Abstract
For the purpose of estimating the epistemic model-form uncertainty in Reynolds-Averaged Navier-Stokes closures, we propose two transport equations to locally perturb the Reynolds stress tensor of a given baseline eddy-viscosity model. The spatial structure of the perturbations is determined by the proposed transport equations, and thus does not have to be inferred from full-field reference data. Depending on a small number of model parameters and the local flow conditions, a ’return to eddy viscosity’ is described, and the underlying baseline state can be recovered. In order to make predictions with quantified uncertainty, we identify two separate methods, i.e. a data-free and data-driven approach. In the former no reference data is required and computationally inexpensive intervals are computed. When reference data is available, Bayesian inference can be applied to obtained informed distributions of the model parameters and simulation output.
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References
Slotnick, J., Khodadoust, A., Alonso, J., Darmofal, D., Gropp, W., Lurie, E., Mavriplis, D.: Cfd vision 2030 study: a path to revolutionary computational aerosciences (2014)
Cheung, S.H., Oliver, T.A., Prudencio, E.E., Prudhomme, S., Moser, R.D.: Bayesian uncertainty analysis with applications to turbulence modeling. Reliab. Eng. Syst. Safe 96(9), 1137–1149 (2011)
Edeling, W.N., Cinnella, P., Dwight, R.P., Bijl, H.: Bayesian estimates of parameter variability in the k– ε turbulence model. J. Comp. Phys. 258, 73–94 (2014)
Hoeting, J.A., Madigan, D., Raftery, A.E., Volinsky, C.T.: Bayesian model averaging: a tutorial. Stat. Sci. 14, 382–401 (1999)
Edeling, W.N., Cinnella, P., Dwight, R.P.: Predictive rans simulations via bayesian model-scenario averaging. J. Comp. Phys. 275, 65–91 (2014)
Emory, M., Larsson, J., Iaccarino, G.: Modeling of structural uncertainties in reynolds-averaged navier-stokes closures. Phys. Fluids 25(11), 110822 (2013)
Gorlé, C., Iaccarino, G.: A framework for epistemic uncertainty quantification of turbulent scalar flux models for reynolds-averaged navier-stokes simulations. Phys. Fluids 25(5), 055105 (2013)
Olsen, M.E., Coakley, T.J.: The lag model, a turbulence model for non equilibrium flows. In: AIAA Computational Fluid Dynamics Conference, 15 th, Anaheim, CA (2001)
Pope, S.B.: Turbulent flows (2001)
Wilcox, D.C.: Turbulence modeling for cfd (1998)
Banerjee, S., Krahl, R., Durst, F., Zenger, C.: Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J. Turbul. 8, N32 (2007)
Kindlmann, G.: Superquadric tensor glyphs. In: Proceedings of the 6th Joint Eurographics-IEEE TCVG conference on Visualization, pages 147–154. Eurographics Association (2004)
Ramachandran, P., Varoquaux, G.: Mayavi: 3d visualization of scientific data. Comput. Sci. Eng. 13(2), 40–51 (2011)
Emory, M., Pecnik, R., Iaccarino, G.: Modeling structural uncertainties in reynolds-averaged computations of shock/boundary layer interactions. AIAA Paper 479, 2011 (2011)
Gorlé, C., Emory, M., Larsson, J., Iaccarino, G.: Epistemic uncertainty quantification for rans modeling of the flow over a wavy wall. Center for Turbulence Research, Annual Research Briefs (2012)
Lillard, R.P., Oliver, A.B., Olsen, M.E., Blaisdell, G.A., Lyrintzis, A.S.: The lagrst model: a turbulence model for non-equilibrium flows. AIAA Paper 444, 2012 (2012)
Rotta, J.C.: Statistische theorie nichthomogener turbulenz. Zeitschrift für Physik 129(6), 547–572 (1951)
Revell, A.J., Benhamadouche, S., Craft, T., Laurence, D.: A stress–strain lag eddy viscosity model for unsteady mean flow. Int. J. Heat Fluid Fl. 27(5), 821–830 (2006)
Ling, J., Templeton, J.: Evaluation of machine learning algorithms for prediction of regions of high reynolds averaged navier stokes uncertainty. Phys. Fluids 27(8), 085103 (2015)
Langley Research Center Turbulence Modeling Resource. Axisymmetric subsonic jet. https://turbmodels.larc.nasa.gov/jetsubsonic_val.html, 2017. [Retrieved February-2017]
Lay, D.C: Linear algebra and its applications (2006)
Schumann, U.: Realizability of reynolds-stress turbulence models. Phys. Fluids 20(5), 721–725 (1977)
Durbin, P.A., Speziale, C.G.: Realizability of second-moment closure via stochastic analysis. J. Fluid Mech. 280, 395–407 (1994)
Edeling, W.N., Iaccarino, G., Cinnella, P.: A return to eddy viscosity model for epistemic uq in rans closures. Center for Turbulence Research, Annual Research Briefs (2016)
Le, H., Moin, P., Kim, J.: Direct numerical simulation of turbulent flow over a backward-facing step. J. fluid Mech. 330(1), 349–374 (1997)
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian data analysis, vol. 2. Chapman & Hall/CRC, Boca Raton (2014)
Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. Roy. Stat. Soc. B 63(3), 425–464 (2001)
Oliver, T.A., Moser, R.D.: Bayesian uncertainty quantification applied to rans turbulence models. In: J Phys Conf Ser, volume 318, page 042032. IOP Publishing (2011)
Chib, S., Greenberg, E.: Understanding the metropolis-hastings algorithm. Am. Stat. 49(4), 327–335 (1995)
Marzouk, Y., Xiu, D.: A stochastic collocation approach to bayesian inference in inverse problems. Commun. Comput. Phys. 6, 826–847 (2009)
Witteveen, J.A.S., Doostan, A., Pecnik, R., Iaccarino, G.: Uncertainty quantification of the transonic flow around the rae 2822 airfoil. Center for Turbulence Research, Annual Briefs, Stanford University (2009)
Edeling, W.N., Dwight, R.P., Cinnella, P.: Simplex-stochastic collocation method with improved scalability. J. Comp. Phys. 310, 301–328 (2016)
Ghanem, R., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57(1), 197–202 (1990)
Witteveen, J.A.S., Iaccarino, G.: Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces. SIAM J. Sci. Comput. 34(2), A814—A838 (2012)
Dongbin, X., Karniadakis, G.E.: The wiener–askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)
Bridges, J., Wernet, M.P.: The nasa subsonic jet particle image velocimetry (piv) dataset (2011)
Le, H., Moin, P.: Direct numerical simulation of turbulent flow over a backward-facing step, Center for Turbulence Research, Annual Briefs, Stanford University (1992)
Jovic, S., Driver, D.M.: Backward-facing step measurements at low reynolds number, r e h = 5000, NASA Technical Memorandum, (108807) (1994)
Thangam, S., Speziale, C.G.: Turbulent flow past a backward-facing step-a critical evaluation of two-equation models. AIAA J. 30(5), 1314–1320 (1992)
Iaccarino, G., Mishra, A.A., Ghili, S.: Eigenspace perturbations for uncertainty estimation of single-point turbulence closures. Phys. Rev. Fluids 2(2), 024605 (2017)
Mishra, A., Iaccarino, G.: Rans predictions fo high-speed flows using enveloping models, Center for Turbulence Research, Annual Research Briefs (2016)
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This investigation was funded by the United States Department of Energy’s (DoE) National Nuclear Security Administration (NNSA) under the Predictive Science Academic Alliance Program II (PSAAP II) at Stanford University.
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Edeling, W.N., Iaccarino, G. & Cinnella, P. Data-Free and Data-Driven RANS Predictions with Quantified Uncertainty. Flow Turbulence Combust 100, 593–616 (2018). https://doi.org/10.1007/s10494-017-9870-6
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DOI: https://doi.org/10.1007/s10494-017-9870-6