Three-dimensional wake structures and aerodynamic coefficients for flow past an inclined square cylinder.pdf
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Three-dimensional wake structures and aerodynamic coefficients for flowpast an inclined square cylinderDong-Hyeog Yoon1, Kyung-Soo Yangn, Choon-Bum ChoiDepartment of Mechanical Engineering, Inha University, Incheon, 402-751, Koreaa r t i c l e i n f oArticle history:Received 23 January 2011Received in revised form20 October 2011Accepted 24 October 2011Available online 3 January 2012Keywords:Inclined square cylinderWakeImmersed boundary methodFloquet stability analysisVortical structuresa b s t r a c tNumerical investigation has been performed for flows past an inclined square cylinder in the range ofRer300 to elucidate the characteristics of the laminar three-dimensional wake behind the cylinderdepending upon Reynolds number and angle of incidence (y). An immersed boundary method was usedfor implementation of the cylinder on a Cartesian grid system. Both Floquet stability analysis and fullthree-dimensional simulation were employed to detect the onset of the secondary instability leading tothree-dimensional flow, and to provide quantitative flow data. It was revealed that mode A becomesmore unstable for symmetric flow configuration, while mode C is dominant for asymmetric flows inthe range of 101ryr251. The most unstable three-dimensional modes predicted by the Floquetstability analysis were well confirmed by the full 3D simulations which were conducted at Re150,200, 250, and 300, with varying angle of incidence. The full 3D simulations also provided the key flowcharacteristics such as mean flow-induced force/moment coefficients and Strouhal number of vortexshedding. It was seen that they are sensitive to slight inclination of the square cylinder, and the Reeffects are insignificant except for mean lift coefficient. Vortical structures in the wake, taken from theresults of the Floquet stability analysis and the full 3D simulations, respectively, were visualized byQ-contours, revealing good agreement between the two.& 2011 Elsevier Ltd. All rights reserved.1. IntroductionRecently, researchers have shown a keen interest in flow past asquare cylinder not only due to its physical significance in fluidmechanics, but also due to its applicability in engineering(Okajima, 1982; Igarashi, 1984; Sohankar et al., 1998; Sohankaret al., 1999; Chen and Liu, 1999; Robichaux et al., 1999; Sahaet al., 2003; Sharma and Eswaran, 2004; Luo et al., 2003, 2007;Tong et al., 2008; Sheard et al., 2009; Yoon et al., 2010). A squarecylinder is regarded as the simplest geometrical model for thestructures immersed in a freestream such as a building, a bridgepier, and a fuel rod, to name a few.In particular, flow-induced forces and vortex-shedding fre-quency have been the key results from the view point ofstructural safety (Okajima, 1982; Chen and Liu, 1999). Approxi-mately for Re4165 (where Re represents Reynolds number basedon the uniform freestream velocity (U) and the projected height ofa square cylinder (h), Fig. 1(a), the wake behind a square cylinderimmersed in the freestream exhibits three-dimensional (3D)behaviour. Thus, understanding flow physics in the Re regime oflaminar 3D wakes is the first step towards elucidating laminar-turbulent transition in the same flow configuration.In the literature, much attention has been paid to flow past acircular cylinder because its geometrical simplicity allows one toperform intensive numerical or experimental studies (Williamson,1996; Barkley and Henderson, 1996). It is well known that in a time-periodic two-dimensional (2D) flow past a circular cylinder, thereexist two distinct instability modes, namely mode A and mode B,leading to 3D flow (Williamson, 1996). Mode A occurs with adominant spanwise wavelength of approximately three to fourcylinder diameters, manifesting as a spanwise distortion of theKa rma n vortices shed from the cylinder. A pair of counter-rotatingstreamwise vortices is alternately and periodically formed in theupper region and in the lower region of the cylinder wake. The senseof those pairs is opposite, which is called odd reflection-translationsymmetry(Barkley and Henderson, 1996) of mode A. On the otherhand, mode B appears with a rather short spanwise wavelength ofabout one diameter (Barkley and Henderson, 1996), and the pairs ofcounter-rotating streamwise vortices exhibit even reflection-trans-lation symmetry. In the literature, the critical Reynolds numbers formode A (ReA) and mode B (ReB) have been consistently reportedeven with diverse investigation methods. The experimental study ofWilliamson (1996) reveals ReAE190 and ReBE230260. Barkleyand Henderson (1996) reported ReAE188 and ReBE259 by using aFloquet stability analysis, while the numerical study of PosdziechContents lists available at SciVerse ScienceDirectjournal homepage: /locate/jweiaJournal of Wind Engineeringand Industrial Aerodynamics0167-6105/$-see front matter & 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jweia.2011.10.012nCorresponding author. Tel.: 82 32 860 7322; fax: 82 32 868 1716.E-mail address: ksyanginha.ac.kr (K.-S. Yang).1Current address: Korea Institute of Nuclear Safety, Daejeon, 305-338, Korea.J. Wind Eng. Ind. Aerodyn. 101 (2012) 3442and Grundmann (2001) found ReAE190.2 and ReBE261. Recently,a new type of instability (mode C) was found in the flow pastimmersed bodies other than a single circular cylinder. Sheard et al.(2003) identified the mode C instability behind slender bluff ringsessentially curved circular cylinders. Carmo et al. (2008) found amode C instability in the flow past two staggered circular cylinders,and reported that mode C is promoted in asymmetric flow config-uration with a period twice that of vortex shedding. Sheard et al.(2009) also noticed a mode-C instability in the flow past an inclinedsquare cylinder for a certain range of inclination angle that incurs anasymmetric flow pattern.Most of research on the flow past a square cylinder has beenperformed on the case of zero angle of incidence (Sohankar et al.,1999; Robichaux et al., 1999; Saha et al., 2003; Luo et al., 2003,2007). Inclination of a rectangular cylinder with respect to the mainflow direction can cause sudden shift of the separation points toother corners, resulting in drastic change of flow topology down-stream of the cylinder (Igarashi, 1984). According to the experi-mental work of Igarashi (1984), the shift of separation points bringssignificant change in flow 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 also reportedin the literature that the angle of incidence greatly affects flowinstability downstream of a rectangular cylinder, altering the criticalReynolds numbers for flow separation, vortex shedding, and bifur-cation to 3D flow, respectively (Sheard et al., 2009; Yoon et al.,2010). Despite this large volume of work, however, full under-standing of the effects of angle of incidence on flow structures in the3D wake, and the associated force loading on the cylinder, is farfrom complete. In the current investigation, we perform a compre-hensive parametric study by means of an immersed boundarymethod to reveal the effects of angle of incidence on flow topology,flow instability, and flow-induced force loading. First of all, weemploy a Floquet stability analysis to detect the onset of flowinstability depending ony. The vortical structures of the mostunstable Floquet modes are presented and discussed. After that, full3D simulations are carried out with various Re andy, to identify theflow structures predicted by the Floquet analysis, and to computetime-averaged force/moment coefficients and St. Time-averagedflow topology is also discussed.2. Formulation and numerical methodsThe computing efforts can be significantly reduced by employ-ing an immersed boundary method (Kim et al., 2001) whichfacilitates implementing the solid surfaces of an inclined squarecylinder on a Cartesian grid system.The governing equations for incompressible flow, modified forthe immersed boundary method, are as follows;rUu?q 01ut ?rUuu?rp1Rer2uf2where u, p, q, and f represent velocity vector, pressure, masssource/sink, and momentum forcing, respectively. All the physicalvariables except p are nondimensionalized by U and h; pressure isnondimensionalized by far-field pressure (PN) and the dynamicpressure. The governing equations are discretized by a finite-volume method on a nonuniform staggered Cartesian grid system(Fig. 1(b). Spatial discretization is second-order accurate. Ahybrid scheme is used for time advancement; nonlinear termsare explicitly advanced by a third-order Runge-Kutta scheme, andthe other terms are implicitly advanced by the Crank-Nicolsonmethod. A fractional step method (Kim and Moin, 1985) isemployed to decouple the continuity and momentum equations.The Poisson equation resulting from the second stage of thefractional step method is solved by a multigrid method. Fordetailed description of the numerical method used in the currentinvestigation, see Yang and Ferziger (1993).Two-dimensional base flows were computed for a Floquetstability analysis with the following boundary conditions. The no-slip condition is imposed on the cylinder surfaces. A Dirichletboundary condition (uU, v0) is used on the inlet boundary ofthe computational domain, while a convective boundary condi-tion is employed at the outlet (Kim et al., 2004). Here, u and vrepresent the velocity components in x and y directions, respec-tively. A slip boundary condition (u/y0, v0) is imposed onthe other boundaries. The entire computational domain is definedas ?33.5hrxr36.5h, and ?50hryr50h. The square cylinder ispositioned at the origin of the coordinate system. The numericalresolution was determined by a grid-refinement study to ensuregrid-independency. Doubling the numerical resolution in eachdirection respectively yields less than 1.0% of error for mean forcecoefficients and St of vortex shedding. The number of cells usedwas 792?448 in x and y directions.Three-dimensional full simulations were performed with aperiodic boundary condition in the spanwise direction (z), and aspanwise domain of 0rzr12h, while the domain size and theboundary conditions in x and y directions remained unchanged.The spanwise domain size was selected in relation to thepredicted spanwise wavelength of 3D instability modes (Sheardet al., 2009; Yoon et al., 2010). The number of cells used was448?480?64 in x, y, and z directions. Further grid refinementshowed little difference in the results reported here.3. Results and discussion3.1. ValidationA large volume of data is available in the literature for the caseof a square cylinder withy01. For validation of our code andx/hy/h-2024-202Fig. 1. Flow configuration and grid system.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344235numerical methodology, rigorous comparisons were made inFig. 2 between our computations of mean drag coefficient (CD),rms (root mean square) of lift-coefficient fluctuation (CL,rms) andStrouhal 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 zeroangle of incidence case. Here, the drag and lift coefficients aredefined asCD Drag=12rU2h,CL Lift=12rU2h3where fluid density is denoted byr. Excellent agreement amongthe data confirms that our numerical methodology and resolutionare adequate and reliable.3.2. Onset of the secondary instability3.2.1. Floquet stability analysisThe description of the Floquet linear stability analysis techni-que below follows Barkley & Henderson (1996). The onset of thesecondary instability leading to a 3D flow can be detected by aFloquet stability analysis in which an instantaneous velocity fieldof the flow past an inclined square cylinder is decomposed into a2D base flow with a period T (U(x,y,t)U(x,y,tT) and a 3Dperturbation velocity (u0(x,y,z,t) as follows,ux,y,z,t Ux,y,tu0x,y,z,t:4Substituting Eq. (4) into the NavierStokes and continuityequations, and then linearizing them, one can obtain the follow-ing governing equations for the perturbation velocity field,rUu0?q0 05u0t ?rUu0UUu0?rp01Rer2u0f0:6Here, the additional terms for the immersed boundary methodare also included. At the inlet, a Dirichlet boundary condition(u00) is imposed, while the convective and slip boundaryconditions are used at the exit and at the lateral boundaries,respectively. Since velocity and pressure fluctuations are assumedto be homogeneous in the spanwise direction, they can beexpressed by an inverse Fourier transform in z as follows,u0p0#x,y,z,t Z1?1up#x,y,b,teibzdb7whereb2p/lrepresents the spanwise wavenumber andlis thecorresponding spanwise wavelength of a disturbance. Since Eqs.(5) and (6) are linear, modes with different 9b9 can be decoupled.The governing equations for each disturbance wave are similar toEqs. (5) and (6), except for the replacement of the gradientoperatorrwithrb(q/qx, q/qy, ib). By defining the operator Lso that L(u) is the right-hand side of the linearized equation, thegoverning equation can be written symbolically as u=t Lu.The general solution of this equation can be expressed as a sum ofFloquet modes of the form,ux,y,b,texpst, wheresis theFloquet exponent, and eachu is a time-periodic function. Instabil-ity of the base flow U is determined by the Floquet multipliers,m?exp(sT); 9m941 indicates exponentially growing perturbation.The Floquet multipliers can be obtained from the eigenvalues of L;u represents the corresponding eigenfunctions. Recently, a one-dimensional(1D)power-typemethodwasintroducedbyRobichaux et al. (1999) to estimate the maximum magnitude ofthe Floquet multipliers by computing the following ratio9m9max? NtT=Nt8where N(t) is the L2norm of the perturbation velocity at an instantof time. This method was verified by Blackburn and Lopez (2003).In this study, we use the method of Robichaux et al. (1999) inconjunction with an immersed boundary method (Kim et al.,2001) to calculate the Floquet instability of the periodic wakepast an inclined square cylinder. For the sake of convenience, theterm Floquet multiplier implies the one that has the maximummagnitude among the Floquet multipliers from now on, and thesubscript, max, is dropped.Eqs. (5) and (6) were temporally and spatially discretized inthe same way as for the base flow. See Section 2. The 2D time-periodic base flow was first computed; thirty-two snapshots weresaved for one period of vortex shedding. They were fed toEqs. (5) and (6), being Fourier interpolated at each time step.3.2.2. Instability modesThe critical Reynolds numbers for the onset of the secondaryinstability are presented depending onyin Fig. 3. The solidsymbols represent the present results, while the hollow onesindicate the results of Sheard et al. (2009). The agreementbetween the two is excellent even though the numerical algo-rithms employed were completely different, confirming therobustness of the Floquet stability analysis. The lower criticalReynolds numbers for modes A or C imply that they are moreunstable than the other modes (B or QP). A quasi-periodic (QP)1.522D present3D present2D Sohankar et al. (1998)2D Sharma and Eswaran (2004)2D De and Dalal (2006)3D Sohankar et al. (1999)CDCL,rms00.20.40.60.82D present3D present2D Sohankar et al. (1999)2D Sharma and Eswaran (2004)2D De and Dalal (2006)3D Sohankar et al. (1999)ReSt0100200300Re0100200300Re01002003000.080.120.16presentpresent 3D2D Sharma and Eswaran (2004)2D De and Dalal (2006)3D Sohankar et al. (1999)Exp. Okajima (1982)Exp. Luo et al. (2007)Fig. 2. Comparison of the current results with those of other authors for squarecylinder withy01; (a) mean drag coefficient, (b) rms of lift-coefficient fluctua-tion, (c) Strouhal number.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344236mode is detected at Re higher than the critical Reynolds numberfor mode B, consistent with the results of Robichaux et al. (1999)and Sheard et al. (2009). Blackburn and Sheard (2010) determinedthat the QP mode smoothly changed into the subharmonic modeC as the incidence angle was increased. It can be seen in Fig. 3 thatmode A becomes prevailing (i.e. the critical Reynolds number formode A becomes lower) as the angle of incidence approaches zeroor 451, while mode C is dominant in the range of 101tyt251.This implies that mode A tends to be more unstable in asymmetric flow configuration, and the opposite is the case formode C. It should be also noted that Sheard (2011) recentlyperformed a detailed stability analysis for inclined square cylin-ders at small incidence angles.Flow past an inclined square cylinder experiences abrupttopological changes around the cylinder asyincreases due to itssharp corners, affecting stability characteristics of the flow. Fig. 4presents time-averaged streamlines of the base flow at Re200for three different values ofy. Foryr51, the flow separated at Breattaches on BC (Fig. 4(a). However, flow separation does notoccur at B whenyis larger than 101 (Fig. 4(b). This topologicalchange incurs asymmetry in the flow, and seems to suppressmode A instability and promote mode C instability (Fig. 3). Forhigher angle of incidence (yZ151), small recirculation bubblesare formed in the vicinity of the corner D, restoring flowsymmetry to some degree (Fig. 4(c). The small bubbles intensifyasyfurther increases. The restored symmetry suppresses mode Cinstability, and enhances mode A instability (Figs. 3 and 4(c).Fig. 5 shows variation of the critical spanwise wavelength ofeach mode withy. The results of Sheard et al. (2009) are alsoincluded in Fig. 5. Agreement between the two is again excellent.It is seen in Fig. 5 that the critical spanwise wavenumber weaklydepends uponyfor each mode.3.2.3. Vortical structures of Floquet modesCharacteristics of instability modes can be elucidated by theFloquet mode corresponding to the spanwise wavenumber of thelargest Floquet multiplier at given Re andy. The fluctuatingvelocity field (u0bx,y,z,t) and its vorticity field (x0bx,y,z,t)corresponding to a Floquet mode can be written as follows;u0bx,y,z,t ucosbz,vcosbz,wsinbz9x0bx,y,z,t oxcosbz,oycosbz,ozsinbz:10In Fig. 6, the streamwise component of vorticity (oxin Eq. (10)of the most unstablebis plotted at x/h2.5 along the verticaldirection with time. Here, time is normalized with T. Temporalperiodicity is clearly noticed in Fig. 6. Fig. 6(a) corresponds to acase of mode A at Re176,b1.35, andy5.11. It is seen that avortical structure of high intensity alternately passes through theupper and lower parts of the wake at x/h2.5 with a time periodof T. Fig. 6(b) shows a similar plot corresponding to a case of modeC at Re167,b3.95, andy15.31, revealing a doubled timeperiod (2T). Finally, Fig. 6(c) presents another case of mode A atRe122,b1.55, andy451, where the flow configurationFig. 3. Critical Reynolds numbers for the instability modes for flow past aninclined square cylinder; K,J, Mode A ; , &, Mode B; , ?,Mode C; m,W,Mode QP, reprinted with permission from Yoon et al. (2010). Copyright (2010),American Institute of Physics. The solid symbols represent the present results, andthe hollow symbols denote the results of Sheard et al. (2009).x/hx/hx/hy/h-1012-101DABCy/h-1012-101BADCy/h-1012-101BADCFig. 4. Time-averaged streamlines for two-dimensional basic flow 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), AmericanInstitute of Physics. For the symbols, see the caption of Fig. 3.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344237becomes perfectly symmetric with respect to the centreline (y0)like flow past a circular cylinder, and an odd reflection-translationsymmetry is noticed at a time interval of T/2. It should be notedthat Fig. 6(a) reveals mode A (as in Fig. 6(c), even though thecylinder geometry in Fig. 6(a) is not symmetric with respect to themain flow direction.To visualize the 3D vortical structure of the most unstablemode, contours of the second invariant of the velocity gradienttensor (Jeong and Hussain, 1995, hereafter, called Q contours)corresponding to Fig. 6(a), (b), and (c) are shown in a plan view inFig. 7(a), (b), and (c), respectively. Combination of the base flowwith the velocity fluctuation (uUu0) corresponding to theselectedbis used in Fig. 7. Therefore, Fig. 7 presents therepresentative vortical structure at given Re andy. Spanwisedistortion of the Ka rma n vortices is clearly seen, revealing thetime periods of T and 2T of mode A and C, respectively.3.3. Three-dimensional simulationsIn this section, the results from full 3D simulations arepresented to clarify the effects of angle of incidence on vorticalstructures downstream of the cylinder as well as flow-inducedforces on the square cylinder. Simulations were performed forRe200, 250, and 300, and ten cases ofyup to 451 werecomputed with an increment of 51 for each Re.3.3.1. Vortical structuresIn Fig. 8, instantaneous contours of Q andoxare presented fory5.11, 10.21, 15.31, 29.71, and 451 at Re200. It should be notedthat both figures are presented in a plan view in which theKa rma n vortices shed from the upper and lower free-shear layersFig. 6. Streamwise vorticity contours of the most unstable Floquet mode on theline x/h2.5; (a)y5.11, Re176,b1.35 (mode A), (b)y15.31, Re167,b3.95 (mode C), (c)y451, Re122,b1.55 (mode A).z/h1050z/h1050z/h1050501015x/hFig. 7. Three-dimensional iso-surface plots of the instantaneous vortical struc-tures (Q contours (Jeong and Hussain, 1995) of the most unstable Floquet modesuperimposed onto the base flow: (a)y5.11, Re176,b1.35, (b)y15.31,Re167,b3.95, (c)y451, Re122,b1.55. The cylinder centre is located atx0. Fluid flows from left to right.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344238are shown altogether. In Fig. 8(a), dominant spanwise waves witha wavelength of approximately 4h, which is the characteristicwavelength of mode A, are clearly identified in the axial directionof Ka rma n vortices as predicted by the Floquet stability analysis.The correspondingoxcontours (Fig. 8(f) reveal three pairs ofcounter-rotating streamwise vortices, again consistent with theFloquet analysis fory5.11 (Figs. 5 and 7). For a slightly largerangle of incidence (y10.21), flow separation shifts from corner Bto corner C (Fig. 4(b), and the flow is stabilized. Consequently, Qcontours in Fig. 8(b) andoxcontours in Fig. 8(g) confirm that theflow is still two-dimensional. We intentionally disturbed theentire flow field with random noise, but the imposed disturbancesquickly decayed, resulting in a 2D flow. This result is consistentwith Fig. 3 in which the critical Reynolds number fory10.21 ishigher than Re200. Tong et al. (2008) reported that the mode Astructures were much weaker at around 101 when compared withthe zero degree case. Fig. 8(c) and (h) correspond toy15.31. It isfound that seven pairs of counter-rotating vortices are dominantin the axial direction, yieldingl1.714h which is in goodagreement with that ofy15.31 in Fig. 5. Furthermore, the signof the vortices is alternate per each vortex shedding, indicatingthat the time period of the dominant spanwise instability is twicethat of the Ka rma n vortex shedding. These are the key character-istics of mode C, as seen in Fig. 6(b) and Fig. 7(b) constructed bythe base flow and the most unstable spanwise mode computedfrom the Floquet stability analysis. Vortical structures in the caseofy29.71 are depicted in Fig. 8(d) and (i). According to theFloquet stability analysis (Fig. 3), both mode A and mode C areunstable at Re200. In Fig. 8(d), one can notice a footprint ofmode A, namely, a dominant spanwise distortion of large wave-length. Fig. 8(i), however, exhibits the alternate sign change pereach vortex shedding, which belongs to the characteristics ofmode C. The mixture of modes A and C in the 3D full simulationconfirms the reliability of the Floquet stability analysis. Thevortical structures in the full simulation are much more complexthan those in Fig. 7 because of nonlinear interaction amongmodes. Nevertheless, the unstable modes predicted by the Flo-quet stability analysis are also detected in the full simulations.Thesamecommentcanbemadeinthecaseofy451(Fig. 8(e) and (j). The vortical structures look disordered, butthe main features of mode A can be identified in these figures.The nonlinear interaction intensifies as Re increases. Fig. 9presents contours of Q and streamwise vorticity in a plan view atRe150 andy451 where Re is slightly above the criticalReynolds number for mode A (Rec121). Three pairs of counter-rotating vortices are clearly seen, and the vortical structures seemto be much more regular compared with those at Re200. Theyeven look like those from the Floquet stability analysis (Fig. 7(c).z/h1050z/h1050z/h1050z/h1050z/h1050z/h1050z/h1050z/h1050z/h1050z/h1050101550x/h101550x/hFig. 8. Instantaneous vortical structures (Q contours (Jeong and Hussain, 1995),(ae) Q0.5) and instantaneous streamwise vorticity contours (f?j),ox 70.1,light contours indicate negative vorticity, while the dark contours indicate positivevalues) from 3D simulations at Re200; (a, f)y5.11, (b, g)y10.21, (c, h)y15.31, (d, i)y29.71, (e, j)y451. The cylinder centre is located at x0. Fluidflows from left to right.z/h1050z/h1050501015x/hFig. 9. Vortical structures from 3D simulations at Re150,y451 (a) Q contours(Q0.5), (b) streamwise vorticity contours (ox 70.1). Dark contours indicatepositive vorticity, while the light colour indicates negative values. The cylindercentre is located at x0. Fluid flows from left to right.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344239The vortical structures at Re250 are shown in Fig. 10 with thesame values ofyas in Fig. 8. Wheny5.11 (Fig. 10(a), a spanwisedistortion of approximately 1h, corresponding to mode B, isidentified in addition to mode A (See Figs. 3 and 5 also). The casewithy10.21 (Fig. 10(b) definitely reveals the alternate char-acteristics of mode C, while the flow remained two-dimensionalin the case of Re200 andy10.21 (Fig. 8(b). Again, theprediction by the Floquet stability analysis turns out to berealized in the full simulation. For higher angles of incidence(y15.31, 29.71, and 451), the critical Reynolds numbers becomelower (Fig. 3), resulting in intensified nonlinear interactions, seeFig. 10(c), (d), and (e). In Fig. 10(d) and (e), the characteristicfeatures of the dominant modes are hardly identifiable, while thealternate sign change of mode C is still visible in Fig. 10(c).3.3.2. Flow-induced forces and momentsChange of flow topology due to inclination of a square cylindersignificantly affects flow-induced forces and moments on thecylinder. Fig. 11 presents mean drag coefficient (CD), mean liftcoefficient (CL), mean moment coefficient (CM), and Strouhalnumber of vortex shedding (St) as a function ofyfor the selectedReynolds numbers (Re200, 250, 300). Strouhal number wasobtained based on time-dependent lift coefficient, and the meanmoment coefficient was defined as follows,CM Moment=12rU2h211where the clockwise direction was set as the positive direction. InFig. 11(a), the mean drag coefficient reaches a local minimum aty5.11, and then monotonically increases up toy451. In thecase of the mean lift coefficient (Fig. 11(b), however, it sharplyincreases to the maximum in magnitude, and then returns to zero,as symmetry in the flow configuration is restored. Assuming thatcontribution of viscous force to flow-induced forces is insignif-icant like any other bluff-body flow (e.g. Fig. 19 of Yoon et al.,2010), these observations of CDand CLcan be explained by thechange of pressure distribution on the faces of the cylinderdepending ony. Fig. 12 shows distributions of pressure coefficientdefined as Cp p?p1=12rU2along the cylinder faces for someselected values ofyat Re200. When the cylinder is inclined, therecirculation bubble on BC disappears (Fig. 4(b) and (c), resultingin high pressure on BC (Fig. 12). On the contrary, pressure on ADand CD drops with increasingy(Fig. 12). The high pressuredistribution on the upstream faces together with the low pressuredistribution on the downstream faces causes an increase of totaldrag. It should be noted that the high pressure force on AB isdecomposed into a horizontal component which is smaller thanthat of the aligned cylinder (y01) and a downward componentwhich was not present in the aligned case. For a slight inclination(e.g.y5.11), these changes are dominant over any other changein pressure forces on the other faces, causing a sudden drop of CD,and a sudden increase of CLin magnitude. The downwardcomponent of high pressure force on AB is also responsible forthe downward lift force for ally. See Fig. 11(b).Asymmetry in pressure distribution along the cylinder facesincurs a net moment. Fig. 11(c) presents CMas a function ofyforthe selected Reynolds numbers, revealing negative mean momentfor all the cases. Asyincreases, the stagnation point on AB and thebase point on CD move towards B and D, respectively (Fig. 12),generating a negative moment. The mean moment reaches themaximum in magnitude in the range of 101ryr151, and mono-tonically approaches zero towardsy451.Variation of St withyis depicted in Fig. 11(d). The Strouhalnumber increases up toy101, and then stays almost constantafterwards. This seems to be related to the change in flowtopology neary101; flow separation occurs at A and B for10501050105010501050101550x/hz/hz/hz/hz/hz/hFig.10. Streamwisevorticitycontoursfrom3DsimulationsatRe250(ox 70.1). Dark contours indicate positive vorticity, while the light colorindicates negative values; (a)y5.11, (b)y10.21, (c)y15.31, (d)y29.71,(e)y451. The cylinder centre is located at x0. Fluid flows from left to right.D.-H. Yoon et al. / J. Wind Eng. Ind. Aerodyn. 101 (2012) 344240yr5.11, but at A and C foryZ101 (Fig. 4). It is also seen that theeffect of Reynolds number is restricted only to the lift coefficient(Fig. 11(b) for the laminar 3D wakes considered in this study.4. ConclusionCharacteristics of laminar three-dimensional wake past aninclined square cylinder have been numerically studied. Firstly,onset of the secondary instability leading to 3D flow waspredicted by performing a Floquet stability analysis on 2D time-periodic base flows. Vortical structures of the dominant modes aswell as their critical Reynolds numbers were presented. It turnedout that mode A becomes more unstable for symmetric flowconfiguration, while mode C is dominant for asymmetric flows inthe range of 101ryr251. Our result is in excellent agreementwith that of Sheard et al. (2009) even though the two results wereindependently obtained by using completely different numericalalgorithms, revealing robustness of the Floquet stability analysis.Secondly, full 3D simulations were conducted for Re150, 200,250, and 300, with varying angle of incidence, and flow-inducedforce/moment coefficients and Strouhal number were presented.It was seen that the Re effects are insignificant except for meanlift coefficient. Flow-induced forces, moments, and Strouhalnumber of vortex shedding are sensitive to slight inclination ofthe square cylinder which causes a shift of separation point. Themagnitude of mean moment coefficient is maximum in the rangeof 101ryr151. Vortical structures in the wake, taken from theresults of the Floquet stability analysis and the full 3D simula-tions, respectively, were visualized by Q-contours. It should bealso noted that the predictions of the Floquet stability analysiswere confirmed by the full 3D simulations in the sense that thedominant mode for each parameter set (Re,y) was also found inthe corresponding 3D simulation. The current study is a first steptowards complete understanding of transition to turbulence inthe wake behind a square cylinder inclined with respect to themain stream.AcknowledgementsThis work was supported by Inha University Research Grant,and also by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry ofEducation, Science and Technology (2011-0004564).ReferencesBarkley, D., Henderson, R.D., 1996. Three-dimensional Floquet stability analysis ofthe wake of a circular cylinder. Journal of Fluid Mechanics. 322, 215241.Blackburn, H.M., Lopez, J.M., 2003. On three-dimensional quasiperiodic Floquetinstabilities of two-dimensional bluff body wakes. Physics of Fluids 15, L57.Blackburn, H.M., Sheard, G.J., 2010. On quasiperiodic and subharmonic Floquetwake instabilities. Physics of Fluids 22, 031701.Carmo, B.S., Sherwin, S.J., Bearman, P.W., Willden, R.H.J., 2008. Wake transition inthe flow around two circular cylinders in staggered arrangements. Journal ofFluid Mechanics 597, 129.Chen, J.M., Liu, C.-H., 1999. Vortex shedding and surface pressures on a squarecylinder at incidence to a uniform air stream. International Journal of Heat andFluid Flow 20, 592597.De, A.K., Dalal, A., 2006. Numerical simulation of unconfined flow past a triangularcylinder. International Journal for Numerical Methods in Fluids 52, 801821.Igarashi, T., 1984. Characteristics of the flow around a square prism. Bulletin ofJSME 27,
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