Modeling particle deposition in a turbulent ribbed channel flow

aM A I Khan, aX Y Luo and bF C G A Nicolleau cP G Tucker dG Lo Iacono

aDepartment of Mathematics, University of GlasgowbDepartment of Mechanical Engineering, University of Sheffield

cCivil and Computational Engineering centre,

University of Wales, Swansea dCentre for Mathematical and Computational Biology,

Rothamsted Research, Harpenden, Hertfordshire

Particle deposition in the human airway

• Targeted drug delivery via inhaled airborne particles require an understanding of the distribution and deposition of aerosol particles in the tracheo-bronchial airway.

• It is also important for risk assessments of contaminant deposition, due to the fact that excessive retention of inhaled particles causes diseases like silicosis, asbestosis etc.

• Detailed knowledge of the flow field pattern is important and necessary for predicting particle transport and deposition.

• Advanced imaging techniques are capable of obtaining deposition pattern in a reasonable detail but the data are averages over many airway branches

•Theoretical calculations provide an alternative but most investigations use simplified models of the flow field

•But flow in the upper airway during heavy breathing approach a high Reynolds number (Re=9300) (Pedley et al. 2004)

•High Re implies flow with turbulent characteristics hence incorporating the effects of turbulence is essential (Luo et al. 2004)

A schematic diagram of a human airway

Why use CFD• In the last decade, computational fluid dynamics

(CFD) has been increasingly used to the study of fluid flow and particle transport in the human airway (Balashazy et al. 1993)

• It is hoped that CFD will contribute to a fuller understanding of the processes involved in precise drug deposition, and in the ways in which inhaled therapies can benefit the suffering person

• A better understanding of the behavior of air flow in the airways will improve treatment, and CFD modeling offers this prospect.

• CFD could be used to study the effects of turbulence on particle transport and deposition.

Turbulence and modelling • Turbulence is a well studied phenomenon and one

of the last unsolved problem of classical physics.• Turbulence is a property of fluid flow not a

property of fluids.• A turbulent flow has a broad range of spatial and

temporal scales.• No analytical techniques exists for solving realistic

turbulent flows.• Numerical methods are limited by the finite power

of computers (N~Re9/4 !!).• Hence modeling is essential and there are plenty of

models, which one to choose?? • LES, RANS etc.

Caricature of turbulenceBig whorls have little whorls Big whorls have little whorls That feed on their velocity,That feed on their velocity,And little whorls have lesser whorlsAnd little whorls have lesser whorlsAnd so on to viscosity--- L F RichardsonAnd so on to viscosity--- L F Richardson

3,2,1;1

ix

u

x

u

xx

puu

xt

u

i

j

j

i

jiji

j

i

0

i

i

x

u

The equation for momentum and continuity (Navier-Stokes equations)

In the large-eddy simulation (LES) approach, one gets rid of the scales of wavelength smaller than the grid mesh x by applying an appropriately chosen low-pass filter characterized by the function G to the flow to eliminate thefluctuations on sub-grid scales

xxxxuxu

dG ),()()(

3,2,1 where

1

i

Tx

u

x

u

xx

puu

xt

uij

i

j

j

i

jiji

j

i

Where the sub-grid scale (SGS) tensor Tij is given by

jijiij uuuuT Most of the effort in turbulence modelling deals with constructing an appropriate model for Tij.

The filtered equation resembles RANS equation forthe mean flow field, but the SGS term is different.

SCST TijTij2 where2

Recent works suggest SGS models can be substituted by numerical diffusion (Boris et al. 1992)

Smagorinsky2

2 ST

Visual representation of LES

Comparison between LES and RANS

Lagrangian model of turbulent dispersion

• Following fluid elements instead of solving the advection diffusion equation

• There is a one to one correspondence between the Lagrangian and Eulerian methods

),(),(),( 2 tttt

xxux

]),([)()( ttttdt

dE xuux

Eulerian

Lagrangian

KS as a model of turbulent dispersion• It is Lagrangian model of turbulent dispersion that takes

into account the effects of spatio-temporal flow structure on particle dispersion (Fung et al. 1998)

• It is a direct Lagrangian purpose built model that is very robust and can be used in fully turbulent flow or transitional flows.

• It is unified Lagrangian model of one, two and indeed multi particle (Khan et al. 2002) turbulent dispersion

• It can be easily used as sub-grid model (Flohr et al. 2000) for LES code thus enabling complex geometry to be taken into account

• Compared to other methods such as DNS and RANS it is a low cost method in terms of computation.

Basics of Kinematic simulation (KS) in 3D

• Velocity field in KS is simulated by a large number of random Fourier modes

• The modes vary in space and time over a large number of realizations.

• Velocity field is incompressible by construction

0),( tE xu

• The energy spectrum is

• Existence of straining and streaming flow structures• Time dependence of these structures determined by

12;~)( pkkE p

)(~)( 2nnnn kEkk

Individual fluid element trajectories x(t) are calculated by integrating

]),([)()( ttttdt

dE xuux

in individual realizations of an Eulerian turbulent-like velocity uE(x,t) which generated as follows (Fung et al. 1998)

)]sin(

)cos([),(1

t

tt

nnnn

N

nnnnnE

k

xkkB

xkkAxu

The positive amplitudes An and Bn are chosen according to

nnnn kkEBA )(22

0 nnnn kBkA

LES in a turbulent ribbed channel flow

• We use a validated LES code to simulate particle deposition in a ribbed channel (Tucker et al. J. Fluid. Eng. 2001)

• We use the Yoshizawa k-l model (Yoshizawa Phy. Fluids. 1986) for sub-grid modelling of the Eulerian velocity field. (=CTlke

½ )• We use KS to model the sub-grid velocity field as

seen by the particles and compute the Lagrangian particle dispersion statistics.

kkkkCkE LK for )( 3/53/2

4/13 / and /1 kkL lke /2/3

Coupling LES and KS

KSLESTot uuu

KSLES uu

SpTotpTotDppp Cdmdt

dLuuuuu 2

8

1

NS KS

)()( ttdt

dpux

3

687.0

2

10Re5.0

Re15.01Re/24

/Re

Re3/4

p

ppd

pTotpfp

pdppp

C

duu

Cd

Geometry and the flow direction in the ribbed channel.

2D section of the Computational grid

Height of rib hh

Channel width LLyy

Stream-wise dimension LLxx

Span-wise dimension LLzz

Reynolds number ReRe==uu00LLyy//Number of grid pointsKolmogorov time scale t t

=T/Re=T/Re(1/2)(1/2) with T=LT=Lxx/u/u00

Time step ttParticle density pp

Particle diameter dd p p

Number of particles tracked

6.35 mm6.35 mm

10h10h

20h20h

10h10h

70007000

121 X 111 X 33 (67)121 X 111 X 33 (67)

4.2 104.2 10-04-04 sec sec

1.0X101.0X10-04-04 sec sec

500,1000 Kg/m500,1000 Kg/m33

87.0,8.787.0,8.7m(10m(10-06-06 m) m)

1000010000

Table I. Computational domain and simulation parameters.

Particle trajectories in a channel with a single rib of height h=0.00635 mm

Mean velocity contours and streamlines in a ribbed channel flow

Velocity vector on a plane parallel to the span-wise direction.

Plot of mean square particle displacements along the x (top), y (middle) and z (bottom) directions, respectively, from a fixed point on the channel mid-plane using LES with KS (circle) and without KS (triangle) sub-grid model. Here p= 500 Kg/m3 (left plots), p=1000 Kg/m3(right plots) and dp=87.0m.

Plot of mean square relative particle separation along the x(x=x1-x2) (top), y(y=y1-y2) (middle) and z (z=z1-z2)(bottom) directions, respectively, from a fixed initial separation on the channel mid-plane (y(t=0)=0) using LES with KS (circle) and without KS (triangle) sub-grid model. Here p= 500Kg/m3 (left plots), p=1000 Kg/m3(right plots) and dp=87.0m.

Particle concentration at x/h=-0.5 (top), 0(middle) and 0.6(bottom) using LES with KS (left plots) and without KS(right). Here p=500 Kg/m3(right plots) and dp=87.0m (p/ t.

Particle concentration at x/h=-0.5 (top), and 0.6(bottom) using LES with KS (left plots) and without KS(right). Here p=500 Kg/m3 and dp=87m (p/ t.

Particle concentration at x/h=-0.5 (top), 0(middle) and 0.6(bottom) using low resolution LES with KS (red) and high resolution LES without KS (Blue). Here p=1000 Kg/m3 and dp=8.70m (p/ t.

Particle concentration at x/h=-0.5 (top), 0(middle) and 0.6(bottom) using low resolution LES with KS (blue) , without KS(green) and high resolution LES (red). Here p=1000 Kg/m3, dp=8.70m and p/ t.

Results and discussions• Our results show that both the deposition and

dispersion of particles are affected by the sub-grid flow structures.

• The smaller the relaxation time the more sensitive the particle statistics to the sub-grid structures.

• KS model introduces flow structures with finite range of scales suitable for finite Re number turbulence

• Our results agree qualitatively with previous studies using a combination of DNS and LES(Armenio et al. Phys. Fluids. 1999)

Work in progress

• Introduce KS model inside FLUENT solver and simulate inertial particle deposition in a simple model of airway

• LES + KS code to simulation of inertial particle deposition in a simple airway model with bifurcating outlets