Modeling particle deposition in a turbulent ribbed channel flow a M A I Khan, a X Y Luo and b F C G A Nicolleau c P G Tucker d G Lo Iacono a Department of Mathematics, University of Glasgow b Department of Mechanical Engineering, University of Sheffield c Civil and Computational Engineering centre, University of Wales, Swansea d Centre 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~Re 9/4 !!). Hence modeling is essential and there are plenty of models, which one to choose?? LES, RANS etc.

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

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 the fluctuations on sub-grid scales

Where the sub-grid scale (SGS) tensor T ij is given by Most of the effort in turbulence modelling deals with constructing an appropriate model for T ij. The filtered equation resembles RANS equation for the mean flow field, but the SGS term is different. Recent works suggest SGS models can be substituted by numerical diffusion (Boris et al. 1992) Smagorinsky

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 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

The energy spectrum is Existence of straining and streaming flow structures Time dependence of these structures determined by

Individual fluid element trajectories x(t) are calculated by integrating in individual realizations of an Eulerian turbulent-like velocity u E (x,t) which generated as follows (Fung et al. 1998)

The positive amplitudes A n and B n are chosen according to

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. (  =C T l  k e ½ ) We use KS to model the sub-grid velocity field as seen by the particles and compute the Lagrangian particle dispersion statistics.

Coupling LES and KS NSKS

Geometry and the flow direction in the ribbed channel. 2D section of the Computational grid

h Height of rib h L y Channel width L y L x Stream-wise dimension L x L z Span-wise dimension L z Re=u 0 L y / Reynolds number Re=u 0 L y / Number of grid points t  =T/Re (1/2) T=L x /u 0 Kolmogorov time scale t  =T/Re (1/2) with T=L x /u 0  t Time step  t  p Particle density  p d p Particle diameter d p Number of particles tracked 6.35 mm 10h20h10h X 111 X 33 (67) sec 1.0X sec 500,1000 Kg/m ,8.7  m( m) Table I. Computational domain and simulation parameters.

Particle trajectories in a channel with a single rib of height h= 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/m 3 (left plots),  p =1000 Kg/m 3 (right plots) and d p =87.0  m.

Plot of mean square relative particle separation along the x(  x =x 1 -x 2 ) (top), y(  y =y 1 -y 2 ) (middle) and z (  z =z 1 -z 2 )(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/m 3 (left plots),  p =1000 Kg/m 3 (right plots) and d p =87.0  m.

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/m 3 (right plots) and d p =87.0  m (  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/m 3 and d p =87  m (  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/m 3 and d p =8.70  m (  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/m 3, d p =8.70  m 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