Tracer in Static Mixer#

In this example, we extend the 3D Static Mixer Using RBF Sharp Immersed Boundary example using a passive tracer to quantify the mixing profile.

We inject two phases through a static mixer to verify how geometry affects their mixing.

../../../_images/static_mixer_stl_casing_arrows.png

Features#

  • Solver: lethe-fluid-sharp

  • Transient problem

  • Tracer physics

  • Accelerated transient auxiliary physics resolution with average_velocity_profile initial condition.

Files Used in this Example#

All mentioned files are found in the example’s folder (examples/multiphysics/tracer-in-static-mixer).

  • Parameter file for solving the flow: flow_in_long_mixer.prm;

  • Parameter file for solving the tracer transport: flow_in_long_mixer_only_tracer.prm;

  • Composite geometry file: mixer_long.composite;

  • RBF geometry file: RBF_helix.output. The extension is .output because it was named from a bitpit perspective. RBF preparation was done in the previous 3D Static Mixer Using RBF Sharp Immersed Boundary example, based on this surface grid from Thingiverse [1] under CC BY 4.0;

  • Python post-processing script: postprocess_tracer.py.

Description of the Case#

In this example, we use the auxiliary tracer physics to quantify the mixing of two fluids in a static mixer. We assume the flow is steady. Hence, we use a two-step simulation, consisting of:

  1. Simulating the transient flow with a small time step until the flow has reached the steady state;

  2. Simulate the flow of the tracer with a larger time step while reusing the computed velocity field from the previous simulation.

This strategy reduces the computational cost of the simulation as the fluid flow solution is reused in the second step.

Parameter File#

Simulation 1: Flow resolution#

Simulation control#

subsection simulation control
  set method    = bdf1
  set time end  = 40e-4
  set time step = 1e-4

  set adapt = false

  set output path      = ./output/
  set output name      = output
  set output control   = iteration
  set output frequency = 5
end

We use a bdf1 time integration scheme with a short time step = 1e-4 \(\text{s}\). After time end = 40e-4 \(\text{s}\), we consider that the velocity field has reached a steady state. According to 3D Static Mixer Using RBF Sharp Immersed Boundary, the pressure drop varies by less than \(5\) % in the range [30, 40] \(\text{s}\).

Restart#

We use Lethe’s checkpoint/restart mechanism to feed the steady-state flow information to the second step of the simulation.

subsection restart
  set checkpoint = true
  set frequency  = 5
  set filename   = restart
  set restart    = false
end

We checkpoint the simulation at every 5 time steps, both for safety (each time step takes a long time to complete) and for reuse in the second simulation.

Multiphysics#

Both fluid dynamics and tracer are enabled for the first simulation to ensure initialization of the tracer field. However, the tracer injection only begins at the second step.

subsection multiphysics
  set fluid dynamics = true
  set tracer         = true
end

Physical Properties#

In this case we consider that we have a passive tracer in water. The units used for the presented case are \(\text{cm}\), \(\text{s}\), and \(\text{g}\).

subsection physical properties
  subsection fluid 0
    set kinematic viscosity      = 0.01
    set tracer diffusivity model = immersed solid tanh
    subsection immersed solid tanh
      set tracer diffusivity inside  = 1e-10
      set tracer diffusivity outside = 1e-5
      set thickness                  = 5e-1
    end
  end
end

  1. The tracer diffusivity model is immersed solid tanh. This model is used in lethe-fluid-sharp for tracer flow percolating immersed solids.

  2. The tracer diffusivity outside is 1e-5 \(\text{cm²/s}\), as this is a typical value for a passive tracer in a liquid.

  3. The tracer diffusivity inside is set to 1e-10 \(\text{cm²/s}\). The low value prevents diffusivity inside the solid while providing numerical stability (\(> 0\)).

  4. The thickness is 5e-1 \(\text{cm}\). At the scale of the problem, this provides a smooth transition without generating oscillations between liquid and solid phases. The thickness is of the order of magnitude of the smallest cell length to restrict the transition to one cell thickness.

Tracer Boundary Conditions#

subsection boundary conditions tracer
  set number         = 6
  set time dependent = true
  subsection bc 0
    set id   = 0
    set type = dirichlet
    subsection dirichlet
      set Function expression = if(y<0,t<11?0:(t<61?1:0),0)
    end
  end
  subsection bc 1
    set id   = 1
    set type = outlet
  end
  subsection bc 2
    set id   = 2
    set type = outlet
  end
  subsection bc 3
    set id   = 3
    set type = outlet
  end
  subsection bc 4
    set id   = 4
    set type = outlet
  end
  subsection bc 5
    set id   = 5
    set type = outlet
  end
end

  1. We use time dependent boundary conditions, defined at \(y<0\) and \(11 < t < 61\) to inject a pulse on the lower half of the inlet.

  2. All other boundary conditions are outlet. This condition is natural for the outlet of the problem. For lateral walls, this condition represents an impermeable wall since velocity is perpendicular.

Post-processing#

subsection post-processing
  set verbosity = verbose

  set calculate average velocities = true
  set initial time                 = 30e-4

  # Tracer post-processing
  set calculate tracer flow rate  = true
  set tracer flow rate name       = tracer_flow_rate
end

  1. calculate average velocities is enabled, beginning at initial time = 30e-4 \(\text{s}\). This means that the last \(25\%\) of the first simulation will be used to calculate the time-averaged fluid velocity profile.

  2. calculate tracer flow rate is enabled to provide data for mixing quantification.

Simulation 2: Tracer transport#

In the second simulation, we reuse the velocity profile from the first simulation and simply transport the passive tracer. This allows us to increase the time step and duration of the simulation while keeping realistic velocity and pressure solutions.

Simulation control#

subsection simulation control
  set method    = bdf1
  set time end  = 500
  set time step = 10
  ...
end

  1. We use a longer time step of 10 \(\text{s}\).

  2. We simulate until time end = 500 \(\text{s}\) to allow the tracer to flow through the entire length of the domain.

Restart#

We enable the restart from the previous simulation.

subsection restart
  ...
  set restart    = true
end

Multiphysics#

We disable fluid dynamics, as we use the time-averaged velocity profile from the previous simulation.

subsection multiphysics
  set fluid dynamics = false
  set tracer         = true
end

Initial conditions#

We use average_velocity_profile from the previous simulation as the initial (and persistent) condition for the fluid dynamics. Since this physics is disabled, the velocity and pressure profiles will remain fixed throughout the simulation.

subsection initial conditions
  set type = average_velocity_profile
end

Running the Simulation#

As previously mentionned, the case is run in two steps:

  1. Simulation to reach a pseudo steady-state of the flow field;

  2. Transient simulation to transport tracer through the domain.

The simulation can be launched on multiple cores using mpirun and the lethe-fluid-sharp executable. Using 6 CPU cores and assuming that the lethe-fluid-sharp executable is within your path, the simulation can be launched by typing:

mpirun -np 6 lethe-fluid-sharp flow_in_long_mixer.prm
mpirun -np 6 lethe-fluid-sharp flow_in_long_mixer_only_tracer.prm

Results#

The following movie shows the flow of the tracer through the static mixer, both as a colored slice and colored streamlines:

The tracer evolution through the inlet and outlet can be monitored by plotting their values in time, extracted from /output/tracer_flow_rate.dat. The Python script postprocess_tracer.py generates the following plot:

  python3 postprocess_tracer.py
../../../_images/tracer_flow_rates.svg

As the Figure shows, the concentration of the tracer flattens as it flows. The gap between between the inlet and outlet peaks is of \(170\) \(\text{s}\). When compared to the theoretical time of \(150\) \(\text{s}\) (\(d_x/u_x\), with \(d_x\) the domain length and \(u_x\) the inlet velocity), this difference can be explained by retention effects and the tortuous paths that the tracer travels through.

Reference#