Bayesian Inference of Cavitation Model Coefficients and Uncertainty Quantification of a Venturi Flow Simulation
<p>The geometry of the venturi tube.</p> "> Figure 2
<p>Void fraction and u-velocity profile: comparisons between experimental data and numerical results. (<b>a</b>) void fraction at X = 5.1 mm, (<b>b</b>) u-velocity at X = 5.1 mm, (<b>c</b>) void fraction at X = 38.4 mm, (<b>d</b>) u-velocity at X = 38.4 mm, (<b>e</b>) void fraction at X = 73.9, (<b>f</b>) u-velocity at X = 73.9 mm.</p> "> Figure 3
<p>(<b>a</b>) Void fractions and (<b>b</b>) u-velocity contour.</p> "> Figure 4
<p>Prior and posterior distributions of the ZGB coefficients. (<b>a</b>) bubble diameter, (<b>b</b>) nucleation site volume fraction, (<b>c</b>) evaporation coefficient, (<b>d</b>) condensation coefficient.</p> "> Figure 5
<p>Void fraction and u-velocity at X = 73.9 mm with the calibrated ZGB model coefficients.</p> "> Figure 6
<p>The void fraction contour with the calibrated ZGB model coefficients.</p> "> Figure 7
<p>Histograms for the inputs of the random variables: inlet velocity and roughness.</p> "> Figure 8
<p>The histogram of the predicted cavitation area.</p> "> Figure 9
<p>Confidence intervals of void fractions and u-velocity profiles at X = 2.5 mm and 5.1 mm. (<b>a</b>) void fraction at X = 2.5 mm, (<b>b</b>) u-velocity at X = 2.5 mm, (<b>c</b>) void fraction at X = 5.1 mm, (<b>d</b>) u-velocity at X = 5.1 mm.</p> "> Figure 10
<p>(<b>a</b>) Void fractions and (<b>b</b>) u-velocity contour.</p> ">
Abstract
:1. Introduction
2. Governing Equation and Numerical Methods
2.1. Cavitation and Turbulence Model
2.2. Grid Convergence Index (GCI)
2.3. Bayesian Inference
2.4. Point-Collocation Nonintrusive Polynomial Chaos (PC-NIPC)
2.5. Source of Uncertainty
2.5.1. Model Inadequacy
2.5.2. Observation Error
2.6. Likelihood Function
2.7. Markov Chain Monte Carlo (MCMC)
3. Deterministic Simulation
3.1. Geometry and Operating Condition
3.2. Validation of the Deterministic Simulation
3.3. Grid Convergence Index (GCI)
4. Bayesian Inference of ZGB Model Coefficients
5. Uncertainty Quantification with Respect to Operating Conditions
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- West, T.K., IV; Hosder, S. Uncertainty quantification of hypersonic reentry flows with sparse sampling and stochastic expansions. J. Spacecr. Rocket. 2015, 52, 120–133. [Google Scholar] [CrossRef]
- Zeng, K.; Hou, J.; Ivanov, K.; Jessee, M.A. Uncertainty quantification and propagation of multiphysics simulation of the pressurized water reactor core. Nucl. Technol. 2019, 205, 1618–1637. [Google Scholar] [CrossRef]
- Sankaran, S.; Kim, H.J.; Choi, G.; Taylor, C.A. Uncertainty quantification in coronary blood flow simulations: Impact of geometry, boundary conditions and blood viscosity. J. Biomech. 2016, 49, 2540–2547. [Google Scholar] [CrossRef] [PubMed]
- 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. Saf. 2011, 96, 1137–1149. [Google Scholar] [CrossRef]
- Schaefer, J.; Hosder, S.; West, T.; Rumsey, C.; Carlson, J.R.; Kleb, W. Uncertainty quantification of turbulence model closure coefficients for transonic wall-bounded flows. AIAA J. 2017, 55, 195–213. [Google Scholar] [CrossRef]
- Bestion, D.; De Crecy, A.; Moretti, F.; Camy, R.; Barthet, A.; Bellet, S.; Munoz Cobo, J.; Badillo, A.; Niceno, B.; Hedberg, P.; et al. Review of uncertainty methods for CFD application to nuclear reactor thermal hydraulics. In Proceedings of the NUTHOS 11-the 11th International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Operation and Safety, Gyeongju, Korea, 9–13 October 2016. [Google Scholar]
- Barre, S.; Rolland, J.; Boitel, G.; Goncalves, E.; Patella, R.F. Experiments and modeling of cavitating flows in venturi: Attached sheet cavitation. Eur. J. Mech.-B/Fluids 2009, 28, 444–464. [Google Scholar] [CrossRef] [Green Version]
- Shih, T.H.; Liou, W.W.; Shabbir, A.; Yang, Z.; Zhu, J. A new k-ϵ eddy viscosity model for high Reynolds number turbulent flows. Comput. Fluids 1995, 24, 227–238. [Google Scholar] [CrossRef]
- Rodio, M.G.; Congedo, P.M. Robust analysis of cavitating flows in the venturi tube. Eur. J. Mech.-B/Fluids 2014, 44, 88–99. [Google Scholar] [CrossRef]
- Schnerr, G.H.; Sauer, J. Physical and numerical modeling of unsteady cavitation dynamics. In Fourth International Conference on Multiphase Flow; ICMF New Orleans: New Orleans, LO, USA, 2001. [Google Scholar]
- Goel, T.; Thakur, S.; Haftka, R.T.; Shyy, W.; Zhao, J. Surrogate model-based strategy for cryogenic cavitation model validation and sensitivity evaluation. Int. J. Numer. Methods Fluids 2008, 58, 969–1007. [Google Scholar] [CrossRef] [Green Version]
- Ge, M.; Sun, C.; Zhang, G.; Coutier-Delgosha, O.; Fan, D. Combined suppression effects on hydrodynamic cavitation performance in Venturi-type reactor for process intensification. Ultrason. Sonochem. 2022, 86, 106035. [Google Scholar] [CrossRef]
- Ge, M.; Zhang, G.; Petkovšek, M.; Long, K.; Coutier-Delgosha, O. Intensity and regimes changing of hydrodynamic cavitation considering temperature effects. J. Clean. Prod. 2022, 338, 130470. [Google Scholar] [CrossRef]
- Ge, M.; Petkovšek, M.; Zhang, G.; Jacobs, D.; Coutier-Delgosha, O. Cavitation dynamics and thermodynamic effects at elevated temperatures in a small Venturi channel. Int. J. Heat Mass Transf. 2021, 170, 120970. [Google Scholar] [CrossRef]
- Lee, G.; Bae, J. Numeircal study of the effect of the surface roughness magnitude on the multi-stage orifice internal flow pattern. Am. Nucl. Soc. Trans. 2021, 124, 748–751. [Google Scholar]
- Lee, G.; Jhung, M.; Bae, J.; Kang, S. Numerical Study on the Cavitation Flow and Its Effect on the Structural Integrity of Multi-Stage Orifice. Energies 2021, 14, 1518. [Google Scholar] [CrossRef]
- ANSYS. ANSYS Fluent Theory Guide. 2019. Available online: http://www.ansys.com/ (accessed on 26 April 2022).
- Zwart, P.J.; Gerber, A.G.; Belamri, T. A two-phase flow model for predicting cavitation dynamics. In Proceedings of the Fifth International Conference on Multiphase Flow, Yokohama, Japan, 30 May–3 June 2004. [Google Scholar]
- Patankar, S.V.; Spalding, D.B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. In Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion; Elsevier: Amsterdam, The Netherlands, 1983; pp. 54–73. [Google Scholar]
- Roache, P.J. Quantification of uncertainty in computational fluid dynamics. Annu. Rev. Fluid Mech. 1997, 29, 123–160. [Google Scholar] [CrossRef] [Green Version]
- Tanaka, M.; Miyake, Y. Numerical simulation of thermal striping phenomena in a T-junction piping system for fundamental validation and uncertainty quantification by GCI estimation. Mech. Eng. J. 2015, 2, 15–134. [Google Scholar] [CrossRef] [Green Version]
- Kaipio, J.; Somersalo, E. Statistical and computational inverse problems. In Applied Mathematical Sciences; Springer: Berlin/Heidelberg, Germany, 2005; p. 160. [Google Scholar]
- Hosder, S.; Walters, R.; Balch, M. Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, USA, 23–26 April 2007. [Google Scholar]
- Marelli, S.; Sudret, B. UQLab: A framework for uncertainty quantification in Matlab. In Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management; American Society of Civil Engineers: Reston, VA, USA, 2014; pp. 2554–2563. [Google Scholar]
- Robert, C.P.; Casella, G.; Casella, G. Monte Carlo Statistical Methods; Springer: New York, NY, USA, 1999. [Google Scholar]
- Metropolis, N.; Rosenbluth, A.W.; Rosenbluth, M.N.; Teller, A.H.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 1953, 21, 1087–1092. [Google Scholar] [CrossRef] [Green Version]
- Goodman, J.; Weare, J. Ensemble samplers with affine invariance. Commun. Appl. Math. Comput. Sci. 2010, 5, 65–80. [Google Scholar] [CrossRef]
- Hakimi, N.; Pierret, S.; Hirsch, C. Presentation and application of a new extended k-ε Model with wall functions. In Proceedings of the ECCOMAS 2000 Conference, Barcelona, Spain, 11–14 September 2000. [Google Scholar]
- Edeling, W.N.; Cinnella, P.; Dwight, R.P.; Bijl, H. Bayesian estimates of parameter variability in the k-ε turbulence model. J. Comput. Phys. 2014, 258, 73–94. [Google Scholar] [CrossRef] [Green Version]
- Dutta, N.; Kopparthi, P.; Mukherjee, A.K.; Nirmalkar, N.; Boczkaj, G. Novel strategies to enhance hydrodynamic cavitation in a circular venturi using RANS numerical simulations. Water Res. 2021, 204, 117559. [Google Scholar] [CrossRef]
- Wang, C.; Ding, H.; Zhao, Y. Influence of wall roughness on discharge coefficient of sonic nozzles. Flow Meas. Instrum. 2014, 35, 55–62. [Google Scholar] [CrossRef]
- Ge, M.; Zhang, X.L.; Brookshire, K.; Coutier-Delgosha, O. Parametric and V&V study in a fundamental CFD process: Revisiting the lid-driven cavity flow. Aircr. Eng. Aerosp. Technol. 2021, 94, 515–530. [Google Scholar]
- Zhang, X.; Xiao, H.; Gomez, T.; Coutier-Delgosha, O. Evaluation of ensemble methods for quantifying uncertainties in steady-state CFD applications with small ensemble sizes. Comput. Fluids 2020, 203, 104530. [Google Scholar] [CrossRef] [Green Version]
10−6 | 5 × 10−4 | 50 | 1 × 10−2 |
Type of Variable | Distribution | Orthogonal Polynomials | |
---|---|---|---|
Uniform | |||
Gaussian | |||
Gamma | |||
Beta |
Convergence Angle (°) | Divergence Angle (°) | ||||
---|---|---|---|---|---|
50.0 | 43.7 | 60.2 | 1512 | 4.3 | 4.0 |
Water Property | Vapor Property | ||
---|---|---|---|
998.2 | 998.2 |
Velocity Inlet | Pressure Outlet | Saturated Pressure |
---|---|---|
10.8 m/s | 38,450 Pa | 2339 Pa |
Coarse | Medium | Fine | |
---|---|---|---|
Number of nodes | 16,027 | 29,536 | 59,492 |
Cavitation area(m2) | 2.57 × 10−4 | 2.52 × 10−4 | 2.51 × 10−4 |
2.53 × 10−4 | 4.22 × 10−6 | 1.12 × 10−5 |
Parameter | Lower Boundary | Upper Boundary |
---|---|---|
~Uniform | 2.0 × 10−7 | 1.0 × 10−6 |
~Uniform | 0.00025 | 0.0006 |
~Uniform | 25 | 55 |
~Uniform | 0.005 | 0.015 |
~Uniform | 0 | 0.3 |
~Uniform | 0 | 0.05 |
ZGB Coefficients | ||||
---|---|---|---|---|
Nominal value | 10−6 | 5 × 10−4 | 50 | 0.01 |
Posterior Mean | 7.9 × 10−7 | 4.5 × 10−4 | 52 | 0.013 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Bae, J.-H.; Chang, K.; Lee, G.-H.; Kim, B.-C. Bayesian Inference of Cavitation Model Coefficients and Uncertainty Quantification of a Venturi Flow Simulation. Energies 2022, 15, 4204. https://doi.org/10.3390/en15124204
Bae J-H, Chang K, Lee G-H, Kim B-C. Bayesian Inference of Cavitation Model Coefficients and Uncertainty Quantification of a Venturi Flow Simulation. Energies. 2022; 15(12):4204. https://doi.org/10.3390/en15124204
Chicago/Turabian StyleBae, Jae-Hyeon, Kyoungsik Chang, Gong-Hee Lee, and Byeong-Cheon Kim. 2022. "Bayesian Inference of Cavitation Model Coefficients and Uncertainty Quantification of a Venturi Flow Simulation" Energies 15, no. 12: 4204. https://doi.org/10.3390/en15124204