particle pair diffusion of inertial particles such as ...€¦ · [email protected] european...

Particle Pair Diffusion of Inertial Particles such

as Atmospheric Dust

Syed M. Usama, Yoseph T. Tereda, and Nadeem A. Malik

King Fahd University of Petroleum and Minerals

Department of Mathematics and Statistics

[email protected]

[email protected]

European Geosciences Union, General Assembly.

Vienna, Austria, 17--22 April 2016

Contents

1. Introduction: importance

2. Mathematical modeling

3. Simulations

4. Discussion

N A Malik 2

Greg Bajor

G. Falkovich

Richard Smith Greg Eyink, J’Hop U.

Turbulent Inertial Particle Diffusion

N A Malik Turbulent Inertial Particle Diffusion 3

Particle motion in the geosciences (particle laden flow) -- clouds

Introduction

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Introduction

Particle motion in the geosciences (particle laden flow) -- pollen


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Introduction

Particle motion in the geosciences (particle laden flow) -- seas


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Introduction

Particle motion in the geosciences (particle laden flow) is ubiquitous in nature.

Important for life: pollination, precipitation, mixing

Also a hazard: dust storms, hale, health

We need to study, understand, and model particle motion.

Ultimately we want to be able to predict such phenomena.


A single inertial particle motion is described by an transport equation.

The most important parameter is the Stoke’s number

𝑆𝑡 =𝜏𝑝

𝜏𝑓

𝜏𝑝 is the particle response time.

𝜏𝑓 is a time scale of fluid particle motion

Stoke’s number characterizes the strength of inertia:

𝑆𝑡 = 0 (𝜏𝑝 = 0) is equivalent to fluid particle.

𝑆𝑡 → ∞ (𝜏𝑝→ ∞) is a very heavy particle.

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Mathematical Modelling


𝑙𝒖𝑝

𝒖𝑓

There are many effects due to:

particle size distribution

particle shape

particle density

pressure gradient

Lift and drag

Groups of particle motion ?

A general theory for inertial particle motion is unknown.

However, some important limiting cases can be studied.

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The Maxey-Riley 1983 theory for small spherical particle of density, 𝜌𝑝, and small

inertia, 𝑆𝑡 ≲ 1, the acceleration of an inertial particle is,

𝑑𝒖𝑝(𝒙,𝑡)

𝑑𝑡= −

1

𝜏𝑝(𝒖𝑝 − 𝒖𝑓) + 6 other terms

𝒖𝑝 is the particle velocity.

𝒖𝑓 is the fluid velocity (wind)

𝜏𝑝 =2𝑎2

9𝜇

𝜌𝑝

𝜌𝑓

In the Stoke’s drag limit only the first term is retained.

𝑑𝒖𝑝(𝒙, 𝑡)

𝑑𝑡= −

1

𝜏𝑝(𝒖𝑝 − 𝒖𝑓)

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𝒖𝑓 is a turbulent velocity field – e.g. atmospheric dust and standstorms.

Turbulence: energy spectrum 𝐸 𝑘 = 𝑘−5/3, 𝑘1 ≤ 𝑘 ≤ 𝑘𝜂 , (Kolmogorov 1941).

𝑘~1/𝑙 is the wavenumber.

Segregation of particles, (Meneguz & Reeks 2011, Murray et al. 2016, others)

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Single particle motion in turbulence


Richardson 1926: diffusivity 𝐾(𝑙) =𝑑( 𝑙2 )

𝑑𝑡. (𝑙 is the pair separation.)

Locality: pair diffusion is governed by 𝐸 1/𝑙 in the inertial subrange

𝐷 depends only upon 𝑙 and 𝜀 (the rate of k.e. dissipation)

𝐾(𝑙) ~ 𝜀2/3𝑙4/3 Richardson 1926

𝑙2 ~ 𝜀𝑡3 Obukov 1941

Malik 2016 (submitted J. Fluid Mech.)

Non-local theory

𝐾(𝑙) ~ 𝑙𝛾𝑛𝑙

𝛾𝑛𝑙 >4

3; data and simulations suggest that 𝛾𝑛𝑙 ≈ 1.56; 𝑙2 ~ 𝑡4.5

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Fluid pair diffusion in turbulence

𝑙


Inertial pair diffusion will be governed by 𝑙 and 𝜀, and also 𝑆𝑡.

𝐾(𝑙) ~𝑓(𝑙, 𝜀, 𝑆𝑡)

Lagrangian method, Kinematic Simulations(Kraichnan 1970; Fung, Hunt, Malik, Perkins 1992; Malik 2016)

Integrate particle motion using,

𝑑𝒖𝑝(𝒙, 𝑡)

𝑑𝑡= −

1

𝜏𝑝(𝒖𝑝 − 𝒖𝑓)

and obtain ensemble of particle trajectories and analyse

the Lagrangian statistics of particle motions.

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Inertial pair diffusion in turbulence

𝑙


A parametric study as a function of 𝑆𝑡, 𝑅 the size of the subrange 𝑅 = 𝑅(𝑅𝑒).Using a Lagrangian diffusion model, KS.

𝜎𝑙2 = 𝑙2 is the ensemble mean square pair separation.

𝜂 is the smallest scale of turbulence (Kolmogorov scale).

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Inertial pair diffusion in turbulence


Inertial Particle Pair Diffusion

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Bec et al JFM 2011: DNS 5123 , 𝑅𝑒𝜆 ≈ 200. 𝑆𝑡 = 𝜏𝑝/𝜏𝜂Local Stokes number 𝑆𝑡 𝑅 = 𝜏𝑝/𝜏𝑅 𝑆𝑡 𝑅∗ = 1 at some separation 𝑅∗

For |Δ| < 𝑅∗, inertia dominant.

For Δ > 𝑅∗, inertia not dominant, behaves like fluid particles

Numerical

Simulation

DNS KS


• Particle laden flow is ubiquitous in nature, important for life and health:

pollination, dust storms.

• General theory for inertia particle motion is unknown, but some limiting cases

are important and solvable.

• Inertial pair diffusion in turbulent flow in the Stoke’s drag limit has been studied.

• KS (Kinematic Simulation) shows promising results, suggesting the existence of

dual regimes for 𝐾(𝑙) governed by the Stoke’s number 𝑆𝑡, and the size of the

inertial subrange 𝑅 𝑅𝑒 .

Thanks to KFUPM ITC department for use of the HPC

[email protected] and [email protected]

Thank you

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Discussion


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