Exemple X : 3D Moving sharp immersed Boundary

In this example, we study the flow around a moving sphere of unit diameter at constant speed for a Reynold number of Re=50. This is done using the Sharp interface immersed boundary method. This example can be solve using the gls_sharp_navier_stokes_3d solver.

NOTE : This simulation only follows the first few moments of the displacement of the particle (0.25 seconde). The flow won't have the time to establish itself in this case. If let the flow establish itself, a longer channel must be given as the domain and a much longer simulation time must be used.

The following schematic describes the simulation.

INSERT IMAGE

We consider slip boundary condition on all the boundaries parallels to the trajectory of the particle and open boundary at each of the boundary perpendicular to the trajectory of the particle.

# --------------------------------------------------
# Boundary Conditions
#---------------------------------------------------
subsection boundary conditions
set number                  = 4

    subsection bc 1
	set id = 2
        set type              = slip
    end
    subsection bc 2
	set id = 3
        set type              = slip
    end
    subsection bc 3
	set id = 4
        set type              = slip
    end
    subsection bc 4
	set id = 5
        set type              = slip
    end

end

The definition of the mesh is done directly through deal.ii tools for mesh generation:

#---------------------------------------------------
# Mesh
#---------------------------------------------------
subsection mesh
    set type                 = dealii
    set grid type            = subdivided_hyper_rectangle
    set grid arguments       = 2,1,1: 0,0,0 : 32,16,16 : true
    set initial refinement   = 5
end

We fix the velocity of the particle to one and varies the Reynolds number by varying the kinematic viscosity of the flow.

#---------------------------------------------------
# Physical Properties
#---------------------------------------------------
subsection physical properties
    set kinematic viscosity            = 0.02
end

The motion of the particle require the use of a transient solver. At the moment only BDF schemes are compatible with moving sharp interface. In this case we use BDF1. In this example we only simulate the first 0.25 seconde. To do so we use a time step of 0.025 seconde.

# --------------------------------------------------
# Simulation and IO Control
#---------------------------------------------------
subsection simulation control
  set method                  = bdf1
  set time step               = 0.025 # Time step
  set time end                = 0.25     # End time of simulation
  set output name             = moving_sphere # Prefix for VTU outputs
  set output frequency        = 1       # Frequency of simulation output
  set subdivision             = 1       # Mesh subdivision when outpu
end

At this point we have defined almost all the elements necessary to this example except the moving sphere. This is done in the IB particles section (See Exemple 10) for more detail on the general parameters of this section). The particle is defined in the same way in 3d as in 2d. To put the particle in motion one only need to specify the particle velocity to be something other than zero. We also add the fact that the boolean variable integrate motion is kept false as this is only need to be true when particle trajectories are subject to the hydrodynamic forces applied to the particle.

# --------------------------------------------------
# IB particles
#---------------------------------------------------
subsection particles
	set number of particles = 1
	set stencil order = 5
	set refine mesh inside radius factor = 0.9
	set refine mesh outside radius factor = 1.1
	set initial refinement = 5
	set integrate motion= false
	set assemble Navier-Stokes inside particles = false

    subsection particle info 0 
            set x = 8
	    set y = 8
	    set z = 8
	    set u = 1
	    set v = 0
	    set w = 0
	    set omega x = 0
	    set omega y = 0
	    set omega z = 0
	    set pressure x =0.00001
	    set pressure y =0.00001
	    set pressure z =0.00001
	    set radius = 0.5
    end
end

We also allow a adaptive mesh refinement to follow the trajectory of the particle. In this section we also fix the minimum mesh size. In this case the minimum mesh size is 1/32 of the diameter of the particle. This give us a particle CFL 0.8 .

# --------------------------------------------------
# Mesh Adaptation Control
#---------------------------------------------------
subsection mesh adaptation
   # Fraction of coarsened elements
  set fraction coarsening  = 0.2

  # Fraction of refined elements
  set fraction refinement  = 0.05

  # How the fraction of refinement/coarsening are interepretedChoices are
  # .
  set fraction type        = number

  # Frequency of the mesh refinement
  set frequency            = 1

  # Maximum number of elements
  set max number elements  = 500000

  # Maximum refinement level
  set max refinement level = 9
 # minimum refinement level
  set min refinement level = 4

  # Type of mesh adaptationChoices are .
  set type                 = kelly

  # Variable for kelly estimationChoices are .
  set variable             = velocity
end

At t=0.25 seconde we get the following velocity field and pressure field.

INSERT IMAGEs

This gives use the following results for the force on the particle.

+------------------------------------------+
|  Force  summary particle 0               |
+------------------------------------------+
particle ID  time     f_x        f_y       f_z    
0           0.0250 -12.763956 -0.000010 -0.001576 
0           0.0500 -2.177713  -0.000027 0.000181  
0           0.0750 -1.491909  -0.000004 0.000017  
0           0.1000 -1.303367  0.000020  -0.000162 
0           0.1250 -1.207774  0.000017  -0.000114 
0           0.1500 -1.088343  0.000015  -0.000106 
0           0.1750 -1.044566  0.000014  -0.000118 
0           0.2000 -0.990040  0.000009  0.000034  
0           0.2250 -0.947240  0.000009  -0.000111 
0           0.2500 -0.932757  0.000008  -0.000078 

We note that the sharp interface immersed boundary is subject to pressure oscillations when it's moving. In this particular case, the pressure oscillation due to the particle moving relative to the mesh will cause an oscillation of the force applied to the particle on the order of 2%-3% of the magnitude of the force.