89 11,3(1):89-94 DOI: 1.116/S11-658(1)69-3 THE CALCULATION OF THE PROFILE-LINEAR AVERAGE VELOCITY IN THE TRANSITION REGION FOR ULTRASONIC HEAT METER BASED ON THE METHOD OF LES * LIU Yong-hu, DU Guang-sheng, TAO L-l, SHEN Fang School of Energy and Power Engneerng, Shandong Unversty, Jnan 561, Chna, E-mal: lu@mal.sdu.edu.cn (Receved October 31, 1, Revsed December 1, 1) Abstract: The measurement accuracy of an ultrasonc heat meter depends on the relatonshp of the profle-lnear average velocty. There are varous methods for the calculaton of the lamnar and turbulence flow regons, but few methods for the transton regon. At present, the tradtonal method to deal wth the transton regon s to adopt the relatonshp for the turbulent flow regon. In ths artcle, a smplfed model of the ppe s used to study the characterstcs of the transton flow wth specfc Reynolds number. The k model and the Large Eddy Smulaton (LES) model are, respectvely, used to calculate the flow feld of the transton regon, and a comparson wth the experment results shows that the LES model s more effectve than the k model, t s also shown that there wll be a large error f the relatonshp based on the turbulence flow s used to calculate the profle-lnear average velocty relatonshp of the transton flow. The profle-lnear average velocty for the Reynolds number rangng from 5 3 to 1 are calculated, and the relatonshp curve s obtaned. The results of ths artcle can be used to mprove the measurement accuracy of ultrasonc heat meter and provde a theoretcal bass for the research of the whole transton flow. Key words: ultrasonc heat meter, Large Eddy Smulaton (LES) model, profle average velocty, lnear average velocty, ppe Introducton At present, the technology level of the homemade ultrasonc heat meter s not hgh and the flux measurement accuracy can not meet varous requrements. The measurement accuracy s manly depend on two factors : (1) the sgnal collecton by the electronc component, () the relatonshp of profle-lnear velocty of flow. Its flow measurement prncple s shown n Fg.1. The ultrasonc Transducer A (TRA) and Transducer B (TRB) not only can emt the ultrasonc mpulse (wth an ncdent angle θ ), but also can receve the ultrasonc sgnal. Because the ultrasonc velocty n the downstream and upstream s dfferent, the tme for the ultrasonc sgnal to reach the two ultrasonc transducers s dfferent, wth a tme dfference Δ t. The lnear average velocty v t s obtaned by capturng ths tme dfference Δtc vl = D tan ( ) 1 θ where D s the dameter of the ppe. (1) * Project supported by the Natonal Natural Scence Foundaton of Chna (Grant Nos. 19713, 184), the Natural Scence Foundaton of Shandong Provnce (Grant No. Y7A4). Bography: LIU Yong-hu (1981-), Male, Ph. D. Canddate Correspondng author: DU Guang-sheng, E-mal: du@sdu.edu.cn Fg.1 Flow measurement prncple of the ultrasonc heat meter
9 The volume flow Q s D kv Q = π l () 4 where k s defned as the flow coeffcent, whch s the rato of the profle average velocty v s to the lnear average velocty v t. k v = s vl (3) In order to nsure the measurement accuracy, the lnear average velocty v l measured by the ultrasonc flowmeter must be converted to the profle average velocty v s. The ultrasonc flowmeter s a much studed topc. Iooss et al. [1], Rašuts [], Wllatzen and Kamath [3] and Ashrafan et al. [4] studed the nfluences of structure, roughness, velocty-profle, temperature on the fully-developed turbulent flow, Inoue et al. [5], Takeda [6] developed a system of ultrasonc flow measurement by usng the Ultrasonc Velocty Profle (UVP) method and studed the nfluence of the ncdence angle on the measurement precson, Wu and Zhang [7], Feng et al. [8] and L et al. [9] also studed the turbulent flow n ppes by usng the DNS method, and measured the flow feld by the Dgtal Partcle Image Velocmetry (DPIV) method. The flow coeffcent k s documented n the Industral Automaton Instrumentaton Manual [1] : When the flow s a lamnar flow 3 k = (4) 4 When the flow s a turbulent flow.3 ( R ) 1 k = 1+.1 6.5 + 431 e (5) These researches are manly focused on the flow measurement of the lamnar and turbulent flows, and not so much on the flow characterstcs n the transton regon. The tradtonal method s to regard the transton flow as the turbulent flow to obtan the relatonshp of the profle-lnear average velocty. Ths smplfed method must lead to measurement errors when the water flow n the ppe s a transton flow. There are many condtons that would lead to the transton flows wth supplyng heat. For example, to satsfy the energy effcency requrement of heat supplyng system for new buldngs, the flux for 1 m s less-than.5 m 3 /h, and the Re = 7 5, whch belongs to the transton regon. In ths artcle, the characterstcs of the transton flow wth the Re = 5 3 n ppes are studed, and the results are compared wth the result of experments to valdate the calculaton. The relatonshp between the profle-lnear average velocty and Reynolds number n the range from 5 3 to 1 are also obtaned va the numercal smulaton, whch can provde a theoretcal bass for the research on the whole transton flow. 1. Physcal model In order to compare our results wth the expermental data of Westerweel [11], the ppe s dameter s.4 m and the length s.5 m n ths artcle. The calculaton s stablzaton s determned by the sze of the smallest mesh n the flow. Accordng to Ref.[1], the smallest mesh scale l needs to satsfy the vscous length scale requrement l =5y (6) where + + y s the vscous length scale, + y = v u (7) u u s the frcton velocty. = u f and f s the fannng frcton factor, (8) 1/4 f =.79Re (9) So, when Re = 5 3, f =.96, u = 9. 3 1 m / s and 4 =1.1 1 m, we have + 4 l =5 y =5.5 1 m. Fg. Cross secton mesh
91 Fgure shows the cross secton mesh. In ths secton, the sze functon method s used to generate the mesh, whch satsfes the requrement of the smallest structures mesh. Cooper method s used for the flow drecton. The number of mesh s about 1 6.. The calculaton model and the valdaton of results.1 Basc governng equatons [13-19] Contnuty equaton: ρ ρ ( u ) + = t x Momentum equaton: ( ρu ) ( ρuu ) (1) p u + = + μ ρuu t xj x x j x j (11) Vscosty coeffcent of turbulence: ρck μ μt = (1) where ρ s the densty of water, u s the velocty of flow, the superscrpt expresses the quantty of average over tme, p s the pressure and μ s the vscosty coeffcent.. The k model k equaton: ( ρk ) ( ρku ) μ t k + = μ + + t x xj σ k xj equaton: ( ) G + G ρ Y + S (13) k b M ρ μ + ( ρu ) = + t μ + t x xj σ xj C1 ( Gk + C3 Gb) C S k k + ρ G k (14) where s the turbulent knetc energy produced by the gradent of the average velocty, s the G b turbulence knetc energy produced by the flotage, Y M represents the pantng acton caused by the dffusvty of the compressble turbulent flow, C 1, C and C 3 are constant coeffcents, σ k and σ are the turbulent Prandtl numbers related wth k and coeffcents, Sk and S are the user-defned source terms. The turbulent knetc energy G b and related coeffcents C3 produced by the flotage are consdered for the compressble flow. When the flud s ncompressble, the user-defned source terms are not consdered, and Gb, Y M, Sk and S are equal to zero. The values of constant coeffcents C1, C, C μ and σ k, σ are: C1 =1.44, C =1.9, C μ =.9, σ k =1., σ =1.3. The boundary condtons: The velocty-nlet s used for the nlet condton. When Re = 5 3, v =.133m/s, the outlet condton s the free outflow. The accuracy of calculaton: The resdual error s less than 1 3..3 The LES model The equaton of the LES model [-] s u u u j 1 p u τ j + = + ν + t x ρ x x x xj j j (15) where τ j = uu uu s defned as the subgrd stress, whch concerns the momentum transport between the fltrated small scale turbulence and the part of the scale whch can be calculated. The subgrd stress s a term not closed n the equaton. So, a certan model s used to deal wth the subgrd stress. In ths artcle, the Smagornsky model s used 1 τ j = μt S j + τkkδ j (16) 3 where μ t s the subgrd turbulent vscous stress, s the transform rato of the tensor S j 1 u u = + x j x S j (17) Ths model s the bass of the subgrd model, whch s advanced by the Samagorn, and the model equaton s μ = ρ L S (18) t s
9 where L s s the mxng length of the mesh, and S = Sj Sj C s, C s =.1. The boundary condtons: The LES model s an unsteady three-dmensonal model, and t requres the nlet nstantaneous velocty profle. In order to obtan a stablzed result as soon as possble, the outlet profle velocty of the k model s used for the nlet velocty of the LES model [18], and the nlet turbulence s provded by the method of Intensty and Hydraulc Dameter. Tme step: The tme step s determned va test calculatons, whch s.5 s. The accuracy of calculaton s the same as the k model..4 Comparson of the results between calculaton and experment The mean streamwse velocty profle obtaned by calculaton and experment wth Re = 5 3 s shown n Fg.3. Y-coordnate s the velocty, X-coordnate s the poston of the measurement ponts n the cross secton and D s the dameter. 3. The profle-lnear average velocty of ppes obtaned wth the LES model 3.1 The method of calculaton of the lnear average velocty The ncdence angle s o[5], as shown n Fg.1. The profle average velocty v s s constant because of the constant flux. When the Re = 5 3, v s =.133m/s. In order to have the statstcal sgnfcance of the results, the tme-average method s used for the lnear average velocty. Fgure 4 shows the lnear average velocty aganst tme. Y-coordnate s the lnear average velocty, and X-coordnate s the tme. Fg.3 Mean streamwse velocty profle obtaned by calculaton and experment wth Re = 5 3 In Fg.3, the results of the k model are compared wth the expermental data by Westerweel [11] va DPIV, whch shows that the velocty profle obtaned by the k model devates from the experment results. The k model s a fully-developed turbulent model, and s not effectve to be used to obtan the profle-lnear average velocty of the transton flow. It s shown that the results obtaned wth the LES model are bascally the same as Westerweel s, whch verfes that the LES model s more effectve to be used to smulate the characterstcs of the transton flow wth Re = 5 3. The mass, momentum and energy are mostly transported by the movng of large eddes, and ths part can be determned by solvng the N-S equaton n the LES model, so the calculaton precson s mproved. When the Reynolds number s larger than 5 3, the LES model s also effectve because of the growng of the turbulent knetc energy. So, n ths artcle the results of the LES model are used to study the profle-lnear average velocty of ppes. Fg.4 The lnear average velocty vs. tme wth Re = 5 3 As s shown, the lnear average velocty fluctuates wth tme, but the tme-average value s constant. The lnear average velocty vs. tme fluctuates round a constant. Ths constant s the tme-average value of the lnear average velocty, and t s.159 m/s. Therefore, the lnear average velocty of ppes v l s ths tme-average value, v l =.159 m / s. 3. A comparson wth the results of the tradtonal coeffcent based on the turbulent flow When Re = 5 3, k can be calculated usng Eq.(5) provded by Ref.[1].3 ( ) 1 Re k = 1+.1 6.5 + 431 =.95 The flow coeffcent model n ths artcle s vs k = =.884 v l The dfference ( ) k obtaned by the LES Δ = k k / k = 4.38%. It shows that the profle-lnear average velocty of the transton flow s dfferent from that of the turbulent flow. Accordng to the ndustry standard of heat meter<cj18-7>, the flow measurement error should not be more than 3%. The error of the method adoptng the relatonshp of the turbulent flow to calculate the transton regon exceeds ths value.
93 Therefore, t s necessary to determne the proflelnear velocty n the transton flow regon. 3.3 The relatonshp of profle-lnear average velocty n transton regon The flow felds wth Re = 6, 7, 8, 9, 1 are calculated by the method of the LES model, and the lnear average veloctes are shown n Fg.5-Fg.9. numbers, and the lnear average velocty s denoted as lnear-v, and the profle average velocty as profle-v, the relatonshp of profle-lnear velocty calculated by the tradton formula (Eq.(5)) s denoted as k, and the relatonshp calculated n ths paper s denoted as k l. The results are shown n Table 1. Fg.9 The lnear average veloctes vs. tme wth Re =1 Fg.5 The lnear average veloctes vs. tme wth Re =6 Fg.6 The lnear average veloctes vs. tme wth Re = 7 Table 1 compared the results of tradton method wth calculaton Re lnear-v profle-v k k l Error 5 3.159.133.95.884 4.38% 6.178.151.96.885 4.45% 7.1994.176.97.884 4.64% 8.75.1.98.885 4.57% 9.566.7.98.883 4.9% 1.835.5.99.888 4.43% The curve of profle-lnear average velocty vs. Reynolds number rangng from 5 3 to 1 s shown n Fg.1. The curve marked as Eq.(5) s the result obtaned by the tradtonal method of dealng wth the transton regon, and the curve marked as Calculaton s the result of ths artcle. Fg.7 The lnear average veloctes vs. tme wth Re = 8 Fg.1 The curve of profle-lnear average velocty vs. Reynolds number rangng from 5 3 to 1 Fg.8 The lnear average veloctes vs. tme wth Re = 9 The tme-average method s used to calculate the lnear average velocty wth dfferent Reynolds It s shown that the dfference between the results obtaned n ths artcle and by the tradtonal formula s 4.38% - 4.9%, whch s more than what s requred by the standard of heat meter. Therefore, the relatonshp of the turbulent flow regon can not be used to calculate the profle-lnear average velocty n the transton regon, and t s necessary to obtan the relatonshp n the transton regon. But the calculaton of ths artcle s not enough to obtan the
94 relatons n the whole transton regon, further work s necessary. 4. Concluson It can be concluded that the LES model s more effectve than the k model to calculate the flow characterstcs of ppes to have a good agreement wth the expermental data of Westerweel when Reynolds number s 5 3, and the profle-lnear average velocty can be obtaned by the LES model. When the Reynolds number s larger than 5 3, the LES model s also effectve because of the growng of the turbulent knetc energy. The relatonshp of profle-lnear average velocty n the transton regon s studed by the LES model, and compared wth the results by the tradtonal method of dealng wth the transton regon, wth 4.38% - 4.9% of dfference, whch s more than what s requred by the standard of heat meter <CJ18-7>; the curve of relatonshp of profle-lnear average velocty wth Reynolds number n the range of 5 3-1 s obtaned. For the next step, one has to confrm the effectveness of the LES model for the flow wth Re < 5 3, so the DNS method or the expermental method wll be used for ths flow regon to obtan the relatonshp of the profle-lnear average velocty. References [1] IOOSS B., LHUILLIER C. and JEANNEAU H. Numercal smulaton of transt-tme ultrasonc flowmeters: Uncertantes due to flow profle and flud turbulence[j]. Ultrasoncs,, 4(9): 19-115. [] RAIŠUTIS R. Investgaton of the flow velocty profle n a meterng secton of an nvasve ultrasonc flow meter[j]. Flow Measurement and Instrumentaton, 6, 17(4): 1-6. [3] WILLATZEN M., KAMATH H. Nonlneartes n ultrasonc flow measurement[j], Flow Measurement and Instrumentaton, 8, 19(): 79-84. [4] ASHRAFIAN A., ANDERSSON H. and MANHART M. DNS of turbulent flow n a rod-roughened channel[j]. Internatonal Journal of Heat and Flud Flow, 4, 5(3): 373-383. [5] INOUE Y., KIKURA H. and MURAKAWA H. et al. A study of ultrasonc propagaton for ultrasonc flow rate measurement[j]. Flow Measurement and Instrumentaton, 8, 19(3-4): 3-3. [6] TAKEDA Y. Velocty profle measurement by ultrasonc Doppler method[j]. Expermental Thermal Flud Scence, 1995, 1(4): 444-453. [7] WU Mn-we, ZHANG Zhao-shun. Numercal research on the structures n turbulent ppe flow[j]. Journal of Hydrodynamcs. Ser. A,, 17(3): 334-34(n Chnese). [8] FENG Bn-chun, CHUI Gu-xang and ZHANG Zhao-shun. Expermental study for fully developed turbulent ppe flow[j]. Acta Mechanca snca,, 34(): 156-167(n Chnese). [9] LI Bu-yang, LIU Nan-sheng and LU X-yun. Drect numercal smulaton of turbulent heat transfer n a wall-normal rotatong channel flow[j]. Journal of Hydrodynamcs, Ser. B, 6,18(1): 6-31. [1] INDUSTRIAL AUTOMATION INSTRUMENTATION MANUAL EDITOR. Industral automaton nstrumentaton manual[m]. Bejng: Mechancal Industry Press, 1988(n Chnese). [11] WESTERWEEL J., DRAAD A. and HOEVEN J. G. Th. Measurement of fully-developed turbulent ppe flow wth dgtal partcle mage velocmetry[j]. Experments n Fluds, 1996, (3): 165-177. [1] Van DOORNE C. W. H., WESTERWEEL J. Measurement of lamnar, transtonal and turbulent ppe flow usng Steroscopc-PIV[J]. Experments n Fluds, 7, 4(): 59-79. [13] DU Guang-sheng, LIU Zheng-gang. Flud characterstc of rotary wng heat meter wth sngle-channel[j]. Journal of Hydrodynamcs, 8, (1): 11-17. [14] DU Guang-sheng, LIU L-nng and LI L et al. The nfluence of nstallaton condtons of heat meters on nteror flud feld and flux measurement accuracy[j]. Journal of Hydrodynamcs, Ser. B, 6, 18(3): 455-46. [15] DU Guang-sheng, WANG Nng and LIU Zheng-gang. Flud chara-cterstc study on rotary wng heat meter wth two-stream[j]. Chnese Journal of scentfc Instrument, 6, 7(9): 171-174(n Chnese). [16] LIU Zheng-gang, DU Guang-sheng and WANG Nng. Research on flud characterstc wthn the rotatng-wng heat meter[j]. Journal of Hydrodynamcs, Ser. B, 6, 18(4): 458-463. [17] LIU Yong-hu, DU Guang-sheng and LIU Zheng-gang. The nfluence of dfferent desgn parameters and workng condtons on characterstcs of heat meters[j]. Journal of Hydrodynamcs, 9, 1(3): 394-4. [18] LI L, DU Guang-sheng and LIU Zheng-gang. The transent aerodynamc characterstcs around vans runnng nto a road tunnel[j]. Journal of Hydrodynamcs, 1, (): 83-88. [19] AN Ru-dong, LI Ja. Characterstc analyss of the plungng of turbdty currents[j]. Journal of Hydrodynamcs, 1, (): 74-8. [] HUANG Zhen-yu, MIAO Guo-png. Large eddy smulaton of ncompressble vscous flow past under water confguraton[j]. Journal of Hydrodynamcs, Ser. A, 6, 1(): 19-197(n Chnese). [1] JIA Xao-he, LIU Hua. Large eddy smulaton of flow around two crcular cylnders[j]. Chnese Journal of Hydrodynamcs, 8, 3(6): 65-63(n Chnese). [] ZOU L-yong, LIU Nan-sheng and LU X-yun. Large eddy smulaton of pulsatng turbulent open channel flow[j]. Journal of Hydrodynamcs, Ser. B, 4, 16(6): 681-686.