A combined DEM-CFD approach
to the simulation of blood flow
Colin Thornton
University of Birmingham
Mos Barigou
Chemical Engineering
Gerard Nash
Medical School
Background
Blood flow
non-Newtonian fluid
• Plasma (92% water)
– Simulated as a continuum:
Newtonian fluid
RBC
WBC
Platelets
SEM image of blood cells
• 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
Methodology - (DEM-CFD Simulations)
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.
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.
PARTICLE DYNAMICS
x B
x1
(2D)
 B
N
T
x A
T
n
N
t
A

x2
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.)
t  tc 
R 
 G
 if R  1mm
then tc  1s
Update contact forces
N  N  N
x B
N  kn

x1
 
A

  x B

x
i
i ni t

n
x A
t
 
 A
A

N  N  kn x B

x
i
i ni t
x2
T  T  T

 
A
 BRB   AR A

  x B

x
t


i
i
i
T  kt 

 B
 
A
 BRB   AR A

T  T  k t x B

x
t


i
i
i
  t
  t
The normal and tangential stiffnesses may be defined by linear or nonlinear springs or by algorithms based on theoretical contact mechanics.
Update particle positions
Fic

i 
x
 gi
 
i t
x i  x i  x
     t
xi  xi  x i t
     t
m
T
Check for new contacts
and contacts lost,
c
N
T
T
R
I
N
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 equidistant 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.
computational fluid cells
7
3
3
3
3
3
3
3
7
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
6
1
1
1
1
1
1
1
4
7
3
3
3
3
3
3
3
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
particle equations of motion
total force acting on particle i
torque applied to particle i
i
fi  fci  ffpi  mig  mix
Ti  Iii
fluid-particle interaction force
ffpi   Vpip  Vpi   f  fdi
fluid continuity and momentum equations
 f 
    u  0
t
f
 fu
   fuu  pf     f  Ffp  f g
t
total particle-fluid interaction force per unit volume
Ffp 
c
n
i1 ffpi
volume of computational fluid cell
Vc
Di Felice (1994)
drag force
2
dpi
1
fdi  CDi f
 2j u j  vi u j  vi  j  1
2
4


fluid drag coefficient for a single unhindered particle
 j   1

4 .8 
CDi  0.63 
0 .5 
Re

pi 

corrects for the presence of other particles
 1.5  log10 Repi 2 

and the dependence on the flow   3.7  0.65 exp  
2


particle Reynolds number
Repi 
 f dpi j u j  vi
s
2
Simulation strategy
y
W = 0.2 mm
H = 0.2 mm
z
x
L = 1 mm
fluid phase
particles
return to the
entrance
Periodic boundary conditions
particles
leave from
the exit
fluid phase
 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
Running procedure
Fluid Only
(Newtonian fluid)
Fluid & RBC
(Non-adhesive RBC)
Fluid & ARBC
(Adhesive RBC)
DEM computational details
particle numbers
N
13000 (2D)
40000 (3D)
1.0e15/m3
particle concentration
same as RBC concentration
in blood flow
particle diameter
dp
8 m
density
p
1050 kg/m3
surface energy
g
2.0e-6 J/m2
friction coefficient

0.1
Poisson's ratio
n
0.25
Young’s modulus
E
1.0e6 Pa
time step
Δt
1.3e-7 sec
solid fraction
1
0.269
fluid density
f
1050 kg/m3
fluid viscosity
f
1.0e-3 kg/m.s
fluid pressure
p
1000 Pa
average fluid velocity
u
0.01, 0.1, 0.001 m/s
L/D/W
1mm / 75m / 175m
2D flow
L/H/W
1 mm / 200m / 200m
3D flow
channel dimension
40 * 3 * 7
(2D)
between RBC and RBC
real value in blood
25 m
computational grid
size
computational grid
number
assumed as spherical !!!
plasma assumed as water
about 3 times the particle
diameter
40 * 8 * 8
(3D)
power law fluid velocity profile
uave = 0.01 m/s

u 3n  1   r

1  
u
n  1   RH




n 1
n




2.0
1.5
W
rH
n/n
Gj,k
1.0
Fluid Only
Fluid & RBC
Fluid & ARBC
Power law fluid (n=1)
Power law fluid (n=0.619)
Power law fluid (n=0.297)
r
O
H
0.5
RH
0.0
0.0
0.2
0.4
0.6
0.8
rH/RH
 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.
1.0
(different flow rates)
Fluid & RBC
2.0
1.5
n/n
1.0
¨ ± =0.01m/s
¨ ± =0.1m/s
¨ ± =0.001m/s
Power law fluid (n=0.604)
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
rH/RH
 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
(different flow rates)
Fluid & ARBC
2.0
1.5
n/n
1.0
¨
¨
¨
¨
¨
¨
0.5
± =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)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
rH/RH
 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
reduce particle concentration by half
( average velocity = 0.01 m/s )
2.0
1.5
n/n
1.0
Fluid Only
Fluid & RBC
Fluid & ARBC
Power law fluid (n=0.974)
Power law fluid (n=0.884)
Power law fluid (n=0.495)
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
rH/RH
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.
uave = 0.01 m/s
clustering
(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)
With adhesion, the particles tend to form one large cluster/agglomerate
spreading all through the flow channel
u0 = 0.1 m/s
Without adhesion,
three largest
clusters (995,
1,777, 2,137
particles)
With adhesion,
three largest
clusters (869,
1,736, 6,658
particles)
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 seeDzwinel, 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.