Machine learning Gene Expression Programming algorithm

1. 1/14
GEP-based RANS modeling of turbulent flows
L. Campoli
The University of Melbourne
Machine Learning Monthly Meeting: September 20, 2022

2. 2/14
Outline
The Body-Of-Revolution (BOR) testcase
On-going work

3. 3/14
Body-of-revolution (BOR) testcase
Configuration
Ï BOR placed on the centerline of a fully-developed turbulent pipe flow
Ï Three body diameters (blockage ratios): d/D = 1/3,
p
2/3,
p
3/3
Ï Fixed Reynolds number: ReD = Ub D / ν = 156000 (Reτ = 3550)
Ï Ub = 4.1 m/s, ν = 10−6, uτ = 0.186 m/s
Ï Inlet profiles, meshes and test cases provided
Computational domain
Bow Center Stern Wake

4. 4/14
Numerical results: velocity in the wake region
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=14.9167
Baseline
CFD-driven GEP
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=15.4167
Baseline
CFD-driven GEP
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=15.9167
Baseline
CFD-driven GEP
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/R Ux/Ub
Axial mean velocity for MEDIUM body at x/R=16.4167
Baseline
CFD-driven GEP
Exp.
Figure: Mean axial velocity profiles in the wake region.

5. 5/14
Numerical results: velocity in other regions
0
0.2
0.4
0.6
0.8
1
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=-0.1
Baseline
JetFlow
CFD-driven GEP
Exp.
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.77
Baseline
JetFlow
CFD-driven GEP
Exp.
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
r/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=11.17
Baseline
JetFlow
CFD-driven GEP
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/R Ux/Ub
Axial mean velocity for MEDIUM body at x/R=14.4167
Baseline
JetFlow
CFD-driven GEP
Exp.
Figure: Mean axial velocity profiles in several regions.

6. 6/14
Numerical results: velocity in the BOW region
0
0.2
0.4
0.6
0.8
1
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=-0.1
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.0
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.1
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8 0.9 1 1.1 1.2 1.3
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.3
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.8 0.9 1 1.1 1.2 1.3
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.4
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.5
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.4
0.5
0.6
0.7
0.8
0.9
1
0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=0.9
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.4
0.5
0.6
0.7
0.8
0.9
1
0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=1.0
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.4
0.5
0.6
0.7
0.8
0.9
1
0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=1.1
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
Figure: Mean axial velocity profiles in the bow region.

7. 7/14
Numerical results: velocity in the CENTRAL region
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.37
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.57
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.77
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=4.97
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=5.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=5.37
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
Figure: Mean axial velocity profiles in the central region.

8. 8/14
Numerical results: velocity in the STERN region
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=8.67
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=9.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=9.67
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=10.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=10.67
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.1 1.2 1.3 1.4 1.5
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=11.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=11.67
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=11.92
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0.5
0.6
0.7
0.8
0.9
1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=12.17
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
Figure: Mean axial velocity profiles in the stern region.

9. 9/14
Numerical results: velocity in the wake region
0
0.2
0.4
0.6
0.8
1
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=12.9167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=13.4167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=13.9167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=14.4167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=14.9167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=15.4167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=15.9167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2
y/R
Ux/Ub
Axial mean velocity for MEDIUM body at x/R=16.4167
Baseline
JetFlow
PH-EVE-MO-11 aij ON Rij OFF
BFS Rij
cube aij ON Rij OFF
SNH2020-Reg-B
Exp.
CFD1
Figure: Mean axial velocity profiles in the wake region.

10. 10/14
Numerical results: shear stresses in the WAKE region
0
0.2
0.4
0.6
0.8
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
y/R
Ux Ur / U2
τ
Shear stresses for MEDIUM body at x/R=16.4167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
y/R
Ux Ur / U2
τ
Shear stresses for MEDIUM body at x/R=15.9167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
y/R
Ux Ur / U2
τ
Shear stresses for MEDIUM body at x/R=15.4167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
y/R
Ux Ur / U
2
τ
Shear stresses for MEDIUM body at x/R=14.9167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
y/R
Ux Ur / U
2
τ
Shear stresses for MEDIUM body at x/R=14.4167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3
y/R
Ux Ur / U
2
τ
Shear stresses for MEDIUM body at x/R=13.9167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-5 -4 -3 -2 -1 0 1 2 3
y/R
Ux Ur / U
2
τ
Shear stresses for MEDIUM body at x/R=13.4167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
0
0.2
0.4
0.6
0.8
1
-6 -5 -4 -3 -2 -1 0 1 2 3
y/R
Ux Ur / U
2
τ
Shear stresses for MEDIUM body at x/R=12.9167
Baseline
GEP squCyl aij ON Rij OFF
GEP Lav’s model
Exp.
CFD1
Figure: Mean shear stress profiles in the wake region.

11. 11/14
GEP-enhanced multi-model framework
OpenFOAM
PythonFOAM
Features
Selection
Dimensionality
Reduction
Clustering
Rules
GEP frozen
training
Anisotropy
Invariant
Maps
Fields
model agnostic methods
LDA, PCA, t-SNE, UMAP
distr./conn./density-based
Figure: GEP-enhanced multi-model framework.

12. 12/14
On-going work
Status and objectives
1. Flexible framework with interchangeable blocks for different types of applications
p
2. Apply Wallin’s RANS model closure1 p
3. Feature selection/model reduction/Clustering (underway)
4. Train GEP models in each cluster (underway)
5. Perform frozen multi-model GEP closure simulations (TODO: post-processing)
Some observations
1. Try to avoid the PythonFOAM interface
2. Use Wallin’s model for in-the-loop or frozen GEP
3. Assess the impact of different cost functions in-the-loop
4. Assess the quality of frozen-trained GEP models a priori
1Wallin S, Johansson AV. An explicit algebraic Reynolds stress model for incompressible and compressible
turbulent flows. Journal of fluid mechanics. 2000 Jan;403:89-132.

13. 13/14
Anisotropy Invariant Maps (AIMs)
Given the tensors Sij and τij , and mean vorticity vector ωi , we can calculate seven angles that can
be used to describe the alignment of the tensor-tensor and tensor-vector eigensystems.2
2Tao B, Katz J, Meneveau C. Statistical geometry of subgrid-scale stresses determined from holographic
particle image velocimetry measurements. Journal of Fluid Mechanics. 2002 Apr;457:35-78.

14. 14/14
Framework overview: Rules
Ï Skope-Rules used to define clusters can be not precise
Ï Some processing is required to use them in OpenFOAM
Cluster 0: [(’nut <= 0.293519 and T1xy > 0.980221 and T6xy <= 0.166932 and T7yy <= 0.332939 and T9xy <= 0.822667’, (1.0,
Cluster 1: [(’I3 <= 0.697363 and Q5 > 2.414920e-05 and T1xy <= 0.015997 and T4zz <= 0.828495 and T5yy <= 0.382479’, (0.99
Cluster 2: [(’nut > 0.462768 and T1xy > 0.893695’, (1.0, 0.692795, 10))]
Cluster 3: [(’Rxx <= 0.186232 and Rxy > 0.609035 and I3 <= 0.533601 and T1xy <= 0.99119’, (0.999143, 0.649415, 2))]
Cluster 4: [(’nut > 0.389866 and T1xx > 0.819800’, (1.0, 0.958994, 10))]