# Taylor Instability Induced by the Suspension

Mécanique

22 févr. 2014 (il y a 4 années et 2 mois)

77 vue(s)

Mixing sediment plumes in Gulf of Mexico (image credit: NASA Earth Observatory
-

http://earthobservatory.nasa.gov)

Numerical Simulation of Turbulent Rayleigh
-
Taylor Instability Induced by the Suspension
of Fine Particles

Yi
-
Ju

Chou

(

)

Multi
-
Scale Flow Physics & Computation Lab.

Institute of Applied Mechanics

National Taiwan University

Sediment River Plume

image
credit: NASA Earth Observatory
-

http://earthobservatory.nasa.gov)

Ocean acidification

Phytoplankton bloom

Subaqueous ecology

Parsons et al, 2001

Outline

Background and motivation

Mathematical formulation

Numerical examples of RT instability

Summary

Physics Background

I
mportant characteristics

Particle diameter ~ O(1
-
10
𝜇
m)

Density ratio~ O(1)

Turbulent flow

Concentration (volume fraction) ~ O(0.01)

I
mportant parameters

Particle Reynolds
n
umber:
𝑅
𝑝
=
𝐮

𝐮

𝑑
0


<
O(100)

Stokes drag

Particle relaxation(response) time:
𝜏
𝑝
=
𝜌

𝜌

𝑑
0
2
18

~
10

5

10

3
sec

Stokes number:
𝜏
𝑝
𝜏
𝑘
<
𝑂
1

(
𝜏
𝑘
:
Kolmogorov

time

scale
)

Bagnold number:
𝑎
=
𝜌
𝑠
𝑑
𝑠

1
/
2
𝛾

~
𝑂
1
<
<
40

Collision negligible

Physics Background

Balachandar

& Eaton,

Annual Rev. Fluid Mech., 2010

Modeling strategy for dispersed two
-
phase
t
urbulent flows :

Review of Existing Method

Single
-
phase method:

Has been employed to study a number of problems
related to fine suspensions using DNS, LES, RANS

Equilibrium state:

A

balance between drag and gravitational force

Scalar limit:

Zero
-
volume for particles

+ Basset history term

+
Saffman

lift force

: Density ratio

-
mass coefficient (1/2 for the sphere)

: Particle relaxation time

Drag

Pressure

Gravity

Motion of a Sediment Grain

Motivation

What are we trying to answer?

How can the equilibrium state be a good approximation?

What else are we missing, and how they effect bulk mixing?

Can we improve the current model without too much extra computational
effort?

Mathematical Formulation

A two
-
way coupled Euler
-
Euler solid
-
liquid system

Mathematical Formulation

A two
-
phase fractional
-
step pressure projection method

Non
-
Boussinesq

pressure Poison solver:

Corrector: Pressure projection

Differs from traditional models for solid
-
gas systems in

T
hree
-
dimensional turbulence
-
resolving two
-
flow system (LES, DNS)

A
dded mass effect (
Auton
, 1988)

Mixture incompressibility (a two
-
phase pressure projection method)

Mathematical Formulation

Mathematical Formulation

What are we missing?
(Chou et al., 2013b)

Non
-
equilibrium particle inertia (NEPI)

NEPI effect in the carrier flow (continuous phase)

Mixture incompressibility

Particle Induced RT Instability

--

A numerical study of two
-
phase effect in suspensions

0.08 m (128)

0.08 m (128)

0.12 m (192)

Simulation setup
(Chou et al., 2013b)

𝜙
0

= 0.0032, 0.0128, 0.0512

0
=
40

𝜇𝑚
;
𝜏
𝑝
=
2
.
4
×
10

4

sec

Direct Numerical Simulation (DNS)

BC: Periodic at horizontal;

Solid wall at bottom;

No sediment supply at top.

Growth of Initial Perturbations

Large
-
Scale Mixing

Growth of mixing zone

Self
-
similar solution: h
=
𝛼
𝑔
𝑡
2

(
α
~
0
.
05

in

the

present

study
)

Energy Spectrum

Energetic

Slightly higher energy release
induced by non
-
equilibrium
particle inertia

Energetic

Feedbacks to the carrier flow
are increasingly important

Energetic

* * 2
-
phase modeling
without mixture
incompressibility

Due to mixture
incompressibility

Summary

We aim to investigate effects of missing mechanisms induced by two
-
phase interactions in the common modeling approach for sediment
suspension problems.

The two
-
phase effects include:

--

non
-
equilibrium particle inertia (NEPI);

--

NEPI in the carrier fluid;

--

mixture incompressibility (MI)

A series of numerical experiments of RT reveals that

--

In low volume fraction
,

NEPI slightly enhances the energy budget.

--

As concentration increases, NEPI and MI become increasing

important, which suppress energy.

--

MI is significant to suppress energy budget at high concentration,

which accounts for almost ¼ of the reduction of the PE release.

Thank you