Numerical Simulation of Transition in Vascular Flows 

                      Paul F. Fischer 

            Mathematics and Computer Science Division 
                 Argonne National Laboratory 

Abstract:

We present algorithms and results for spectral element based simulation 
of transitional (weakly turbulent) flows in general complex domains, with
applications to the simulation of vascular flows.

In the first part of this talk, we present the numerical approach.
The spatial discretization is based upon the spectral element method,
which is a rapidly convergent, high-order weighted residual technique 
employing tensor-product polynomial bases of degree N in each of K 
hexahedral elements.
Because of its low numerical dissipation and dispersion, the method is 
well suited to transitional Reynolds number applications where the 
physical viscosity is small.  For high Reynolds numbers, the Galerkin 
formulation is stabilized using a recently developed filter that is local
and preserves both interelement continuity and spectral accuracy. 
Time advancement is based on a consistent third-order operator splitting
that permits large time steps (typ. a convective CFL of 1--5) and yields
independent convective, viscous, and pressure subproblems to be solved
at each step.  The elliptic viscous and pressure subproblems are solved
iteratively using Jacobi and Schwarz-based preconditioners, respectively.
Key aspects regarding parallelism are also discussed.

In the second part of this talk, we discuss issues that are specific
to the simulation of vascular flows.   We present parallel simulation
results for transitional flow in a carotid artery bifurcation at Re=1000, 
and in an AV graft at Re=1060 and 1820.