| Speaker: | C. I. Christov, University of Louisiana at Lafayette |
| Title: | An Absolutely Stable Operator-Splitting Scheme for the Stream-Function Formulation of Unsteady Navier-Stokes Equations |
| Time: | 3:00-4:00 pm |
| Place: | Blocker 628 |
By introducing a fictitious time in the unsteady stream-function
equation, the latter becomes a higher-order ultra-parabolic
equation. Convergence with respect to the fictitious time
("internal iterations") is obtained, which results in a
fully-implicit nonlinear scheme with respect to the physical time.
For a particular choice of the (fictitious time) step size used in
the internal iterations, the scheme with respect to the physical
time is of second order of accuracy. The internal time stepping is
done via a fractional-step scheme based on a splitting of the
combination of Laplace, bi-harmonic and advection operators. A
judicious time-staggering of the nonlinear advective terms allows
us to prove that the internal time stepping scheme is
unconditionally stable and convergent. We prove that when the
internal iterations converge, the nonlinear, implicit scheme with
respect to the physical time is also unconditionally stable and
second-order accurate in time and space. The performance of the
scheme is demonstrated for the flow created by the oscillatory
motion of the lid of a square cavity. All theoretical findings are
demonstrated practically.
Moreover, the scheme is applied to the Boussinesq approximation for
the convective flow in a vertical slot with differentially heated
walls, in the presence of a vertical temperature gradient. The
bifurcation of the trivial solution, which leads to the appearance
of traveling and stationary modes, is studied for very large
Rayleigh numbers and different values of the vertical temperature
gradient. The role of the dimensionless wavelength is investigated,
and the issue of the most dangerous wave is addressed
numerically.
Last revised: 10/05/06 By: christov@math