Air Bubble Compression

This example simulates the compression of an air bubble by surrounding liquid. The problem is inspired by the test case of Caltagirone et al. [1]

Features

• Solver: lethe-fluid (with Q1-Q1)

• Volume of fluid (VOF)

• Isothermal compressible fluid

• Unsteady problem handled by an adaptive BDF2 time-stepping scheme

• Usage of a python script for post-processing data

Files Used in This Example

Both files mentioned below are located in the example’s folder (examples/multiphysics/air-bubble-compression).

• Parameter file: air-bubble-compression.prm

• Postprocessing python script: air-bubble-compression-postprocessing.py

Description of the Case

A circular air bubble of diameter \(D=0.06\) lies at the center of a square-shaped domain with sides of length \(L=0.1\). On all four sides of the domain, water penetrates with a constant velocity norm of \(||\mathbf{v}||=0.0025\) causing the compression of the air bubble. The initial configuration of this example is illustrated below.

Note

In this example, gravity and surface tension forces are not considered.

Parameter File

Simulation Control

Time integration is handled by a 2nd-order backward differentiation scheme (bdf2) with a variable time step. The initial time step is set to \(0.1 \, \text{s}\) and the simulation lasts \(2.5 \, \text{s}\).

subsection simulation control
  set method           = bdf2
  set time end         = 2.5
  set time step        = 0.1
  set adapt            = true
  set max cfl          = 1
  set output name      = air-bubble-compression
  set output frequency = 5
  set output path      = ./output/
end

Multiphysics

The multiphysics subsection is used to enable the VOF solver.

subsection multiphysics
  set VOF  = true
end

VOF

In the VOF subsection, the compressible, the interface sharpening, and the phase filtration features are enabled. The enabled compressible parameter allows interface compression by adding the term \(\phi (\nabla \cdot \mathbf{u})\) to the VOF equation. The interface sharpening method and its parameters are explained in the Dam-Break example. The phase filtration filters the phase field used for the calculation of physical properties by stiffening the value of the phase fraction. We refer the reader to The Volume of Fluid (VOF) Method theory guide for further explanation on the phase filtration.

subsection VOF
  set compressible = true
  subsection interface sharpening
    set enable              = true
    set threshold           = 0.5
    set interface sharpness = 2.2
    set frequency           = 8
  end
  subsection phase filtration
    set type      = tanh
    set beta      = 10
  end
end

Initial Conditions

In the initial conditions subsection, we define the initial air bubble with a radius of \(D/2=0.03\) surrounded by water. An initial velocity field is used to avoid discontinuities in the solution.

subsection initial conditions
  set type = nodal
  subsection uvwp
    set Function expression = 0.0025*-sin(2*pi*x/0.2); 0.0025*-sin(2*pi*y/0.2);0
  end
  subsection VOF
    set Function expression = if (x^2 + y^2 < 0.03^2, 0, 1)
  end
end

Boundary Conditions

On all four sides of the domain, water which is associated with the phase fraction \(\phi=1\) is injected. This is done in the simulation by setting the velocities of the fluid in the boundary conditions subsection and by selecting the correct fluid in the boundary conditions VOF subsection with a dirichlet boundary condition on the phase fraction as shown below.

Boundary Conditions - Fluid Dynamics

subsection boundary conditions
  set number = 4
  subsection bc 0
    set id   = 0
    set type = function
    subsection u
      set Function expression = 0.0025
    end
  end
  subsection bc 1
    set id   = 1
    set type = function
    subsection u
      set Function expression = -0.0025
    end
  end
  subsection bc 2
    set id   = 2
    set type = function
    subsection v
      set Function expression = 0.0025
    end
  end
  subsection bc 3
    set id   = 3
    set type = function
    subsection v
      set Function expression = -0.0025
    end
  end
end

Boundary Conditions - VOF

subsection boundary conditions VOF
  set number = 4
  subsection bc 0
    set id   = 0
    set type = dirichlet
    subsection dirichlet
      set Function expression = 1
    end
  end
  subsection bc 1
    set id   = 1
    set type = dirichlet
    subsection dirichlet
      set Function expression = 1
    end
  end
  subsection bc 2
    set id   = 2
    set type = dirichlet
    subsection dirichlet
      set Function expression = 1
    end
  end
  subsection bc 3
    set id   = 3
    set type = dirichlet
    subsection dirichlet
      set Function expression = 1
    end
  end
end

Physical Properties

In the physical properties subsection, we define the properties of the fluids. For air, represented by fluid 0, the isothermal_ideal_gas density model is used to account for the fluid’s compressibility. We refer the reader to the Physical Properties - Density Models documentation for further explanation on the isothermal compressible density model. The properties of air and water at \(25 \, \text{°C}\) are used in this example.

subsection physical properties
  set number of fluids = 2
  subsection fluid 0
    set density model       = isothermal_ideal_gas
    subsection isothermal_ideal_gas
      set density_ref = 1.18
      set R           = 287.05
      set T           = 298.15
    end
    set kinematic viscosity = 0.0000156
  end
  subsection fluid 1
    set density             = 1000
    set kinematic viscosity = 0.000001
  end
end

Mesh

In the mesh subsection, we define a hyper cube with appropriate dimensions. The mesh is initially refined \(7\) times to ensure adequate definition of the interface.

subsection mesh
  set type               = dealii
  set grid type          = hyper_cube
  set grid arguments     = -0.05 : 0.05 : true
  set initial refinement = 7
end

Mesh Adaptation

In the mesh adaptation subsection, adaptive mesh refinement is defined for the phase. min refinement level and max refinement level are \(7\) and \(9\), respectively. Since the size of the bubble changes, we choose a rather large fraction refinement (\(0.99\)) and moderate fraction coarsening (\(0.01\)).

subsection mesh adaptation
  set type                     = kelly
  set variable                 = phase
  set fraction type            = fraction
  set max refinement level     = 9
  set min refinement level     = 7
  set frequency                = 1
  set fraction refinement      = 0.99
  set fraction coarsening      = 0.01
  set initial refinement steps = 6
end

Running the Simulation

We can call lethe-fluid by invoking the following command:

mpirun -np 8 lethe-fluid air-bubble-compression.prm

to run the simulation using eight CPU cores. Feel free to use more.

Warning

Make sure to compile lethe in Release mode and run in parallel using mpirun. This simulation takes \(\sim\) 1.5 minute on 8 processes.

Results

We compare the density (\(\rho_{\text{air}}\)) and pressure (\(p_{\text{air}}\)) in the air bubble with their analytical values. The density is given by:

\[\rho_{\text{air}}=\frac{\rho_{\text{air,}\;\! \text{initial}}}{1-\frac{4qt}{\pi D^2}}\]

where \(\rho_{\text{air,}\;\! \text{initial}}=1.18\) is the initial density of air, \(q = 4 \cdot ||\mathbf{v}|| \cdot L = 0.001\) is the volumetric flow rate, and \(t\) is the time.

From the ideal gas law, we obtain the following expression for the pressure:

\[p_{\text{air}} = (\rho_{\text{air}}-\rho_{\text{air,}\;\! \text{initial}}) \cdot R \cdot T\]

where \(R=287.05\) is the specific gas constant of air and \(T=298.15\) is the temperature of the fluid in Kelvin.

The results can be post-processed by invoking the following command from the folder of the example:

python3 air-bubble-compression-postprocessing.py . air-bubble-compression.prm

Important

You need to ensure that lethe_pyvista_tools is working on your machine. Click here for details.

The following figures present the comparison between the analytical results and the simulation results for the density and pressure evolutions evaluated at the center of the bubble. A pretty good agreement between the simulation and analytical results is observed.

References

[1]

J.-P. Caltagirone, S. Vincent, and C. Caruyer, “A multiphase compressible model for the simulation of multiphase flows,” Comput. Fluids, vol. 50, no. 1, pp. 24–34, Nov. 2011, doi: 10.1016/j.compfluid.2011.06.011.