Rising Bubble

This example shows how to model two immiscible fluids, tracking the fluid-fluid interface. An oil bubble rises through water and merges with oil already residing at the top of the container. Initially three different regions exist: the initially still oil bubble, the oil at the top of the container, and the water surrounding the bubble (see Figure 1). The container is cylindrical with a diameter of 1·10−2 m and a height of 1.5·10−2 m. The oil phase has a viscosity of 0.0208 Pa·s and has a density of 879 kg/m3. For water the viscosity is 1.01·10−3 Pa·s and the density is 1000 kg/m3. Buoyancy effects cause the oil bubble to rise through the water phase. As the bubble reaches the liquid-liquid interface, it merges with the oil phase.

Figure 1: Initial bubble position. The geometry is axisymmetric.

As outlined above, the topology of the fluid interface changes with time. You start with three separate fluid regions and end up with two. The level set method as well as the phase field method are both well suited for modeling moving boundaries where topology changes occur. Both methods are available in the CFD Module as predefined physics interfaces. This example shows you how to use the Laminar Two-Phase Flow, Level Set interface.

Model Definition

Representation and Convection of the Fluid Interface

The Level Set interface finds the fluid interface by tracing the isolines of the level set function,

. The level set or isocontour

determines the position of the interface. The equation governing the transport and reinitialization of

where u (SI unit: m/s) is the fluid velocity, and γ (SI unit: m/s) and ε (SI unit: m) are reinitialization parameters. The ε parameter determines the thickness of the layer around the interface where

goes from zero to one. When stabilization is used for the level set equation, you can typically use an interface thickness of ε = hc/2, where hc is the characteristic mesh size in the region passed by the interface. The γ parameter determines the amount of reinitialization. A suitable value for γ is the maximum velocity magnitude occurring in the model.

Because the level set function is a smooth step function, it is also used to determine the density and dynamic viscosity globally by

and

Here ρw, μw, ρo, and μo denote the constant density and viscosity of water and oil, respectively.

Mass and Momentum Transport

In the Laminar Two-Phase Flow, Level Set interface, the transport of mass and momentum is governed by the incompressible Navier-Stokes equations, including surface tension:

In the above equations, ρ (SI unit: kg/m3) denotes the density, u is the velocity (SI unit: m/s), t equals time (SI unit: s), p is the pressure (SI unit: Pa), and μ denotes the viscosity (SI unit: Pa·s). The momentum equations contain gravity, ρg, and surface tension force components, denoted by Fst.

Surface Tension

The surface tension force is defined by

where σ is the surface tension coefficient, I is the identity matrix, n is the interface unit normal, and δ is a Dirac delta function, nonzero only at the fluid interface. The interface normal is calculated from

The level set parameter

is also used to approximate the delta function by a smooth function defined by

Initial Condition

At t = 0, the velocity is zero. Figure 2 shows the initial level set function. This is automatically computed using a Phase Initialization study step by solving for the geometrical distance to the initial interface, Dwi. The initialized level set function is then defined from the analytical steady state solution for a straight fluid-fluid interface:

in the domains initially filled with Fluid 1 and Fluid 2 respectively,

Figure 2: A surface and contour plot of the initialized level set function.

Boundary Conditions

Use no slip conditions, u = 0 at the top and bottom and a wetted wall condition on the right boundary. The left boundary corresponds to the symmetry axis.

Results and Discussion

Figure 3 and Figure 4 contain snapshots of the fluid interface. The snapshots show how the bubble travels up through the water and merges with the oil above. As the bubble rises, its shape remains spherical due to the surface tension and the high viscosity of the oil. As the droplet hits the water surface, it merges with the oil above and creates waves on the surface.