The profile-linear average velocity for the transition in pipes based on the method of LES *

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1 9 h Inernaonal Conference on Hydrodynamcs Ocober 11-15, 010 Shangha, Chna , (5), supplemen : DOI: /S (09)600-1 The profle-lnear average velocy for he ranson n ppes based on he mehod of LES * Yong-hu Lu, Guang-sheng Du, Zheng-gang Lu, l L, L-l Tao School of energy and power engneerng, Shandong Unversy, Jnan, Chna E-mal: du@sdu.edu.cn ABSTRACT : The ulrasonc flowmeer has been wdely used n he ndusral flow measuremen. The flow measuremen accuracy depends on he relaonshp of he profle-lnear average velocy. Bu hs relaonshp of he ranson zone s no avalable a presen. In hs paper, he characerscs of ranson flow wh specfc Re number n ppes are researched. The model and LES(Large Eddy Smulaon) model are respecvely used o calculae he flow feld of he ranson zone, and he expermen resuls show ha he LES model s more effecve han he model. The relaonshp of he profle-lnear average velocy for he ranson zone n ppes s obaned by he calculaed resuls of he LES model and s proved ha here s a bg error usng he radon relaonshp based on he urbulence flow o calculae he profle-lnear average velocy relaonshp of he ranson flow. The research resuls of hs paper can mprove he measuremen accuracy of ulrasonc flowmeers and provde heorecal bass for he research on he whole ranson flow. EY WORDS: Ulrasonc flowmeer; LES model; profle average velocy; lnear average velocy; ppe. 1 INTRODUCTION The ulrasonc flowmeer has been wdely used n he ndusral flow measuremen, and s measuremen prncple s shown n Fg.1. The ulrasonc Transducers TRA and TRB can no only send he ulrasonc mpluse(θ s he ncdence angle), bu also receve he ulrasonc sgnal. Because he ulrasonc velocy n he downsream and upsream s dfferen, so he me when he wo ulrasonc ransducers receved he ulrasonc sgnal s Fg.1 Measuremen prncple of he ulrasonc flowmeer dfferen, and here s a me dfference Δ. The lnear average velocy v l s obaned by capurng hs me dfference: v /( cg ) l =Δ c D θ (1) Where, D s dameer of he ppe. The formula of volume flow Q s: Q =π D v l /4 () where, s defned as he flow coeffcen, whch s he rao of he profle average velocy v s and he lnear average velocy v l. =v s / v l (3) In order o nsure he measuremen accuracy, he lnear average velocy v l measured by he ulrasonc flowmeer mus be convered o he profle average velocy v s. Many researchers have done a lo of wors on he ulrasonc flowmeer. B.Iooss,C, Renaldas Rasus [1~4] researched he nfluences of srucure, roughness, velocy-profle, emperaure on he fully-developed urbulen flow; Yuo Inoue, TaedaY [5-6] used UVP (Ulrasonc Velocy Profle) mehod o develop a * Proec suppored by he Naonal Scence Foundaon of Chna(Gran No ) Correspondng auhor: DU Guang-sheng

2 356 9 h Inernaonal Conference on Hydrodynamcs Ocober 11-15, 010 Shangha, Chna sysem of ulrasonc flow measuremen, and researched he nfluence of he ncdence angle on he measuremen precson; Jeff Dooma, Jaca Lu, S. [7-11] Srsup used he DNS(Drec Numercal Smulaon) mehod o calculae he urbulen flow; Wu mn-we,zhang zhao-shun [1-14] also used he DNS mehod o sudy he urbulen flow n ppes, and measured he flow feld by he DPIV(Dgal Parcle Image Velocmery) mehod; The flow coeffcen s suppled by he Indusral Auomaon Insrumenaon Manual [15] : when he flow s lamnar flow: =3/4 (4) when he flow s urbulen flow: = ( Re ) (5) These researches are manly focused on he flow measuremen of he lamnar and urbulen flow, bu he flow characerscs of he ranson zone are rarely suded. The radanal mehd s o regard he ranson flow as he urbulen flow o oban he relaonshp of he profle-lnear average velocy. Ths smplfed mehod mus lead o measure error whle waer flow of he ppe s ranson flow. In hs paper, he characerscs of he ranson flow wh he Re number 5300 n ppes are suded, and he proflelnear average velocy s obaned va he numercal smulaon, whch can provde heorecal bass for he research on he whole ranson flow. PHYSICAL MODEL As shown n Fg., n order o compare wh he expermenal daa of Weserweel [16], he ppe s dameer s 0.04m and he lengh s 0.5m n hs paper. Fg. Physcal model The calculaon s sablzaon s deermned by he sze of he smalles mesh n he flow. Accordng o he reference[17], he smalles mesh scale l demand: + l=5 y 0 (6) Where, y + 0 s he vscous lengh scale, + y 0 = v / u * (7) u* s he frcon velocy. u* = u f / (8) f s he fannng frcon facor, f=0.079re 1/4 (9) So, when Re=5300, f=0.0096, u*= m/s, y + 0 = m, and l=5 y + 0 = m. Fg.3 shows he cross secon mesh. In hs secon, he sze funcon mehod s used o generae he mesh, and sasfed he demand of he smalles srucures mesh. Cooper mehod s used for he flow drecon. 3 NUMERICAL SIMULATION WITH THE MODEL 3.1 The Model The mahemacal model [18~] are: Connuy equaon: ρ ρ ( u ) + = 0 (10) x Momenum equaon: ρ u ) ( ρuu ) p u + = + μ ρu u x x x (11) Vscosy coeffcen of urbulence: ρcμ μ = (1) Where, ρ s he densy of waer, u s he velocy of flow; he superscrp expresses he quany of average me, P s pressure, μ s he vscosy coeffcen. equaon: ρ) ρu ) μ + = ( μ + ) + G + Gb ρ YM + S σ (13) equaon: ρ μ ) + ( ρu ) = ( μ+ + C1 ( G + C3 Gb ) C ρ + S σ (14) Wheere, G s urbulen nec energy produced by Fg.3 Cross secon mesh he graden of average velocy, G b s urbulence nec energy produced by he floage, Y M expresses

3 9 h Inernaonal Conference on Hydrodynamcs Ocober 11-15, 010 Shangha, Chna 357 he panng acon caused by he dffusvy of compressble urbulen flow, C 1 C C 3 are consan coeffcens, σ and σ are he urbulen, Prandl numbers of and coeffcen, S and S are source em user-defned. The urbulen nec energy G b and relaed coeffcens C 3 produced by he floage are consdered a he compressble flow. When he flud s ncompressble and he source em user-defned are no consdered, G b, Y M, S and S are equal o zero. The value of consan coeffcen C 1 C C μ σ, σ are: and C =, C = 1.9, C μ = 0.09, σ = 1.0, σ = 1.3 The velocy-nle s used for he nle condon. When Re=5300,v=0.133m/s. The oule condon s he free ouflow. 3. The resuls of he Model The mean sreamwse velocy profle s shown n Fg.4. Y-coordnae s he velocy, and X-coordnae s he poson of he measuremen pons n he cross secon(d s he dameer). Fg.4 Mean sreamwse velocy profle of he model In Fg.4, compared he resuls of he model wh he expermenal daa by Weserweel [16] va DPIV, s shown ha he velocy profle obaned by he model has error wh he expermen resuls. The calculaon resuls show ha he model s a fully-developed urbulen model, and can no smulae he ranson flow correcly. So, he model s no f for he research of he profle-lnear average velocy of he ranson flow. The exac mehod o smulae ranson flow s DNS mehod, bu he DNS needs large CPU resources and long calculaon me. The LES model s beween he model and he DNS mehod, he large eddy s calculaed va N-S equaon and he nfluence of he small eddy on he large eddy s consdered by he oher model. Ths paper used he LES model o smulae he ranson flow. The LES model s a non-seady hree-dmenson model, and needs he nle velocy profle. In hs paper, he calculaed oule velocy profle by he model s used as he nle-velocy boundary condon of he LES model [3]. The oule velocy profle vecor obaned by he model s shown n Fg.5. Fg.5 Oule velocy profle vecor of he model 4 NUMERICAL SIMULATION WITH THE LES MODEL 4.1 The LES model The equaon of LES model [3~5] s: u u u 1 p u τ + = + υ + ρ (15) Where, τ = u u uu, s defned o he subgrd sress, and s he momenum ranspor beween he flraed small scale urbulence and he scale whch can be calculaed. The subgrd sress s he no closed ems of he equaon. So, a ceran model s used o deal wh he subgrd sress. In hs paper, he Smagornsy model s used: 1 τ = μ S + τ δ (16) 3 Where, μ s he subgrd urbulen vscous sress, S s he ranform rao of ensor: 1 u u S = ( + ) (17) Ths model s he basc of he subgrd model, whch s advanced by he Samagorn, and he model equaon s: μ = ρ S (18) L s Where, S S L s s he mxng lengh of mesh, and = S Cs, s C =0.1. (19)

4 358 9 h Inernaonal Conference on Hydrodynamcs Ocober 11-15, 010 Shangha, Chna The boundary condon: The nle velocy profle s shown n Fg5, and he nle urbulence s provded by he mehod of Inensy and Hydraulc Dameer Tme sep: he me sep s confrmed va es calculaon, and s s. 4. The resuls of he LES model The resuls of he calculaon by he LES model are shown n Fg.6. Fg.6 Mean sreamwse velocy profle of he LES model I s shown ha he resuls wh he LES model are he same wh Weserweel s, and verfed ha he LES model s more effecve o smulae he characerscs of he ranson flow wh he Re So, n hs paper he resuls of he LES model are used o sudy he profle-lnear average velocy of ppes. 5 THE PROFILE-LINEAR AVERAGE VELOCITY OF PIPES WITH THE LES MODEL 5.1 The profle-lnear average velocy The ncdence angle s 0 [5], as shown n Fg.1. The profle average velocy v s s consan because of he consan flux. The Re number of hs paper s 5300, so v s =0.133m/s. In order o mae he resul sascal sgnfcance, he me-average mehod s used for he lnear average velocy. Fg.7 shows he lnear average velocy n he dfferen me. Y-coordnae s he lnear average velocy, and X-coordnae s he dfferen me. As shown n Fg.7, he lnear average velocy s flucuang, bu he me-average value s consan. The lnear average veloces he dfferen me flucuae round a consan. Ths consan s he meaverage value of he lnear average velocy, and s 0.151m/s. Therefore, he lnear average velocy of ppes v l s hs me-average value, v l =0.151m/s. 5. Compared wh he resul of he randonal coeffcen based on he urbulen flow When Re=5300, can be calculaed usng he Eq. 5 provded by he reference [15]: = ( Re ) =0.95. The flow coeffcen obaned by he LES model n hs paper s: =v s /v l = The error =(- )/ =4.99%. I s shown ha he profle-lnear average velocy of he ranson flow s dfferen from ha of he urbulen flow. Therefore, s sgnfcan for he sudy of he profle-lnear velocy of he ranson flow. 6 CONCLUSIONS AND PROSPECT I can be concluded ha he model s no effecve o calculae he flow characerscs of ppes as Re=5300, bu he resul of he calculaon wh he LES model s more closer o he expermenal daa of Weserweel. The profle-lnear average velocy of he ranson zone can be obaned by he LES model. When he Re number s larger han 5300, he LES model s also effecve because of he growng of he urbulen nec energy, and he profle-lnear average velocy of hs flow regon can also be calculaed by he LES model. Ths paper proved ha here s a bg error when usng he radonal profle-lnear average velocy relaonshp of he urbulence flow o calculae of he ranson flow. I ndcaed he mporance of he profle-lnear average velocy relaonshp of he ranson zone. Accordngly, he nex sep worng s o sudy he characerscs of he whole ranson flow: (1) Usng he LES model o calculae he characerscs of he flow wh Re>5300, and obanng he relaonshp of he profle-lnear average velocy. () Because can no denfy he effecveness of he LES model for he flow wh Re<5300, so he DNS mehod should be used for hs flow regon o oban. REFERENCES Fg.7 The lnear average veloces he dfferen me [1] B Iooss, C Lhuller. Numercal smulaon of rans-me ulrasonc flowmeers:unceranes due o flow profle and flud urbulence[j]. Ulrasoncs, 00():

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