Environmental Flow Physics& Computation Group

Influence of Basset History Force on Preferential Concentration of Small Particles and Bubbles in Turbulence
S. Yokojima, Y. Shimada, and K. Mukaiyama (Shizuoka University, Japan)
Time averaged PV vs. b

Abstract

Numerical experiment

Computation of the Basset history force can be very expensive, and hence,in past studies, (at least some fraction of) the history has been neglectedeven when its contribution may be important. Here the effects of the Basset
force on preferential concentration (PC) of small particles in a
homogeneous isotropic turbulence under a wide range of the particle-tofluid mass density ratio g have been investigated. The Basset force isapproximated by the method of van Hinsberg et al. Compared with thetraditional window method where the Basset integral is evaluated only overthe latest period of size twin, van Hinsberg’s method that approximates thetail of the Basset force kernel by exponential functions gives a better result
by using a much shorter (typically, two-order-of-magnitude smaller) twin.
The presence of the Basset force weakens the level of PC to some extent,
especially at g of around 1.5-10 for heavy particles and smaller than 0.7 forlight particles.

Flow field✓ Forced homogeneous isotropic turbulence at Rel = 42
✓ Spectral DNS with 483 Fourier modes• Particles✓ Particle-to-fluid mass density ratio g˗ Light particles – 0, 0.001, 0.01, 0.1, 0.3, 0.5, 0.7, 0.9
˗ Heavy particles – 1.005, 1.1, 3, 10, 100, 1000, 10000✓ Stokes number Sh = 1.0✓ One-way coupling, ignoring gravitational effects✓ # of particles: 1.5*483, initially seeded at random locations•Window method vs. van Hinsberg’s method
Window methodvan Hinsberg’s method

Maxey-Riley eq.
✓ For a small isolated rigid spherical particle in nonuniform flow

Pressure gradient
Gravity force
Added mass force

Table 1. Relation among b, b1/2, and the particle-to-fluid mass density ratio g.

1000
1001030.0015 0.0149 0.1429 0.4286
0.0387 0.1222 0.3780 0.6547

111

g=0.01

g=0.1

g=0.3

g=0.5

g=0.7

g=0.9

g=1.005

g=1.1

g=1.5

g=3

g=10g=100g=1000

g=10000

Figure 3. Influence of g on instantaneous spatial distribution of particles inside a thin layer.

Figure 1. Influence of parameter twin that defines the extent of contribution fromfB,win, on the particle void fraction Pv.

Basset history force∞00g=0.001

with fB

Stokes drag force

gbb1/2g=0

without fB

Introduction•Influence of g on instantaneous particle distribution

without fB•with fB

✓ Dt = 0.03th, total period of simulation time 180th

Figure 2. Continued.

0.90.50.10.01
01.0714 1.5000 2.5000 2.9412
31.0351 1.2247 1.5811 1.7150 1.7321

Particle void fraction Pv˗ represents the ratio of # of computational cells containing no particles
˗ describes the extent of nonuniformity in particle distribution•

Relation between particle distribution and coherent vortical structures
Without fB

With fB

Impact of Basset force on PC of light/heavy particles•

Influence on particle void fraction Pv
Light particles, with fB

Approximation to Basset force(van Hinsberg et al. JCP 230 (2011) 1465)
Light particles, without fB
tail part: fB,tailFigure 4. Relation between spatial distribution of particles (greel: very heavy particle g = 10000;
yellow: ultimately light particle g = 0) and coherent vortical structures in homogeneous isotropic
turbulence visualized by color contours of the second invariant of velocity-gradient tensor normalized
by Kolmogorov scale.

window part: fB,win✓ Kernel (t - t)-1/2 in tail part is approximated by superposition ofexponential functions:

Heavy particles, without fB

Heavy particles, with fBConclusion•✓ It enables us to compute tail part of the integral in a recursive way:
tail part

window parti.e., the yellow part of the integral can be easily calculated using the valueof the orange part at the previous time step

Figure 2. Influence of mass density ratio g on Pv.

Computation of Basset history force✓ van Hinsberg’s approximation is very efficient and accurate.
✓ Traditional window method requires great caution in accuracy.
• Influence of Basset history force on PC (preferential concentration)✓ Presence of Basset force weakens PC to some extent.
✓ It is especially noticeable at g = 1.5 - 10 for heavy particles (i.e., solidparticles in liquid) and smaller than 0.7 for light particles (i.e., bubbles inliquid).
✓ Further investigation is necessary to clarify effects of two-way coupling andgravitational acceleration.