Paul F. Fischer

High-order methods for incompressible fluid flow

Preface 1. Fluid mechanics and computation: an introduction 2. Approximation methods for elliptic problems 3. Parabolic and hyperbolic problems 4. Mutidimensional problems 5. Steady Stokes and Navier-Stokes equations 6. Unsteady Stokes and Navier-Stokes equations 7. Domain decomposition 8. Vector and parallel implementations Appendix A. Preliminary mathematical concepts Appendix B. Orthogonal polynomials and discrete transforms.

Wall-induced forces on a rigid sphere at finite Reynolds number

We perform direct numerical simulations of a rigid sphere translating parallel to a flat wall in an otherwise quiescent ambient fluid. A spectral element method is employed to perform the simulations with high accuracy. For

Filter-based stabilization of spectral element methods.

Re\,{ , we observe the lift coefficient to decrease with both Reynolds number and distance from the wall. In this regime the present results are in good agreement with the low-Reynolds-number theory of Vasseur & Cox (1977), with the recent experiments of Takemura & Magnaudet (2003) and with the simulations of Kim et al. (1993). The most surprising result from the present simulations is that the wall-induced lift coefficient increases dramatically with increasing

Direct numerical simulation of stenotic flows, Part 1: Steady flow

Re

DIRECT NUMERICAL SIMULATION OF TRANSITIONAL FLOW IN A STENOSED CAROTID BIFURCATION

above about 100. Detailed analysis of the flow field around the sphere suggests that this increase is due to an imperfect bifurcation resulting in the formation of a double-threaded wake vortical structure. In addition to a non-rotating sphere, we also simulate a freely rotating sphere in order to assess the importance of free rotation on the translational motion of the sphere. We observe the effect of sphere rotation on lift and drag forces to be small. We also explore the effect of the wall on the onset of unsteadiness.

Direct numerical simulation of stenotic flows. Part 2. Pulsatile flow

Abstract We present a simple filtering procedure for stabilizing the spectral element method (SEM) for the unsteady advection–diffusion and Navier–Stokes equations. A number of example applications are presented, along with basic analysis for the advection–diffusion case.

Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

Direct numerical simulations (DNS) of steady and pulsatile flow through 75% (by area reduction) stenosed tubes have been performed, with the motivation of understanding the biofluid dynamics of actual stenosed arteries. The spectral-element method, providing geometric flexibility and high-order spectral accuracy, was employed for the simulations. The steady flow results are examined here while the pulsatile flow analysis is dealt with in Part 2 of this study. At inlet Reynolds numbers of 500 and 1000, DNS predict a laminar flow field downstream of an axisymmetric stenosis and comparison to previous experiments show good agreement in the immediate post-stenotic region. The introduction of a geometric perturbation within the current model, in the form of a stenosis eccentricity that was 5% of the main vessel diameter at the throat, resulted in breaking of the symmetry of the post-stenotic flow field by causing the jet to deflect towards the side of the eccentricity and, at a high enough Reynolds number of 1000, jet breakdown occurred in the downstream region. The flow transitioned to turbulence about five diameters away from the stenosis, with velocity spectra taking on a broadband nature, acquiring a -5/3 slope that is typical of turbulent flows. Transition was accomplished by the breaking up of streamwise, hairpin vortices into a localized turbulent spot, reminiscent of the turbulent puff observed in pipe flow transition, within which r.m.s. velocity and turbulent energy levels were highest. Turbulent fluctuations and energy levels rapidly decayed beyond this region and flow relaminarized. The acceleration of the fluid through the stenosis resulted in wall shear stress (WSS) magnitudes that exceeded upstream levels by more than a factor of 30 but low WSS levels accompanied the flow separation zones that formed immediately downstream of the stenosis. Transition to turbulence in the case of the eccentric stenosis was found to be manifested as large temporal and spatial gradients of shear stress, with significant axial and circumferential variations in instantaneous WSS.

Spectral Element Methods for Transitional Flows in Complex Geometries

The blood flow dynamics of a stenosed, subject-specific, carotid bifurcation were numerically simulated using the spectral element method. Pulsatile inlet conditions were based on in vivo color Doppler ultrasound measurements of blood velocity. The results demonstrated the transitional or weakly turbulent state of the blood flow, which featured rapid velocity and pressure fluctuations in the post-stenotic region of the internal carotid artery (ICA) during systole and laminar flow during diastole. High-frequency vortex shedding was greatest downstream of the stenosis during the deceleration phase of systole. Velocity fluctuations had a frequency within the audible range of 100-300Hz. Instantaneous wall shear stress (WSS) within the stenosis was relatively high during systole ( approximately 25-45Pa) compared to that in a healthy carotid. In addition, high spatial gradients of WSS were present due to flow separation on the inner wall. Oscillatory flow reversal and low pressure were observed distal to the stenosis in the ICA. This study predicts the complex flow field, the turbulence levels and the distribution of the biomechanical stresses present in vivo within a stenosed carotid artery.

Fast Parallel Direct Solvers for Coarse Grid Problems

Direct numerical simulations (DNS) of stenotic flows under conditions of steady inlet flow were discussed in Part 1 of this study. DNS of pulsatile flow through the 75 % stenosed tube (by area) employed for the computations in Part 1 is examined here. Analogous to the steady flow results, DNS predicts a laminar post-stenotic flow field in the case of pulsatile flow through the axisymmetric stenosis model, in contrast to previous experiments, in which intermittent disturbed flow regions and turbulent breakdown were observed in the downstream region. The introduction of a stenosis eccentricity, that was 5 % of the main vessel diameter at the throat, resulted in periodic, localized transition to turbulence. Analysis in this study indicates that the early and mid-acceleration phases of the time period cycle were relatively stable, with no turbulent activity in the post-stenotic region. However, towards the end of acceleration, the starting vortex, formed earlier as the fluid accelerated through the stenosis at the beginning of acceleration, started to break up into elongated streamwise structures. These streamwise vortices broke down at peak flow, forming a turbulent spot in the post-stenotic region. In the early part of deceleration there was intense turbulent activity within this spot. Past the mid-deceleration phase, through to minimum flow, the inlet flow lost its momentum and the flow field began to relaminarize. The start of acceleration in the following cycle saw a recurrence of the entire process of a starting structure undergoing turbulent breakdown and subsequent relaminarization of the post-stenotic flow field. Peak wall shear stress (WSS) levels occurred at the stenosis throat, with the rest of the vessel experiencing much lower levels. Turbulent breakdown at peak flow resulted in a sharp amplification of instantaneous WSS magnitudes across the region corresponding to the turbulent spot, accompanied by large axial and circumferential fluctuations, even while ensemble-averaged axial shear stresses remained mostly low and negative. WSS levels dropped rapidly after the mid-deceleration phase, when the relaminarization process took over, and were almost identical to laminar, axisymmetric shear levels through most of the acceleration phase.

Transitional Flow at the Venous Anastomosis of an Arteriovenous Graft: Potential Activation of the ERK1/2 Mechanotransduction Pathway

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.