Two new papers published on the inclined plate problem in JFM and the Journal of Fire Sciences

Two new publications were recently published, both focused on studying inclined flames.

1. On the heat transferred to the air surrounding a semi-infinite inclined hot plate (Final PDF)
   Gollner, M. J., Sanchez, A. L., and Williams, F. A., Journal of Fluid Mechanics, 732, 2013, pp. 304-315.
2. Burning on Flat Wicks at Various Orientations (PDF Pre-print)
   Zhang, Y., Bustamante, M.J., Gollner, M.J., Sunderland, P.B., Quintiere, J.G., Journal of Fire Sciences, 0(0), 2013, pp. 1-20.

The first paper, published in the Journal of Fluid Mechanics presents an asymptotic analysis of heat transfer over an inclined plate – a simplification of the inclined burning problem. We initially expected that by incorporating density variations within the boundary layer, we would be able to resolve differences between the top and bottom surfaces of the plate, explaining burning behavior observed in our previous combustion institute publication. As it turns out, incorporating density differences does not change the solution very much, even to very high orders. Interestingly, we have found that the effect on different sides of the surface may likely be due to effects near the leading edge of the plate, following work by Messiter and Linan. This provides a nice analytical basis for us to extend our research in inclined boundary layer flames. More detailed experimental measurements on the problem should come out in the next year!

The second paper was a collaboration with Dr.’s Quintiere and Sunderland and their students on inclined plates burning. We modified Ahmad and Faeth’s upward burning integral solution for steady burning and incorporated side entrainment which has modified the solution to be applied somewhat more accurately to inclined plates. The solution works better than previous, but many unresolved problems still exist. Particularly, the fact that different size wicks will modify burning leaves many open questions for us to continue to investigate!

On the heat transferred to the air surrounding a semi-infinite inclined hot plate

An asymptotic analysis of laminar free convection in a boundary layer over an isothermal semi-infinite flat plate inclined at some angle to the vertical has been performed. Existing analytical solutions show no difference in the heat-transfer rate between the upper and lower surfaces of the plate, contrary to observations. To investigate this, higher-order perturbations of the non-dimensional temperature, velocity and pressure across the boundary layer were computed and found to show only small variations from first-order perturbations previously reported. Unexpectedly, third-order perturbations of all functions were found to be identical to those of the vertical plate, indicating that differences in temperature between both sides of the plate are limited to exceedingly small terms of order ${x}^{- 9/ 4} $ or smaller, $x$ being the distance from the leading edge, non-dimensionalized by the buoyancy length scale. Dominant differences between heat-transfer rates on the upper and lower surfaces were therefore concluded to be due to near-leading-edge effects. In applying an integral form of the conservation equations to the near-leading-edge region, it was found that, up to terms of order unity in $x$, the total heat-exchange rate for the inclined plate is identical to that of the vertical plate, so that the heat-transfer gain on one side balances exactly the loss occurring on the other. This simplification allowed determination of an upper bound for differences in heat-transfer rates between the upper and lower sides, even though complete determination of the differences would require a numerical integration of the full Navier–Stokes equations near the leading edge of the plate.

Burning on Flat Wicks at Various Orientations

Burning on flat plates was studied at various orientations with respect to gravity. Flat wicks of ceramic (Kaowool PM) board (10 cm wide and 1–10 cm long) were saturated with methanol or ethanol. Steady flames were obtained that ranged from boundary layer flames to plume-type burning. The onset of unsteady flow and transition to turbulence commenced at Grashof numbers of 106 –107 , increasing with decreasing angle (toward underside burning). The average burning rate per unit area was recorded along with the flame location. Experiments on polymethylmethacrylate were used for comparison with the liquid-wick results. The results roughly correlated with laminar pure convective theory, and improved results were indicated when the gravity term associated with the pressure gradient normal to the plate was included. Theoretical results by the integral method to reduce the partial differential equations to ordinary differential equations are presented

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