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|Title:||Thermal Instability of Wedge Flows|
|Department / Program:||Mechanical Engineering|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||Thermal instability of wedge flow boundary layers over a heated surface, including those of stagnation flows, has been investigated for moderate to high Prandtl numbers. These flows are studied because they serve as the prototypes of general forced convection boundary layers in heat transfer. Attention is focused on the streamwise vortex made, which had been shown to be the dominant mode of thermal instability for all shear flows. It is shown that the equations for the neutrally stable disturbances can be solved by similarity transformation. In contrast, previous studies of related problems had to resort to neglecting the streamwise dependence of the disturbances, an assumption which is not self-consistent. The problem reduces to an eighth-order eigenvalue problem with four parameters: the Prandtl number, the wedge parameter, and Rayleigh and wave numbers based on a suitable boundary layer thicknesses. All the x-dependences are vested in the last two parameters. The Reynolds number is found to have no influence, except indirectly through its influence on the boundary layer thickness. It is further shown that a reference length related to the thermal boundary layer is more relevant than that for the viscous boundary layer, especially for very large Prandtl numbers. For the limiting case of infinite Prandtl number the streamwise velocity perturbation vanishes, reducing the problem to a sixth-order eigenvalue problem.
Two numerical methods of solution, both based on the shooting method, are developed for solving the eigenvalue problem for a semi-infinite domain. In one method, computations are carried out for a finite domain, whose outer boundary is made to approach infinity by fixed increments. Successive improvements are found to decrease geometrically. A final correction is derived by summing the infinite geometric series. In the second method, the numerical integration is matched to an approximate analytical solution valid around a large, though finite z. This approximate solution is obtained by assuming the coefficients of the differential equation to be nearly constant within the interval where the eigenfunction has significant non-zero values. This amounts to an zeroth-order WKB approximation. Both of the numerical schemes are found to be comparable in accuracy and efficiency, though the extrapolation scheme appears to be more versatile and is used for most of the production runs in this research.
Neutral stability curves of Rayleigh number versus wave number are presented for wedge angles between 0 and (pi), and Prandtl numbers between 1 and (INFIN). Corresponding eigenfunctions show that the disturbances extend only slightly beyond the thermal boundary layer. The convection cells for the critical wave number retain approximately the unity aspect ratio familiar to thermal instability problems in confined fluid layers. It is particularly interesting to note that when the conduction thickness of the boundary layer is used as the reference length for Rayleigh number and wave number, the results are nearly independent of the wedge angle and the Prandtl number. These results are: Ra('c)(,c) = 200 (+OR-) 10%; K('c)(,c) = 1.2 (+OR-) 15% for wedge angles between 0 and (pi) and Prandtl numbers between 1 and (INFIN). Physically, these results suggest that thermal instability is only sensitive to the thermal structure of the boundary layer, and is quite insensitive to the details of the velocity distribution, except in terms of the latter's influence on the thermal boundary layer thickness. Practically, the results raise the hope that a "universal" marginal stability condition may be approximately applicable to all forced convection boundary layers. One direct consequene of this is that boundary layers above a critical thickness are thermally unstable. Thus there exists a "minimum heat transfer coefficient", which is equal to
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
for all the cases considered.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981.
|Date Available in IDEALS:||2014-12-13|
This item appears in the following Collection(s)
Dissertations and Theses - Mechanical Science and Engineering
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois