Ninth International PHOENICS User Conference September 23 – 27 2002, Moscow, Russia

1. National Tunisian Engineering School (ENIT) LAMSIN Two-dimensionalfree surfacemodelling fora non-dimensionalDam-Breakproblem M. Ben Haj , Z. Hafsia , H. Chaker and K. Maalel Ninth International PHOENICS User Conference September 23 – 27 2002, Moscow, Russia

2. Initial state 0 Dam-site 0 Height (m) Length (m) Dam break profile at t = 20 s t t Height (m) Length (m) Problemposition

3. 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

4. High (H) h P = Center of the cell North (N) South (S) P s n z y l Low (L) Control volume The fluid flow equations the continuity equation ; the momentum equation In discrete and implicit formulation:

5. The free surface model } single-phase treatments gas cell ; liquid cell Boundary conditions

6. The Scalar Equation Method (SEM) Governing Equation: Van Leer discretisation of the scalar-convection terms: CFL condition:dt = min (dy/v, dz/w )

7. P = Center of the cell S P N s n z North face y

8. 7 z 6 5 MT - ML 4 ML 3 2 1 Definition of variables for HOL • The Height of Liquid Method (HOL)

9. 10 m U1 = 0 y = 0 300 m 300 m The problem position • 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.

10. Non-dimensional analytical solution of Dam-Break Problem { Where and h1 is the initial upstream flow depth in the reservoir. y* = 0 y* = -1 y* = 2

11. Non-dimensional Free Surface Profiles for SEM method

12. Non-dimensional Free Surface Profiles for HOL method

13. a) b) Non-dimensional Free Surface Profiles: a) For SEM method b) For HOL method

14. Non-dimensional Front Location for SEM method

15. Non-dimensional Front Location for HOL method

16. a) b) Non-dimensional Front Location: a) For SEM method b) For HOL method

17. Time Variation of Flow Depth at Dam Site for SEM method

18. Time Variation of Flow Depth at Dam Site for HOL method

19. a) b) Time Variation of Flow Depth at Dam Site: a) For SEM method b) For HOL method

20. Pressure History at Dam Site for SEM method

21. Pressure History at Dam Site for HOL method

22. a) b) Pressure History at Dam Site a) For SEM method b) For HOL method

23. Evolution of Pressure Distribution at Dam Site for SEM method

24. Evolution of Pressure Distribution at Dam Site for HOL method

25. Some Conclusions • 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. • 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. • 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. • 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.