No abstract

The effects of Reynolds and Womersley numbers on the hemodynamics of two simplified intracranial aneurysms (IAs), that is, sidewall and bifurcation IAs, and a patient-specific IA are investigated using computational fluid dynamics. For this purpose, we carried out three numerical experiments for each IA with various Reynolds (Re = 145.45 to 378.79) and Womersley (Wo = 7.4 to 9.96) numbers. Although the dominant flow feature, which is the vortex ring formation, is similar for all test cases here, the propagation of the vortex ring is controlled by both Re and Wo in both simplified IAs (bifurcation and sidewall) and the patient-specific IA. The location of the vortex ring in all tested IAs is shown to be proportional to Re/Wo2 which is in agreement with empirical formulations for the location of a vortex ring in a tank. In sidewall IAs, the oscillatory shear index is shown to increase with Wo and 1/Re because the vortex reached the distal wall later in the cycle (higher resident time). However, this trend was not observed in the bifurcation IA because the stresses were dominated by particle trapping structures, which were absent at low Re = 151.51 in contrast to higher Re = 378.79.

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 – 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80–90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

Non-dimensional parameters are routinely used to classify different flow regimes. We propose a non-dimensional parameter, called Aneurysm number (An), which depends on both geometric and flow characteristics, to classify the flow inside aneurysm-like geometries (sidewalls and bifurcations). The flow inside aneurysm-like geometries can be widely classified into (i) the vortex mode in which a vortex ring is formed and (ii) the cavity mode in which a stationary shear layer acts similar to a moving lid of a lid-driven cavity. In these modes, two competing time scales exist: (a) a transport time scale, Tt, which is the time scale to develop a shear layer by transporting a fluid particle across the expansion region, and (b) the vortex formation time scale, Tv. Consequently, a relevant non-dimensional parameter is the ratio of these two time scales, which is called Aneurysm number: An = Tt/Tv. It is hypothesized, based on this definition, that the flow is in the vortex mode if the time required for vortex ring formation Tv is less than the transport time Tt (An ≳ 1). Otherwise, the flow is in the cavity mode (An ≲ 1). This hypothesis is systematically tested through numerical simulations on simplified geometries and shown to be true through flow visualizations and identification of the main vortex and shear layer. The main vortex is shown to evolve when An ≳ 1 but stationary when An ≲ 1. In fact, it is shown that the flows with An ≲ 1 (cavity mode) are characterized by much smaller fluctuations of wall shear stress and oscillatory shear index relative to flows with An ≳ 1 (vortex mode) because of their quasi-stationary flow pattern (cavity mode) compared to the evolution and breakdown of the formed vortex ring (vortex mode).

A simple parameter, called the Aneurysm number (An) which is defined as the ratio of transport to vortex time scales, has been shown to classify the flow mode in simplified aneurysm geometries. Our objective is to test the hypothesis that An can classify the flow in patient-specific intracranial aneurysms (IA). Therefore, the definition of this parameter is extended to anatomic geometries by using hydraulic diameter and the length of expansion area in the approximate direction of the flow. The hypothesis is tested using image-based flow simulations in five sidewall and four bifurcation geometries, i.e., if An ≲ 1 (shorter transport time scale), then the fluid is transported across the neck before the vortex could be formed, creating a quasi-stationary shear layer (cavity mode). By contrast, if An ≳ 1 (shorter vortex time scale), a vortex is formed. The results show that if An switches from An ≲ 1 to An ≳ 1, then the flow mode switches from the cavity mode to the vortex mode. However, if An does not switch, then the IAs stay in the same mode. It is also shown that IAs in the cavity mode have significantly lower An, temporal fluctuations of wall shear stress and oscillatory shear index (OSI) compared to the vortex mode (p < 0.01). In addition, OSI correlates with An in each flow mode and with pulsatility index in each IA. This suggests An to be a viable hemodynamic parameter which can be easily calculated without the need for detailed flow measurements/ simulations.