η2 8
a Kg/m3
a
x x
γ = 4.4 / π = 6.10 ρ = 1000
Lz Nz Water Column
a= 0.06m η2=2
BROKEN DAM PROBLEM J.C. MARTIN AND W.J. MOYCE (1952)
z/η2 a
x/a L =0.5m
= 0.15m N = 200
= 60
Fig. 15. Initial configuration for the broken dam problem.
Fig. 16. Isovalues of the volume fraction for the broken dam problem.
A. Murrone, H. Guillard / Journal of Computational Physics 202 (2005) 664–698 695
totally identical with those obtained in [12], a good agreement with these results is reached. Again, we emphasize that in the limit where the space step goes to zero, for this interface problems between compress- ible fluids, the two models have to give exactly the same results. For this computation, a mesh of 400· 400 thus seems sufficient for this purpose.
5.3.3. The broken dam problem
Finally, we present here a computation of the well known broken dam problem of Martin and Moyce [11]. This test consists of the simple configuration represented in Fig. 15. Initially a water column with a = 0.06 m wide and g2a = 0.12 m high is a rest. All the boundaries are solid walls. Under the effect of the gravity g = 9.81 m s2, the column collapses. The computation is made with the acoustic solver. The mesh contains 200· 60 nodes and the CFL number is equal to 0.8. The experimental results of[11], for the front position x=a¼ F1ðg2; t ffiffiffiffiffiffiffiffiffiffi
p2g=a
Þ and the height of the column z=ðg2aÞ ¼ F2ðg2; t ffiffiffiffiffiffiffiffi pg=a
Þ are used for comparison (note that in the experimental results, the non-dimensional times are different for the front position and the water height: we have used the same non-dimensional units in the presentation of the numerical results to allow a direct comparison with the original publication of[11]). InFig. 17, are com- pared the numerical results and the experimental ones for the case g2= 2. Although, the Mach number is
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6
Dimensionless front position x/a
Dimensionless time t*sqrt(2g/a)
Experimental data Numerical Solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5 2 2.5 3 3.5
Dimensionless height z/(2a)
Dimensionless time t*sqrt(g/a)
Experimental data Numerical Solution
Fig. 17. Comparison between numerical solution and experimental results for the broken dam problem. Front position (top) and height of the column (bottom).
696 A. Murrone, H. Guillard / Journal of Computational Physics 202 (2005) 664–698
very small, we note a good agreement with the experimental results.Fig. 16shows the isovalues of the vol- ume fraction at the different dimensionless time t ffiffiffiffiffiffiffiffiffiffi
p2g=a
¼ 0; 1:19; 1:98; 2:97; 4:02; 5:09 corresponding to the physical times t = 0, 0.066, 0.109, 0.164, 0.222, 0.281 s.
6. Conclusion
We have derived a five equation reduced model from an asymptotic analysis in the limit of zero relax- ation time of a seven equation two velocity, two pressure model. Although, this model cannot be cast in conservative form, the mathematical structure of the model have been analyzed and shown to be very close to the structure of the Euler equations of fluid dynamics. This model presents an interesting alternative to the use of the seven equation model: it is cheaper, simpler to implement and is easily extensible to an arbi- trary number of materials. For instance, in three dimensions, for a number k of different material, the two velocity, two pressure model uses 6k 1 variables while the reduced model will use only 2k + 3 variables.
From a numerical point of view, we have proposed two different approximation schemes of this system.
The first one (VFRoe-ncv) relies on an approximate linearized Riemann solver. The second one, that is use- ful for the simulation of interface problems in low Mach number flows, uses the mathematical structure of the model and relies on the linearization of characteristic relations.
The numerical results show that the reduced five equation model is able of accurate computations of interface problems between compressible material as well as of some two-phase flow problems where pres- sure and velocity equilibrium between the phases is reached. The numerical methods have been shown to be efficient and robust for a large range of Mach number from almost zero to 1.9 and for density ratio as large as 1000. In the future, we plan to improve the efficiency and accuracy of the numerical method for low Mach number flows by designing implicit and preconditioned schemes for this model. Preliminary results in this direction are promising[13].
Acknowledgements
The work of Angelo Murrone has been supported by a grant of the ‘‘Conseil Regional’’ of region PACA and the CEA Cadarache. Angelo Murrone acknowledges the support of M. Grandotto (CEA Cadarache), P. Gubernatis (CEA Cadarache) and R. Saurel (IUSTI Marseille). He would also like to thank his family for their constant support. We also address a warm thank to M.H. Lallemand for her help with the relax- ation procedures in the numerical solution of the seven equation model.
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