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Level Set method for fluid/fluid flows

Among the various numerical techniques to model fluid/fluid interfaces, namely Volume-Of-Fluid, (VOF) Level Set and Front Tracking, we have selected and implemented the Level Set method in PeliGRIFF for its asset in the calculation of the geometric features of the interface and the fact that it does not require any interface reconstruction algorithm. Conversely, the classical Level Set method, the one implemented in PeliGRIFF, does not conserve mass while the VOF method does. The outcome is then the following: there are both assets and drawbacks in these two methods.

The principle of the Level Set method implies to define a smooth distance function to the interface whose 0 value corresponds to the interface. This distance function obeys a classical purely convective unsteady equation (cf Figure 1).

Fig 1. Principle of the Level Set method

In our implementation, we employ standard ingredients to improve the accuracy of the computed solutions. These improvements comprise:

• Surface tension is treated by a Continuum Surface Force (CSF) method,
• WENO-type schemes for the spatial discretization of the convective term,
• High order Runge-Kutta time schemes,
• PDE-based redistanciation.

Recently, new conservative variants of the Level Set method have been suggested in the literature which solves the crucial mass loss issue related to the non-conservative classical formulation. This new version looks most promising. Otherwise, more advanced techniques that couple VOF and Level Set, usually named CLSVOF for Coupled Level Set Volume Of Fluid, thus enabling one to benefit from the good properties of the two methods, are also available in the literature.

So far, our Level Set method has essentially been used to improve the understanding of the dynamics of water-oil separation. In practice, the mass loss can be controlled by employing a fine enough mesh, as shown in Figure 2 below.

(a) Water-oil interface time evolution as a function of the grid size

(b) Separation time and mass loss as a function of the grid size

Fig 2. A 2D Level Set simulation of a water-oil separation process at Reynolds number 85 and Bond number 33

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