Numerical Methods for Two-Phase Jet Flow

Two numerical methods are developed and analyzed for studying two-phase jet flows. The first numerical method solves the eigenvalue problem for the matrix system that is constructed from the pseudo-spectral discretization of the 3D linearized, incompressible, perturbed Navier-Stokes (N-S) equations for two-phase flows. This first numerical method will be denoted as LSA for “linear stability analysis.” The second numerical method solves the 3D (nonlinear) N-S equations for incompressible, two-phase flows. The second numerical method will be denoted as DNS for “direct numerical simulation.” In this thesis, predictions of jet-stability using the LSA method are compared with the predictions using DNS. Researchers have not previously compared LSA with DNS for the co-flowing two-phase jet problem. Researchers have only recently validated LSA with DNS for the simpler RayleighCapillary stability problem [77] [20] [103] [26]. In this thesis, a DNS method has been developed for cylindrical coordinate systems. Researchers have not previously simulated 3D, two-phase, jet flow, in cylindrical coordinate systems. The numerical predictions for jet flow are compared: (1) LSA with DNS (2) DNS-CLSVOF with DNS-LS, and (3) 3D rectangular with 3D cylindrical. “DNS-CLSVOF” denotes the coupled level set and volumeof-fluid method for computing solutions to incompressible two-phase flows [99]. “DNS-LS” denotes a novel hybrid level set and volume constraint method for simulating incompressible two-phase flows [89]. The following discoveries have been made in this thesis: (1) the DNS-CLSVOF method and the DNS-LS method both converge under grid refinement to the same results for predicting the break-up of a liquid jet before and after break-up; (2) computing jet break-up in 3D cylindrical coordinate systems is more efficient than computing jet breakup in 3D rectangular coordinate systems; and (3) the LSA method agrees with the DNS method for the initial growth of instabilities (comparison method made for classical Rayleigh-Capillary problem and co-flowing jet problem). It is found that for the classical Rayleigh-Capillary stability problem, the LSA prediction differs from the DNS prediction at later times.

[1]  M. Rudman.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 24, 671–691 (1997) VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS , 2022 .

[2]  Marcus Herrmann,et al.  A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids , 2008, J. Comput. Phys..

[3]  Chung-Jen Tam,et al.  Gaseous and Liquid Injection into High-Speed Crossflows , 2005 .

[4]  M. Arienti,et al.  Dynamics of Pulsed Jet in Crossflow , 2007 .

[5]  M. J. McCarthy,et al.  Review of stability of liquid jets and the influence of nozzle design , 1974 .

[6]  Mehdi R. Khorrami,et al.  A Chebyshev spectral collocation method using a staggered grid for the stability of cylindrical flows , 1991 .

[7]  Alfonso M. Gañán-Calvo,et al.  Linear stability of co-flowing liquid–gas jets , 2001, Journal of Fluid Mechanics.

[8]  Jack Dongarra,et al.  Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems , 1995 .

[9]  J. Lasheras,et al.  Liquid Jet Instability and Atomization in a Coaxial Gas Stream , 2000 .

[10]  M. Lessen,et al.  Stability of Pipe Poiseuille Flow , 1968 .

[11]  Stéphane Popinet,et al.  An accurate adaptive solver for surface-tension-driven interfacial flows , 2009, J. Comput. Phys..

[12]  F. Jaberi,et al.  Large-eddy simulation of a dispersed particle-laden turbulent round jet , 2008 .

[13]  T. N. Stevenson,et al.  Fluid Mechanics , 2021, Nature.

[14]  P. Weidman,et al.  Helical instability of a rotating viscous liquid jet , 2007 .

[15]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[16]  S. Zaleski,et al.  A Quasi-Direct 3D Simulation of the Atomization of High-Speed Liquid Jets , 2005 .

[17]  C. Grosch,et al.  Linear stability of Poiseuille flow in a circular pipe , 1980, Journal of Fluid Mechanics.

[18]  Raul Payri,et al.  Measurements of Spray Momentum for the Study of Cavitation in Diesel Injection Nozzles , 2003 .

[19]  Dvoralai Wulfsohn,et al.  PA—Precision Agriculture: Agricultural Sprays in Cross-flow and Drift , 2001 .

[20]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[21]  M. Arienti,et al.  Towards an Efficient, High-Fidelity Methodology for Liquid Jet Atomization Computations , 2010 .

[22]  Heinz Pitsch,et al.  An accurate conservative level set/ghost fluid method for simulating turbulent atomization , 2008, J. Comput. Phys..

[23]  Joseph B. Keller,et al.  Spatial instability of a jet , 1973 .

[24]  S. P. Lin,et al.  Role played by the interfacial shear in the instability mechanism of a viscous liquid jet surrounded by a viscous gas in a pipe , 1998, Journal of Fluid Mechanics.

[25]  Raoyang Zhang,et al.  A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability , 1998 .

[26]  J. Abraham,et al.  Dissipative-particle-dynamics model for two-phase flows. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[28]  Mark Sussman,et al.  A parallelized, adaptive algorithm for multiphase flows in general geometries , 2005 .

[29]  Zhu Guo-xiang.  Stability Analysis of Nonparallel Boundary Layers on Wings , 2005 .

[30]  Ronald Fedkiw,et al.  A Boundary Condition Capturing Method for Multiphase Incompressible Flow , 2000, J. Sci. Comput..

[31]  M. Malik.  Numerical methods for hypersonic boundary layer stability , 1990 .

[32]  Isidore Rigoutsos,et al.  An algorithm for point clustering and grid generation , 1991, IEEE Trans. Syst. Man Cybern..

[33]  Mark Sussman,et al.  A sharp interface method for incompressible two-phase flows , 2007, J. Comput. Phys..

[34]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part II: software and applications , 1993, TOMS.

[35]  Steven J. Ruuth,et al.  A Simple Scheme for Volume-Preserving Motion by Mean Curvature , 2003, J. Sci. Comput..

[36]  Jean-François Remacle,et al.  Transient adaptivity applied to two-phase incompressible flows , 2008, J. Comput. Phys..

[37]  T. Bridges,et al.  Differential eigenvalue problems in which the parameter appears nonlinearly , 1984 .

[38]  M. Arienti,et al.  Time-resolved proper orthogonal decomposition of liquid jet dynamics , 2009 .

[39]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

[40]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[41]  H. Seifert,et al.  Rocket Propulsion Elements , 1963 .

[42]  G. M. Faeth,et al.  Surface Properties During Primary Breakup of Turbulent Liquid Jets in Still Air , 2003 .

[43]  T. Colonius,et al.  Numerical Treatment of Polar Coordinate Singularities , 2000 .

[44]  Parviz Moin,et al.  The Primary Breakup of a Round Liquid Jet by a Coaxial Flow of Gas , 2007 .

[45]  A. D. Gosman,et al.  Large Eddy Simulation of Primary Diesel Spray Atomization , 2004 .

[46]  S. Tanguy,et al.  Application of a level set method for simulation of droplet collisions , 2005 .

[47]  Sébastien Tanguy,et al.  A Level Set Method for vaporizing two-phase flows , 2007, J. Comput. Phys..

[48]  Matthew W. Williams,et al.  A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework , 2006, J. Comput. Phys..

[49]  Osamu Tatebe,et al.  The multigrid preconditioned conjugate gradient method , 1993 .

[50]  Jean-François Remacle,et al.  A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows , 2006, J. Comput. Phys..

[51]  P. Spalart,et al.  Linear and nonlinear stability of the Blasius boundary layer , 1992, Journal of Fluid Mechanics.

[52]  David P. Schmidt,et al.  Adaptive tetrahedral meshing in free-surface flow , 2005 .

[53]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[54]  Robert L. Ash,et al.  Application of spectral collocation techniques to the stability of swirling flows , 1989 .

[55]  R. Reitz,et al.  DROP AND SPRAY FORMATION FROM A LIQUID JET , 1998 .

[56]  Heinz Pitsch,et al.  A spectrally refined interface approach for simulating multiphase flows , 2009, J. Comput. Phys..

[57]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I: theory and algorithms , 1993, TOMS.

[58]  Shiyi Chen,et al.  On the three-dimensional Rayleigh–Taylor instability , 1999 .

[59]  Christoph Hassa,et al.  Modelling of the atomization of a plain liquid fuel jet in crossflow at gas turbine conditions , 2002 .

[60]  Mark Sussman,et al.  An Improved Sharp Interface Method for Viscoelastic and Viscous Two-Phase Flows , 2008, J. Sci. Comput..

[61]  W. T. Rouleau,et al.  Linear spatial stability of pipe Poiseuille flow , 1972, Journal of Fluid Mechanics.

[62]  M. Sussman.  A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles , 2003 .

[63]  Djamel Lakehal,et al.  Direct numerical simulation of turbulence in a sheared air–water flow with a deformable interface , 2003, Journal of Fluid Mechanics.

[64]  E. Villermaux,et al.  Physics of liquid jets , 2008 .

[65]  Xiaofeng Yang,et al.  An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids , 2006, J. Comput. Phys..

[66]  J. Gordillo,et al.  Aerodynamic effects in the break-up of liquid jets: on the first wind-induced break-up regime , 2005, Journal of Fluid Mechanics.

[67]  Michael Gaster,et al.  A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability , 1962, Journal of Fluid Mechanics.

[68]  Manolis Gavaises,et al.  Investigation of Cavitation in a Vertical Multi-Hole Injector , 1999 .

[69]  S. Osher,et al.  Spatially adaptive techniques for level set methods and incompressible flow , 2006 .

[70]  Mark Sussman,et al.  A hybrid level set-volume constraint method for incompressible two-phase flow , 2012, J. Comput. Phys..

[71]  Brad Lee Holian,et al.  Nanohydrodynamics simulations: an atomistic view of the Rayleigh-Taylor instability. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[72]  Chester E. Grosch,et al.  The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions , 1981, Journal of Fluid Mechanics.

[73]  M. R. Malik,et al.  Nonparallel and nonlinear stability of supersonic jet flow , 2000 .

[74]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[75]  James A. Sethian,et al.  A coupled quadrilateral grid level set projection method applied to ink jet simulation , 2005 .

[76]  Gian Marco Bianchi,et al.  3D Large Scale Simulation of the High-Speed Liquid Jet Atomization , 2007 .

[77]  David L. Chopp,et al.  Some Improvements of the Fast Marching Method , 2001, SIAM J. Sci. Comput..

[78]  Marco Badami,et al.  Cavitation in real-size, multi-hole diesel injector nozzles , 2000 .

[79]  A. Kalaaji,et al.  Breakup length of forced liquid jets , 2003 .

[80]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

[81]  Said Elghobashi,et al.  A two‐equation turbulence model for two‐phase flows , 1983 .

[82]  P. Smereka,et al.  A Remark on Computing Distance Functions , 2000 .

[83]  何国威,et al.  A SECOND-ORDER DYNAMIC SUBGRID-SCALE STRESS MODEL , 2000 .

[84]  Mark Sussman,et al.  A Discontinuous Spectral Element Method for the Level Set Equation , 2003, J. Sci. Comput..

[85]  Emil J. Hopfinger,et al.  Break-up and atomization of a round water jet by a high-speed annular air jet , 1998, Journal of Fluid Mechanics.

[86]  C. Shiyi,et al.  A second-order dynamic subgrid-scale stress model , 2000 .

[87]  Zoltan Farago,et al.  MORPHOLOGICAL CLASSIFICATION OF DISINTEGRATION OF ROUND LIQUID JETS IN A COAXIAL AIR STREAM , 1992 .

[88]  Enrico O. Purisima,et al.  A new tetrahedral tesselation scheme for isosurface generation , 1998, Comput. Graph..

[89]  V. Theofilis.  Advances in global linear instability analysis of nonparallel and three-dimensional flows , 2003 .

[90]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[91]  R. Reitz,et al.  Mechanism of atomization of a liquid jet , 1982 .

[92]  Ludovic Raynal,et al.  Instabilité et entrainement à l'interface d'une couche de mélange liquide-gaz , 1997 .

[93]  Pierre Saramito,et al.  Improving the mass conservation of the level set method in a finite element context , 2010 .

[94]  Olivier Boutin,et al.  Laminar jet dispersion and jet atomization in pressurized carbon dioxide , 2005 .

[95]  Gunilla Kreiss,et al.  A conservative level set method for two phase flow II , 2005, J. Comput. Phys..

[96]  R. Donnelly,et al.  Experiments on the capillary instability of a liquid jet , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[97]  M. Sussman,et al.  A Coupled Level Set and Volume-of-Fluid Method for Computing 3D and Axisymmetric Incompressible Two-Phase Flows , 2000 .

[98]  S. P. Lin.  Breakup of liquid sheets and jets , 2003 .

[99]  Chester E. Grosch,et al.  The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions , 1976, Journal of Fluid Mechanics.

[100]  P. Woodward,et al.  SLIC (Simple Line Interface Calculation) , 1976 .

[101]  C. Sleicher,et al.  The instability of capillary jets , 1975, Journal of Fluid Mechanics.

[102]  R. G. Jacobs,et al.  SHEAR SHELTERING AND THE CONTINUOUS SPECTRUM OF THE ORR-SOMMERFELD EQUATION , 1998 .