Transition to turbulence in an oscillatory flow through stenosis

Abstract

Onset of flow transition in a sinusoidally oscillating flow through a rigid, constant area circular pipe with a smooth sinusoidal obstruction in the center of the pipe is studied by performing direct numerical simulations, with resolutions close to the Kolmogorov microscales. The studied pipe is stenosed in the center with a 75% reduction in area in two distinct configurations—one that is symmetric to the axis of the parent pipe and the other that is offset by 0.05 diameters to introduce an eccentricity, which disturbs the flow thereby triggering the onset of flow transition. The critical Reynolds number at which the flow transitions to turbulence for a zero-mean oscillatory flow through a stenosis is shown to be nearly tripled in comparison with studies of pulsating unidirectional flow through the same stenosis. The onset of transition is further explored with three different flow pulsation frequencies resulting in a total of 90 simulations conducted on a supercomputer. It is found that the critical Reynolds number at which the oscillatory flow transitions is not affected by the pulsation frequencies. The locations of flow breakdown and re-stabilization post-stenosis are, however, respectively shifted closer to the stenosis throat with increasing pulsation frequencies. The results show that oscillatory physiological flows, while more stable, exhibit fluctuations due to geometric complexity and have implications in studies of dispersion and solute transport in the cerebrospinal fluid flow and understanding of pathological conditions.

Appendices

Appendix 1: Mesh sensitivity analysis

Spatial and temporal resolutions in the present study were chosen based on the outcomes of previous studies (Jain 2016; Jain et al. 2016), but due to the oscillating nature of the flow, mesh-dependent changes are nevertheless quantified here.

Simulations with three different resolutions were conducted on the eccentric stenosis with the highest Re of 2100 and pulsation frequency $\omega _1$. Table 1 lists the 3 different resolutions of the eccentric stenosis, $\delta x$, $\delta t$, the resulting number of cells along the stenosis throat and corresponding ratios ($l^+$ and $t^+$) against Kolmogorov scales. The number of cores of the Hazel Hen utilized for each case and the time taken for the computation of each cycle are also listed.

Table 1 Three different resolutions and corresponding number of cells for eccentric stenosis, and the Kolmogorov microscales. Compute time for each cycle is shown in minutes

Fig. 17

Axial centerline velocity at $x=4D$ in eccentric stenosis with $Re=2100$ and pulsation frequency $\omega _1$ for two cycles computed by HR, NR and LR simulations

Instantaneous velocity during 9th and 10th cycles computed by HR, NR and LR simulations is shown in Fig. 17. Qualitatively similar fluctuations are captured by all the resolutions, but it should be noted that the LR results contain some spurious fluctuations even during the acceleration phase that are numerical instabilities in LBM. The simulation of $Re=2100$ at LR brings lattice Mach number to the stability limit that results in compressibility errors in the LBM BGK scheme. With a different tuning of parameters and with the employment of multiple time relaxation schemes, these errors may reduce but the simulation would be marginally stable. The flow computed by NR and HR resolutions is similar, and whether resolutions down to the Kolmogorov scales should be employed or not for a particular case will follow the recommendations provided by Jain (2016) and Moin and Mahesh (1998).

Appendix 2: Influence of pipe length and outflow boundary conditions

The onset of transition in an oscillatory flow can be influenced by the length of the pipe upstream and downstream from stenotic throat, and the prescription of outflow boundary conditions can also affect transition. The pre- and post-stenotic regions of the vessel in this study were extended by 12 diameters, and a zero pressure gradient was maintained at the outlet. An additional sensitivity analysis was performed to ensure the sufficiency of this extension and the outflow boundary condition. Three additional simulations were performed on the eccentric case with highest Reynolds number of 2100 and oscillation frequency $\varOmega _1$² with 25 diameter extension on each side of the stenosis, with 3 different outflow boundary conditions, namely zero pressure, zero pressure gradient and convective outflow.

Fig. 18

Vorticity field during observation point P3 in the eccentric stenosis at ${Re}=2100$ and $\varOmega _1$ and 25 diameter extensions on each side of the stenosis throat

Fig. 19

Cross-sectional pressure contours one diameter to the left of the outlet (${x}=24{D}$) with three different outflow boundary conditions in eccentric stenosis with ${Re}=2100$ and pulsation frequency $\varOmega _1$ during observation point P3 with 25 diameters extension of pipe on each side of the stenosis throat

The highest resolutions were employed resulting in about 1.6 billion cells, and simulations were conducted utilizing all the capacity of the SuperMUC-NG ($304\,128$ cores), a petascale system recently installed at the LRZ in Munich, Germany. Figure 18 shows the vorticity field across xz plane of the eccentric stenosis during observation point P3 with zero pressure gradient outflow condition. Beyond the turbulent activity, a laminar flow is observed up to whole length of the pipe. Figure 19 shows pressure contours with three different outflow conditions, at a yz cross-sectional plane one diameter left of the outflow, i.e., ${x}=24{D}$. Subtle differences are seen in zero pressure outlet, whereas zero pressure gradient and convective outflow look identical except for minor curvature differences at places. The negligible differences are attributed to the difference in extrapolation schemes of the LB algorithm. Because of their similar accuracy and stabilization of flow beyond $x=\pm 6D$ in all the cases, the boundary conditions do not play a role in the onset of transition, and an extension of 12D from stenosis throat is thus considered sufficient. It is however remarked that in a configuration with smaller pipe lengths, where flow exhibits fluctuations or even minor vortices near the outlet, the outflow boundaries are expected to influence the results, and such an analysis is left for future studies.