is very small.
Fig.3 Mean pressure coefficients and mean velocity vector distributions for wavy and circular cylinders in the x-y plane at Re = 3000
The mean pressure coefficient distributions in different planes of the wavy cylinder are also captured (see Fig.3). The maximum negative mean pressure coefficient near the wake of the wavy cylinder is smaller than that of the circular cylinder and its center is far away behind the wavy cylinder. In general, the
312
distributions of the mean pressure coefficient for the wavy cylinder have the same trend as the mean velocity vector distributions in both nodal and saddle planes. Considering the discussed above, we conclude that a wavy cylinder with smaller pressure difference can lead to the reduction of drag.
smaller than that of the circular cylinder.
Fig.4 Surface streamlines and the topology of the 3-D eparation lines for wavy and circular cylinders at Re = 3000
6.3 Turbulent kinetic energy Figure 5 shows the contour plots of the normalized Turbulent Kinetic Energy (TKE) in the x-y planes ( TKE = (u c2 + vc2 + wc2 ) / 2U f2 ) of the wake behind the wavy and circular cylinders at Re = 3000 . The region of negligible kinetic energy is noticeably larger behind the wavy cylinder than that of a circular cylinder. The maximum TKE and the high-value TKE regime in the nodal plane are larger than those in the saddle plane. However, all the TKE values in different cross-section planes of the wavy cylinder are smaller than those of the circular cylinder. The position of maximum TKE is farther away from the back of the wavy cylinder than that of the circular cylinder. Therefore, the overall TKE of the flow behind the wavy cylinder is lower than that of the circular cylinder at such subcritical Reynolds number. Figure 6 shows that in the x-z plane with y/Dm = 0 , the maximum TKE regime behind the circular cylinder is around the position x/Dm = 2 and the values of TKE are uniform along the spanwise direction. For the wavy cylinder at the downstream location of approximately x/Dm = 3.5 , however, the TKE exhibits local maxima behind the nodes (see Fig.6(a)). This is possibly due to the converging flow of the free shear layer from the saddle to the node (the significant spanwise flow moves from the saddle plane) and additional transport of kinetic energy near the node by the streamwise vortices. In the regimes behind the saddles, the values of TKE are smaller than those behind the nodes. In general, the spanwise variation of TKE shows periodic repetition for the wavy cylinder as contrast to the uniform distribution of the circular cylinder. Furthermore, the maximum of TKE along the spanwise direction for wavy cylinder is
Fig.5 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the x-y plane at
Re = 3000
Figure 7 shows the contours of the normalized TKE in the y-z plane at the downstream location x/Dm = 2 for the circular and wavy cylinders. For a circular cylinder, the TKE contours are nearly parallel to the cylinder axis. The significantly large value of TKE is in the wake center regime due to active momentum exchange, while it decreases along the crosswise direction. The TKE distribution of the flow behind the wavy cylinder shows periodic variation along the spanwise direction. Compared with the circular cylinder, the values of TKE in the near wake behind the wavy cylinder are smaller, especially at the saddle positions. It indicates that for the wavy cylinder, the strength of TKE decreases substantially in the near wake region which is the physical reason for the reduction of mean drag and fluctuating lift acting on the cylinder. In general, the flow around a wavy cylinder shows an evidently periodic characteristic in
313
the flow patterns along the spanwise direction. This kind of wavy surface leads to a significant reduction of TKE and in turn, the reduction of the mean drag force and the suppression of the fluctuating lift.
drag force reduction and the weakening of the lift force vibration.
Fig.7 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the y-z plane at x/Dm = 2 Fig.6 Normalized turbulent kinetic energy distributions for wavy and circular cylinders in the x-z plane at y/Dm = 0
and Re = 3000
and Re = 3000
6.4 3-D vortex structures Figure 8 shows the iso-surfaces of streamwise vorticity ( Z x ) of the wavy cylinder. It exhibits the periodic patterns along the spanwise direction with alternating negative and positive vortices and symmetrically distributed with respect to the central axis of the cylinders. The iso-surface of streamwise vorticity behind the wavy cylinder shows distinctively the regimes of zero vorticity (the recovery region) centered adjacent to the saddle planes. While no clear zero vorticity patterns can be found in other ranges of wavelength and no zero vorticity regimes can be observed for the circular cylinder. Zhang et al.[7] investigated a wavy cylinder with O /Dm = 2 at the same Reynolds number using the PIV method also observed this phenomenon. The periodic patterns of near-wake streamwise vorticity appear to be well organized and coherent. It means that the 3-D free shear layer generated from the wavy surface is more difficult to roll up to form vortex structures at the near wake positions. Finally, it rolls up to mature vortex at a further downstream position in the cylinder wake. It also means that a large value of vortex formation length is generated which leads to the significantly
Fig.8 Instantaneous streamwise vorticity iso-surfaces ( Z x = r1 ) for wavy and circular cylinders at Re = 3000
6.5 Reynolds number effect Table 2 shows the force coefficients and Strouhal numbers for wavy and circular cylinders at low Reynolds numbers of Re = 60, 100 and 150, respectively. The periodic vortex structures along the spanwise direction of the wavy cylinder are also observed in Fig.9. In laminar flow condition, the values of CD , C Lc and St for wavy cylinder are close
314
to those of a corresponding circular cylinder[15]. No significant force reduction can be obtained in such low Reynolds number regime. In general, the reduction ratios of CD and C Lc increase with the increase of the Reynolds number. Considering the discussion above in turbulent flow condition (refer to Table 1), we conclude that the force reduction character of wavy cylinder has a strong relationship with the Reynolds number.
Fig.9 Instantaneous vorticity iso-surfaces for wavy and circular cylinders in laminar flow condition Table 2 Force coefficients and Strouhal numbers of wavy and circular cylinders in laminar flow condition
CD
Case
Re=60
100
150
CY (Henderson[15])
1.42
1.34
1.35
CY (Present)
1.43
1.32
1.34
7. Conclusions The cross-flow past a wavy cylinder has been simulated using LES at Re = 3000 in the present study. The present LES results show good agreement with the experimental measurements. Due to the wavy flow separation line along the spanwise direction of the wavy cylinder, the wake width expands in the region behind the saddles of cylinders and shrinks behind the nodes. As a result, the near wake vortex structures exhibit a periodic variation along the spanwise direction. Furthermore, the free shear layer is more difficult to roll up and develop to a mature vortex at a further downstream position by a longer vortex formation length. Hence, the effects of pressure and pressure fluctuation are less strongly felt by the cylinder. It significantly modifies and controls the 3-D vortex structures behind the wavy cylinder and weakens the strength of near wake vortex. All of the above results have explained why such wavy cylinder can produces a significant drag reduction and suppression of the cylinder vibration. Moreover, the simulations in laminar flow condition have also been carried out to examine the Reynolds number effect. In general, we conclude that the force reduction character of wavy cylinder has a strong relationship with the Reynolds number. A large value of Reynolds number may give rise to a high force reduction of the wavy cylinder. Acknowledgement This work was supported by the Research program of the Wuhan University of Technology, China (Grant No. 471-38650324). References [1]
C Lc
WY (present)
1.55
CY (Present)
0.087
0.233
0.381
WY (Present)
0.115
0.212
0.183
1.43
1.31 [2] [3] [4] [5]
CY (Present)
0.137
St
0.164
0.185 [6]
0.158 WY (Present)
0.138
0.171 [7]
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