## Low Mach number flows: Broken dam problem

Objective

1. Explore various numerical strategies to achieve a fast and also accurate computation for low Mach multiphase flows
2. Well-balanced treatment for gravitational source terms

Problem description

• We consider a well-known broken dam problem of Martin and Moyce . The initial configuration is shown as below with a rectangular column of water of size a=0.06m width and h=0.12m high. The computational domain is a rectangular region of 0.5m width and 0.15m high. All the boundaries are treated as solid walls numerically. Under the gravity where we have g= 9.8 m/s^2, the water column collapses. Note that there exist many mathematical models to describe the physical feature of this problem; see a sample of them in below. No matter what mathematical model and numerical method are used in practice, your numerical results should be in comparison with the experimental data on the time history of

for validation, see the sample results shown below on how this is done. Since the Mach number during the computation is low and is of the order of 0.1, it is important to show the CPU time that is used for a run.

Mathematical model

• A five reduced equation model for compressible two-phase flow proposed in 

• A preconditioned five reduced equation model for compressible two-phase flow proposed in 

• A fluid-mixture type algorithm for compressible two-phase flow proposed in 

• Model based on isentropic compressible Navier-Stokes equations

Sample computations

• Numerical results shown in  obtained using a inviscid compressible five equation model

• Numerical results shown in  obtained using a preconditioned method

• Numerical results obtained using a fluid-mixture type method proposed in Related references

1. J. C. Martin and W. J. Moyce, Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane , Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 244, No. 882 (Mar. 4, 1952), pp. 312-324

2. J. C. Martin and W. J. Moyce, Part V. An Experimental Study of the Collapse of Fluid Columns on a Rigid Horizontal Plane, in a Medium of Lower, but Comparable, Density , Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 244, No. 882 (Mar. 4, 1952), pp. 325-334

3. M. A. Cruchaga, D. J. Celentano, and T. E. Tezduyar, Collapse of a liquid column: numerical simulation and experimental validation , Comput. Mech. (2007) 39: 453ĪV476

4. A. Murrone and H. Guillard, A five reduced equation model for compressible two-phase flow problems , J. Comput. Phys. (2004) 202: 664-698

5. A. Murrone and H. Guillard, Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model , Computers and Fluids (2008) 37: 1209-1224

6. K.-M. Shyue, An efficient shock-capturing algorithm for compressible multicomponent problems , J. Comput. Phys. (1998) 142: 208-242