Effects of wavelength and amplitude of a wavy cylinder in cross-flow at low Reynolds numbers

Transcription

1 J. Fluid Mech. (2009), vol. 620, pp c 2009 Cambridge Universit Press doi: /s Printed in the United Kingdom 195 Effects of wavelength and amplitude of a wav clinder in cross-flow at low Renolds numbers K. L A M AND Y. F. L I N Department of Mechanical Engineering, The Hong Kong Poltechnic Universit, Hung Hom, Kowloon, Hong Kong (Received 2 Januar 2008 and in revised form 11 September 2008) Three-dimensional numerical simulations of laminar flow around a circular clinder with sinusoidal variation of cross-section along the spanwise direction, named wav clinder, are performed. A series of wav clinders with different combinations of dimensionless wavelength (λ/d m ) and wave amplitude (a/d m ) are studied in detail at a Renolds number of Re = U D m /ν = 100, where U is the free-stream velocit and D m is the mean diameter of a wav clinder. The results of variation of mean drag coefficient and root mean square (r.m.s.) lift coefficient with dimensionless wavelength show that significant reduction of mean and fluctuating force coefficients occurs at optimal dimensionless wavelengths λ/d m of around 2.5 and 6 respectivel for the different amplitudes studied. Based on the variation of flow structures and force characteristics, the dimensionless wavelength from λ/d m =1toλ/D m = 10 is classified into three wavelength regimes corresponding to three tpes of wake structures. The wake structures at the near wake of different wav clinders are captured. For all wav clinders, the flow separation line varies along the spanwise direction. This leads to the development of a three-dimensional free shear laer with periodic repetition along the spanwise direction. The three-dimensional free shear laer of the wav clinder is larger and more stable than that of the circular clinder, and in some cases the free shear laer even does not roll up into a mature vorte street behind the clinder. As a result, the mean drag coefficients of some of the tpical wav clinders are less than that of a corresponding circular clinder with a maimum drag coefficient reduction up to 18 %. The r.m.s. lift coefficients are greatl reduced to practicall ero at optimal wavelengths. In the laminar flow regime (60 Re 150), the values of optimal wavelength are Renolds number dependent. 1. Introduction Flows past bluff bodies are important in man engineering applications, for eample in the designs for heat echangers, offshore structures, high-rise buildings, chimnes, and cables. The periodic vorte shedding and fluctuating velocit fields behind the bluff bodies can cause structural damage as a result of periodic surface loading which increases the vibration of the bodies and shortens the life of the structures. In fundamental research, flow past clindrical bodies has been etensivel investigated. It is a challenge to control the vorte-shedding phenomenon and hence to reduce the potential for flow-induced vibration (FIV). Over the past ears, man eperimental and numerical investigations had been carried out on the control of FIV of the address for correspondence: mmklam@polu.edu.hk

2 196 K. Lam and Y. F. Lin PC-A1 U U PC-A2 U PC-B2 PC-B1 U PC-B3 U U Figure 1. Simple model tpes for different wav clinders. PC-C U U U U AC-A1 AC-A2 AC-B1 AC-B2 Figure 2. Simple model tpes for active control of the flow induced vibrations. clindrical structures. It is well known that the free shear laers generated b bluff bodies have a profound influence on the wake structure and hence on the force characteristic of the bluff bod. In the recent few ears, several tpes of clinders with surface profile varing sinusoidall along their spanwise direction, named wav clinders, were introduced (see figure 1). Ahmed & Bas-Muchmore (1992), Ahmed, Khan & Bas-Muchmore (1993), Tombais & Bearman (1997), Bearman & Owen (1998), Owen, Sewck & Bearman (2000), Darekar & Sherwin (2001a, b), Keser et al. (2001), Lam et al. (2004), Lam, Wang & So (2004), Zhang, Dai & Lee (2005), Dobre, Hangan & Vicker (2006), Lee & Nguen (2007) and Lam & Lin (2007, 2008) have investigated various tpes of wav objects eperimentall and numericall under different flow conditions. Their studies are summaried in table 1. Tombais & Bearman (1997) introduced a spanwise wav surface to the trailing face of the clindrical bodies (figure 1; PC-A1), while Bearman & Owen (1998) and Dobre et al. (2006) investigated the effects of a spanwise wav surface at the leading face (figure 1; PC-A2) of the clindrical bodies. At Renolds number Re = , the maimum drag reduction of about 30 % was obtained b Bearman & Owen (1998). Owen et al. (2000) studied flow past a sinuous bluff bod (figure 1; PC-B1), using flow visualiation method and found that the Kármán vorte shedding was effectivel suppressed, and a periodic variation in the wake width across the spanwise direction was observed. Darekar & Sherwin (2001a, b) numericall investigated the flow past a square clinder with a wav stagnation face at low Renolds numbers (figure 1; PC-B2, PC-B3). The showed that the unstead and staggered Kármán vorte wake could be suppressed to a stead and smmetric wake structure due to the waviness of the square clinder. The maimum drag reduction of about 16 % was obtained at a Renolds number of 100 compared with the straight, non-wav square clinder. Moreover, at higher Renolds numbers, the drag reduction increases substantiall. Furthermore, active flow control method based on the idea of wav surface effect is also pursued (see figure 2 and table 1). Kim et al. (2004) and Kim & Choi (2005) used

3 Authors Tpes Cross-section Ke features Re Methods AC Kim et al. (2004) AC-A1, A2 Half-ellipse shape Three-dimensional vorte structures; Drag 4200, Num., Ep. reduction Kim & Choi (2005) AC-B1, B2 Circular shape Optimal regime of wave parameters , 3900 Num. PC Tombais & Bearman (1997) PC-A1 Half-ellipse shape Shedding frequenc, wake patterns 2500, Ep. Bearman & Owen (1998) PC-A2 Square shape Drag reduction; wake stabilit 100, Ep. Dobre et al. (2006) PC-A2 Square shape Drag reduction Ep. Owen et al. (2000) PC-B1 Circular shape Vorte shedding suppression 100 Ep. Keser et al. (2001) PC-B1, C Circular shape Three-dimensional vorte structures NA Num. Darekar & Sherwin (2001a, b) PC-B2, B3 Square shape Optimal regime of wave parameters , 500 Num. Ahmed & Ba-Muchmore (1992) PC-C Circular shape Three-dimensional separation line, wake Ep. Ahmed et al. (1993) PC-C Circular shape topolog Wake patterns Ep. Lam et al. (2004a, b) PC-C Circular shape Drag reduction; wake patterns Ep. Zhang et al. (2005) PC-C Circular shape Periodic spanwise wake patterns 3000 Ep. Lee & Nguen (2007) PC-C Circular shape Drag reduction Ep. Lam & Lin (2007,2008) PC-C Circular shape Drag reduction; three-dimensional vorte 100, 3000 Num. structures Present stud PC-C Circular shape Optimal regime of wave parameters Num. Table 1. Summar of tpes of the wav clinder for the control of FIV: AC = active control; PC = passive control; Ep. = eperimental measurement; Num. = numerical simulation. Effects of wavelength and amplitude of a wav clinder in cross-flow 197

4 198 K. Lam and Y. F. Lin an active open-loop control method (called distributed forcing) for the reduction of drag of a half-ellipse shaped bod (figure 2; AC-A1, AC-A2) and a circular clinder (figure 2; AC-B1, AC-B2), respectivel. The forcing profile was achieved b sinusoidal variation of blowing and suction from slots on the upper and lower surfaces of the object in the spanwise direction. The Kármán vorte shedding was attenuated for both laminar flow condition and turbulent flow condition, and the drag was also reduced substantiall. Moreover, the vorte shedding was significantl suppressed for some tpical cases at Re = 100 (Kim & Choi 2005). Alternativel, a clinder whose diameter varied sinusoidall along its spanwise direction was studied (figure 1; PC-C). Such clinder is omnidirectional to the free oncoming flow. In fundamental research, eperimental measurements on the surface pressure distributions of such wav clinders with different spanwise wavelengths were carried out b Ahmed & Bas-Muchmore (1992). The found that the separated flow structures near the geometric nodes are distinctl asmmetric for a large fraction of time, and the sectional drag coefficients at the geometric nodes are greater than that at the geometric saddle. Ahmed et al. (1993) further investigated eperimentall the turbulent wake behind a wav clinder. The described the topolog of the boundar laer separation lines and the subsequent three-dimensional development of turbulent structure of the wake. The formation of trailing streamwise vortices behind the nodal points of separation gives rise to a locall narrower wake, a rapid wake velocit recover and a suppression of the turbulence development within the separated boundar laer. However, the phenomena of the drag reduction and suppression of lift fluctuating were not discussed. Keser et al. (2001) used a three-dimensional discrete vorte method to simulate the separated flow around wav clinders. Lam et al. (2004a) started to focus attention on the effects of different wav clinders on drag and lift reduction and the related effects on FIV. It was anticipated that such tpe of clinder could lead to a better control of the vorte shedding and hence a better suppression of FIV b changing the geometric wavelength and wave amplitude of the clinder. For constant values of wave amplitude, the found eperimentall that with the value of wavelength ratio from 1.45 to 2.27, a large value of drag reduction up to 20 % can be obtained in subcritical Renolds numbers range of It was also found that the root mean square (r.m.s.) lift coefficients of the wav clinders are much lower than those of the circular clinder. Furthermore, Lam et al. (2004b) measured the near wake velocit distributions of a wav clinder. The found that the wav geometr plaed an important role in vorte formation length, drag reduction and vorte shedding suppression. The vorte formation length of the wav clinder is longer than that of the circular clinder. Flow visualiation studies were also performed, and in the near wake of the wav clinder, the rib structures were detected near the saddle planes at the Renolds number of 600. At the nodal and saddle planes, the streamwise velocit distributions are ver different compared with those of a circular clinder. Recentl, Zhang et al. (2005) investigated the three-dimensional near wake structures behind a wav clinder b using the particle image velocimetr (PIV) technique at the Renolds number of Along the spanwise direction of the wav clinder, well-organied streamwise vortices with alternating positive and negative vortices were observed. The suppress the formation of the large-scale spanwise vortices and decrease the overall turbulent kinetic energ in the near wake of the wav clinder. Eperiments b Lee & Nguen (2007) and others, using hot-wire anemometer and flow visualiation method, over a wide range of Renolds number from 5300 to , showed that when the values of wavelength ratio equal 2, wav clinders can give rise to larger drag reductions

5 Effects of wavelength and amplitude of a wav clinder in cross-flow 199 than when those ratios equal 1. The maimum drag reduction of up to 22 % is obtained at wavelength ratio of 2 and Re = The also showed that the longer vorte formation region of the wav clinder seems to be related to the effect of drag reduction. At similar Renolds numbers, the vorte formation length is longer and the turbulent intensit smaller than those of the circular clinder. From these results, it can be seen that an optimal value of spanwise wavelength ratio eists in the lower wavelength ratio range of for drag reduction at higher Renolds numbers. Recentl, Lam & Lin (2007, 2008) carried out computational studies and captured the detailed three-dimensional vorte structures of wav clinders for laminar flow at low Renolds number Re = 100 and in turbulent flow at Re = 3000; the optimal values of spanwise wavelength for drag reduction were obtained over a small range of wavelength variation. For turbulent flow, the confirmed the findings above b using the large edd simulation for wav clinders with wavelength ratio of at Re = The optimal value of wavelength ratio for turbulent flow was around λ/d m =1.9. The relationships between the vorte formation length and drag were also given. All investigations above with respect to the wav clinder of model tpe PC-C (figure 1) have onl been performed over a small region of the wavelength ratio λ/d m along the spanwise direction. The optimal value of wavelength which can well control the vorte formation is still not full revealed over a wider range of wavelength ratio. The amplitude ratio a/d m of the wav clinder should also pla an important role in the control of the vorte structure from the wav clinders. The aim of the present work is to appl a well-established three-dimensional numerical simulation method to carr out etensive investigations on laminar flows past wav clinders. It is anticipated that numerical simulations can well capture the instantaneous threedimensional vorte structures and other valuable data, such as drag, lift, pressure and vorte shedding frequenc and thus provide a fuller picture for the understanding of the comple flow phenomenon. The relationship between wavelength, wave amplitude and force reduction can be full investigated, and the effect of Renolds number can also be eamined. 2. Parametric definition of the bod geometr As shown in figure 3, the diameter of the wav clinder varies sinusoidall along its spanwise direction. The geometr of the wav clinders is described b the following equation: D = D m +2acos(2π/λ), (2.1) where D denotes the local diameter of the wav clinder and varies in the spanwise direction. The mean diameter D m is defined b D m =(D min + D ma )/2. (2.2) The amplitude of the wav surface a is equal to half peak-to-peak distance. The spanwise wavelength λ is also indicated in figure 3. The aial location with maimum local diameter D ma is called node, while the aial location of the minimum diameter D min is called saddle. The middle is also defined at the midpoint position between nodal and saddle planes. The diameter of the middle cross-section is equal to the mean diameter D m. The nodal position of the wav clinder along the spanwise direction is denoted b /λ = 0, while /λ = 0.5 denotes the saddle position. Furthermore, a

6 200 K. Lam and Y. F. Lin 2a D min Saddle U a λ D m Middle Saddle θ D ma Node Middle Node Figure 3. Schematic diagram of a wav clinder. circular clinder with diameter D m is used for comparison stud. In the present stud, all geometrical lengths are normalied with the mean diameter D m. 3. Numerical method In the present stud, unstead three-dimensional laminar flow of a viscous incompressible fluid is considered. The finite volume method (FVM) with an unstructured heahedral grid is emploed to solve the unstead three-dimensional incompressible Navier Stokes equations Governing equations The three-dimensional dimensionless Navier Stokes equations governing the flow of a Newtonian fluid can be written in vector form as u t 1 + u u = p + Re 2 u, (3.1) u =0, (3.2) where u is the non-dimensional flow velocit vector in the Cartesian coordinate sstem (,, ) with its three velocit components u, v and w. The non-dimensional static pressure is denoted b p. Re = U D m /ν is the Renolds number, with U being the free-stream velocit and ν being the fluid kinematic viscosit. In solving the governing equations, the different phsical quantities are changed to non-dimensional form; i.e. all the variables are normalied. The flow time t is normalied b the free-stream velocit and the clinder mean diameter, such that t = tu /D m. In the above equations the velocities are non-dimensionalied b the free-stream velocit U. The present FVM code calculates the solution of the incompressible Navier Stokes equations. The spatial discretiation of the equations is based on an unstructured heahedral mesh. The primitive variables are all discretied in a cell-centred fashion, with one value for each variable in each element. The semi-implicit pressure linked equations (SIMPLE) method is used to deal with the pressure velocit coupling between the momentum and the continuit equations. The second-order upwind differencing scheme is used for convective terms. Second-order central differencing scheme is adopted for diffusion terms, while the second-order implicit scheme is emploed to advance the equations with time.

7 Effects of wavelength and amplitude of a wav clinder in cross-flow Computational domain and boundar conditions In the present stud, the computational domain used for the simulations is set at 24D m 36D m λ in,, directions of a fied Cartesian coordinate sstem (,, ), respectivel as shown in figure 15 in the appendi. The domain parameters are similar to that chosen b Darekar & Sherwin (2001a, b). A wav clinder model is placed inside the flow domain. The -ais is chosen to be along the inlet main flow direction (streamwise direction); the -ais is parallel to the clinder ais (spanwise direction); and the -ais is perpendicular to both -ais and -ais (crosswise direction). The origin (0, 0, 0) of the coordinate is set at the centre of the nodal cross-section of the wav clinder. The velocit at upstream boundar of the computational domain is set at a distance of 12D m from the centreline of the clinder, while the downstream boundar is 24D m awa from the wav clinder. Each lateral surface is 12D m awa from the ais of the clinder (blockage 4.2 %). Furthermore, for the computational domain along the spanwise direction, onl one wavelength λ is adopted due to periodic boundar condition being used in the simulation. The boundar conditions are summaried in the following manner: At the inlet boundar, a uniform velocit profile (u = 1,v = w = 0) is imposed. A convective boundar condition ( u/ t + U c ( u/ ) = 0) is used at the outlet boundar, where U c is the characteristic convective velocit (surface-averaged streamwise velocit) at the plane of the outlet. A periodic boundar condition is emploed at the boundaries in the spanwise direction, and no-slip boundar condition (u = v = w =0) is prescribed at the surface of the wav clinders. The lateral surfaces are treated as slip surfaces, using smmetr boundar conditions ( u/ = w/ = v = 0). The accurac of computational results is highl dependent on the grid sie and number of cells. Details of the grid independence tests and the validation of the numerical models are provided in the Appendi Definition of dimensionless phsical quantities The important dimensionless phsical quantities are summaried below. In the present calculations, the drag coefficient C D and lift coefficient C L are defined b C D = 2F D ρu D 2 m λ, (3.3) C L = 2F L ρu D 2 m λ, (3.4) where ρ is the fluid densit; F D and F L are the total drag force and total lift force, respectivel; C D is the mean drag coefficient; and C L is the r.m.s. lift coefficient. The Strouhal number is the non-dimensional vorte shedding frequenc f s and is epressed as St = f sd m. (3.5) U The frequenc of vorte shedding f s is obtained b fast Fourier transform (FFT) of the lift fluctuating-time histor. The pressure coefficient C P on the clinder surface is defined as C p = 2(P P ), (3.6) ρu 2 where P is the oncoming flow static pressure, and P is the static pressure on the clinder surface. The mean base pressure coefficient C pb is the value of negative

8 202 K. Lam and Y. F. Lin non-dimensional mean pressure at the rear stagnation point (180 from the frontal point) of the clinder. The definition of the velocit recover length (vorte formation length L f )blam et al. (2004b) is adopted in the present stud. The location of the closure point (U/U = 0) of the time-averaged streamwise velocit U on the wake centerline ( =0 plane) has been used to define the vorte formation length L f. The flow separation point in the boundar laer is determined at the position on the clinder wall at which the shear stress is ero. The separation angle is the angle measured from the frontal stagnation point to the separation point. 4. Results and discussions The present stud focuses attention on investigating the characteristics of flow past wav clinders at low Renolds numbers. A series of computational investigations have been carried out mainl at Renolds number equal to 100 for wav clinders with wavelength ratio λ/d m = 1 to 10 and wave amplitude ratio a/d m =0.02 to Comparisons are made with a straight circular clinder (λ/d m =, a/d m =0). The three-dimensional vorte structures are captured and classified. Other phsical parameters such as velocit, pressure, drag, lift and vorte formation length are obtained to provide better understanding of the phenomena Force distributions (1 λ/d m 10, 0.05 a/d m 0.25,Re= 100) Figure 4 summaries the variations of the mean drag coefficient C D and the r.m.s. lift coefficient C L with different wave amplitude ratios and wavelength ratios of the wav clinders. The values of C D and C L of a corresponding straight circular clinder with diameter equal to the mean diameter (D m ) are also incorporated for comparison. The value of C D and C L of a straight circular clinder at Re = 100 for the present simulation are 1.34 and 0.234, respectivel. In figure 4, two troughs are found for both the mean drag coefficient curve and the r.m.s. lift coefficient curve at positions around λ/d m =2.5 andλ/d m = 6 for wave amplitude a/d m > 0.1. The drop at the second trough at λ/d m = 6 is severer than that at the first trough with a maimum drag reduction of about 18 % and the r.m.s. lift approaching ero. For a constant a/d m and an increased λ/d m, the values of C D and C L reduce quickl when λ/d m 2 but increase to a maimum again at λ/d m = 3. Compared with a circular clinder, the mean drag coefficient and the r.m.s. lift coefficient start to drop below the values of a circular clinder at the positions λ/d m =1.75 and 1.25, respectivel. At this lower range of wavelength ratio (1 λ/d m 3), although the maimum drag coefficient reduction is onl up to 9 %, the maimum r.m.s. lift coefficient is reduced to which is much smaller than the reduction of a circular clinder. The values of C D and C L begin to rise at λ/d m =2.5 to until the reach the maimum at λ/d m = 3. It can be seen that around λ/d m = 3, no drag reduction is obtained for the wav clinder, while there are still slight reduction in r.m.s. lift coefficient for large amplitude ratio a/d m. Further increasing the wavelength, the values of C D and C L drop again to ver low values at λ/d m = 5. The minimum values of C D and C L are found at λ/d m = 6 (the value of the second trough point). After that, C D and C L start to increase again moderatel with the wavelength for λ/d m 6. The maimum drag coefficient reduction of up to 18 % is found for the wav clinder at the values λ/d m =6 and a/d m =0.25 (see figure 4a), while the r.m.s. lift coefficient is equal to ero at λ/d m =6 and a/d m =0.15. The ero r.m.s. lift implies that there is no vorte shedding at the near wake of such wav clinders. Considering the previous eperimental and numerical results b Lam et al. (2004a), Lee & Nguen (2007) and

9 Effects of wavelength and amplitude of a wav clinder in cross-flow 203 (a) C D 1.4 a/d m = a/d m = a/d m = a/d m = a/d m = Circular (b) C L λ/d m Figure 4. Mean drag coefficients C D and r.m.s. lift coefficients C L forwavandcircular clinders at Re = 100 with different wavelengths λ/d m and wave amplitudes a/d m. Lam & Lin (2008), the optimal wavelength λ/d m of the present results in the lower range of wavelength ratio (1 λ/d m 3, Re = 100) appears to be in the range 2 2.5, which is slightl higher than that for turbulent flow at high Renolds number (around 1.9). In previous investigations for both laminar and turbulent flows, all the curves of C D and C L for different λ/d m with constant a/d m show a trough characteristic, since the results have onl been obtained over a small range of wavelength around the first trough. It can be seen from the present simulations in a broader range of wavelength ratio (1 λ/d m 10) that there is a first trough followed b a bigger second trough. At least for Re = 100, the true optimal wavelength appears to be close to λ/d m = 6. This result is similar to that found b Darekar & Sherwin (2001b) for a square circular with wav stage face. The optimal wavelength found b them is 5.7 at Re = 100 with a drag reduction up to 16 %. It is also close to that of the circular clinder, found using active control method (Kim & Choi 2005). The mean drag coefficient was reduced about 20 % with the optimal wavelength between 4 and 5atRe = 100. Furthermore, a larger value of a/d m will give a more significant reduction of drag and suppression of r.m.s. lift. With a large value of wave amplitude a, the mean drag coefficients of wav clinders are significantl reduced for the same λ/d m when λ/d m is larger than 1.75, while the r.m.s. lift coefficients are also drasticall reduced when

10 204 K. Lam and Y. F. Lin I (C) I (B) II (C) 0.20 II (B) III (B) a D m I (A) II (A) III (A) λ/d m Figure 5. Flow pattern regimes of wav clinders with different spanwise wavelengths λ/d m and wave amplitudes a/d m at Re = 100. The smbol represents each wav clinder case in the present simulations. λ/d m is larger than However, with the values λ/d m 1.75 for drag coefficient and λ/d m 1.25 for r.m.s. lift coefficient, a large value of a/d m, on the contrar, leads to an increase of C D and C L in such Renolds number range Flow pattern regimes (1 λ/d m 10, 0.05 a/d m 0.25,Re= 100) To understand the variations in drag and r.m.s. lift and the occurrence of the first trough and the second trough, the flow patterns for different wav clinders are captured. The wav clinders over the whole range of wavelength and wave amplitude investigated and classified into three wavelength regimes (I, II and III) with three different flow patterns (A), (B) and (C) as shown in figure 5). The spanwise vorticit ω = v/ u/ at different spanwise sections was used as a flow indicator. The three-dimensional vorte structures corresponding to different flow patterns are shown in figure 6. The time histories of some tpical wav clinders are also incorporated for comparison of flow patterns and force characteristics. Flow pattern (A) is similar to the wake structure behind a straight circular clinder ecept with some distortions in the spanwise direction. Flow pattern (B) is characteried b an increase of threedimensional wake vorte distortion and increase of vorte formation length with a progressive weakening of vorte shedding, while for flow pattern (C), the free shear laer does not roll up into a vorte, and hence no vorte shedding occurs. The flow pattern regimes I, II and III correspond to different wavelength regimes in the range of 1 λ/d m 2.5, 2.5 λ/d m 6and6 λ/d m 10, respectivel. It is to be noted that the division lines are drawn at the positions of maimum reduction in drag and r.m.s. lift at λ/d m =2.5 and 6, so that the change of flow patterns around these positions can be clearl illustrated. It can be seen from figure 6 that the reduction of drag and r.m.s. lift in the three regimes is due to the transformation of flow pattern from pattern (A) to pattern (B) and then to pattern (C) in the three regimes. The prominent features for the wav clinders in different flow pattern regimes are summaried in table 2. The flow structures in the nodal plane are not identical with those in the saddle plane due to the three-dimensional effect. In regime I(1 λ/d m < 2.5) and regime II (2.5 λ/d m < 4), the vortices epand along both the

11 (a) (b) (c) (d) D m (e) (f) (g) (h) Nodal plane /D m Saddle plane (2, 0.1) (2, 0.25) (2, 0.3) (3, 0.075) (4, 0.25) (6, 0.2) (9, 0.1) (9, 0.25) I (A) I (B) I (C) II (A) II (B) II (C) III (A) III (B) Three-diamentional vorte structure Regime I Regime II Regime III 1.3 C D C D C D 1.2 (2, 0.25) (2, 0.1) (λ/dm, a/dm)(2, 0.3) (4, 0.25) (9, 0.1) (6,0.2) (3, 0.075) (9, 0.25) t ~ t ~ + 25 t ~ + 50 Time histor of drag and lift coefficients (t ~ = tu /D m ) Figure 6. Flow patterns for different wav clinders at Re = 100: λ/d m and a/d m are respectivel denoted b the values in parentheses. C L C L C L Effects of wavelength and amplitude of a wav clinder in cross-flow 205

12 Ke features Flow mode Near wake flow structures Force reduction Regime I I (A) Quasi-two-dimensional wake, similar to that of a straight No drag reduction, r.m.s. lift is evident; 1 λ/d m 2.5, circular clinder; slight spanwise distortions. St (WY) St (CY) 0.1 a/d m 0.3 I (B) Noticeable free shear laer elongation towards the Drag reduction with the weakening of r.m.s. lift; downstream direction. St (WY) St (CY) I (C) Stead and smmetrical wake; first disappearance of Significant drag reduction and full suppression in periodic vorte shedding phenomenon. r.m.s. lift; St (WY) = 0; first optimal wavelength ratio. Regime I II (small optimal wavelength λ/d m =2.5) Regime II II (A) Similar to that of in regime I (A); more spanwise Drag and lift are close to or slightl smaller than 2.5 λ/d m 6, distortions. those of circular clinders; St (WY) St (CY) 0.02 a/d m 0.3 II (B) Full three-dimensional wake; evident spanwise distortions; Evident drag reduction and suppression in r.m.s. the strength of vortices is weakening. lift; St (WY) < St (CY) II (C) Well controlled, stead and smmetrical wake patterns; no Maimum drag reduction and no r.m.s. lift; vorte shedding occurs. St (WY) = 0; Real optimal wavelength ratio. Regime II III (large optimal wavelength λ/d m =6) Regime III III (A) Prominent complicated three-dimensional wake; vorte Drag reduction and the weakening of r.m.s. lift; 6 λ/d m 10, dislocation appears with the weakening of vortices strength. St (WY) St (CY) 0.05 a/d m 0.25 III (B) Full complicated three-dimensional wake; spanwise Evident drag reduction and suppression in r.m.s. direction epansion of the regime II (B) wake patterns. lift; St (WY) < St (CY) Table 2. Summar of flow patterns of different wav clinders at Re = 100: WY = wav clinder; CY = circular clinder. 206 K. Lam and Y. F. Lin

13 Effects of wavelength and amplitude of a wav clinder in cross-flow 207 streamwise direction and the crosswise direction. The wake width in the saddle plane is increased, giving rise to a wide wake further downstream (figure 6a b, d e). On the contrar, the vortices in the nodal plane appear to be elongated in the streamwise direction and noticeabl compressed in the crosswise direction. The flow separation occurs later at the nodal position than that at the saddle position. As a result, it produces a narrower wake downstream in the nodal plane. In such smaller wavelength regime, similar characteristic of flow pattern (B) was found b Ahmed et al. (1993) for a wav clinder with λ/d m =1.2 anda/d m =0.1 atre from 5000 to Lam et al. (2004b) showed similar flow characteristics of pattern (A) at Re 400 and pattern (B) at Re 600 for a wav clinder with λ/d m =2.27 and a/d m =0.09. Lam & Lin (2008) also found the flow pattern (B) with 1.14 λ/d m 3.33 and 0.09 a/d m In regime II with 4 < λ/d m 6 and regime III, the wake pattern in the nodal plane is wider than that in the saddle plane (see figure 6f h). It is opposite the regimes discussed above. Due to the effect of vorte dislocation along the spanwise direction, the drag coefficient C D varies in a larger time scale ccle (λ/d m =9 and a/d m =0.1, in regime III (A); see figure 6g). It is evidentl different from other wav clinders in regimes I and II. Similar results were observed b Darekar & Sherwin (2001b; λ/d m = 10, wav square clinder) and Kim & Choi (2005; λ/d m = 7, circular clinder b active control method) at the similar Renolds number Re = 100. With the increase of a/d m, such phenomenon disappears. For constant values of λ/d m, a large value of a/d m means a steeper wav surface along the spanwise direction. A steeper wav surface will then give rise to different angle of flow separation along the spanwise direction. In the nodal plane, the far downstream wake width is much different from that in the saddle plane. It makes a large degree of distortion of the near wake structures formed in regime II (B) and regimes III (A) and III (B). It is interesting to note that no periodic vorte shedding appears in the saddle plane in regime III (B) (see figure 6h). The wake of the saddle plane shows a stead plane vorte sheet similar to the saddle plane flow pattern in regime II (C). For pattern (C), the flow behind the wav clinder is close to a stead plane vorte sheet. The force reduces sharpl with the r.m.s. lift of the wav clinder being greatl suppressed. In general, onl weak vorte shedding can be found at the ver far downstream positions. For the optimal wav clinder models (λ/d m =2, a/d m =0.3 and λ/d m =6, a/d m =0.15), the periodic vorte shedding cannot be discerned. The smmetrical and stead plane vorte sheets do not roll up into vorte structures. The value of Strouhal number becomes ero. These characteristics are also reflected b the distributions of different time histories of drag and lift coefficients (see figure 6). As the value of a/d m, in regime II increases, the value of C L becomes discernible again (C L =0.0006), and slight vorte shedding appears ver far downstream with St = for a wav clinder for λ/d m =6anda/D m =0.2(see figure 6f). To understand wh free shear laer will not roll up into a vorte easil at optimal wavelength, the spanwise variation of vorticit in the shear laers which leads to additional vorticit components in the streamwise direction is studied. Figure 7 shows the streamwise vorticit isosurface for flow around the optimal wav clinder (λ/d m =6 and a/d m =0.15). Two tpes of stable streamwise vorticities can be observed an outer streamwise vorticit and an inner streamwise vorticit. The central position of the outer streamwise vorticit is closer to the surface of the wav clinder than that of the inner one. Both the outer and inner streamwise vorticities are well organied and ehibit smmetrical distributions of vorticit in the plane with = 0, while at the same time showing a periodic repetition along the spanwise

14 208 K. Lam and Y. F. Lin (a) Outer streamwise vorticit (b) Inner streamwise vorticit λ/d m = 6 a/d m = 0.15 (c) Figure 7. Streamwise vorticit isosurface plot for the flow around a wav clinder at optimal wavelength for Re = 100: (a) overview; (b) the same structures viewed in the plane and (c) the plane. (a) (b) 0.5 θ = 0 θ = λ λ/d m = Circular clinder λ/d m = C p C p Figure 8. Spanwise distributions of the mean pressure coefficients for different wav clinders at Re = 100: (a), (b) a/d m =0.175; (c) (d) a/d m =0.15. The position of /λ = 0 denotes the node point, while /λ =0.5 is the saddle point. direction. This kind of additional streamwise component tends to stabilie the twodimensional spanwise vorticit of the free shear laers. The free shear laers separated from the surface of the wav clinder are pushed to further downstream positions as shown in figure 6 (patterns B and C). This stable three-dimensional redistribution of vorticit appears to prevent the free shear laer interaction in the near wake behind the clinder and hence suppresses the formation of Kármán vorte Flow characteristics of tpical wav clinders at Re = 100 To deepen our understanding of the effects of change of wavelength and wave amplitude, the surface pressure distribution, the vorte formation length, the surface velocit separation angles and the three-dimensional vorte structures of some tpical wav clinders are obtained and investigated Base pressure coefficient and vorte formation length Figure 8 shows the spanwise direction mean pressure coefficient C p distributions of several wav clinders at the positions of θ =0 and 180. At the position θ =0, the values of C p are generall smaller than those for a circular clinder along the spanwise direction at Re = 100, ecept at the positions of nodal and saddle planes (see

15 Effects of wavelength and amplitude of a wav clinder in cross-flow 209 (a) (b) λ/d m = 6 λ λ/d m = L f Circular clinder L f Figure 9. Spanwise distributions of the vorte formation length (L f ) for different wav clinders at = 0 with the mean steamwise velocit equal to ero (Re = 100): (a) a/d m =0.175; (b) a/d m =0.15. The position of /λ = 0 denoted the node, while /λ =0.5 is saddle point. figure 8a). At the middle section (/λ =0.25), the mean pressure coefficients at θ =0 of all the wav clinders are smaller than those in other planes of the wav clinders. The value of C p decreases with the decrease in λ/d m for the same wave amplitude (a/d m =0.175, 1 λ/d m 3anda/D m = 0.15, λ/d m = 6 and 9). This means larger wave steepness a/λ can increase the variation of the spanwise pressure coefficients and also that it gives rise to a strong three-dimensional effect on the near wake of wav clinders. Similar results were also obtained b Ahmed & Bas-Muchmore (1992), Lam et al. (2004a) and Lam & Lin (2008) with λ/d m 2.27 for turbulent flow. Figure 8(b) shows the mean pressure coefficients C p of different wav and circular clinders at the rear position θ = 180. In general, the maimum value of C p (the minimum value of base pressure coefficient C pb ) along the spanwise direction is obtained at λ/d m =2.5 (1 λ/d m 3, a/d m =0.175), while the value of C p shows little variation along the spanwise direction for wavelength λ/d m = 1, and it is smaller than that of a circular clinder. A stronger base suction leads to a higher drag (figure 4a). Increase in the wavelength (1 λ/d m 2.5) leads to increase in the value of C p from the nodal position (/λ = 0) to the saddle position (/λ =0.5). Similar results were also obtained b Ahmed & Bas-Muchmore (1992), Lam et al. (2004a) and Lam & Lin (2008) with λ/d m 2.27 for turbulent flow. When λ/d m = 3, the value of C p is close to that of a circular clinder. As the value of the wavelength further increases, however, the value of C p decreases from the nodal position to the saddle position. The maimum C p is obtained at λ/d m = 6. It means that the minimum value of C pb corresponds to the maimum drag reduction. Apart from the wav clinders with λ/d m = 1, 1.5 and 3, all the value of C p along the spanwise direction are larger than that of a corresponding circular clinder. The length of the reversed flow region behind the bluff bodies has a strong relationship with the mean base pressure coefficient C pb.lamet al. (2004b) and Lam & Lin (2008) showed that the values of the velocit recover lengths of both wav and circular clinders are inversel proportional to the values of the mean base pressure coefficient C pb in turbulent flow. Similar trend can be found in laminar flow condition. Figure 9 shows that the general trend of spanwise distribution of vorte formation length is similar to the spanwise distribution of the mean pressure coefficient C p at θ = 180 (see figure 8) for all the wav clinders. Along the spanwise direction of wav clinders, the values of L f ehibit a large variation. For the tpical

16 210 K. Lam and Y. F. Lin (a) λ 2 a/d m = Node Saddle (b) Node λ 2 a/d m = Saddle Figure 10. Mean surface streamlines of wav clinders with different wave amplitude at Re = 100: (a) λ/d m =2; (b) λ/d m =6. wav clinder with λ/d m =1.5 anda/d m =0.175, the value of L f shows in two line distributions along the spanwise direction due to the definition adopted (see figure 9a). The line close to the back surface (θ = 180 ) of the wav clinder is not the real vorte formation length. The second line is the real vorte formation length, considering the relationship between L f and C pb. Furthermore, we also found that at the nodal position and the mean values of streamwise velocit U/U in the = 0 plane are not equal to ero for the wav clinders with λ/d m = 2 and 2.5. Along the spanwise direction, the value U/U = 0 can be captured at /λ 0.08 for λ/d m =2 and at /λ 0.13 for λ/d m = 2.5. The central position of the streamwise velocit recover appears around the saddle position of the wav clinder. However, it is far from the back surface of the wav clinder and leads to a small value of C pb. In figure 9(b), the values of the vorte formation length L f of all wav clinders are evidentl longer than that of a circular clinder. At the nodal positions, the vorte formation lengths are longer than that in the saddle plane in the range 3.5 λ/d m 10. In general, the vorte formation length increases from the nodal plane to the saddle plane. Unlike that of the wav clinders with 1.5 λ/d m 2.5, the streamwise velocit recover centre moves to the nodal sections. An evidentl different vorte structure is generated along the spanwise direction. The wake pattern in the nodal plane is much wider than that in the saddle plane (figure 5g, h). Despite the difference of wake structures, velocit and pressure distributions, it can be concluded that longer vorte formation region can provide more reduction of the drag coefficient and a much greater suppression of the r.m.s. lift coefficient Velocit separation angles Figure 10 shows the time-averaged surface streamlines with the variation of wave amplitude 0.1 a/d m 0.3 (λ/d m =2)and0.02 a/d m 0.3 (λ/d m =6)atRe = 100. The velocit separation lines are also drawn along the spanwise direction. In general, the separation position varies from the nodal plane to the saddle plane. Lam et al. (2004b) pointed out the significant spanwise flow motion from the saddle plane towards the nodal plane when λ/d m =2.27forRe between 3000 and 9000, while Lam & Lin (2008) also found such phenomenon for turbulent flow with 1.14 λ/d m 3.33 at Re = Similar phenomenon is observed in laminar flow condition with λ/d m =2 and 0.1 a/d m 0.3 (see figure 10a). As shown in figure 11(a), the mean

17 Effects of wavelength and amplitude of a wav clinder in cross-flow 211 (a) 160 θ sep (deg.) 120 λ/d m = (b) 120 θ sep (deg.) 110 λ/d m = a/d m Figure 11. Mean velocit separation angles of wav clinders with different wave amplitude at Re = 100: (a) λ/d m =2; (b) λ/d m = , circular clinder;, nodal plane;, saddle plane. velocit separation angle θ sep in the nodal plane is larger than that in the saddle plane λ/d m = 2. It means that flow separation occurs earlier at the saddle position and later at the nodal position and hence gives rise to the distortion of free shear laer and the formation of three-dimensional vorte structures. It also leads to the flow motion from the saddle plane to the nodal plane and the generation of a sinusoidal wave line along the spanwise direction. The wav velocit separation line becomes progressivel clear with the increase in a/d m. It means that the wave steepness value a/λ plas an important role in the determination of the separation point. The stabilit of the free shear laer and the resulting vorte structure are highl dependent on the separation point of the wav clinder. For a wav clinder with a larger value of wavelength (λ/d m = 6; see figure 10b), the separation line shows little waviness with the small value a/d m = With the increase in a/d m, the variation becomes clearer. All the values of θ sep in the nodal planes are less than those in the saddle planes (see figure 11b). It means that flow separation at the nodal position is much earlier than that at the saddle position for all wav clinders with λ/d m =6. The spanwise flow motion moves from the nodal plane towards the saddle plane, opposite to those shown in figure 10(a) with λ/d m =2. At a/d m < 1.5 the values of the separation angle in both the nodal and saddle planes are less than that of circular clinder, while at the saddle plane, it is larger than that of the circular clinder when a/d m > 1.5. Similar results were also obtained using an active control method at the same Renolds number with λ/d m = 5 b Kim & Choi (2005). Because of the wav velocit separation line, the two-dimensional vorte sheet is distorted, and the vorte structures become highl three-dimensional in the wake of the wav clinder. That means a longitudinal vorticit component is produced resulting from the wav separation line. Thereafter, the three-dimensional vorte sheet rolls up into comple three-dimensional vorte structure at further downstream positions Spanwise wake structure topolog Figure 12 shows the spanwise vorte structure distributions for several tpical wav clinders. The wake structures are plotted with several periodic repetitions along the spanwise direction. With a small value of spanwise wavelength, the

18 212 K. Lam and Y. F. Lin (a) (b) (c) (d) Quasi-two-dimensional unstead strucutre N M S N M S M N Riblike structure N S M Riblike structure (e) ( f ) N S N M M Stead strucutre (St = 0) Riblike structure N S M S M N M S Riblike structure Figure 12. The instantaneous spanwise vorte structure distributions with several periodic repetitions along the spanwise direction (Re = 100): (a) λ/d m =1.5 and a/d m = 0.175; (b) λ/d m =2.5 and a/d m = 0.175; (c) λ/d m =3.5 and a/d m = 0.175; (d) λ/d m =5 and a/d m = 0.15; (e) λ/d m =6 and a/d m = 0.15; (f) λ/d m =7.5anda/D m = The smbols N, M and S denote the nodal, middle and saddle planes, respectivel. three-dimensional feature is not prominent at the further downstream positions (see figure 12a, λ/d m =1.5 and a/d m = 0.175, in the flow pattern (A) in regime I). The wake pattern is close to that of a circular clinder at Re = 100 (two-dimensional unstead wake vorte shedding flow pattern). With λ/d m = 2.5, the three-dimensional vorte structures become clearer. The riblike vorte structures captured are similar to the spanwise vorte structures of the wav clinder (λ/d m =2.27 and a/d m = 0.09) at Re = 600 obtained b the laser-induced fluorescence (LIF) method (Lam et al. 2004b). Furthermore, the riblike vorte structure in the near wake of the wav clinder appears at the positions of saddle planes (see figure 12b, regime I II (B)). It also shows a periodic characteristic along the spanwise direction. As shown in figure 12(c, d) in regimes II (A, B), as the values of λ/d m further increased from 3.5 to 5.0, the three-dimensional feature became more and more prominent. The riblike vorte structures along the spanwise direction were also observed. However, the appear at the positions near the nodal plane, different from that in figure 12(b). Considering the flow separation characteristics for wav clinders with different wavelength (figures 10 and 11), the riblike vorte structures appear at the positions at which the separation of the flow occurs first. The free shear laers behind the wav clinder at such positions epand along both the streamwise direction and the crosswise direction while shrink at other positions. It distorts the spanwise wake structures, and the evidentl three-dimensional wake structures are thus generated. With a larger waviness of the wav clinder, the riblike structures become more prominent. As discussed in 4.3.2, there is a significant spanwise flow motion from the saddle plane towards the nodal plane for smaller wavelength (λ/d m = 2), while the flow motion is from the nodal plane to the saddle plane for larger wavelength (λ/d m = 6). It implies that the different spanwise flow motion with the variation of wavelength ma lead to

19 Effects of wavelength and amplitude of a wav clinder in cross-flow 213 different additional vorticit components which hence give rise to two tpes of riblike structures (at the saddle or nodal position) being generated as shown in figure 12. For the optimal spanwise wavelength λ/d m = 6 corresponding to the flow pattern regime II (C), the spanwise vorte structures appear as a smooth plane vorte sheet along the spanwise direction from the nodal position to the saddle position. The riblike structures disappear (see figure 12e). Onl two wide and smooth vorte sheets are generated behind the wav clinder (the flow pattern in figure 6f). Unlike a two-dimensional vorte sheet, the three-dimensional vorte sheet behind the optimal wav clinder is ver stable and will not roll up into a mature vorte at this Renolds number even at ver far downstream position. The periodic vorte shedding is well suppressed at λ/d m =6 and a/d m = Therefore, the largest reduction of the mean drag coefficient and the r.m.s. lift coefficient is obtained. At λ/d m =7.5 and a/d m = 0.15, the riblike structures appear again (figure 12f) at the positions of nodal planes, while structures are ver different from that of figure 12(b d) for such large value of spanwise wavelength. The flow pattern characteristic in regime III (B) is full demonstrated Renolds number effect Renolds number plas an important role in the effect of optimal spanwise wavelength in controlling the vorte shedding, drag reduction and suppression of the vorteinduced vibrations. Several tpical wav clinders are used to eamine the effects of Renolds numbers from 60 to 150. A corresponding circular clinder is also simulated at the same Renolds numbers for comparison. The mean drag coefficient C D,the r.m.s. lift coefficient C L and the Strouhal number St for the wav and circular clinders at different Renolds numbers are obtained. Figure 13(a c) shows that the maimum drag reduction of up to 21.5 % is obtained at Re = 150 with λ/d m = 2.5 with the r.m.s. lift coefficient C L = 0 and the Strouhal number St = 0 in the range 1.5 λ/d m 3.0 and a/d m = In this spanwise wavelength range at Re = 60, the minimum value of C D =1.426 is obtained for λ/d m = 3 and is close to the value of a circular clinder at the same Renolds number ( C D =1.428). The value of C D decreases as the value of λ/d m increases. The lift fluctuation is not significantl weakened with little change of the Strouhal number. At Re = 80, the minimum value of C D =1.36 is obtained with λ/d m = 2.5 which is slightl lower than that of the wav clinder with λ/d m = 3 and the circular clinder. Increasing the Renolds number to Re = 100, the minimum value of C D is obtained with λ/d m = 2.5. Compared with a circular clinder, the lift fluctuation is significantl weakened for λ/d m = 2.0 and 2.5, and the Strouhal numbers are lower. At Re = 150, the values of C D for the wav clinders with λ/d m = 2.0 and 2.5 are lower than that of the circular clinder. However, the values of C D are close to the values of a circular clinder for λ/d m = 1.5 and 3.0. In general, the optimal spanwise wavelength is slightl reduced with the increase in Renolds numbers in the range 1.5 λ/d m 3.0. But it is still within the lower wavelength range 2.0 λ/d m 2.5. The spanwise vorte structures of the wav clinders (λ/d m = 2.0 and 2.5, a/d m = 0.175) at Re =80 and 150 are shown in figure 14. Longer near wake vorte structures generated behind the wav clinder at Re = 150 contrast with the shorter vorte structures generated at Re = 80 for the wav clinder with λ/d m =2 and a/d m = A well-organied stead and smmetrical wav vorte sheet structure is found for λ/d m =2.5 and a/d m = at Re = 150. In the wavelength range 5.0 λ/d m 7.5, the value of optimal wavelength for C D and C L reduction is also reduced with the increase in Renolds numbers (see

20 214 K. Lam and Y. F. Lin (a) 1.50 (d) 1.30 C D (b) (e) C L (c) (f ) 0.16 St Re Figure 13. Mean drag coefficients, r.m.s. lift coefficients and Strohual numbers of wav and circular clinders at the Renolds number values of : (a) (c) a/d m = with, λ/d m = 1.5;, λ/d m =2;, λ/d m =2.5;, λ/d m = 3; (d) (f) a/d m = 0.15 with, λ/d m = 5;, λ/d m =6;, λ/d m = 7.5; Here,, circular clinder. Re figure 13d f). The maimum drag reduction appears at λ/d m =7.5 for Re = 60, while it appears at λ/d m =6 for Re = 100. At Re = 150, the optimal value is further reduced to λ/d m = 5. These phenomena are consistent with the results of Kim & Choi (2005), which used the active control method. As shown in figure 13(d), at Re = 60, 80 and 100, the drag reduction of a wav clinder with the optimal wavelength (λ/d m =6, a/d m = 0.15) is 5.7 %, 11.8 % and 16.4 %, respectivel, compared with that of a circular clinder at the same Renolds number. The magnitude of reduction decreases at Re = 150. The drag reduction is 17 % which is onl slightl higher than that at Re = 100. The wake pattern of this wav clinder at Re = 80 and 100 is stead, and no vorte shedding appears (see figure 15a,b). At Re = 150, the wake vorte structure behind the wav clinder becomes unstead, and the wake pattern reverts back to the vorte shedding characteristic (see figure 15c). The continuous distortion

21 Effects of wavelength and amplitude of a wav clinder in cross-flow 215 (a) (b) Unstead periodic structure Unstead periodic structure (c) (d) Unstead periodic structure Stead smmetrical structure Figure 14. Instantaneous spanwise vorte structures for the wav clinders with a/d m = 0.175: (a) Re =80 (λ/d m =2); (b) Re = 150 (λ/d m =2); (c) Re =80 (λ/d m =2.5); (d) Re = 150 (λ/d m =2.5). (a) (b) (c) Stead smmetrical structure Unstead periodic structure Figure 15. Instantaneous spanwise vorte structures for the wav clinder (λ/d m =6 and a/d m = 0.15) at Renolds number values of : (a) Re = 80; (b) Re = 100; (c) Re = 150. of the vorte sheet leads to ultimate rolling up of three-dimensional vorte structure in the near wake with the occurrence of periodic vorte shedding at downstream position. This means that the effects of drag reduction and the suppression of r.m.s. lift still occur. Along the spanwise direction, the riblike vorte structures are formed. Figure 13(e, f) also confirms that the r.m.s. lift is discernible again and not full suppressed for the wav clinder with λ/d m =6 and a/d m =0.15 at Re = 150. In the range 5.0 λ/d m 7.5 at Re = 150, all the wake patterns of the wav clinder become unstead with three-dimensional vorte shedding at far downstream. From the discussion above, we can conclude that the vibration control b modifing the spanwise wavelength of the clinder and the values of optimal wavelength are also Renolds number dependent.

22 216 K. Lam and Y. F. Lin 5. Conclusions In the present investigations, the three-dimensional numerical simulations based on the finite volume method are used to calculate the laminar flow around wav clinders with different combinations of spanwise wavelength λ/d m and wave amplitude a/d m at low Renolds numbers. The three-dimensional near wake vorte structures of wav clinders mainl at the Renolds number of 100 were captured and analsed. It was found that such kind of wav surfaces with certain optimal values of λ/d m and a/d m can significantl modif the free shear laer development and control the three-dimensional vortices formation behind the wav clinder. The wav clinders are classified into three wavelength regimes (I, II and III) with different flow patterns (A), (B) and (C). For flow pattern (A), the flow is unstead and shows a quasi-twodimensional unstead wake structures similar to the straight circular clinder. Flow pattern (B) is characteried b an increase of three-dimensional wake vorte distortion and vorte formation length with a progressive weakening of vorte shedding. In pattern (C), the flow pattern behind the wav clinder is ver stead. A threedimensional wav vorte sheet is generated, but it does not roll up into a vorte even at far downstream position. The r.m.s. lift reduces to ero with the full suppression of vorte-induced vibration. The velocit separation line of the wav clinders varies along the spanwise direction. This leads to the distortion of the two-dimensional smmetrical vorte structures along the spanwise direction into three-dimensional structures. In regimes I and II with flow pattern (A) or (B), the wake width epands in the region behind the saddles of clinders and shrinks behind the nodes of clinders. On the contrar, in regimes II III and III with the flow pattern (A) or (B), the wake width shrinks behind the saddle positions and epands at the nodal position. As a result, the near wake vorte structures ehibit a periodic variation along the spanwise direction. In both regimes with flow pattern (B), it is difficult for the free shear laer to roll up. Hence it can onl develop into mature vorte at a further downstream position. A wav clinder with certain optimal values of λ/d m and a/d m can significantl modif and control the three-dimensional vorte structure behind itself. It weakens and even suppresses the vorte shedding. The base pressure coefficient decreases due to the formation of a longer vorte formation length. The mean drag coefficients of a tpical wav clinder are reduced compared with those of a corresponding straight circular clinder at the same Renolds numbers. At an optimal wavelength λ/d m around 6, the mean drag coefficient and the r.m.s. lift of a wav clinder drop sharpl with the increase of wave amplitude a/d m and the maimum drag coefficient reduction of up to 18 % is obtained. The reduction of the r.m.s. lift coefficients to ero indicates that such a wav clinder is an ideal device for the suppression of FIV. With a large value of wave amplitude, the three-dimensional near wake vorte structures are coherent and well organied. Furthermore, the value of optimal spanwise wavelength of the wav clinder for the control of FIV varies with the Renolds number range In the wavelength range 5.0 λ/d m 7.5, the value of optimal wavelength reduces from λ/d m =6 at Re = 100 to λ/d m =5 at Re = 150. In the wavelength regime 2.0 λ/d m 2.5, the variation of C D and C L is small. The maimum drag reduction of up to 21.5 % is obtained at Re = 150 for λ/d m =2.5 and a/d m = In conclusion, the value of optimal wavelength decreases with the increase in Renolds number. The authors wish to thank the Research Grants Council of the Hong Kong Special Administrative Region, China, for its support through grant no. PolU 5311/04E.

23 Effects of wavelength and amplitude of a wav clinder in cross-flow 217 Case N c H/D m (N ) Cells tu /D m C D C L St C pb CY (16) CY (16) CY (16) CY-4a (32) CY-4b (32) CY (56) CY (16) CY (16) Table 3. Grid independence test of a circular clinder at Re = 100: CY = circular clinder; N c = mesh numbers around the clinder circumference; N = mesh laers along the clinder spanwise direction. (a) 10 (b) 1.0 D m (c) /D m /D m D m /D m Figure 16. Computational domain and mesh distribution: (a) computational mesh distribution in the plane; (b) mesh around the circumference of a clinder; (c) spanwise direction mesh distribution. Appendi. Grid independence test and the validation of numerical models Grid independence test and the validation of numerical models are etremel important prior to etensive numerical simulations. Figure 16 shows the detailed diagram of the grid sstem. The mesh distribution in the plane is non-uniform, while it is uniform along the -direction (the spanwise direction of the clinder). A finer grid was generated near the clinder and graduall became coarser in the wake and the far field. The grid was clustered near the clinder with an epansion rate of 1.2 in the radial direction from the clinder surface. Table 3 shows that seven circular clinder mesh models were constructed for the present simulations at Re = 100. The distance between the first grid and the clinder surface was 0.01D m. The dimensionless time steps tu /D m = and were