Acta Mechanica

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

Two-dimensional modeling of viscous liquid jet breakup

Article

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.

List of Symbols

a

Unperturbed jet radius

a1, a2, . . . , a8

Coefficients used in grid velocity equations

c1, c2c3

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

Rd

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

Subscripts

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

Superscripts

n

Marching time

t

Transpose

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Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAssiut UniversityAssiutEgypt

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