Tesi di Laurea Break-up of inertial aggregates in turbulent channel flow Frammentazione di aggregati inerziali in flusso turbolento Relatore: Dott. Ing

Tesi di Laurea

Break-up of inertial aggregatesin turbulent channel flow

Frammentazione di aggregati inerzialiin flusso turbolento

Relatore: Dott. Ing. Cristian Marchioli

Correlatore: Prof. Alfredo Soldati

Candidato: Marco Svettini

UNIVERSITÀ DEGLI STUDI DI UDINEFacoltà di Scienze Matematiche Fisiche e Naturali

CdLS in FISICA COMPUTAZIONALE

Anno Accademico 2011/2012

Premise

What is turbulence?

Turbulent flux characteristics:

• Unstable and unstationary (Reynolds)

• Tridimensional

• Diffusive

• Dissipative (h Kolmogorov l.s.)

• Rotational: w=rot(u)≠0

• Coherent

Jet flow

η= ඨν3ε4 τ= ට

νε

Δx,Δy,Δz < 𝜂 Δt < 𝜏= η2ν

DNS solver req.:

Length scale:

CFL condition:

Random nature of turbulent flow:u = U+u’(Reynolds decomp.)

𝐶𝑖 = 𝑢𝑖 ∙∆𝑡 ∆𝑥𝑖Τ < 0.1

Premise

Aggregate Break-up in Turbulence

What kind of application?

Processing of industrial colloids• Polymer, paint, and paper industry

Premise




Environmental systems• Marine snow as part of the oceanic carbon sink

Premise




Environmental systems• Marine snow as part of the oceanic carbon sink

Aerosols and dust particles• Flame synthesis of powders, soot, and nano-particles• Dust dispersion in explosions and equipment breakdown

Premise


What kind of aggregate?

Aggregates consisting ofcolloidal primary particles

Schematic of an aggregate

What kind of aggregate?

Aggregates consisting ofcolloidal primary particles

Break-up due toHydrodynamics stress (Dp << h)

Schematic of break-up

Premise


Problem Definition

Description of the Break-up Process

Focus of this work!

SIMPLIFIEDSMOLUCHOWSKIEQUATION (NOAGGREGATIONTERM IN IT!)

• Turbulent flow laden with few aggregates (one-way coupling)

• Aggregate size < O(h) with h the Kolmogorov length scale

• Heavy aggregates:

• Aggregates break due to hydrodynamic stress

• Tracer-like aggregates:

• Brittle and deformable aggregates

Problem Definition

Further Assumptions

𝜎~𝜕𝑢𝑖′𝜕𝑥𝑗 , 𝜀~𝜕𝑢𝑖′𝜕𝑥𝑗𝜕𝑢𝑖′𝜕𝑥𝑗 ⇒ 𝜎∝ 𝜀

𝜌𝑝 > 𝜌𝑓

Problem Definition

Strategy for Numerical Experiments

• Consider a fully-developed statistically-steady flow

• Seed the flow randomly with aggregates of mass x at a given location

• Neglect aggregates released at locations where > e ecr

• Follow the trajectory of remaining aggregates until break-up occurs

• Compute the exit time, = t tecr (time from release to break-up)

Problem Definition







Problem Definition







Problem Definition







Problem Definition


t

For jth aggregatebreaking afterNj

time steps:

x0=x(0)

x t =x(tcr)

dtn n+1

tj=tcr,j=Nj·dt






Problem Definition


• Break up of brittle aggregates occours

instantly if > e ecr (power per unit

mass)

• The break-up rate is the inverse of the

ensemble-averaged exit time:𝑓ሺ𝜀𝑐𝑟ሻ= 1 𝑃𝜀𝑐𝑟ሺ𝜏ሻ∞0 𝜏𝑑𝜏= 1𝜏ሺ𝜀𝑐𝑟ሻۃ ۄ

t

For jth aggregatebreaking afterNj

time steps:

x0=x(0)

x t =x(tcr)

dtn n+1

tj=tcr,j=Nj·dt

Problem Definition


• The break-up rate is the inverse of the ensemble-averaged exit time:

Break-up of deformable aggregates occour if:

• Aggregate start to deform ( > e ecr): deformation activation

• Deformation exceed the maximum allowed deformation (E > Ecr)

Aggregate returns to a relaxed state when < e ecr

Deformation is proportional to the dissipated energy per unit mass:𝐸=න 𝜀ሺ𝑡ሻ𝑑𝑡𝜀>𝜀𝑐𝑟

𝑓ሺ𝜀𝑐𝑟ሻ= 1 𝑃𝜀𝑐𝑟,𝐸𝑐𝑟ሺ𝜏ሻ∞0 𝜏𝑑𝜏= 1𝜏ሺ𝜀𝑐𝑟,𝐸𝑐𝑟ሻۃ ۄ

Flow Instances and Numerical Methodology

Channel Flow

• Pseudospectral DNS of 3D

time-dependent turbulent gas flow

• 256x256x257 grid resolution

• Shear Reynolds number:

Ret = uth/n = 150

Near wall particles transfer model

Navier-Stokes equations:𝜕𝑢𝑗𝜕𝑥𝑗 = 0 𝜕𝑢𝑖𝜕𝑡 = ቆ−𝜕൫𝑢𝑖𝑢𝑗൯𝜕𝑥𝑗 + 𝛿1𝑖ቇ+ 1𝑅𝑒𝜏∇2𝑢𝑖 − 𝜕𝑝𝜕𝑥𝑖

Particle Tracer and Numerical Methodology

Stokes number

Equation of motion for a small spherical particle in a nonuniformflow (Maxey & Riley, 1983) – wall units:𝑑𝒙𝑑𝑡 = 𝒗

𝑑𝒗𝑑𝑡 = ሺ𝒖− 𝒗ሻ𝑆𝑡 ൫1+ 0.15∙𝑅𝑒𝑝0.687൯ 𝑆𝑡 = 𝜏𝑝𝜏𝑓 = 𝜌𝑝𝜌 ൫𝐷𝑝+൯218 , 𝑅𝑒𝑝 = ȁA𝒖− 𝒗ȁA𝐷𝑝𝜈

• Time-integration: 4th order

Runge-Kutta scheme

• Fluid velocity interpolation: 6th

order Lagrange polynomials

Inertial particles behaviour:Stokes number dependence

Channel Flow

Choice of Critical Energy Dissipation

Characterization of the local energydissipation in bounded flow:

Wall-normal behavior ofmean energy dissipation

𝜀= 12 ቆ𝜕𝑢𝑗𝜕𝑥𝑖 + 𝜕𝑢𝑖𝜕𝑥𝑗ቇ

23𝑖,𝑗=1

Tracers dissipation plot

Inertial aggregates dissipation plot

Break-up analysis

Tracer aggregates

Spanwise channel view

Streamwise channel view

Break-up analysis

Tracer aggregates

Brittle aggregate

Break-up analysis

Tracer aggregates

Brittle aggregate

Deformable aggregate

Break-up analysis

Inertial aggregates

Spanwise channel view

Streamwise channel view

Break-up analysis

Inertial aggregates

Brittle aggregate

Break-up analysis

Inertial aggregates

Brittle aggregate

Deformable aggregate

Break-up analysis (brittle aggr.)

Choice of Critical Dissipation

Distribution is strongly affected by flow anisotropy (skewed shape)

Whole channel dissipation

Wall-normal behavior ofmean dissipation

0 < z+ < 150



Bulk dissipation


Bulk ecr


40 < z+ < 150



Intermediate dissipation


Intermediate ecr


10 < z+ < 40



Wall dissipation


Wall ecr


0 < z+ < 10



Distribution of local dissipation

Inertia affect very much the dissipation distribution

• Inertial aggregates

sample higher dissipation

channel region and empty

the bulk

• Increasing Stokes cause

higher dissipation events

(near wall region)

• e+=0.2 is Stokes invariant


Where does aggregates break-up?

Tracers released in themiddle of the channel

Inertial aggregates released in the middle of the channel


Where does aggregates break-up?

Tracers released in themiddle of the channel

Inertial aggregates released in the middle of the channel

Bulk dissipation Intermediate dissipation

Wall dissipation

Generally speaking, inertial aggregates have lower probability to reach regions far from the middle of the channel due to segregation process,

inertial aggregates sample regions with higher dissipation values


Break-up frequency

Break-up frequency of brittle aggregates as function of inertia and

critical dissipation threshold

• Break-up frequency increase

with Stokes number as

consequence of the segregation

process

• Break-up estimation is over

estimated for ecr > 0.02 due to

simulation finite length

• For low threshold values break-

up decreasing function can be

fitted with a power law


Break-up frequency




process







First exit time break-up

events distribution

Bulk dissipation

Intermediate dissipation


Break-up frequency

Break-up frequency of brittle aggregates as function of inertia and

critical dissipation threshold




process







Break-up analysis (deformable aggr.)

Choice of Critical Deformation

Distribution of deformation values as a function of dissipation and

deformation threshold

• Dissipated energy per unit

mass goes from 0.01 to 100

• For increasing dissipation

threshold we observe:

Higher energy events

Shorter event duration

• Events number depend on

the Stokes number


Break-up frequency

• Break-up frequency tends to the

brittle case if Ecr reduce to zero

• Bulk dissipation region is

particularly affected by

deformation (red curve on the

right plot)

• Brittle and deformable cases

overlap if:

Ecr = 0.04 and ecr > 0.008

Ecr = 0.4 and ecr > 0.12

Ecr = 2.8 and ecr > 0.7


Break-up frequency

Inertia affect break-up deformation plot only quantitatively but not qualitatively

Documents

Tesi di Laurea Break-up of inertial aggregates in turbulent channel flow Frammentazione di aggregati inerziali in flusso turbolento Relatore: Dott. Ing