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A combined DEM-CFD approach to the simulation of blood flow

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

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A combined DEM-CFD approach to the simulation of blood flow

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  1. A combined DEM-CFD approach to the simulation of blood flow Colin Thornton University of Birmingham Mos Barigou Chemical Engineering Gerard Nash Medical School

  2. Background Blood flow non-Newtonian fluid • Plasma (92% water) – Simulated as a continuum: Newtonian 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 RBC WBC Platelets SEM image of blood cells

  3. 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) 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.

  4. PARTICLE DYNAMICS (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.)

  5. Update contact forces The normal and tangential stiffnesses may be defined by linear or non-linear springs or by algorithms based on theoretical contact mechanics.

  6. Update particle positions If the distance between the centres of two particles is equal or less than the sum of the two radii then there is contact. Check for new contacts and contacts lost, Repeat cyclic calculations of updating contact forcesand particle motions.

  7. 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)

  8. 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.

  9. computational fluid cells 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

  10. particle equations of motion total force acting on particle i torque applied to particle i fluid-particle interaction force fluid continuity and momentum equations total particle-fluid interaction force per unit volume volume of computational fluid cell

  11. Di Felice (1994) drag force fluid drag coefficient for a single unhindered particle corrects for the presence of other particles and the dependence on the flow particle Reynolds number

  12. fluid phase particles return to the entrance particles leave from the exit fluid phase Simulation strategy y W = 0.2 mm H = 0.2 mm z x L = 1 mm Periodic boundary conditions • 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 Runningprocedure Fluid Only (Newtonian fluid) Fluid & RBC (Non-adhesive RBC) Fluid & ARBC (Adhesive RBC)

  13. DEM computational details

  14. W Gj,k rH r H O RH power law fluid velocity profile uave = 0.01 m/s • 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.

  15. (different flow rates) Fluid & RBC • 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

  16. (different flow rates) Fluid & ARBC • 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

  17. 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.

  18. 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

  19. 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)

  20. 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.

  21. 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.

  22. Thank you for your attention.

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