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Two-dimensional modeling of viscous liquid jet breakup

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Abstract

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.

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Abbreviations

a :

Unperturbed jet radius

a 1, a 2, . . . , a 8 :

Coefficients used in grid velocity equations

c 1, c 2, c 3 :

Coefficients used in governing equations of fluid flow

d :

Diameter of drops

f, g :

Flux vectors

h :

Perturbed jet radius

h*:

Dimensionless perturbed jet radius (h/a)

J :

Jacobian of the coordinate transformation

K :

Dimensionless surface tension parameter

k :

Dimensionless oscillation wave number

L :

Nozzle length

n :

Outward unit normal vector on the boundary surface

P(η, ξ):

ξ-forcing function for grid and grid velocity equations

p :

Pressure

p*:

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

Q :

Primitive variable vector

q :

Conserved variable vector

R(η, ξ):

η-forcing function for grid and grid velocity equations

R d :

Dimensionless drop radius (d/2a)

Re :

Reynolds number (a v m ρ/μ)

r :

Radial direction

r*:

Dimensionless variable (r/a)

S :

Source term in curvilinear coordinate

s :

Source term vector

t :

Time

t*:

Dimensionless time (t v m /a)

U :

Contravariant velocity component

u :

Velocity component in r direction

u*:

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

V :

Contravariant velocity component

v :

Velocity component in z direction

v*:

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

We :

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

z :

Axial direction

z*:

Dimensionless parameter (z/a)

Z :

Dimensionless Ohnesorge number [μ/(ρσa)0.5]

α, β, γ :

Coefficients used in grid generation equations

η, ξ, τ :

General curvilinear coordinates (computational coordinates)

ε :

Dimensionless oscillation magnitude

μ :

Dynamic viscosity

ν :

Kinematic viscosity

σ :

Surface tension

i, j :

Marching indices

m :

Mean

s :

Free surface

v :

Viscous flux

t, r, z, η, ξ, τ :

Partial differentiation with respect to each respective variable

Δ:

Increment value

n :

Marching time

t :

Transpose

References

  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. Hilbing J.H., Heister S.D.: Droplet size control in liquid jet breakup. Phys. Fluids 8(6), 1574–1581 (1996)

    Article  MATH  Google Scholar 

  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. Rump, K.M.: Modeling the Effect of Unsteady Chamber Conditions on Atomization Process. M.S. Thesis, Purdue University, W. Lafayette, IN, USA (1996)

  5. Hilbing, J.H.: Nonlinear Modeling of Atomization Process. Ph.D. Thesis, Purdue University, W. Lafayette, IN, USA (1996)

  6. Sirignano W.A., Methring M.: Review of theory of distortion and disintegration of liquid streams. Prog. Energy Combust. Sci. 26, 609–655 (2000)

    Article  Google Scholar 

  7. Moses, M.P.: Visualization of Liquid Jet Breakup and Droplet Formation. M.S. thesis, Purdue University, USA (1995)

  8. Scardovelli R., Zaleski S.: Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603 (1999)

    Article  MathSciNet  Google Scholar 

  9. Maronnier V., Picasso M., Rappaz J.: Numerical simulation of free surface flows. J. Comp. Phys. 155, 439–455 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sellens R.W.: A one-dimensional numerical model of the capillary instability. At. Sprays 2(4), 239–251 (1992)

    Google Scholar 

  11. Lundgren T.S., Mansour N.N.: Oscillations of drops in zero gravity with weak viscous effect. J. Fluid Mech. 194, 479–510 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Setiawan E.R., Heister S.D.: Nonlinear modeling of an infinite electrified jet. J. Electrostat. 42, 243–257 (1997)

    Article  Google Scholar 

  13. Hirt C.W., Nichols B.D.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201 (1981)

    Article  MATH  Google Scholar 

  14. Floryan J.M., Rasmussen H.: Numerical methods for viscous flows with moving boundaries. Appl. Mech. Rev. 42(12), 323–341 (1989)

    Article  MathSciNet  Google Scholar 

  15. Eggers J., Dupont T.F.: Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  20. Eggers J.: Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929 (1997)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Gorokhovski M., Herrmann M.: Modeling primary atomization. Annu. Rev. Fluid Mech. 40, 343:34 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  27. Demirdzic I., Peric M.: Space conservation law in finite volume calculations of fluid flow. Int. J. Numer. Methods Fluids 8, 1037–1050 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  31. Uchikawa S.: Generation of boundary- fitted curvilinear coordinate systems for a two dimensional axisymmetric flow problem. J. Comp. Phys. 99, 39–55 (1992)

    Article  Google Scholar 

  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. Thompson J.F., Warsi Z.U.A., Mastin C.W.: Numerical Grid Generation, Foundation and Applications. North- Holland, New York (1985)

    Google Scholar 

  34. Christodoulou K.N., Scriven L.E.: Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99, 39–55 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  37. Ferziger J.H., Peric M.: Computational Methods for Fluid Dynamics. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  38. Ashgriz N., Mashayek F.: Temporal analysis of capillary jet breakup. J. Fluid Mech. 291, 163–190 (1995)

    Article  MATH  Google Scholar 

  39. Papageorgiou D.T.: On the breakup of viscous liquid threads. Phys. Fluids 7(7), 1529–1544 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Tjahjadi M., Stone H.A., Ottino J.M.: Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317 (1992)

    Article  Google Scholar 

  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. Shinjo J., Umemura A.: Simulation of liquid jet primary breakup: dynamics of ligament and droplet formation. Int. J. Multiph. Flow 36, 513–532 (2010)

    Article  Google Scholar 

  43. Gonzalez H., Garcia F.J.: The measurement of growth rates in capillary jets. J. Fluid Mech. 619, 179–212 (2009)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  47. Lafrance P.: Nonlinear break-up of a laminar liquid jet. Phys. Fluids 18, 428–432 (1975)

    Article  MATH  Google Scholar 

  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. 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. Bousfied D.W., Stockel I.H.: The breakup of viscous jets with large velocity modulations. J. Fluid Mech. 218, 601–617 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mechanical Engineering Department, Assiut University, Assiut, 71516, Egypt

    M. Ahmed, M. Youssef & M. Abou-Al-Sood

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  1. M. Ahmed
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  2. M. Youssef
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  3. M. Abou-Al-Sood
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Correspondence to M. Ahmed.

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Ahmed, M., Youssef, M. & Abou-Al-Sood, M. Two-dimensional modeling of viscous liquid jet breakup. Acta Mech 224, 499–512 (2013). https://doi.org/10.1007/s00707-012-0766-1

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  • DOI: https://doi.org/10.1007/s00707-012-0766-1

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