Numerical simulation of non-viscous liquid pinch off using a coupled levelset boundary integral method

Maria Garzon1,∗, James A. Sethian2,∗∗, and Len Gray3,∗∗∗

1 Universidad de Oviedo, Calle Calvo Sotelo s/n. 33007 Oviedo. Spain.2 University of California, Berkeley, CA 94720, USA.3 Oak Ridge National Laboratory, Oak Ridge, TN 37831-6367, USA.

The pinch off of an inviscid fluid column is described using a potential flow model with capillary forces. The interface velocityis obtained via a Galerkin boundary integral method for the 3D axisymmetric Laplace equation, whereas the interface locationand the velocity potential on the free boundary are both approximated using level set techniques on a fixed domain. Thealgorithm is validated computing the Raleigh-Taylor instability for liquid columns which provides an analytical solution forshort times. The simulations show the time evolution of the fluid tube and the algorithm is capable of handling pinch-off andafter pinch-off events.

1 Introduction

The coupling of level set methods (LSM) [1] with boundary integral methods (BIM) can be used to approximate free boundaryproblems where the boundary condition for the BIM has to be obtained front the solution of a PDE defined on the front. Thisis the case of potential flow models with moving boundaries, see [2], [3]. Here we study surface tension driven potential flowsand we apply the method to simulate the pinch off of an inviscid fluid column.

2 The model equations

Consider an incompressible, inviscid and irrational flow. The model equations with axial symmetry around the z axis are:

u = ∇φ in Ωd(t)

∂2φ

∂r2+

∂2φ

∂z2+

1

r

∂φ

∂r= 0 in Ωd(t)

Rt = u on Γt(s)

Dφ

Dt=

1

2(∇φ · ∇φ) −

γ

ρ(

1

R1+

1

R2) on Γt(s)

φ |Γ1= φ |Γ2

,

where R(s, t) = (r(s, t), z(s, t)) is the position vector of a fluid particle on the front, u(r, z, t) is the fluid velocity fieldand φ(r, z, t) the velocity potential function. Ωd(t), Γt(s) denote the fluid domain and the free boundary in 2D. This is theEulerian-Lagrangian formulation of the model equations.

To embed the equations in a complete Eulerian framework, two new functions are defined over a fixed domain Ω1 thatcontains the free boundary for all times:

1. The level set function Ψ(r, z, t) which satisfies Ψ(R(s, t), t) = 0, ∀t.

2. The fictitious potential function G(r, z, t) on Ω1, with the property G(R(s, t), t) = φ(r, z, t) |Γt(s)= Φ(s, t), ∀t.

Deriving both equations with respect to time we get: Ψt+u·∇Ψ = 0 on Γt(s) and DΦDt

= Gt+u·∇G = f on Γt(s),where f = 1

2 (∇φ · ∇φ) − γρκ, κ = ( 1

R1

+ 1R2

) and R1, R2 are the principal radii of curvature.Thus, the Eulerian formulation of the model equations is

u = ∇φ in Ω(t) (1)

∆φ = 0 in Ω(t) (2)

Ψt + uext · ∇Ψ = 0 in Ω1 (3)

Gt + uext · ∇G = fext in Ω1 (4)

φ |Γ1= φ |Γ2

(5)

∗ Corresponding author E-mail: maria@math.lbl.gov, Phone: +001 510 486 5296, Fax: +001 510 486 6199∗∗ sethian@Math.Berkeley.EDU∗∗∗ (865) 574-8189, ljg@ornl.gov

PAMM · Proc. Appl. Math. Mech. 7, 1024801–1024802 (2007) / DOI 10.1002/pamm.200700180

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Here uext, fext on Ω1 are defined such that uext |Γt(s)= u(R(s, t), t) and fext |Γt(s)= f(R(s, t), t).

3 Numerical approximation and results

The integral formulation of equation (2) is solved using a BEM with an axisymmetric Galerkin approximation of the BI [5],with linear and quadratic shape functions [4], and an interface velocity obtained by postprocessing the BI solution ∂φ

∂non the

interface using a Galerkin technique. For equations (3) and (4), first or second order upwind finite differences in space andfirst order in time are used. To validate our numerical results we used the analytical solution of the linearized model, theso-called Raleigh-Taylor instability. For the initial perturbation, we choose a cosine wave with wave number λ = 0.698 whichcorresponds to the maximum growth rate. Agreement with the analytical solution is very good. We continued the simulationnear pinch off and after pinch off and some of the profiles for various times are presented in the following figures.

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Acknowledgements This work was supported by the Director, Office of Science, Computational and Technology Research, U.S. Depart-ment of Energy under Contract No. DE-AC02-05CH11231. First author was also supported by the Spanish project MTM2004-05417.

References

[1] J.A. Sethian, Level set methods and fast marching methods: Evolving intefaces in computational geometry, fluid mechanics, computervision, and material science. (Cambridge University press, Cambridge 1999).

[2] M. Garzon, D. Adalsteinsson, L. J Gray and J. A Sethian, Interfaces and Free Boundaries 7, 277-302 (2005).[3] M. Garzon, J.A. Sethian, International Series of Numerical methods 154, 189-198, (2006).[4] L. J. Gray, M. Garzon, Computers and Structures 83, 889-894 (2005).[5] G J Gray, M. Garzon, V. Mantic, and E. Graciani, International Journal For Numerical Methods in Engineering 66, 2014-2034, (2005).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Minisymposia – 02 Numerical Analysis 1024802

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