Static mixer example

This example simulates flow through a static mixer - a duct containing a fixed solid insert that passively mixes two fluid streams. The geometry is the design by Casper Schousboe Andreasen. The example demonstrates the use of the Brinkman immersed boundary method in Neko to represent the solid mixer structure on a structured background mesh.

Physics

The simulation solves the incompressible Navier-Stokes equations coupled with a passive scalar transport equation (labelled temperature) using the pN/pN spectral-element scheme with third-order time integration. The key non-dimensional parameters are the Reynolds number Re and the Peclet number Pe (set equal to Re by default).

Domain and mesh

Parameter	Value
Domain	Rectangular duct, 4 x 1 x 1
Mesh resolution	Nx x Ny x Nz elements, where Ny = Nz = Nx/4
Polynomial order	7 (GLL nodes per element)

Boundary and initial conditions

Velocity

Boundary	Condition
Inlet (x = 0, zone 1)	Smooth parabolic profile - smooth-step blending in y and z, u_max = 1, v = w = 0. The boundary layer thickness is max(1.0 / sqrt(Re), element_size), ensuring the blending region is at least one mesh element wide.
Outlet (x = 4, zone 2)	Pressure outflow (outflow+dong)
Walls (y = 0/1, z = 0/1, zones 3-6)	No-slip

Temperature (passive scalar)

Boundary	Condition
Inlet (x = 0, zone 1)	Dirichlet: smooth-step transition from T = 0 (bottom, z < 0.45) to T = 1 (top, z > 0.55)
All remaining boundaries (zones 2-6)	Zero-flux Neumann

Initial conditions

• Velocity: u = 1 everywhere outside the solid (the Brinkman indicator is subtracted so that velocity is zero inside the mixer structure), v = w = 0.
• Temperature: uniform T = 0.

Brinkman immersed boundary method

The solid mixer geometry is represented using a Brinkman penalization source term rather than body-conforming mesh boundaries. A scalar indicator field \( \chi(\mathbf{x}) \in [0, 1] \) is constructed from the signed distance to the mixer surface (provided as an STL file), where \( \chi = 1 \) marks solid and \( \chi = 0 \) marks fluid.

The indicator is mapped to a local penalization coefficient \( k(\chi) \) using the RAMP (Rational Approximation of Material Properties) scheme:

\[ k(\chi) = k_0 + (k_1 - k_0)\, \chi\, \frac{q + 1}{q + \chi} \]

where \( k_0 \) = limits[0] (minimum, fluid region), \( k_1 \) = limits[1] (maximum, solid region), and \( q \) = penalty is the RAMP convexity parameter. The Brinkman source term then adds a volumetric resistance force to the Navier-Stokes momentum equation:

\[ \mathbf{f} = -k(\chi)\, \mathbf{u} \]

When \( k \to \infty \) inside the solid region the velocity is driven to zero, recovering a no-slip condition on the embedded surface. In practice \( k_1 \) is set to 1e6 / Re following the rule-of-thumb that the Brinkman layer thickness scales as \( 1/\sqrt{k_1} \).

Implementation in Neko

The Brinkman term is added to the fluid solver through the source_terms array in the case file:

"source_terms": [
    {
        "type": "brinkman",
        "limits": [0.0, <eta_max>],
        "penalty": 3.0,
        "output": { "enable": true, "format": "vtkhdf" },
        "filter": {
            "type": "PDE",
            "radius": <r>
        },
        "objects": [
            {
                "type": "boundary_mesh",
                "name": "<path/to/mixer.stl>",
                "cache": true,
                "cache_file": "<path/to/cache_>",
                "distance_transform": {
                    "type": "smooth_step",
                    "value": 0.01
                }
            }
        ]
    }
]

Key parameters:

Parameter	Description
limits	[k_0, k_1] - RAMP penalization range; k_1 = 1e6 / Re
penalty	RAMP convexity parameter \( q \) controlling the sharpness of the solid-fluid transition (default 3.0)
filter.radius	PDE filter length scale; set to 4.0 / Nx so it spans one mesh element in the x direction
distance_transform.value	Half-width of the smooth-step interface layer

At startup, Neko reads the STL, computes the signed distance field, applies the smooth-step transform to obtain \( \chi \), smooths it with a PDE filter, and caches the result. The indicator field is subsequently available in user Fortran code via:

brinkman_indicator => neko_registry%get_field("brinkman_indicator")

This allows the initial condition and any other user routines to mask off the fluid region inside the solid, as shown in petsc/mixer.f90.

Note

The STL file for the mixer geometry (Test_mixer_095_smooth.stl) is not distributed with this repository. Please open an issue or contact the ExtremeFLOW team directly to request a copy.

Generating the case files

The script create_examples.sh generates case files and (optionally) meshes for all combinations of resolution and flow parameters. It must be run from the project root:

./examples/static_mixer/create_examples.sh        # generate case files only
./examples/static_mixer/create_examples.sh mesh   # also generate meshes and seed caches

The parameter space iterated by the script is:

Variable	Values	Description
N	64, 128	Number of elements in the longest (x) direction
Re	1000, 1500, 3000	Reynolds number
Pe	mirrors Re	Peclet number

For each combination the script:

1. Computes Nx = N, Ny = Nz = N/4 and derives the PDE filter radius as r = 4.0 / Nx (one element width in x).
2. Sets the Brinkman upper limit to k_1 = 1e6 / Re.
3. Copies petsc/case.template and substitutes all derived values (mesh path, Re, Pe, filter radius, Brinkman limits, cache path).
4. Writes a LUMI-G job script to scripts/jobscripts/LUMI-G/static_mixer/petsc/.
5. When the mesh argument is given, calls mesh.sh to generate the background mesh and runs a short warm-up simulation to populate the Brinkman distance cache. If the cache is not found at the end of this step the generated case and job files are removed automatically.

Generated case files are written to examples/static_mixer/petsc/ with names of the form <N>_re_<Re>_pe_<Pe>.case.

Running the example

Use the top-level run.sh from the project root. The script locates the case file under examples/, sets up the environment and launches Neko. A Neko installation must be available; set the NEKO_DIR environment variable if it is not in the default location (external/neko).

# Run a single case
./run.sh static_mixer/petsc/32_re_1000_pe_1000.case
 
# Run all static mixer cases
./run.sh static_mixer/petsc
 
# Run with a specific number of MPI ranks
./run.sh --procs 16 static_mixer/petsc/128_re_1000_pe_1000.case
 
# Submit to a cluster (e.g. LUMI)
./run.sh --submit LUMI-G static_mixer/petsc/128_re_1000_pe_1000.case

Useful run.sh flags:

Flag	Description
-p N / --procs N	Number of MPI ranks
-r / --re-run	Re-run even if results already exist
-c / --clean	Remove logs from previous runs
-d / --delete	Delete result directories before running
--dry-run	Print the commands that would be executed without running them
-s CLUSTER / --submit CLUSTER	Submit job scripts to a cluster queue

Expected output and visualisation

During the simulation Neko writes two types of VTKHDF output files (by default into a fields/ subdirectory next to the case file):

File	Contents
fields/field_0.vtkhdf	Time-series of velocity (Velocity) and temperature (temperature) fields
brinkman_0.vtkhdf	Static Brinkman indicator field \( \chi \) (written once at startup)
Checkpoint files	Restart snapshots written every checkpoint_value simulation time units

The visualise.py script post-processes the output with ParaView (pvbatch). It reads both VTKHDF files, clips the domain at the mid-plane, shows cross- sections at the inlet (x ~= 0) and outlet (x = 4), and renders two separate views - one coloured by velocity magnitude, one by temperature. A PNG frame is saved for each timestep.

# Single-process post-processing
pvbatch examples/static_mixer/visualise.py <path/to/fields/parent>
 
# Parallel post-processing (use MPICH-compatible launcher)
mpiexec.mpich -n 8 pvbatch examples/static_mixer/visualise.py <path/to/fields/parent>

Available options:

Option	Default	Description
--stride N	1	Process every Nth timestep
--brinkman-clip VALUE	0.5	Threshold on \( \chi \) used to surface the solid geometry
--text-color black\|white	black	Color of annotations; each variant is saved in its own sub-folder
--output-dir DIR	visualisation	Output directory (relative to input_dir)
--overwrite	false	Re-render frames that already exist

Output PNG files are written to <input_dir>/visualisation/<text-color>/ with names Velocity_XXXXXX.png and Temperature_XXXXXX.png where XXXXXX is the zero-padded timestep index. A well-mixed case at steady state should show a nearly uniform temperature field at the outlet cross-section, confirming that the mixer structure has homogenised the two inlet streams.