COMPARATIVE STUDY OF 4(8)-PATH AND 5(10)-PATH CONFIGURATIONS FOR ATT FLOW MEASUREMENTS IN CIRCULAR CONDUITS INTRODUCTION

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1 COMPARATIVE STUDY OF 48-PATH AD 5-PATH COFIGURATIOS FOR ATT FLOW MEASUREMETS I CIRCULAR CODUITS Sergey Marushchenko serg.maruschenko@hslu.ch Lucerne Unversty of Appled Scences and Arts Technkumstr., CH-648, Horw, Swterland Peter Gruber peter.gruber@hslu.ch Lucerne Unversty of Appled Scences and Arts Technkumstr., CH-648, Horw, Swterland & Rttmeyer Ltd, CH Baar, Swterland ITRODUCTIO The IEC4 norm [] on hydraulc effcency s under a major revson. The Acoustc Transt Tme ATT method for flow measurement, up to now classfed only as a secondary method lsted n the Annex, wll move forward and wll become a prmary method for determnng the flow n turbnes and pump-turbnes. Major upgrades nclude the ntroducton of the OWICS method for flow ntegraton and the extenson of recommended values of the number of acoustc paths. The OWICS method s expected to mprove for fully developed velocty profles the flow ntegraton accuracy n comparson wth the Gauss-Jacob method by the use of a more realstc velocty profle. An ncrease of the number of acoustc paths s also expected to mprove the accuracy of flow ntegraton. Comparsons of the OWICS and Gauss-Jacob methods have already been conducted n [, 3]. However, t was done only for =4 and for the analytcally defned flow velocty profles. The ntegraton error of the OWICS and the Gauss-Jacob methods as a functon of have also been carred out n []. However, these calculatons were made only for fully developed flow velocty profles. Therefore, the man objectve of ths study s to compare the performance of the Gauss-Jacob and the OWICS methods, and the beneft of an ncrease of under more realstc dsturbed flow condtons.. GEERAL THEORY AD PROBLEM STATEMET Fg. : Conventonal ATT flow meter schematc Fgure shows the schematc of a typcal ATT flow meter nstallaton. Conventonally, t conssts of several acoustc paths whch form two equal measurng planes. The most common number of acoustc paths n one plane s 4, however, accordng to the latest IEC4 code revson, the recommended number of acoustc paths s allowed to vary from 4 to 9. Two paths on the same level form a measurng layer,.e. total number of measurng layers equals. Double plane desgn s used to decrease the possble nfluence of cross flow on the accuracy of measurements.

2 In order to prevent possble msunderstandngs, n the framework of the present paper, f nothng else s specfed, the value means the number of measurng paths n each measurement plane,.e. the total number of measurng paths n a -path flow meter s. Therefore also the abbrevaton 48-path n the ttle.. The area flow functon The total flow rate through a unt crcle cross secton can be approxmated as a sum of the elementary flow rates passng through the number of horontal layers Fg.., left: Q Q v ax l, where v ax : average axal velocty at -th layer l : projected path length Fg.., rght : wdth of -th layer Fg. : Schematc of flow measurement n a regular ppe usng the ATT technque wth normaled radus of R= Tendng the number of layers to nfnty, the sum n eq. can be replaced by the ntegral: Q lm v ax l F d, wth F vax l 3 The functon F s called area flow functon and descrbes the dstrbuton of the partal flow rates through the elementary layers of the cross secton and has dmenson [m /sec]. A flowmeter has a fnte number of measurng layers, therefore, the ntegral n eq. has to be replaced by a fnte sum of dscrete values of F. Conventonally, for that purpose the Gaussan quadrature method s used, because t has the hghest polynomal degree of - of all the exstng quadrature methods The polynomal degree of a quadrature method ndcates the hghest order of a polynomal whch can be ntegrated wth ero error. Replacng the ntegral n eq. by the -pont Gaussan quadrature leads to eq. 4 []:

3 3 l v w R Q,.4 wth w : weghtng factor for -th measurng layer v : averaged axal velocty at -th measurng layer l : projected path length R: radus of crcular secton. Weghtng and postonng The hghest polynomal degree of the Gaussan quadrature among exstng quadratures obtaned thanks to the specfcally calculated postons of the quadrature nodes correspondng to the heghts of the measurng layers. The postons of the -path Gaussan quadrature can be found as the eros roots of the polynomal p generated by the recurrence relaton: p b p a p, 5 wth d p F d p F a, d p F d p F b. The weghtng factors w are calculated accordng to the equaton: d L F F w 6 where j j j j L. As the postons and weghts are defned durng the stage of the meter nstallaton, the future flow condtons and, partcularly, the real area flow functon Freal n the measurement secton are, obvously, unknown. Therefore, n order to calculate the weghts and postons an assumed are flow functon Fassum s conventonally used n eqs. 5-6 for F. Ths assumed area flow functon Fassum s determned based on the measurement secton s geometry and an assumed velocty profle r v assum. In the conventonal Gauss-Jacob method for crcular secton ths profle s assumed unform, whch leads to a devaton of the ntegrated dscharge from the true dscharge n the case of fully developed flow. Therefore, Voser [] proposed the OWICS method whch uses the profle of eq. 7 for calculatng the postons and weghts. The -path Gaussan quadrature method guarantes that polynomal devatons of order - of the real area flow functon Freal from the assumed area flow functon Fassum can stll be ntegrated wth ero. R r r v assum 7 Table shows the Gauss-Jacob postons R / for =..5 and correspondng weghts w calculated accordng to the Gauss-Jacob and OWICS methods. In practce, for OWICS method,

4 Due to practcal reasons, usually the Gauss-Jacob postons are used for the OWICS method and the weghts are recalculated accordng the OWICS method takng nto account the actual paths postons. The senstvty nvestgaton of Tresch & al. [] shows the neglgble nfluence on the accuracy of such mnor paths shft. Gauss-Jacob OWICS /R [-] w [-] /R [-] w [-].57796, ,48795, , ,69568, ,33783, ,799639, , ,49366, ,858534,65433 Table : Path postons and weghts for =,..5 [].3 Problem statement The OWICS method wll probably be approved for the flow ntegraton by the latest revson of the IEC4 norm []. However, there s no data publshed whch allows to conclude whether and under what crcumstances the new OWICS method outperforms the conventonal Gauss-Jacob method n terms of accuracy. Hence, the frst objectve of ths study s the comparson of the OWICS and Gauss-Jacob methods n terms of ntegraton accuracy. One of the man features of numercal quadrature s that the ntegraton accuracy grows wth the number of quadrature nodes. Ths means that for the dscharge ntegraton the accuracy of the flow rate ntegraton s expected to mprove wth the number of acoustc paths. The proposed revson of the IEC4 code extends the range of recommended number of acoustc paths n prevous verson the =4 meter was consdered as basc. The frst queston s therefore: how much can be ganed n terms of accuracy f s ncreased from 4 to 5? It was mentoned above that n practce the OWICS method often uses the Gauss-Jacob postons, whch means, the only dfference between the methods conssts n the values of weghts used. Unlke to ths fact, an ncrease of the number of acoustc paths leads n many cases to the need of new sensors. Consequently, t s of hgh practcal nterest to the manufacturers to understand the benefts of changng the exstng ndustry standards. Hence, the second objectve of ths study s to ncrease the number of measurng layers by one and compare the former basc 4-path arrangement wth a 5-path. Here hgher number of paths are not nvestgated here although nstallatons of 9-path arrangements exst. Drlled n acoustc transducers for ATT flow meters n crcular secton are usually desgned for exact path postons / R. At the moment transducers for 4 and =9 at Gauss-Jacob postons are common. However, for a 5-path Gauss-Jacob arrangement Table the transducer for the poston of / R needs to be developed the transducers for / R =;.5 are already used n - and -path arrangements correspondngly. If therefore the outer path of a 5-path arrangement could be shfted from the / R to the / R. 897 poston Fg. 3 whch corresponds to the outer path of a 4-path arrangement Table, the necessty to develop a new transducer could be avoded. Ths not neglgble shft affects the accuracy of the flow rate ntegraton as the Gaussan quadrature requres specfcally calculated quadrature nodes. The thrd objectve of ths study s therefore to evaluate the performance of a 5-path arrangement wth shfted outer paths, n the framework of ths paper referred to as 54-path arrangement. 4

5 Fg. 3: Gauss-Jacob 5-path sold lnes outer path shft to 4-path poston dashed lne. THE ACCURACY OF FLOW ITEGRATIO IVESTIGATIO. Research methodology Consderng the objectves stated n the prevous secton, the followng arrangements were selected for a comparatve study: Gauss-Jacob 4-path Gauss-Jacob 5-path Gauss-Jacob 54-path OWICS 4-path OWICS 5-path In the OWICS confguratons the acoustc paths are arranged accordng to the Gauss-Jacob method and the weghts are recalculated accordng to the OWICS method for the Gauss-Jacob postons. The summary of the path postons / R and the weghts w for selected confguratons are presented n Table. Gauss-Jacob 4-path Gauss-Jacob 5-path Gauss-Jacob 45-path OWICS 4-path OWICS 5-path /R[-] w [-] /R[-] w [-] /R[-] w [-] /R[-] w [-] /R[-] w [-] Path -,897, ,8665,6799 -,897, ,897,365 -,8665,65433 Path -,397, ,5, ,5,3464 -,397, ,5, Path 3,397,597566,53598,599994,397,598639,55768 Path 4,897,36937,5,453449,5,3464,897,365,5, Path 5,8665,6799 -,897,35435,8665,65433 Table : The summary of the path postons / R and the weghts w for selected confguratons The followng ntegraton error eq. 8 s used as performance crteron: Q Qref % 8 Q ref 5

6 For the nvestgaton of the ntegraton error, the measured velocty values v and the reference flow rate value Qref are needed as nput data. Ths data s obtaned from numercal smulaton. The study of the selected confguraton s performances s dvded nto two logcal parts. In the frst part the performance of the selected confguratons s studed for the example of the flow downstream of elbow-type dsturbers: straght ppe for reference, sngle 9 elbow, double 9 elbow n plane and double 9 elbow out of plane Fg. 4. For each dsturber two nstallaton postons are studed: D and 5D downstream the dsturber And three dfferent Reynolds numbers are consdered: 5, 6 and 7. Addtonally, at each nstallaton poston the amuthal nstallaton angle α of the meter s vared from to 8 n 5 steps n order to nvestgate the nstallaton angle effect. In the second part of study the performance of the selected confguratons s studed for the example of the smulated flow n the Aratata hydraulc power plant water ntake Hug & al. [4] Fg. 5. Ths part nvestgates the meter performance for dfferent hydraulc condtons n a real plant. a b c d Fg. 4: Investgated dsturbers` schematcs and 3D models: a: straght ppe; b: sngle 9 elbow; c: double 9 elbow n plane; d: double 9 elbow out of plane 6

7 Fg. 5: Aratata hydraulc power plant schematc. Performance nvestgaton on example of elbow-type dsturbers.. umercal smulaton of flow downstream the selected dsturbers The man data of the smulated flow are: Ppe dameter: D =.5 m Reynolds number Re: 5, 6, 7 Flow velocty:.,, m/s The numercal smulaton was performed wth the ASYS Fluent software. The computatonal O-grd mesh wth x 6 cells was developed n ASYS ICEM CFD The flow was smulated mplementng the SST Shear Stress Transport model. For the nlet boundary condton of the computatonal doman the fully developed velocty profle shown n Fg..3 s used. Ths profle was obtaned from numercal smulaton of the flow n a straght ppe of 5D length wth translatonal perodc boundary condtons. Fg. 6: Fully developed velocty profle used for nlet boundary condton defnton 7

8 .. Results and dscusson Fgure 7 shows the smulated velocty contours n a perodc straght ppe. The ntegraton error values for a straght ppe are presented n Table.. Fg. 7: Smulated mean velocty contours n straght ppe 4-path 5-path 54-path G-J OWICS G-J OWICS G-J [%],7,4,,3,3 Table 3: Integraton error values for fully developed flow n the straght ppe Fgures 8- show the smulaton results for the selected dsturber together wth the ntegraton error values of each studed arrangement as a functon of the nstallaton angle α and a Reynolds number of 5. In the appendx the smulaton results are shown for the Gauss-Jacob methods and Re= 6 and 7. Addtonally, each graph contans marked wth colored crcles eps_average - values, the error value averaged over all the nstallaton angle. In order to smplfy the percepton of the data presented n Fg. 8- and Table 3, the summary of the average ntegraton error and the error dsperson dfference between mnmal and maxmal error values over all angles, for dsp each dsturber and confguraton are shown n Table 4. From the summary presented n Table.3, Fg. 8- and the appendx, the followng observatons can be made:. The OWICS method has a smaller average error value than the Gauss-Jacob method. The error dsperson value dsp, however, s vrtually ndependent on the method used.. A ncrease of the number of acoustc paths from 4 to 5 sgnfcantly up to 5% decreases the average ntegraton error. 3. The meter nstallaton at the 5D downstream poston does not mprove the average error value, however, the error dsperson dsp sgnfcantly up to 6% decreases n comparson wth D downstream the elbow. 4. For each dsturber and poston D, 5D a specfc range of angle of nstallaton can be defned whch mnmes the ntegraton error under the assumpton that the smulaton s correct. 5. The varaton of the Reynolds number does not show a clear trend. The Gauss-Jacob 54-path arrangement usually performs worse than the Gauss-Jacob 4-path arrangement n terms of average ntegraton error. The error dsperson value of ths arrangement s not defntely better or worse than the Gauss-Jacob 4-path arrangement s, however s sgnfcantly worse than Gauss-Jacob 5-path arrangement s. 8

9 ε [%] Gauss-Jacob and OWICS at D ε [%],5 Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D OWICS 4-path at D OWICS 5-path at D eps_average=.4 % eps_average=.34 % eps_average=.7 % eps_average=.5 %,5 Gauss-Jacob and OWICS at 5D Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D OWICS 4-path at 5D OWICS 5-path at 5D eps_average=.36 % eps_average=.3 % eps_average=. % eps_average=.6 %,5,5 -, , ε [%],5 Gauss-Jacob at D and 5D Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D eps_average=.4 % eps_average=.34 % eps_average=.36 % eps_average=.3 %,5 -, ,5 Gauss-Jacob at D nkl. 54-path ε [%] ε [%],5 Gauss-Jacob at 5D nkl. 54-path,5,5 -, Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 54-path at D eps_average=.4 % eps_average=.34 % eps_average=.68 % Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D -,5 Gauss-Jacob 54-path at 5D eps_average=.36 % eps_average=.3 % eps_average=.58 % - Fg. 8: Integraton error values for sngle elbow, Re= 5 9

10 ε [%] Gauss-Jacob and OWICS at D ε [%] Gauss-Jacob and OWICS at 5D,5 Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D OWICS 4-path at D OWICS 5-path at D eps_average=.54 % eps_average=.4 % eps_average=.36 % eps_average=.5 %,5 Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D OWICS 4-path at 5D OWICS 5-path at 5D eps_average=.38 % eps_average=.3 % eps_average=. % eps_average=.3 %,5, ,5 -,5 ε [%],5 Gauss-Jacob at D and 5D Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D eps_average=.54 % eps_average=.4 % eps_average=.38 % eps_average=.3 %,5 -, Gauss-Jacob at D nkl. 54-path ε [%] ε [%],5 Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 54-path at D eps_average=.54 % eps_average=.4 % eps_average=.7 %,5 Gauss-Jacob at 5D nkl. 44-path Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D Gauss-Jacob 54-path at 5D eps_average=.38 % eps_average=.3 % eps_average=.64 %,5,5 -, , Fg. 9: Integraton error values for double elbow n plane, Re= 5

11 ,5,5 ε [%] Gauss-JAcob and OWICS at D ε [%] Gauss-Jacob and OWICS at 5D,5,5 -, ,5 Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D OWICS 4-path at D OWICS 5-path at D eps_average=.5 % eps_average=.3 % eps_average=.3 % eps_average=.4 % -,5 - Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D OWICS 4-path at 5D OWICS 5-path at 5D eps_average=.39 % eps_average=.8 % eps_average=.3 % eps_average=.9 %,5 ε [%] Gauss-Jacob at D and 5D,5 -,5 - -, Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D eps_average=.5 % eps_average=.3 % eps_average=.39 % eps_average=.8 %,5 Gauss-Jacob at D nkl. 54-path ε [%] ε [%],5 Gauss-Jacob at 5D nkl. 54-path,5,5 -,5 - -, Gauss-Jacob 4-path at D Gauss-Jacob 5-path at D Gauss-Jacob 54-path at D eps_average=.5 % eps_average=.3 % eps_average=.4 % -, Gauss-Jacob 4-path at 5D Gauss-Jacob 5-path at 5D Gauss-Jacob 54-path at 5D eps_average=.39 % eps_average=.8 % eps_average=.55 % Fg. : Integraton error values for double elbow out of plane, Re= 5

12 4-path 5-path 54-path Gauss-Jacob OWICS Gauss-Jacob OWICS Gauss-Jacob D 5D D 5D D 5D D 5D D 5D Straght ppe Re= 5 dsp [%] [%],7,4,,3,3 Re= 5 [%],4,36,7,,34,3,5,6,68,58 dsp [%],,63,,63,,43,,43,56,69 Sngle Elbow Re= 6 dsp [%] [%] Re= 7 dsp [%] [%] Re= 5 [%],54,38,36,,4,3,5,3,7,64 dsp [%],83,59,9,59,63,47,63,45,8,73 Double elbow n plane Re= 6 dsp [%] [%] Re= 7 dsp [%] [%] Re= 5 [%],5,39,3,3,3,8,4,9,4,55 dsp [%],,8,,,9,63,8,65,9,67 Double elbow out of plane Re= 6 dsp [%] [%] Re= 7 dsp [%] [%] Table 4: Average ntegraton error and error dsperson dsp values summary.3 Performance nvestgaton for the Aratata power plant.3. umercal flow smulaton at Aratata power plant The smulatons were performed wth ASYS CFX.. Three dfferent operatng condtons were smulated: Test : 3x9 m3/s all three turbnes n operaton Test : x9 m3/s turbne and n operaton, turbne 3 shutdown Test 3: x5 m3/s turbne n operaton, turbne and 3 shutdown The boundary condtons for these operatng ponts are lsted below: Inlet: The straght ntake tunnel of the power plant s about 5 dameters long wth constant crcular cross secton. The range of the Reynolds numbers les between Re=7 6 and for the nvestgated operatng ponts. Consequently, a fully developed velocty profle can be assumed at

13 the nlet to the smulaton doman. Ths profle s calculated beforehand for each of the gven flow rates n a separate smulaton wth a dameter short straght secton wth translatonal perodc boundary condtons. These velocty dstrbutons as well as the turbulence quanttes are then used as nlet condtons for the man smulatons. Outlets: The number of outlets vares from to 3 outlets dependng on the operatng condtons. The mass flow s set for each of the outlets. The outlet mass flows then are lnked to the mass flow at the nlet n order to satsfy the mass flow balance. The expressons n the brackets show the boundary condtons, whch are set at the outlet. Test : All three outlets have the same mass flow. Test : Outlet and have the same mass flow. Outlet 3 s defned as a no slp wall. Ths s a smplfcaton, n realty the doman ends n the spral casng upstream of the closed gude vanes. Test 3: Outlet and nlet have the same mass flow. Outlet and 3 are defned as no slp walls. Wall. The wall s specfed as a no slp wall assumng hydraulcally smooth surfaces. Free surface: The free surface of the surge tank s defned as a free slp wall. Ths means that the water level s constant and the water has no frcton at ths boundary..3. Meter performance at Aratata power plant In Table 5 the ntegraton errors for dfferent smulated cases are shown. The code for the cases s as follows: the frst number, or 3 ndcates how many penstocks are n operatng, the second number 5, 9 or ndcates the flow rate, the term n brackets + or +3 ndcates whch of 3 penstocks are n operatng, the last part of the code P or P tells whch penstock s evaluated. The best result n the strng s marked wth the green color, the worst wth the red color. Gauss-Jacob 4-path [%] Gauss-Jacob 5-path [%] Integraton error [%] Gauss-Jacob 54-path [%] OWICS 4-path [%] OWICS 5-path [%] x5_p,9 -,46,46,9 -,56 x5_p,6,34 -,,,3 x9+_p -,99,86 -,7 -,7,75 x9+_p,3, -,4,4 -,8 x9+3_p,7,8 -,36,,8 3x5_P, -,8,54,84 -,37 3x5_P,4 -,3 -,9,9 -, 3x9_P,97 -,35,6,8 -,45 3x9_P,9,8 -,8,77, 3x_P,96 -,37,6,78 -,47 3x_P,86, -,7,7,3 [%] Table 5: Integraton error summary for Aratata nstallaton As can be seen from the Table 5, the worst confguraton s the Gauss-Jacob 4-path. The best performance exhbts the Gauss-Jacob 5-path confguraton. The OWICS 5-path confguraton has the best result n 4 cases out of. So the OWICS method superorty over the Gauss-Jacob s not confrmed. If the penstocks are operated n a symmetrcal mode, OWICS outperforms the Gauss- Jacob confguraton ndcatng that OWICS s more sutable for less dsturbed flow profles. The 3

14 accuracy mprovement by ncreasng s confrmed, as well as that a 45-path confguraton s useless. COCLUSIO In the frst part the performance of dfferent measurng confguratons was nvestgated wth the example n the presence of dfferent elbow-type dsturbers upstream of the measurement secton. The obtaned results show a defnte superorty of the OWICS ntegraton method over the Gauss- Jacob method n terms of the average error. Addtonally, the results confrm that the ncrease of the number of acoustc paths from 4 to 5 decreases the average ntegraton error and the error dsperson. Furthermore, the effect of the flow dsturbance ntensty on the error dsperson dsp value s observed: at 5D nstallaton locaton the ntegraton error dsperson values dsp are sgnfcantly smaller than the correspondng values at D. However, a larger dstance from the dsturber does not mprove the average error. It s cause could be the result of a systematc naccuracy n the weghtng and postonng procedures, as even n case of the straght perodc ppe, where the flow s fully developed, the ntegraton error s stll present Table.3. For each dsturber and poston D, 5D a specfc range of angle of nstallaton can be defned whch mnmes the ntegraton error under the assumpton that the smulaton s correct. If no smulaton s avalable or f t s too naccurate, the average ntegraton error together wth ts dsperson s a good estmate of what knd of accuracy range can be expected. The frst part of the study dsp also demonstrates that the Gauss-Jacob 54-path arrangement s worthless, as n terms of the average error value t performs even worse than Gauss-Jacob 4-path arrangement. In the second part, the performance of a specfc measurng confguratons was studed on the smulated flow of the Aratata power plant n ew Zealand. The results from the Table 5, unlke the ones of Table 4 don t show a defnte superorty of the OWICS over the Gauss-Jacob method. However, they also confrm that by ncreasng the number of measurng paths from 4 to 5 the ntegraton error value decreases. It s also demonstrated, that the 54-path arrangement s nferor. In the cases wth symmetrcal operaton of the power plant, the OWICS ntegraton error for 5-paths for the mddle ppe P s on the average superor to the other methods. REFERECES [] IEC 64 Internatonal Standard / Internatonal Electrotechncal Commsson, Geneva, Swterland, 99 [] T. Tresch, P. Gruber, T. Staubl: Comparson of ntegraton methods for multpath acoustc dscharge measurements, IGHEM 6, Portland, Paper6/d6. pdf [3] A.Voser Analyse und Fehleroptmerung der mehrpfadgen akustschen Durchflussmessung n Wasserkraftanlagen, ETH Zürch Dssertaton r. 3, 999 [4] S. Hug, T. Staubl, P. Gruber: Comparson of measured path veloctes wth numercal smulatons for heavly dsturbed velocty dstrbutons, IGHEM, Trondhem 4

15 Appendx: Velocty dstrbutons after the dsturber and ntegraton errors for the Gauss- Jacob methods for Reynolds number of 6 and 7 5 s already n man body Fg. 8- Sngle elbow Re= 6 Re= 7 5

16 Double elbow n plane Re= 6 Re= 7 6

17 Double elbow out of plane Re= 6 Re= 7 7