CHANSON, H. (1995). "Air Concentration Distribution in Self-Aerated Flow - Discussion." Jl of Hyd. Res.,
IAHR, Vol. 33, No. 4, pp. 586-588 (ISSN 0022-1686).
Air Concentration Distribution in Self-Aerated Flow
by N.R. AFSHAR, G.L. ASAWA, and K.G. RANGA RAJU
Jl of Hyd. Res., IAHR, Vol. 32, No. 4, pp. 623-631
Discussion by
Hubert CHANSON
Lecturer in Fluid Mechanics, Hydraulics and Environmental Engineering
Department of Civil Engineering, The University of Queensland
Brisbane QLD 4072, AUSTRALIA.
The authors presented interesting new data. For completeness on the subject, the writer must state that
the authors omitted two important studies. These are the works of WOOD (1984) and CAIN (1978).
Air concentration distribution
Downstream of the point of inception of air entrainment, WOOD (1984) developed a simple model to
represent the turbulent diffusion of the entrained air within the flow. The model gives the shape of the air
concentration distribution for all mean air concentrations :
C
=
B'
B' + exp(-G' cosα y'2)
(A1)
where B' and G' are functions of the mean air concentration only (table A1), y' = y/Y90, and Y90 is the
characteristics depth where C = 90%. The mean air concentration Cmean is defined in terms of Y90 and dw:
(1 - Cmean) Y90
=
dw
(A2)
where dw is the equivalent clear water flow depth.
Although equation (A1) was initially developed for fully-developed (or equilibrium) aerated flows, the
equation was validated with model and prototype data in both the developing and fully-developed aerated
flow regions (WOOD 1984,1985,1991, CHANSON 1993). An example is shown on figure A1. Note that, next
to the channel bottom, the air content profile departs from equation (A1) as discussed by CHANSON (1994).
Equation (A1) is a simple expression based upon a physical analysis, taking into account the turbulent
diffusion process and the buoyancy effects. It is more meaningful than the empirical fitting proposed by the
authors (eq. (1)).
Table A1 - Air concentration parameters in self-aerated flows (after CHANSON 1993)
Cmean
G'*cosα
( a)
B'
(a)
(1)
0.0
(2)
+
infinite
7.999
5.744
4.834
3.825
2.675
2.401
1.8942
1.5744
(3)
0.00
0.161
0.241
0.310
0.410
0.569
0.622
0.680
0.721
Note : (a)
0.003021
0.028798
0.07157
0.19635
0.62026
0.8157
1.3539
1.8641
computed from STRAUB and ANDERSON's (1958) data
Page 1
CHANSON, H. (1995). "Air Concentration Distribution in Self-Aerated Flow - Discussion." Jl of Hyd. Res.,
IAHR, Vol. 33, No. 4, pp. 586-588 (ISSN 0022-1686).
Model and prototype data
Several researchers (e.g. ISACHENKO 1965, CAIN 1978, XI 1988) performed also air concentration
measurements in the developing and fully-developed flow regions. The writer re-analysed several sets of
data (CHANSON 1993) and, in each case, these data matched very closely equation (A1) (e.g. fig. A1).
Further, the work of CAIN (1978) provides unique field data.
CAIN (1978) performed experiments on a prototype spillway (i.e. Aviemore dam spillway, New Zealand
- α = 45 deg., smooth concrete) downstream of the inception point of air entrainment. Some results are
shown on figure A1. CAIN's (1978) work, "free" of scale effects, is an important milestone in the analysis of
self-aerated flows. It is regrettable that the authors ignored CAIN's (1978) data.
List of symbols
B'
Cmean
integration constant of the equilibrium air concentration distribution;
depth averaged air concentration defined as : (1 - Y90) Cmean = dw ;
dw
equivalent clear water depth (m) defined as :
C=90%
dw =
⌠ (1 - C) dy
⌡
C=0%
G'
qw
Y90
y'
integration constant of the equilibrium air concentration distribution;
water discharge per unit width (m2/s);
characteristic depth (m) where the air concentration is 90%;
dimensionless depth : y' = y/Y90.
List of references
CAIN, P. (1978). "Measurements within Self-Aerated Flow on a Large Spillway." Ph.D. Thesis, Ref. 78-18,
1978, Univ. of Canterbury, Christchurch, New Zealand.
CHANSON, H. (1993). "Self-Aerated Flows on Chutes and Spillways." Jl of Hyd. Engrg., ASCE, Vol. 119,
No. 2, pp. 220-243. Discussion : Vol. 120, No. 6, pp. 778-782.
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.
ISACHENKO, N.B. (1965). "Effect of Relative Roughness of Spillway Surface on Degree of Free-Surface
Flow Aeration." Izv. VNIIG, Vol. 78, pp. 350-357 (in Russian).
WOOD, I.R. (1984). "Air Entrainment in High Speed Flows." Proc. of the 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.
WOOD, I.R. (1991). "Air Entrainment in Free-Surface Flows." IAHR Hydraulic Structures Design Manual
No. 4, Hydraulic Design Considerations, Balkema Publ., Rotterdam, The Netherlands, 149 pages.
XI, Ruze (1988). "Characteristics of Self-Aerated Flow on Steep Chutes." Proc. Intl Symp. on Hydraulics for
High Dams, IAHR, Beijing, China, pp. 68-75.
Page 2
CHANSON, H. (1995). "Air Concentration Distribution in Self-Aerated Flow - Discussion." Jl of Hyd. Res.,
IAHR, Vol. 33, No. 4, pp. 586-588 (ISSN 0022-1686).
Fig. A1 - Air concentration distributions on Aviemore dam spillway (CAIN 1978) - Comparison with
equation (A1) - α = 45 degrees, qw = 2.23 m2/s
y/Y90
1
DATA - Cmean = 0.27
0.8
DATA - Cmean = 0.36
0.6
DATA - Cmean = 0.43
DATA - Cmean = 0.50
0.4
0.2
C
0
0
0.2
0.4
0.6
0.8
Page 3
1
EQ. (A1) - Cmean =
0.27
EQ. (A1) - Cmean =
0.36
EQ. (A1) - Cmean =
0.43
EQ. (A1) - Cmean =
0.50