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 Prfessrs, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt, E-mail: amnegm85@yah.cm Prfessr f Civil Engineering, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt 3 Assistant Prfessr, Dept. f Water & Water Structures Eng., Faculty f Engineering, Zagazig University, Zagazig, Egypt ABSTRACT Presence f steps in radial stilling basins affect psitively r negatively the characteristics f the free hydraulic jumps depending upn the type f the step, the height f the step, the psitin f the step in the basin and the dminant flw cnditins. The effects f all these factrs n the characteristics f the free radial hydraulic B-jump are addressed thrugh the present experimental investigatin. It is fund that the depth rati and the length f jump rati increase by the increase f the relative step height while the energy lss is decreased. A theretical mdel fr the cmputatin f the energy lss is develped. Als, statistical mdels are prpsed fr predicting the characteristics f the frmed free hydraulic B-jump at the negative step in the radial stilling basin. The experimental results are cmpared with the theretical nes fr energy lss and with the staistical nes fr the length f jump. Gd agreements are btained between the results f the develped mdels and the experimental nes. Keywrds: Hydraulic jumps, Stilling basin, Nn-prismatic basins, Expanding channels, Negative steps, Sudden drp, Statistical mdeling

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 1. INTRODUCTION Hydraulic jumps are ne f the mst frequently used energy dissipatrs. It may be free r submerged depending n bth the lcatin and the initial depth f the jump relative t the gate. Mst f the studies n different types f hydraulic jump are presented in Hager [1]. The hydraulic jump may be als frmed in prismatic r in nnprismatic channels, and may be frced r nn-frced. Based n studies f Khalifa and McCrqudale [] and Abdel-Aal et al. [3], it was fund that the relative depth f free radial jump as well as the length f the jump was shrter than thse frmed in rectangular channels, while the rate f energy lss increases thrugh the the jump in radial basin cmpared t that in rectangular ne. A drp r negative step is used when the dwnstream depth is larger than the sequent depth fr a classical jump t insure the jump ccurrence and t prvide mre stability f the jump fr a wide range f the dwnstream values. The available studies regarding the frmatin f hydraulic jumps at steps are fr nes frmed in rectangular basins. Hager [4] perfrmed experimental and theretical investigatin n B-type jumps at abrupt drps. Hager and Bretz [5] discussed the characteristics f A and B jumps at negative steps. The ranges f relative depth and length representative f these types f jump were analyzed with particular attentin t the design f stilling basins. Ohatsu and Yasuda [6] presented a systematic investigatin n the characteristics f the hydraulic jump ver a wide range f negative steps. Many f the bserved cases were studied theretically by the use f mmentum equatin. Als, measurements f the pressure distributin ver the face f the step were analyzed. Negm [7] studied theretically and experimentally the hydraulic jump frmed in slping and hrizntal rectangular channel with psitive r negative step. Armeni et al. [8] investigated the pressure fluctuatins beneath a hydraulic jump that develped ver a negative step. The study was carried ut experimentally using tw different drps, an abrupt drp and a runded ne. The inflw and the utflw cnditins were varied t btain B- jump and a wave jump. Recently, a few studies were cnducted n the frmatin f the submerged hydraulic jump at negative steps in radial basins. Negm et al [9,10,11] investigated experimentally and theretically the effect f submergence, the relative height f the step, the relative height f end sill and the relative psitin f negative step n the characteristics f the submerged radial jump. They als studied the effect f the relative height f end sill with r withut the negative step. It was fund that the ratis f depth and length f the jump slightly decrease with the increase f the relative height f end sill while the energy lss rati is increased. This paper presents the results f a theretical and experimental study n the characteristics f B-jump frmed at negative step in radail basin. A theretical equatin is develped (and verified using experimental data) t predict the energy lss rati f the free radial hydraulic jump in the presence f negative step in the radial

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 basin. The effects f the initial Frude number, the relative height f step and the relative ppsitn f step n the characteristics f the B-jump are addressed.. THEORETICAL CONSIDERATION.1 Dimensnal Analysis Figure 1 shws a definitin sketch fr the B-jump at negative step in the radial stilling basin. The fllwing functin can be frmed: f ρ,g, µ,d,d,d,v,z,r,r,r,l (1) ( ) 0 1 3 1 1 r j = in which ρis the density f water, g is the gravitatinal acceleratin, µ is the dynamic viscsity f water, d 1 is the initial depth f jump, d is the sequent depth f jump, d 3 is the depth f water at the step, V 1 is the average velcity f flw at the beginning f jump, z is the height f the negative step, r 1 is the radius where the jump begins, r is the radius where the jump ends, r 3 is the radius where the step is cnstructed and L j is the length f jump. Using the dimensinal analysis principle based n the three repeating variables ρ, d 1 and V 1, Eqn. (1) becmes L j d d3 z r1 r r3 f,,,,,,,f 1,R1 = 0 d () 1 In Eqn. (), bth d /d 1 and d 3 /d 1 are functin f F 1, r /d 1 and r 1 /d 1 gives r /r 1 and r 3 /d 1 and r 1 /d 1 gives r 3 /r 1 while the effect f R 1 is neglected as the viscsity has a negligible effect n the hydraulic jump characteristics in the present study because the temperature was fixed during the curse f the experimental wrk. Als, r /r 1 is a functin f the length f the jump. Equatin () becmes L j z r = 3 f,, (3) r1 Similar relatinships fr the depth rati and fr the energy lss rati culd be btained. The nature f the functin presented in Eqn. (3) will be determined based n the experimental data using the multiple linear regressin analysis... Mmentum Apprach Based n the use f the 1-D mmentum and cntinuity equatins as applied n the cntrl vlume shwn in Figure 1 fr B- jump frmed at negative step in radial basin. Negm et al. [1] develped the fllwing theretical equatin fr B jump based n the shwn frces in Figure 1. d 3 (r + r r) d Z(4rd+ r d) + d [(d+ Z)(r (r r r)] d r [ + Z 1) + d(r 1) + r] 6F 1 (d (r+ r r ) + 1) = 0.0 (4) in which d =d /d 1, r =r /r 1, d=d 3 /d 1, r=r 3 /r 1 and Z=z/d 1.

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 Equatin (4) can be slved easily if rearranged t take the fllwing explicit frms as in Eqn. (5) F 1 d 3 (r + rr ) d [(d + Z)(r rr )] [ + Z (r+ r ) + Z(4rd+ r d) + d (r 1) + d(r 1) + r] dr = (5) 6(d r 1) These equatins are verified by Negm et al.[1] using experimental cllected data n the same labratry flume, Habib [13]. U.S. 1 d 3 d P 1 d 1 P 3 z P L j P s P 1 P 3 P r 1 r 3 r P s Figure 1. Definitin sketch shwing the frmatin f the B-jump at negative step in radial stilling basin. Shwn als the frces used in applying the mmentum equatin by Negm et al. [1]

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003.3. Energy Apprach Applying the energy equatin at sectins 1 and where the jump begins and ends and assuming unifrm velcity distributin and hydrstaic pressure distriutin, ne btains V1 V E + L = E1 E = d + + z d 1 (6) g g Keeping in mind that F 1 =V 1 /(gd 1 ) 0.5 and V =V 1 /r d, ne btains: 1 3 E1 = 1+ + Z and r d + = E (7) r d Knwing that E L /E 1 =1-E /E 1, ne gets E E L 1 r d ( + d + Z) = (8) r d ( + F + Z) 1 3. EXPERIMENTAL WORK The experimental wrk f this study was cnducted using a re-circulating adjustable flume f 15.0 m lng, 45 cm deep and 30 cm wide, Habib [13]. The discharges were measured using pre-calibrated rifice meter fixed in the feeding pipeline. The tailgate fixed at the end f the flume was used t cntrl the tail-waterdepth f flw. The radial basin was made frm a clear prespex t enable visual inspectin f the phenmenn being under investigatin. The mdel length was kept cnstant at 130 cm and the angle f the divergence was kept cnstant t 5.8. The mdel was fixed in the middle third f the flume between its tw side-walls as shwn in Figure. A smth blck f wd was frmed t fit well inside the basin mdel extending frm upstream the gate by 5.0 cm t the psitin where the step was desired. The wd was painted very well by a waterprf material (plastic) t prevent wd frm changing its vlume by absrbing water. A fixed height f the step f.5 cm was used at tw different psitins f the step (r 3 =r 1 and r 3 =1.17r 1 ) dwnstream frm the gate pening were tested under almst the same flw cnditins. Testing the ther psitins f the step at this height (.5 cm) did nt prduce B-jump. The range f the experimental data were as fllws: Frude numbers (.0-7.0), r (1.-1.4), relative psitin f the step, r (1.0-1.17), and relative height f the step, z/d 1 (0.0 1.35). Each mdel was tested using five different gate penings and five discharges fr each gate pening. The measurements were recrded fr each discharge. A typical test prcedure cnsisted f (a) a gate pening was fixed and a selected discharge was allwed t pass. (b) the tailgate was adjusted until a free hydraulic jump is frmed. (c) nce the stability cnditins were reached, the flw rate, length f the jump, water depths upstream and at the vena cntracta dwnstream f the gate in additin t the tail water depth and the depth f water abve the step were recrded. The length f jump was taken t be the sectin at which the flw depth becmes almst level. These

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 steps were repeated fr different discharges and different gate penings and s n till the required ranges f the parameters being under investigatin were cvered. Vertical gate θ b=18cm B=3 0cm 50cm Plan L b=130cm Figure. General sketch f the smth radial stilling basin mdel 4. VERIFICATION OF THEORETICAL EQUATIONS Figure 3a presents the cmparisn between the theretical values f the energy lss and the experimental nes fr all data at the tw different relative psitins f the step (r=1.0 and r=1.17). The crrelatin cefficient between bth values is 0.979 and the mean relative abslute errr, MRE, is 0.034. Figure 3b shws the cmparisn fr typical values f the relative height f the step at the first psitin (r=1.0) as an example. Clearly, gd agreement is btained in bth cases. 0.8 0.7 0.6 EL/E1The. 0.5 0.4 0.3 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 EL/E1Exp. Figure 3a. Equatin (8) versus experimental results fr B-jump at negative step in radial stilling basins Fig. (6.130) Cmparisn between EL/E1 Exp. and EL/E1The.accrding

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 0.7 0.6 EL/E1 0.5 0.4 0.3 0. The. Eqn.(8) The. Eqn.(8) The. Eqn.(8) z/=1.9 z/=1.0 z/=0.6.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 3b. Typical relatinship between E L /E 1 and F 1 fr different z/d 1 at r=1.0, Shwing bth experimental results and theretical nes based n Eqn.(8). 5. EXPERIMENTAL RESULTS AND DISCUSSIONS Figures 4a and 4b present typical variatins f the relatinship between d /d 1 and F 1 fr B-jump frmed at negative step psitined at r=1.0 (L s /L b =0.0) and r=1.17 (L s /L b =0.5). Als, Figures 5a and 5b present the typical variatins f L j /d 1 with F 1 fr the same tw psitins while Figure 6a and 6b shw the typical variatins f E L /E 1 with F 1. These figures indicate similar trend f variatin fr the same relatinship, i.e. linear variatin fr d /d 1 and nnlinear variatins fr bth L j /d 1 and E L /E 1. In all these figures, the jump prperty (d /d 1, L j /d 1 r E L /E 1 ) increases with the increase f F 1 at particular z/d 1 due t the increase in the crrespnding flw rate. At particular F 1, bth d /d 1 and L j /d 1 increase with the increase f z/d 1 with a higher rate f increase at the psitin r=1.17 cmpared t rate f increase at the psitin r=1.0. Regarding the energy lss rati, it decreases with the increase f z/d 1 at particular F 1 as shwn frm Fgures 6a and 6b. The rate f reductin in E L /E 1 increases with the increase f r frm r=1.0 t r=1.17. Table 1 shws the percentage increase r dcrease in the jump prperty fr unit increase in z/d 1 at the different relative psitins f the negative step. Table 1. Percentages f increase r decrease in value f hydraulic jump characteristics fr each unit increase in z/d 1 r r L s /L b % increase in d /d 1 % increase in L j /d 1 % decrease in E /E 1 1.0 r 0.0 7.0 5.0 5.7 1.17 r 0.5 11.0 6.8 15.0

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 The increase in the depth rati is due t the increase in the depth f flw which is turn is due t the increase in the specific energy at the sectin where the jump ends while the incming flw is subcritical leading t a rise in the water surface. This rise in the water surface increases the weight f the jump which makes the supercritical jet f flw takes lng distance t decay and in turn lnger length f jump. This increase in the depth f flw and in turn in the weight f jump decreases the rate f the frmed eddies and reverse mtin that are created within the jump leading t a reductin in the rate f the energy dissipatin cmpared t the case f n step (free bed). 10 9 8 d/ 7 6 5 4 Ls/Lb=0.0 z/=1.9 z/=1.3 z/=1.0 z/=0.8 z/=0.6 Free bed 3.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 4a. Typical experimental relatinship between d/ and fr different z/ at r=1.0 11 10 9 8 d/ 7 6 5 4 3.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 4b. typical experimental relatinship between d/ and fr different z/ at r=1.17

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 45 40 35 Lj/ 30 5 0 15 Ls/Lb=0.0 z/=1.9 z/=1.3 z/=1.0 z/=0.8 z/=0.6 Free bed 10.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 5a. Typical experimental relatinship between L j /d 1 and F 1 fr different z/d 1 at r=1.0 (L s /L b =0.0) 45 40 35 Lj/ 30 5 0 15 Ls/Lb=.5 z/=1.9 z/=1.3 z/=1.0 z/=0.8 z/=0.6 Free bed 10.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 5b. Typical experimental relatinship between L j /d 1 and F 1 fr different z/d 1 at r=1.17 (L s /L b =0.5)

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 0.8 0.7 0.6 EL/E1 0.5 Ls/Lb=0.0 0.4 0.3 z/=1.9 z/=1.3 z/=1.0 z/=0.8 z/=0.6 Free bed 0..5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 6a. Typical experimental relatinship between EL/E1 and fr different z/ at r=1.0 0.8 0.7 0.6 0.5 EL/E1 0.4 0.3 0. 0.1 Ls/Lb=.5 z/=1.9 z/=1.3 z/=1.0 z/=0.8 z/=0.6 Free bed 0.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 6b. Typical experimental relatinship between EL/E1 and fr different z/ at r=1.17

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 6. STATISTICAL MODELING OF JUMP CHCRACTERISTICS Since it is difficult t derive a pure theretical equatin fr the length f jump, the nature f functin f Eqn. (1) shuld be determined t enable the predictin f the length f jump. This culd be dne by prpsing many different frms and trying t fit each frm t the experimental data using the statistical techniques. The multiple linear regressin is used fr this purpse. Amng the different trials, the fllwing Eqn. (9) is fund t be suitable. It has a determinatin cefficient f R =0.941 and mean relative abslute errr, MRE, f 0.056. The residuals are uncrrelated with the predicted values as the crrelatin cefficient between them apprches zer (R=7.16E-08). Figure 7a presents the cmparisn between the predicted and the measured values f L j /d 1 fr all measured data. The values are very clse t the line f perfect agreement indicating gd agreement between predicted and measured values f L j /d 1. The variatins f the residuals with the predicted values f L j /d 1 are shwn in Figure 7b. Clearly, the reiduals are small, symmetrically distributed arund the line f zer errr and uncrrelated indicating the validity f the develped mdel. L j /d 1 = -0.135+5.55 F 1-4.407 r+3.74 z/d 1 (9) Figure 8 presents a typical cmparisn between measured and predicted L j /d 1 as a functin f F 1 fr different z/d 1 f 0.6, 1.6 and 1.9 at r=1.0. Clearly, the trends f the predicted values are the same as thse f the measured ne indicating the merits f Eqn. (9) in the predictin f L j /d 1. Similar mdels culd be develped fr bth depth rati d /d 1 and fr the energy lss rati E L /E 1 as given by equatins (10) and (11) respectively. d /d 1 = -3.31+1.115 F 1 +.59 r+1.143 z/d 1 (10) E L /E 1 = -0.818-0.175 F 1 +1.1 F 1 0.5-0.195 r-0.037 z/d 1 (11) Equatins (10) and (11) have R f 0.970 and 0.977 and MRE f 0.03 and 0.01 respectively. The crrelatin cefficients f the residuals with the predicted values are 9.59E-07 and 4.47E-0 respectively. Similar figures as thse f 7a, 7b and 8 are prepared fr the depth rati and the energy lss rati but nt presented here t reserve space and t avid repetitin.

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 50 45 40 35 Lj/(Predicted) 30 5 0 15 10 Lj/(Exp.) 10 15 0 5 30 35 40 45 50 a Residuals 8 6 4 0 - -4-6 Lj/(Predicted) -8 10 15 0 5 30 35 40 45 50 Fig. (7.5) Results f statistical mdel f (Eqn 7.19) Figure 7. Results f the predictin mdel Eq. (9), (a) predicted versus measured and (b) residuals versus predicted values 45 b 40 35 Lj/ 30 5 0 15 Eqn (9) z/=1.9 Eqn (9) z/=1.0 Eqn (9) z/=.6 z/=1.9 z/=1.0 z/=0.6 10.0.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 Figure 8. Cmparisn between predictin f equatin (9) and measured Lj/ fr typical values f z/ at r=1.0

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 7. CONCLUSIONS An experimental prgram was cnducted t investigate the effect f presence f negative step in radial stilling basin n the hydraulic characteristics f the free radial hydraulic B-jump. The effects f the initial Frude number, the relative psitin f the step in the basin and the relative height f the step n the chracteristics f the hydraulic B-jump were addressed. The experimental results were cmpared the develped theretical and statistical mdels fr the energy lss rati and the length f jump rati respectively. The presence f negative step in the first half f the radial stilling basin affects the depth rati and the length f jump rati psitively while the energy lss is affected negatively. Amng the investigated psitins f the negative step, the ne at the psitin r=1.17 (1/4 f the basin length) has the maximum effects n the jump chcracteristics than the ther psitins. At this psitin a unit increase in z/d 1 prduces abut 7% increase in L j /d 1. NOTATIONS b = cntracted width f the channel; B = width f the channel; d 1 = water depth at vena cntracta dwnstream the gate (initial depth); d = sequent water depth; d = the relative water depth f the jump, d /d 1 ; d 3 = depth f water abve the step; d = the rati f d 3 t d 1 ; F 1 = Frude s number at the initial depth; L b = length f stilling basins; L j = the length f the hydraulic jump; L s =length frm the gate t the end f the step in the basin; Q = rate f flw; r 1 = radius at the beginning f the jump; r = radius at the end f the jump; r = the rati f r t r 1 ; r 3 = radius at the end f the step in the basin; r = the rati f r 3 t r 1 ; R = the cefficient f determinatin; R = crrelatin cefficient; P 1 = the hydrstatic pressure at the beginning f the jump; P = the hydrstatic pressure just at the end f the jump; P s = channel side pressure frce; P 3 = = the hydrstatic pressure n the face f the step; V 1 = average velcity at the initial depth; V = average velcity at the sequent depth; z = the drp height; Z = the rati f z t d 1 ; and = the angle f divergence.

Seventh Internatinal Water Technlgy Cnference Egypt 1-3 April 003 REFERENCES [1] Hager, W.H., Energy Dissipatrs and Hydraulic Jumps. Kluwer Academic Publicatins, Drdrecht, The Netherlands, 199. [] Khalifa, A.M. and McCrqudale, J.A., Radial Hydraulic Jump. Jurnal f the Hydraulic Divisin, ASCE: 105(HY9), 1979, pp. 1065-1078. [3] Abdel-Aal, G.M., El-Saiad, A.A., and Saleh, O.K., Hydraulic Jump within a Diverging Rectangular Channel. Engineering Research Jurnal, Faculty f Engineering, Helwan University, Mataria, Cair, Vl. 57, June, 1998, pp. 118-18. [4] Hager, W. H., B-Jumps at Abrupt Channel Drps. Jurnal f Hydraulic Eng., Vl.111, N.5, 1985, pp. 861-866. [5] Hager, W.H. and Bretz, N.V., Hydraulic Jumps at Psitive and Negative Step. Jurnal f Hydraulic Research, Vl. 4, N. 4, 1986, pp. 37-53. [6] Ohtsu, I., and Yasuda, Y., Transitin frm Supercritical t Subcritical Flw at an Abrupt Drp, Jurnal f Hydraulic Research, Vl. 9, N. 3, 1991, pp.309-37. [7] Negm, A.M., (1996). Hydraulic Jumps at Psitive and Negative Steps n Slping Flrs. Jurnal f Hydraulic Research, Vl. 34, N. 3, 1996, pp. 409-40. [8] Armeni, V., Tscand, P., and Firtt, V., The Effects f a Negative Step in Pressure Fluctuatins at the Bttm f a Hydraulic Jump. Jurnal f Hydraulic Res., Vl.38, N. 5, 000, pp. 359-368. [9] Negm, A.M., Abdel-Aal, G.M., Elfiky, M.M., and Mhmed, Y.A., Theretical and Experimental Evaluatin f the Effect f End Sill n Characteristics f Submerged Radial Hydraulic Jump. Sc. Bulettin, Faculty f Engineering, Ain Shams Univ., Cair, Egypt, 00a, (Accepted). [10] Negm, A.M., Abdel-Aal, G.M., Elfiky, M.M., and Mhmed, Y.A., Characteristics f Submerged Hydraulic Jump in Radial basins with a Vertical Drp in the Bed. AEJ, Faculty f Eng., Alex. Univ., Egypt, 00b (Accepted). [11] Negm, A.M., Abdel-Aal, G.M., Elfiky, M.M., and Mhmed, Y.A., Hydraulic Characteristics f Submerged Flw in Nn-prismatic basins. Prc. f 5 th Int. Cnf. n Hydrscience and Engineering, ICHE00, Sept. 18-1, Warsaw, Pland, 00c. [1] Negm, A.M., Abdel-Aal, G.M., Owais, T.M. and Habib, A.A., Theretical Mdeling f Hydraulic Jumps at Negative Step in Radial Stilling Basin. Prc. f 6 th Int. Cnf. n River Engineering, Published n CD, Jan. 8-30, Ahvaz, Iran, 003. [13] Habib, A.A. Characteristics f Flw in Diverging Stilling Basins, Ph. D. Thesis, Submitted t the Faculty f Engineering, Zagazig University, Zagazig, Egypt.