Key words: Hydraulic jump, Theoretical modeling, Stilling basin, Non-prismatic basins, Expanding channels, Negative steps

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1 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) Theretical meling f free hyraulic jumps at negative step in raial stilling basins Abelazim M. Negm, G.M. Abel-Aal, T.M. Owais, an A.A. Habib Steps may be use t cntrl the lcatin f the hyraulic jump wnstream f hyraulic structures. Extensive stuies have been cnucte t investigate the effect f steps in rectangular basins but nt yet in raial basins. In the present paper, theretical mels are evelpe t preict the epth ratis f the raial hyraulic jumps when steps present wnstream f the cntrl structures. Bth the mmentum an cntinuity equatins in ne imensin are applie t the cntrl vlume where the jump begins an ens. Bth frms f the hyraulic jump (A an B) at negative step in the raial stilling basins are ealt in thrugh the present research. An experimental prgram is cnucte t cllect experimental ata t enable verificatin f the evelpe theretical mels. G agreement between theretical an experimental results is btaine encuraging the use f the evelpe mel in fiel applicatins where measurements are ifficult an cstly r nt pssible. Key wrs: Hyraulic jump, Theretical meling, Stilling basin, Nn-prismatic basins, Expaning channels, Negative steps Intructin Hyraulic jumps are ne f the mst frequently use energy issipatrs. It may be free r submerge epening n bth the lcatin an the initial epth f the jump relative t the gate. Mst f the stuies n ifferent types f hyraulic jump are presente in Hager (99). The hyraulic jump may be als frme in prismatic r in nn-prismatic channels, an may be frce r nn-frce. Base n stuies f Khalifa an McCrquale (979) an Abel-Aal (998), it was fun that the relative epth f free raial jump as well as the length f the jump was shrter than thse frme in rectangular channels. While the rate f energy lss increases thrugh the the jump in raial basin cmpare t that in rectangular ne. A rp r negative step is use when the wnstream epth is larger than the sequent epth fr a classic jump t insure the jump ccurrence an t prvie mre stability f the jump fr a wie range f the wnstream values. The available stuies regaring the frmatin f hyraulic jumps at steps are fr nes frme in rectangular basins. Hager (985) perfrme experimental an theretical investigatin n B-type jumps at abrupt rps. Hager an Bretz (98) iscusse the characteristics f A an B jumps at negative steps. The ranges f relative epth an length representative f these types f jump were analyze with particular attentin t the esign f stilling basins. Ohatsu an Yasua (99) presente a systematic investigatin n the characteristics f the hyraulic jump ver a wie range f negative steps. All the cases were stuie theretically by the use f mmentum equatin with measurements f the pressure istributin ver the face f the step. Negm (99) stuie Assciate Prfessrs, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt, amnegm85@htmail.cm Prfessr f Civil Engineering, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt. Pst grauate stuent, Assistant Lecturer, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt

2 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) theretically an experimentally the hyraulic jump frme in slping an hrizntal rectangular channel with psitive r negative step. Armeni et al (000) investigate the pressure fluctuatins beneath a hyraulic jump that evelpe ver a negative step. The stuy was carrie ut experimentally using tw ifferent rps, an abrupt rp an a rune ne. The inflw an the utflw cnitins were varie t btain B-jump an a wave jump. Recently a few stuies were cnucte n the frmatin f the submerge hyraulic jump at negative steps in raial basins. Negm et al (00a,b,c) investigate experimentally an theretically the effect f submergence, an the relative psitin f negative step n the characteristics f the submerge raial jump. They als stuie the effect f the relative height f en sill with r withut the negative step. It was fun that the ratis f epth an length f the jump slightly ecrease with the increase f the relative height f en sill while the energy lss rati is increase. This paper presents a theretical stuy t preict the epth ratis f the free raial hyraulic jump in the presence f negative step wnstream f a cntrl structure. Tw theretical mels are btaine fr the cases f B an A jumps, by applying the ne imensinal mmentum an cntinuity equatins. The evelpe theretical mels are verifie with the cllecte experimental ata. Develpment f the theretical mels B-jump at negative step Figures a presents a efinitin sketch fr negative B-jump that cul be frme in raial stilling basin prvie with a vertical negative step. This type f jump is frme such that it begins upstream f the step an ens wnstream f the step. The assume pressure istributins are als shwn. Bth the -D mmentum an cntinuity equatins are use t evelp a theretical esign mel fr cmputing the epth rati fr the free hyraulic jump frme in raial stilling basin prvie with a rp. The present evelpment is base n the fllwing assumptins: (a) the flw is raial an steay (b) the liqui is incmpressible (c) the channel is hrizntal an has smth bunaries, () hyrstatic pressure istributin alng the cnsiere reach f flw (e) unifrm velcity istributin, i.e. the values f the kinetic energy crrectin factr an the mmentum crrectin factr, α an β are cnsiere unity, an (f) the effects f air entrainment an turbulence are neglecte. In the present stuy, the cntrl vlume where the mmentum equatin is applie starts at the beginning f the jump an ens at the en f the jump in the irectin f the flw, Figure (a). The mmentum equatin fr the case f B-jump at negative step is written as fllws: γq P P PSsin P = (βv β V ) g () in which : P : hyrstatic pressure at the beginning f the jump. P : hyrstatic pressure at the en f the jump. P : hyrstatic pressure n the face f the step. P S : channel sie pressure. U.S

3 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) P P P z L j P s P P P r r r P s Figure a. Definitin sketch shwing the frmatin f the jump an the assume pressure istributins in case f B-jump at negative step γ b γ b P =, P =, an γz( z)b P = () γ Ps = [(r r )( ) (r r )( ( z) ( z)) ] () Substituting fr P,,P, P an P S frm Eqs. () an () int Eq. () t btain γb γ b γ (r r )( ) (r r)( γ γq z( z)b = (βv β V ) g Applying the cntinuity equatin: z z z) sin ()

4 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) Q= b V = b V (5) Where b, an b are the channel withes at the beginning, an the en f the jump. b =r sin /, b =r sin /, an b =r sin / () Where b, is the channel with at the step. Substituting frm Eq. () in Eq. (5) an slving fr V then: V = V r /r =V / r (7) Where / =, an r /r = r Substituting frm () an (7) in () an assuming β =β =.0, t get γr [ )( ) (r r)( z z z) ] γ (r sin γr z( z)sin r γr sin γ V = V r sin (V g r (8) Diviing equatin (8) by γ. r.sin/ t get ) sin r r ( r )( ) (r r )( r z z z ) rz r ( V z) = g r ( ) r (9) Let r /r =r, z/ =Z, an / = then multiplying Eq. (9) by r an simplify t get (r rr) [( Z)(r rr)] r[ Z (r r ) Z(r r ) (r ) (r) r] F ( r ) = 0.0 (0) Equatin (0) can be rearrange t take an explicit frm as fllws F ( r rr ) [( Z)(r rr )] [ Z ( r r ) Z( r r ) (r ) (r ) r] r = () ( r ) Fr n step case where Z=0.0, r=r, an =, Eq. (0) tens t the previusly evelpe equatin by Abel-Aal et al (998). Furthermre, if the basin is rectangular an cntains

5 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) negative step, i.e., case f B-jump where r=r =.0, an =.0, Eq. (0) tens t the previusly evelpe equatins by Hager (985) an by Hager an Bretz (98). A-jump at negative step Fr the case f A-jump at negative step that is frme entirly upstream f the step as shwn in Figure (b), the sie pressure frce cul be expresse as fllws: γ PS = [(r r )( ) (r r )( ) ] () U.S. P z P P L j P s P P P r r r P s Figure b. Definitin sketch shwing the frmatin f the A-jump in raial basin with negative step an the assume pressure istributin The mmentum equatin in the irectin f the flw may be written in the fllwing frm

6 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) γq P P P PSsin = (βv β V ) g () in which P is given as fllws γ( z) b P = () By substituting frm equatins (), (), an () in equatin () : γ( z) b γ b γ z( sin. γq [[(r r )( ) (r r )( ) ] = (β V β V ) (5) z)b Similarly t the case f B-jump, ne cul btain r( Z)( r) r( Z) γ [ r (rr )] [ rz( Z) r r r r( Z) ] F [ r( Z) ] = 0. 0 () g r( Z) Equatin () may be rearrange t take t the fllwing explicit frm F = r ( Z) ( r) r ( Z) [ r (r r )] [ rz( Z) r r r r ( Z) ] [ r( Z) ] r ( Z). (7) Fr n step case where Z=0.0, r=r, an =, Eq.() tens t the previusly evelpe equatin by Abel-Aal et al (998). Furthermre if the basin is rectangular an cntains negative step, i.e, case f A-jump where r=r =.0, an = -Z, Eq. () tens t the previusly evelpe equatin by Hager an Bretz (98). Experimental wrk The experimental wrk f this stuy is cnucte using a re-circulating ajustable flume f 5.0 m lng, 5 cm eep an 0 cm wie. The ischarges were measure using pre-calibrate rifice meter fixe in the feeing pipeline. The tailgate fixe at the en f the flume was use t cntrl the tail-water-epth f flw. The raial basin was mae frm a clear prespex t enable visual inspectin f the phenmenn being uner investigatin. The mel length was kept cnstant at 0 cm an the angle f the ivergence was kept cnstant t 5.8. The mel was fixe in the mile thir f the flume between its tw sie-walls, Figure (). A smth blck f w was frme t fit well insie the basin mel extening frm upstream the gate by 5.0 cm t the psitin where the rp was esire. The w was painte very well by a waterprf material (plastic) t prevent w frm changing its vlume by

7 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) absrbing water. A fixe height f the rp f.5 cm was use at ifferent psitins f the rp (r =r, r =.r, r =.r, r =.5r, an r =.7r ) wnstream frm the gate pening were teste uner the same flw cnitins. Als a rp f variable heights (.5,.5, an 5.5) was use at tw psitins f the rp (r =.r, an.5r ). The range f the experimental ata were as fllws: Frue numbers (.0-7.0), r (.-.), relative psitin f the rp, r (-.7), an relative height f the rp, z/ (0..). Each mel was teste using five ifferent gate penings an five ischarges fr each gate pening. The measurements were recre fr each ischarge. The ttal number f runs was 75. A typical test prceure cnsiste f (a) a gate pening was fixe an a selecte ischarge was allwe t pass. (b) the tailgate was ajuste until a free hyraulic jump is frme. (c) nce the stability cnitins were reache, the flw rate, length f the jump, water epths upstream an at the vena cntracta wnstream f the gate in aitin t the tail water epth an the epth f water abve the step were recre. The length f jump was taken t be the sectin at which the flw epth becmes almst level. These steps were repeate fr ifferent ischarges an ifferent gate penings an s n till the require ranges f the parameters being uner investigatin were cvere. Vertical gate b=8cm B= 0cm 50cm Plan L b=0cm Figure. General sketch f the experimental mel Verificatin f the evelpe mels Equatins () an (7) are slve an pltte against the experimental values as inicate in Figure fr the B-jump an in Figure () fr the A-jump. Clearly, g agreement between theretical values (F The ) an the values frm measurements (F Exp ). The values f the cefficient f eterminatin (R ) in bth figures are 0.99 an 0.98 respectively while the values f mean relative abslute errr (MRE) are 0.0 an 0.09 respectively. Als, Figure (5) an () present the cmparisn between the theretical values an the experimental nes fr typical values f relative step height. Figure (5) shws the verificatin fr the first psitin f the step at L s /L b =0.0 (r=) where the B-jumps are frme in mst f the case. Figure () shws the same fr the thir psitin at L s /L b =0.5 (r=.) where the A-jumps are frme in mst f the cases. Clearly, g agreement is btaine in bth cases. Cnclusins Theretical mels, equatins () an (7) are evelpe fr the preictin f the epth ratis f the ifferent frms f hyraulic jumps that cul be frme in raial stilling basins when negative steps are existe in the basins. The hyraulic jumps may be negative B- jump r negative A-jump. An experimental prgram is cnucte t cllect experimental ata n the tw types f jumps using raial basins f cnstant ivergence angle an fixe length but variable psitins f the step in the basin. The evelpe theretical mels are verifie using

8 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) the cllecte experimental ata. G agreement is btaine with mean relative abslute errr (MRE) f 0.0 an 0.09 fr B an A jumps respectively encuraging the use f the evelpe mel in the preictin f the epth ratis f the cnsiere jumps. 9 8 Line f perfect agreement 7 F The. 5 F Exp. r=.00 r=. r= Figure. Verificatin f Eq. () fr negative B-jump 9 8 Line f perfect agreement 7 F The. 5 F Exp. r=. r=.50 r= Figure Verificatin f Eq. (7) fr negative A-jump

9 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) / 7 5 Eqn. () Z=.9 Z=.0 F Z= Figure 5. Relatinship between / an F fr ifferent relative height f step, Z, shwing experimental values versus Eq. () fr r= an negative B-jump / 5 Eqn. (7) Z=.9 Z=.0 F Z= Figure 5. Relatinship between / an F fr ifferent relative height f step, Z, shwing experimental values versus Eq. () fr r=. an negative A-jump.

10 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) References - Abel-Aal, G.M., El- Saia, A.A., an Saleh, O.K. (998). Hyraulic Jump within a Diverging Rectangular Channel. Engineering Research Jurnal, Faculty f Engineering, Helwan University, Mataria, Cair, Vl. 57, June, 998, PP Armeni, V., Tscan, P., an Firtt, V. (000). The Effects f a Negative Step in Pressure Fluctuatins at the Bttm f a Hyraulic Jump. Jurnal f Hyraulic Res., Vl.8, N. 5, PP Hager, W. H., (985). B-Jumps at Abrupt Channel Drps. Jurnal f Hyraulic Eng., Vl., N.5, PP Hager, W.H. an Bretz, N.V. (98). Hyraulic Jumps at Psitive an Negative Step. Jurnal f Hyraulic Research, Vl., N., PP Hager, W.H. (99). Energy Dissipatrs an Hyraulic Jumps. Kluwer Acaemic Publicatins, Drrecht, The Netherlans. - Khalifa, A.M. an McCrquale, J.A. (979). Raial Hyraulic Jump. Jurnal f the Hyraulic Divisin, ASCE: 05(HY9), PP Negm A.M., (99). Hyraulic Jumps at Psitive an Negative Steps n Slping Flrs. Jurnal f Hyraulic Research, Vl., N., PP Negm, A.M., Abel-Aal, G.M., Elfiky, M.I., an Mhme, Y.A. (00a). Theretical an Experimental Evaluatin f the Effect f En Sill n Characteristics f Submerge Raial Hyraulic Jump. Sc. Bulettin, Faculty f Engineering, Ain Shams Univ., Cair, Egypt (Accepte). 9- Negm, A.M., Abel-Aal, G.M., Elfiky, M.I., an Mhme, Y.A.(00b). Characteristics f Submerge Hyraulic Jump in Raial basins with a Vertical Drp in the Be. AEJ, Faculty f Eng., Alex. Univ., Egypt (Accepte). 0- Negm, A.M., Abel-Aal, G.M., Elfiky, M.I., an Mhme, Y.A. (00c). Hyraulic Characteristics f Submerge Flw in Nn-prismatic basins. Prc. Of 5 th Int. Cnf. On Hyrscience an Engineering, ICHE00, Sept. 8-, Warsw, Plan. - Ohtsu, I., an Yasua, Y., (99), Transitin Frm Supercritical t Subcritical Flw at an Abrupt Drp, Jurnal f Hyraulic Research, Vl. 9, N. : pp Ntatins b = cntracte with f the channel ; B = with f the channel; F = Frue s number at the initial epth; z = the rp height; Z= the rati f z t ; L j = the length f the hyraulic jump; P = the hyrstatic pressure at the beginning f the jump; P = the hyrstatic pressure at the en f the jump; P s = channel sie pressure frce; P = pressure frce ue t step; P = pressure frce just wnstream f the step in the case f A-jump; Q = rate f flw; r = raius at the beginning f the jump ; r = raius at the en f the jump ; r = the rati f r t r ; r = raius at the en f the step;

11 Prc. f th Int. River Engineering Cnf., 8-0 Jan 00, Shahi-Chamran Univ., Ahvaz, Iran (CD ROM) r = the rati f r t r ; R = the cefficient f eterminatin: V = average velcity at the initial epth; V = average velcity at the sequent epth; V = average velcity just wnstream f the step in case f A-jump; = water epth at vena cntracta wnstream the gate (initiaal epth); = sequent water epth; = the relative water epth, / ; = epth f water abve the step; = the rati f t ; γ = the specific weight, an = the angle f ivergence.

INVESTIGATION OF B-JUMP NEGATIVE STEP IN RADIAL STILLING BASINS

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 INVESTIGATION OF B-JUMP NEGATIVE STEP IN RADIAL STILLING BASINS A.M. Negm 1, G. M. Abdel-Aal 1, T.M. Owais and A.A. Habib 3 1 Assciate