Self-aeration in the rapidly- and gradually-varying flow regions

20 Pages·2015·1.61 MB·English

by  Gangfu Zhang

EnvironFluidMech(2017)17:27–46
DOI10.1007/s10652-015-9442-z
ORIGINAL ARTICLE
Self-aeration in the rapidly- and gradually-varying flow
regions of steep smooth and stepped spillways
Gangfu Zhang1 • Hubert Chanson1
Received:9August2015/Accepted:15December2015/Publishedonline:24December2015
(cid:2)SpringerScience+BusinessMediaDordrecht2015
Abstract In high-velocity chute flows, free-surface aeration is often observed. The
phenomenoniscalledself-aerationorwhitewaters.Whentheturbulentshearstressesnext
to the free-surface are large enough, air bubbles are entrained throughout the entire air–
water column. A rapidly-varied flow region is observed immediately downstream of the
inceptionpointoffree-surfaceaeration.Ananalyticalsolutionoftheairdiffusionequation
is proposed and the results compare well with new experimental data. Both experiments
and theory indicate that the flow bulking spans over approximately 3–4 step cavities
downstream of the inception point of free-surface aeration on a stepped chute. Further
downstream the void fraction distributions follow closely earlier solutions of the air dif-
fusion equation. The application of the diffusion equation solution to prototype and lab-
oratory data shows air bubble diffusivities typically larger than the momentum transfer
coefficient. The result highlights however a marked decrease in the ratio of air bubble
diffusivitytoeddyviscositywithincreasingReynoldsnumber.Thefindingmightindicate
somelimitationoflaboratoryinvestigationsofairbubblediffusionprocessinself-aerated
flows and of their extrapolation to full-scale prototype applications.
Keywords Air entrainment (cid:2) Self-aeration (cid:2) Smooth chutes (cid:2) Stepped spillways (cid:2)
Rapidly-varied flow
1 Introduction
In high-velocity open channel flows, free-surface aeration, also called self-aeration, is
commonlyobserved[8,20,37,40].Figure 1illustratestypicalexamplesofself-aerationin
prototype spillways. The process of free-surface aeration in chutes and spillways was
& HubertChanson
[email protected];
http://www.uq.edu.au/*e2hchans/
1 TheUniversityofQueensland,SchoolofCivilEngineering,Brisbane,QLD4072,Australia
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28 EnvironFluidMech(2017)17:27–46
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EnvironFluidMech(2017)17:27–46 29
bFig.1 Free-surfaceaerationonsmoothandsteppedprototypespillways,includingdetailsoftherapidly-
variedflowregionimmediatelydownstreamoftheinceptionpointoffree-surfaceaeration.aSomersetdam
(Australia) on 28 January 2013—Gates fully-opened: Q*450m3/s (q*8m2/s), Re 39107, smooth
chute,shutterspeed:1/400s(left)and1/3,200s(right).bHinzedam(Autralia)on29January2013—Q
203m3/s, d/h*2.53, Re*6.69107, stepped chute (h=51.3(cid:3), h=1.2m)—left general view from
c
downstream (shutter speed: 1/4000s); right view from upstream looking at the large-scale free-surface
structures (boils) next to the inception point of free-surface (shutter speed: 1/1000s). c Paradise dam
(Australia) on 5 March 2013—Q 2300m3/s, d/h=2.86, Re 2.79107, stepped chute (h=57.48,
c
h=0.62m)—left general view from right bank (shutter speed: 1/1250s); right details of free-surface
structuresnexttotheinceptionpointoffree-surface(shutterspeed:1/1250s)
originally investigated because of the induced air–water flow bulking, the drag reduction
inducedbybubblesnexttotheinvert,thereductionincavitationdamageandtheair–water
masstransfer[22,36,47].Onbothsmoothandsteppedchutes,aturbulentboundarylayer
develops at the upstream end. When the boundary layer outer edge is close to the free-
surface,theturbulentshearstressnexttothefree-surfaceislargeenoughtoovercomeboth
the surface tension and buoyancy effects and free-surface aeration occurs [15, 21]. This
locationiscalledtheinceptionpointoffree-surfaceaeration[12,27,48].Itisclearlyseen
in prototype chutes (Fig. 1). On smooth spillways, surface waves are seen immediately
upstream of the inception, as well as longitudinal streaks [2, 30]. On stepped spillways,
prototypeandlaboratoryobservations showedthe presenceof largesurface scarsdirectly
upstream of the inception point and some surface flapping [9, 16] (Fig. 1b). The flow
measurements about the inception point showed intense free-surface aeration in this very
rapidly-varied flow region: ‘‘at the point of inception […] the mean concentration is
approximately 0.200 [32]. The finding was confirmed experimentally for a range of slopes
[3, 6, 18, 33, 39].
Afterabriefbibliography,thepaperpresentsanewanalysisoftheairbubblediffusion
in the rapidly-varied flow region immediately downstream of the inception point of free-
surfaceaeration.Ananalyticalsolutioniscomparedwithexistingsolutionsofthediffusion
equation in uniform equilibrium flow. The results are compared with new experimental
data obtained in the rapidly-varied flow region and in the gradually-varied flow down-
stream. The results are also presented in term of the air bubble diffusion coefficient. The
outcomesarediscussedintermsoftheratioofturbulentdiffusivitytomomentumtransfer
coefficient.
1.1 Bibliography
Since the first successful air–water flow measurements of Ehrenberger [20], a few early
models of air bubble diffusion were derived from sediment-laden flow studies assuming
uniform equilibrium: e.g., the model of Straub and Anderson [38] (Table 1). Straub and
Andersonproposedatwo-layermodel,withaninner/lowerregionconsistingofairbubbles
distributedthroughwaterbyturbulenttransportfluctuationsandanouter/upperflowregion
with a heterogeneous mixture of water droplets ejected from the flowing stream [26, 38].
The model was extended numerically, taking into account a non-uniform velocity profile
[28].Incontrast,RaoandGangadharaiah[35]andWood[45]developedtheconservation
equation for the air–water mixture density assuming a homogeneous air–water mixture
between0andY ,whereY isthedistancenormaltotheinvertwherethevoidfraction
90 90
C = 0.90. Measured void fraction and velocity distributions indeed showed that the air–
water flow motion behaves as a homogeneous gas–liquid mixture between 0 and Y [8,
90
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30 EnvironFluidMech(2017)17:27–46
Y)90 Y)90 Y)90
\ \ \ n
o
)dTd)T \y \y \y epti
Discussion \Uniformequilibriumflow—lowerflowregion(y[Uniformequilibriumflow—upperflowregion(y \\Gradually-variedflow(0yY)90 Uniformequilibriumandgradually-variedflows(0 Uniformequilibriumandgradually-variedflows(000K=0.32745?0.5/D:101040D=0.756999CmeanUniformequilibriumandgradually-variedflows(0D0D¼o(cid:5)(cid:6)2 y112(cid:4)(cid:3)(cid:4)Y39000K=0.32745?0.40123/DoD=-0.27679Ln(1.0434-1.3129C)omean Rapidly-variedflowimmediatelydownstreamofinc?\\pointoffree-surfaceaeration(0y?)
s
w
o
fl
g)
n
mi
m
ki
s
nsforself-aeratedsmoothandsteppedchute( Voidfractiondistribution 0(cid:2)(cid:3)zyCC¼(cid:3)Tdy(cid:4)T(cid:5)(cid:6)þ1(cid:2)(cid:3)2R1C2u(cid:4)expdu¼(cid:3)(cid:4)(cid:3)pffiffiC00hh1(cid:4)p(cid:3)Tyd(cid:4)T(cid:2)(cid:3)(cid:2)(cid:3)y1C1(cid:4)1erf¼(cid:3)(cid:4)CbExL21(cid:4)(cid:3)(cid:3)ð(cid:4)ÞDi0BC¼(cid:5)(cid:6)(cid:5)(cid:6)200BGyhexpcosþ(cid:4)(cid:3)(cid:3)Y90(cid:2)(cid:3)=yY20C1tanhK90¼(cid:4)(cid:4)0D2(cid:3) (cid:2)(cid:3)3=yY==yY13ð(cid:4)Þ200C1tanhK90¼(cid:4)(cid:4)þ90DD23(cid:3)(cid:3)oo 01Yy(cid:4):C05erfc¼(cid:3)50@AqffiffiffiffiffiffiffiffiffiffiffiffiffiffiDxL(cid:3)ð(cid:4)Þti2(cid:3)V
o
uti
ol
s
el
d
o
m
n
o
si
u
diff
Air udy
Table1 Reference [38] [35] [45] [13] [18] Presentst
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EnvironFluidMech(2017)17:27–46 31
11, 14, 46, 47]. The model of Rao and Gangadharaiah [35] was proposed for gradually-
varied flows, but required the determination of a significant number of semi-empirical
parameters.Althoughdevelopedfortheequilibriumregion,Wood’s[45]modelprovideda
simplerexpressionandanexcellentfitofsmoothinvertdatainbothgradually-variedand
equilibriumflowregions.Butthemodelusedempiricalconstantsderivedfromtheconcept
of‘‘diffusivityoftheaveragedensity’’andtheneedtoestimatea‘‘fallvelocityofwater’’.
Two newer models were developed based upon analytical solutions of the diffusion
equation for air bubbles in the equilibrium region [13, 18] (Table 1). Chanson [13]
assumed a constant dimensionless diffusivity D’ across the water column, with D’ being
solely a function of the depth-averaged void fraction C . Chanson and Toombes [18]
mean
proposedanon-constantdiffusivityD’acrossthewatercolumn(Table 1,column3).Both
models compared well with experimental data in the equilibrium and gradually-flow
regions,althoughChanson’s[13]modeltendedtofitbettersmoothchuteflowdata,andthe
model of Chanson and Toombes [18] skimming flow data on stepped chutes. This is
illustrated in Fig. 2, showing both smooth and stepped chute data compared to these two
solutions.
2 Basic considerations
Considering a two-dimensional flow as illustrated in Fig. 3 and assuming the advective
transporttermstobesmall,theequationofconservationofmassforairmaybeexpressed
as [14]:
oC¼div(cid:2)D (cid:3) g(cid:7)(cid:7)r!adC (cid:4) C (cid:3) !u (cid:3) ð1Þ
ot t r
whereCisthetime-averagedvoidfraction,tisthetime,D istheturbulentdiffusivityand
t
u is the bubble rise velocity.
r
Considering the rapidly-varied flow region immediately downstream of the inception
point, the upstream boundary conditions at the inception point (x = L) are: C = 1 for
i
y[d and C = 0 for y\d, where x is the longitudinal coordinate with x = 0 at the
i i
spillway crest, L is the longitudinal distance where the inception of free-surface aeration
i
takes place, y is the distance normal to the chute invert, d is the flow depth at inception
i
(Fig. 3). For x[L, the time of travel between L and any streamwise position x is:
i i
t = (x - L)/V ,whereV isthemeanflowvelocity,averagedoverthedistancebetweenx
i a a
and L. In first approximation, Eq. (1) may be reduced to a one-dimensional diffusion
i
equation in terms of void fraction governed by Fick’s second law:
oC o2C
¼D (cid:3) ð2Þ
ot t oy2
assumingthatqC/qx (cid:5) qC/qyandneglectingbuoyancyeffects.Thisscenarioisanalogous
tothatofaninter-diffusionbetweentwosemi-infinitemassesaslongasthediffusionlayer
remainsthincomparedtotheflowdepth(Fig. 3).Ifthediffusionlayerextendstotheinvert
(orpseudo-bottom formedby thestep edges),a complete solutionmay beobtainedusing
the method of images, although preliminary calculations showed that the invert had vir-
tually no effect on the accuracy of solution under current experimental conditions (see
below).TheboundaryconditionsforEq. (2)are:C(y = ??,t[0) = 1andC(y = -?,
t[0) = 0. The solution of Eq. (2) is:
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32 EnvironFluidMech(2017)17:27–46
A
1.2
1.1
1
0.9
0.8
0.7
0
9
Y 0.6
Y/
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C
B
Slope q (m2/s) Re Step C Reference
mean
8.9 0.099 21 0.33 Felder and Chanson (2013)
26.6 0.090 13 0.34 Bung (2009)
50 0.35 45 0.59 Boes (2000)
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
9
Y 0.8
y/ 0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C
Fig.2 Void fraction distributions in self-aerated chute flows. a Uniform equilibrium flows in a smooth
chute for h=22.5(cid:3), 45(cid:3) and 75(cid:3)—Data: Straub and Anderson [38], q=0.136m2/s, Re=5.49105—
comparison with analytical solutions by Chanson [13] (red) and Chanson and Toombes [18] (blue).
bGradually-variedanduniformequilibriumskimmingflowsonsteppedchutes—comparisonwithanalytical
solutionsbyChanson[13](red)andChansonandToombes[18](blue)forh=26.6(cid:3)and50(cid:3)
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EnvironFluidMech(2017)17:27–46 33
Fig.3 Longitudinaldevelopmentofairentrainmentinasteppedchuteflow
0 1
1 1 Y (cid:4) y
C¼ (cid:3) erfcB (cid:3) 50 C ð3Þ
2 @2 qDffiffiffiaffiffi(cid:3)ffiffiffiðffixffiffiffi(cid:4)ffiffiffiLffiffiiffiÞffiA
V
a
whereerfcisthecomplementaryerrorfunction,Y istheelevationwhereC = 0.50,and
50
D is an apparent time-averaged diffusivity:
a
T
1 Z
D ¼ (cid:3) D (cid:3) dt ð4Þ
a T t
0
with D = D in uniform flow with homogeneous turbulence and the integration time T
a t
beinglargerthanthecharacteristictimescalesofturbulentfluctuations.Thisnewsolution
(Eq. 3)isvalidfortherapidly-varyingflowimmediatelydownstreamoftheinceptionand
presentssomesimilaritywiththatobtainedforawaterjetdischargingintoatmosphere[4,
10, 14]. Equation (3) is compared to experimental data in Fig. 4a, with the diffusivity
being selected based upon a best fit and t = 0.01 s at the location of inception. A good
agreementwasobservedoverthefirsttwostepedgesdownstreamoftheinceptionpointof
free-surface aeration which corresponded to the rapidly-varied flow region. In the gradu-
ally-varied flow region further downstream, Eq. (3) tended to underpredict the void frac-
tion next to the invert (step edge 12, Fig. 4a) because the buoyancy effects were
unaccounted for, although the agreement with experimental data remained good for
0.2\C\0.9. The analytical solutions of Chanson [13] and Chanson and Toombes [18]
might be more appropriate in this case.
Inthegradually-variedanduniformequilibriumair–waterflowregions,Eq. (1)maybe
simplified into [13, 14]:
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34 EnvironFluidMech(2017)17:27–46
A
00..66 Step 7
Step 8
Step 9
00..44 Step 10
Step 11
Step 12
Step 7 (theory)
00..22
Step 8 (theory)
c Step 12 (theory)
d
)/
50 00
Y
00..22 00..44 00..66 00..88 11
-
y
(
--00..22
--00..44
--00..66
C
B
1.6
dc/h = 1.3, step edge 12
Chanson (1995)
1.4
Chanson and Toombes (2002)
Equation (3)
1.2
1
0
9
Y 0.8
y/
0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
C
Fig.4 Void fraction distributions in skimming flows on a stepped spillway (h=45(cid:3), h=0.1m, d/
c
h=1.3,inceptionpointatstepedge7).aVoidfractiondistributionsintherapidly-variedflowregion—
comparisonbetweenEq.(3)anddata.bComparisonbetweenexperimentaldataatstepedge12andvarious
voidfractionmodels[13,18],Presentstudy(Eq.3)
o (cid:5) oC(cid:6) o (cid:2) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (cid:3)
0¼ D (cid:3) (cid:4) cosh (cid:3) C (cid:3) 1 (cid:4) C (cid:3) ðuÞ ð5Þ
oy t oy oy r o
wherehisthespillwayinvertslopeand(u) isthebubblerisevelocityinclear-waterwith
r o
a hydrostatic pressure gradient. Analytical solutions were developed for y\Y where
90
Y is the elevation where C = 0.90 and they are reported in Table 1. Three diffusion
50
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EnvironFluidMech(2017)17:27–46 35
models were compared to experimental data recorded at the downstream end of a steep
steppedchuteinFig. 4b.ThemodelofChanson[13]providedthebestfitinthelowerpart
of the flow, while both Eq. (3) and Chanson and Toombes’ [18] solution underpredicted
the void fraction in that region. Note that the present model is applicable for y\??.
3 Experimental observations
Newexperimentswereperformedina5 mlong0.985 mwidetestsection,consistingofa
1.2 m high broad crested weir followed by 12 steps, 0.10 m long and 0.10 m high each
(Fig. 5).Waterwasfedfromalargeintakebasinthougha2.8 mlongsidewallconvergent
withacontractionrationof5.08:1ensuringasmoothandwavelessinflow.Air–waterflow
propertiesweremeasuredwithadouble-tipphasedetectionprobe.Eachneedlesensorhad
Fig.5 Free-surfaceaerationinalarge-sizesteppedspillwaymodel(h=45(cid:3),h=0.10m,W=0.985m).
a General view—d/h=0.9, Re=3.39105, shutter speed: 1/80s. b Details of free-surface structures
c
nexttotheinceptionpointoffree-surfaceaeration,viewedfromupstream—d/h=1.7,Re=8.79105,
c
shutterspeed:1/80s
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36 EnvironFluidMech(2017)17:27–46
an inner electrode of 0.25 mm diameter and sampled at 20 kHz for 45 s. Herein a nappe
flowwasobservedford /h\0.4,whered isthecriticalflowdepth:d = (q2/g)1/3,qisthe
c c c
discharge per unit width, g is the gravity acceleration and h is the vertical step height. A
transition flow was observed for 0.4\d /h\0.9. The present study is focused on the
c
skimming flow regime observed for d /h C 0.9. The overflow skimmed over the pseudo-
c
bottom formed by the step edges and the streamlines were approximately parallel to the
pseudo-bottom,althoughthefree-surfaceexhibitedawavyprofileapproximatelyinphase
withthesteps(Fig. 5).Downstreamoftheinceptionpointofaeration,somecomplexair–
waterinteractionswereobserved.Theflowinstepcavitiesexhibitedastablerecirculation
motioncharacterised byself-sustainingvortices,albeitirregularejectionoffluidfromthe
cavity into the mainstream flow next to the upper vertical step face, and replacement of
cavityfluidnexttothestepedge.Presentobservationsindicatedstrongmainstream-cavity
flow interactions as previously reported. Figure 5 presents two photographs of skimming
flow conditions.
Ateachstepedgedownstreamoftheinceptionpoint,thevoidfractiondatashowedan
inverted S-shape typically observed in self-aerated flows (Figs. 2, 4b and 6a). The air
bubbleconcentrationgradientqC/qywasmaximuminthemid-air–watercolumnbetween
0.2\y/d \0.5. At step edges, the data showed some self-similarity and a close agree-
c
mentwithanalyticalsolutions(Fig. 4b),despitesmallscatterunderlyingvoidfractionand
height measurements uncertainties. Typical void fraction distributions at step edges are
presented in Fig. 6a.
ThebubblecountrateF,definedashalfthenumberofair–waterinterfacesdetectedby
the probe sensor per unit time, provides some information on the air–water flow frag-
mentationand isproportionaltothe specificinterface area. Typicaldimensionlessbubble
count rate F 9 d /V distributions are shown in Fig. 6a, where F is the bubble count rate
c c
definedasthenumberofbubblesdetectedperunittimeandV isthecriticalflowvelocity:
c
V = (g 9 q)1/3.Thedatashowedadistinctshape,withamaximumF atapproximately
c max
0.3\y/d \0.4 corresponding to a void fraction between 0.4 and 0.5, as previously
c
reported [5, 18, 25, 50]. The bubble count rate is linked to the flow fragmentation, char-
acterised by characteristic length scales of a series of discrete water and air elements,
denoted k and k respectively. A simplistic theoretical model may be developed by
w a
consideringeachair–waterinterfacerepresentedbytwoconsecutiveelementsindifferent
states [43]:
F C (cid:3) ð1 (cid:4) CÞ
¼ ð6Þ
Fmax a (cid:3) b (cid:3) C2Fmax
where F is the maximum bubble count rate at a cross-section, C is the corre-
max Fmax
spondingvoidfraction,aisaparameterthataccountsfordifferencesbetweentheaverage
sizes of k and k , and b is a parameter allowing for variations of length scales:
w a
(cid:5)k (cid:6)
a¼1 þ C (cid:3) w (cid:4) 1 ð7Þ
k
a
b¼1 (cid:4) b (cid:3) ð1 (cid:4) 2 (cid:3) CÞ4 ð8Þ
with b a coefficient that characterises the maximum variations of b with (1-b) B b B 1
[41, 43]. Note that the right hand side of Eq. (6) is proportional to the probability of
occurrence of one pair of consecutive air–water elements. Equation (6) is compared to
presentdatainFig. 6b.Agoodagreementwasfoundhereinfork /k = 2.2andb = 0.4.
w a
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