Axisymmetric compact finite-difference lattice Boltzmann method for blood flow simulations

An axisymmetric compact finite-difference lattice Boltzmann method is proposed to simulate both Newtonian and non-Newtonian flow of blood through a lumen. The curvature of the arteries could be accurately resolved using body-fitted mesh owing to the proposed finite-difference formulation. The axisymmetric nature of the flow, as well as the non-Newtonian nature of blood, are incorporated into the lattice Boltzmann equation using separate source terms. Using Chapman-Enskog expansion it is shown that the resulting lattice Boltzmann equation with these additional source terms recovers the macroscopic axisymmetric hydrodynamic equations. The solver is verified for (1) steady inflow of a Newtonian fluid through a stenosed lumen, (2) temporally developing pulsatile flow (Womersley flow) through a straight lumen with Newtonian fluid, and (3) steady inflow of a non-Newtonian fluid through a straight lumen. The solver is then applied to simulate the steady flow of a non-Newtonian fluid through a stenosed lumen, and it was found that a smaller recirculation zone and lower WSS values are obtained when compared with the flow of a Newtonian fluid. The capability of the solver to simulate spatially developing (velocity-driven) pulsatile flow is then demonstrated by simulating physiological pulsatile flow through an axisymmetric abdominal aortic aneurysm. From this simulation, the cycle-averaged wall shear stress is observed to have a steep gradient going from a minimum (negative) to a maximum (positive) value towards the distal end of the aneurysm, which is prone to the risk of rupture. An iterative procedure to select the geometric and flow parameters for unsteady inflow condition in the lattice Boltzmann method framework is demonstrated that accurately resolves all the timescales to achieve incompressibility. Overall, the present solver seems to be promising to simulate axisymmetric flow of blood with steady and pulsatile inflows while considering the blood rheology.

Figure 1

Geometry of the axisymmetrically stenosed lumen.

Figure 2

Grid convergence study for the present numerical simulation of the steady axisymmetric flow through the stenosis with $S_0=D, {\alpha }_s=75\%$ at $\text{Re}=500$.

Figure 3

The body-fitted mesh, with a resolution of $360\times 60$ in the $z$ and $r$ directions, respectively, for the case of $\text{Re}=500$. In the figure, every third grid point in the $z$ direction and every sixth grid point in the $r$ direction are shown.

Figure 4

Comparison of axial velocity profiles at different axial locations obtained from the present LBM (lines) with the results of Varghese et al. [70] (points) and experimental data of Ahmed and Giddens [71] (dashed lines) for steady axisymmetric flow through the stenosis with $S_0=D, {\alpha }_s=75\%$ at $\text{Re}=500$.

Figure 5

The axial variation of WSS obtained from the present LBM for steady axisymmetric flow through the stenosis with $S_0=D, {\alpha }_s=75\%$ at $\text{Re}=500$.

Figure 6

Comparison of normalized axial velocity profiles obtained from the present LBM (lines) with the analytical solution from Eq. (46) (circles) at different times $t=n{T}_p/16$ over a period for Womersley flow at $\text{Re}=1200$ and $\text{Wo}=7.927$.

Figure 7

(a) Volumetric flow-rate wave form as recorded by Suh et al. [74] at the infrarenal (IR) location of the abdominal aorta under rest condition. (b) Inlet axial velocity profiles obtained using Eq. (47) at various instants of one cardiac cycle that correspond for the wave form given in panel (a).

Figure 8

Geometry of the model abdominal aortic aneurysm (AAA).

Figure 9

Temporal variation of contours of vorticity over a cardiac cycle at $\text{Re}=264$ and $\text{Wo}=12$. The geometric parameters considered are $W=0.5D$ and $H=0.5D$. The nondimensional vorticity values range from $-20$ (blue) to 20 (red), and the axial range is $-3<z/D<7$.

Figure 10

(a) Axial variation of cycle-averaged WSS. (b) Axial variation of OSI. The values are compared with the results of Gopalakrishnan [27]. The simulation parameters considered are $W=0.5D, H=0.5D, \text{Re}=264, \text{Wo}=12$.

Figure 11

Comparison of observed order of accuracy with theoretical value at different mesh levels for the present axisymmetric CFDLBM. In this figure the values are plotted on a log-log scale.

Figure 12

Comparison of normalized axial velocity profiles from present model (lines) with the power-law solution (symbols) for different value of $n$ with $\text{Re}=100$.

Figure 13

Comparison of axial velocity profiles at different axial locations obtained from the present LBM for Newtonian flow (solid line) and shear-thinning flow (dashed lines) for steady axisymmetric flow through a stenosed lumen with $S_0=D, {\alpha }_s=75\%$ at $\text{Re}=500$.

Figure 14

Comparison of WSS obtained from the present LBM for the Newtonian flow (solid line) and shear-thinning flow (dashed lines) for steady axisymmetric flow through a stenosed lumen with $S_0=D, {\alpha }_s=75\%$ at $\text{Re}=500$.