Reynolds Stress Perturbation for Epistemic Uncertainty Quantification of RANS Models
4. Summary and Conclusions
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-5 0 5 10 15
10-3 0
1 2 3
y/h
(a) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=1.5 nk=1/1.5
-5 0 5 10 15
10-3 0
1 2 3
y/h
(b) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=1.5 nk=1/1.5
-2 0 2 4 6 8
10-3 0
1 2 3
y/h
(c) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=1.5 nk=1/1.5
-5 0 5 10 15
10-3 0
1 2 3
y/h
(d) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=2 nk=1/2
-5 0 5 10 15
10-3 0
1 2 3
y/h
(H) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=2 nk=1/2
-2 0 2 4 6 8 10 12
10-3 0
1 2 3
y/h
(f) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST nk=2 nk=1/2
-5 0 5 10 15
10-3 0
1 2 3
y/h
(g) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
-5 0 5 10 15
10-3 0
1 2 3
y/h
(h) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
-2 0 2 4 6 8
10-3 0
1 2 3
y/h
(i) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
-5 0 5 10 15
10-3 0
1 2 3
y/h
(j) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
-5 0 5 10 15
10-3 0
1 2 3
y/h
(k) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
-2 0 2 4 6 8
10-3 0
1 2 3
y/h
(l) DNS, Le, Moin & Kim, 1997
Experiment, Jovic & Driver, 1994 RANS k- SST X1c X2c X3c
Figure 9.Reynolds shear stress calculated atx/h=4,x/h=6 andx/h=10 and normalized by the inlet mean velocity (−uv/u20). (a)kperturbation withnk=1.5 atx/h=4, (b)kperturbation with nk=1.5 atx/h=6, (c)kperturbation withnk=1.5 atx/h=10, (d)kperturbation withnk=2 atx/h=4, (e)kperturbation withnk=2 atx/h=6, (f)kperturbation withnk=2 atx/h=10, (g) eigenvalue perturbation withδB=0.1 atx/h=4, (h) eigenvalue perturbation withδB=0.1 at x/h=6, (i) eigenvalue perturbation withδB=0.1 atx/h=10, (j) eigenvalue perturbation with δB =0.25 atx/h =4, (k) eigenvalue perturbation withδB =0.25 atx/h=6 and (l) eigenvalue perturbation withδB=0.25 atx/h=10.X1c,X2candX3crepresent perturbations toward the three corners of the barycentric map. DNS data [15] (−·); experimental data [9] (•); RANSk-ωSST model (−). The uncertainty bounds are shown with gray areas.
that the amount of the perturbation applied to the turbulent kinetic energy is not chosen systematically, thus it is left for future investigation.
In view of the results of this article, it is recommended for future work to focus on applying the perturbation to other parameters of the Reynolds stress such as the eigenvectors or a combination that optimizes the range that captures the most precise solution [7]. Recent studies [23,24] suggested that machine learning can be used as a powerful tool to predict discrepancies in the magnitude, anisotropy and orientation of the Reynolds stress tensor. It would be useful to continue applying this methodology to a new set of cases to monitor its behavior.
Author Contributions:Conceptualization, L.F.C.R., D.F.H. and M.A.; methodology, L.F.C.R., D.F.H. and M.A.;
software, L.F.C.R.; validation, D.F.H., L.F.C.R. and M.A.; formal analysis, L.F.C.R., D.F.H. and M.A.; investigation, L.F.C.R., D.F.H. and M.A.; resources, D.F.H.; writing—original draft preparation, L.F.C.R.; writing—review and editing, D.F.H. and M.A.; visualization, L.F.C.R.; supervision, D.F.H. and M.A.; project administration, D.F.H. and M.A.
Funding:This research received no external funding.
Acknowledgments:This research was supported by Kamstrup A/S (Group Quality—Fluid Dynamics Center) and Aarhus University. This research is solidly based on the final Master’s Thesis of Luis F. Cremades Rey [25].
The authors also acknowledge the comments by three anonymous reviewers.
Conflicts of Interest:The authors declare no conflict of interest. Kamstrup A/S had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Appendix A. Implementation of the Reynolds Stress Perturbation in OpenFOAM
The application of the Reynolds stress perturbation results in the appearance of an extra term (i.e.,ΔRij) in the right-hand side of Equation (2) as
∂tui+uj∂jui=∂ip+ν∂j∂jui−∂j(Rij+ΔRij), (A1) whereΔRijis the discrepancy between the perturbed Reynolds stress tensor shown in Equation (14) and the original one. This extra term is evaluated separately and added in OpenFOAM as a new line of code into the momentum equations as highlighted in FigureA1.
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#
$$
%&
'
Figure A1.Modification to the Momentum predictor subroutine implemented in OpenFOAM.
FigureA2shows the mean velocity field obtained from the baseline RANS solver. To better describe the procedure of the Reynolds stress perturbation, three different locations are chosen and their positions in the barycentric map are shown in FigureA3. This figure shows how the extracted Reynolds stress tensor from the baseline case is perturbed toward different corners in the barycentric map. This procedure is done for all the points in the domain and the perturbed Reynolds stress tensor is computed following Equation (14). Next, the discrepancy between the perturbed Reynolds stress tensor and the original one from the baseline case (i.e.,ΔRij) is evaluated (here using a separate Matlab code) and added to the right-hand side of the momentum equations in OpenFOAM. The
Fluids2019,4, 113
simulations are run again until the results are converged. The corresponding flowchart is also plotted in FigureA4.
Figure A2.Mean velocity field in thex-direction, normalized with respect to the inlet velocity ( ¯u/u0) obtained from the baseline case. Three different locations have been chosen in the flow domain to study their positions in the barycentric map. The points 1, 2 and 3 are respectively located at (h, 0.7h), (10h, 0.7h) and (8h, 5h).
Figure A3.Locations of the chosen points shown in FigureA2in the barycentric map. The points were perturbed towards the three limiting states and the amount of perturbation isδB=0.25.
Figure A4.A flowchart describing the implementation of Reynolds stress perturbation in OpenFOAM.
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