Evolution of vortical structures in a curved artery model with non-Newtonian blood-analog fluid under pulsatile inflow conditions

Abstract

Steady flow and physiological pulsatile flow in a rigid 180° curved tube are investigated using particle image velocimetry. A non-Newtonian blood-analog fluid is used, and in-plane primary and secondary velocity fields are measured. A vortex detection scheme (d ₂-method) is applied to distinguish vortical structures. In the pulsatile flow case, four different vortex types are observed in secondary flow: deformed-Dean, Dean, Wall and Lyne vortices. Investigation of secondary flow in multiple cross sections suggests the existence of vortex tubes. These structures split and merge over time during the deceleration phase and in space as flow progresses along the 180° curved tube. The primary velocity data for steady flow conditions reveal additional vortices rotating in a direction opposite to Dean vortices—similar to structures observed in pulsatile flow—if the Dean number is sufficiently high.

Appendix: Analytical solution for physiological pulsatile flow a straight pipe

Equations (3–6) are the governing momentum equation in cylindrical coordinates, boundary and initial conditions and mass flux, respectively. The nonzero initial condition (Eq. 5) is obtained from a parabolic fit to the flow rate at the end of the cycle and is used to eliminate the transient effect faster. The velocity, u _bulk, on right-hand side of Eq. (6) was calculated from a 28-Fourier mode fit to experimental flow rate obtained by an ultrasonic flowmeter. This equation allows omission of the pressure term from calculations.

$$\frac{\partial u}{\partial t} = - \frac{1}{\rho }\frac{\partial p}{\partial x} + \nu \left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r}} \right)$$

(3)

$$u\left( {R, t} \right) = 0\quad \left. {\frac{\partial u}{\partial r}} \right|_{r = 0,t} = 0$$

(4)

$$u(r,0) = a + cr^{2} \quad a = 0.1366,\quad c = - a/R^{2}$$

(5)

$$\int_{0}^{R} {2\pi ru(r,t){\text{d}}r} = u_{\text{bulk}} (t)\pi R^{2}$$

(6)

A Laplace transform is applied to Eqs. (3–6) to convert partial differential equation to ordinary differential equation. For brevity, only transformed form of Eqs. (3 and 6) is provided in Eqs. (7 and 8). The overbar ($\overline{\,\,\,}$) denotes Laplace transform.

$$\frac{{{\text{d}}^{2} \bar{u}(r,s)}}{{{\text{d}}r^{2} }} + \frac{1}{r}\frac{{{\text{d}}\bar{u}(r,s)}}{{{\text{d}}r}} - \frac{s}{\nu }\bar{u}(r,s) = \frac{1}{\mu }\frac{{{\text{d}}\bar{p}(x,s)}}{{{\text{d}}x}} - \frac{1}{\nu }u(r,0)$$

(7)

$$\int_{0}^{R} {r\bar{u}(r,s){\text{d}}r} = \frac{1}{2}\bar{u}_{\text{bulk}} (s)R^{2}$$

(8)

The solution to Eq. (7) is given in Eq. (9)

$$\bar{u}(r,s) = C_{1} \,I_{0} \left( {r\sqrt {s/r} } \right) + Ar^{2} + D$$

(9)

where $C_{1} = - \frac{{AR^{2} + D}}{{I_{0} \left( {R\sqrt {s/r} } \right)}}$, $A = \frac{c}{s}$, $D = \frac{4c\nu }{{s^{2} }} - \frac{\phi \nu }{s} + \frac{a}{s}$, $\phi = \frac{1}{\mu }\frac{{{\text{d}}\bar{p}(x,s)}}{{{\text{d}}x}}$ and $I_{0}$ is the modified Bessel function of zeroth-order first kind. The only unknown in Eq. (9) is ϕ. By substituting $\bar{u}(r,s)$ from Eq. (9) into Eq. (8), ϕ can be calculated. Therefore, Eq. (9) gives the Laplace transform of the velocity profile. In order to invert the $\bar{u}(r,s)$ to obtain an expression for the physical velocity profile, a numerical approach was employed because of complexity of the analytical form. An open-source MATLAB code “Numerical inverse Laplace transform” McClure (2013) based on the Talbot algorithm (Abate and Whitt 2006) is used to perform the inversion. The optimum value for the parameter M (precision) in the code was chosen to ensure that calculated velocity profile satisfies the flow rate at each phase. The slight differences ($\left| {\Delta u} \right|/u_{\text{center}} = 0.3{-}8\,\%$) between the experimental result and the analytical solution in Fig. 7 are mainly due to errors implicit in the Laplace inversion numerical scheme.