Three-dimensional wake structures and aerodynamic coefficients for flow past an inclined square cylinder.pdf

1、Three-dimensional wake structures and aerodynamic coeffi cients for fl ow past an inclined square cylinder Dong-Hyeog Yoon 1, Kyung-Soo Yangn, Choon-Bum Choi Department of Mechanical Engineering, Inha University, Incheon, 402-751, Korea a r t i c l e i n f o Article history: Received 23 January 2011

2、 Received in revised form 20 October 2011 Accepted 24 October 2011 Available online 3 January 2012 Keywords: Inclined square cylinder Wake Immersed boundary method Floquet stability analysis Vortical structures a b s t r a c t Numerical investigation has been performed for fl ows past an inclined sq

3、uare cylinder in the range of Rer300 to elucidate the characteristics of the laminar three-dimensional wake behind the cylinder depending upon Reynolds number and angle of incidence (y). An immersed boundary method was used for implementation of the cylinder on a Cartesian grid system. Both Floquet

4、stability analysis and full three-dimensional simulation were employed to detect the onset of the secondary instability leading to three-dimensional fl ow, and to provide quantitative fl ow data. It was revealed that mode A becomes more unstable for symmetric fl ow confi guration, while mode C is do

5、minant for asymmetric fl ows in the range of 101ryr251. The most unstable three-dimensional modes predicted by the Floquet stability analysis were well confi rmed by the full 3D simulations which were conducted at Re150, 200, 250, and 300, with varying angle of incidence. The full 3D simulations als

6、o provided the key fl ow characteristics such as mean fl ow-induced force/moment coeffi cients and Strouhal number of vortex shedding. It was seen that they are sensitive to slight inclination of the square cylinder, and the Re effects are insignifi cant except for mean lift coeffi cient. Vortical s

7、tructures in the wake, taken from the results of the Floquet stability analysis and the full 3D simulations, respectively, were visualized by Q-contours, revealing good agreement between the two. Igarashi, 1984; Sohankar et al., 1998; Sohankar et al., 1999; Chen and Liu, 1999; Robichaux et al., 1999

8、; Saha et al., 2003; Sharma and Eswaran, 2004; Luo et al., 2003, 2007; Tong et al., 2008; Sheard et al., 2009; Yoon et al., 2010). A square cylinder is regarded as the simplest geometrical model for the structures immersed in a freestream such as a building, a bridge pier, and a fuel rod, to name a

9、few. In particular, fl ow-induced forces and vortex-shedding fre- quency have been the key results from the view point of structural safety (Okajima, 1982; Chen and Liu, 1999). Approxi- mately for Re4165 (where Re represents Reynolds number based on the uniform freestream velocity (U) and the projec

10、ted height of a square cylinder (h), Fig. 1(a), the wake behind a square cylinder immersed in the freestream exhibits three-dimensional (3D) behaviour. Thus, understanding fl ow physics in the Re regime of laminar 3D wakes is the fi rst step towards elucidating laminar- turbulent transition in the s

11、ame fl ow confi guration. In the literature, much attention has been paid to fl ow past a circular cylinder because its geometrical simplicity allows one to perform intensive numerical or experimental studies (Williamson, 1996; Barkley and Henderson, 1996). It is well known that in a time- periodic

12、two-dimensional (2D) fl ow past a circular cylinder, there exist two distinct instability modes, namely mode A and mode B, leading to 3D fl ow (Williamson, 1996). Mode A occurs with a dominant spanwise wavelength of approximately three to four cylinder diameters, manifesting as a spanwise distortion

13、 of the Ka rma n vortices shed from the cylinder. A pair of counter-rotating streamwise vortices is alternately and periodically formed in the upper region and in the lower region of the cylinder wake. The sense of those pairs is opposite, which is called odd refl ection-translation symmetry(Barkley

14、 and Henderson, 1996) of mode A. On the other hand, mode B appears with a rather short spanwise wavelength of about one diameter (Barkley and Henderson, 1996), and the pairs of counter-rotating streamwise vortices exhibit even refl ection-trans- lation symmetry. In the literature, the critical Reyno

15、lds numbers for mode A (ReA) and mode B (ReB) have been consistently reported even with diverse investigation methods. The experimental study of Williamson (1996) reveals ReAE190 and ReBE230260. Barkley and Henderson (1996) reported ReAE188 and ReBE259 by using a Floquet stability analysis, while th

16、e numerical study of Posdziech Contents lists available at SciVerse ScienceDirect journal homepage: Journal of Wind Engineering and Industrial Aerodynamics 0167-6105/$-see front matter fax: 82 32 868 1716. E-mail address: ksyanginha.ac.kr (K.-S. Yang). 1 Current address: Korea Institute of Nuclear S

17、afety, Daejeon, 305-338, Korea. J. Wind Eng. Ind. Aerodyn. 101 (2012) 3442 and Grundmann (2001) found ReAE190.2 and ReBE261. Recently, a new type of instability (mode C) was found in the fl ow past immersed bodies other than a single circular cylinder. Sheard et al. (2003) identifi ed the mode C ins

18、tability behind slender bluff rings essentially curved circular cylinders. Carmo et al. (2008) found a mode C instability in the fl ow past two staggered circular cylinders, and reported that mode C is promoted in asymmetric fl ow confi g- uration with a period twice that of vortex shedding. Sheard

19、et al. (2009) also noticed a mode-C instability in the fl ow past an inclined square cylinder for a certain range of inclination angle that incurs an asymmetric fl ow pattern. Most of research on the fl ow past a square cylinder has been performed on the case of zero angle of incidence (Sohankar et

20、al., 1999; Robichaux et al., 1999; Saha et al., 2003; Luo et al., 2003, 2007). Inclination of a rectangular cylinder with respect to the main fl ow direction can cause sudden shift of the separation points to other corners, resulting in drastic change of fl ow topology down- stream of the cylinder (

21、Igarashi, 1984). According to the experi- mental work of Igarashi (1984), the shift of separation points brings signifi cant change in fl ow characteristics such as Strouhal number (St) of vortex shedding, drag, and lift forces on the cylinder, depending upon the angle of incidence (y). It has been

22、also reported in the literature that the angle of incidence greatly affects fl ow instability downstream of a rectangular cylinder, altering the critical Reynolds numbers for fl ow separation, vortex shedding, and bifur- cation to 3D fl ow, respectively (Sheard et al., 2009; Yoon et al., 2010). Desp

23、ite this large volume of work, however, full under- standing of the effects of angle of incidence on fl ow structures in the 3D wake, and the associated force loading on the cylinder, is far from complete. In the current investigation, we perform a compre- hensive parametric study by means of an imm

24、ersed boundary method to reveal the effects of angle of incidence on fl ow topology, fl ow instability, and fl ow-induced force loading. First of all, we employ a Floquet stability analysis to detect the onset of fl ow instability depending ony. The vortical structures of the most unstable Floquet m

25、odes are presented and discussed. After that, full 3D simulations are carried out with various Re andy, to identify the fl ow structures predicted by the Floquet analysis, and to compute time-averaged force/moment coeffi cients and St. Time-averaged fl ow topology is also discussed. 2. Formulation a

26、nd numerical methods The computing efforts can be signifi cantly reduced by employ- ing an immersed boundary method (Kim et al., 2001) which facilitates implementing the solid surfaces of an inclined square cylinder on a Cartesian grid system. The governing equations for incompressible fl ow, modifi

27、 ed for the immersed boundary method, are as follows; rUu?q 01 u t ?rUuu?rp 1 Rer 2uf 2 where u, p, q, and f represent velocity vector, pressure, mass source/sink, and momentum forcing, respectively. All the physical variables except p are nondimensionalized by U and h; pressure is nondimensionalize

28、d by far-fi eld pressure (PN) and the dynamic pressure. The governing equations are discretized by a fi nite- volume method on a nonuniform staggered Cartesian grid system (Fig. 1(b). Spatial discretization is second-order accurate. A hybrid scheme is used for time advancement; nonlinear terms are e

29、xplicitly advanced by a third-order Runge-Kutta scheme, and the other terms are implicitly advanced by the Crank-Nicolson method. A fractional step method (Kim and Moin, 1985) is employed to decouple the continuity and momentum equations. The Poisson equation resulting from the second stage of the f

30、ractional step method is solved by a multigrid method. For detailed description of the numerical method used in the current investigation, see Yang and Ferziger (1993). Two-dimensional base fl ows were computed for a Floquet stability analysis with the following boundary conditions. The no- slip con

31、dition is imposed on the cylinder surfaces. A Dirichlet boundary condition (uU, v0) is used on the inlet boundary of the computational domain, while a convective boundary condi- tion is employed at the outlet (Kim et al., 2004). Here, u and v represent the velocity components in x and y directions,

32、respec- tively. A slip boundary condition (u/y0, v0) is imposed on the other boundaries. The entire computational domain is defi ned as ?33.5hrxr36.5h, and ?50hryr50h. The square cylinder is positioned at the origin of the coordinate system. The numerical resolution was determined by a grid-refi nem

33、ent study to ensure grid-independency. Doubling the numerical resolution in each direction respectively yields less than 1.0% of error for mean force coeffi cients and St of vortex shedding. The number of cells used was 792?448 in x and y directions. Three-dimensional full simulations were performed

34、 with a periodic boundary condition in the spanwise direction (z), and a spanwise domain of 0rzr12h, while the domain size and the boundary conditions in x and y directions remained unchanged. The spanwise domain size was selected in relation to the predicted spanwise wavelength of 3D instability mo

35、des (Sheard et al., 2009; Yoon et al., 2010). The number of cells used was 448?480?64 in x, y, and z directions. Further grid refi nement showed little difference in the results reported here. 3. Results and discussion 3.1. Validation A large volume of data is available in the literature for the cas

36、e of a square cylinder withy01. For validation of our code and x/h y/h -2024 -2 0 2 Fig. 1. Flow confi guration and grid system. D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344235 numerical methodology, rigorous comparisons were made in Fig. 2 between our computations of mean drag coef

37、fi cient (CD), rms (root mean square) of lift-coeffi cient fl uctuation (CL,rms) and Strouhal number (St), and those of other authors (Okajima, 1982; Sohankar et al., 1998; Sohankar et al., 1999; Sharma and Eswaran, 2004; De and Dalal, 2006; Luo et al., 2003, 2007) for the zero angle of incidence ca

38、se. Here, the drag and lift coeffi cients are defi ned as CD Drag=1 2rU 2h, CL Lift=1 2rU 2h 3 where fl uid density is denoted byr. Excellent agreement among the data confi rms that our numerical methodology and resolution are adequate and reliable. 3.2. Onset of the secondary instability 3.2.1. Flo

39、quet stability analysis The description of the Floquet linear stability analysis techni- que below follows Barkley 9m941 indicates exponentially growing perturbation. The Floquet multipliers can be obtained from the eigenvalues of L; u represents the corresponding eigenfunctions. Recently, a one- di

40、mensional(1D)power-typemethodwasintroducedby Robichaux et al. (1999) to estimate the maximum magnitude of the Floquet multipliers by computing the following ratio 9m9max? NtT=Nt8 where N(t) is the L2norm of the perturbation velocity at an instant of time. This method was verifi ed by Blackburn and L

41、opez (2003). In this study, we use the method of Robichaux et al. (1999) in conjunction with an immersed boundary method (Kim et al., 2001) to calculate the Floquet instability of the periodic wake past an inclined square cylinder. For the sake of convenience, the term Floquet multiplier implies the

42、 one that has the maximum magnitude among the Floquet multipliers from now on, and the subscript, max, is dropped. Eqs. (5) and (6) were temporally and spatially discretized in the same way as for the base fl ow. See Section 2. The 2D time- periodic base fl ow was fi rst computed; thirty-two snapsho

43、ts were saved for one period of vortex shedding. They were fed to Eqs. (5) and (6), being Fourier interpolated at each time step. 3.2.2. Instability modes The critical Reynolds numbers for the onset of the secondary instability are presented depending onyin Fig. 3. The solid symbols represent the pr

44、esent results, while the hollow ones indicate the results of Sheard et al. (2009). The agreement between the two is excellent even though the numerical algo- rithms employed were completely different, confi rming the robustness of the Floquet stability analysis. The lower critical Reynolds numbers f

45、or modes A or C imply that they are more unstable than the other modes (B or QP). A quasi-periodic (QP) 1.5 2 2D present 3D present 2D Sohankar et al. (1998) 2D Sharma and Eswaran (2004) 2D De and Dalal (2006) 3D Sohankar et al. (1999) CD CL,rms 0 0.2 0.4 0.6 0.8 2D present 3D present 2D Sohankar et

46、 al. (1999) 2D Sharma and Eswaran (2004) 2D De and Dalal (2006) 3D Sohankar et al. (1999) Re St 0100200300 Re 0100200300 Re 0100200300 0.08 0.12 0.16 present present 3D 2D Sharma and Eswaran (2004) 2D De and Dalal (2006) 3D Sohankar et al. (1999) Exp. Okajima (1982) Exp. Luo et al. (2007) Fig. 2. Co

47、mparison of the current results with those of other authors for square cylinder withy 01; (a) mean drag coeffi cient, (b) rms of lift-coeffi cient fl uctua- tion, (c) Strouhal number. D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344236 mode is detected at Re higher than the critical Rey

48、nolds number for mode B, consistent with the results of Robichaux et al. (1999) and Sheard et al. (2009). Blackburn and Sheard (2010) determined that the QP mode smoothly changed into the subharmonic mode C as the incidence angle was increased. It can be seen in Fig. 3 that mode A becomes prevailing

49、 (i.e. the critical Reynolds number for mode A becomes lower) as the angle of incidence approaches zero or 451, while mode C is dominant in the range of 101tyt251. This implies that mode A tends to be more unstable in a symmetric fl ow confi guration, and the opposite is the case for mode C. It shou

50、ld be also noted that Sheard (2011) recently performed a detailed stability analysis for inclined square cylin- ders at small incidence angles. Flow past an inclined square cylinder experiences abrupt topological changes around the cylinder asyincreases due to its sharp corners, affecting stability

51、characteristics of the fl ow. Fig. 4 presents time-averaged streamlines of the base fl ow at Re200 for three different values ofy. Fory r51, the fl ow separated at B reattaches on BC (Fig. 4(a). However, fl ow separation does not occur at B whenyis larger than 101 (Fig. 4(b). This topological change

52、 incurs asymmetry in the fl ow, and seems to suppress mode A instability and promote mode C instability (Fig. 3). For higher angle of incidence (yZ151), small recirculation bubbles are formed in the vicinity of the corner D, restoring fl ow symmetry to some degree (Fig. 4(c). The small bubbles inten

53、sify asyfurther increases. The restored symmetry suppresses mode C instability, and enhances mode A instability (Figs. 3 and 4(c). Fig. 5 shows variation of the critical spanwise wavelength of each mode withy. The results of Sheard et al. (2009) are also included in Fig. 5. Agreement between the two

54、 is again excellent. It is seen in Fig. 5 that the critical spanwise wavenumber weakly depends uponyfor each mode. 3.2.3. Vortical structures of Floquet modes Characteristics of instability modes can be elucidated by the Floquet mode corresponding to the spanwise wavenumber of the largest Floquet mu

55、ltiplier at given Re andy . The fl uctuating velocity fi eld (u0b x,y,z,t) and its vorticity fi eld (x0bx,y,z,t) corresponding to a Floquet mode can be written as follows; u0bx,y,z,t ucosbz,vcosbz,wsinbz9 x0bx,y,z,t oxcosbz,oycosbz,ozsinbz:10 In Fig. 6, the streamwise component of vorticity (oxin Eq

56、. (10) of the most unstablebis plotted at x/h2.5 along the vertical direction with time. Here, time is normalized with T. Temporal periodicity is clearly noticed in Fig. 6. Fig. 6(a) corresponds to a case of mode A at Re176,b1.35, andy5.11. It is seen that a vortical structure of high intensity alte

57、rnately passes through the upper and lower parts of the wake at x/h2.5 with a time period of T. Fig. 6(b) shows a similar plot corresponding to a case of mode C at Re167,b3.95, andy15.31, revealing a doubled time period (2T). Finally, Fig. 6(c) presents another case of mode A at Re122,b1.55, andy 45

58、1, where the fl ow confi guration Fig. 3. Critical Reynolds numbers for the instability modes for fl ow past an inclined square cylinder; K,J, Mode A ; , , ?,Mode C; m,W, Mode QP, reprinted with permission from Yoon et al. (2010). Copyright (2010), American Institute of Physics. The solid symbols re

59、present the present results, and the hollow symbols denote the results of Sheard et al. (2009). x/h x/h x/h y/h -1012 -1 0 1 DA BC y/h -1012 -1 0 1 B A D C y/h -1012 -1 0 1 B A D C Fig. 4. Time-averaged streamlines for two-dimensional basic fl ow at Re200; (a) y01, (b)y10.21, (c)y34.81. Fig. 5. Normalized critical spanwise wavelengths (l/h) of the instability modes, reprinted with permission from Yoon et al. (2010). Copyright (2010), American Institute of Physics. For the symbols, see the caption of Fig. 3. D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344237 becomes