| 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