THE NEAR-WALL INFLUENCE ON THE FLOW AROUND A SINGLE SQUARE CYLINDER. Campregher, Rubens Faculty of Mechancal Engneerng, FEMEC Federal Unversty of Uberlânda, UFU 38400-902 Uberlânda - Brazl campregher@mecanca.ufu.br Slvera Neto, Arsteu Federal Unversty of Uberlânda, UFU arsteus@mecanca.ufu.br Mansur, Sérgo S. Mechancal Engneerng Department UNESP Câmpus Ilha Soltera 15385-000 Ilha Soltera - SP - Brazl mansur@dem.fes.unesp.br Abstract. The flow around a sold body s very nfluenced by a wall placed at close dstance, due to the strong nteracton between the boundary layer on the wall and the vortex sheddng behnd the cylnder. The ntensty of ths teraton depends on the dsplacement between the wall and the body, characterzed by the crtcal gap, where the vortex sheddng loses ts perodcty. In order to smulate ths problem, a numercal code was developed n FORTRAN employng the Fnte Volume technque, and the Large Eddy Smulaton (LES) for turbulence modelng. A set of dfferent dsplacements and Reynolds numbers were performed, amng to better understand the phenomena nvolved Keywords. square cylnder, numercal smulaton, near-wall, vortex sheddng, LES. 1. Introducton The capablty of mantanng a constant vortex sheddng frequency s a very mportant characterstc for many applcatons, such as vortex sheddng flowmeters and electronc devces coolng systems. Furthermore, the proxmty of walls and other obstacles, the flow free stream turbulence level, the flow regme and the velocty dstrbuton could have strong nfluence on the vortex sheddng frequency. In ths work, numercal smulatons of a square cylnder close to a wall, and mmersed n a 2D, ncompressble, turbulent flow, were performed n order to reproduce the crtcal gap encountered n the expermental tests. For a square cylnder, havng the sde dmenson B and at dstance y from a plane wall, the crtcal gap s the rato y/b where there s no regular vortex sheddng. However, ths crtcal gap may vary accordngly to the shape of the body and ts dstance from the boundary layer. In cases where the boundary layer thckness s large enough, or the dstance from the body to the plane wall s too close, the vortex street may come n contact wth t, suppressng the vortex sheddng regular pattern the Von Karman street. It s expected that the break-up n the regular vortex sheddng frequency would be due the nteracton between the separated shear layer, behnd the cylnder, and the boundary layer developed on the plane wall. The hgh turbulence ntensty presented n the boundary layer s nected nto the recrculaton zone, avodng the nverson n the velocty profle as an effect of the shear layer. In order to better reproduce the expermental results found n lterature (Durão et al., 1991), several numercal smulatons were performed for dfferent Reynolds number, rangng from 2000 to 18000, and three y/b ratos: 0.25, 0.35, and 0.55 were tested. Due to the hgh Re regme, n some cases a Large Eddy Smulaton turbulence methodology were employed. For ths flow regme and square cylnders, the crtcal gap was found to be around y/b < 0.35 (Durão et al., 1991). The Reynolds number s gven by: ρ U = B Re (1) µ It s worth notng that no free stream turbulence ntensty was ntroduced n the numercal smulaton, as well as no wall frcton effect was modeled, such as presented n the expermental tests (Durão et al., 1991). In that expermental work, a hgh free stream turbulence ntensty - about 6% - was found due to the lack of screens and honeycombs, and the skn frcton was evaluated to be C f = 0.0046. Furthermore, n both expermental and numercal tests, no blockage correcton had been done.
2. Numercal method An unsteady, ncompressble, and sothermal flow of a Newtonan flud can be modeled by the Naver-Stokes equatons assocated wth the contnuty equaton. These equatons are stated, respectvely, as follows: t ( u ) + ( u u ) = + ν + + Sc 1 p ρ u x u, (2) ( u ) = 0. (3) The numercal smulatons were performed over a non-unform regular orthogonal mesh and an example for an y/b = 0.35 confguraton s depcted n Fg. (1). Due to the hgh Reynolds numbers, a hghly densty of cells around the cylnder lne was needed n order to capture the smaller scales vortexes. The mesh generaton had been done usng the pre-processng routnes of the smulaton software. The Large Eddy Smulaton methodology was used wth the Smagornsky sub-grd model (Smagornsky, 1963). The advectve terms were nterpolated by the Consstent QUICK scheme (Hayase et al., 1992) and the pressurevelocty couplng algorthm were solved by the SIMPLEC scheme (Patankar, 1980) n fully mplct frst order tme dscretzaton. In order to solve the lnear system generated by the Eq. (2) and Eq. (3) dscretzaton, a TDMA algorthm was employed. The doman dmenson has been changed for each confguraton. In the Fg. (2), the dmensons for y/b = 0.35 confguraton s depcted as well as the boundary condtons employed for all confguratons. Fgure 1. Numercal mesh (228x135 cells) for y/b = 0.35 confguraton. 6.45B [70] u = v = 0 u v = = 0 v = 0 u = U ** y B u = v = 0 5B [60] * 15B [120] (**) 1.35B [65] (*) 1B [48] Fgure 2. Doman dmensons and boundary condtons. Numercal probes were employed n order to capture the velocty oscllatons caused by the vortex emssons. The power spectrum analyss of ths sgnals helps to evaluate the regular vortex sheddng frequency break-up n the crtcal gap. A more detaled descrpton about the numercal smulaton software can be found n Campregher (2002).
As stated above, the man goal was to reproduce the expermental data, gven by Durão et al. (1991). The dmensons used n ths present work were the same as that employed by Durão et al. (1991),.e., the numercal square cylnder had B = 20mm, and the doman was 420 x 156 mm. 3. Results and dscusson The tests were performed for three dfferent gaps dstance y, see Fgure (2 and, for each gap, several Reynolds numbers and Smagornsky constant (Cs). The Smagornsky constant was adusted to the numercal code. The sovortcty contours and ther related spectral densty power for Re = 8500 are presented n Fg (3). For the frst geometrc confguraton, y/b = 0.55, the power spectrum shows a well-defned peak, representng the man vortex sheddng frequency. Furthermore, there s no secondary peak comng up n any other frequency. Ths scenaro s very nterestng and shows that the wake behnd the cylnder had suffered no nterference from the wall. Hence, some authors had reported that the crtcal gap ncreases as the boundary layer heght ncreases. In the sovortcty contours lnes, one can see several well-defned vortces shed behnd the cylnders. The accelerated flow below the cylnder pushes down the boundary layer, decreasng ts heght n that regon. The second confguraton smulated, y/b = 0.35, had presented the nteracton between the boundary layer and the vortex street behnd the cylnder. As expected, ths contact had produced some nterference n the vortex sheddng pattern, suppressng the fundamental frequency. At the same tme, more turbulence s nected nto the shear layer. As the gap becomes smaller, the boundary layer faces some resstance n flowng throughout t. In the sovortcty contours, n the Fg (3) rght column, t s possble to see the stagnaton regon before the cylnder, as a consequence of the boundary layer detachment. In the thrd confguraton, y/b = 0.25, one can see that the stagnaton pont n front of the cylnder has ncreased ts heght. Ths was expected due to the narrow gap between the cylnder and the wall. One can see that upwnd the cylnder, the boundary layer had detached. 0,08 B/y = 0.55 Spectral Densty Power 0,06 0,04 0,02 0,00 0 2 4 6 8 10 0,020 B/y = 0.35 Spectral Densty Power 0,015 0,010 0,005 0,000 0 5 10 0,020 B/y = 0.25 Spectral Densty Power 0,015 0,010 0,005 0,000 0 2 4 6 8 10 Fgure 3. Spectral densty power (left) and vortcty contours for the three confguratons tested. The Fgure (4) depcts the flow behavor around the cylnder n more detals for the three confguratons and the early detachment of the boundary layer. One can see that the velocty vectors have devated almost symmetrcally around the cylnder front face n the y/b = 0.55 confguraton. Ths characterstc tends to make the shear layer n both
upper and lower sdes of the cylnder to present the same pattern as the encountered n the cylnder mmersed n a wall free stream. In the other confguratons, the pctures renforce the presence of a stagnaton regon n front of the obstacle and the lower face vortex break-up due to the flud necton from the boundary layer. Y/B = 0.25 Y/B = 0.35 Y/B = 0.55 Fgure 4. Velocty vectors for the dfferent gap confguratons. The frequency results for dfferent confguratons are compared aganst the expermental data from Durão et al. (1991) and depcted n the Fgure (5). One can see n the Fgure that the frequency and, as a consequence, the Strouhal number, behave n a smlar manner for both expermental and numercal tests. In both works, the frequency value ncreases n a lnear behavor as a functon of Reynolds number, although the error stays around 20%. Furthermore, t s possble to realze that, the Smagornsky constant (Cs) of 0.10 has presented almost the same error as for Cs = 0.15. It s mportant to menton that the calculaton s two-dmenson and the measure was performed n a three-dmensonal flow. Fnally, t s worth notng that the results for the crtcal gap, n ths work, have presented a good agreement wth the expermental results.
6 5 4 3 2 1 Y/B: 0.35 (Durão et al. - 1991) 0.55 (Durão et al. - 1991) 0.35 (Ths work wthout LES) 0.35 (Ths work Cs = 0.10) 0.35 (Ths work Cs = 0.15) 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Reynolds (Re) Fgure 5. Vortex sheddng frequency versus Reynolds number for numercal and expermental tests. 4. Conclusons The farly large dfferences, especally for the man vortex sheddng frequency, obtaned between the two approaches, numercal and expermental, could be related to several characterstcs of the numercal smulaton. Among them could be cted the b-dmensonal doman, whch s well known as napt of gettng the cross-flow nfluence behnd the cylnder. The cross-flow s a great responsble for the onset of the transton effects. It worth notng that no wall treatment was employed n the smulaton. The wall treatment could nfluence on the boundary layer heght, nfluencng drectly on the crtcal gap value and on the vortex sheddng frequency. Accordngly to the results presented above, the value of the former seemed not be nfluenced at all there was great concordance wth the expermental data. Despte the dfferences presented n ths prelmnary analyss, the results are promsng, that encourages performng addtonal tests n order to explore the phenomena n more detals. 5. References Campregher, R., 2002, Smulação Numérca de Escoamentos Transconas e Turbulentos ao Redor de Geometras Cartesanas, MSc. Thess, UNESP - Ilha Soltera, SP, Brazl. Durão, D.F.G., Gouvea, P.S.T., Perera, J.C.F., 1991, Velocty Characterstcs of the Flow Around a Square Cross Secton Cylnder Placed Near a Channel Wall, Experments n Fluds, Vol. 11, pp. 341-350. Hayase, T., Humphrey, J.A.C., Gref, R.,1992, A Consstently Formulated QUICK Scheme for Fast and Stable Convergence Usng Fnte-Volume Iteratve Calculaton Procedures, Journal of Computatonal Physcs, vol. 98, p.018-118. Patankar, S.V., 1980, Numercal Heat Transfer and Flud Flow, Hemsphere. Smagornsky, J.,1963, General Crculaton Experments Wth the Prmtve Equatons I. The Basc Experment, Mon. Weather Rev., Vol. 91, pp. 99-164. 6. Copyrght Notce The author s the only responsble for the prnted materal ncluded n hs paper.