EXPERIMENTAL STUDY OF THE AIR-WATER SHEAR FLOW IN A HYDRAULIC JUMP

Transcription

1 EXPERIMENTAL STUDY OF THE AIR-WATER SHEAR FLOW IN A HYDRAULIC JUMP b H. CHANSON an T. BRATTBERG Department of Civil Engineering, The Universit of Queenslan, Brisbane QLD 4072, Australia Abstract : Although the hraulic jump has been investigate experimentall for nearl two centuries, little information is known of the air-water flow properties in the shear region. New experiments were performe in a horizontal channel with partiall-evelope inflow conitions. Distributions of air concentration, mean air-water velocit an bubble frequenc were recore an presente herein. The results inicate an avective iffusion of air in the shear laer. The velocit profiles have a similar shape as wall jet flows but ifferent quantitative parameters must be introuce. The relationship between air content an bubble frequenc has a parabolic shape which is not et unerstoo but was observe previousl in open channel flows. Ke wors : air bubble entrainment, hraulic jump, voi fraction, air-water velocit, bubble frequenc, air-water shear flow, experimental ata. INTRODUCTION In open channels, the transition between supercritical an subcritical flow (i.e. a hraulic jump) is characterise b a sharp rise in free-surface elevation, strong turbulence, splashing an air entrapment in the roller. Historicall air entrainment in hraulic jump was investigate in terms of the air eman : i.e., the total quantit of entraine air (e.g. WOOD 99, CHANSON 997a). A 'milestone' contribution was the work of RESCH an LEUTHEUSSER (972) who showe first that the air entrainment process, the transfer of momentum an the energ issipation are strongl affecte b the inflow conitions. Recentl the first author (CHANSON an

2 QIAO 994, CHANSON 995a,b) stuie particularl the air-water properties in partiall-evelope hraulic jumps an he showe a similarit with plunging jet entrainment. Past investigations were usuall performe with Prantl-Pitot tubes, propeller, LD anemometer an hot-film probes (table ). Most measurement evices coul be significantl affecte b the air bubble entrainment an some hot-film probe ata were ver cruel processe (e.g. RESCH an LEUTHEUSSER 972). Few stuies provie accurate air-water flow measurements (e.g. CHANSON 995a, MOSSA an TOLER 998). Up to ate, the air bubble iffusion process an the mechanisms of momentum transfer in the air-water flow of hraulic jumps are not et full unerstoo. It is the purpose of this work to present new experimental results, to compare these with existing ata (table ), an to present new compelling conclusions regaring momentum an voi fraction evelopment of jumpentraine air-water flows. The stu is focuse in the eveloping air-water flow region (i.e. (x-x )/ < 50) of hraulic jumps with partiall-evelope inflow conitions. EXPERIMENTAL APPARATUS The experiments were performe in a 3.2-m long horizontal channel of uniform rectangular section (CHANSON an QIAO 994, CHANSON 995a) (Fig. ). The flume is 0.25-m wie, the siewalls are 0.30-m high an both walls an be are mae of glass. Regulate flows are supplie through an ajustable vertical sluice gate. During the experiments, the gate opening was fixe at 20 mm. The experimentall observe values for the coefficient of contraction were about 0.6. Tailwater levels were controlle b an overshoot sharp-creste gate at the ownstream en of the channel. The water was supplie b a constant hea tank. The ischarge was measure b a 90-egree -notch weir, previousl calibrate. The percentage of error is expecte to be less than 2%. The air-water flow properties were recore using a ual-tip conuctivit probe, the two tips being aligne in the flow irection. Each tip has an internal concentric electroe ( = 25 µm, Platinum electroe) an an external stainless steel electroe of 200 µm iameter. The probe was excite b an air bubble etector (Ref. AS25240) an

3 the signals were scanne at 20 khz per channel for 0 secons. The analsis of the ata provie the voi fraction, mean air-water interface velocit an air bubble frequenc. In aition, clear water jet velocities an turbulent velocit fluctuations (in clear-water) were measure with a Pitot tube (external iameter Ø = 3.3 mm) connecte to a pressure transucer (aline DP5). The transucer was scanne at 500 Hz an the accurac of the clear-water velocit ata was normall estimate as : / = %. The translation of the probes in the irection perpenicular to the channel bottom was controlle b a fine ajustment travelling mechanism connecte to a Mitutoo igimatic scale unit (Ref. No ). The error on the vertical position of the probes (i.e. Pitot tube an conuctivit probes) was less than 0.0 mm. The longituinal an transversal translations of the probes were controlle manuall : the probes an the igimatic scale unit were fixe to a stiff L-shape aluminium beam fixe on a trolle sstem. The error on the longituinal location of the probes was less than 5 mm. The error on the transverse location of the probes was less than 0.5 mm. Note that most measurements were taken on the channel centreline. Discussion Previous stuies (CHANSON an QIAO 994, CHANSON 995a) were conucte with a single-tip conuctivit probe ( = 0.35 mm) an analog sampling times ranging from 60 to 300 secons. During the present stu, the ata were igitall sample at 20,000 Hz per channel for 0 secons. Initial tests were conucte for the same flow conitions as CHANSON an QIAO (994) an CHANSON (995), an the showe no ifference in air concentration istributions. Higher bubble frequencies were observe consistentl because the probe ha a smaller sensor size (i.e. = 25 µm). Two series of experiments were performe (table ). In each case, the jump toe was locate at x = 0.5-m ownstream of the gate (fig. ) an the inflow was partiall-evelope : i.e., δ/ 0.65 for both experiments, where δ is the bounar laer thickness an is the upstream flow epth. The result was obtaine with Pitot tube measurements an it is consistent with previous results (CHANSON an QIAO 994, fig. 4-2). Full etails of the experimental results are reporte in CHANSON an BRATTBERG (997).

4 EXPERIMENTAL RESULTS : OID FRACTION DISTRIBUTION Air-water flow regions The air-water flow of the hraulic jump is characterise b a turbulent shear region an a recirculating flow region above (fig. ). In the turbulent shear region, momentum is exchange between the impinging flow an the recirculating region. The recirculation region is characterise b strong unstea recirculation, large bubbles an air packets, an the liqui becomes reuce to a foam structure (i.e. thin films separating the air bubbles) near the free-surface (e.g. CHANSON 995b, 997a). oi fraction profile in partiall-evelope jumps For hraulic jumps with partiall evelope inflow conitions, several stuies (e.g. RESCH an LEUTHEUSSER 972, CHANSON 995a,b) showe that the air concentration istributions exhibit a voi fraction peak in the turbulent shear region (fig. 2 an 3). The air concentration ata are best correlate b a solution of the iffusion equation (CHANSON 995a,997a) : 2 Y C max - 4 * U * D * - t x -x for : /Y shear < () C = C max * exp where C is the voi fraction, C max is the maximum air content in the turbulent shear laer region measure at a istance Y C max from the bottom, U is the free-stream velocit of the inflow, is the inflow epth, x an are the longituinal an vertical istances measure from the channel intake an be respectivel, x is the location of the jump toe, D t is a turbulent iffusivit an Y shear is the upper limit of the turbulent shear region (fig. 3). Equation () is compare with experimental ata in figure 2. A goo agreement is note between theor (eq. ()) an ata, but when approaching /Y shear =. Y shear correspons to the transition between the shear region an the recirculation region in which air bubble entrainment is not an avective iffusion process. For the present stu, the upper limit of the turbulent shear region is best correlate b : Y shear = * x - x for : (x-x )/ 28.7 (2)

5 In the shear laer region, the maximum voi fraction ecreases with istance from the jump toe an the ata are correlate b : C max m x - x for : (x-x )/ 28.7 (3) with m = an for Fr = 6.3 an 8.5 respectivel. The position of the maximum air content is inepenent of the inflow Froue number an it is best correlate b : Y C max = * x - x for : (x-x )/ 28.7 (4) Equation (4) is shown in figure 4 where it is compare with the ata (Present stu) an some re-analse ata. The results inicate a goo agreement between all experiments performe with partiall-evelope inflow conitions. Note that equation (4) is close to CHANSON's (995b) correlation valiate with both plunging jet an hraulic jump ata (table 2). The values of turbulent iffusivit D t were estimate for each experiment b fitting equation () to the ata of C (table 3). Overall the are comparable with a previous investigation performe in the same flume an this suggests a goo repeatabilit of the experiments. EXPERIMENTAL RESULTS : AIR-WATER ELOCITY PROFILE Upstream flow With partiall-evelope inflow conitions, the upstream flow consists of a eveloping bottom bounar laer an an ieal-flui flow region above (fig. ). In the bounar laer, the velocit istribution ma be approximate b a power law : /N U = Upstream bounar laer (5) where U is the free-stream velocit an is the upstream flow epth. For both experiments, the authors observe : N = This result is close to the finings of CHANSON an QIAO (984, fig. 4-5) in the same flume.

6 elocit istribution in the jump The authors measure the velocit istributions in the jump using a Pitot tube in the clear water region an a ualtip conuctivit probe in the air-water region. The latter technique gives mean air-water interfacial velocities. Figure 2 presents tpical results. Note the scatter of conuctivit probe ata which is cause b the 'boiling' nature of the jump roller. The ouble-tip conuctivit probe is esigne to have the two tips aligne along the streamline. In the recirculation region, the cross-correlation between the probe tips becomes low because of the unstea an fluctuating nature of the flow, an the ata scatter is large. At each cross-section, the velocit ata are best correlate b : = * max /N max = exp max - 2 *.765 * 2 - max 0.5 for : / max < (6) for : < / max < 3 to 4 (7) where is the local velocit, max is the maximum velocit measure at a istance max from the bottom an 0.5 is the location where = 0.5* max (fig. 3). For their experiments, the authors obtaine N = 6.43 for Fr = 6.3 an N = 5.24 for Fr = 8.5. Note that equation (7) was first evelope b OHTSU et al. (990) (table 4). For the present stu, the characteristic parameters of the velocit profiles are best correlate b : max = * x - x for : (x-x )/ 2.4 (8) 0.5 = * x - x for : (x-x )/ 2.4 (9) max 0.5 = 0.25 for : (x-x )/ 2.4 (0) 0.5 Y 90 = m' for : (x-x )/ 2.4 () where = q w /, m' = an for Fr = 6.3 an 8.5 respectivel, an Y 90 is the upper limit of the roller efine as the istance normal to the be where C = 90%.

7 Comparison with previous stuies RAJARATNAM (965) evelope a ver interesting analog between the hraulic jump an the wall jet. He suggeste that the transfer of momentum an the velocit profiles in the jump shoul be similar to plane turbulent wall jet results (table 5). His experiments confirme partiall the hpothesis. It is, however, unlikel that his velocit measurement evice (i.e. Pitot tube) was accurate in air-water flow (table ). Since then, several researchers propose empirical correlations for the velocit profile (table 4). But most stuies use clear-water velocit measurement evices (e.g. Pitot tube, LD) (table ) an little accurate information is available in the air-water flow region. The present ata confirm RAJARATNAM's (965) analog of velocit profile between hraulic jump an wall jet. The results suggest however that the characteristic parameters of the air-water velocit istribution (i.e. eq. (6) to ()) iffer quantitativel from monophase flow results (table 5). The main characteristics of the velocit profiles are summarise in figures 5 to 7. In each figure, the ata (Present stu) are compare with the re-analsis of previous stuies (table ) an equations (8) to (0). Altogether the maximum velocit ecreases linearl with the istance from the jump toe an for (x-x )/ < 30 (fig. 5). Figure 6 presents the imensionless istance 0.5 / where = 0.5* max. The ata (Present stu) are consistentl larger than past results. Base upon their own experience (e.g. CHANSON an BRATTBERG 997,998), the writers believe that previous stuies coul not estimate accuratel 0.5 because of measurements errors : the air content is substantial at the location where = 0.5* max an clear-water instrumentation woul be inaccurate. Figure 7 suggests that the ratio max / 0.5 is basicall inepenent of the longituinal istance although the ata exhibit some scatter. EXPERIMENTAL RESULTS : AIR BUBBLE FREQUENCY DISTRIBUTION The authors investigate also the istributions of air bubble frequenc. The ata provie aitional information on the structure of the air-water flow. The experimental results exhibit a characteristic profile (fig. 2 an 3) : i.e., a triangular profile in the turbulent shear region, a brusque change of slope at the upper ege of the shear region an a flatter shape in the

8 recirculation region. The authors believe that the brusque change of shape of the bubble frequenc istribution is relate to a change of air-water flow structure. isual observations through the siewalls an high-spee photographs (e.g. CHANSON 995b, 997a pp &80-8) showe that the turbulent shear region is characterise b small bubble sizes (millimetric size tpicall) while the recirculating region inclues both small an large size bubbles, an air-water packets, with a foam structure next to the free-surface. There is some similarit with the transition from bubbl flow to plug or slug flow in horizontal circular pipes. In the turbulent shear region, the bubble frequenc istributions follow a simple triangular shape which might be approximate b : F ab (F ab ) max = Y F max F ab (F ab ) max = 2 - Y F max for : /Y F max < (2a) for : < /Y F max < Y shear /Y Fmax (2b) where F ab is the bubble count rate (bubble frequenc), (F ab ) max is the maximum bubble frequenc observe at a istance Y F max from the bottom an Y shear is the upper limit of the turbulent shear region (eq. (2)). The location of the maximum bubble frequenc is best correlate b : Y F max = *.7 x - x for : (x-x )/ 28.7 (3) The maximum bubble frequenc was observe to eca exponentiall with the istance from the jump toe : (F ab ) max * U = 0.7 * Fr * exp * x - x for : (x-x )/ 28.7 (4) Remarks The bubble frequenc istribution ma be presente also as a function of the air content. The ata (fig. 8) exhibit a characteristic parabolic shape which is best fitte b : F ab (F ab ) = - - C 2 max C o for :.4 (x-x )/ 28.7 (5) where C o is the air content at the maximum bubble frequenc (fig. 3). C o ma be correlate as : C o C max = * x - x for : 3.6 (x-x )/ 28.7 (6)

9 where C max is the maximum air content in the turbulent shear laer (fig. 3, eq. (3)). Note that such a parabolic shape (i.e. eq. (5)) was observe also in high-velocit water jets (BRATTBERG et al. 998) an in open channel flows (CHANSON 997b). The result suggests a similarit of air-water flow patterns between the three flow situations. DISCUSSION A hraulic jump is an unstea namic process characterise b longituinal fluctuations of the jump toe. MOSSA an TOLE (998) presente flow pictures, suggesting that the jump fluctuations are associate with a vortex pairing mechanism. During the experiments, the probes were fixe an i not follow the longituinal oscillations. The present ata (e.g. fig. 2, 5 an 8) exhibit a greater scatter than the probe accurac, reflecting the fluctuating nature of the investigate flow. In the air-water region, the position of the air iffusion laer ma be compare with the region of momentum transfer. The locations of the maximum velocit, maximum bubble frequenc an maximum air content (in the turbulent shear region) satisf consistentl: max Y F max Y C max < < < 0.5 for : (x-x )/ 28.7 (7) The relationship is illustrate in figure 9 where the ata are plotte with the empirical correlations. Figure 9 an equation (4) impl that most air entraine in the shear laer is avecte in the high-velocit region (i.e. max /2 < < max ). Note the similarit with plunging jet flows (CHANSON 995a,997a) in which experimental observations inicate : Y C max < (= max /2). Further equation (7) implies that the location of maximum voi fraction ( = Y C max ) is associate with larger bubble sizes (an/or lower velocities) than the location of maximum bubble count ( = Y F max ).

10 Local aeration: analog between hraulic jumps an plunging jets CHANSON (995b) evelope a complete analog between vertical plunging jet flows an hraulic jumps in horizontal channel with partiall-evelope inflow (fig. 0). The present stu confirms the similarit an it ientifies some notable ifferences. In the eveloping shear region, the istributions of air bubble concentration follow the same relationship, both qualitativel an quantitativel (i.e. eq. ()). The location of the smmetr line of the air iffusion laer is nearl ientical: Y C max = * x - x Hraulic jump flow ((x-x )/ 28.7) (4) Y C max = * x - x Plunging jet flow {ata : CHANSON an BRATTBERG 997} (8) alues of the turbulent iffusivities D t are close between the two tpes of air-water flows. For example, D t /(U * ) = 0.04 an 0.02 for a hraulic jump flow with U = 3.47 m/s an for a plunging jet flow with U = 3 m/s respectivel (CHANSON an BRATTBERG 997). In both flow situations, the maximum air concentration in the air iffusion laer ecas exponentiall with the longituinal istance: C max -m' x - x Hraulic jump flow an plunging jet flow (3) with m' = 0.4 to 0.7. Hraulic jumps an supporte plunging jets are eveloping shear flows (fig. 0). The mixing laer centreline (i.e. streamline where = 0.5* max ) correspons approximatel to the location of maximum shear stress, an its location is almost ientical for both tpes of local aeration : 0.5 = * x - x Hraulic jump flow ((x-x )/ 2.4) (9) 50 = * x - x Plunging jet flow {ata : CHANSON an BRATTBERG 997} (9) The transfer of momentum between the jet core an the flui at rest at infinit is affecte b the flow geometr an some ifferences are expecte between a horizontal hraulic jump an a vertical plunging jet (fig. 0). In a plunging jet flow, the flui entrainment into the shear laer causes a 90-egrees change in momentum irection of

11 surrouning flui. In a hraulic jump, the entrainment of the recirculating flui into the shear flow inuces a 80-egrees change in momentum irection of the roller flow. It was thought that the ifferent moe of flui entrainment into the shear flow coul have affecte the air iffusion process. This is not the case an the fining suggests that the air entrainment process is preominantl an avective ispersion. Note that, at a given cross-section, the relationship between bubble frequenc an air concentration iffers between hraulic jump flow an plunging jet flow. In a plunging jet flow, the bubble frequenc an air concentration are not relate b an unique parabolic shape (fig. 8). CONCLUSION The authors have escribe an new stu of the air-water flow properties in a hraulic jump flow. The stu is focuse on the eveloping shear laer of hraulic jumps with partiall-evelope inflow conitions an new correlations were evelope for x/ 20 to 25. The present investigation highlights that, with partiall-evelope inflow conitions, a hraulic jump is characterise b two air-water flow region with significantl ifferent properties. In the air-water turbulent shear region, the voi fraction istribution follows a solution of the iffusion equation an the bubble frequenc profile exhibits a triangular shape with a maximum value. In the recirculating region, the air content increases towar 00% (at the free-surface) an the bubble frequenc profile follow a ifferent tren which is relate to a ifferent air-water flow structure an bubble size composition. An interesting result is the relationship between the air content an the bubble frequenc in the turbulent shear region. The present results suggest a parabolic relationship in the shear region as in self-aerate open channel flows an high-velocit water jets ischarging into air. The velocit istribution has a similar shape as wall jet flows (RAJARATNAM 965) but the quantitative parameters iffer. It is believe that the are affecte significantl b the air entrainment process. The results confirms the air-water shear laer analog between horizontal hraulic jumps an vertical plunging jets. The suggest that the air-water iffusion process an the momentum transfer in the eveloping shear flow are little affecte b gravit in first approximation.

12 In the authors' opinion, the stu emphasises the complexit of the air-water region of hraulic jump. Further experimental investigations are require to gain a better unerstaning of the complete flow fiel, incluing with full-evelope inflow conitions. ACKNOWLEDGMENTS The authors wants to thank particularl Professor C.J. APELT, Universit of Queenslan, who supporte this project since its beginning. The thank also Dr T. NAKAGAWA, Kanazawa Institute of Technolog (Japan) for proviing information of interest. The authors acknowlege the support of the Department of Civil Engineering at the Universit of Queenslan which provie the experimental facilities an the financial support of Australian Research Council (Ref. No. A893359). The secon author was supporte b an APA scholarship sponsore b the Australian Research Council (Ref. No. A894296). The authors thank the reviewers for their helpful comments. REFERENCES BABB, A.F., an AUS, H.C. (98). "Measurements of Air in Flowing Water." Jl of H. Div., ASCE, ol. 07, No. HY2, pp BRATTBERG, T., TOOMBES, L., an CHANSON, H. (998). "Developing Air-Water Shear Laers of Two- Dimensional Water Jets Discharging into Air." Proc. 998 ASME Fluis Eng. Conf., FEDSM'98, Washington DC, USA, June 2-25, Paper FEDSM , 7 pages. CHANSON, H. (995a). "Air Bubble Entrainment in Free-surface Turbulent Flows. Experimental Investigations." Report CH46/95, Dept. of Civil Engineering, Universit of Queenslan, Australia, June, 368 pages. CHANSON, H. (995b). "Air Entrainment in Two-imensional Turbulent Shear Flows with Partiall Develope Inflow Conitions." Intl Jl of Multiphase Flow, ol. 2, No. 6, pp

13 CHANSON, H. (997a). "Air Bubble Entrainment in Free-surface Turbulent Shear Flows." Acaemic Press, Lonon, UK, 40 pages. CHANSON, H. (997b). "Air Bubble Entrainment in Open Channels. Flow Structure an Bubble Size Distributions." Intl Jl of Multiphase Flow, ol. 23, No., pp CHANSON, H., an BRATTBERG, T. (997). "Experimental Investigations of Air Bubble Entrainment in Developing Shear Laers." Report CH48/97, Dept. of Civil Engineering, Universit of Queenslan, Australia, Jul. CHANSON, H., an BRATTBERG, T. (998). "Air Entrainment b Two-Dimensional Plunging Jets : the Impingement Region an the er-near Flow Fiel." Proc. 998 ASME Fluis Eng. Conf., FEDSM'98, Washington DC, USA, June 2-25, Paper FEDSM , 8 pages. CHANSON, H., an QIAO, G.L. (994). "Air Bubble Entrainment an Gas Transfer at Hraulic Jumps." Research Report No. CE49, Dept. of Civil Engineering, Universit of Queenslan, Australia, Aug., 68 pages. HAGER, W.H. (992). "Energ Dissipators an Hraulic Jump." Kluwer Acaemic Publ., Water Science an Technolog Librar, ol. 8, Dorrecht, The Netherlans, 288 pages. IMAI, S., an NAKAGAWA, T. (992). "On Transverse ariation of elocit an Be Shear Stress in Hraulic Jumps in a Rectangular Open Channel." Acta Mechanica, ol. 93, pp MOSSA, M., an TOLE, U. (998). "Flow isualization in Bubbl Two-Phase Hraulic Jump." Jl Fluis Eng., ASME, ol. 20, March, pp MYERS, G.E., SCHAUER, J.J., an EUTIS, R.H. (96). "The Plane Turbulent jet. I. Jet Development an Friction Factor." Technical Report, No., Dept. of Mech. Eng., Stanfor Universit, USA. NAKAGAWA, T. (996). Private Communication, 2 Oct., 4 pages. OHTSU, I.O., YASUDA, Y., an AWAZU, S. (990). "Free an Submerge Hraulic Jumps in Rectangular Channels." Report of Research Inst. of Science an Technolog, No. 35, Nihon Universit, Japan, Feb., 50 pages. RAJARATNAM, N. (965). "The Hraulic Jump as a Wall Jet." Jl of H. Div., ASCE, ol. 9, No. HY5, pp Discussion : ol. 92, No. HY3, pp & ol. 93, No. HY, pp

14 RAJARATNAM, N. (976). "Turbulent Jets." Elsevier Scientific, Development in Water Science, 5, New York, USA. REIF, T.H. (978). "The Effects of Drag Reucing Polmers on the Turbulence Characteristics of the Hraulic Jump." Report EW--78, US Naval Acaem, Annapolis, USA, 50 pages. RESCH, F.J., an LEUTHEUSSER, H.J. (972). "Le Ressaut Hraulique : mesure e Turbulence ans la Région Diphasique." ('The Hraulic Jump : Turbulence Measurements in the Two-Phase Flow Region.') Jl La Houille Blanche, No. 4, pp (in French). SCHWARZ, W.H., an COSART, W.P. (96). "The Two-Dimensional Wall-Jet." Jl of Flui Mech., ol. 0, Part 4, pp THANDAESWARA, B.S. (974). "Self Aerate Flow Characteristics in Developing Zones an in Hraulic Jumps." Ph.D. thesis, Dept. of Civil Engrg., Inian Institute of Science, Bangalore, Inia, 399 pages. WOOD, I.R. (99). "Air Entrainment in Free-Surface Flows." IAHR Hraulic Structures Design Manual No. 4, Hraulic Design Consierations, Balkema Publ., Rotteram, The Netherlans, 49 pages. WU, S., an RAJARATNAM, N. (996). "Transition from Hraulic Jump to Open Channel Flow." Jl of H. Engrg., ASCE, ol. 22, No. 9, pp

15 Table - Experimental investigations of hraulic jump flows Reference Flow conitions Measurement Comments (measurement technique) () (2) (3) (4) RAJARATNAM (965) 2.68 Fr 9.78 elocit (Prantl-Pitot tube) W = m m/s Pitot tube : 3-mm external iameter m P/D inflow conitions RESCH an LEUTHEUSSER (972) Fr = 2.98 & 8.04 =.84 & 2.78 m/s = & 0.02 m x = 0.39 & 0.22 m P/D inflow conitions Fr = 3.26 & 7.32 = 2.5 & 2.0 m/s = & 0.02 m x = 2.44 & 7.8 m F/D inflow conitions THANDAESWARA Fr = 7.6 to 3.3 (974) = 2.8 to 4.60 m/s = to 0.52 m x = 0.23 m P/D inflow conitions REIF (978) Fr = 2.0 x = 0. m P/D inflow conitions Air content, velocit, velocit fluctuations (hotfilm) Air content (conuctivit probe), velocit (Pitot tube an conuctivit probe) elocit, velocit fluctuations (LD) W = 0.39 m. Conical hot-film probe DISA 55A87 (0.6-mm sensor size). W = m. Pitot tube : 3.2-mm external iameter. Conuctivit probe : ouble tip. W = 0. m. LD DISA-55L (5mW He- Ne laser tube). Polmer aitive : polacrlamie Calgon TRO-375 (0 & 00 ppm). W = 0.46 m. Conical hot-film probe DISA 55R42 (0.4-mm sensor size) BABB an AUS (98) Fr = 6.0 Air content, velocit, = 3.5 m/s velocit fluctuations (hotfilm) = m P/D inflow conitions OHTSU et al. (990) 2.5 Fr 9.5 elocit (Prantl-Pitot tube?) Case (a). W = 0.5 m. P/D inflow conitions IMAI an NAKAGAWA Fr = 3.7 an 6.5 elocit (Pitot tube an W = 0.3 m. (992) ( a ) =.94 & 2.76 m/s propeller) Pitot tube : 3-mm external iameter. = & m Propeller : 3-mm external x =.4 & 0.65 m iameter. HAGER (992) 4.3 Fr 8.9 elocit (Propeller?) W = 0.5 m.

16 Table - Experimental investigations of hraulic jump flows Reference Flow conitions Measurement Comments (measurement technique) () (2) (3) (4) CHANSON an QIAO Fr = 5.0 to 8. elocit (Pitot tube), voi W = 0.25 m. (994), CHANSON =.975 to 3.9 m/s fraction (conuctivit probe) Pitot tube : 3.3-mm external (995a,b) iameter. = 0.06 to 0.07 m Conuctivit probe : single x = 0.7 to 0.96 m P/D inflow conitions tip (0.35-mm inner electroe). WU an RAJARATNAM Fr = 3.87 & 0.48 elocit (Prantl-Pitot tube) W = m. (996) =.56 & 4.22 m/s Pitot tube : 3-mm external iameter. = m P/D inflow conitions MOSSA an TOLER Fr = 6.42, 6.45 & 7.33 oi fraction (vieo-camera W = 0.40 m. (998) = 2.85, 2.87 & 3.2 m/s = 0.02, 0.02 & m P/D inflow conitions image processing) Present stu Fr = 6.33 & 8.48 oi fraction, air-water W = 0.25 m. = 2.34 & 3.4 m/s velocit, bubble frequenc Conuctivit probe : ouble (conuctivit probe) tip (25-µm inner electroe). U = 2.58 & 3.47 m/s = 0.04 m x = 0.5 m P/D inflow conitions Notes : LD : laser Doppler velocimeter; ( a ) : also NAKAGAWA (996); P/D : partiall evelope inflow conitions; F/D : full-evelope inflow conitions.

17 Table 2 - oi fraction istribution in hraulic jump flows Reference Correlation Range Comments () (2) (3) (4) C = C max *exp * - Y 2 aliate with author's C max Y Y shear plunging jet an hraulic 50% P/D inflow conitions jump ata. Page aliate with author's C max hraulic jump ata. Page 7. CHANSON (995b) Present stu x - x Y C max x - x = * Y 50% x - x = * C = C max *exp - * 4 * D t * 2 Y C max - x -x Y shear x-x 28.7 P/D inflow conitions C max m x-x x - x 28.7 Y C max x - x x-x = * 28.7 Y shear x - x x-x = * 28.7 aliate with plunging jet an hraulic jump ata. Page 7. aliate with plunging jet an hraulic jump ata. Page 7. aliate with authors' ata. m = an for Fr = 6.3 an 8.5 respectivel Notes : Y 50% : 50%-ban with (i.e. where C = 0.5*C max ).

18 Table 3 - Turbulent iffusivit in the turbulent shear region of hraulic jumps with partiall-evelope inflow conitions Reference Run D t * m/s m x - x () (2) (3) (4) (5) (6) CHANSON (995a) ( a ) C E-2 < 7.6 C E-2 < 9.5 P E-2 < 23 C E-2 < 2.6 C E-2 < 9 Present stu T6_ E-2 < 4.3 T8_ E-2 < 2.4 Note : ( a ) analsis b CHANSON (997a).

19 Table 4 - Empirical correlations of hraulic jump flow velocit istributions Reference Correlation Range Comments () (2) (3) (4) RAJARATNAM (965) P/D inflow conitions aliate with author's max = 0.8 ata (free jump). Page OHTSU et al. (990) HAGER (992) max = * = exp max = exp max max = * max - 2 = max = 2* /2 < max max P/D inflow conitions /7 < max max F/D inflow conitions - 2 max < max P/D inflow conitions - 2 max < max F/D inflow conitions x-x 0. L r - 2 *.765* - 2 *.84* max 0.5 = max 0.5 = *log 0 x - x L r aliate with authors' ata. Page 34. Page 34. Page 34. Page 34. Page Fr 9.5 P/D an F/D inflow conitions P/D inflow conitions Page 34. F/D inflow conitions Page = x - x 3 Fr 9.5 * Fr x-x 0. L 7 r P/D inflow conitions 0.5 = x - x 3 Fr 9.5 * Fr x-x 0. L 7 r F/D inflow conitions 5* *exp - 5* 0.2 x-x Fr 9.05 max = 42 * x-x 45 - x-x = + x-x x-x 5 * 30 Page 35. Page 35. aliate with ata from RAJARATNAM (965). Page 20.

20 Table 4 - Empirical correlations of hraulic jump flow velocit istributions Reference Correlation Range Comments () (2) (3) (4) HAGER (992) - min max - = min cos Fr aliate with author's max 00* ata. Page max max = exp -2* x-x.8 x-x Page 22. L 0 L.4 r r min x-x 0. + L r = - sin 2. * max 2 - = 0.06* + 5* x-x L - r 4 CHANSON (995b) 0.5 x-x = * Present stu 2 x-x 0.05 L.4 r x-x 0.05 L.2 r Page 22. Page 23. Re-analsis of ata from RAJARATNAM (965). 0.5 x-x P/D inflow conitions Re-analsis of ata from = * OHTSU et al. (990). 0.5 x-x F/D inflow conitions Re-analsis of ata from = + 0.4* OHTSU et al. (990). /N / = * max < max P/D inflow conitions max = exp - max 2 *.765 * 2 < / - max < 3 to 4 max (x-x )/ P/D inflow conitions max x - x (x-x )/ 2.4 = * 0.5 x - x (x-x )/ 2.4 = * max 0.5 = Y 90 = m' (x-x )/ 2.4 (x-x )/ 2.4 aliate with the authors' ata. Correlation evelope b OHTSU et al. (990). m' = an for Fr = 6.3 an 8.5 respectivel. Notes : 2 : ownstream flow epth; erf(u) = 2 π * 0 u exp(- t 2 ) * t; L r : roller length; 2 : ownstream flow velocit; P/D : partiall evelope inflow conitions; F/D : full-evelope inflow conitions.

21 Table 5 - Empirical correlations of wall jet velocit istributions Reference Correlation Range Comments () (2) (3) (4) RAJARATNAM (976) /4 aliate with wall jet ata = < max (MYERS et al. 96, max max SCHWARZ an COSART 96). Page 26. /7 =.48* max * - erf 0.68* Page > 0.5 max max 3.5 x Page 29. = 00 x 0.5 Page 29. x = Note : erf(u) = u 2 π * exp(- t 2 ) * t 0

22 Fig. - Sketch of the hraulic jump flow experiment Sluice gate Recirculating region Outer ege of bounar laer Impingement point U C 2 ho δ C, Bounar laer Turbulent shear regio n x x

23 Fig. 2 - Tpical istributions of voi fraction, imensionless velocit an imensionless bubble frequenc (Present stu) (A) Fr = 6.33, U = 2.58 m/s, x = 0.5 m, x - x = 0.05 m / 3.5 x-x = 0.05 m Fr = 6.33 U = 2.58 m/s = 0.04 m C Data fab Data /U (Con. Probe) /U (Pitot) Tu (Pitot) C Theor /U Correlation Note : fab = Fab*/U (B) Fr = 6.33, U = 2.58 m/s, x = 0.5 m, x - x = 0.0 m / Fr = 6.33 U = 2.58 m/s 4.5 x-x = 0.0 m = 0.04 m C Data fab Data /U (Con. Probe) /U (Pitot) Tu (Pitot) C Theor /U Correlation Note : fab = Fab*/U

24 (C) Fr = 6.33, U = 2.58 m/s, x = 0.5 m, x - x = 0.20 m / 6.5 x-x = 0.20 m Fr = 6.33 U = 2.58 m/s = 0.04 m C Data fab Data /U (Con. Probe) /U (Pitot) Tu (Pitot) C Theor /U Correlation Note : fab = Fab*/U (D) Fr = 8.48, U = 3.47 m/s, x = 0.5 m, x - x = 0.05 m / Fr = 8.48 x-x = 0.05 m = 0.04 m U = 3.47 m/s C Data fab (Data) /U (Con. Probe) /U (Pitot) Tu (Pitot) /U Correlation

25 (E) Fr = 8.48, U = 3.47 m/s, x = 0.5 m, x - x = 0.0 m 6 / x-x = 0.0 m Fr = 8.48 U = 3.47 m/s = 0.04 m C Data fab Data /U (Con. Probe) /U (Pitot) Tu (Pitot) C Theor /U Correlation (F) Fr = 8.48, U = 3.47 m/s, x = 0.5 m, x - x = 0.5 m / 7 x-x = 0.5 m Fr = 8.48U = 3.47 m/s = 0.04 m C Data fab Data /U (Con. Probe) /U (Pitot) Tu (Pitot) C Theor /U Correlation Note : fab = Fab*/U

26 Fig. 3 - Definition sketch of the air-water flow properties in hraulic jump with partiall-evelope inflow conitions Fig. 4 - Location of the maximum air content Y C max / in hraulic jump with partiall evelope inflow conitions : comparison between equation (4) an ata (present stu, CHANSON 995a, RESCH an LEUTHEUSSER 972, THANDAESWARA 974)

27 5.00 YCmax/ EQ. (4) Present stu CHANSON RESCH THANDAESWAR A (x-x)/

28 Fig. 5 - Dimensionless maximum velocit max / : comparison between equation (8) an ata max/ EQ. (8) Present stu (x-x)/ BABB an AUS (98) IMAI an NAKAGAWA (992) OHTSU et al. (990) RAJARATNAM (965) REIF (978) RESCH an LEUTHEUSSER (972) WU an RAJARATNAM (996) Fig. 6 - Dimensionless istance 0.5 / : comparison between ata (table ) an equation (9) / (x-x)/ EQ. (9) Present stu IMAI an NAKAGAWA (992) OHTSU et al. (990) RAJARATNAM (965) REIF (978) RESCH an LEUTHEUSSER (972) WU an RAJARATNAM (996)

29 Fig. 7 - Dimensionless istance max / 0.5 : comparison between ata (table ) an equation (0) max/0.5 (x-x)/ EQ. (0) Present stu IMAI an NAKAGAWA (992) OHTSU et al. (990) RAJARATNAM (965) REIF (978) RESCH an LEUTHEUSSER (972) WU an RAJARATNAM (996)

30 Fig. 8 - Dimensionless bubble frequenc (f ab = F ab * /U ) istribution in the turbulent shear region as a function of the local air content C - Comparison with equation (5) (A) Fr = 6.33, U = 2.58 m/s, x = 0.5 m fab = Fab*/U Fr = (x-x)/=.4 (x-x)/=3.6 (x-x)/=7. (x-x)/=0.7 (x-x)/=4.3 Correlation (x-x)/=.4 Correlation (x-x)/=7. Correlation (x-x)/= Note : < Yshear C (B) Fr = 8.48, U = 3.47 m/s, x = 0.5 m fab = Fab*/U. 0.9 Fr = Note : < Yshear C (x-x)/=.4 (x-x)/=3.6 (x-x)/=7. (x-x)/=0.7 (x-x)/=4.3 (x-x)/=2.4 Correlation (x-x)/=3.6 Correlation (x-x)/=7. Correlation (x-x)/=0.7

31 Fig. 9 - Dimensionless characteristic parameters as functions of the imensionless istance from the jump toe - Comparison with empirical correlations (eq. (2), (4), (9), (0), (3)) / upper limit of turbulent shear region 0.5/ YCmax/ YCmax/ YFmax/ max/ YFmax/ 0.5/ Yshear/ Correlations max/ (x-x)/

32 Figure 0 - Transfer of momentum an flui entrainment process in eveloping shear laers at hraulic jump an vertical plunging jet