Bardiaux J.B.

Bonakdari H.

Larrarte F.

Mosé R.

Vazquez J.

I-Revues

SESSION 7.2 
NOVATECH 2007  1441 
Assessment of velocity fields through open-
channels: creation of empirical laws 
Evaluation des champs de vitesses dans un canal ouvert: création 
de lois empiriques 
 
Bardiaux J.B.1, Vazquez J.1,Mosé R.1, Bonakdari H. 2, Larrarté F. 2 
1 Ecole Nationale du Génie de l'Eau et de l'Environnement de Strasbourg  
1 quai Koch 67070 Strasbourg cedex France bardiaux@engees.u-strasbg.fr 
2 Laboratoire Central des Ponts et Chaussées, Division eau et environnement 
route de Bouaye 44341 Bouguenais Cedex France frederique.larrarte@lcpc.fr 
RESUME 
L’exploitation des réseaux d’assainissement exige désormais une connaissance fine 
des flux polluants véhiculés. Dans ce contexte, connaître la distribution spatiale des 
vitesses dans la section étudiée rend l’estimation du débit plus réaliste. L’article 
défendu ici présente deux relations mathématiques simples permettant d’établir une 
cartographie des vitesses dans une section donnée, pour un écoulement développé. 
L’une, issue des équations de Navier-Stokes, requiert la connaissance de la vitesse 
de cisaillement et la position de la vitesse maximale quand l’autre utilise des 
paramètres empiriques issus de traitement statistiques. Les deux sont validées sur 
des cas expérimentaux. 
ABSTRACT 
The exploitation of the sewers networks requires now a fine knowledge of conveyed 
polluting flows. In this context, to know the spatial distribution velocities in the studied 
section returns the estimate of the flow more realistic. The article defended here 
presents two mathematical relations making it possible to establish a cartography of 
the velocity in a cross section, for a full-developed flow. One, resulting from the 
Navier-Stokes equations, requires the shear velocity and the maximal velocity 
position, when the other uses empirical parameters resulting from statistics treatment. 
Both are validated on experimental cases. 
KEY WORDS 
Laws, open-channels flow, velocity fields
SESSION 7.2 
1442 NOVATECH 2007  
INTRODUCTION 
Since about fifteen years now, the lawful constraints on the sewer networks were 
developed. Indeed the January 1992 national water policy law transposed from the 
May 1991 European Community Directive stipulates that any town, producing a daily 
pollutant load of more than 900 kg, has to be equipped with a wastewater collection 
network. Moreover, according to the decree of the 22nd December, relating to the self 
survey, in order to better evaluate the depollution rate of a system, it is necessary to 
install, today, measuring equipment able to estimate the flows moving towards the 
plants or the discharge system. Besides the fact that the usual hypothesis about 
spatial homogeneity for flow rates and pollutant loads is false (Wohrle and Brombach, 
1991), the dual behaviour of some wastewater collection networks linked to the rain 
water catching gives difficult the flow rates measurement. A better assessment of the 
spatial distribution of the velocities in a cross section is, therefore, one way to improve 
the flow acquisition, but also pollutant fluxes. The present paper presents two 
empirical laws to calculate, for a fully developed flow in a given cross section, the 
spatial distribution of the velocity. 
 
1 METHODOLOGIC APPROACH 
1.1 Simple model based on Navier-Stokes equations 
The Reynolds Averaged Navier-Stokes (RANS) momentum equations can be written 
as: 
i i i
j i j
j i j j
U U 1 P UU u u
t x x x x
⎛ ⎞∂ ∂ ∂ ∂ ∂
+ = − + ν −⎜ ⎟⎜ ⎟∂ ∂ ρ ∂ ∂ ∂⎝ ⎠
   (1) 
where iU  are the mean velocity in the x (streamwise), y (lateral) and z (vertical) 
directions, P  is the pressure, ρ  is the fluid density and i ju u  are the components of 
the Reynolds stress tensor. In a steady, uniform fully developed turbulent flow, 
equation (1) in the streamwise (x-) direction can be written: 
( ) ( )
2 2
0 2 2
U U U UV W gS uv uw
y z y z y z
∂ ∂ ∂ ∂ ∂ ∂
+ = + − + − + ν + ν
∂ ∂ ∂ ∂ ∂ ∂
 (2) 
where g is the acceleration due to gravity, 0S  the energy slope ( 0S sin= θ ). Yang et 
al. (2004) have reported that in the near bed region, the vertical gradient ( .
z
∂
∂
) should 
be dominating and the horizontal gradient ( .
y
∂
∂
) could be neglected. Therefore 
equation (2) in the near bed region can be written as follows: 
( )
2
0 2
U UW gS uw
z z z
∂ ∂ ∂
= + − + ν
∂ ∂ ∂
 (3) 
Nezu et al. (1989) have highlighted the pattern of secondary currents in open 
channels, the transverse flow (V>0) near the free surface is directed from the side 
wall towards the channel centre and this current flows down (W<0) along the channel 
SESSION 7.2 
NOVATECH 2007  1443 
centre from the free surface to the bottom. This means, that in the channel centre, 
there is only the vertical mean velocity that contributes to the secondary current. In 
order to assess such constraint, the vertical velocity must satisfy 3 conditions: W 
equals to 0 at the bed and free surface level, and W negative in the channel centre. 
After integration and simplification, by using the no-slip boundary condition at 
0
0
z
h
ξ = ξ = , where z0 is the roughness length of the surface, the vertical 
distribution of the velocity in the channel central region and out of the inner region (z/h 
> 0.2) can be determined by: 
2
2
0
U 1 0.25 C(A)ln
u (0.5 C(A))∗
⎛ ⎞α ξ
= ξ + ξ +⎜ ⎟
κ ξ + ξ + ξ⎝ ⎠
    (4) 
where κ is the Von-Karman constant, u* the shear velocity, 2
ghS 1
u∗
α = − , z
h
ξ =  relative 
distance from the bottom and C(A) a parameter in function of Ar (the aspect ratio) that 
can be evaluated as follows: 
( )EdipC(A)=D. ξ  (5) 
where D = 9.3 and E = 1.7 are two constants deduced from the analytical resolution 
of equation (4).  When dipξ is unknown, we can use the sigmoid relation developed by 
Morgan and al (1975) with adjusted parameters. In the inner region, the logarithmic 
law remains valid. 
1.2 Simple model based on fitted empirical parameters  
The following approach comes within the framework of the development of a new 
sensor measuring flow rates in sewers. First, a transductor, using ultrasonic signal, 
gives a Doppler response transformed in a velocity profile in a cross-section. Then, as 
each velocity measurements have identified positions, it could be interesting to insert 
this chord in the whole velocity field and to deduct from this the flow rate. The 
challenge is clearly here to know this field only from the hydraulic characteristics, i. e. 
the geometry channel and the water level. 
1.2.1 First step: use numerical modelling 
The method starts with the numerical simulations of different flows in several classical 
sections (in sewer networks): rectangular, circular and egg-shape. The table 1 shows 
the list of all the simulations realised with the Fluent software. 
Shape rectangular circular egg-shape 
Dimensions  
 (mm) 
width x height : 
600 x 800, 1000 x 
500, 2000 x 500 
diameter: 
500,1000,1500, 
2000 
height: 
1000,1500,2000 
Filling 20%, 30%, 40%, 50%, 70%, 90% 
20%, 30%, 40%, 
50%, 70%, 80% 
20%, 30%, 40%, 
50%, 70%, 80% 
Mean Velocity 
(m/s) 
0,25-0,5-0,75-1,00-
1,25-1,5-2 
0,25-0,5-0,75-1,00-
1,25-1,5-2 
0,25-0,5-0,75-1,00-
1,25-1,5-2 
Table 1 : Fluent simulation framework 
SESSION 7.2 
1444 NOVATECH 2007  
The first work were the definition of the modelling way. The first problem with the open 
channels remains the free surface and the anisotropy of the turbulence. The many 
studies realised (Larrarte and al 2005) showed that the best way to have a good 
agreement with the reality (Bonakdari and al 2006)) were the V.O.F method (diphasic 
model) combined with an R.S.M. model for the turbulence (Bardiaux and al 2006).  
1.2.2 Second step: Have a representative field of the type of channel 
The library of velocity fields obtained, for each kind of channel allows us to generate a 
representative field. After having transformed the fields within a dimensionless 
channel (the position is now given by ξ and the relative distance from the wall y
B
) 
divided each local velocity by the associated mean velocity, the local values for the 
medium field come are the average of all the local velocities for this position.  
1.2.3 Last step: fit an algebraic formulation to the medium field 
The formulation chosen here is clearly based on the known description of classical 
velocity fields in open-channels (Nezu (1993)). End, logarithmic (for the “vertical “ 
behaviour) and power (for the horizontal one) laws have been crossed. Moreover, in 
the aim of taking into account the dip-phenomenon, a trigonometric one has been 
added. At last, to limit the asymmetric effects, we use absolute value. Then, the 
formulation proposed is the following one (equation (6)): 
( ) ( )
f g
mean
U y y= a.ln ξ +b+c.sin d.ξ +e. .
U B B
⎡ ⎤⎛ ⎞ ⎛ ⎞
⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
 (6) 
where a, b, c, d, e, f, g are empiric parameters stemmed to the fitting, depending on 
the water level. 
2 RESULTS 
2.1 Circular open-channel:  
2.1.1 Description of the data 
The data presented in the figure 1 correspond to a flow through a circular pipe            
(Q=11.7l/s and diameter=123 mm, water level=123.5 mm) (Knight and Sterling (2000)) 
width (m) 
H
eight (m
) 
 
Figure 1 : Isolevels of velocity in a circular pipe (Knight and Sterling (2000)) 
SESSION 7.2 
NOVATECH 2007  1445 
2.1.2 Profiles on the central axis 
 
a -0.45
b -2.19
c 23.56
d 0.03
e -0.79
f 3.62
g 0.19
 
 
Figure 2 : Velocity profiles on central axis 
The two approaches present a good agreement with the experimental data in the 
central region according the figure 2. In fact, unlike Coles relation, the two models are 
able to simulate the dip-phenomenon on the right position. 
2.1.3 Velocity distribution in the section 
 
Figure 3 : Isolevel of 
mean
U
U
 from eq(6) 
The figures 3 and 4 show the ability of the equation (7) to simulate the velocity field 
on around 80% (less than 5%) of the section with a good agreement, but around the 
wall the model is inadequate. The right position of the Dip-phenomenon is clearly on 
the figures 2 and 3, despite of a weaker maximal velocity. 
 
SESSION 7.2 
1446 NOVATECH 2007  
  
Figure 4 : Spatial distribution of the error (%) 
2.2 Experimental site in a sewer network 
The experimental site (figure 6) is located in an area called Cordon Bleu, a few 
kilometres upstream the treatment plant on the main sewer line of the city of Nantes 
(France). In the egg-shape pipe, the mean velocity was 0.88 m/s, the water height 
1.20 m. No measurement could be realised over the walkway but a wide of flow 
conditions have been experimentally investigated (Larrarte (2006)).  
 
a -0.40
b -1.93
c 20.72
d 0.03
e -0.70
f 3.62
g 0.19
 
Figure 5 : Velocity profiles on central axis 
 
 
SESSION 7.2 
NOVATECH'2007  1447 
  
a)                                               b) 
Figure 6 : spatial distribution of the error (%) a) from eq(4) and b) from eq(6) 
The figures 5 and 6 raise, as in 3.1, the good description in the central region, less 
than 5 %. But, the walk-away generate special flow (re-circulations) which are not 
correctly reproduced. 
3 CONCLUSION 
The two models presented here allow a good representation of a velocity field in a 
fully developed flow. The advantage is to be independent of the measurement 
conditions. Indeed, geometry and water level are sufficient to reach an assessment of 
the flow rate. Thus, if a sensor gives a partial chord (range limit), it is possible to build 
up the whole field and estimate the flow rate. Furthermore, if the sensor has to be 
fasten sideways, the localisation of the sampled volume can be used even if the 
maximal velocity area is not scanned.  
Certainly the nearness of the walls remains a problem but this is not the area, which 
contributes the most to the flow rate. But the main work to add is the treatment of the 
walk-away area. Indeed, local deformation of the velocity field cannot be simulated 
with the expressions (4) and (6).  Data acquisition on a new physical model will help 
us to improve this. 
 
BIBLIOGRAPHIE 
Bardiaux, J.B., Bonakdari,H. , Larrarte, F. , Mose, R. , Vazquez, J. (2006) Velocity profiles 
through a sewer channel using CFD to obtain velocity fields Strömungssimulation in 
Wasserbau, Dresdner Wasserbaukolloquium, 9-10 March 2006 
BonakdariI, H. , Larrarte, F. ,Bardiaux, J.B (2006) Experimental and computational study of 
velocity fields in narrow or compound section sewers 7th International Conference on Urban 
Drainage Modelling, Melbourne, Australia, 3-7 
Czernuszenko W. and Rylov A., (2002) « Modélisation des champs de vitesses 
tridimensionnelles en canal à surface libre », Journal of Hydraulic Research, Vol. 40, No. 2 
  F ree  surface 
SESSION 7.2 
1448 NOVATECH'2007  
Knight,D.W.,and Sterling,M., (2000).”boundary shear in circular pipes running partially full.” 
Journal of hydraulic Engineering 126(4), 263-275 
Larrarte, F., ,Bardiaux, J.B., BonakdariI, H. , (2005) Computational and experimental study of 
velocity profiles in sewers 10th International Conference on Urban Drainage, Copenhagen, 
Denmark, 21-26 
Larrarte F. (2006), Velocity fields in sewers : an experimental study, Flow Measurement and      
 Instrumentation 17 N°5, 282–290 
Morgan, P. H., Preston, L., Mercer, and Flodin, N.W. (1975), General Model for Nutritional 
Responses of Higher Organisms, Proc. Nat. Acad. Sci. USA, Vol. 72, No. 11, pp. 4327-4331, 
Nezu I. and Nakagawa H. (1993) « Turbulence in open - channel Flows », IAHR, 281p. 
Nezu I., Nakagawa H., Rodi W., (1989), Significant difference between secondary currents in 
closed channels and narrow open channels, Proc., 23rd IAHR Congress, Vol. A, Delft, The 
Netherland, pp 125-132. 
Wohrle, C. Brombach, H. (1991), Probenahme im abwasserkanal, Wasserwirtschaft, Vol. 81, pp. 
60-65.  
Yang S. Q., Tan S. K., Lim S. Y., (2004), Velocity distribution and dip phenomenon in smooth 
uniform open channel flow, Journal of Hydraulic Engineering, Vol. 130, No. 12, pp 1179-
1186. 
 


Evaluation des champs de vitesses dans un canal ouvert: création de lois empiriques

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Evaluation des champs de vitesses dans un canal ouvert: création de lois empiriques

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