Improvement of numerical stability of non-orthogonal mesh simulations only via changes in the diffusion operator handling.
(Theory) What is face non-orthogonality:
"The angle between the centroid vector and the unit normal vector is the orthogonality angle." (Wimhurst, 2019)
(Theory) Why is face non-orthogonality important:
Can affect the discrete diffusion operator:
Numerical stability (hence cost)
Level of errors
Numerical stability
Larger explicit treatment -> less stability
Level of errors
Larger corrections -> larger truncation errors
Cannot be reduced by mesh refinement
(Theory) What is the technical challenge in face non-orthogonality:
"The difficulty is to evaluate the dot product of the normal vector and the velocity gradient on the cell face." (Wimhurst, 2019)
(Theory) Treatments in OpenFOAM:
Explicit correction by using an over-relaxed-type unit-normal vector decomposition
Resolved bugs
N/A
Methodology
Problem definition:
Convergence rate
Explicit component can become limiting factor for numerical stability
During initial iterations, especially for steady-state runs or when starting from uniform fields in transient runs
Yet its removal reduces accuracy
Convergence order
Non-orthogonality treatments are neglected at boundaries (except cyclic conditions)
Even minor non-orthogonality on patches reduce the convergence order for cases where Dirichlet/Neumann boundary conditions are applied. (Noriega et al., 2018)
A solution: Field relaxation
Field relaxation for explicit non-orthogonality components
Under-relaxation for numerical stability
Over-relaxation for numerical cost
Implementation: a Laplacian scheme
Based on an idea from nextFoam
Named relaxedNonOrthoLaplacian
Implementation: a surface-normal gradient scheme:
Named relaxedSnGrad
Test case:
The DNS study of a smooth-wall plane channel flow by (Moser et al., 1999) where the friction Reynolds number of the flow is ReTau=395.
Metrics:
Numerical stability
Simulation crashes or not
Prediction fidelity
Flow field predictions vs DNS statistics
Control variable:
Results
Discussion
What happened?
New schemes
Did not dampen initial residuals unlike nextFoam observations (at least for this particular case)
Improved prediction fidelity for above 50 degrees
Prevented simulations crashing after a certain non-orthogonality angle (above 50)
Numerical cost
For below-50-degree non-orthogonality cases, the runtime remained similar to that of the base
For above-50-degree cases, the numerical cost seems to be reduced
What do the results mean in practise? (How will it change in how we do things?)
New schemes may be used
to avoid any simulation crashes due to non-orthogonality treatments
without affecting the fidelity of numerical predictions
without affecting numerical cost estimations
more tests are needed
Risks
No changes in existing output.
No changes in existing user input.
Constraints
No finite-area numerics
Needs extensive tests to assume safe for single-phase, multiphase and overset finite-volume applications
No boundary treatments for Dirichlet and Neumann conditions
Tests
Compilation (incl. submodules):
linux64ClangDPInt32Opt (clang11)
linux64GccDPInt32Opt
linux64GccSPDPInt64Debug
Alltest: No change in output with respect to the develop HEAD + no error
