Example X 1 : 3D particle sedimentation with sharp interface

In this example we study the sedimentation of a particle in a fluid. This case is based on the case E4 of the experiment realized by Cate et al (click here for a link to the article describing this case). This case model the sedimentation of a particle (D=1.5 cm) in a container of 10x10x16 cm. The fluid choice and the relative density of the particle and the fluid lead to approximated maximal Reynold number of RE=31.9. This case is simulated using the centimeter as the base length unit.

This example will only model the first few moments of the sedimentation (0.025 seconde). The motion of immersed boundary also require the use of a BDF scheme in this case we use BDF 1 and a time step of 0.0025 seconde. These elements are defined in the

# --------------------------------------------------
# Simulation and IO Control
#---------------------------------------------------
subsection simulation control
  set method                  = bdf1
  set time step               = 0.0025 # Time step
  set time end                = 0.025  # End time of simulation
  set output name             = sedimentation_ib # Prefix for VTU outputs
  set output frequency        = 1       # Frequency of simulation output
  set subdivision             = 1       # Mesh subdivision when output
end

We first defined the container using a subdivided hyper rectangle from the deal.ii mesh generation tool.

#---------------------------------------------------
# Mesh
#---------------------------------------------------
subsection mesh
    set type                 = dealii
    set grid type            = subdivided_hyper_rectangle
    set grid arguments       = 5,8,5: 0,0,0 : 10 , 16 ,10 : true
    set initial refinement   = 1
end

We applied no slip boundary condition on all the sides except at the top where we let the boundary open.

# --------------------------------------------------
# Boundary Conditions
#---------------------------------------------------
subsection boundary conditions
set number                  = 5
    subsection bc 0
	set id = 0
        set type              = noslip
    end
    subsection bc 1
	set id = 1
        set type              = noslip
    end

    subsection bc 2
	set id = 2
        set type              = noslip
    end
    subsection bc 4
	set id = 4
        set type              = noslip
    end
    subsection bc 5
	set id = 5
        set type              = noslip
    end

end

The fluid kinematics viscosity of the case E4 is in this case 0.604167. This is set as usual in the Physical Properties of the fluid.

#---------------------------------------------------
# Physical Properties
#---------------------------------------------------
subsection physical properties
    set kinematic viscosity            = 0.604166666666667

end

Finally, we can define the particle and its property in the section IB particles. The generic parameters are defined as usual and their description can be found in the Exemple 10. To the we add 2 new parameters. The fluid density and the gravity and its 3 components.

• Fluid density: The fluid density use in the calculation of the buoyancy force applied to the particle.
• gravity_x : The gravitational acceleration in the x direction. This is useful in order to defined the buoyancy force applied to the particle.
• gravity_y : Idem but for the y direction.
• gravity_z : Idem but for the z direction.

For the specific parameter applied to the particle individually, we have to define the density of the particle. This is useful in order to defined the buoyancy force applied to the particle. This parameter assumes that the particle is full and has a uniformly distributed density.

For this case we have the following parameter.