Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 4, No. 3, August 2016 Numercal Analyss and Optmzaton on Vortex Shedder by the Lnearty of Strouhal Number aganst Reynold Number Su Myat Nyen and He Xu Abstract The desgn parameters and blocage rato of the bluff body called vortex shedder of the vortex flowmeter strongly nfluence on the flowmeter performance. The released frequency only depends on the sze of vortex shedder and flow rate of flud whle ndependent of the flud propertes. In the condton of constant Strouhal number, the flowmeter s not affected by the propertes such as temperature or pressure of measurng fluds. Thus, optmzaton a flowmeter desgn by consderng the lnearty of Strouhal number aganst the Reynold number becomes one of the deas of desgnng the flowmeter. In the present wor, the unsteady flow past a bluff body of damond-shape wth slt has been carred out to analyss on the Strouhal number usng FLUENT, a CFD code. The numercal analyss was conducted for non-crcular cross-secton ppe of 7.457mm hydraulc dameter n the Reynold number range of 2400 to 60000. The blocage rato from 0.16 (about 1/6) to 0.30 (about 1/3) were consdered and for the slt rato, 0.10-0.16 were used. We too only three angle, 45o, 60o, and 90o for ths wor as longer and more slender sheddng bodes result n weaer and less coherent sheddng. The optmzed desgn was acheved by the lnearty of Strouhal number aganst Reynold number. And, the lnearty s determned by the error percentage of Strouhal number. Accordng to the results, t has been found that the bluff body wth the parameters of 0.24 blocage rato, 0.14 slt rato and 60o apex angle s the best desgn. Index Terms Lnearty of Strouhal number aganst Reynold number, damond-shape wth slt, non-crcular cross-secton ppe, numercal analyss. I. INTRODUCTION Vortex flowmeter s one nd of flowmeter based on Karman vortex street theory. It uses a sensor to detect the frequency of the flud caused by flowng through the vortex shedder n the ppe and then measures the flow of flud. The frequency depends on the sze of vortex shedder and flow rate of flud, whle ndependent of the flud parameters, such as the temperature, pressure. It can be expressed by the followng formula: Manuscrpt receved March 23, 2015; revsed July 23, 2015. Ths wor was supported by the Natonal Natural Scence Foundaton of Chna under Grant 60775060, Specalzed Research Fund for the Doctoral Program of Hgher Educaton under Grant 200802171053, 20102304110006, 20122304110014, Natural Scence Foundaton of the Helongjang Provnce of Chna under Grant F200801 and Harbn Scence and Technology Innovaton Talents Specal Fund under Grant 2012RFXXG059. The authors are wth the College of Mechancal and Electrcal Engneerng, Harbn Engneerng Unversty, Harbn, 150001, P.R. Chna (e-mal: sumyatnyen.me@gmal.com, ralway_dragon@sohu.com). f St U / d (1) where, f s the frequency of Karman Vortex Street, St s Strouhal number, U s flud velocty, and d s wdth of vortex shedder. The bluff body called vortex shedder s a prmary element of the vortex flowmeter. The geometrcal desgn parameters and blocage rato of the vortex shedder strongly affect on the vortex flowmeter performance. In the frst desgn, crcular cylnder was used as the vortex shedder. However, t was found that the crcular cylnder cannot gve the fxed separaton pont n the range of Reynold number [1]. The lnearty of Strouhal number requres the locaton of the separaton pont to be fxed regardless of Reynolds number. Wth ths reason, most modern flowmeter have used the non-crcular cross-secton bluff bodes and sharp-edges bluff bodes n order to remove the shfts resultng from changes n the boundary layer. Attempts of many researchers to desgn the bluff body wth sharp edges can be seen n the prmary element secton of revews by [2], [3]. Although the vortex sheddng flowmeters are wdely used n many felds of ndustry, t can be seen that nether deal desgn of flowmeter nor unversal bluff body was found. The dea of desgnng a flowmeter based on Strouhal number aganst Reynolds number as constant was frst suggested by [4]. The constant Strouhal number means that the flowmeter s not affected by the propertes such as temperature or pressure of measurng fluds. Ref. [5] carred out an experment wth trapezodal-shape bluff body to optmze a flowmeter desgn by consderng the lnearty of Strouhal number aganst the Reynold number. The expermental study by [6] wth three dfferent shape (trapezodal cylnder, crcular wth slt and trangular-semcylnder consstng of a slt between these two parts) hghlghted that the two shedder wth a slt can sgnfcantly reduced the pressure loss. Moreover, these two shedders are superor n lnearty to the trapezodal cylnder. [7] nvestgated numercally on the effect of bluff body shape on vortex flowmeter performance. The vortex flowmeter requres a body shape wth sharp corners to generate stable vortex sheddng frequency. Thus, they selected a trangular shape of 60 o angle wth spltter plate or a damond shape as optmum desgns that generate stable vortex sheddng frequency wth low coeffcents of permanent pressure loss and drag. However, the Strouhal number dd not evaluated. In ths paper, the unsteady flow past a bluff body of damond shape wth slt have been analyzed usng FLUENT, a computatonal flud dynamcs (CFD) code. The wors DOI: 10.7763/IJMMM.2016.V4.257 204
Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 4, No. 3, August 2016 [5]-[7] force us to choose the damond shape wth slt (or two trangular shapes ther base edges are facng wth each other) as the nterest body shape for ths smulaton. The numercal analyss was conducted for non-crcular ppe of 7.457mm hydraulc dameter n the Reynold number range of 2400 to 60000. The optmzed desgn was acheved by the lnearty of Strouhal number aganst Reynold number. And, the lnearty s determned by the error percentage. II. NUMERICAL ANALYSIS METHODOLOGIES FLUENT s a computatonal dynamc code capable of modelng flud flow and heat transfer. The supportng numercal methods adopt the governng equatons for conservaton n ntegral form and use the control-volume-based technque. Ths technque conssts of dvson of the doman nto dscrete control volumes usng a computatonal grd and ntegraton the governng equatons on each control volumes [8]. A. Mean Flow Equatons The contnuty equaton and unsteady momentum equaton are: The turbulent vscosty s calculated as: 2 t c (7) The model constant are use, by default, as C 0.09, C 1 1.44, C 2 1.92, 1.0, 1.3 and the effectve vscosty C. Geometry of Analyss s obtaned by: eff eff (8) t Fg. 1 shows the geometry of nterested body taen for ths smulaton. A bluff body wth apex angle (), wdth (d) and slt (s) s placed at a dstant 2D from the nlet and the outlet s suffcent far downstream from the bluff body n order to avod the unnecessary effects of nlet and ext, where D s the ppe dameter. x 0 U (2) Inlet d Outlet D p U U U U j t x x x eff j j x j (3) s 2D 5D Fg. 1. Flow doman and Interested Bluff Body Shape (not to scale). B. Turbulent Model Used The standard turbulence model s used n ths study. The unsteady transport equatons for and can be wrtten as: U t P t x x x t U x t C 1 P C 2 x x where s turbulent netc energy (Nm), t s eddy/turbulent vscosty (Nsm-2), (4) (5) s turbulent Prandtl number for, s turbulent Prandtl number for, P s generaton term, s turbulent dsspaton rate (Nm2g-1), C 1 and C 2 are turbulence model constants. The generaton term P has the followng form: U U U j P t x j x j x (6) D. Meshng The preprocessor software pacage GAMBIT s employed n meshng. The total number of grd cells s about 150000 and was obtaned by usng Quad/Tr Pave meshng scheme. E. Boundary Condtons In the present case, velocty at the nlet and outflow were specfed as the boundary condtons. The velocty nlet profle was set to a constant value and the hydraulc dameter ( D h ) and turbulent ntensty ( I ) were used as turbulent quanttes. The turbulent ntensty ( I ) s calculated from the followng formula. -1/8 I 0.16(Re D ) (9) h No-slp boundary condtons are appled to the surface of bluff body as well as to the top and the bottom of the doman. F. Valdaton of the Code As the shedder desgn used n ths smulaton s new, there are no data n the lterature about the flow analyss wth exactly same bluff body desgn. Johnsson et al. [9] have done the numercal smulaton usng a turbulence model. They used the trangular cylnder wth 0.33 blocage rato as bluff body and found that the flow behnd trangular-shaped body s unsteady wth Strouhal number of 0.27 numercally 205
Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 4, No. 3, August 2016 (an expermental value of 0.25). In the present wor, the Strouhal number of dual trangular-shaped body wth blocage 0.3 at Re=50000 s 0.28 whch s reasonable agreement wth the Johnsson s data. III. SIMULATION RESULTS Numercal nvestgaton on unsteady flow past a bluff body of damond shape wth slt has been carred out to analyss on the Strouhal number. To accurately calculate the sheddng frequency, an average value of 10 sheddng cycles was taen. The tme dervatve scheme s the same n Johnsson s paper [9]. The frst order fully mplct and suffcent small tme steps (90 tme steps per one sheddng cycle) are used n order to mnmze the dscretzaton error. Ths study was made by changng the rato of vortex shedder dameter (d) to ppe dameter (D), the rato of slt (s) to shedder dameter (d) and the apex angel (). The lnearty of the Strouhal number s determned by the error percentage and the error ( e ) has a functon of 1 N St St e (10) N 1 St A. Analyss on Blocage Rato (d/d Rato) At frst, the blocage rato was analyzed whle the other parameters such as slt and apex angle are ept constant. The bluff bodes wth blocage rato of from 0.16 to 0.30 were consdered n the present wor. The Lnearty of Strouhal number (St) aganst Reynold number (Re) wth dfferent blocage ratos s shown n Fg. 2. Here, the Reynold number s based on hydraulc dameter of ppe. It was found that 0.25 s the most lnearty among these consdered ratos. Thus, ths blocage rato 0.25 was selected as optmum n general. Also t s reasonable wth the reported blocage ratos [10]. S t r o u h a l N u m b e r 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 d/d=0.30 d/d=0.25 d/d=0.20 d/d=0.16 2.4E+03 5.0E+03 1.0E+04 2.0E+04 3.0E+04 4.0E+04 5.0E+04 6.0E+04 Re y n o l d Nu m b e r Fg. 2. Lnearty of St aganst Re wth dfferent blocage rato (d/d=0.16 ~ d/d=0.30). TABLE I: MEAN STROUHAL NUMBER AND ERROR (%) FOR MORE PRÉCISED VALUES Blocage rato d / D Mean Strouhal number 0.23 0.2592 0.458* 0.24 0.2598 0.404* 0.25 0.2631 0.518 0.26 0.2640 0.474* 0.27 0.2674 1.075 * The selected ratos for further studyng on slt effect. St Error (%) e To get the more précsed rato, we tred to observe on four rato proxmty to 0.25 whch s the most lnear n the past secton. Table I presents the mean Strouhal number ( St ) and the error (%). Mean Strouhal number ( St ) s the average value of Strouhal numbers for dfferent Reynold numbers at the fx blocage rato and slt rato. It can be notced that the blocage 0.24 s mnmum n error percentage. The three blocages: 0.23, 0.24, and 0.26, that gve least error were selected for further studyng on the effect of slt to dameter rato (s/d). B. Analyss on Slt to Dameter Rato (s/d Rato) The bluff body used n ths case s slt-body shape. It s needed, therefore, to analyss on the effect of slt to Strouhal number. The slt ratos from 0.10 to 0.16 were used for the present wor. The d/d ratos that gve mnmum error by Table I have been analyzed wth dfferent slt rato. As shown n Fg. 3, the Strouhal number vares not only by the blocage but also by the slt. Although d/d=0.26 s least error for s/d=0.10, d/d=0.23 gves mnmum error at s/d=0.16. Overall, d/d=0.24 wth s/d=0.14 has the least n error among the tested ratos. On the other hand, the desgn wth blocage rato 0.24 and slt rato 0.14 has the hgher lnearty than other desgn ratos. So, we choose ths desgn as optmum by the lnearty of Strouhal number aganst Reynold number. e r r o r ( % ) 0.7 0.6 0.5 0.4 0.3 0.1 0.12 0.14 0.16 s/d d/d=0.23 d/d=0.24 d/d=0.26 Fg. 3. Varaton n error (%) at dfferent slt to shedder dameter rato (s/d) for blocages 0.23, 0.24, & 0.26. C. Analyss on Apex Angle ( ) It s remaned to analyze the effect of apex angle to Strouhal number. Apex angle means the angle facng wth ncomng flow or outgong flow. Whle the other ratos are ept constant, just changng n apex angle seems very senstve to vortex sheddng. The smaller the angle, the more slender the shedder body s. It s well now that longer and more slender sheddng bodes result n weaer and less coherent sheddng [10]. Therefore, we too only three angle, 45 o, 60 o, and 90 o for ths wor. Fg. 4. Vortcty contour for dfferent at Re=60000 (a) =45 o (b) =60 o (c) =90 o 206
Internatonal Journal of Materals, Mechancs and Manufacturng, Vol. 4, No. 3, August 2016 The smulaton results show that 45 o angle bluff body s poor n lnearty n Strouhal number wth an error of over 5%. For 90 o shedder, t can generate strong vortex sheddng wth good accuracy of 1.4266% error. However, ts percentage s stll hgh compared wth the 60 o shedder (0.3%) that also generates the strong vortex sheddng. The vortcty contour lnes at Re = 60000 for all cases are presented n Fg. 4 as a better representaton of the Von Kármán vortex street. By the smulaton results above, we chose the nterested bluff body wth the parameters of 0.24 blocage rato, 0.14 slt rato and 60 o apex angle as the best desgn by the lnearty n Strouhal number. IV. CONCLUSION The unsteady flow past a new vortex shedder desgn has been nvestgated numercally. The optmzaton was acheved by the lnearty of Strouhal number aganst Reynold number. The lnearty s determned by the error (%) calculated from equaton (10). To obtan the accurate sheddng frequency, we tae the average values from the 10 sheddng cycles. For one sheddng cycle, 90 tme steps are used so as to mnmze the tme dscretzaton error. In the present wor, the bluff bodes wth blocage rato (0.16-0.30) were consdered and the slt to shedder dameter rato (s/d) 0.10-0.16 were used. And, we too only three apex angle, (45 o, 60 o and 90 o ) to analyss as longer and more slender sheddng bodes result n weaer and less coherent sheddng. By the numercal results, t has found that the slt-damond shape wth blocage rato (d/d) 0.24, slt to shedder dameter rato (s/d) 0.14 and apex angle () 60 o s the best desgn by the pont of lnearty n Strouhal number aganst Reynold number. Furthermore, ths bluff body desgn can gve the followng advantages: The desgn s smple. So, t s easy to manufacture. As there are no curve surfaces, manufacturng errors can be reduced. As t s a sharp-edge vortex shedder, t wll be sure for the fxed separaton pont rrespectve of Reynold number range. It can be used n the flow rate as low as Reynold number of 2400 wth well accuracy. REFERENCES [1] E. Achenbach, "Dstrbuton of local pressure and sn frcton around a crcular cylnder n cross-flow up to Re = 5X106," Journal of Flud Mechancs, vol. 34, no. 625-639, 1968. [2] G. L. Panann, "The vortex flowmeter: Varous methods of nvestgatng phenomena," Measurement Scence and Technology, vol. 16, pp. R1-R16, 2005. [3] A. Venugopal and S. V. Prabhu, "Revew on vortex flowmeter Desgner perspectve," Sensors and Actuators, vol. 170, pp. 8-23, 2011. [4] A. Rosho, "Experments on the flow past a crcular cylnder at very hgh Reynolds number," Journal of Flud Mechancs, vol. 10, pp. 345-356, 1961. [5] M. T. Y. Terao, "Standardzaton of vortex sheddng flowmeter," presented at 4th Internatonal Symposum on Flud Flow Measurement, Denver, Colorado USA, June 27-30, 1999 [6] T. Igarash, "Flow resstance and strouhal number of a vortex shedder n a crcular ppe," JSME Internatonal Journal, vol. 42, no. 4, 1999. [7] B. K. Gandh, S. N. S. V. Seshadr, and J. Sngh, "Effect of bluff body shape on vortex flowmeter performance," Indan Journal of Engneerng & Materals Scences, vol. 11, pp. 378-384, 2004. [8] FLUENT 6.3 User's Gude. Fluent Inc., 2006. [9] S. H. Johnsson et al., "Numercal smulaton of vortes sheddng past trangular cylnders at hgh Reynold numbers usng a -e turbulence model," Internatonal Journal for Numercal Methods n Fluds, vol. 16, pp. 859-878, 1993. [10] R. C. Baer, "Flow measurement handboo: Industral desgn, operaton prncples," Performance, and Applcatons, Cambrdge Unversty Press, 2000. Su Myat Nyen was born n 1984, n Magway regon, Republc of the Unon of Myanmar. She has been graduated wth the bachelor of engneerng (mechancal) degree n 2006 and the master of engneeng (mechancal) degree n 2009 from Technologcal Unversty (Paou), Republc of the Unon of Myanmar. At present, she s pursung her Ph.D degree at College of Mechancal and Electrcal Engneerng, Harbn Engneerng Unversty, Chna. Her current research nterests are parametrc optmzaton and flud flow analyss. He Xu was born n 1964, Helongjang provnce, Chna. He receved hs BS n flud transmsson and pressure control from Zhejang Unversty, Hangzhou, Chna n 1987, and the MSc and PhD degrees n mechatronc control and automaton from Harbn Insttute of Technology, Harbn, Chna n 1990 and 2006, respectvely. He has partcpated n some mass transt projects at home and abroad. He s now a professor n Harbn Engneerng Unversty. Hs research nterests nclude specal mcro robots n extreme envronment, analyss of mult-physcs couplng n water hydraulc system, mechatronc and vson systems. He has publshed more than 30 refereed papers n techncal journals and conference proceedngs. 207
Industral Engneerng and Optmzaton Management