Mathematics and Computer Science Division

Argonne National Laboratory, Argonne, IL

**Henry M. Tufo**

Mathematics and Computer Science Division

Argonne National Laboratory, Argonne, IL

We are interested in parallel algorithms and software to enable accurate simulation of unsteady incompressible fluid flows in general three-dimensional geometries. The difficulty with this class of problems stems from a number of features of the governing Navier-Stokes equations. First, the highest order derivative is typically multiplied by a small number (the inverse of the Reynolds number). This singular perturbation results in thin boundary layers and gives rise to disparate length scales to be resolved by the numerical grid. Second, the equations are nonlinear, which leads to thin internal layers having unknown and possibly time varying positions. Finally, the incompressibility constraint must be satisfied at all times, implying global coupling of degrees-of-freedom at each time step.

The following figures illustrate numerical solutions computed
with the spectral element method,
which is a high-order weighted residual technique in which the solution,
data, and geometry within each element are represented by tensor product
polynomials.
Inter-element function continuity is determined by the governing equations.
For incompressible flows, the velocity is *C*^{0} continuous,
while the pressure may be discontinuous.
The spectral element method method is naturally block-structured and enjoys
excellent data re-use on modern cache-based architectures. In fact, most
of the spectral element residual evaluations can be cast as matrix-matrix
products and therefore attain good serial performance.
The relatively low communication requirements resulting from the
*C*^{0} continuity yield a high degree of parallel efficiency.

The above movie illustrates the interaction of a flat-plate boundary layer flow
with an isolated hemispherical roughness element, as studied experimentally
by Acalar and Smith (*JFM*, ** 175 **, 1987).
Above Reynolds number 450, the flow transitions from a steady state to the
steady-periodic, illustrated on the left for Re=700.
A time trace of the vertical
velocity signal at (x,y,z) = (20,0,1.7)R is shown on the right for Re=800.
The plan and profile views (top) show pressure mapped onto the vortex
surfaces at Re=850, and were generated with the help of
Mike Papka
in the
MCS Futures Lab

Of interest is the creation of the hairpin vortices that form an
interlacing pattern in the wake of the hemisphere and lift away from the wall.
The vortices are stretched by the shearing action of the boundary layer
since the tails remain in the low-speed (near-wall) region of the flow
while the heads are entrained in the high-speed region.
This simulation employed 1021 elements of order 13, (2.2 million gridpoints)
and was computed on the 512-node Intel Paragon at Caltech. Simulation time
for a single shedding cycle is about 3 hours.
The Reynolds number is 700, based upon free-stream velocity and hemisphere
radius. The boundary layer thickness is roughly equal to the radius.
The vortices are identified using the definition of a vortex
developed by Jeong and Hussain in (*JFM*, ** 285 **, 1995).

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Additional hairpin vortex animations.
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Additional hairpin vortex images.
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Transition in a stenosed carotid artery.
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Vortex generation in an oscillatory boundary layer.
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Animation of heat transfer augmentation.
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Nonconforming spectral element results.
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Spatio-temporal chaos collaboration with Caltech-Duke.
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Related Articles:

- H.M. Tufo, P.F. Fischer, M.E. Papka, and K. Blom “ Numerical Simulation and Immersive Visualization of Hairpin Vortices ,”
- H.M. Tufo and P.F. Fischer, “ Fast Parallel Direct Solvers For Coarse Grid Problems ,”
- P.F. Fischer, N.I. Miller, and H.M. Tufo, “ An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows,” to appear in IMA Volumes in Mathematics and its Applications, "Parallel Solution of Partial Differential Equations" Petter Bjorstad and Mitchell Luskin (eds.), Springer-Verlag New York, Inc. (2000)
- P.F. Fischer, An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations, J. of Comp. Phys. 133 84-101 (1997).

Paul F. Fischer, MCS Division

Argonne National Laboratory

Argonne, Illinois 60439

E-mail: fischer@mcs.anl.gov

(630) 252-6018

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Last update: March 3, 2000 (pff)