THE EFFECT OF SHAPE FACTOR ON THE AVERAGE BED SHEAR STRESS IN OPEN CHANNEL FLOW

Transcription

1 roceegs o the 4 th Iteratoal oerece o Evrometal cece a Techology hoes, Greece, 3-5 eptemer 05 THE EFFET OF HE FTO ON THE EGE BED HE TE IN OEN HNNEL FLOW HDDIN N. Departmet o vl Egeerg, The Uversty o Jora, mma 94, Jora E-mal:[email protected], [email protected] BTT Ths paper presets a aalytcal metho to eterme the eect o se all correcto actor o the semet trasport. The aalyss s ase o the computato o ouary shear stress strutos actg o the lo permeter o prsmatc ope chaels. The applcato epes o the prcple o steay a uorm lo smooth trapezoal ope chaels th eret se slopes. The average e shear stress s calculatg y utlzg Darcy-Wesach, Mag a hézy roughess coecets. Base o ths cocept, a ovel metho s evelope or the parttog o the lo cross-sectoal area to varous parts correspog to the chael shape a roughess composto o the ette permeter. The techque permts the struto o the ouary shear over the ette permeter to e evaluate th each su-lo area. alytcal equatos goverg the ouary shear stress strutos have ee erve a are val or all chael aspect ratos. The al equato as apple to calculate ouary shear struto trapezoal ope chaels. The results sho that as the chael se slope creases, the rcto actor o the e ll crease. osequetly, oth the e shear stress a semet trasport ll also crease. Keyors: e shear stress, semet trasport, se all correcto actor, ope chael. Itroucto The shape o the chael cross secto s oe o the actors luecg the e loa ucto. I ths secto s ot luece ether structurally or y vegetato t s oly a ucto o the semet a o the lo, Este (950). The shape actor or the reach must eter to the aalyss o atural rver ecause the shape actor aects the eergy losses. Ths s ue to es a aks a the eect o the shape actor o the velocty struto, causg varato velocty, th, epth, ouary shear a secoary crculato, mos a etürk, 99. The cross secto ll e more trapezoal or rectagular a crossg. ross-secto shape ca e escre y a umer o varales, such as the area, th, a maxmum epth. I e chaels th W/ greater tha approxmately 0, the hyraulc raus a the mea epth are approxmately equal. The coveyace or capacty o a chael s relate to the area a hyraulc raus a s ee as /3 (Watso et al., 999). Bagol s (980) research mples that or a gve scharge a slope, larger semet loa requres a er chael. Whereas Heerso s (966) threshol chael equatos mply the cotrary, semet trasport creases as epth creases; thereore, the th/epth rato ecreases. resoluto o these argumets s acheve y the extremal hypothess a comparsos mae y Whte et al. (98) sho that the maxmzato o semet scharge a mmzato o stream poer, or slope or gve ater scharge, ll gve the same result semet scharge s a varale that ca e maxmze. propose y Whte et al. (98), hch maxmum trasportg capacty s attae at some termeate th. Ths hypothess as apple ET05_0064

2 to the precto o hyraulc a geometrc characterstcs o alluval chaels th sa a gravel es. The aalyses The all rcto actor or turulet lo over smooth ouary e/ 0 ca e calculate rom (Jule, 998): e log e 0.048log The eyols umer s calculate rom the average velocty 4 h e v here v s the ketc vscosty o the lu. The hézy ormula escres the mea lo velocty o steay, turulet ope chael lo. The hézy equato s erve rom hyroyamcs theory. Ulke The Mag ormula s a emprcal ormula estmatg the average velocty o a lqu log ope chaels rve y gravty. Darcy equato s erve ase o the equlrum o orces the recto o lo or uorm a steay state lo. These equatos are ell-ko equatos that utlze to compute the lo scharge a lo velocty ope chael. These equatos epeet o may actors, clug surace roughess, logtual slope a chael cross sectoal geometry. These equatos are use ths research to aalyze the eect o shape actor o the average e shear stress ope chael lo.. reset pproach To calculate the average e shear stress Este assume that the cross secto ca e ve to separate susectos th each susecto havg the same eergy le slope, a the same average lo velocty a the lo ormula ca e apple to each susecto vually. The total resstace to lo ca e escre term o the hézy coecet, the Darcy- Wesach rcto actor, or the Mag coecet. The ollog etty etee three actors has ee estalshe Jule (998): Q 8g h 6 ( I uts) here, 6.49 h ( Eglsh uts) h s the hyraulc raus g s the gravtatoal accelerato The uametal mesos are as ollos: s 3 L / T, s mesoless, a s T / L. To calculate the average e shear stress,, t s ecessary to calculate the hyraulc raus,. ET05_0064

3 Where, g s the lo esty, s the chael logtual slope, a s the hyraulc raus relate to the chael e By usg the Darcy-Wesach, the Mag or the hézy equato, as the lo ormula, the average rcto actor o the total cross secto ca e calculate. alyss the eect o chagg cross sectoal geometry such as th epth rato a the clato o se slope o the semet trasport, usg Darcy-Wesach, Mag a hézy equatos are presete ths research. 3. alyss a esults 3.. The Darcy-Wesach equato here 8g () h h s the hyraulc raus o the total cross secto a s equal to, s the total area o the cross secto, a s the ette permeter o the total cross secto. here, s the lo epth, a s the chael th pplyg the lo ormula to each susecto yels: 8g h h here s the hyraulc raus o susecto, a s the rcto actor o susecto. The total area o the cross secto,, s the sum o the area o susecto,, hece () h 8g (3) The average rcto actor,, ca e rtte the orm: (4) For rectagular cross secto (5) ET05_0064

4 here e, s the average rcto actor or the total cross secto, s the rcto actor o the s the rcto actor o the all, s the chael th, a s the chael epth. rragg the aove equato yels, Itroucg a o-mesoal chael shape actor th-epth rato s the shape actor creases the rcto actor o the e ecreases, ecause the oth the e shear stress a semet trasport ll ecrease. (6) (7), 8 For trapezoal cross secto th se slope measure rom the vertcal, the rcto actor o the e, s: cos s the chael se slope creases the rcto actor o the e creases. Thereore, oth the e shear stress a semet trasport ll crease. 3.. The Mag Equato / 3 h ( I uts) (9) / 3 h h / 3 (0) 3/ 3 / 3 / 4 () (8) h 3/ 3/ 4 3/ () / 3 h (3) The average Mag coecet,, ca e rtte the orm: ET05_0064

5 / 3 3 / (4) For rectagular cross secto / 3 3 / 3 / (5) here s the Mag coecet or the total cross secto, s the Mag coecet o the e, s the Mag coecet o the all, s the chael th a s the chael epth. rragg the aove equato yels 3 / 3 / 3/ 3/ s the shape actor creases the Mag coecet o the e / 3 (6) ecreases, / 3 ecause g, h, ecreases the the value o ll ecrease, cosequetly, ll ecrease, a semet trasport ll ecrease. For trapezoal cross secto th se slope measure rom the vertcal, the Mag coecet o the e s 3 / 3 / cos 3/ 3 / s the chael se slope creases the Mag coecet o the e creases, oth a semet trasport ll crease The hézy Equato h (8) h h h (9) (0) (7) h ave h () The average hézy coecet,, ca e rtte the orm: ET05_0064

6 ET05_0064 () For rectagular cross secto (3) here s the hézy coecet or the total cross secto, s the hézy coecet o the e, s the hézy coecet o the all, s the chael th,a s the chael epth. rragg the aove equato yels (4) (5) (6) (7) s the shape actor creases the hézy coecet o the e creases. Because g, h, creases, the value o ll ecreaseas a result, oth a semet trasport ll ecrease. For trapezoal cross secto th se slope measure rom the vertcal, the hézy coecet o the e s

7 cos (8) s the chael se slope creases the hézy coecet o the e susequetly, a semet trasport ll crease. ecreases, 4. ocluso For rectagular cross secto ope chael lo t s ou that as the shape actor creases oth the e shear stress a semet trasport ll ecrease. For trapezoal cross secto as the chael se ak slope creases, the rcto actor o the e creases the oth the e shear stress a semet trasport ll crease. EFEENE. Bagol,.. (980). emprcal correlato o eloa trasport rates lumes a atural rvers. roceeg o the oyal ocety, Loo, Egla, 37: Este, H.. (950). The e-loa ucto or semet trasport ope chael los. Techcal Bullet No. 06, U.. Departmet o grculture, ol oservato ervce, Washgto, D, eptemer. 3. Heerso, F. M. (966). Ope hael Flo. Ne York, NY: MacMlla, 5 pp. 4. Jule,. Y. (998). Eroso a emetato. Ne York, NY: amrge Uversty ress, 80 pp. 5. mos, D. B., a etürk, F. (99). emet Trasport Techology, Water a emet Dyamcs. Lttleto, O: Water esources ulcatos. 6. Watso,.., Beehar D.., a Thore,.. (999). Demostrato eroso cotrol esg maual. eport to U.. rmy orps o Egeers, Egeer esearch a Developmet eter, cksurg, M, 74 pp. 7. Whte,. J., Bettess,., a ars, E. (98). alytcal approach to rver regme. J. Hyraulc Dvso, E, 08: ET05_0064