Numerical Simulation of Turbulent RayleighTaylor Instability Induced by the Suspension
of Fine Particles
Yi-Ju Chou (周逸儒)
Multi-Scale Flow Physics & Computation Lab.
Institute of Applied Mechanics
National Taiwan University
Mixing sediment plumes in Gulf of Mexico (image credit: NASA Earth Observatory - http://earthobservatory.nasa.gov)
Sediment River Plume
• Ocean acidification
• Phytoplankton bloom
• Subaqueous ecology
Parsons et al, 2001
image credit: NASA Earth Observatory - http://earthobservatory.nasa.gov)
Outline
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Background and motivation
Mathematical formulation
Numerical examples of RT instability
Summary
Physics Background
Important characteristics
• Particle diameter ~ O(1-10 𝜇m)
• Density ratio~ O(1)
• Turbulent flow
• Concentration (volume fraction) ~ O(0.01)
Important parameters
𝐮𝑐 −𝐮𝑑 𝑑0
< O(100)
Stokes drag
𝜈
𝜌 𝑑 𝑑0 2
Particle relaxation(response) time: 𝜏𝑝 =
~10−5 − 10−3 sec
𝜌𝑐 18𝜈
𝜏𝑝
Stokes number: < 𝑂 1 (𝜏𝑘 : Kolmogorov time scale)
𝜏
• Particle Reynolds number: 𝑅𝑒𝑝 =
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•
𝑘
• Bagnold number: 𝐵𝑎 =
𝜌𝑠 𝑑𝑠 𝜆1/2 𝛾
~𝑂
𝜇
1 << 40
Collision negligible
Physics Background
Modeling strategy for dispersed two-phase turbulent flows :
Balachandar & Eaton,
Annual Rev. Fluid Mech., 2010
Review of Existing Method
• Single-phase method:
Equilibrium state:
A balance between drag and gravitational force
Scalar limit:
Zero-volume for particles
Has been employed to study a number of problems
related to fine suspensions using DNS, LES, RANS
Motion of a Sediment Grain
Drag
Pressure
Added mass
Gravity
: Density ratio
: Added-mass coefficient (1/2 for the sphere)
: Particle relaxation time
+ Basset history term
+ Saffman lift force
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
Mathematical Formulation
Differs from traditional models for solid-gas systems in
• Three-dimensional turbulence-resolving two-flow system (LES, DNS)
• Added mass effect (Auton, 1988)
• Mixture incompressibility (a two-phase pressure projection method)
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
Simulation setup (Chou et al., 2013b)
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𝜙0 = 0.0032, 0.0128, 0.0512
𝑑0 = 40 𝜇𝑚; 𝜏𝑝 = 2.4 × 10−4 sec
Direct Numerical Simulation (DNS)
BC: Periodic at horizontal;
Solid wall at bottom;
No sediment supply at top.
0.12 m (192)
0.08 m (128)
0.08 m (128)
Growth of Initial Perturbations
Large-Scale Mixing
Growth of mixing zone
• Self-similar solution: h= 𝛼𝐴𝑔𝑡 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
Due to mixture
incompressibility
* * 2-phase modeling
without mixture
incompressibility
Summary
• We aim to investigate effects of missing mechanisms induced by twophase 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