This code was developed by Quentin Chevalier (aka hawkspar) as part of his PhD from 2020 to 2023.

The acronym means "Swirling Parallel Yet another jet code" as an hommage to Chuhan Wang who gave me the first building blocks for this code. His contribution was then extended to 3D perturbations and parallel framework.

It contains two subparts, SPYB ("SPY Baseflow") and SPYP ("SPY Perturbations").

This little README is intended as a small quick documentation, for details on the science, you're welcome to read the relevant chapter of my thesis !

SPY handles :

• arbitrary eddy viscosities
• 2D mesh
• cylindrical coordinates (x-r is provided by the user)
• arbitrary Dirichlet boundary conditions
• stress-free boundary conditions

It is a fully parallel code.

Using a Newton solver, SPYB can compute a baseflow satisfying :

• incompressibility
• 3 components of velocity
• axisymmetric baseflow

SPYP can do global temporal stability analysis or resolvent calculations on perturbations satisfying :

• incompressibility
• 3 components of velocity
• azimuthally decomposition

The code also contains routines for 2D or 3D visualisation.

The code uses a memoisation paradigm all around and provides utilities as well as handlers.

This is a memory leak when using a large amount of EPS.solve with unassembled dolfinx objects. - keep an eye out when performing overly large computations !

Depends under the hood on PETSc and SLEPc, and as capricious as them.

General BCs.

This code requires python3.

This code depends on the FEniCsX library. This library is included as dolfinx in the code. It is strongly recommanded to use the docker image of this library available at https://hub.docker.com/r/dolfinx/dolfinx

Parts of the project are also dependant on the meshio library that doesn't come in the docker image. It can be readily installed from the command line inside the container by using pip : pip3 install h5py meshio

Plotly has to be installed as well to use 3D visualisation routines.

The tree of the project separates cases which represents user application, and src, where most of the magic happens.

src/helpers.py provides a couple useful stuff like operators, convenient file finders, but spy.py, spyb.py and spyp.py provide only classes.

The reference case should be nozzle. This case uses most of the interesting features currently implemented.

Every case folder must include a mesh folder with the desired meshes in .xdmf format.

It is advised to regroup utilities in a setup.py which can be used to prescribe solver parameters, mesh directions, boundary conditions and boxing functions...

During computations, SPY will try to create directories in your case file. It will save every result it can using complicated strings formats like q_Re=2000_S=0,2_St=1,2e-04_n=12_p=0.npy. Don't panic ! It knows what it's doing. If you want to go parallel, be aware that SPY's saving scheme is number of processors dependent - pick one and stick to it !

First things first - SPY is direction agnostic. This means that you must tell SPY which direction is which in your mesh. This is done by providing a direction_map to it at creation. Something like direction_map={'x':0,'r':1,'th':2}.

Another key step is to choose computation parameters. It goes without saying that some computations are very sensitive to this, especially the relaxation factor for a Newton solver. This is expected to be specified as another dictionary:

params = {"rp":.97,     # Relaxation parameter
		  "atol":1e-12, # Absolute tolerance
		  "rtol":1e-9,  # Relative tolerance
		  "max_iter":10}

Then you'll need to import SPY classes like so:

import sys
sys.path.append('/home/shared/src')
from spy  import SPY
from spyb import SPYB
from spyp import SPYP

Next step is defining your geometry. SPY doesn't use anything on a mesh other than points and edges, therefore it is advised to use indicator functions to build geometry.

import numpy as np
def symmetry(x:ufl.SpatialCoordinate) -> np.ndarray: return np.isclose(x[params['r']],0,params['atol']) # Axis of symmetry at r=0

In order to compute a baseflow, one must use the SPYB module. This module is a child of SPY and inherits all its convenience functions.

Its call signature goes like SPYB(params,data_path,base_mesh,direction_map) with computation parameters, path to file, name of mesh inside mesh/, and dictionary with directions.

Axisymmetric baseflows should be computed in real mode, running source /usr/local/bin/dolfinx-real-mode before anything else. This improves storage space and performance.

The first thing to do is define boundary conditions. The simplest of these, homogeneous boundary conditions, can be specified directly using SPY.applyHomogeneousBCs, providing a geometry indicator function, a list of directions and optionally a value. By defaults it sets the relevant velocity component at the provided boundary to zero.

More involved custom Dirichlet boundary conditions can be specified using dolfinx.fem.locate_dofs_geometrical, dolfinx.fem.dirichletbc and SPY.applyBCs. Like all functions supplied to dolfinx.fem.Function.interpolate, the boundary condition must take as argument a single array containing two lines for the evaluation points and return an appropriately sized value for every point.

The following snippet gives the procedure for imposing the BC profile on component dir at geo_indic:

subspace=spy.TH.sub(0).sub(dir)
subspace_collapsed,_=subspace.collapse()
u=Function(subspace_collapsed)
u.interpolate(profile)
# Degrees of freedom
dofs = dfx.fem.locate_dofs_geometrical((subspace, subspace_collapsed), geo_indic)
bcs  = dfx.fem.dirichletbc(u, dofs, subspace) # Same as OpenFOAM
# Actual BCs
spy.applyBCs(dofs[0],bcs)

nozzle\src\baseflow implements an additional trick - it is possible to change a boundary condition on the fly during a computation by providing profile as a class with a __call__ method and changing the attributes of the profile object. This is useful to avoid recomputing boundary conditions from scratch during a computation where a single parameter is varying. This trick is what prevents more encapsulation into SPY.applyBCs.

If nothing is specified for a component on a boundary, it will default to stress-free $Pn=(grad U+grad U^T)n/Re$ one.

Right now, SPY does not accomodate non-Dirichlet boundary conditions, you'll have to change the equations weak form for more.

SPYB can handle an eddy viscosity but is unable to compute one ! The only way to include one right now is to compute one using OpenFOAM, convert it to meshio friendly format using a custom Paraview macro and feed it to cases/nozzle/src/OpenFOAM_reader. This will create a baseflow directory with a nut subfolder containing .npy files that SPY can use. Note that OpenFOAM and SPY can operate on different meshes, as long as the SPY one is included in the OpenFOAM one.

cases/nozzle/src/OpenFOAM_reader contains a couple of surprises that are essential to the nozzle case but which may prove detrimental to your case. For instance SPYB.smootenNu includes a smoothing of the eddy viscosity field as well as a cut-off.

Given a list of OpenFOAM times of interest times and associated parameters Res and Ss, providing appropriate paths to the OpenFOAM and SPY cases, the following Paraview macro will convert OpenFOAM .vtk output into .xmf that meshio can read.

from paraview.simple import *

association_dict={}

path_openFOAM='./'
path_SPY_case='SPY/cases/nozzle/'

for time in range(1): association_dict[str(time)]={'Re':Res[time], 'S':Ss[time]}

for vtk in association_dict:
    # find source
    vtm = XMLMultiBlockDataReader(registrationName='case_'+vtk+'.vtm', FileName=[path_OpenFOAM+'VTK/case_'+vtk+'.vtm'])

    # create a new 'Append Location Attributes'
    vtm_with_loc = AppendLocationAttributes(registrationName='case_loc', Input=vtm)

    save_str=path_SPY_case+'baseflow/OpenFOAM/Re='+str(association_dict[vtk]['Re'])+'_S='+str(round(association_dict[vtk]['S'],3)).replace('.',',')+'.xmf'
    # save data
    SaveData(save_str, proxy=vtm_with_loc, ChooseArraysToWrite=1, PointDataArrays=['U', 'p', 'nut'],  CellDataArrays=['CellCenters', 'U', 'p', 'nut'])
    # clear the pipeline
    for name in ('case_'+vtk+'.vtm','case_loc'):
        source = FindSource(name)
        SetActiveSource(source)
        Delete(source)
        del source

    with open(save_str, 'r+') as fp:
        # Read all lines
        lines = fp.readlines()
        fp.seek(0) # Move file pointer to the beginning of a file
        fp.truncate() # Erase the file
        # Cut out clumsy Times and Grids
        fp.writelines(lines[:3]+lines[5:12]+lines[15:37]+lines[39:])

    print("Handled case_"+vtk+'.vtm')

It is possible to supply an initial guess to the Newton solver using a function or to use a previous computation as a starting point. To use a result of a previous computation, simply run SPY.loadBaseflow (notice that since SPYB is a child of SPY, it inherits this method) with the required parameters. SPYB will try to load the closest run to the parameters you provide according to a naive L1 norm - check the printed output that it got it right.

SPYB.Re must be set by hand prior to computation - it may be a number or a dolfinx.fem.Function, allowing easy implementation of sponge zones. This is somewhat redundant with a custom SPY.nut.

SPYB.baseflow computes the baseflow. Right now, parameters Re and S are only used in the saving string scheme. It might cause problems to have a non-integer Re. refinement should not be used. save means that the baseflow must be written to file. baseflow_init allow for the velocities of the baseflow to be initialised by an arbitrary provided function satisfying, again, FEniCsX interpolation constraints.

A useful tool is SPYB.smoothenU that uses a pseudo-Laplace equation to smoothen out a velocity - it can handle only a component or the whole vector. It should only ever be called with small e and operates in place on SPY.Q where the baseflow is to be found.

SPY.saveBaseflow should handle all the loading for you. It can optionally save eddy viscosity and velocity fields in .xdmf format - currently, FEniCsX can write .xdmf in parallel not read them. Remember, in parallel these functions are number of processors dependant !

SPY.sanityCheck and SPY.printStuff are good places to start debugging.

A couple SPYB.computeX methods exist to produce 3D Plotly objects that can be put in relation to SPYP 3D visualisations. Otherwise, just go digging in the baseflow/print folder of the case.

This requires going complex with source /usr/local/bin/dolfinx-complex-mode and use of the SPYP object.

The call signature is identical to SPYB. It can operate on a different mesh than SPYB, but then its mesh has to be included in the SPYB mesh.

/!\ Reading meshes in parallel FEniCsX is an order-dependant process ! That means that extra care should be taken to always load meshes in the same order - for instance initialising SPYB in the setup file so that it is always read first ! /!\

Boundary conditions are applied like before.

There's a necessary subtlety when using MUMPS in parallel over a large mesh. It seems to require extra memory, which is configured in helpers.configureKSP, setting the icntl_14 parameter for working matrix size to be significantly higher than default.

The slowest process here is always the building of a LU preconditioner for the linearised Navier-Stokes L matrix (which doesn't include a time derivative).

Use SPYP.interpolateBaseflow given a SPYB object to accomodate different meshes more easily.

For performance reasons and to accomodate loops, it is necessary to assemble matrices prior a run. Use SPYP.assembleMMatrix to compute SPYP.M so that $qMq$ with $q$ a Taylor-Hood vector including pressure gives the velocity $L^2$ norm. SPYP.assembleLMatrix for the linearised Navier-Stokes equations given an m, again without a time derivative.

Eigenvalue computations are performed using SPYP.eigenvalues around a provided shift sigma, in number k. Again, Re, S and m are purely for saving purposes.

To read the eigenvalues and plot them one should use cases/nozzle/src/print/eig.py.

Eigenvalue calculations is highly dependant on the shift, and sadly it is not possible to leverage both the nice matrix-free properties of the current code and targets like LARGEST_REAL in SLEPc.

The process is very similar to global temporal stability. The only difference is that an additional routine should be used prior launch SPYP.assembleWBRMatrices. $W$ is so that $uWu$ gives the $L^2$ norm of $u$, so it is identical to $M$ when removing columns and rows of zeros and operates on a smaller subspace. $B$ is the extensor that prevents forcing of the incompressibility equations. $R$ is the matrix-free resolvent operator to avoid actually inverting $L-i\omega M$.

The only reason SPYP.assembleMMatrices and SPYP.assembleWBRMatrices are kept seperate is that one needs only the first to perform temporal stability analysis, whereas resolvent analysis needs the two of them to run.

Two indicator functions can be used. indic_f acts on $B$ to constrain forcing. Usually this is a fonction going smoothly from 0 to 1 which allow for concentration of forcing in a specific area of space. This actually quite critical to cases where noise in the baseflow values may be exploited to grow spurious modes. indic_u acts on $W_\psi$ and penalises response in a specific area.

Launch is done using SPYP.resolvent specifying number of required modes, list of frequencies and again Re, S and m to print saving strings.

Resolvent analysis is a slow process even in parallel. It's normal to have computations taking more than a minute for several hundred thousands of points, even with over ten processors.

There are more options to print modes. Loading modes is easy with SPYP.readMode, 2D cuts are named SPYP.saveX and 3D stuff SPYP.computeX. Inspiration on how to use these can be found in cases/nozzle/src/print

Careful using isosurfaces, Plotly is capricious and can only handle regular meshes - i.e. stuff from numpy.meshgrid.

Because the nondimensionalisation of case cases/nozzle involves the radius of the jet not its diameter, all printing routines include a multiplication by two of the Strouhal number (dimensionless frequency) to make comparaison with the literature easier.