Boundary Integral Methods
In the numerical simulation of free-surface flows a considerable simplification is possible in the opposite limits of very weak or very strong viscous effects. This chapter focuses for the most part on the former case and provides an overview of several b
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Abstract. In the numerical simulation of free-surface flows a considerable simplification is possible in the opposite limits of very weak or very strong viscous effects. This chapter focuses for the most part on the former case and provides an overview of several boundary integral methods that have been developed to handle it. The last section gives a brief description of the basis for boundary integral methods suitable for the Stokes equations.
The numerical calculation of general viscous free-surface ftows is a task well known for its difficulty as well as for its considerable practical importance. Several techniques have been developed to tackle it, the simplest ones of which are applicable when viscous effects are negligible or, at the opposite extreme, dominant. The former situation often arises during the impact of liquid masses, the oscillations of drops in air, the motion ofbubbles in low-viscosity liquids, and many others. The latter situation might occur, for instance, when a drop in an immiscible liquid slowly approaches a liquid/liquid interface, as in the instability of an emulsion due to drop coalescence. For the situations of concern in this chapter, the low-viscosity situation is of greater relevance and, accordingly, we shall devote most of this chapter to the inviscid case. Weak viscous effects are briefty described in section 5, and an introduction to the analogous methods when viscosity dominates is provided in section 6. Later, an overview of methods applicable at finite Reynolds numbers will be presented.
1 Governing equations The velocity field u of an incompressible fluid is determined by the momentum balance (NavierStokes equation): 2 1 8u (1) 8t +(u·V) u = -PVp+v'\1 u+g, where p is the density, v the kinematic viscosity, arid g the body force per unit mass; the pressure field p ensures that the ftow is incompressible (continuity equation):
V·u = 0.
(2)
Because of the general vector identities
(u · V) u = V ( ~u · u) + w x u,
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M. Rein (ed.), Drop-Surface Interactions © Springer-Verlag Wien 2002
(3)
A. Prosperetti
220 where w = V x u is the vorticity, and because of (2), an equivalent form of ( 1) is
(12
-au +V -u · u + -p + g · x )
at
P
= u
xw-vV xw,
(4)
V
(5)
from which the vorticity equations follows as
aw at
=
V
X
(u
X
w) - vV
X
X
w.
Since this equation has no sources, vorticity can only be generated at boundaries away from which it is transported by convection and viscous diffusion. For a transient flow characterized by a characteristic time T a simple scaling analysis shows that the characteristic thickness of this vortex layer is of the order of ..jliT, while w rapidly decreases further away. Thus, if the characteristic length ofthe problern (e.g., the drop radius) is L » ..jliT, vorticity is confined to a very small fraction of the fluid volume; outside this vorticallayer the flow admits a potential
u = Vc/J,
(6)
and the right-hand side of (4) vanishes so that
8cjJ
1
p
at + 2u · u + p+ g · x = F(t),
(7)
where F (t)
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