ninth international phoenics user conference september 23 – 27 2002, moscow, russia
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Ninth International PHOENICS User Conference September 23 – 27 2002, Moscow, Russia
Two-dimensional free surface modelling for a non-dimensional Dam-Break
problem M. Ben Haj , Z. Hafsia , H. Chaker and K. Maalel
National Tunisian Engineering School (ENIT)
LAMSIN
0
0
Dam-site
Length (m)
Hei
ght (
m)
Initial state
He i
g ht (
m)
Length (m)
t
t
Dam break profile at t = 20 s
Problem position
The mathematical model will need to:
• Locate the unknown inter-fluid boundaries;• Satisfy the field equations governing conservation of mass, momentum;• Be consistent with the boundary conditions.
Free Surface Equation
: High of a point from the free surface to a reference plan
: High of a point to the same reference plan
0),( sztxzF
),( txzsz
The fluid flow equations
South (S) North (N)
High (H)
Low (L)
P
P = Center of the cell
z
y
Control volume
h
n
l
s
PTLHSN
TTLLHHSSNNP aaaaaa
Saaaaa
In discrete and implicit formulation:
S ) . ( μ ) u . (ρ ρt
the continuity equation ; 1 u the momentum equation
The free surface model
)( ρ ) (1ρ ρ LG
)( ) (1 LG } single-phase treatments
0 1 gas cell ; liquid cell
Boundary conditions
surface free at the aPP bottom at the n.u un
• The Scalar Equation Method (SEM)
0 ) u . (Φ. tΦ
Governing Equation:
Van Leer discretisation of the scalar-convection terms:
ni(U)
Faces
i P
iinin
PnP ΦV
ΔtAn . u - Φ Φ
1
11
0 for 2)/ ( . )/( nnPPn dtudydyd
0 for 2)/ ( . )/( nnNNn dtudydyd
CFL condition: dt = min (dy/v, dz/w )
S NP
P = Center of the cell
z
yNorth face
ns
0 n
z7
6
5
1
2
3
4
Definition of variables for HOL
MT - ML
ML
TL
T
V M
LMΦ
• The Height of Liquid Method (HOL)
y = 0
The problem position
300 m 300 m
10 m U1 = 0
• NY1 = 60 for SEM and NY1 = 300 for HOL (upstream);
• NY2 = 60 for SEM and NY2 = 300 for HOL (downstream);
• NZ1 = 20 for both SEM and HOL;
• The computations are performed for a time of 15 s and with a time step ∆t = 0,2 s for SEM and ∆t = 0,04 s for HOL.
1 * if 1 1
yhh
2 * 1 if *2 91 )( 2
1 yyh
h
2 * if 0 1
yhh
1ghy *
ty
Non-dimensional analytical solution of Dam-Break Problem
{Where and h1 is the initial upstream flow depth in the reservoir.
y* = -1 y* = 2y* = 0
1 1h
h
Non-dimensional Free Surface Profiles for SEM method
Non-dimensional Free Surface Profiles for HOL method
Non-dimensional Free Surface Profiles: a) For SEM method b) For HOL method
a) b)
Non-dimensional Front Location for SEM method
Non-dimensional Front Location for HOL method
Non-dimensional Front Location: a) For SEM method b) For HOL method
a) b)
Time Variation of Flow Depth at Dam Site for SEM method
Time Variation of Flow Depth at Dam Site for HOL method
Time Variation of Flow Depth at Dam Site: a) For SEM method b) For HOL method
a) b)
Pressure History at Dam Site for SEM method
Pressure History at Dam Site for HOL method
a) b)
Pressure History at Dam Site a) For SEM method b) For HOL method
Evolution of Pressure Distribution at Dam Site for SEM method
Evolution of Pressure Distribution at Dam Site for HOL method
1. The location of the tip in the cases of SEM and HOL is under predicted by the analytical model, as compared with the numerical result.
2. The two dimensional effects reduce the rate at which the tip advances on a dry bed for SEM and HOL which is smaller than a value of 2 as suggested by Ritter 1892. These results indicate a significant long-term effect of non-hydrostatic pressure distribution, in the case of dry-bed condition.
3. Ritter’s (1892) solution, which use the hydrostatic assumption, predict that the flow depth at the dam site attains a constant value of 4/9 instantaneously upon the dam break. However, with the SEM and HOL methods, the flow depth at the dam site takes some times to attain this constant value.
4. In both cases of SEM and HOL, the pressure is not equal but greater than the hydrostatic pressure at the beginning due to the streamline curvature. It eventually approaches the hydrostatic value as time progress.
Some Conclusions
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