# A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Incompressible Multiphase Flows

#### 2.1. Preconditioning Dual-Time Stepping Method

**g**denotes the gravity vector.

#### 2.2. Numerical Method and Body-Fitted Curvilinear Coordinate System

**,**$\widehat{Q}=\frac{1}{J}Q$ is the state vector, $\widehat{S}=\frac{1}{J}S$, and convective flux vector ${\widehat{F}}_{1}$ and viscous flux vector ${\widehat{G}}_{1}$ given as;

#### 2.3. Coupling Artificial Density-Based Dual-Time Method and Sharp Interface Methods

#### 2.4. Simulations of Incompressible Multiphase Flows

#### 2.4.1. Modeling Two-Phase Flows with Sharp Interface

#### 2.4.2. Modeling Cavitating Flows

## 3. Compressible Multiphase Flows

#### 3.1. Dual-Time Preconditioning Method for Compressible NS Equation System

#### 3.2. Numerical Solution Procedures

#### 3.3. Simulations of Compressible Multiphase Flows

## 4. Conclusions

- (i)
- Incorporating more accurate and efficient numerical schemes for solving dual-time, pseudo-compressibility NS equations.
- (ii)
- Considering the effects of turbulence by implementing various accurate turbulence models and other physical phenomena that may affect the multiphase flow behaviors.
- (iii)
- Additionally, the method can be extended to handle more complex and realistic boundary conditions and geometries. Accordingly, efforts should be focused on reducing the computational cost of the method while preserving its accuracy and robustness.

## Funding

## Conflicts of Interest

## Nomenclature

$\alpha $ | Phasic volume fraction |

$Y$ | Phasic mass fraction |

$\beta $ | Preconditioning parameter |

$\rho $ | Density |

${\gamma}_{tt}$ | Exponential factor of mixture density |

$\Gamma $ | Preconditioning matrix |

${\Gamma}_{e}$ | Jacobian of physical time derivatives |

$\mu $ | Molecular viscosity |

$\tau $ | Pseudo time |

$\mathsf{\Delta}\tau $ | Pseudo time step |

$\mathsf{\Delta}\mathrm{t}$ | Physical time step |

$\mathsf{\sigma}$ | Cavitation number |

$g$ | Gravity |

$p$ | Pressure |

$t$ | Physical time |

$T$ | Transformation matrix |

$u$ | x-direction velocity of fluid |

$U$ | Contravariant velocity in x-direction |

${U}_{0}$ | Parameter reference to velocity |

${U}_{\infty}$ | Free-stream velocity |

$v$ | y-direction velocity of fluid |

$V$ | Contravariant velocity in y-direction |

$w$ | z-direction velocity of fluid |

$W$ | Contravariant velocity in z-direction |

$\xi ,\eta ,and\zeta $ | computational space in the general curvilinear coordinate system |

$\dot{m}$ | Mass transfer rate |

$\tilde{\rho}$ | Artificial density |

$\infty $ | Reference to free stream |

## References

- Nguyen, V.-T.; Park, W.-G. Enhancement of Navier–Stokes solver based on an improved volume-of-fluid method for complex interfacial-flow simulations. Appl. Ocean. Res.
**2018**, 72, 92–109. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Vu, D.-T.; Park, W.-G.; Jung, C.-M. Navier–Stokes solver for water entry bodies with moving Chimera grid method in 6DOF motions. Comput. Fluids
**2016**, 140, 19–38. [Google Scholar] [CrossRef] - Palomino Solis, D.A.; Piscaglia, F. Toward the Simulation of Flashing Cryogenic Liquids by a Fully Compressible Volume of Fluid Solver. Fluids
**2022**, 7, 289. [Google Scholar] [CrossRef] - Yao, J.; Yao, Y. Transient CFD Modelling of Air–Water Two-Phase Annular Flow Characteristics in a Small Horizontal Circular Pipe. Fluids
**2022**, 7, 191. [Google Scholar] [CrossRef] - Fayed, H.; Bukhari, M.; Ragab, S. Large-Eddy Simulation of a Hydrocyclone with an Air Core Using Two-Fluid and Volume-of-Fluid Models. Fluids
**2021**, 6, 364. [Google Scholar] [CrossRef] - Harlow, F.H.; Welch, J.E. Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface. Phys. Fluids
**1965**, 8, 2182–2189. [Google Scholar] [CrossRef] - Puckett, E.G.; Almgren, A.S.; Bell, J.B.; Marcus, D.L.; Rider, W.J. A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows. J. Comput. Phys.
**1997**, 130, 269–282. [Google Scholar] [CrossRef] [Green Version] - Yu, J.D.; Sakai, S.; Sethian, J.A. A coupled level set projection method applied to ink jet simulation. Interface Free. Bound.
**2003**, 5, 459–482. [Google Scholar] [CrossRef] [Green Version] - Merkle, C. Time-accurate unsteady incompressible flow algorithms based on artificial compressibility. In Proceedings of the 8th Computational Fluid Dynamics Conference, Honolulu, HI, USA, 9–11 June 1987. [Google Scholar]
- Chorin, A.J. A Numerical Method for Solving Incompressible Viscous Flow Problems. J. Comput. Phys.
**1997**, 135, 118–125. [Google Scholar] [CrossRef] - Štrubelj, L.; Tiselj, I. Modeling of Rayleigh-Taylor instability with conservative level set method. In Proceedings of the ICMF 2007, 6th International Conference on Multiphase Flow, Leipzig, Germany, 9–13 July 2007. [Google Scholar]
- Elmahi, I.; Gloth, O.; Hänel, D.; Vilsmeier, R. A preconditioned dual time-stepping method for combustion problems. Int. J. Comput. Fluid Dyn.
**2008**, 22, 169–181. [Google Scholar] [CrossRef] - Lin, H.; Jiang, C.; Hu, S.; Gao, Z.; Lee, C.-H. Disturbance region update method with preconditioning for steady compressible and incompressible flows. Comput. Phys. Commun.
**2023**, 285, 108635. [Google Scholar] [CrossRef] - Li, X.-s.; Gu, C.-w.; Xu, J.-z. Development of Roe-type scheme for all-speed flows based on preconditioning method. Comput. Fluids
**2009**, 38, 810–817. [Google Scholar] [CrossRef] - Muradoglu, M.; Gokaltun, S. Implicit Multigrid Computations of Buoyant Drops Through Sinusoidal Constrictions. J. Appl. Mech.
**2005**, 71, 857–865. [Google Scholar] [CrossRef] - Kinzel, M.P. Computational Techniques and Analysis of Cavitating-Fluid Flows; A Dissertation in Aerospace Engineering; The Pennsylvania State University: State College, PA, USA, 2008. [Google Scholar]
- Maia, A.A.G.; Kapat, J.S.; Tomita, J.T.; Silva, J.F.; Bringhenti, C.; Cavalca, D.F. Preconditioning methods for compressible flow CFD codes: Revisited. Int. J. Mech. Sci.
**2020**, 186, 105898. [Google Scholar] [CrossRef] - Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Nguyen, V.-T.; Phan, T.-H.; Duy, T.-N.; Kim, D.-H.; Park, W.-G. Fully compressible multiphase model for computation of compressible fluid flows with large density ratio and the presence of shock waves. Comput. Fluids
**2022**, 237, 105325. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Phan, T.-H.; Duy, T.-N.; Park, W.-G. Unsteady cavitation around submerged and water-exit projectiles under the effect of the free surface: A numerical study. Ocean. Eng.
**2022**, 263, 112368. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Park, W.-G. A free surface flow solver for complex three-dimensional water impact problems based on the VOF method. Int. J. Numer. Methods Fluids
**2016**, 82, 3–34. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Park, W.-G. A volume-of-fluid (VOF) interface-sharpening method for two-phase incompressible flows. Comput. Fluids
**2017**, 152, 104–119. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Thang, V.-D.; Park, W.-G. A novel sharp interface capturing method for two- and three-phase incompressible flows. Comput. Fluids
**2018**, 172, 147–161. [Google Scholar] [CrossRef] - Ansari, M.R.; Nimvari, M.E. Bubble viscosity effect on internal circulation within the bubble rising due to buoyancy using the level set method. Ann. Nucl. Energy
**2011**, 38, 2770–2778. [Google Scholar] [CrossRef] - Komrakova, A.E.; Eskin, D.; Derksen, J.J. Lattice Boltzmann simulations of a single n-butanol drop rising in water. Phys. Fluids
**2013**, 25, 042102. [Google Scholar] [CrossRef] [Green Version] - Magnini, M.; Pulvirenti, B.; Thome, J.R. Numerical investigation of the influence of leading and sequential bubbles on slug flow boiling within a microchannel. Int. J. Therm. Sci.
**2013**, 71, 36–52. [Google Scholar] [CrossRef] - Tomar, G.; Biswas, G.; Sharma, A.; Agrawal, A. Numerical simulation of bubble growth in film boiling using a coupled level-set and volume-of-fluid method. Phys. Fluids
**2005**, 17, 112103. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Park, W.-G. Numerical study of the thermodynamics and supercavitating flow around an underwater high-speed projectile using a fully compressible multiphase flow model. Ocean. Eng.
**2022**, 257, 111686. [Google Scholar] [CrossRef] - Ding, H.; Zhang, Y.; Yang, Y.; Wen, C. A modified Euler-Lagrange-Euler approach for modelling homogeneous and heterogeneous condensing droplets and films in supersonic flows. Int. J. Heat Mass. Tran.
**2023**, 200, 123537. [Google Scholar] [CrossRef] - Chen, J.; Huang, Z.; Li, A.; Gao, R.; Jiang, W.; Xi, G. Numerical simulation of carbon separation with shock waves and phase change in supersonic separators. Process. Saf. Environ.
**2023**, 170, 277–285. [Google Scholar] [CrossRef] - Heydar Rajaee Shooshtari, S.; Honoré Walther, J.; Wen, C. Combination of genetic algorithm and CFD modelling to develop a new model for reliable prediction of normal shock wave in supersonic flows contributing to carbon capture. Sep. Purif. Technol.
**2022**, 309, 122878. [Google Scholar] [CrossRef] - Munz, C.D.; Roller, S.; Klein, R.; Geratz, K.J. The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids
**2003**, 32, 173–196. [Google Scholar] [CrossRef] - Kajzer, A.; Pozorski, J. A weakly Compressible, Diffuse-Interface Model for Two-Phase Flows. Flow Turbul. Combust.
**2020**, 105, 299–333. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Nguyen, N.T.; Phan, T.-H.; Park, W.-G. Efficient three-equation two-phase model for free surface and water impact flows on a general curvilinear body-fitted grid. Comput. Fluids
**2019**, 196, 104324. [Google Scholar] [CrossRef] - Phan, T.-H.; Shin, J.-G.; Nguyen, V.-T.; Duy, T.-N.; Park, W.-G. Numerical analysis of an unsteady natural cavitating flow around an axisymmetric projectile under various free-stream temperature conditions. Int. J. Heat Mass. Tran.
**2021**, 164, 120484. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Ha, C.-T.; Park, W.-G. Multiphase Flow Simulation of Water-entry and -exit of axisymmetric bodies. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Diego, CA, USA, 15–21 November 2013. [Google Scholar]
- Phan, T.-H.; Kadivar, E.; Nguyen, V.-T.; el Moctar, O.; Park, W.-G. Thermodynamic effects on single cavitation bubble dynamics under various ambient temperature conditions. Phys. Fluids
**2022**, 34, 023318. [Google Scholar] [CrossRef] - Phan, T.-H.; Nguyen, V.-T.; Duy, T.-N.; Kim, D.-H.; Park, W.-G. Numerical study on simultaneous thermodynamic and hydrodynamic mechanisms of underwater explosion. Int. J. Heat Mass. Tran.
**2021**, 178, 121581. [Google Scholar] [CrossRef] - Kunz, R.F.; Boger, D.A.; Stinebring, D.R.; Chyczewski, T.S.; Lindau, J.W.; Gibeling, H.J.; Venkateswaran, S.; Govindan, T.R. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction. Comput. Fluids
**2000**, 29, 849–875. [Google Scholar] [CrossRef] - Kunz, R.F.; Boger, D.A.; Chyczewski, T.S.; Stinebring, D.; Gibeling, H.; Govindan, T. Multi-phase CFD analysis of natural and ventilated cavitation about submerged bodies. In Proceedings of the 3rd ASME-JSME Joint Fluids Engineering Conference, San Francisco, CA, USA, 18–23 July 1999; Volume 99. [Google Scholar]
- Owis, F.M.; Nayfeh, A.H. Numerical simulation of 3-D incompressible, multi-phase flows over cavitating projectiles. Eur. J. Mech. B-Fluid.
**2004**, 23, 339–351. [Google Scholar] [CrossRef] - de Jouëtte, C.; Laget, O.; Le Gouez, J.M.; Viviand, H. A dual time stepping method for fluid–structure interaction problems. Comput. Fluids
**2002**, 31, 509–537. [Google Scholar] [CrossRef] - Helluy, P.; Golay, F.; Caltagirone, J.-P.; Lubin, P.; Vincent, S.; Drevard, D.; Marcer, R.; Fraunié, P.; Seguin, N.; Grilli, S.; et al. Numerical simulations of wave breaking. ESAIM Math. Model. Numer. Anal. Model. Math. Anal. Numer.
**2005**, 39, 591–607. [Google Scholar] [CrossRef] - Nourgaliev, R.R.; Dinh, T.N.; Theofanous, T.G. A pseudocompressibility method for the numerical simulation of incompressible multifluid flows. Int. J. Multiphas Flow.
**2004**, 30, 901–937. [Google Scholar] [CrossRef] - Ha, C.-T.; Lee, J.H. A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows. Comput. Fluids
**2020**, 197, 104345. [Google Scholar] [CrossRef] - Metcalf, B.; Longo, J.; Ghosh, S.; Stern, F. Unsteady free-surface wave-induced boundary-layer separation for a surface-piercing NACA 0024 foil: Towing tank experiments. J. Fluid. Struct.
**2006**, 22, 77–98. [Google Scholar] [CrossRef] - GeolLee, C.; YoubLee, S.; Ha, C.-T.; HwaLee, J. Bursting jet in two tandem bubbles at the free surface. Phys. Fluids
**2022**, 34, 083309. [Google Scholar] - Alotaibi, H.; Abeykoon, C.; Soutis, C.; Jabbari, M. Numerical Simulation of Two-Phase Flow in Liquid Composite Moulding Using VOF-Based Implicit Time-Stepping Scheme. J. Compos. Sci.
**2022**, 6, 330. [Google Scholar] [CrossRef] - Nguyen, V.-T.; Phan, T.-H.; Duy, T.-N.; Park, W.-G. Numerical simulation of supercavitating flow around a submerged projectile near a free surface. In Proceedings of the 11th International Symposium on Cavitation, Daejon, Korea, 10–13 May 2021. [Google Scholar]
- Merkle, C.L.; Feng, J.; Buelow, P. Computational modeling of the dynamics of sheet cavitation. In Proceedings of the Third International Symposium on Cavitation, Grenoble, France, 7–10 April 1998. [Google Scholar]
- Schnerr, G.H.; Sauer, J. Physical and numerical modeling of unsteady cavitation dynamics. In Proceedings of the Fourth International Conference on Multiphase Flow, New Orleans, LA, USA, 27 May–1 June 2001. [Google Scholar]
- Ullas, P.K.; Chatterjee, D.; Vengadesan, S. Prediction of unsteady, internal turbulent cavitating flow using dynamic cavitation model. Int. J. Numer. Methods Heat
**2022**, 32, 3210–3232. [Google Scholar] [CrossRef] - Singhal, A.K.; Athavale, M.M.; Li, H.; Jiang, Y. Mathematical Basis and Validation of the Full Cavitation Model. J. Fluids Eng.
**2002**, 124, 617–624. [Google Scholar] [CrossRef] - 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–4 June 2004. [Google Scholar]
- Tauviqirrahman, M.; Jamari, J.; Susilowati, S.; Pujiastuti, C.; Setiyana, B.; Pasaribu, A.H.; Ammarullah, M.I. Performance Comparison of Newtonian and Non-Newtonian Fluid on a Heterogeneous Slip/No-Slip Journal Bearing System Based on CFD-FSI Method. Fluids
**2022**, 7, 225. [Google Scholar] [CrossRef] - Hejranfar, K.; Ezzatneshan, E.; Fattah-Hesari, K. A comparative study of two cavitation modeling strategies for simulation of inviscid cavitating flows. Ocean. Eng.
**2015**, 108, 257–275. [Google Scholar] [CrossRef] - Lindau, J.W.; Kunz, R.F.; Boger, D.A.; Stinebring, D.R.; Gibeling, H.J. High Reynolds number, unsteady, multiphase CFD modeling of cavitating flows. J. Fluid. Eng.
**2002**, 124, 607–616. [Google Scholar] [CrossRef] - Lindau, J.W.; Kunz, R.F.; Mulherin, J.M.; Dreyer, J.J.; Stinebring, D.R. Fully coupled, 6-DOF to URANS, modeling of cavitating flows around a supercavitating vehicle. In Proceedings of the Fifth International Symposium on Cavitation (CAV2003), Osaka, Japan, 1–4 November 2003. [Google Scholar]
- Vrionis, Y.-P.G.; Samouchos, K.D.; Giannakoglou, K.C. Implementation of a conservative cut-cell method for the simulation of two-phase cavitating flows. In Proceedings of the 10th International Conference on Computational Methods (ICCM2019), Singapore, 9–13 July 2019; pp. 440–452. [Google Scholar]
- Coutier-Delgosha, O.; Fortes-Patella, R.; Reboud, J.L.; Hakimi, N.; Hirsch, C. Stability of preconditioned Navier–Stokes equations associated with a cavitation model. Comput. Fluids
**2005**, 34, 319–349. [Google Scholar] [CrossRef] - Hejranfar, K.; Fattah-Hesary, K. Assessment of a central difference finite volume scheme for modeling of cavitating flows using preconditioned multiphase Euler equations. J. Hydrodyn. Ser. B.
**2011**, 23, 302–313. [Google Scholar] [CrossRef] [Green Version] - Hajihassanpour, M.; Hejranfar, K. A high-order nodal discontinuous Galerkin method to solve preconditioned multiphase Euler/Navier-Stokes equations for inviscid/viscous cavitating flows. Int. J. Numer. Methods Fluids
**2020**, 92, 478–508. [Google Scholar] [CrossRef] - Tian, B.; Chen, J.; Zhao, X.; Zhang, M.; Huang, B. Numerical analysis of interaction between turbulent structures and transient sheet/cloud cavitation. Phys. Fluids
**2022**, 34, 047116. [Google Scholar] [CrossRef] - Kinzel, M.P.; Lindau, J.W.; Kunz, R.F. Gas entrainment from gaseous supercavities: Insight based on numerical simulation. Ocean. Eng.
**2021**, 221, 108544. [Google Scholar] [CrossRef] - Saurel, R.; Pantano, C. Diffuse-Interface Capturing Methods for Compressible Two-Phase Flows. Annu. Rev. Fluid. Mech.
**2018**, 50, 105–130. [Google Scholar] [CrossRef] [Green Version] - Weiss, J.M.; Smith, W.A. Preconditioning applied to variable and constant density flows. AIAA J.
**1995**, 33, 2050–2057. [Google Scholar] [CrossRef] - Venkateswaran, S.; Lindau, J.W.; Kunz, R.F.; Merkle, C.L. Computation of multiphase mixture flows with compressibility effects. J. Comput. Phys.
**2002**, 180, 54–77. [Google Scholar] [CrossRef] - Lindau, J.; Kunz, R.; Venkateswaran, S.; Merkle, C. Development of a fully-compressible multi-phase Reynolds-averaged Navier-Stokes model. In Proceedings of the 15th AIAA Computational Fluid, Dynamics Conference, Anaheim, CA, USA, 11–14 June 2001. [Google Scholar]
- Braconnier, B.; Nkonga, B. An all-speed relaxation scheme for interface flows with surface tension. J. Comput. Phys.
**2009**, 228, 5722–5739. [Google Scholar] [CrossRef] - Pelanti, M. Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model. Appl. Math. Comput.
**2017**, 310, 112–133. [Google Scholar] [CrossRef] - Murrone, A.; Guillard, H. Behavior of upwind scheme in the low Mach number limit: III. Preconditioned dissipation for a five equation two phase model. Comput. Fluids
**2008**, 37, 1209–1224. [Google Scholar] [CrossRef] [Green Version] - Gupta, A. Preconditioning Methods for Ideal and Multiphase Fluid Flows. Ph.D. Thesis, The University of Tennessee at Chattanooga, Chattanooga, TN, USA, 2013. [Google Scholar]
- Yoon, S.; Jameson, A. Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations. AIAA J.
**1988**, 26, 1025–1026. [Google Scholar] [CrossRef] [Green Version] - Housman, J.A.; Kiris, C.C.; Hafez, M.M. Time-Derivative Preconditioning Methods for Multicomponent Flows—Part I: Riemann Problems. J. Appl. Mech.
**2009**, 76, 021210. [Google Scholar] [CrossRef] - Ha, C.-T.; Park, W.-G. Evaluation of a new scaling term in preconditioning schemes for computations of compressible cavitating and ventilated flows. Ocean. Eng.
**2016**, 126, 432–466. [Google Scholar] [CrossRef] - Kim, H.; Kim, C. A physics-based cavitation model ranging from inertial to thermal regimes. Int. J. Heat Mass. Tran.
**2021**, 181, 121991. [Google Scholar] [CrossRef] - Yoo, Y.-L.; Sung, H.-G. A hybrid AUSM scheme (HAUS) for multi-phase flows with all Mach numbers. Comput. Fluids
**2021**, 227, 105050. [Google Scholar] [CrossRef] - Kadioglu, S.Y.; Sussman, M.; Osher, S.; Wright, J.P.; Kang, M. A second order primitive preconditioner for solving all speed multi-phase flows. J. Comput. Phys.
**2005**, 209, 477–503. [Google Scholar] [CrossRef] - Shin, B.R.; Yamamoto, S.; Yuan, X. Application of Preconditioning Method to Gas-Liquid Two-Phase Flow Computations. J. Fluids Eng.
**2004**, 126, 605–612. [Google Scholar] [CrossRef] - Goncalves, E.; Patella, R.F. Numerical simulation of cavitating flows with homogeneous models. Comput. Fluids
**2009**, 38, 1682–1696. [Google Scholar] [CrossRef] [Green Version] - Phan, T.-H.; Nguyen, V.-T.; Duy, T.-N.; Kim, D.-H.; Park, W.-G. Influence of phase-change on the collapse and rebound stages of a single spark-generated cavitation bubble. Int. J. Heat Mass. Tran.
**2022**, 184, 122270. [Google Scholar] [CrossRef] - Kinzel, M.P.; Lindau, J.W.; Kunz, R.F. A multiphase level-set approach for all-Mach numbers. Comput. Fluids
**2018**, 167, 1–16. [Google Scholar] [CrossRef] - Cassidy, D.A.; Edwards, J.R.; Tian, M. An investigation of interface-sharpening schemes for multi-phase mixture flows. J. Comput. Phys.
**2009**, 228, 5628–5649. [Google Scholar] [CrossRef] - Kakumanu, N.; Edwards, J.R.; Choi, J.-I. Numerical Simulation of Underwater Burst Events Using Sharp Interface Capturing Methods. In Proceedings of the AIAA Propulsion and Energy 2019 Forum, Indianapolis, IN, USA, 19–22 August 2019; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2019. [Google Scholar]
- Gouin, C.; Junqueira-Junior, C.; Goncalves Da Silva, E.; Robinet, J.-C. Numerical investigation of three-dimensional partial cavitation in a Venturi geometry. Phys. Fluids
**2021**, 33, 063312. [Google Scholar] [CrossRef] - Yoo, Y.-L.; Kim, J.-C.; Sung, H.-G. Homogeneous mixture model simulation of compressible multi-phase flows at all Mach number. Int. J. Multiphas Flow.
**2021**, 143, 103745. [Google Scholar] [CrossRef] - Huber, G.; Tanguy, S.; Béra, J.-C.; Gilles, B. A time splitting projection scheme for compressible two-phase flows. Application to the interaction of bubbles with ultrasound waves. J. Comput. Phys.
**2015**, 302, 439–468. [Google Scholar] [CrossRef] [Green Version] - LeMartelot, S.; Nkonga, B.; Saurel, R. Liquid and liquid–gas flows at all speeds. J. Comput. Phys.
**2013**, 255, 53–82. [Google Scholar] [CrossRef] - Meng, H.; Yang, V. A unified treatment of general fluid thermodynamics and its application to a preconditioning scheme. J. Comput. Phys.
**2003**, 189, 277–304. [Google Scholar] [CrossRef] - Huang, D.; Wang, Q.; Meng, H. Modeling of supercritical-pressure turbulent combustion of hydrocarbon fuels using a modified flamelet-progress-variable approach. Appl. Therm. Eng.
**2017**, 119, 472–480. [Google Scholar] [CrossRef] - Neaves, M.D.; Edwards, J.R. All-Speed Time-Accurate Underwater Projectile Calculations Using a Preconditioning Algorithm. J. Fluids Eng.
**2005**, 128, 284–296. [Google Scholar] [CrossRef] - De Wilde, J.; Heynderickx, G.J.; Vierendeels, J.; Dick, E.; Marin, G.B. An extension of the preconditioned advection upstream splitting method for 3D two-phase flow calculations in circulating fluidized beds. Comput. Chem. Eng.
**2002**, 26, 1677–1702. [Google Scholar] [CrossRef] - De Wilde, J.; Vierendeels, J.; Heynderickx, G.J.; Marin, G.B. Simultaneous solution algorithms for Eulerian–Eulerian gas–solid flow models: Stability analysis and convergence behaviour of a point and a plane solver. J. Comput. Phys.
**2005**, 207, 309–353. [Google Scholar] [CrossRef]

**Figure 2.**Simulation of a 3D falling water column in a container (This figure has been adapted from [21]).

**Figure 3.**Simulated and experimental free-surface wave profiles around a surface-piercing NACA0024 foil (This figure has been adapted from [1]).

**Figure 4.**Water entry of an oblique cylinder (This figure has been adapted from [1]).

**Figure 5.**Simulation of cavitating flows over a hemispherical body with different angles of attack (This figure has been adapted from [2]).

**Figure 6.**Cavitation evolution during oblique water exits of a projectile (This figure has been adapted from [20]).

**Figure 7.**Cavitation about a 3D NACA66 hydrofoil fixed at an 8° angle of attack (This figure has been adapted from [63]).

**Figure 8.**Ventilated cavitating flow over a body at Fr

_{N}= 26.7, Re

_{N}= 56,000 (This figure has been adapted from [64]).

**Figure 9.**The contours of pressure, density, and temperature for a transonic flow over a projectile, where the free stream velocity is 1540 m/s (This figure has been adapted from [36]).

**Figure 10.**Thermodynamic effects on single cavitation bubble dynamics (This figure has been adapted from [37]).

**Figure 11.**Evolution of vapor volume fraction in an aerated-liquid injector. Numerical computation using the preconditioning method and interface sharpening technique (This figure has been adapted from [83]).

**Figure 12.**Dynamics of partial cavitation developing in a 3D Venturi geometry and the interaction with sidewalls. Time-averaged comparison at midspan at stations S1 (top) and S2 (bottom) (This figure has been adapted from [85]).

**Figure 13.**Shock/water-column interaction problem. The z-vorticity and vorticity magnitude contour results at 194 μs (This figure has been adapted from [86]).

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Nguyen, V.-T.; Park, W.-G.
A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows. *Fluids* **2023**, *8*, 100.
https://doi.org/10.3390/fluids8030100

**AMA Style**

Nguyen V-T, Park W-G.
A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows. *Fluids*. 2023; 8(3):100.
https://doi.org/10.3390/fluids8030100

**Chicago/Turabian Style**

Nguyen, Van-Tu, and Warn-Gyu Park.
2023. "A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows" *Fluids* 8, no. 3: 100.
https://doi.org/10.3390/fluids8030100