Orientation and distribution of highly elongated and inertial fibres in turbulent flow: a comparison of

experimental and numerical data

Stella Dearing, Cristian Marchioli, Alfredo Soldati

Dipartimento di Energetica e Macchine, Università di Udine, Italy

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12th Workshop on two phase flow predictions, Halle-Wittenberg, Germany, 22-25 March 2010.

• Motivating factors for work

• “State of art” – DNS*

• Objectives

• Methodology

• Number density & mean orientations

• Summary

Motivation

Methodology

Results

Summary

Contents

DNS

Objectives

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*Marchioli C., Fantoni M. & Soldati A., Orientation, distribution, and deposition of elongated, inertial fibres in turbulent channel flow, Phys. Fluids , 22,(2010).

• Fibres as an alternative to polymers as a drag reducing additives • Examples include the TAPs,

medical application, firehouse• Fibres provide more modest

reductions but improved shear degradation and filterability

Introduction

• Pulp and paper processing• Controlling rheological behaviour and

fibre orientation distribution crucial to optimise operations

• Furniture Industry• Pneumatic transport of fibres

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DNS simulations – incompressible turbulent channel flow drive by streamwise pressure gradient

Solved using a pseudo-spectral method: 128x128x129 modes in Fourier-

Chebyshev space. (x, y, z- respectively )

Periodic boundary conditions in x & y

No slip condition at the wall: turbulent boundary.

Equations of continuity & Navier-Stokes:

DNS – “State of Art”: methodology

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Langrangian particle tracking.Each particle path, resulting from the

forces acting on it by the turbulent flow, is calculated for each time step.

200,000 particles are tracked: initial position and orientation of particles are

random; Initial particle velocity = to fluid

3 frames of reference (for orientations) Eulerian inertial frame of reference, x,y,z A Lagrangian fibre frame of reference: x’,

y’ , z’, attached to the fibre with origin at the fibre center of mass;

A co-moving frame of reference, x’’, y’’, z’’ attached to the fibre with origin at the fibre center of mass and axes parallel to the inertial frame.

Euler’s angles: φ,ψ,θ

Euler angles: e0, e1, e2,e3

Rotation matrix:

, …

DNS – “State of Art”: orientation behaviour

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Translational motion is strongly dependent on hydrodynamic drag

Newton’s law:

Hydrodynamic drag (Brenner)

(particle reference system)

In channel reference frame:

Resistance tensor

Coupling of translational motion and rotational motion

(particle trajectory)

(assume other forces are negligible)

DNS – “State of Art”

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Relative density: F

PS

Particle and flow parameters:

DNS – “State of Art”

Aspect Ratio:

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“State of the art”: macroscopic fibre motion

8y

z

“State of the art”: macroscopic fibre motion

9y

x

“State of the art”: Concentration data

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Instantaneous concentration profiles computed as volumetric fiber number density

Near wall peak - behaviour of fibre build up is complex and largely dependent on wall normal fibre translational velocity

Decrease of concentration at z+ approximately 1- after which point λ has little or no effect on concentration profiles

State of the art: mean orientations

Figure 16

λ=1.001

λ=3

λ=10

λ=50

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• Fibres tend to align in the streamwise direction

• Preferential orientation increases with aspect ratio and decreases with inertia

b Up

Spherical particles have no preferential alignment

• Fibres align in regions of high shear

• In regions of small velocity gradients: random orientations

Objectives

• One way coupling• Inertia is concentrated in fibre centre of mass (CoM)• Rotation is computed according to shear in CoM • Dilute

Numerical models

• Macroscopic fibres• Fibres may affect flow• Fibres may interact (increased local concentration)• Very limited literature available*

Real world

12*Bernstein, O., Shapiro, M. Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows. Journal of Aerosol Science,25, 113-136, (1994)

• Complete existing literature• Justify DNS assumptionsObjectives

Experimental Set-Up : Imaging for visualisations

Figure (a)

Laser

Cameraz

x

Mirror

Flow Direction

x

z

• System details:• PCO sensicam 1280 x1024• ND Yag laser 1000mJ

Figure (b) 13

A

• Pipe length – 30m; Pipe diameter- 0.1m; Max Re ͠͠͠͠ 300,000

Experimental set-up: Fibres

• Uniform vs non uniform size distribution

• Synthetic plastic fibres (nylon) • Shredded wood fibres

Experimental set-up: Fibres

Cum

ula

tive

frequency

%

Frequenc

y %

Most probable length

Spurious

Fibre diameter

Experimental set-up :Fibre and flow parameters

[2]

Flow Velocity, m/s Re τ+ Reτ

0.71 71 938 0.108 1737

1.08 111 568 0.46 2507

1.25 145 804 0.5 3131

1.60 178 218 0.9 3513

2.13 226 531 1.1 4514

Fibre type

Specific Gravity

Fibre length, microns

Fibre diameter, (microns)

Aspect ratio (λ)

Mass fraction, %

wppm Concentration parameter nb3

Nylon 1.14 300 25 13 0.01 100 0.018

0.02 200 0.035

Dilute suspensions: based on concentration parameter nb3 << 1

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Phase discrimination: fibre identificationPre-

processing

Object identification

Discriminate objects based on length and aspect ratio

Fitting object to an ellipse using least

squares method

Adjust intensity Dilate Remove

noise

Erode back to normal

size

Figure (b)

Figure 6

Figure (a)

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Phase discrimination: orientation calculations

Special formulation of a general conic

Least square fit to data point (centre locations of pixels that make up object):

• Fit an ellipse to fibre using least square fitting algorithm

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Phase discrimination: statistics

• Mean orientations

• Normalised number densityi

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Results: Normalised mean number density

Lower fibre mass fraction

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Higher fibre mass fraction

Results: Normalised mean number density

(a)

(b)

(c)

(d)

Re 71 000 Re 145 804

21Re 178 000 Re 226

531

Results: Mean orientations

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Lower fibre mass fraction Higher fibre mass fraction

Results: Mean orientations

(a)

(b)

(c)

(d)

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Re 71 000 Re 145 804

Re 178 000 Re 226 531

Artificial Images of fibres

Figure (b) 24

Discussion, Conclusions, Future Work

• Good agreement with mean statistics from DNS data

• Differences can be accounted for due to projection of a 3D body onto a 2D plane:

• We plan to calculate mean statistics using “phase discrimination” DNS slices

• Validate using 3D model

• Fibre velocities using PTV

• Validation in process of calculation of phase velocity

• Measurements of suspension viscosity

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Thank you for your attention.

Comments, suggestions and questions welcome!

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