View
228
Download
0
Tags:
Embed Size (px)
Citation preview
A combined DEM-CFD approach to the simulation of blood flow
Colin ThorntonUniversity of Birmingham
Mos BarigouChemical Engineering
Gerard NashMedical School
SEM image of blood cells
Blood flownon-Newtonian fluidnon-Newtonian fluid
• Plasma (92% water)– Simulated as a continuum:
Newtonian fluidNewtonian fluid
• Blood cells– Erythrocytes (red blood cells RBC):5x1012/L– Thrombocytes(platelets):3x1011/L– Leukocytes (white blood cells WBC): 9x109/L – Simulated as a discrete particulate phase
Background
RBC
WBC
Platelets
The technique can be used both for dispersed systems in which the particle-particle interactions are collisional and compact systems of particles with multiple enduring contacts. Consequently, although particle systems may have the superficial appearance of behaving like a gas, a liquid or a solid when observed at the macroscopic scale, all these different states can be investigated using DEM.
The Discrete Element Method (DEM) is a numerical simulation technique appropriate to systems of particles in which the interactions between contiguous particles are modelled as a dynamic process and the time evolution of the system is advanced by applying a simple explicit finite difference scheme to obtain new particle positions and velocities.
Methodology - (DEM-CFD Simulations)
PARTICLE DYNAMICS
Ax
Bx
A
B
1x
2x
n
t
N
N
TT
(2D)
Use a small timestep (based on the Rayleigh wave speed of the solid particles) to advance the simulation in time, with t some fraction of the critical timestep. (We do not want to transfer energy across a system faster than nature.)
s1tthenmm1RifG
Rtt cc
Ax
Bx
A
B
1x
2x
Update contact forces
nkNNNN
tnxx iAi
Bi
tnxxkNN iAi
Bin
tkTTTT tRRtxx AABBi
Ai
Bi
tRRtxxkTT AABBi
Ai
Bit
The normal and tangential stiffnesses may be defined by linear or non-linear springs or by algorithms based on theoretical contact mechanics.
n
t
N
N
TT
Update particle positions
i
ci
i gm
Fx
I
RTc
txxx iii t
txxx iii t
Check for new contacts and contacts lost,
If the distance between the centres of two particles is equal or less than the sum of the two radii then there is contact.
Repeat cyclic calculations of updating contact forces and particle motions.
contact interactions
non-adhesive spheres
normal stiffness – Hertz (1896)
tangential stiffness – Mindlin and Deresiewicz (1953)
auto-adhesive spheres
normal stiffness – Johnson, Kendal and Roberts (1971)
tangential stiffness – Thornton (1991), Savkoor and Briggs (1977)
What about the fluid ?
A semi-implicit finite difference technique, employing a staggered grid, is used for discretising the compressible Navier-Stokes equation on an equi-distant Cartesian grid.
A staggered grid is used because the pressure and porosity (scalars) are defined at the centre of each computational fluid cell but the fluid velocity components (vectors) are defined at the cell faces.
A standard, first-order accurate, upwind scheme is used to discretise the convective momentum fluxes.
The solution of each time step Δt, using the voidage and particle velocity field from the discrete particle scheme, evolves through a series of computational cycles consisting of (i) explicit calculations of fluid velocity components for all fluid cells and (ii) implicit determination of pressure distributions using an iterative procedure.
7 6 6 6 6 6 6 6 7
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
3 1 1 1 1 1 1 1 3
7 4 4 4 4 4 4 4 7
1 interior fluid cell, no boundary conditions
2 impermeable wall, free slip boundaries
3 impermeable wall, no slip boundaries
4 specified gas velocity influx wall cell
5 prescribed pressure outflow wall cell, free slip
6 continuous gas outflow wall cell, free slip
7 corner cell, no boundary conditions
8 periodic boundary cell
computational fluid cells
particle equations of motion
0ut f
f
gFpuut
uffpfff
f
total force acting on particle i
torque applied to particle i
iiifpicii xmgmfff
iii IT
fluid-particle interaction force difpipifpi fVpVf
fluid continuity and momentum equations
c
n1i fpi
fp V
fF
c
total particle-fluid interaction force per unit
volume
volume of computational fluid cell
1jijij
2j
2pi
fDidi vuvu4
dC
21
f
2
Relog5.1exp65.07.3
2pi10
2
5.0pi
Di Re
8.463.0C
Di Felice (1994)
drag force
1j
corrects for the presence of other particles
fluid drag coefficient for a single unhindered particle
and the dependence on the flow
particle Reynolds numbers
ijjpifpi
vudRe
Simulation strategy
Particles leaving from the exit are returned into the entrance with same velocities Fluid velocities are duplicated between relevant grid layers to ensure the continuity of blood flow
particles return to the entrance
fluid phase
fluid phase
particles leave from the exit
Periodic boundary conditions
Running procedure
Fluid Only(Newtonian fluid)
Fluid & RBC(Non-adhesive RBC)
z
y
x
W = 0.2 mm
H = 0.2 mm
L = 1 mm
Fluid & ARBC(Adhesive RBC)
DEM computational details
u
particle numbers N 13000 (2D) 40000 (3D)
particle concentration
1.0e15/m3 same as RBC concentration in blood flow
particle diameter dp 8 m assumed as spherical !!!
density p 1050 kg/m3
surface energy 2.0e-6 J/m2 between RBC and RBC
friction coefficient 0.1
Poisson's ratio 0.25
Young’s modulus E 1.0e6 Pa
time step Δt 1.3e-7 sec
solid fraction 0.269 real value in blood
fluid density f 1050 kg/m3 plasma assumed as water
fluid viscosity f 1.0e-3 kg/m.s
fluid pressure p 1000 Pa
average fluid velocity 0.01, 0.1, 0.001 m/s
channel dimension L/D/W 1mm / 75m / 175m 2D flow
L/H/W 1 mm / 200m / 200m 3D flow
computational grid size
25 m about 3 times the particle diameter
computational grid number
40 * 3 * 7 (2D)
40 * 8 * 8 (3D)
The fluid cells whose centre points are on the dashed square have the same hydraulic radius. The fluid velocity profiles show good agreement with the power law fitting curves by using rH.
W
O H
rH
RH
r
Gj,k
0.0 0.2 0.4 0.6 0.8 1.0
rH/RH
0.0
0.5
1.0
1.5
2.0
Fluid OnlyFluid & RBCFluid & ARBCPower law fluid (n=1)Power law fluid (n=0.619)Power law fluid (n=0.297)
n/n
uave = 0.01 m/s
n
n
HR
r
n
n
u
u1
11
13
power law fluid velocity profile
(different flow rates)
All data points can be fitted by one power law curve with the power index of 0.604, which indicates that the power index for this case is independent of the average flow rate
Fluid & RBC
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
¨ ± =0.01m/s¨ ± =0.1m/s¨ ± =0.001m/sPower law fluid (n=0.604)
rH/RH
n/n
(different flow rates)
The power index is very dependent on the average fluid velocity When the average flow rate decreases, the power index decreases and at low flow rates the velocity profile corresponds to plug flow
Fluid & ARBC
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
¨ ± =0.01m/s, DEM results¨ ± =0.01m/s, power law fluid (n=0.297)¨ ± =0.1m/s, DEM results¨ ± =0.1m/s, power law fluid (n=0.492)¨ ± =0.001m/s, DEM results¨ ± =0.001m/s, power law fluid (n=0.006)
rH/RH
n/n
0.0 0.2 0.4 0.6 0.8 1.0
rH/RH
0.0
0.5
1.0
1.5
2.0
Fluid OnlyFluid & RBCFluid & ARBCPower law fluid (n=0.974)Power law fluid (n=0.884)Power law fluid (n=0.495)
n/n
reduce particle concentration by half
( average velocity = 0.01 m/s )
For Fluid & RBC, the power index has increased to 0.884 from 0.619.For Fluid & ARBC, the power index has increased to 0.495 from 0.297.
(a) Without adhesion (different colours indicate different sizes of clusters)
(b) Without adhesion, three largest clusters (1,405, 1,827, 2,078 particles)
(c) With adhesion, only one large cluster/agglomerate (21,114 particles)
clustering
With adhesion, the particles tend to form one large cluster/agglomerate spreading all through the flow channel
uave = 0.01 m/s
Without adhesion, three largest clusters (995, 1,777, 2,137 particles)
With adhesion, three largest clusters (869, 1,736, 6,658 particles)
u0 = 0.1 m/s
Conclusions
The power law index (n) is independent of average flow rate if the particles are non-adhesive.
For autoadhesive particles the power law index is very dependent on flow rate and at low flow rates plug flow occurs.
The power law index increases with reduced particle concentration for both non-adhesive and autoadhsive particles.
More work needs to be done in the context of general suspension rheology.
Further work
Perform simulations in a cylindrical tube which has flexible walls. This can be done using the Immersed Boundary Method. (Done.)
Consider particle shape. In terms of a soft solid, RBC’s can be considered as biconcave discs. However, a red blood cell is essentially a viscoelastic membrane filled with a concentrated solution of haemoglobin. For this approach to modelling RBC’s see-
Dzwinel, Boryczko & Yuen (2003) J. Colloid and Interface Sci. 258, 163-173.
Liu & Liu (2006) J. Comp. Physics 220, 139-154.
Freund (2007) Phys. Fluids 19,023301.
Thank you for your attention.