Numerical simulation of flow past circular duct

Transcription

1 Water Scence and Engneerng, 010, 3(): do:10.388/.ssn e-mal: Numercal smulaton of flo past crcular duct Ze-gao YIN* 1, Xan-e Cao 1, Hong-da SHI 1, Jan MA 1. Ocean Engneerng Key aboratory of Shandong Provnce, Ocean Unversty of Chna, Qngdao 66100, P. R. Chna. Cvl Engneerng epartment, Zheang Unversty, Hangzhou 31007, P. R. Chna Abstract: The Renormalzaton Group (RNG) k turbulence model and Volume of Flud (VOF) method ere employed to smulate the flo past a crcular duct n order to obtan and analyze hydraulc parameters. Accordng to varous upper and bottom gap ratos, the force on the duct as calculated. When the bottom gap rato s 0, the drag force coeffcent, lft force coeffcent, and composte force reach ther maxmum values, and the azmuth reaches ts mnmum. Wth an ncrease of the bottom gap rato from 0 to 1, the drag force coeffcent and composte force decrease sharply, and the lft force coeffcent does not decreases so much, but the azmuth ncreases dramatcally. Wth a contnuous ncrease of the bottom gap rato from 1 upard, the drag force coeffcent, lft force coeffcent, composte force, and azmuth vary lttle. Thus, the bottom gap rato s the key factor nfluencng the force on the crcular duct. When the bottom gap rato s less than 1, the upper gap rato has a remarkable nfluence on the force of the crcular duct. When the bottom gap rato s greater than 1, the varaton of the upper gap rato has lttle nfluence on the force of the crcular duct. Key ords: crcular duct; RNG k turbulence model; VOF method; numercal smulaton 1 Introducton Ppelnes are the man ay that ol and gas are transported through the sea. The exteror flo past horzontal crcular ducts s an mportant factor that many scholars and engneers are concerned about. In order to nvestgate flo near the crcular cylnder, physcal models have been ntensvely employed. Bearman and Zdravkovch (1978) used a nd tunnel test to nvestgate flo around a crcular cylnder hen the Reynolds numbers (Re) ere and , and the gap rato beteen the crcular cylnder and the plate all vared from 0 to 3.5. Then, the smoke-re method as used to express the flo vsualzaton and pressure dstrbuton, and t as found that the Strouhal number remaned constant hen the gap rato as greater than 0.3. On the bass of Bearman s research, Fredsøe and Hansen (1987) conducted flume experments to study flo separaton, drag force, and lft force of the cylnder. Accordng to the varable dameter cylnder, flo past the cylnder near the bottom boundary as nvestgated th the experments. Q et al. (006) analyzed the characterstcs of the Ths ork as supported by the Shandong Provnce Natural Scence Foundaton (Grant No. ZR009FQ003). *Correspondng author (e-mal: ynzegao@ouc.edu.cn) Receved Mar. 1, 010; accepted May 10, 010

2 vortex of flo past the cylnder on the bass of physcal model data hen the gap rato as 0.5. Wth the ncrease of computatonal poer, mathematcal models have been utlzed to smulate flo past the crcular duct. Wllamson (1996), Travn et al. (000), Kravchenko and Mon (000), and Sarpkaya (004) used the Reynolds-averaged Naver-Stokes equatons to study the flo near a crcular cylnder. Kalro and Tezduyar (1997), Breuer (000), Jordan (00), and Catalano et al. (003) used the large eddy smulaton method to study the flo around a cylnder. Ths research focused manly on the hydraulc characterstcs and the flud structure near the cylnder. As far as drect numercal smulaton for Reynolds numbers from 10 to 1000 s concerned, Henderson (1997) mproved the capabltes of research on flo transton from to dmensons to three dmensons. Mttal and Balachandar (1995) nvestgated the three-dmensonal effect on smulaton accuracy at Re = 55. Although these scholars have concentrated on flo past the cylnder, there have been fe studes on the flo force on crcular ducts th varous upper and bottom gap ratos. In ths study, the Renormalzaton Group (RNG) k turbulence model and Volume of Flud (VOF) method ere employed to calculate the hydraulc parameters around a crcular duct th varous upper and bottom gap ratos, and the forces on the crcular duct ere also computed and analyzed. Mathematcal model The crcular duct boundary causes the streamlne to curve ntensvely. Compared th the standard k model, the RNG k model more accurately smulates complex boundary flo. Therefore, the RNG k model as employed n ths study. The tensor form of the Naver-Stokes equatons and the RNG k equatons can be rtten as follos (u et al. 004; Baumal et al. 1998; Bazdd et al. 004): Contnuty equaton: u 0 (1) t x Momentum equaton: u uu u p u u pt, t F t x x x x x x k equaton: equaton: k k k u u u u t t t x x x x x x () (3) Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No.,

3 * c1 t u u u c u t t x x x k x x x k here s flud densty; s molecular vscosty; u s the velocty component n the (4) x ( = 1, ) coordnate drecton, denotng x and y coordnates, respectvely, hen s equal to 1 and ; p s pressure n the x drecton; k s turbulent knetc energy; s the turbulent dsspaton rate; F s the body force n the x drecton: F1 0 and F g; s the Kronecker delta; hen =, c k 1 * 0 1, and hen, 0 ; t ; c ; pt k ; c1 c1 ; c1 1.4 ; 1/ k E, E, ; E 1 u u ; x x ; 0.01 ; and c Numercal methods and boundary condtons 3.1 Numercal methods The VOF method as adopted to handle the free surface. Its concept can be expressed as follos: a ater volume fracton functon (WVFF),, s defned to present the relatve ater volume fracton n every computatonal grd cell. In each cell, the WVFF plus the volume fracton of ar, F a, s 1: Fa 1F (5) For a gven cell, three stuatons exst: F 1, hen the cell s completely flled th ater; F 0, hen the cell s completely flled th ar; and 0 <F 1, hch ndcates that the cell s flled th both ater and ar, and an nterface beteen the to exsts n the cell. Free surface flo conforms to the thrd stuaton. The control equaton of F can be expressed as follos: F F 0 (6) t x y F Fa a (7) F Fa a (8) here and a are the denstes of ater and ar, respectvely; and a are the molecular vscosty coeffcents of ater and ar, respectvely; and and can be obtaned by Eqs. (7) and (8) after F has been computed. After the governng equatons of the turbulence model consstng of the coupled RNG k model and the VOF method, ncludng Eqs. (1) through (4) and Eqs. (6) through (8), are solved, the hydraulc factors can be obtaned. In ths study, the mesh generaton as performed th GAMBIT..30. In the Fluent softare, a to-dmensonal, sngle-precson, mplct, segregated solver as selected. The PIC-VOF algorthm as used for the reconstructon of the nterface, and the sem-mplct method for a pressure-lnked equaton algorthm as adopted for the soluton of dscrete F 10 Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No., 08-16

4 equatons. The solutons ere consdered to have converged hen resduals ere less than The smulatons ere performed on a PC th a.8 GHz processor and one GB of RAM. 3. Boundary condtons The nlet boundary as dvded nto a ater velocty nlet and an ar pressure boundary. The ater velocty at the nlet boundary as defned as. The ar boundary as regarded as atmospherc pressure. The k and at the nlet boundary ere evaluated accordng to the follong formulas: k k 0.4 U 0 0 U 0 (9) (10) H here H 0 s the ater depth at the nlet boundary. The outlet boundary as regarded as the atmospherc pressure boundary. At the all, a no-slp boundary as specfed, and flud veloctes ere set to zero. The all functon as adopted to treat the vscous sub-layer. 4 Numercal results and analyss 4.1 Model valdaton In Fg. 1, e can see that s the dameter of the crcular duct; e s the dstance beteen the loer end of the crcular duct and the bottom plane; H s the ater depth; U s the mean flo velocty; Re U, th beng the knematc vscosty; E s defned as the bottom gap rato: E e ; and G s defned as the upper gap rato: G H e. In order to valdate the mathematcal model, a physcal model and a mathematcal model ere establshed here Re = , E = 1, G = 3, and = 0. m. Velocty measurement as conducted th an acoustc oppler velocmeter (AV), as n Fg.. Pressure as measured by pressure sensors. The computatonal area as parttoned by unstructured grds, hose cell sze as 0.1 m, and the total number of grd cells as ue to the hydraulc complexty near the crcular duct boundary, the area near the duct as parttoned th refned grds (Fg. 3). Then, the data of the physcal model ere compared th computed results. Fg. 1 Sketch map of computatonal case Fg. Velocty measurement sketch of physcal model th AV Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No.,

5 Fg. 3 Computatonal grds of case here E = 1 and G = 3 From Fg. 4, e can see that the measured free surface agrees th the computatonal results for the most part. From Table 1, e can see that all the relatve errors beteen physcal model veloctes (U p ) and computatonal veloctes (U c ) are less than 7%, and measured data approach computatonal results. From Table, e can see that all the relatve errors beteen physcal model pressures (p p ) and computatonal pressures (p c ) are less than 6%, and measured data also approach computatonal results. Thus, computatons of flo past the crcular duct th the RNG k turbulent model and the VOF method are vald and relable. Fg. 4 Comparson beteen computatonal free surface and measured free surface (E = 1, G = 3, x s the horzontal dstance from the duct center, and H s the ater depth) Table 1 Comparson beteen U p and U c NRP U c (m/s) U p (m/s) E U (%) NRP U c (m/s) U p (m/s) E U (%) Note: NRP denotes the number of representatve ponts n a typcal secton, at a dstance of four tmes the damater from the center of the crcular duct; E U = (U p U c )/U p 100%. Table Comparson beteen p p and computatonal p c NRP p c (Pa) p p (Pa) E p (%) NRP p c (Pa) p p (Pa) E p (%) Note: E P = (p p p c )/p p 100%. 1 Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No., 08-16

6 4. Computatonal veloctes and pressure When Re = , E vares among the values 0, 1, and ; G vares among the values 1,, 3, and 4; and e obtan 1 condtons th dfferent values of E and G (Table 3). The mathmatcal model as used to compute the hydraulc factors assocated th the 1 condtons. The results of the 7th condton, here E = 1 and G = 3, ere chosen to be analyzed. Table 3 Telve computatonal condtons G E st condton nd condton 3rd condton 4th condton 1 5th condton 6th condton 7th condton 8th condton 9th condton 10th condton 11th condton 1th condton From Fg. 5 e can see that the velocty drecton s sgnfcantly deflected near the front of the crcular duct boundary ong to the squeezng acton that the crcular duct boundary exerts on the flo, and the magntude of the velocty becomes larger. The velocty dstrbuton s symmetrcal n the ake area, and the magntudes of veloctes are less than those of other areas. From Fg. 6 e can see that the pressure along the duct boundary s unevenly dstrbuted. The pressure on the duct close to the nflo drecton and the bottom plane s the largest, and the pressure on the duct close to the free surface s the smallest. The pressure recovers faster than the velocty does. Pressure contours approxmately parallel each other n the secton that s about fve tmes the duct dameter aay from crcular duct, and the contour drecton s almost horzontal, shong that the nfluence of the duct on pressure can be gnored here. Fg. 5 Computatonal flo veloctes (E = 1, G = 3) Fg. 6 Computatonal pressure (Pa) (E = 1, G = 3) 4.3 Computatonal turbulent knetc energy and turbulence dsspaton rate It can be seen n Fg. 7 and Fg. 8 that the dstrbuton of turbulent knetc energy approxmately agrees th that of the turbulence dsspaton rate. The largest area les on a duct close to the nflo and the free surface, and the turbulent stress s the largest here. In the ake area, the turbulent ntensty s the loest. The turbulent knetc energy and the Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No.,

7 turbulence dsspaton rate are dstrbuted symmetrcally, and the horzontal lne across the duct center s the symmetry axs. Fg. 7 Turbulent knetc energy (m /s ) (E = 1, G = 3) Fg. 8 Turbulence dsspaton rate (m /s 3 ) (E = 1, G = 3) 4.4 Analyss of force on duct The force on the duct can be decomposed nto a horzontal drag force and a vertcal lft force. The drag force coeffcent ( C ), lft force coeffcent ( C ), composte force (F), and azmuth ( ) can be defned as follos: F x F y C, C, 1 1 U U F F x F, arctan F y y F x here F x and F y are the horzontal drag force and vertcal lft force, respectvely. In Fg. 9, Fg. 10, Fg. 11, and Fg. 1, e can see that, hen E = 0, C, C, and F are at ther largest values, but s at ts smallest. If e do not consder the duct dead eght and the nternal flo force, the crcular duct s most lkely to break hen E = 0. Wth an ncrease of E from 0 to 1, C and F decrease sharply, and C does not decreases so much, but ncreases dramatcally. Wth an ncrease of E from 1 to, C, C, F, and vary slghtly. Wth an ncrease of G from 1 to 4, C, C, and F decrease, but ncreases hen E = 0. More specfcally, th the ncrease of G from 1 to hen E = 0, C, C, and F decrease sharply, but ncreases quckly, shong that lft force contrbutes more and more to composte force. Wth the contnung ncrease of G from to 4 hen E = 0, C, C, and F decrease sloly. The value of, hoever, also ncreases sloly. Wth an ncrease of G from 1 to 4 hen E = 1 and E =, C, C, F, and vary slghtly. More specfcally, vares from 1.5 to 1.7, C vares from 60 to 61, F vares from N to 7 60 N, and vares from 88.3 to 88.6, shong that the varaton of G has lttle nfluence on C, C, F, and hen E = 1 and E =. C 14 Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No., 08-16

8 Fg. 9 Relaton curve of C and G Fg. 10 Relaton curve of C and G Fg. 11 Relaton curve of F and G Fg. 1 Relaton curve of and G 5 Conclusons (1) Usng Fluent softare, the RNG k turbulence model and the VOF method ere employed to smulate the flo past a crcular duct. The computatonal results regardng related hydraulc parameters, ncludng surface poston, veloctes, pressure, turbulent knetc energy, and the turbulence dsspaton rate, ere obtaned and analyzed. () The forces on the duct ere computed and analyzed th dfferent bottom gap ratos and upper gap ratos. When E = 0, the drag force coeffcent, lft force coeffcent, and composte force are at ther largest, but the azmuth s at ts smallest, compared to other condtons. When E = 0, the drag force coeffcent, lft force coeffcent, and composte force decrease, but the azmuth ncreases along th the upper gap rato. Wth an ncrease of the bottom gap rato from 0 to 1, the drag force coeffcent and the composte force decrease sharply, and the lft force coeffcent does not decrease so much, but the azmuth ncreases dramatcally. (3) The bottom gap rato s the most mportant factor nfluencng the force on the crcular duct. When the bottom gap rato s less than 1, the upper gap rato has a sgnfcant nfluence on the force of the crcular duct. When the bottom gap rato s greater than 1, the varaton of the Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No.,

9 upper gap rato has lttle nfluence on the force of the crcular duct. References Baumal, A. E., McPhee, J. J., Calama, P. H., and Mompean, G Numercal smulaton of a turbulent flo near a rght-angled corner usng the Spezale non-lnear model th RNG k equatons. Computers and Fluds, 7(7), [do: /s (98) ] Bazdd, T. F, Shahmr., A, and Haghparast, K. A Numercal analyss of a sngle ro of coolant ets nected nto a heated crossflo. Journal of Computatonal and Appled Mathematcs, 168(1-), [do: /.cam ] Bearman, P. W, and Zdravkovch, M. M Flo around a crcular cylnder near a plane boundary. Journal of Flud Mechancs, 89(1), [do: /s x] Breuer, M A challengng test case for large eddy smulaton: Hgh Reynolds number crcular cylnder flo. Internatonal Journal of Heat and Flud Flo, 1(5), Catalano, P., Wang, M., Iaccarno, G., and Mon, P Numercal smulaton of the flo around a crcular cylnder at hgh Reynolds numbers. Internatonal Journal of Heat and Flud Flo, 4(4), [do: /S014-77X(03) ] Fredsøe, J., and Hansen, E. A ft forces on ppelnes n steady flo. Journal of Wateray, Port, Coastal and Ocean Engneerng, 113(), [do: /(asce) x(1987)113:(139)] Henderson, R Nonlnear dynamcs and pattern formaton n turbulent ake transton. Journal of Flud Mechancs, 35(1), [do: /s ] Jordan, S. A. 00. Investgaton of the cylnder separated shear-layer physcs by large-eddy smulaton. Internatonal Journal of Heat and Flud Flo, 3(1), 1-1. Kalro, V., and Tezduyar, T Parallel 3 computaton of unsteady flos around crcular cylnders. Parallel Computng, 3(9), [do: /s (97) ] Kravchenko, A. G., and Mon, P Numercal studes of flo over a crcular cylnder at Re = Physcs of Fluds, 1(), [do: / ] u, P. Q., Qu, Q.., Wang, Z. G., and Zhang, H. M Numercal smulaton on hydrodynamc characterstcs of bfurcaton ppe th nternal crescent rb. Journal of Hydraulc Engneerng, 35(3), (n Chnese) Mttal, R., and Balachandar, S Effect of three-dmensonalty on the lft and drag of nomnally to-dmensonal cylnders. Physcs of Fluds, 7(8), Q, E. R., G. Y.,, W., Wu, J., and Zhang, X Study of vortex characterstcs of the flo around a horzontal crcular cylnder at varous gap-ratos n the cross-flo. Journal of Hydrodynamcs, Seres B, 18(3), Sarpkaya, T A crtcal reve of the ntrnsc nature of vortex-nduced vbratons. Journal of Fluds and Structures, 19(4), [do: /s (04) ] Travn, A., Shur, M., Strelets, M., and Spalart, P etached-eddy smulatons past a crcular cylnder. Flo, Turbulence and Combuston, 63(1-4), [do:10.103/a: ] Wllamson, C. H. K Vortex dynamcs n the cylnder ake. Annual Reve of Flud Mechancs, 8(1), [do: /annurev.fl ] 16 Ze-gao YIN et al. Water Scence and Engneerng, Jun. 010, Vol. 3, No., 08-16