AIR BUBBLE ENTRAINMENT IN HYDRAULIC JUMPS: PHYSICAL MODELING AND SCALE EFFECTS

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1 AIR BUBBLE ENTRAINMENT IN HYDRAULIC JUMPS: PHYSICAL MODELING AND SCALE EFFECTS Hubert CHANSON Professor in Civil Engineering The University of Queenslan Brisbane QLD 4072 Australia Ph.: (6 7) Fax: (6 7) h.chanson@uq.eu.au Abstract : A hyraulic jump is characterise by strong energy issipation an air entrainment. In the present stuy new air-water flow measurements were performe in hyraulic jumps with partiallyevelope flow conitions in relatively large-size facilities with phase-etection probes. The experiments were conucte with ientical Froue numbers but a range of Reynols numbers an relative channel withs. The results showe rastic scale effects at small Reynols numbers in terms of voi fraction an bubble count rate istributions. The voi fraction istributions implie comparatively greater etrainment at low Reynols numbers leaing to a lesser overall aeration of the jump roller while imensionless bubble count rates were rastically lower especially in the mixing layer. The experimental results suggeste also that the relative channel with ha little effect on the air-water flow properties for ientical inflow Froue an Reynols numbers. Keywors : Hyraulic jumps Air bubble entrainment Physical moelling Scale effects Aspect ratio. INTRODUCTION A hyraulic jump is the suen transition from a high-velocity supercritical open channel flow into a slow-moving sub-critical flow (Fig. ). It is characterise by a suen rise of the free-surface with strong energy issipation an mixing large-scale turbulence air entrainment waves an spray. Air entrainment in hyraulic jumps was investigate originally in terms of the air eman: i.e. the total quantity of entraine air. A significant contribution was the work of RESCH an LEUTHEUSSER (972) who showe first that the air entrainment process momentum transfer an energy issipation are affecte by the inflow conitions. Recent stuies stuie particularly the airwater properties in partially-evelope hyraulic jumps (e.g. MOSSA an TOLE 998 CHANSON an BRATTBERG 2000 MURZYN et al. 2000). Despite such stuies the air bubble iffusion process an the mechanisms of momentum transfer in hyraulic jumps are not completely unerstoo. In the present stuy the air bubble entrainment in the eveloping region of hyraulic jump flows is investigate. Basic imensional consierations are evelope. New experiments were conucte in two geometrically-similar flumes to assess the scale effects affecting air entrainment. It is the purpose of this stuy to present new compelling conclusions regaring air bubble entrainment an scale effects affecting hyraulic jumps with partially-evelope inflow conitions. DIMENSIONAL ANALYSIS AND SIMILITUDE Analytical an numerical stuies of air bubble entrainment in hyraulic jumps are complicate because the large number of relevant equations. Some physical moelling is often preferre using geometrically similar moels but the scale moel stuies must be esigne base upon a soun similitue. For a hyraulic jump in a horizontal rectangular channel a simplifie imensional analysis shows that the parameters affecting the air-water flow properties at a position (x y z) inclue the flui properties incluing the air an water ensities ρ air an ρ w the air an water ynamic viscosities µ air an µ w an the surface tension σ the gravity acceleration g the channel properties incluing the with W an the inflow properties such as the inflow epth the inflow velocity the characteristic turbulent velocity u' an the bounary layer thickness δ. In aition biochemical properties of the water solution may be consiere. If the local voi fraction C is known the ensity an viscosity of the air-water mixture may be expresse in terms of the water properties an voi fraction only an the air ensity an viscosity may be ignore. The imensional analysis yiels an expression of the air-water flow properties as : C F u'... = F (x y z u' x δ W g ρ w µ w σ...) ()

2 (A) High-spee photograph of a hyraulic jump Fr = 0.2 Re = 8.9 E+4 = m W/ =25 (/000 s) (B) Definition sketch Fig. - Air bubble entrainment in a hyraulic jump with partially-evelope inflow conitions where F is the bubble count rate is the velocity u' is a characteristic turbulent velocity x is the coorinate in the flow irection measure from the upstream gate y is the vertical coorinate z is the transverse coorinate measure from the channel centreline x is the istance from the upstream gate an δ is the upstream bounary layer thickness (Fig. B). Since the relevant length scale is the upstream flow epth Equation () may be rewritten in imensionless terms as :... u' g F C = δ µ ρ σ ρ... W u' g x z y x x F w w 2 w 2 (2a) In Equation (2a) the imensionless air-water flow properties at a imensionless position (x/ y/

3 z/ ) are expresse as functions of the imensionless inflow properties an channel geometry. In the right hansie term the fifth sixth an seventh terms are the inflow Froue Weber an Reynols numbers respectively. Any combination of these numbers is also imensionless an may be use to replace one of the combinations. One parameter can be replace by the Morton number Mo = gµ 4 w /(ρ w σ 3 ) which is a function only of flui properties an gravity constant. The Morton becomes an invariant if the same fluis (air an water) are use in both moel an prototype an hence: F u' C... = g 4 x x y z x u' δ W g µ w F ρ... 3 w (2b) g µ 3 w ρw σ Discussion In a geometrically similar moel a ynamic similarity is achieve if an only if each imensionless parameter has the same value in both moel an prototype. Scale effects may exist when one or more imensionless terms have some ifferent value between the moel an prototype. In a stuy of free-surface flows incluing the hyraulic jump a Froue similitue is commonly use (e.g. HENDERSON 966 CHANSON 2004). The moel an prototype Froue numbers must be equal. The air bubble entrainment an the mechanisms of bubble breakup an coalescence are ominate by surface tension effects. The turbulent processes in the shear region are riven by viscous forces. A ynamic similarity of air entrainment in hyraulic jumps becomes impossible because of too many relevant parameters (Froue Reynols an Morton numbers) in Equation (2). To ate no systematic stuy was conucte to assess the extent of scale effects affecting the air bubble entrainment in hyraulic jump flows. It is worth aing that the above analysis oes not account for the characteristics of the instrumentation. With phase-etection intrusive probes the probe sensor size an shape the scanning rate an possibly other probe characteristics affect the minimum bubble size etectable by the metrology. At present all systematic stuies of scale effects affecting air entrainment processes were conucte with the same instrumentation an sensor size in all experiments. The probe sensor size was never scale own in the small size moels. The present stuy is no exception but it is acknowlege that this aspect might become a limitation. EXPERIMENTAL CHANNELS AND INSTRUMENTATION New experiments were performe in the Goron McKAY Hyraulics Laboratory at the University of Queenslan (Table ). The first channel was horizontal 3.2 m long an 0.25 m wie. Both bottom an siewalls were mae of 3.2 m long glass panels. This channel was previously use by CHANSON (995) an CHANSON an BRATTBERG (2000). The secon channel was horizontal 3.2 long an 0.5 m wie. The siewalls were mae of 3.2 m long glass panels an the be was mae of 2 mm thick PC sheet. Both channels were fe by a constant hea tank. Further etails on the experiments were reporte in CHANSON (2006). Instrumentation In the small flume the flow rate was measure with a 90º -notch weir which was calibrate onsite with a volume-per-time technique. In the large channel the water ischarge was measure with a enturi meter which was calibrate in-situ with a large -notch weir. The percentage of error was expecte to be less than 2%. The water epths were measure using rail mounte pointer gauges with an accuracy of 0.2 mm. The air-water flow properties were measure with a single-tip conuctivity probe (neele probe esign). The probe consiste of a sharpene ro ( = 0.35 mm) which was insulate except for its tip. It was excite by an electronic system (Ref. UQ82.58) esigne with a response time less than 0 µs an calibrate with a square wave generator. The probe vertical position was controlle by a fine ajustment system with an accuracy of 0. mm.

4 A sensitivity analysis was performe to assess the effects of sampling uration T scan an sampling rate F scan on the hyraulic jump air-water properties (CHANSON 2006). The sensitivity tests were conucte with sampling times within 0.7 T scan 300 s an a sampling frequency between 600 F scan Hz. The results showe that the bubble count rate was rastically unerestimate for sampling rates below 5 to 8 khz. The sampling uration ha little effect on both voi fraction an bubble count rate for scan perios longer than 30 to 40 s. In the present stuy the probe sensor was scanne at 20 khz for 45 s at each sampling location. Experimental proceure The channels an the inflow conitions were esigne to be geometrically similar base upon a Froue similitue with a geometric scaling ratio L r = 2.0 between the narrow an wie channels where L r is the ratio of prototype to moel imensions. Similar experiments were conucte for ientical inflow Froue numbers Fr relative channel with W/ an relative gate-to-jump toe istance x /. In aition the effects of the relative channel with were teste for ientical inflow Froue an Reynols numbers (Table ). The air-water flow measurements were performe at ientical cross-sections (x-x )/ in both channels but the present stuy was focuse in the eveloping air-water flow region: i.e. (x-x )/ 25. Preliminary clear water velocity measurements were performe in both flumes using a Prantl- Pitot tube ( = 3.3 mm). The results showe that the supercritical inflow was partially-evelope for all investigate flow conitions (Table ). The relative bounary layer thickness δ/ was about 0.5 to 0.6 epening upon the inflow conitions. Table - Summary of experimental flow conitions (Present stuy) Channel x W/ Fr Re Comments m m m/s Small flume Glass bottom an siewalls. (W = 0.25 m) E+4 Run E+4 Run E+4 Run E+4 Run Large flume Glass siewalls an PC be. (W = 0.5 m) E+4 Run E+4 Run Notes : Fr : upstream Froue number; Re : upstream Reynols number. BASIC FLOW PATTERNS A hyraulic jump is a flow singularity characterise by a rapi turbulent transition between super- an sub-critical flow an substantial energy issipation an air entrainment. At the jump toe air bubbles an air packets were entraine into a free shear layer characterise by intensive turbulence prouction preominantly in large-scale vortices with horizontal axes perpenicular to the flow irection (Fig. ). Air entrainment occurre in the form of bubbles an pockets entrappe at the impingement of the upstream jet flow with the roller. The air packets were broken up into very small air bubbles as they were subsequently avecte in the shear region. Further ownstream bubble collisions an coalescence le to larger bubble pockets that were riven by buoyancy towars the freesurface. In the recirculating region unsteay flow reversal an recirculation were observe with significant spray an splashing. The location of the jump toe was consistently fluctuating aroun its mean position an some vortex sheing was observe in the mixing layer. The jump toe pulsations were believe to be cause by the growth avection an pairing of large scale vortices in the eveloping shear layer of the jump (LONG et al. 99 HABIB et al. 994). The position of the hyraulic jump toe fluctuate with time within a 0.2 to 0.4 m range epening upon the flow conitions. Pulsation frequencies F toe of the

5 jump toe were typically about 0.5 to 2 Hz. Effects of Reynols number an aspect ratio on air-water flow patterns When experiments with ientical inflow Froue numbers were repeate in both channels the hyraulic jump flow appeare visually more energetic in the large flume at the larger Reynols number. This was seen also using high-shutter spee photographs an movies (e.g. /000 s). Further the amount of air-water projections above the jump roller was larger at the higher Reynols numbers in the wier channel. This was associate with significant spray splashing an waves that sometimes overtoppe the channel walls. Spray roplets were seen commonly reaching height of more than 0.5 to m above the invert in the large channel. In contrast little spray was observe in the small channel for an ientical inflow Froue number. The aspect ratio or relative channel with W/ was foun to have no influence on the basic flow patterns within the range of the experiments DISTRIBUTIONS OF OID FRACTION AND BUBBLE COUNT RATE A hyraulic jump with partially-evelope inflow is characterise by a turbulent shear layer with an avective iffusion region in which the air concentration istributions exhibit a peak in the turbulent shear region (RESCH an LEUTHEUSSER 972 CHANSON 995 CHANSON an BRATTBERG 2000 MURZYN et al. 2005). This feature is sketche in Figure. The bubble iffusion region is similar to that observe in two-imensional plunging jet flows (e.g. CUMMINGS an CHANSON 997ab). The avective iffusion layer was observe in the present stuy an it is ocumente experimentally in Figures 2 an 3. Figure 2A presents some longituinal variation in voi fraction istributions for one experiment. In the air iffusion layer the peak voi fraction C max ecrease with increasing istance (x-x ) from jump toe while the iffusion layer broaene (Fig. 2A). The interactions between eveloping shear layer an air iffusion layer are complicate an they are believe to be responsible for the existence of a peak F max in bubble count rate seen in Figure 2B. Experimental observations showe that the location where F = F max i not coincie with the locus of maximum voi fraction. In the air iffusion layer the voi fraction istribution in the avective layer followe closely an analytical solution of the avective iffusion equation for air bubbles (CHANSON 995). In the present stuy the voi fraction ata were best preicte by an approximate expression : 2 y Y Cmax C = C max exp Avective iffusion layer (3) # 4 D x x where C max is the maximum air content in the turbulent shear layer region measure at y = Y C max above the bottom (Fig. ) D # is a imensionless iffusivity: D # = D t /( ) an D t is the turbulent iffusivity which averages the effects of turbulent iffusion an of longituinal velocity graient.. Equation (3) is compare with experimental ata in Figures 2 an 3. The values of C max an D # were euce from the best ata fit. Overall the orer of magnitue of the results was consistent with the earlier stuies of CHANSON (995) an CHANSON an BRATTBERG (2000). Note that Equation (3) was observe only for Re > 2.5 E+4 in the present stuy. For lower inflow Reynols numbers the rate of air entrainment was weak an rapi air etrainment estroye any organise avective iffusion layer (Fig. 3A Re = 2.4 E+4)). Effects of Reynols number an aspect ratio Similar experiments were repeate with ientical inflow Froue numbers Fr an relative channel with W/ but ifferent inflow Reynols numbers Re. The results showe systematically that the voi fraction istributions ha a similar shape in the avective iffusion layer but for Re = 2.5 E+4. The longituinal variations in voi fraction istributions showe some e-aeration associate with an upwar shift of the avective iffusion layer (Fig. 2A). The e-aeration rate was greater for a

6 given inflow Froue number in the small flume as illustrate in Figure 3 which present results for ientical Froue numbers an aspect ratio W/ but ifferent inflow Reynols numbers. Further lesser imensionless bubble count rates were recore in the small channel at the smaller Reynols numbers particularly in the air-water mixing layer. For Fr = 8.5 the imensionless bubble count rates in the small channel were about half of those recore at larger Reynols number in the large flume. Figure 3 illustrates the effects of the inflow Reynols number Re on the imensionless istributions of voi fractions an bubble count rates. In the avective iffusion layer voi fraction ata are compare with Equation (3). In aition some experiments were performe with ientical inflow Froue an Reynols numbers but with a ifferent relative channel with W/. A comparison showe little effect of the channel with on voi fraction an bubble count rate istributions an bubble chor time istributions within 8 W/ 22 an for 0.25 W 0.50 m. In summary the present experimental results emonstrate consistently some scale effects in terms of voi fraction an bubble count rate istributions in the small channel with Re < 4 E+4 for ientical Froue numbers Fr (5 Fr 8.5) an aspect ratio W/. DISCUSSION: CHARACTERISTICS OF THE ADECTIE DIFFUSION LAYER The measure locations of maximum voi fraction C max an bubble count rate F max an associate air-water flow properties are summarise in Table 2 an Figure 4. In Figure 4 the experimental flow conitions are ocumente in the legen. The experimental results showe that the maximum air content in the shear layer region C max ecrease with istance from the jump toe an the ata followe closely both power law an exponential ecay functions as shown by CHANSON an BRATTBERG (2000) an MURZYN et al. (2005). Similarly the maximum bubble frequency F max was observe to ecay exponentially with the istance from the impingement point. In Figure 4B the ata are compare with the empirical correlation of CHANSON an BRATTBERG (2000): Fmax x x = x x 0.7 Fr exp 0.04 for < 30 (4) 9 y/ Data (x-x)/=4.2 9 y/ 8 Data (x-x)/=8.4 Data (x-x)/= Data (x-x)/=6.8 Theory (x-x)/=8.4 Theory (x-x)/= Theory (x-x)/= C F / (A) oi fraction istributions. Comparison with Eq. (3) (B) Bubble count rate istributions Fig. 2 - Dimensionless istributions of voi fraction an bubble count rate for Fr = 8.6 Re = 9.8 E+4 W = 0.50 m x-x = m

7 y/ F. / C C Re= 6.8E+4 C Re= 2.5E+4 C Theory Re=6.8E+4 F./ Re= 6.8E+4 F// Re = 2.5E+4 Run Fr Re (m) x (m) W (m) x-x (m) E E (A) Fr = 5 W/ = 9 (x-x )/ = Comparison with Equation (3) y/ F. / C Re=9.8E+4 C Re=3.8E+4 C theory shear layer Re=9.8E+4 C Theory shear layer Re=3.8E+4 F./ Re=9.8E+4 F./ Re=3.8E C Run Fr Re (m) x (m) W (m) x-x (m) E E (B) Fr = 8.5 W/ = 9 (x-x )/ = 2 - Comparison with Equation (3) Fig. 3 - Effects of the inflow Reynols number on the imensionless istributions of voi fraction an bubble count rate for two inflow Froue numbers Fr

8 Despite some general agreement with earlier ata sets an empirical correlations Figures 4A an 4B illustrate some effect of the inflow Reynols number on the air-water flow properties. In both Figures 4A an 4B the ata in the upper part of the graphs correspon to the largest Reynols numbers while the fastest ecay in maximum voi fraction an count rate occurre for the experiments with the lowest Reynols numbers in the small channel. The experimental observations showe systematically that the locus of maximum voi fraction Y Cmax was always higher than the location of maximum bubble count rate Y Fmax. Such a result was previously observe in hyraulic jumps in vertical supporte plunging jets an in vertical circular plunging jets (CHANSON an BRATTBERG 2000 CHANSON et al. 2004). These stuies suggeste that the fining was relate to a ouble iffusion process whereby vorticity an air bubbles iffuse at a ifferent rate an in a ifferent manner ownstream of the impingement point. In turn there woul be some issymmetry in turbulent shear stress across the bubbly flow region which woul influence the characteristic bubble size an hence the number of bubbles for a given voi fraction in the avective iffusion region C max Fr=5 Re=2.5E+4 Run 055 Fr=5 Re=7.7E+4 Run 057 Fr=5 Re=6.8E+4 Run Fr=8 Re=9.4E+4 Run Fr=8.5 Re=3.8E+4 Run 0522 Fr=8.5 Re=9.8E+4 Run (x-x )/ CHANSON&BRATTBERG Fr=6.3 Re=3.3E+4 CHANSON&BRATTBERG Fr=8.5 Re=4.4E+4 F max. / (A) Maximum voi fraction C max Fr =5 Fr = (x-x )/ Fr=5 Re=2.5E+4 Run 055 Fr=5 Re=7.7E+4 Run 057 Fr=5 Re=6.8E+4 Run Fr=8 Re=9.4E+4 Run Fr=8.5 Re=3.8E+4 Run 0522 Fr=8.5 Re=9.8E+4 Run CHANSON&BRATTBER G Fr=6.3 Re=3.3E+4 CHANSON&BRATTBER G Fr=8.5 Re=4.4E+4 Correlation (B) Maximum imensionless bubble count rate F max / - Comparison between experimental ata an Equation (4) for Fr = 5 an 8.5 Fig. 4 - Longituinal variations of maximum voi fraction an bubble count rate in the avective iffusion layer of hyraulic jump with partially-evelope inflow

9 Table 2 - Experimental observations of air iffusion layer characteristics in hyraulic jump with partially-evelope inflow (Present stuy) Run Fr Re W/ x / (x-x )/ F max / Y F max C max Y C max () (2) (3) (4) (5) (6) (7) (8) (9) (0) () E N/A N/A N/A E N/A E N/A E E E E E N/A N/A N/A E E E E E E E N/A N/A N/A E E E N/A N/A N/A E E E Notes : D # : imensionless iffusivity satisfying Equation (3); N/A : not applicable; Italic ata : possibly incorrect ata; (--) : ata not available. Present ata were compare successfully with the experimental ata of CHANSON an BRATTBERG (2000) an their empirical correlations: YC max x x x x = + 0. < 30 (5).7 YFmax x x x x = < 30 (6) where Y Cmax an Y Fmax are the vertical elevations where the voi fraction an bubble count rate are maximum respectively (Table 2). Lastly in Table 2 the last column (column ) lists the values of imensionless air bubble iffusivity euce from the best ata fit in the avective iffusion region. The orer of magnitue is consistent with the earlier stuies of CHANSON (995) an CHANSON an BRATTBERG (2000). # D CONCLUSION New air-water flow measurements were performe in hyraulic jumps with partially-evelope flow conitions. The experiments were performe in two channels in which similar experiments were performe with ientical Froue numbers an relative with but ifferent inflow Reynols numbers. Ientical experiments were also performe with ientical Froue an Reynols numbers but ifferent channel withs. The experimental range of the investigations was 5 Fr E+4 Re 9.8 E+4 an 8 W/ 22.

10 The voi fraction istributions showe the presence of an avection/iffusion shear layer in which the air concentration istributions followe an analytical solution of the iffusion equation for air bubbles. The present results emonstrate however that the avective iffusion layer was observe only for Re > 2.5 E+4. For smaller inflow Reynols numbers the air entrainment rate was weak an air etrainment tene to ominate the air-water flow pattern. Similar experiments with ientical inflow Froue number an aspect ratio were conucte with a true geometric scaling ratio of 2:. The results showe rastic scale effects in the smaller channel in terms of voi fraction an bubble count rate. The voi fraction istribution results implie comparatively greater etrainment at low Reynols numbers yieling to lesser overall aeration of the jump roller in the small channel. The imensionless bubble count rates were significantly lower in the smaller channel especially in the mixing layer. That is they were not scale accoring to a Froue similitue. The experimental results suggeste also that the relative channel with ha little effect on the airwater flow properties for ientical inflow Froue an Reynols numbers within 8 W/ 22 an for 0.25 W 0.50 m. ACKNOWLEDGEMENTS The writer acknowleges the technical assistance of Graham ILLIDGE an the helpful iscussions with Drs C. GUALTIERI (University of Napoli Feerico II) an F. MURZYN (ESTACA France). REFERENCES CHANSON H. (995). "Air Entrainment in Two-Dimensional Turbulent Shear Flows with Partially Develope Inflow Conitions." Intl Jl of Multiphase Flow ol. 2 No. 6 pp CHANSON H. (2004). "The Hyraulics of Open Channel Flow : An Introuction." Butterworth- Heinemann Oxfor UK 2n eition 630 pages. CHANSON H. (2006). "Air Bubble Entrainment in Hyraulic Jumps. Similitue an Scale Effects." Report No. CH57/05 Dept. of Civil Engineering The University of Queenslan Brisbane Australia Jan. 9 pages. CHANSON H. an BRATTBERG T. (2000). "Experimental Stuy of the Air-Water Shear Flow in a Hyraulic Jump." Intl Jl of Multiphase Flow ol. 26 No. 4 pp CHANSON H. AOKI S. an HOQUE A. (2004). "Physical Moelling an Similitue of Air Bubble Entrainment at ertical Circular Plunging Jets." Chemical Engineering Science ol. 59 No. 4 pp CUMMINGS P.D. an CHANSON H. (997a). "Air Entrainment in the Developing Flow Region of Plunging Jets. Part Theoretical Development." Jl of Fluis Eng. Trans. ASME ol. 9 No. 3 pp CUMMINGS P.D. an CHANSON H. (997b). "Air Entrainment in the Developing Flow Region of Plunging Jets. Part 2 : Experimental." Jl of Fluis Eng. Trans. ASME ol. 9 No. 3 pp HENDERSON F.M. (966). "Open Channel Flow." MacMillan Company New York USA. MOSSA M. an TOLE U. (998). "Flow isualization in Bubbly Two-Phase Hyraulic Jump." Jl Fluis Eng. ASME ol. 20 March pp MURZYN F. MOUAZE D. an CHAPLIN J.R. (2005). "Optical Fibre Probe Measurements of Bubbly Flow in Hyraulic Jumps" Intl Jl of Multiphase Flow ol. 3 No. pp RESCH F.J. an LEUTHEUSSER H.J. (972). "Le Ressaut Hyraulique : mesure e Turbulence ans la Région Diphasique." Jl La Houille Blanche No. 4 pp