Mosé R.

Vazquez J.

Wertel J.

I-Revues

SESSION 7.2 
NOVATECH 2007  1465 
Modélisation des profils tridimensionnels de 
vitesse et de turbulence pour améliorer 
l’instrumentation des réseaux d’assainissement 
Model for the numerical simulation of 3D turbulent flow to improve 
sewer net instrumentation 
J. WERTEL**, J. VAZQUEZ* , R. MOSE* 
* Maître de Conférences à l’ENGEES 
** En thèse au SHU (johnwertel@hotmail.com) 
RESUME 
L’objectif de ce travail est l’interpolation d’un champ de vitesse à partir de données 
discrètes fournies par un vélocimètre Doppler. Pour cela une modélisation précise de 
l’écoulement est indispensable. Ainsi ce travail porte sur la description d’un modèle 
RSM en vue de la modélisation 3D des écoulements turbulents en canal 
rectangulaire à surface libre. Nous avons implémenté ce modèle en utilisant la 
méthode des éléments finis de Galerkine pour écrire un code sous Matlab. Les 
courants secondaires du champ moyen ont été mis ainsi en évidence à travers une 
section tranversale ainsi que leur sensibilité aux parois. Ils ne peuvent être modélisés 
qu’en tenant compte de l’anisotropie de la turbulence. D’où l’intérêt d’un modèle 
complet comme le RSM. 
ABSTRACT 
 
The aim of this work is the interpolation of a mean flow stream from discreet data 
supplied by a velocimeter Doppler. That is why a precise flow modelling in needed. 
This work deals so with the description of a Reynolds stress model for the numerical 
simulation of uniform 3D turbulent open-channel flows. The finite element method 
(Galerkine‘s method) is used for the numerical solution (with Matlab) of the flow 
equations and transport equations of the Reynolds stress components. It is found that 
both production terms by anisotropy of Reynolds normal stress and by Reynolds 
shear stress contribute to the generation of secondary currents. The results 
presented here are the calculation for full section flow in a rectangular channel, it is a 
step in the overall project. 
KEY WORDS  
Open channel flow, reynold stress model, secondary currents, turbulent. 
SESSION 7.2 
1466 NOVATECH 2007  
INTRODUCTION 
 
A great number of natural receiving waters are polluted because of urban net 
discharges. The protection of these natural sites needs a real time management of 
urban sanitation net. This is the general context of the RITEAU project, whose aim is 
the conception of a sensor which is able to measure both flow and suspended 
material concentration at the same time. SHU laboratory is taking part in this project, 
has been asked for 3D hydraulic modelling and experiments in laboratory. 
 
This main aim of this study is the modelling of 3D turbulent open channel flows. The 
anisotropy of the turbulence, brought about the wall and the free surface, is known to 
generate secondary currents in open channel flows. k-ε model is isotropic closure so 
it cannot be used here. The RSM model has been implemented, directly solving 
Reynolds stresses transport equations thanks to the finite element method (Galerkine 
method). This code has been specially designed for sewer net conduit flow modelling.   
  
1 TURBULENT FLOW MODELLING 
 
1.1 Bibliography 
 
Clarification of secondary currents is essential for the understanding of 3D turbulent 
flow structure. Secondary currents are defined as currents which occur in a plane 
normal to the local primary flow axis. These secondary currents have a great effect on 
the mean flow stream. There are two kinds, the first one, which is brought about by 
the channel’s curves, is not considered here. The second one, called ‘secondary 
currents of Prandtl’s second kind as pointed out by Rodi (1986) is due to the 
anisotropy of turbulence and consists of large opposed spin vortices.  
The secondary currents cause the ‘Dip phenomenon’. The maximum velocity is not at 
the free surface but below where z/H ≈ 80%. The k-ε model does not allow to get this 
phenomenon. Furthermore the walls geometry (solid boundaries) and the free surface 
influence the second currents. 
Figure 3 :  Illustration of Dip 
phenomenon 
 
Figure 2 :  Secondary currents and 
angle effects. (Nezu and Rodi 1985)Figure 1 : 
Secondary currents 
SESSION 7.2 
NOVATECH 2007  1467 
The issue at the core of this modelling is the turbulent closure model to solve the 
Reynolds averaged equations. A common model is the k-ε one which was originally 
developed by Jones and Launder (1972) for the modelling of homogeneous isotropic 
turbulence. This model does not correctly predict near-wall flow and the Dip 
phenomenon. 
 
An other range of models deals with the transport of all Reynolds-stress components 
in addition to Reynolds averaged equations like the RSM developed by Hanjalić and 
Launder (1972). In that way, the anistropy of turbulence is taken into account. 
Moreover the RSM requires the ε equation. Here is what Christophe BAILLY wrote 
about it (2003)[1]: “These second order models have never had the development of 
the models k-ε, in particular because of the number of constants to be determined 
[…]. There is no standard model.”. That is why a RSM based on former works is 
developed here. 
 
The steady flow is studied in an 
X infinite rectangular channel. So 
all x spatial derivatives are 
ignored except for the mean 
pressure; there is charge loss. In 
that way, the mean velocity 
stream is calculated in a cross 
section thanks to a 2D mesh. 
  
It is the appearance of the Reynolds stress that makes the problem so difficult. To 
calculate it, it is necessary to close Reynolds averaged equations. Reynolds stress 
transport equations are obtained by subtracting Reynolds averaged equations to 
Navier Stockes equations and then multiplying by turbulent velocity and averaging. 
 
There are numerous  degrees of freedom due, in particular, to turbulent velocity triple 
correlations. The following part deals with models for the different term of RSM 
equations, the aim is to reduce the number of unknowns. 
Figure 4 : the channel 
² ²
² ²
² ²
² ²
² ²
² ²
U U P U U uv uwV W
y z x y z y z
V V V V vv vwV W
y z y z y z
W W W W vw wwV W
y z y z y z
ρ ρ μ μ ρ ρ
ρ ρ μ μ ρ ρ
ρ ρ μ μ ρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ = − + + − −
∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
+ = + − −
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
+ = + − −
∂ ∂ ∂ ∂ ∂ ∂
0V W
y z
∂ ∂
+ =
∂ ∂
DP/dx = cst : 
charge loss 
Reynolds averaged 
equations:
SESSION 7.2 
1468 NOVATECH 2007  
1.1.1 Modelling Turbulent Diffusive Transport 
 
Here the diffusion of Reynolds stresses is modelled using Daly and Harlow triple 
correlations model.  
( )j k i ji ki j k i l j l k l
l l l
u u u uu uk
u u u Cs u u u u u u
x x xε
∂ ∂∂
= − + +
∂ ∂ ∂
 This model looks like 
Boussinesq approximation that assumes that the principal axes of the Reynolds 
stress tensor are coincident with those of the mean strain rate, applied for triple 
turbulent velocity correlation tensor. 
²
Boussinesq:     
i j
i j t t
j i
U U k
u u with C
x x
μ
ν ν
ε
∂ ∂
− = + =
∂ ∂
⎛ ⎞
⎜ ⎟
⎝ ⎠
,  
²
i l t
k k
u u ν
ε ε
≈ ≡ shows the 
similarities between the two models. 
 
1.1.2 Modelling the Dissipation Rate 
 
( ) 2
2 2
3
i jjk j ji i
i ij ij
k k k k k k k
u u u uu u
u with
x x x x x x x
τ
μ μ μ ε δ ρε
∂∂ ∂ ∂∂ ∂∂
= − = =
∂ ∂ ∂ ∂ ∂ ∂ ∂
⎛ ⎞
⎜ ⎟⎜ ⎟
⎝ ⎠
 
The scalar dissipation rate, ε, is computed with a model transport equation similar to 
the one that is used in the standard k-ε model. 
 
 
 
1.1.3 Modelling the Pressure-Strain Term 
 
The pressure-strain term plays a pivotal role in modelling the turbulent flow. The 
following decomposition is used: 
( )1 2 ,' ' 23i j i j ij ij ij ij wallj i
P Pu u C u u k C P C
x x k
ερ δ φ∂ ∂ ⎛ ⎞− − = − − − − +⎜ ⎟∂ ∂ ⎝ ⎠
 
Φij, wall redistributes turbulent kinetic energy among the Reynolds stresses. Here is the 
model proposed par Speziale et al [6]. 
 
0 1 2 3,
54                           
2  : anisotropy tensor  
3
1( )3
2( ) ( )3
ij
ij ij
mn nmij ij ij ijij wall ik jk k
ijik jk jk ik kl kl ik jk jk ik
R
b
k
b b b b b kS P b
k b S b S b S k b W b W
δ
φ α ε α ε δ α α
α δ α
= −
= + − + +
+ + − + +
 
1 22
t kk
k
k k t k
PU C C
t x x x k ε ε
με ε ε ερ μ ρε
σ
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ⎛ ⎞+ = + + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ⎝ ⎠⎝ ⎠⎝ ⎠
SESSION 7.2 
NOVATECH 2007  1469 
1/ 2
0 1 2 3
1 1 : rate of strain tensor         : rotation tensor     
2 2
: rate of production of turbulent energy 
3.4,  4.2,  0.8 1.3( ) , 1.8,
j ji i
ij ij
j i j i
k
mn nm
U UU US W
x x x x
P
b bα α α α
⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂
= + = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
= − = = − = − 4 5 1.25 and 0.4 α α= =
 
Speziale’s model transports anisotropy tensor and so is necessary to generate 
second currents vortices. Because it is very difficult, as the Speziale model has not 
yet been totally implemented in order to see if the last term Φij, wall, the more 
complicate, is indispensable. 
 
Reynolds stresses transport equation follows: 
 
 
1.2 Flow conservation 
 
The overall continuity equation is added to impose flow (Q), to calculate charge loss 
(dP/dx) that has been considered as a constant in the whole channel. Kang and Choi 
2006 [4] impose charge loss writing dP/dx = g.I with g gravitational acceleration and I 
the channel slope. Maybe Q is imposed that way considering that dP/dx ≡ Q².  
The overall equation is not integrated at each node but on the whole section every 
iteration. Here is the equation: UdS Q=∫ . 
 
2 FINITE ELEMENT MODELING: CALCULUS CODE CREATION 
 
2.1 Presentation 
 
In the present study, the finite element (Galerkine ‘s method) is used to solve 
Reynolds averaged and Reynolds stresses transport equations. Because the system 
is non-linear, the numeric scheme is iterative by substitution. The code has been 
optimised by reducing the calculus number for each iteration (using preintegration). 
Moreover Matlab is a powerful tool for matrix manipulations, that is why logical 
operators (if, for, while…) have been replaced by matrix operations as much as 
possible.  
 
This work has not been completed. One step has been to work with full section flow to 
get rid of free surface influence and to verify that the result given is symmetric. This is 
what is presented here. 
( )
2
2 1
(1 ) ( )
                                    ( 1) ( ) 2 ( )
i j k j k i ji k
i l j l k l
k l l l
ji
j k i k i j i j
k k k
u u U u u u uu uk
C Cs u u u u u u
x x x x
UU
C u u u u C u u u u
x x k
ρ
ε
ε
ρ ρ μ
∂ ∂ ∂∂
+ = + +
∂ ∂ ∂ ∂
∂∂
+ − + + + Δ
∂ ∂
∑
SESSION 7.2 
1470 NOVATECH 2007  
2.2 Modelling results 
 
2.2.1 Laminar flow 
 
To test the part  of the code which calculates mean velocities, the laminar flow has 
been modelled; the 6 Reynold stresses are equivalent to zero.  The water depth is 
10m, channel width is 1m, slope is 0, stream-wise mean velocity is 1 m.s-1, and Q = 
10 m3.s-1. The channel is narrow to get rid of ceiling and bottom roughless wall effect. 
V and W are also 
modelled but the 
result is close to 0 
(<<10-300).  
Here results are 
compared with 
Poiseuille mean 
velocity profile. 
The correlation 
between the two 
curves is very 
good. 
 
 
 
 
2.2.2 Turbulent flow 
 
Here is turbulent flow modelling, the water depth is 4m, channel width is 4m, slope 
is 0, stream wise mean velocity is 1 m.s-1, and Q = 16 m3.s-1. 
 
Figure 6 : Turbulent flow modelling, mean 
Figure 1 : Laminar flow modelling stream
SESSION 7.2 
NOVATECH 2007  1471 
 
The mean stream obtained is clearly not symmetric but close to it. U looks a lot like  
the laminar solution. Without free surface there is no Dip phenomenon, so calculated 
profile and average value appear to be quite good. 
 According to Nezu and Nakagawa [5], magnitude of V and W is only about 2 % of the 
maximum stream wise velocity. Results are thus coherent. Secondary currents form 
is not good, calculations do not end up converging. Moreover Speziale model’s Φij, wall 
has no physical meaning but is a mathematic function that reproduce secondary 
currents. This term seems to be necessary. 
 
On this other modelling, secondary 
currents are better, opposed spin vortices 
can be seen but the angle effect is not 
visible. On the following figure, the angle 
effect appears to be good, that is to say 
vortices close to the bottom corner are 
divided along the angle bisector. 
 
About double turbulent velocity 
correlations, shown below, overall values 
are quite good according to the Kang and 
Choi article [4]. Nevertheless, vw > 0 it 
should be the other way around. As a 
matter of fact, Reynolds tensor terms not 
from diagonal are negative like the results 
for uv and uw. Here the symmetry is quite 
good. 
Figures 7-8 : Opposed spin vortices illustration 
 
Figure 9 : Turbulent flow modelling, double turbulent velocity correlations 
SESSION 7.2 
1472 NOVATECH 2007  
In the aim of understanding the difficulties there are for calculation, here an analyse 
on the studied dynamic system. 
 
There are three kinds of dynamic system. The first one takes place on an attractor 
such as the Poiseuille flow. These systems are very stable and always converge to 
the same final stage. On the contrary, chaotic systems are very unstable and almost 
impossible to foresee as they are sensitive to all governing factors. A tiny change of 
the initial conditions changes the behaviour of the system completely.  
The last kind is composed of a chaos order transition system which gives rise to 
strange attractors. It is greatly detailed and complex in order to manage a large 
energy contribution. For instance, the flow studied has a high Re which implies high 
kinetic energy. At the beginning, this energy is consumed in a ‘common’ way thanks 
to transport and viscosity dissipation but it is not enough and the energy balance is 
not obtained. As the fluid’s behaviour takes place on a complex structure, secondary 
currents are generated (see 1.2). These secondary currents consist of large opposed 
spin vortices. In this way, the system consumes more energy and is energy balanced. 
Actually the studied flow is complex structure hydrodynamic system, close to chaotic, 
which implies difficulties to converge and high sensibily to boundary conditions. 
 
CONCLUSION 
 
The aim of the present study was precise modelling of 3D turbulent flow for the 
conception of a sensor. The RSM model is the only one taking into account 
anisotropy of turbulence that generates secondary currents and the Dip phenomenon. 
On one hand, a very precise RSM has been developed, on the other hand the 
problem has been simplified so that the calculations are lead into 2D mesh. The RSM 
equation has been rewritten with a finite element formulation, then integrated and 
implemented. The computed mean flow and turbulent velocity correlations were 
compared with Kang and Choi’s results. Model predictions are quite good but not 
totally successfully because of convergence problems. Inner secondary currents are 
reproduced but significant improvements will be made when the Speziale model is 
properly implemented. 
 
 
REFERENCES 
[1] C. BAILLY & G. COMTE-BELLOT (2003), Turbulence, CNRS editions 
[2]  P. CHASSAING (1994), Mécanique des fluides: Eléments d’un premier parcours, Cepadues 
editions  
[3] G. DHATT, G. TOUZOT & E. LEFRANCOIS (1983) Méthode des éléments finis HERMES 
édition. 
[4] H. KANG & S. CHOI (2006) Reynolds stress modelling of rectangular open-channel flow 
parue dans la revue internationale INTERNATIONAL FOR NUMERICAL METHODS IN 
FLUIDS 
[5] I. NEZU & H.NAKAGAWA (1993), Turbulence in Open-Channel flows, IAHR Monograph 
[6] CG. SPEZIALE, S. Sarkar, T. Gatski Modeling the pressure strain correlation of turbulence: an 
invariant dynamical systems approach. Journal of Fluid Mechanics 1991; 227:245–272. 


Modélisation des profils tridimensionnels de vitesse et de turbulence pour améliorer l'instrumentation des réseaux d'assainissement

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Modélisation des profils tridimensionnels de vitesse et de turbulence pour améliorer l'instrumentation des réseaux d'assainissement

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