Air bubble entrainment in open channels is called 'white waters'. It is observed in supercritical turbulent flows. The air bubble diffusion process is analysed both analytically and experimentally in uniform equilibrium flows. The solution of the diffusion equation is compared with model and prototype data. The results indicate that the turbulent diffusivity is of the same order of magnitude as the momentum transfer coefficient (i.e. eddy viscosity.). However D_t/v_T is larger on models than on prototype, suggesting that model investigations might not reproduce accurately the air bubble diffusion process

Chanson, Hubert

University of Queensland eSpace

Twelfth Australasian Fluid Mechanics Conference 707 
The University of Sydney, Australia 1995 
 
 
ABSTRACT 
Air bubble entrainment in open channels is called 'white 
waters'. It is observed in supercritical turbulent flows. The 
air bubble diffusion process is analysed both analytically 
and experimentally in uniform equilibrium flows. The 
solution of the diffusion equation is compared with model 
and prototype data. The results indicate that the turbulent 
diffusivity is of the same order of magnitude as the 
momentum transfer coefficient (i.e. eddy viscosity.). 
However Dt/νT is larger on models than on prototype, 
suggesting that model investigations might not reproduce 
accurately the air bubble diffusion process. 
INTRODUCTION 
In supercritical open channel flows, free-surface 
aeration is frequently observed. It is also called 'self-
aeration'. Air entrainment occurs when the turbulence 
acting next to the free-surface is large enough to overcome 
both the surface tension and buoyancy effects. 
The process of self-aeration in chutes and storm 
waterways was initially studied because of the effects of 
entrained air on the thickness of the flow and the possible 
reduction in cavitation damage. Recently self-aeration on 
chutes has been recognised also for its contribution to the 
air-water transfer of atmospheric gases such as oxygen 
and nitrogen. 
After a brief bibliography, the paper presents a new 
analysis of the air bubble diffusion. An analytical solution 
of the diffusion equation in uniform equilibrium flow is 
deduced. The result are compared with existing and new 
experimental data. The results are analysed also in term of 
the air bubble diffusion coefficient. 
Bibliography 
Self-aerated open channel flows have been studied only 
recently. Although some researchers observed 'white 
waters' and discussed possible effects, no experimental 
investigation was conducted successfully before the first 
quarter of the 20-th century. EHRENBERGER (1926) 
presented probably the first set of conclusive data. 
Another milestone was the experimental work lead by 
L.G. STRAUB (e.g. STRAUB and ANDERSON 1958). 
The early models of air bubble diffusion derived from 
sediment-laden flow studies. Two original models were 
developed by STRAUB and ANDERSON (1958) and 
WOOD (1984). STRAUB and ANDERSON described the 
structure of self-aerated open channel flows as consisting 
of the inner region consisting of air bubbles distributed 
through the water flow by turbulent transport fluctuations 
and the outer flow region with a heterogeneous mixture of 
water droplets ejected from the flowing liquid stream. In 
the writer's opinion, the model of STRAUB and 
ANDERSON does not reflect the physical nature of the 
air-water flow. Measured air concentration and velocity 
distributions show clearly that the air-water flow behaves 
as a homogeneous mixture between 0 and Y90 as shown 
by WOOD (1985) and CHANSON (1993). 
WOOD (1984) developed the conservation equation for 
the air-water mixture density in the equilibrium region. 
His model provides a very good fit with the experimental 
data for mean air contents between 10% and 75% (WOOD 
1984,1985). It fits well also air concentration distributions 
in the gradually-varied flow region. But the model is 
based upon empirical constants derived from the concept 
of "diffusivity of the average density" and the need to 
estimate a "fall velocity of water". 
 
AIR BUBBLE DIFFUSION 
IN SUPERCRITICAL OPEN CHANNEL FLOWS 
 
 
 
Hubert CHANSON 
Department of Civil Engineering 
The University of Queensland 
Brisbane QLD 4072 
AUSTRALIA 
 708 
 
AIR BUBBLE DIFFUSION 
In a spillway chute, the upstream flow region is non-
aerated, followed by a gradually-varied flow region and 
eventually an uniform equilibrium flow region (fig. 1). In 
the uniform equilibrium flow region, the flow properties 
(including the air concentration distribution) are 
independent of the distance x along the channel. The 
turbulence diffusion normal to the bottom counterbalances 
exactly the buoyancy effect. For a small control volume, 
the continuity equation for air in the air-water flow yields 
: 
0  =  
d
dy⎝⎛ ⎠⎞Dt * d Cdy   -  cosα * ddy(ur * C) (1) 
where C is the air content, y the distance normal to the 
channel, Dt the diffusion coefficient, ur the bubble rise 
velocity (in air-water mixture) and α the slope. The 
bubble rise velocity ur  in a fluid of density ρw*(1 - C) 
can be related to the rise velocity in hydrostatic pressure 
gradient as : ur
2 = (ur)Hyd
2 *(1 - C) (see Appendix). 
Assuming a homogenous turbulence, a solution of 
equation (1) is then : 
C  =  1  -  tanh2⎝⎛ ⎠⎞K'  -  y'2 * D'  (2) 
where D' = Dt/((ur)Hyd*cosα*Y90), y' = y/Y90, Y90 the 
distance where C = 0.9, α is the channel slope and tanh is 
the hyperbolic tangent function. K' and D' are integration 
constants satisfying : 
 C(y' = 1) = 0.9 
and Cmean = ⌡⌠
0 
 1
C * dy' 
K' and D' are dimensionless functions of the mean air 
concentration only (table 1). Full details of the integration 
are reported in CHANSON (1995b). 
Equation (2) is compared firstly with uniform 
equilibrium flow data (STRAUB and ANDERSON 1958) 
(fig. 2). Figure 3 compares equation (2) with experimental 
data obtained in a 4.0-degree slope chute (CHANSON 
1995b). The data were recorded in the gradually-varied 
flow region. WOOD (1985) showed that the longitudinal 
variation of air concentration is gradual. Hence equation 
(2) can be used with local rather than equilibrium value. 
On figures 2 and 3, the agreement between equation (2) 
and data is good in both equilibrium and gradually-varied 
flows. 
DIFFUSION COEFFICIENTS 
In open channel flows, the relationship between the air 
bubble diffusion coefficient and the momentum transfer 
coefficient (νT) is : 
Dt
νT  =  
2
Κ * D* (3) 
where Κ is the Von Karman constant (Κ = 0.40) and 
D* = Dt/(V**Y) where V* is the shear velocity and Y is 
the characteristic flow depth. D* is a classical 
dimensionless expression of the diffusion coefficient in 
sediment-laden flows and for the vertical dispersion of 
matters (e.g. dye, salt). In non-aerated flows, Y is the flow 
depth while Y90 is the characteristic depth in self-aerated 
flows. 
The ratio Dt/νT describes the combined effects of : 1- 
the difference in the diffusion of a discrete particle (e.g. 
air bubble, sediment) and the diffusion of a small coherent 
fluid structure, and 2- the influence of the particles on the 
turbulence field (e.g. turbulence damping or drag 
reduction). Values of Dt/νT for several self-aerated flow 
experiments (uniform equilibrium flow conditions) are 
summarised in table 2 and on figure 4. They are compared 
with sediment diffusion coefficients and diffusion 
coefficients of matter in open channels. 
First note that Dt is of the same order of magnitude as 
the eddy viscosity. 
In equilibrium self-aerated flows, equation (3) can be 
rewritten as : 
Dt
νT  =  
2
Κ * 
ur * cosα
V*
 * D' (3b) 
where D' is a function of the mean air content (table 1). 
Equation (3b) implies that Dt/νT depends not only upon 
ur*cosα/V* (in a similar form as for sediment-laden 
flows) but also upon the mean air content. 
On figure 4, the reader shall note that, on large 
prototypes, the ratio of the turbulent diffusivity over the 
eddy viscosity is less than unity while it is larger than one 
on models. Such a result (fig. 4) suggests that scale-model 
studies of self-aerated flows might not describe accurately 
the air bubble diffusion process in uniform equilibrium 
self-aerated flows. The trend shown on figure 4 should 
however be confirmed with additional uniform 
equilibrium field data. 
 
Table 1 - Relationship between Cmean, D' and K 
 
Cmean D' K' 
(1) (2) (3) 
0.01 0.007312 68.70445 
0.05 0.036562 14.0029 
0.10 0.073124 7.16516 
0.15 0.109704 4.88517 
0.20 0.146489 3.74068 
0.30 0.223191 2.567688 
0.40 0.3111 1.93465 
0.50 0.423441 1.508251 
0.60 0.587217 1.178924 
0.70 0.878462 0.896627 
 
Cmean : mean air concentration defined in term of Y90 
DISCUSSION 
1- The air concentration distribution (eq. (2)) has been 
obtained assuming : a homogeneous turbulence and a 
constant bubble rise velocity (ur)Hyd (from 0 to Y90). The 
latter is an approximation. The author (CHANSON 1995a) 
showed that the air bubble size varies across the flow from 
micro-sizes next to the bottom up to large air packets in 
the upper flow region. For such a wide range of bubble 
sizes, the rise velocity (ur)Hyd is not a constant (e.g. 
COMOLET 1979). 
For sediment-laden flows, several studies (e.g. 
COLEMAN 1970, GRAF 1971) indicated that the 
sediment diffusion coefficient Dt is not a constant across 
the flow. However model experiments (COLEMAN 1970) 
 709 
 
and river data (ANDERSON 1942) showed that Dt is 
constant in the outer flow region (i.e. typically y/Y > 
0.15). 
2- The relationship between the dimensionless diffusivity 
D' and the mean air concentration is unique and 
independent of the discharge and bottom friction. 
3- Note that equation (2) does not describe the air bubble 
diffusion next to the wall (i.e. within the air concentration 
boundary layer). Next to the wall, the interactions between 
the air bubbles and the turbulent shear layers result in a 
different profile and induce some drag reduction. A 
detailed review can be found in CHANSON (1994). 
 
Table 2 - Ratio Dt/νT in open channels 
 
Reference Dt/νT Comments 
   
(1) (2) (3) 
Self-aerated flows   
STRAUB and ANDERSON 
(1958) 
0.88 to 3.62 
(a) 
Model data. 
AIVAZYAN (1986) 1.85 to 4.1 
(a) 
Model data. 
 0.6 to 2.6 (a) Prototype data. 
Sediment-laden flows   
LANE and KALINSKE 
(1941) 
0.335  
ANDERSON (1942) 0.4 to 1.5 (b) Enoree river, 
USA. 
COLEMAN (1970) 0.25 to 2 (b) Model data. 
Matter diffusion in open channel  
FISCHER et al. (1978) 0.335 Vertical mixing. 
 
Notes : 
(a) Calculations performed assuming (ur)Hyd = 0.4 m/s 
(b) Sediment diffusion coefficient in outer flow region. 
CONCLUSION 
The air bubble diffusion in self-aerated supercritical 
flows is analysed both analytically and experimentally. A 
new solution of the diffusion equation (eq. (2)) compares 
favourably with experimental data in both the equilibrium 
flow region and gradually-flow region (fig. 2 and 3). 
The solution of the advective diffusion equation implies 
turbulent diffusivities of the same order of magnitude as 
the momentum transfer coefficient (table 2). The ratio 
Dt/νT is however larger with model experiments than with 
prototype data. If the trend is confirmed, it would suggest 
that model experiments overestimate the air bubble 
diffusion coefficient. 
ACKNOWLEDGMENTS 
The author wants to thank particularly Professor I.R. 
WOOD who initiated his interest on the topic. The 
financial support of Australian Research Council (Ref. 
No. A89331591) is acknowledged. The paper was 
prepared during a Special Study Program at the National 
Cheng Kung University (Taiwan R.O.C.). The support of 
the Department of Hydraulics and Ocean Engineering, 
National Cheng Kung University and of Professor LEE 
J.F. (NCKU) is acknowledged. 
APPENDIX - RISE BUBBLE VELOCITY IN NON-
HYDROSTATIC PRESSURE GRADIENT 
The buoyant force on a submerged body (e.g. an air 
bubble in a fluid) is the difference between the vertical 
components of the pressure force on its underside and on 
its upper side. For a single bubble rising at a constant 
velocity in a quiescent surrounding fluid, the drag force 
counterbalances the resultant of the weight force and the 
buoyant force. Neglecting the weight of an air bubble, the 
rise velocity squared is proportional to the pressure 
gradient : 
ur
2  ~  - 
d P
dy  (A-1) 
In an air-water flow (fig. 1), the local pressure and the 
pressure gradient at any position y are : 
P(y)  =  ⌡⌠
y 
 +oo
ρw * (1 - C) * g * cosα * dξ (A-2) 
d P
dy (y)  =  ρw * (1 - C) * g * cosα (A-3) 
Considering the bubble rise velocity in a hydrostatic 
pressure gradient (i.e. dP/dy = ρw*g*cosα), the 
expression of the bubble rise velocity in a fluid of density 
ρw*(1-C) becomes : 
ur
2  =  [(ur)Hyd]
2 * (1 - C) (A-4) 
At the limit the rise velocity is zero in air (i.e. C = 1). 
REFERENCES 
AIVAZYAN, O.M. (1986). "Stabilized Aeration on 
Chutes." Gidrotekhnicheskoe Stroitel'stvo, No. 12, pp. 33-
40 (in Russian). 
ANDERSON, A.G. (1942). "Distribution of Suspended 
Sediment in a Natural Stream." Trans. Amer. Geophys. 
Union, Vol. 23, Pt. 2, pp. 678-683. 
CHANSON, H. (1993). "Self-Aerated Flows on Chutes 
and Spillways." Jl of Hyd. Engrg., ASCE, Vol. 119, No. 2, 
pp. 220-243. 
CHANSON, H. (1994) "Drag Reduction in Open 
Channel Flow by Aeration and Suspended Load." Jl of 
Hyd. Res., IAHR, Vol. 32, No. 1, pp. 87-101. 
CHANSON, H. (1995a). "Predicting Oxygen Content 
Downstream of Weirs, Spillways and Waterways." Proc. 
Instn Civ. Engrs Wat. Marit. & Energy, UK, Vol. 112, 
Mar., pp. 20-30. 
CHANSON, H. (1995b). "Air Bubble Entrainment in 
Free-surface Turbulent Flows. Experimental 
Investigations." Report CH46/95, Dept. of Civil Eng., Uni. 
of Queensland, Australia. 
COLEMAN, N.L. (1970). "Flume Studies of the 
Sediment Transfer Coefficient." Water Ressources Res., 
Vol. 6, No. 3, pp. 801-809. 
COMOLET, R. (1979). "Vitesse d'ascension d'une bulle 
de gaz isolée dans un liquide peu visqueux." Jl de 
Mécanique Appliquée, Vol. 3, No. 2, pp. 145-171. 
EHRENBERGER, R. (1926). "Wasserbewegung in 
steilen Rinnen (Susstennen) mit besonderer 
Berucksichtigung der Selbstbelüftung." Zeitschrift des 
Österreichischer Ingenieur und Architektverein, No. 15/16 
and 17/18 (in German). 
FISCHER, H.B., LIST, E.J., KOH, R.C.Y., 
IMBERGER, J., and BROOKS, N.H. (1979). "Mixing in 
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Inland and Coastal Waters." Academic Press, New York, 
USA. 
GRAF, W.H. (1971). "Hydraulics of Sediment 
Transport". McGraw-Hill, New York, USA. 
LANE, E.W., and KALINSKE, A.A. (1941). 
"Engineering Calculations of Suspended Sediment." 
Trans. Amer. Geophys. Union, Vol. 20. 
STRAUB, L.G., and ANDERSON, A.G. (1958). 
"Experiments on Self-Aerated Flow in Open Channels." Jl 
of Hyd. Div., ASCE, Vol. 84, No. HY7, pp. 1890-1 to 
1890-35. 
WOOD, I.R. (1984). "Air Entrainment in High Speed 
Flows." Proc. Intl. Symp. on Scale Effects in Modelling 
Hydraulic Structures, IAHR, Esslingen, Germany, H. 
KOBUS editor, paper 4.1. 
WOOD, I.R. (1985). "Air Water Flows." Proc. 21st 
IAHR Congress, Melbourne, Australia, Keynote address, 
pp. 18-29. 
 
 
 Fig. 1 - Air entrainment on an open channel chute Fig. 2 - Air concentration distributions in uniform equilibrium 
  flows - STRAUB and ANDERSON (1958) 
 
 
 
Fig. 3 - Air concentration distributions in gradually-varied Fig. 4 - Ratio Dt/νT in uniform equilibrium self-aerated flow 
  self-aerated flows - CHANSON (1995b) and in sediment-laden flows 
 
 
 


Air Bubble Diffusion in Supercritical Open Channel Flow

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Air Bubble Diffusion in Supercritical Open Channel Flow

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