multiUQ: an intrusive uncertainty quantification tool for gas-liquid multiphase flows. (English) Zbl 1453.76181

Turnquist, Brian; Owkes, Mark

J. Comput. Phys. 399, Article ID 108951, 30 p. (2019).

Summary: Uncertainty quantification (UQ) of fluid flows offers the ability to understand the impact of variation in fluid properties, boundary conditions, and initial conditions on simulation results. In this work, an open-source program called multiUQ is developed which performs UQ using an intrusive approach applied to gas-liquid multiphase flows. Intrusive methods require modifying the governing equations by incorporating stochastic (uncertain) variables. This adds complexity but reduces computational cost compared to non-intrusive methods (e.g. Monte Carlo). To date, much of the work on intrusive UQ has focused on single phase flows. We extend this work by adding capabilities for gas-liquid flows which include a stochastic conservative level set method to capture the location of the phase interface, computing a stochastic curvature, and development of a stochastic surface tension force. Several test cases are presented which illustrate the strength of the framework. Both deterministic and stochastic channel flow cases converge to analytic results and demonstrate the accuracy of the level set transport. Zalesak’s disk and the deformation test cases further highlight the abilities of the transport method as well as the robustness of the reinitialization equation, which maintains the level set profile. Deterministic and stochastic oscillating droplet test cases paired with analytic results, solve a true multiphase flow problem, and highlight the abilities of the UQ framework. Finally, results from a stochastic atomizing jet show droplet breakup and merging for cases of uncertainty about the surface tension coefficient and incoming velocity.

MSC:

76M35	Stochastic analysis applied to problems in fluid mechanics
76T10	Liquid-gas two-phase flows, bubbly flows
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
76D05	Navier-Stokes equations for incompressible viscous fluids
76M12	Finite volume methods applied to problems in fluid mechanics

Keywords:

stochastic; polynomial chaos; conservative level set; atomization; Navier-Stokes equations; multiphase flow

Software:

multiUQ

References:

[1]	Le Maître, O. P.; Knio, O. M., Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, With Applications to Computational Fluid Dynamics (2010), Springer Netherlands: Springer Netherlands Dordrecht · Zbl 1193.76003
[2]	Lee, C. S.; Reitz, R. D., Effect of liquid properties on the breakup mechanism of high-speed liquid drops, At. Sprays, 11, 1-19 (2001)
[3]	Metropolis, N.; Ulam, S., The Monte Carlo method, J. Am. Stat. Assoc., 44, 335-341 (1949) · Zbl 0033.28807
[4]	Cowles, M. K.; Carlin, B. P., Markov chain Monte Carlo convergence diagnostics: a comparative review, J. Am. Stat. Assoc., 91, 883-904 (1996) · Zbl 0869.62066
[5]	Tatang, M. A.; Pan, W.; Prinn, R. G.; McRae, G. J., An efficient method for parametric uncertainty analysis of numerical geophysical models, J. Geophys. Res., Atmos., 102, 21925-21932 (1997)
[6]	Liao, Q.; Zhang, D., Constrained probabilistic collocation method for uncertainty quantification of geophysical models, Comput. Geosci., 19, 311-326 (2015) · Zbl 1395.65105
[7]	Liu, Z.; Wang, X.; Kang, S., Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations, Energy, 73, 126-136 (2014)
[8]	Mathelin, L.; Hussaini, M. Y.; Zang, T. A., Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms, 38, 209-236 (2005) · Zbl 1130.76062
[9]	El-Beltagy, M. A.; Wafa, M. I., Stochastic 2d incompressible Navier-Stokes solver using the vorticity-stream function formulation, J. Appl. Math., 1-14 (2013) · Zbl 1397.76027
[10]	Xiu, D.; Karniadakis, G. E., Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187, 137-167 (2003) · Zbl 1047.76111
[11]	Kröker, I.; Nowak, W.; Rohde, C., A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems, Comput. Geosci., 19, 269-284 (2015) · Zbl 1396.65126
[12]	Le Maître, O.; Reagan, M. T.; Najm, H. N.; Ghanem, R. G.; Knio, O. M., A stochastic projection method for fluid flow: II. Random process, J. Comput. Phys., 181, 9 (2002) · Zbl 1052.76057
[13]	Knio, O.; Le Maître, O., Uncertainty propagation in CFD using polynomial chaos decomposition, Recent Topics in Computational Fluid Dynamics, 38, 616-640 (2006) · Zbl 1178.76297
[14]	Tryggvason, G.; Scardovelli, R.; Zaleski, S., Direct Numerical Simulations of Gas-Liquid Multiphase Flows (2011), Cambridge University Press · Zbl 1226.76001
[15]	Hughes, T. J.; Liu, W. K.; Zimmermann, T. K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Appl. Mech. Eng., 29, 329-349 (1981) · Zbl 0482.76039
[16]	Hirt, C.; Amsden, A.; Cook, J., An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys., 135, 203-216 (1997) · Zbl 0938.76068
[17]	Dettmer, W.; Saksono, P. H.; Perić, D., On a finite element formulation for incompressible Newtonian fluid flows on moving domains in the presence of surface tension, Commun. Numer. Methods Eng., 19, 659-668 (2003) · Zbl 1112.76387
[18]	Rudman, Murray, Volume-tracking methods for interfacial flow calculations, Int. J. Numer. Methods Fluids, 24, 671-691 (1998) · Zbl 0889.76069
[19]	Hirt, C.; Nichols, B., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201-225 (1981) · Zbl 0462.76020
[20]	Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech., 31, 567-603 (1999)
[21]	Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49 (1988) · Zbl 0659.65132
[22]	Olsson, E.; Kreiss, G., A conservative level set method for two phase flow, J. Comput. Phys., 210, 225-246 (2005) · Zbl 1154.76368
[23]	Desjardins, O.; Moureau, V.; Pitsch, H., An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys., 227, 8395-8416 (2008) · Zbl 1256.76051
[24]	Wiener, N., The homogeneous chaos, Am. J. Math., 60, 897-936 (1938) · JFM 64.0887.02
[25]	Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169, 708-759 (2001) · Zbl 1047.76574
[26]	Osher, S.; Fedkiw, R. P., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 463-502 (2001) · Zbl 0988.65093
[27]	Olsson, E.; Kreiss, G.; Zahedi, S., A conservative level set method for two phase flow II, J. Comput. Phys., 225, 785-807 (2007) · Zbl 1256.76052
[28]	Romberg, Werner, Vereinfachte numerische Integration, Forh. - K. Nor. Vidensk. Selsk., 28, 30-36 (1955) · Zbl 0065.35901
[29]	Richardson, L. F., IX. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character, 210, 307 (1911) · JFM 42.0873.02
[30]	Harten, Amiram, The artificial compression method for computation of shocks and contact discontinuities. I. Single conservation laws, Commun. Pure Appl. Math., 30, 611-638 (1977) · Zbl 0343.76023
[31]	Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146-159 (1994) · Zbl 0808.76077
[32]	Chopp, D. L., Computing Minimal Surfaces Via Level Set Curvature Flow (1991), Elsevier, Technical Report
[33]	Le Maître, O.; Knio, O. M.; Najm, H. N.; Ghanem, R. G., A stochastic projection method for fluid flow, J. Comput. Phys., 173, 481-511 (2001) · Zbl 1051.76056
[34]	Brackbill, J.; Kothe, D.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354 (1992) · Zbl 0775.76110
[35]	Crank, J.; Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. Proc. Camb. Philos. Soc., 43, 50-67 (1947) · Zbl 0029.05901
[36]	Gutiérrez, E.; Favre, F.; Balcázar, N.; Amani, A.; Rigola, J., Numerical approach to study bubbles and drops evolving through complex geometries by using a level set - Moving mesh - Immersed boundary method, Chem. Eng. J., 349, 662-682 (2018)
[37]	Nourgaliev, R.; Theofanous, T., High-fidelity interface tracking in compressible flows: unlimited anchored adaptive level set, J. Comput. Phys., 224, 836-866 (2007) · Zbl 1124.76043
[38]	Owkes, M.; Desjardins, O., A discontinuous Galerkin conservative level set scheme for interface capturing in multiphase flows, J. Comput. Phys., 249, 275-302 (2013) · Zbl 1427.76218
[39]	McCaslin, J. O.; Desjardins, O., A localized re-initialization equation for the conservative level set method, J. Comput. Phys., 262, 408-426 (2014) · Zbl 1349.76506
[40]	Chiodi, R.; Desjardins, O., A reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows, J. Comput. Phys., 343, 186-200 (2017) · Zbl 1380.76054
[41]	Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257-283 (1989) · Zbl 0681.76030
[42]	Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. Comput. Phys., 183, 83-116 (2002) · Zbl 1021.76044
[43]	Owkes, M.; Desjardins, O., A computational framework for conservative, three-dimensional, unsplit, geometric transport with application to the volume-of-fluid (VOF) method, J. Comput. Phys., 270, 587-612 (2014) · Zbl 1349.76636
[44]	Herrmann, M., A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids, J. Comput. Phys., 227, 2674-2706 (2008) · Zbl 1388.76252
[45]	Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362 (1979) · Zbl 0416.76002
[46]	Lord Rayleigh VI, F. R.S., On the capillary phenomena of jets, Proc. R. Soc. Lond., 29, 71-97 (1879)
[47]	Garrick, D. P.; Owkes, M.; Regele, J. D., A finite-volume HLLC-based scheme for compressible interfacial flows with surface tension, J. Comput. Phys., 339, 46-67 (2017) · Zbl 1375.76099
[48]	Fyfe, D.; Oran, E.; Fritts, M., Surface tension and viscosity with Lagrangian hydrodynamics on a triangular mesh, J. Comput. Phys., 76, 349-384 (1988) · Zbl 0639.76043
[49]	Shin, S.; Juric, D., Modeling three-dimensional multiphase flow using a level contour reconstruction method for front tracking without connectivity, J. Comput. Phys., 180, 427-470 (2002) · Zbl 1143.76595