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.