Multi-time-lag PIV analysis of steady and pulsatile flows in a sidewall aneurysm

Abstract

The effect of inflow waveform on the hemodynamics of a real-size idealized sidewall intracranial aneurysm (IA) model was investigated using particle imaging velocimetry (PIV). For this purpose, we implemented an error analysis based on several PIV measurements with different time lags to ensure high precision of velocity fields measured in both the IA and the parent artery. The relative error measured in the main part of the circulating volume was <1 % despite the three orders of magnitude difference of parent artery and IA dome velocities. Moreover, important features involved in IA evolution were potentially emphasized from the qualitative and quantitative flow pattern comparison resulting from steady and unsteady inflows. In particular, the flow transfer in IA and the vortical structure were significantly modified when increasing the number of harmonics for a typical physiological flow, in comparison with quasi-steady conditions.

Appendix: Laser thickness correction

Due to its finite thickness, h, the laser sheet enlighten particles located in different out-of-planes distances from the center of the measurement plane. The computation of the velocity field from the recorded pictures by means of cross-correlations is therefore influenced by all the velocities in the illuminated sheet. Assuming a cylindrical symmetry of the actual velocity profile $v_x(y,z) = v_x(s)$ with $s = \sqrt{y^2+z^2}$, the measured velocity profile in the symmetry plane at the $j^{\text {th}}$ radial measured location, $v^{\text {mes}}_z(y_j)$, can be written as a weighted sum

$$\begin{aligned} v_x^{\text {mes}}(y_j)& = \int v_x\left( \sqrt{y_j^2+z^2}\right) w(z)\text {d}z\nonumber \\& = \sum _{n\geqslant 0} \frac{w_n}{n!}\left. \frac{\partial ^n v_x}{\partial z^n}\right| _{(z,y) = (0,y_j)} . \end{aligned}$$

(10)

w(z) weights the effect of each illuminated particle on the computed velocity field according to their out-of-plane distance, z. The second equality comes from a Taylor expansion of $v_x$ at fixed $y = y_j$ and for $z = 0$. $w_n = \int z^nw(z)\text {d}z$ is the nth momentum of w(z).

For a uniform weighted function¹¹ $w(z) = h^{-1}\text {rect}\left( z/h\right)$ (i.e. assuming that all the enlighten particles have the same effect on the computed velocity fields) and keeping only the first $n\leqslant 2$ components of the Taylor expansion in (10), we can write

$$\begin{aligned} v_x^{\text {mes}}(y_j)\approx v_x(y_j)+\left. \frac{h^2}{24s}\frac{\text {d} v_x}{\text {d} s}\right| _{s = y_j}. \end{aligned}$$

(11)

Using the finite difference approximation $\text {d} v_x /\text {d} s|_{s = y_j}\approx [v_x(y_{j+1})-v_x(y_{j-1})] /(y_{j+1}-y_{j-1})$, Eq. (11) becomes a linear system which can be solved for¹² $v_x(y_j)$. The results shown in Figs. 3 and 8a assume a measured laser thickness $h = 1\,\hbox {mm}$.