Acta Mechanica

, Volume 224, Issue 3, pp 499–512 | Cite as

Two-dimensional modeling of viscous liquid jet breakup



A two-dimensional computational model has been developed to study the evolution and breakup of a viscous laminar liquid jet, using a boundary-fitted curvilinear coordinate system. A system of elliptic partial differential equations for coordinate transformations has been developed to map the moving boundaries’ physical domain of the jet to a simple rectilinear computational domain. The equations developed for the model comprise the transformed two-dimensional unsteady Navier–Stokes equations for the liquid jet, grid velocity equations, kinematic boundary conditions, and the Geometric Conservation Law. The resulting systems of equations are solved using an implicit finite difference scheme. Effects of inflow oscillation magnitude, wave number, Weber number, and Reynolds number on the breakup process of jets have been studied. The model predicts the instantaneous shape of the jet surface, formation of the main and satellite drops, and the breakup length and time. These results are compared with available experimental data. The comparisons show a good agreement between measured and computed values of drop sizes and breakup lengths for different Reynolds and Weber numbers. However, at a relatively high Reynolds number of 1,254, the model slightly overpredicts the main drop sizes and underpredicts the satellite drop sizes at a wave number of 0.4. At a low Reynolds number of 587, the model overpredicts the main drop sizes at a lower wave number of 0.3. Moreover, the model underpredicts the satellite drop sizes at a lower wave number of about 0.4 and overpredicts the satellite drop sizes at a wave number of 0.8.

List of Symbols


Unperturbed jet radius

a1, a2, . . . , a8

Coefficients used in grid velocity equations

c1, c2c3

Coefficients used in governing equations of fluid flow


Diameter of drops

f, g

Flux vectors


Perturbed jet radius


Dimensionless perturbed jet radius (h/a)


Jacobian of the coordinate transformation


Dimensionless surface tension parameter


Dimensionless oscillation wave number


Nozzle length


Outward unit normal vector on the boundary surface

P(η, ξ)

ξ-forcing function for grid and grid velocity equations




Dimensionless pressure \({(p{/}\rho{v}_{m}^{2})}\)


Primitive variable vector


Conserved variable vector

R(η, ξ)

η-forcing function for grid and grid velocity equations


Dimensionless drop radius (d/2a)


Reynolds number (a v m ρ/μ)


Radial direction


Dimensionless variable (r/a)


Source term in curvilinear coordinate


Source term vector




Dimensionless time (t v m /a)


Contravariant velocity component


Velocity component in r direction


Dimensionless velocity component in r-direction (u/v m )


Contravariant velocity component


Velocity component in z direction


Dimensionless velocity component in z-direction (v/v m )


Weber number (a \({{v}_{m}^{2}\rho/\sigma)}\)


Axial direction


Dimensionless parameter (z/a)


Dimensionless Ohnesorge number [μ/(ρσa)0.5]

α, β, γ

Coefficients used in grid generation equations

η, ξ, τ

General curvilinear coordinates (computational coordinates)


Dimensionless oscillation magnitude


Dynamic viscosity


Kinematic viscosity


Surface tension


i, j

Marching indices




Free surface


Viscous flux

t, r, z, η, ξ, τ

Partial differentiation with respect to each respective variable


Increment value



Marching time




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hilbing J.H., Heister S.D., Spangler C.A.: A boundary-element method for atomization of a finite liquid jet. At. Sprays 5, 621–638 (1995)Google Scholar
  2. 2.
    Hilbing J.H., Heister S.D.: Droplet size control in liquid jet breakup. Phys. Fluids 8(6), 1574–1581 (1996)MATHCrossRefGoogle Scholar
  3. 3.
    Hilbing J.H., Heister S.D.: Nonlinear simulation of a high-speed, viscous liquid jet. At. Sprays 8, 155–178 (1998)Google Scholar
  4. 4.
    Rump, K.M.: Modeling the Effect of Unsteady Chamber Conditions on Atomization Process. M.S. Thesis, Purdue University, W. Lafayette, IN, USA (1996)Google Scholar
  5. 5.
    Hilbing, J.H.: Nonlinear Modeling of Atomization Process. Ph.D. Thesis, Purdue University, W. Lafayette, IN, USA (1996)Google Scholar
  6. 6.
    Sirignano W.A., Methring M.: Review of theory of distortion and disintegration of liquid streams. Prog. Energy Combust. Sci. 26, 609–655 (2000)CrossRefGoogle Scholar
  7. 7.
    Moses, M.P.: Visualization of Liquid Jet Breakup and Droplet Formation. M.S. thesis, Purdue University, USA (1995)Google Scholar
  8. 8.
    Scardovelli R., Zaleski S.: Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Maronnier V., Picasso M., Rappaz J.: Numerical simulation of free surface flows. J. Comp. Phys. 155, 439–455 (1999)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Sellens R.W.: A one-dimensional numerical model of the capillary instability. At. Sprays 2(4), 239–251 (1992)Google Scholar
  11. 11.
    Lundgren T.S., Mansour N.N.: Oscillations of drops in zero gravity with weak viscous effect. J. Fluid Mech. 194, 479–510 (1988)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Setiawan E.R., Heister S.D.: Nonlinear modeling of an infinite electrified jet. J. Electrostat. 42, 243–257 (1997)CrossRefGoogle Scholar
  13. 13.
    Hirt C.W., Nichols B.D.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201 (1981)MATHCrossRefGoogle Scholar
  14. 14.
    Floryan J.M., Rasmussen H.: Numerical methods for viscous flows with moving boundaries. Appl. Mech. Rev. 42(12), 323–341 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eggers J., Dupont T.F.: Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205 (1994)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Brenner M.P., Eggers J., Joseph K., Nagel S.R., Shi X.D.: Breakdown of scaling in droplet fission at high Reynolds number. Phys. Fluids 9, 1573 (1997)CrossRefGoogle Scholar
  17. 17.
    Hanchak, M.S.: One Dimensional Model of Thermo-Capillary Driven Liquid Jet Break-up with Drop Merging. Ph.D. Thesis, University of Dayton, Dayton, Ohio, USA (2009)Google Scholar
  18. 18.
    Ahmed M., Abou Al-Sood M.M., Ali A.: A one dimensional model of viscous liquid jets breakup. ASME J. Fluid Eng. 133, 11450 (2011)CrossRefGoogle Scholar
  19. 19.
    Ambravaneswaran B., Wilkes E.D., Basaran O.A.: Drop formation from a capillary tube: comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys. Fluids 14(8), 2606–2621 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Eggers J.: Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929 (1997)MATHCrossRefGoogle Scholar
  21. 21.
    Adams R.L., Roy J.: A one dimensional numerical model of a drop-on-demand ink jet. J. Appl. Mech. 53, 193–197 (1986)CrossRefGoogle Scholar
  22. 22.
    Dravid V., Songsermpong S., Xue Z., Corvalan C.M., Sojka P.E.: Two-dimensional modeling of the effects of insoluble surfactant on the breakup of a liquid filament. Chem. Eng. Sci. 61, 3577–3585 (2006)CrossRefGoogle Scholar
  23. 23.
    Desjardins O., Moureau V., Pitsch H., Sch H., Tsch H., Pitsch V.: Set/ghost fluid method for simulating turbulent atomization. J. Comput. Phys. 227, 8395–8416 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Yu Pan Y., Suga K.: A numerical study on the breakup process of laminar liquid jets into a gas. Phys. Fluids 18, 052–101 (2006)Google Scholar
  25. 25.
    Gorokhovski M., Herrmann M.: Modeling primary atomization. Annu. Rev. Fluid Mech. 40, 343:34 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Debuc L., Cantariti F., Woodgate M., Gribben B., Badcock K.J., Richards B.E.: A grid deformation technique for unsteady flow computations. Int. J. Numer. Methods Fluids 32, 285–311 (2000)CrossRefGoogle Scholar
  27. 27.
    Demirdzic I., Peric M.: Space conservation law in finite volume calculations of fluid flow. Int. J. Numer. Methods Fluids 8, 1037–1050 (1988)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Demirdzic I., Peric M.: Finite volume methods for prediction of fluid flow in arbitrarily shaped domains with moving boundaries. Int. J. Numer. Methods Fluids 10, 771–779 (1990)MATHCrossRefGoogle Scholar
  29. 29.
    Thomas P.D., Lombard C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Thompson J.F., Thames F.C., Mastin C.W.: TOMCAT-a code for numerical generation of boundary-fitted curvilinear coordinate systems on field containing any number of arbitrary two dimensional bodies. J. Comp. Phys. 50, 316–321 (1983)CrossRefGoogle Scholar
  31. 31.
    Uchikawa S.: Generation of boundary- fitted curvilinear coordinate systems for a two dimensional axisymmetric flow problem. J. Comp. Phys. 99, 39–55 (1992)CrossRefGoogle Scholar
  32. 32.
    Thompson J.F.: Numerical Solution of Flow Problem Using Body-Fitted Coordinate system for a Two Dimensional Axisymmetric Flow Problem, pp. 1–98. Computational Fluid Dynamics, Hemisphere, Washington (1980)Google Scholar
  33. 33.
    Thompson J.F., Warsi Z.U.A., Mastin C.W.: Numerical Grid Generation, Foundation and Applications. North- Holland, New York (1985)Google Scholar
  34. 34.
    Christodoulou K.N., Scriven L.E.: Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99, 39–55 (1992)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    The J.L., Raithby G.D., Stubley G.D.: Surface-adaptive finite-volume method for solving free surface flows. Numer. Heat Transf. Part B 26, 367–380 (1994)CrossRefGoogle Scholar
  36. 36.
    Hindman R.G., Kulter P., Anderson D.: Two-dimensional unsteady euler-equation solver for arbitrarily shaped flow regions. AIAA J. 19(4), 424–431 (1981)CrossRefGoogle Scholar
  37. 37.
    Ferziger J.H., Peric M.: Computational Methods for Fluid Dynamics. Springer, Berlin (1999)MATHCrossRefGoogle Scholar
  38. 38.
    Ashgriz N., Mashayek F.: Temporal analysis of capillary jet breakup. J. Fluid Mech. 291, 163–190 (1995)MATHCrossRefGoogle Scholar
  39. 39.
    Papageorgiou D.T.: On the breakup of viscous liquid threads. Phys. Fluids 7(7), 1529–1544 (1995)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Tjahjadi M., Stone H.A., Ottino J.M.: Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317 (1992)CrossRefGoogle Scholar
  41. 41.
    An-Cheng Ruo., Min-Hsing Chang., Falin Chen.: On the nonaxisymmetric instability of round liquid jets. Phys. Fluids 20, 062–105 (2008)Google Scholar
  42. 42.
    Shinjo J., Umemura A.: Simulation of liquid jet primary breakup: dynamics of ligament and droplet formation. Int. J. Multiph. Flow 36, 513–532 (2010)CrossRefGoogle Scholar
  43. 43.
    Gonzalez H., Garcia F.J.: The measurement of growth rates in capillary jets. J. Fluid Mech. 619, 179–212 (2009)MATHCrossRefGoogle Scholar
  44. 44.
    Spangler C.H., Hilbing J.H., Heister S.D.: Nonlinear modeling of jet atomization in the wind induced regime. Phys. Fluids 7(5), 964–971 (1995)MATHCrossRefGoogle Scholar
  45. 45.
    Moses M.P., Collicott S.H., Heister S.D.: Detection of aerodynamic effcts in liquid jet breakup and droplet formation. At. Sprays 9, 331–342 (1999)Google Scholar
  46. 46.
    Rutland D.E., Jameson G.J.: Theoretical prediction of the sizes of droplets formed in the breakup of capillary jets. Chem. Eng. Sci. 25, 1689–1698 (1970)CrossRefGoogle Scholar
  47. 47.
    Lafrance P.: Nonlinear break-up of a laminar liquid jet. Phys. Fluids 18, 428–432 (1975)MATHCrossRefGoogle Scholar
  48. 48.
    Mansour A., Chigier N.: Effect of turbulence on the stability of liquid jets and resulting droplet distribution. At. Sprays 4, 583–604 (1994)Google Scholar
  49. 49.
    Karasawa M., Tanaka M., Abe k., Shiga S., Kuraboyashi T.: Effect of nozzle configuration on the atomization of a steady spray. At. Sprays 2, 411–426 (1992)Google Scholar
  50. 50.
    Bousfied D.W., Stockel I.H.: The breakup of viscous jets with large velocity modulations. J. Fluid Mech. 218, 601–617 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAssiut UniversityAssiutEgypt

Personalised recommendations