# A Framework for Characterizing Structural Uncertainty in Large-Eddy Simulation Closures

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## Abstract

Motivated by the sizable increase of available computing resources, large-eddy simulation of complex turbulent flow is becoming increasingly popular. The underlying filtering operation of this approach enables to represent only large-scale motions. However, the small-scale fluctuations and their effects on the resolved flow field require additional modeling. As a consequence, the assumptions made in the closure formulations become potential sources of incertitude that can impact the quantities of interest. The objective of this work is to introduce a framework for the systematic estimation of structural uncertainty in large-eddy simulation closures. In particular, the methodology proposed is independent of the initial model form, computationally efficient, and suitable to general flow solvers. The approach is based on introducing controlled perturbations to the turbulent stress tensor in terms of magnitude, shape and orientation, such that propagation of their effects can be assessed. The framework is rigorously described, and physically plausible bounds for the perturbations are proposed. As a means to test its performance, a comprehensive set of numerical experiments are reported for which physical interpretation of the deviations in the quantities of interest are discussed.

## Keywords

Large-eddy simulation Predictive science Turbulence modeling Uncertainty quantification## Notes

### Acknowledgments

This work was funded by the United States Department of Energy’s (DoE) National Nuclear Security Administration (NNSA) under the Predictive Science Academic Alliance Program (PSAAP) II at Stanford University.

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to help improve the quality of the paper.

### Compliance with Ethical Standards

### Conflict of interests

The authors declare that they have no conflict of interest.

## References

- 1.Hermeth, S., Staffelbach, G., Gicquel, L.Y.M., Poinsot, T.: LES Evaluation of the effects of equivalence ratio fluctuations on the dynamic flame response in a real gas turbine combustion chamber. Proc. Combust. Inst.
**34**, 3165–3173 (2013)CrossRefGoogle Scholar - 2.Bulat, G., Fedina, E., Fureby, C., Stopper, U.: Reacting flow in an industrial gas turbine combustor: LES and experimental analysis. Proc. Combust. Inst.
**35**, 3175–3183 (2015)CrossRefGoogle Scholar - 3.Masquelet, M., Yan, J., Dord, A., Laskowski, G., Shunn, L., Jofre, L., Iaccarino, G.: Uncertainty quantification in large eddy simulations of a rich-dome aviation gas turbine. In: Proceeding of the ASME Turbo Expo 2017, GT2017-64835, pp 1–11 (2017)Google Scholar
- 4.Ang, J., Evans, K., Geist, A., Heroux, M., Hovland, P., Marques, O., Curfman, L., Ng, E., Wild, S.: Workshop on Extreme-Scale Solvers: Transition to Future Architectures. Tech. Rep., U.S, Department of Energy, Office of Advanced Scientific Computing Research (2012)Google Scholar
- 5.Ghosal, S.: An analysis of numerical errors in large-eddy simulations of turbulence. J. Comput. Phys.
**125**, 187–206 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Meyers, J., Geurts, B.J., Baelmans, M.: Database analysis of errors in large-eddy simulation. Phys. Fluids
**15**, 2740 (2003)CrossRefzbMATHGoogle Scholar - 7.Meldi, M., Lucor, D., Sagaut, P.: Is the Smagorinsky coefficient sensitive to uncertainty in the form of the energy spectrum? Phys. Fluids
**23**, 125,109 (2011)CrossRefGoogle Scholar - 8.Meyers, J., Sagaut, P.: Evaluation of Smagorinsky variants in large-eddy simulations of wall-resolved plane channel flows. Phys. Fluids
**19**, 095,105 (2007a)CrossRefzbMATHGoogle Scholar - 9.Meyers, J., Sagaut, P.: Is plane-channel flow a friendly case for the testing of large-eddy simulation subgrid-scale models? Phys. Fluids
**19**, 048,105 (2007b)CrossRefzbMATHGoogle Scholar - 10.Dunn, M.C., Shotorban, B., Frendi, A.: Uncertainty quantification of turbulence model coefficients via latin hypercube sampling method. J. Fluids. Eng.
**133**, 041,402 (2011)CrossRefGoogle Scholar - 11.Lucor, D., Meyers, J., Sagaut, P.: Sensitivity analysis of large-eddy simulations to subgrid-scale-model parametric uncertainty using polynomial chaos. J. Fluid. Mech.
**585**, 255–280 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Cheung, S., Oliver, T., Prudencion, E., Pridhomme, S., Moser, R.: Bayesian uncertainty analysis with applications to turbulence modeling. Reliab. Eng. Syst. Saf.
**96**, 1137–1149 (2011)CrossRefGoogle Scholar - 13.Völker, S., Moser, R., Venugopal, P.: Optimal large eddy simulation of turbulent channel flow based on direct numerical simulation statistical data. Phys. Fluids
**14**, 3675–3691 (2002)CrossRefzbMATHGoogle Scholar - 14.Phillips, N.A.: Models for weather prediction. Annu. Rev. Fluid. Mech.
**2**, 251–292 (1970)CrossRefGoogle Scholar - 15.Leith, C.E.: Objective methods for weather prediction. Annu. Rev. Fluid. Mech.
**10**, 107–128 (1978)CrossRefzbMATHGoogle Scholar - 16.Gorlé, C., Iaccarino, G.: A framework for epistemic uncertainty quantification of turbulent scalar flux models for Reynolds-averaged Navier-Stokes simulations. Phys. Fluids
**25**, 055,105 (2013)CrossRefGoogle Scholar - 17.Emory, M., Larsson, J., Iaccarino, G.: Modeling of structural uncertainties in Reynolds-averaged Navier-Stokes closures. Phys. Fluids
**25**, 110,822 (2013)CrossRefGoogle Scholar - 18.Vasilyev, O.V., Lund, T.S., Moin, P.: A general class of commutative filters for LES in complex geometries. J. Comput. Phys.
**146**, 82–104 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Marsden, A.L., Vasilyev, O.V., Moin, P.: Construction of commutative filters for LES on unstructured meshes. J. Comput. Phys.
**175**, 584–603 (2002)CrossRefzbMATHGoogle Scholar - 20.Leonard, A.: Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. A
**18**, 237–248 (1974)CrossRefGoogle Scholar - 21.Lund, T.S.: The use of explicit filters in large eddy simulation. Comput. Math. Appl.
**46**, 603–616 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Carati, D., Winckelmans, G.S., Jeanmart, H.: On the modelling of the subgrid-scale and filtered-scale stress tensors in alrge-eddy simulation. J. Fluid. Mech.
**441**, 119–138 (2001)CrossRefzbMATHGoogle Scholar - 23.Rogallo, R.S., Moin, P.: Numerical simulation of turbulent flow. Annu. Rev. Fluid. Mech.
**16**, 2150 (1984)CrossRefzbMATHGoogle Scholar - 24.Clark, R.A., Ferziger, J.H., Reynolds, W.C.: Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid. Mech.
**91**, 1–16 (1979)CrossRefzbMATHGoogle Scholar - 25.Bardina, J., Ferziger, J.H., Reynolds, W.C.: Improved subgrid scale models for large eddy simulation. In: Proceeding of the AIAA 13th Fluid & Plasma Dynamics Conference, pp 1–10 (1980)Google Scholar
- 26.Zang, Y., Street, R.L., Koseff, J.R.: A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids. A
**5**, 3186–3195 (1993)CrossRefzbMATHGoogle Scholar - 27.Meneveau, C., Katz, J.: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid. Mech.
**32**, 1–32 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Smagorinsky, J.: General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather. Rev.
**91**, 99–164 (1963)CrossRefGoogle Scholar - 29.Germano, M., Piomelli, U., Moin, P., Cabot, W.: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids. A
**3**, 1760–1765 (1991)CrossRefzbMATHGoogle Scholar - 30.Nicoud, F., Ducros, F.: Subgrid-scale stress modelling based on the square of the velocity gradient. Flow. Turbul. Combust.
**62**, 183–200 (1999)CrossRefzbMATHGoogle Scholar - 31.Nicoud, F., Toda, H.B., Cabrit, O., Bose, S., Lee, J.: Using singular values to build a subgrid-scale model for large eddy simulations. Phys. Fluids
**23**, 085,106 (2011)CrossRefGoogle Scholar - 32.Rozema, W., Bae, H.J., Moin, P., Verstappen, R.: Minimum-dissipation models for large-eddy simulation. Phys. Fluids
**27**, 085,107 (2015)CrossRefGoogle Scholar - 33.Jofre, L., Lehmkuhl, O., Ventosa, J., Trias, F.X., Oliva, A.: Conservation properties of unstructured finite-volume mesh schemes for the Navier-Stokes equations. Numer. Heat. Transfer, Part. B
**65**, 53–79 (2014)CrossRefGoogle Scholar - 34.Yoshizawa, A.: Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Phys. Fluids
**29**, 2152–2164 (1986)CrossRefzbMATHGoogle Scholar - 35.Moin, P., Squires, K., Cabot, W., Lee, S.: A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids. A
**3**, 2746–2757 (1991)CrossRefzbMATHGoogle Scholar - 36.Schumann, U.: Realizability of Reynolds-stress turbulence models. Phys. Fluids
**20**, 721–725 (1977)CrossRefzbMATHGoogle Scholar - 37.Vreman, B., Geurts, B., Kuerten, H.: Realizability conditions for the turbulent stress tensor in large-eddy simulation. J. Fluid. Mech.
**278**, 351–362 (1994)CrossRefzbMATHGoogle Scholar - 38.Lumley, J.L., Newman, G.: The return to isotropy of homogeneous turbulence. J. Fluid. Mech.
**82**, 161–178 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 39.Choi, K.S., Lumley, J.L.: The return to isotropy of homogeneous turbulence. J. Fluid. Mech.
**436**, 59–84 (2001)CrossRefzbMATHGoogle Scholar - 40.Banerjee, S., Krahl, R., Durst, F., Zenger, C.: Presentation of anisotropy properties of turbulence, invariants versus eigenvalues approaches. J. Turbul.
**8**, 1–27 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 41.Kindlmann, G.: Superquadric tensor glyphs. In: Proceeding of the 6th Joint Eurographics-IEEE TCVG Conference, pp 147–154 (2004)Google Scholar
- 42.Teem: Tools to process and visualize scientific data and images. http://teem.sourceforge.net (2003)
- 43.Stolz, S., Adams, A.: An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids
**11**, 1699–1701 (1999)CrossRefzbMATHGoogle Scholar - 44.Pope, S.B.: Turbulent Flows. Cambridge University Press (2000)Google Scholar
- 45.Piomelli, U., Cabot, W., Moin, P., Lee, S.: Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids
**3**, 1766–1771 (1991)CrossRefzbMATHGoogle Scholar - 46.Lasserre, J.B.: A trace inequality for matrix product. IEEE Trans. Autom. Control.
**40**, 1500–1501 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 47.Lund, T.S., Ghosal, S., Moin, P.: Numerical experiments with highly variable eddy viscosity model. Eng. Appl. LES
**162**, 7–11 (1993)Google Scholar - 48.Domino, S.P.: Sierra low mach module: Nalu theory manual 1.0. Tech. Rep. SAND2015-3107w, Sandia National Laboratories, Unclassified Unlimited Release (UUR). https://github.com/NaluCFD/NaluDoc (2015)
- 49.Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to
*R**e*_{τ}= 590. Phys. Fluids**11**, 943–945 (1999)CrossRefzbMATHGoogle Scholar - 50.Chapman, D., Kuhn, G.: The limiting behavior of turbulence near a wall. J. Fluid. Mech.
**170**, 265–292 (1986)CrossRefzbMATHGoogle Scholar - 51.Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes. Cambridge University Press (2007)Google Scholar
- 52.Gullbrand, J., Chow, F.K.: The effect of numerical errors and turbulence models in large-eddy simulations of channel flow, with and without explicit filtering. J. Fluid. Mech.
**495**, 323–341 (2003)CrossRefzbMATHGoogle Scholar