Mccaughan, Frances E.

NASA NTRS1995

The Benard problem: A comparison of finite difference and spectral collocation eigen value solutions

The application of spectral methods, using a Chebyshev collocation scheme, to solve hydrodynamic stability problems is demonstrated on the Benard problem. Implementation of the Chebyshev collocation formulation is described. The performance of the spectral scheme is compared with that of a 2nd order finite difference scheme. An exact solution to the Marangoni-Benard problem is used to evaluate the performance of both schemes. The error of the spectral scheme is at least seven orders of magnitude smaller than finite difference error for a grid resolution of N = 15 (number of points used). The performance of the spectral formulation far exceeded the performance of the finite difference formulation for this problem. The spectral scheme required only slightly more effort to set up than the 2nd order finite difference scheme. This suggests that the spectral scheme may actually be faster to implement than higher order finite difference schemes.

Skarda, J. Raymond Lee↗

NASA NTRS1994

Coupled Marangoni-Benard/Rayleigh-Benard Instability with Temperature Dependent Viscosity

The onset of convection induced by coupled surface tension gradient and buoyancy forces is investigated with temperature dependent viscosity. Both surface tension and viscosity are assumed to vary linearly with temperature. The limiting case, Ma=0, is the buoyancy driven convection problem typically referred to as the Rayleigh-Benard stability problem. The other limiting case, Ra=0, is the surface tension (gradient) driven flow problem referred to as the Marangoni-Benard problem. The equations and boundary conditions obtained from the linear analysis are solved numerically as a generalized eigenvalue problem. Neutral stability curves for different viscosity slopes have been generated for the Marangoni-Benard and Rayleigh-Benard problems. It is shown that the curves can be collapsed to a single curve by appropriately scaling the results for each of the limiting cases. The critical Marangoni number is determined as a function of the slope of the viscosity temperature variation, epsilon, for different values of the Rayleigh number. When the viscosity decreases linearly with temperature, the coupled buoyancy-surface tension problem, including the limiting cases of Ra=0 and Ma=0, is found to be more stable than the constant viscosity case.

Skarda, J. Raymond Lee↗