Computers and Chemical Engineering 28 (2004) 1873–1879
Wavelets and fast summations for particle simulations of
gravitational flows of miscible drops
Ludwig C. Nitsche
a,∗
, Gunther Machu
b
, Walter Meile
c
a
Department of Chemical Engineering, University of Illinois at Chicago, 810 South Clinton Street, Chicago, IL 60607, USA
b
Hoerbiger Kompressortechnik Services, Braunhubergasse 23, A-1110 Vienna, Austria
c
Institute of Fluid Mechanics and Heat Transfer, Technical University Graz, Inffeldgasse 25F, A-8010 Graz, Austria
Received 10 June 2003; received in revised form 2 March 2004; accepted 3 March 2004
Available online 26 April 2004
Abstract
For sedimentation of miscible drops through quiescent liquid of the same viscosity, a recent paper [J. Fluid. Mech. 447 (2001) 299] has
shown the effectiveness of computer simulations based upon a swarm of point forces in tracking coalescence, mixing and rupture. Robustness
of the approach was offset by the slow O(N
2
p
) summations needed to calculate the mutual viscous interactions between all N
p
particles.
Motivated by applications of wavelets to linear operators, this paper develops a conceptually simple scheme for dramatically accelerating the
simulations. After lumping the particles together into N local clusters (three-dimensional “pixels”), a Haar discrete wavelet transform (DWT)
is used to “compress” the “bitmapped” six-dimensional image of pixel–pixel hydrodynamic interactions. Depending upon the criterion of
accuracy, the numerically observed scaling of the operation count seems to be either O(N) or O[N(log N)
α
]. The DWT also works without
modification for hydrodynamic wall effects, where the kernel is not purely translational and therefore fast convolutions (FFT) cannot be used.
© 2004 Elsevier Ltd. All rights reserved.
PACS: 47.11.+j; 47.15.Gf; 47.20.Bp; 47.55.Dz; 47.15.Pn
Keywords: Wavelets; Fast summations; Particle methods; Miscible drops; Creeping flow; Buoyancy-driven instability
1. Introduction: a swarm of point forces
This article describes the use of wavelets to substantially
accelerate particle simulations of low-Reynolds-number
drop flows. As a physical point of reference, within the
realm of common experience and having also geophysical
analogs in mantle plumes (Manga, 1997; Manga, Stone,
& O’Connell, 1993) consider the rising (or sedimentation)
of a miscible drop of ink (or suspension of glass beads)
through a quiesent bath of solvent (Fig. 1). The resulting
nonlinear shape instability has been described and studied
extensively (Adachi, Kiriyama, & Koshioka, 1978; Arecchi,
Buah-Bassuah, Francini, Pérez-Garcia, & Quercioli, 1989;
Joseph & Renardy, 1993; Kojima, Hinch, & Acrivos, 1984;
Machu, Meile, Nitsche, & Schaflinger, 2001; Schaflinger
& Machu, 1999; Thomson & Newall, 1885) and includes
∗
Corresponding author. Tel.: +1-312-996-3469;
fax: +1-312-996-0808.
E-mail addresses: lcn@uic.edu (L.C. Nitsche), ma@hkts.hoerbiger.com
(G. Machu), meile@fluidmech.tu-graz.ac.at (W. Meile).
(i) entrainment of solvent and formation of a tail; (ii) inter-
mediate mushroom and ring shapes; and (iii) an eventual
cascade of successive breakups. All of these phenomena
are observed in the theoretically simplest case of inertialess
flow with equal viscosity of the ink and solvent.
The work of Machu et al. (2001) formalized an analogy
between a miscible drop of homogeneuos liquid and a swarm
of point forces (cf. Adachi et al., 1978; Kojima et al., 1984;
Nitsche & Batchelor, 1997), and solved numerically a (non-
linear) dynamical system for the Stokeslet positions:
dr
i
dt
=
j
G(r
i
- r
j
)δV (1)
with G the (dimensionless) disturbance field produced by a
unit point force acting in the direction of gravity,
G(r) = (||r||
-1
I + ||r||
-3
rr) · e
g
(2)
and δV the equivalent volume of ink associated with one
particle. The resultant error in the local liquid velocity scales
with the number N
p
of particles like N
-1/2
p
log N
p
.
0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compchemeng.2004.03.001