Post-Processing of Air Entrainment on NASIR Flow Solver Results for Skimming Flow over Stepped Chutes

Transcription

1 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 Post-Proessng of Ar Entranment on NASIR Flow Solver Results for Skmmng Flow over Stepped Chutes SAEED REZA SABBAGH-YAZDI *, ROSITA SAFAHIEH * * Assoate Professor, Department of Cvl Engneerng K.N.TOOSI Unverst of Tehnolog Cvl Engneerng Department, KN Toos Unverst of Tehnolog, No.346 Valasr Street, Tehran IRAN * SYazd@kntu.a.r * * RSafah and NIKOS E. MASTORAKIS Mltar Instutes of Unverst Eduaton (ASEI) Hellen Naval Aadem Terma Chatzkrakou 8539, Praues, GREECE mastor@wseas.org Abstrat: - Ar entranment plas an mportant role n the reduton of avtatons rsk potental n hutes and spllwas. The advantages of stepped spllwas over onventonal smooth spllwas, the ar entranment phenomenon, and the flow regmes ourrng on stepped hutes have been revewed. Then the numeral analss of skmmng flow usng shallow water equatons n nlned oordnate sstem s desrbed. The numeral omputatons of ar entranment n the skmmng flow over stepped spllwas are performed n two stages. In the frst stage, the water flow equatons are onverted to dsrete form usng the overlappng ell vertex fnte volume method on trangular unstrutured mesh, and then, n the seond stage, the flow feld soluton results are used for omputng ar onentraton profles b a separate post-proessor. Ths postproessor uses expermental relatons for omputng the nepton pont haratersts and ar onentraton dstrbuton profles along the stepped hute. The omputed results of ar entranment nto the superrtal skmmng flow on stepped spllwas have been ompared wth laborator expermental measurements. Ke-Words: Ar Conentraton Dstrbuton, Skmmng Flow, Stepped Spllwa, Fnte Volume Soluton, Depth-Averaged Equatons Introduton Over the last two deades, stepped spllwas have ganed more attenton wth the development of RCC tehnque n dam onstruton. The are also used as an overtoppng proteton sstem n embankment dams. One of the most mportant advantages of stepped hutes s the reduton of avtaton rsk potental beause of sgnfant ar entranment and smaller velotes. Another advantage s large amount of energ dsspaton n omparson wth smooth spllwas. Stepped spllwas an be easl and eonomall onstruted and attahed to the downstream fae of RCC dams. Smulaton of flow ondton onsderng the effet s of the ar entranment would help desgners of the step spllwas to have better understandng about the onsequenes of the geometr desgn of ths tpe of hdraul strutures. The frst author suessfull developed a fnte volume depth averaged flow solver, NASIR, whh solves depth and velot felds n dam reservor as well as over spllwa wth steep slopes, and then, omputes velot profles b laered treatment of the flow feld, Sabbagh-Yazd (7). In present work the qualt of the results of a post proessor developed for omputaton of ar onentraton n skmmng flow s presented. Numeral result of NASIR software, whh uses ell vertex fnte volume soluton of depth average flow equatons on unstrutured meshes, s used for Numeral Analzer for Sentf and Industral Requrements

2 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 3 proessng the ar onentraton profles along the spllwa as well as n flow depth. Flow Regmes on Stepped Chutes There are two dfferent flow regmes on stepped hutes: nappe flow and skmmng flow. Nappe flow whh happens n low dsharges onssts of a suesson of over-falls jumpng from one step to another. The steps at as a seres of drops plunged one b one b the fallng nappes. One part of energ dsspaton n the nappe flow regme ours when the jet breaks n the ar. Another part of t s aused b the plungng of nappes on the step surfaes. In ths ase a hdraul jump whh ontrbutes to energ dsspaton mght take plae. Skmmng flow whh happens n large dsharges ats as a oherent stream over the outer edges of the steps. These step edges make a pseudo surfae over whh the flow skms beng dsspated b the maro roughness of the steps. The zones formed b the step surfaes and the pseudo bottom ontan rerulaton vortes whh ause part of the energ dsspaton n skmmng flow regme. The shear transfer from the man stream to the rerulaton zone auses the mantenane of the vortes. The exhanges of momentum between the man and the avt flows results n sgnfant energ dsspaton. Ths paper deals wth the skmmng flow regme. A transton flow regme has been observed between the nappe and the skmmng flow regmes b Esser and Horner (978). At the downstream part of the nepton pont of ar entranment, transton flow exhbts strong splashng and droplet ejetons, and ar pokets from small to large szes an be observed. In hgh velot flows on spllwas and steep hutes, the turbulent boundar laer reahes the flow surfae at the pont of nepton, ntatng ar entranment nto the flow stream. Observatons have shown that there s a developng flow regon after the nepton pont of ar entranment. For some dstane n the developng regon, there s a regon of partall aerated flow, untl the ar bubbles penetrate to ther maxmum depth n the water and the flow beomes full aerated. After the developng regon, there s a full developed aerated flow regon where unform ondtons have been obtaned. 3 Assumed Flow Condton The numeral smulaton results have been ompared wth those obtaned from the expermental model stud on the two-phase skmmng flow down a stepped hute onduted at the laborator of Hdrauls, Hdrolog and Glaolog (VAW) of the Swss Federal Insttute of tehnolog (ETH), Zurh. The experments have been onduted n a prsmat retangular hannel of wdth.5 m and length 5.7 m. The bottom angleφ = 3 and the step heght s=46.mm have been used for omparson. A shema of the laborator model s shown n Fg.. Fg.: Shemat geometr of the laborator model and observed flow feld, Frzell et al () In present paper a set of laborator measurements of above menton researhers are used for the verfaton of the numeral model whh orresponds to skmmng flow wth unt wdth dsharge of q=.573 m /s and Froude Number of Fr=.6.For ths ase, the laborator measurements for the nepton pont the non-dmensonal poston and water depth are reported h / s =. 6 and z / s = 9.5, respetvel. Hene, onsderng the step heght s=46.mm, the depth at the nepton pont and the vertal dstane measured from the spllwa rest to the pont of nepton are alulated as h =.8m and z =39m, respetvel 4 Mathematal Equatons The water phase mathematal equatons used n the NASIR flow solver are shallow water equatons Numeral Analzer for Sentf and Industral

3 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 4 modfed for a oordnate sstem wth an axs normal and two axes(x' and ) parallel to the bed surfae aordng to Sabbagh-Yazd (6). ( ) ( v) + + = t x ( ) ( ) (v u ) g osα h = g Snα t x x g S fx ( v) ( v) (v v) g osα h = g S f t x () n whh: n + v n v + v Sfx = ;S 4 3 f = () 4 3 In these equatons x' s the axs tangental to the hute slope and s the same as the axs n the global oordnate sstem; u' and v are the velot omponents n x' and dretons, respetvel; h' s the flow depth perpendular to the hute bed surfae and g s gravt aeleraton; α s the hute angle; S fx and S f are the bed surfae frton slopes n x' and dretons, respetvel and n s Mannng s frton oeffent. In order to use the above desrbed mathematal model for skmmng flow over stepped spllwa, the effet of the steps has been smulated as bed roughness. For smulatng the equvalent bed roughness the followng expermental relatonshp has been used: K f [.5 sn( )] b = φ ; 9 φ 55 (3) Dh, w where s the hdraul dameter. D h,w 5 Computaton of Vertal Velot Profle The depth averaged velot feld parallel to the bed surfae that an be omputed b solvng mathematal model desrbed n aforementoned seton, ma be used for omputaton of the velot profle n flow depth. Suh a velot profle an be omputed usng empral relatons suggested for skmmng flow over stepped spllwas. The relaton used for defnng velot profles n flow depth s the Boes and Hager s relaton (3), as follows: Requrements u = 4.3.5( ) ;.4. 8 u u u = ; >. 8 (4) where s the haraterst mxture depth wth loal ar onentraton of C=. and u s the mxture surfae velot. 6 Expermental Relatons for Ar Conentraton Dstrbuton Followng expermental relatons are used n the post-proessor for smulatng ar entranment. Frzell et al. () gave a relatonshp for defnng mean ar onentraton: 6 x L C = (5) mean Matos () derved: x L x L C mean = + 97 exp 97 ln( ).97 ; < < 3 = x L Cmean ; 3 x L (6) where C mean s the mean ar onentraton, x s the dstane from the rest along the hute slope, L s the blak water length from spllwa rest to nepton pont, and s the flow depth at the pont of nepton. Ar onentraton profles, ompared wth an ar bubble dffuson model proposed b Chanson (), are expressed b: C() = tanh (K ) (7) D h To defne K' and D', the followng relatonshps have been ft to expermental data:.993 K =.79C mean (8) 3 D = Cmean.6Cmean +.59Cmean.7 (9) The followng relatons have been used for defnng the nepton pont dstane from the rest along the hute. Boes and Mnor () derved: h L = (9) (snφ) s Chanson (994) derved: L = 9.79(snφ) F () h K

4 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 5 Chaman (997) derved: L = 58 8.F K () Matos et al. () derved: L = Fs K () where, K = s.osφ, F h =q / ( g snφ K 3 ) /, F =q/(g.(s/l).k 3 ) /, F s =q/(g s 3 ) /, l and s are the step length and heght, respetvel and φ s the hute angle. The followng relatons have been used for defnng the mxture flow depth at the pont of nepton. Boes and Mnor () derved:.. s h = (3) ( snφ). 3 Chanson (994) derved: = F (4).4 h K (snφ) 7 Numeral Soluton of Flow Feld In the frst stage, the flow feld s numerall solved wth NASIR software. Ths software uses fnte volume method for soluton of the modfed shallow water equatons for a oordnate sstem wth an axs tangent to the slope dreton (x') and another axs n transversal horzontal dreton (), Sabbagh-Yazd (7). The depth and velot values are depth-averaged values omputed on a trangular unstrutured mesh usng the fnte volume method. The equatons have been onverted to dsrete form usng the overlappng ell vertex method. For the numeral modelng of the problem, free slp boundar ondtons n wall boundares are appled b mposng zero normal velot. At boundar, super rtal nflow ondton s mposed b enforng depth and velot omponents. The flow depth and velot values mposed at the nflow boundar are.63 m and.9 m/s, respetvel.the outflow boundar ondton s onsdered free from mplementaton of an varables. 8 Velot Profle Computaton As the frst post-proessng, the depth averaged velot feld parallel to the bed surfae whh s solved b NASIR software s used to ompute the velot profle n flow depth. The velot profle s omputed usng empral relatons suggested for skmmng flow over stepped hutes. Fg. shows normalzed velot (u/u ) n terms of relatve flow depth (/ ). The RMS values for the three statons are almost the same; beause steps dsspate energ and the flow keeps an almost onstant velot after a short dstane from the rest. So the relatonshp used for velot dstrbuton n the numeral smulaton reates small RMS values. /. numeral expermental. v/v Fg.: normalzed velot n terms of relatve flow depth; omparson between numeral and expermental results, RMS=.3 9 Ar Conentraton Computaton For proessng the ar onentraton all through water flow feld, expermental relatons have been used for omputng the nepton pont dstane from the rest, the flow depth n ths seton and the depth-averaged ar onentraton n eah pont. Then the depth values obtaned from the frst stage have been modfed onsderng the bulkng aused b the entraned ar. Fnall, the ar dstrbuton profles n flow depth have been obtaned from the omputed depth-averaged values usng expermental relatons. If the non-dmensonal vertal dstane from the nepton pont (Fg.) s defned as Z = ( z z ) / h, the statons wth Z =-.5, Z =.9 and Z =4.3 are used for omparson of the omputed results of present numeral model wth expermental measurements reported for ths phsal model b Frzell et al (). The atual vertal dstanes of the statons from the spllwa rest are lsted n the followng table. Table : vertal dstanes of the statons measured from the rest

5 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 6 Staton 3 Z z (m) The errors alulated for the relatons used for omputng the loaton of the nepton pont are lsted n the table for omparson. Table : errors n loaton of nepton pont relaton error Boes and Mnor () 37.8% Chanson (994) 39.4% Chaman (997).9% The errors alulated for the relatons used for omputng the flow depth at the nepton pont are lsted n the table 3 for omparson. Table 3: errors n flow depth at the nepton pont relaton error Boes and Mnor () 3.6% Chanson (994) 3.6% Matos et al. () a small value reated In order to hoose the best relaton for defnng mean ar onentraton, the program has been run for the ase of usng Chanson s relaton for nepton pont loaton and flow depth at that seton. The root mean square (RMS) of the ar onentraton profles n terms of relatve flow depth (/ ) has been ompared for the two relatons used for defnng mean ar onentratons. The results are lsted n table 4. Table 4: RMS values of ar onentraton profles (usng Chanson s relatons for nepton pont) RMS values of ar relaton onentraton profles Z=-.5 Z=.9 Z=4.3 Frzell et al. () Matos et al. () 8.5 It an be onluded from table 4 that the relaton proposed b Frzell et al. () reates smaller values of RMS when ompared to expermental results. Now the program s run for the ase of usng Chaman s relaton for nepton pont loaton, Chanson s relaton for flow depth at the pont of nepton, and Frzell et al. s relaton for mean ar onentraton. Fg.s 3 to 5 show ar onentraton (C) n terms of relatve flow depth (/ ) for the three statons mentoned before. /. numeral expermental. Fg.3: ar onentraton as funton of relatve flow depth at staton ; omparson between numeral and expermental results, RMS=.375 /. numeral expermental. Fg.4: ar onentraton as funton of relatve flow depth at staton ; omparson between numeral and expermental results, RMS=.6

6 Proeedngs of the 9th WSEAS Internatonal Conferene on Automat Control, Modelng & Smulaton, Istanbul, Turke, Ma 7-9, 7 7 /. numeral expermental. Fg.5: ar onentraton as funton of relatve flow depth at staton 3; omparson between numeral and expermental results, RMS=.479 Conluson The numeral analss of skmmng flow on stepped spllwas has been arred out at two stages. At the frst stage, numeral soluton of modfed shallow water equatons for nlned oordnate sstem s performed usng NASIR fnte volume flow solver. At the seond stage, a post-proessor routne s developed, whh suessfull omputes the velot profle n vertal dreton and uses varous empral relatons for alulatng ar onentraton profles. The omparson of the numeral results wth the expermental measurements shows that the followng relatons for omputng ar onentraton profles produe better results when the are used as means of post-proessng for ar entranment n skmmng flow over stepped hutes: Relaton proposed b Chaman (997) for defnng nepton pont Relatons suggested b Boes and Mnor () as well as Chanson (994) for defnng flow depth at the nepton pont Relaton proposed b Frzell et al. () for defnng mean ar onentraton Relaton suggested b Chanson () for defnng ar onentraton profles Referenes: [] James C. S., Ohtsu I., Yasuda Y., Takahas M., Tatewar P., Ingle N. and Pore D. (), Dsusson of On set of Skmmng Flow on Stepped Spllwas. B M. R. Chaman, N. Rajaratnam., J. Hdraul Eng., Vol.7, No.6, pp [] Dewals B.J., Andre S., Shless A.J. and Protton M. (4), Valdaton of a Quas-D Model for Aerated Flows over Mld and Steep Stepped Spllwas, 6th Internatonal Conferene on Hdro-nformats, pp [3] Chanson H. (4), Hdrauls of Stepped Chutes: The Transton ffow, J. Hdraul Res., Vol.4, No., pp [4] Chanson H. and Toombes L., (), Expermental Investgaton of Ar Entranment n Transton and Skmmng Flows Down a Stepped Chute, Researh Report No. CE 58, [5] Chaman M.R. (997), Skmmng Flow n a large Model of a Stepped Spllwa, Thess for the Degree of Dotor of Phlosoph n Water Resoures Engneerng, [6] Rajaratnam N. (9), Skmmng Flow n Stepped Spllwas., J. Hdraul Eng., Vol.6, No.4, pp [7] Khatsura R.M. (5), Hdrauls of Spllwas and Energ Dsspaters, pp [8] M. Boes, H. Hager, Hdraul Desgn of Stepped Spllwas, J. Hdraul Eng., Vol.9, No.9, 3, pp [9] Boes M. and Hager H. (3), Two-Phase Flow Charatersts of Stepped Spllwas, J. Hdraul Eng., Vol.9, No.9, pp [] Sabbagh-Yazd, S.R. (6), Spllwa Flow Modelng b Fnte Volume Soluton of Sloppng Depth Averaged Equatons on Trangular Mesh; Applaton to KAROUN-4 Spllwa, th WSEAS Internatonal Conferene on Appled Mathemats, Dallas (Texas), USA [] Sabbagh-Yazd, S.R., N.E. Mastoraks and Zounemat M. (7), Velot Profle over Spllwa b Fnte Volume Soluton of Sloppng Depth Averaged Flow, nd IASME/WSEAS Internatonal Conferene on Contnuum Mehans, Slovena (to be appeared n proeedng CD) [] Yoon T.H. and Kang S.K. (4), Fnte Volume Model for Two-Dmensonal Shallow Water Flows on Unstrutured Grds, J. Hdraul Eng., Vol. 3, No.7, pp