Curvature and its effects on heat transfer with thermal radiation in the free jet boundary

This investigation was conducted to determine the effects of curvature on the thermal boundary layer in curved jet-type flow fields. Particular interest was directed to examining these effects in the presence of thermal radiation. The investigation was initiated by first determining a set of transformation variables for the boundary layer equations which would allow their reduction to ordinary differential form. Solution of these equations based on assumptions of local similarity in the flow field provided the mechanism through which the effects of curvature could be studied.

The chief applications involved in solution of the problem included the principles and theory of fluid mechanics and heat transfer as related to incompressible, viscous flows where properties such as viscosity and thermal conductivity were assumed invariant. .Techniques of numerical analysis utilized to generate the solution sets for various values of the curvature parameter, R, employed the Runge-Kutta algorithm with Gill Coefficients. This algorithm was employed in a program designed to generate solutions based on automatic corrections to initial guesses for starting values given to the slopes of the velocity and temperature profiles. Solution of the momentum equation first provided values for the dimensionless stream function which were needed in solution of the energy equation. Additionally, analysis of the asymptotic boundary conditions at both extremes of the thermal boundary layer provided the information needed to define a "feasible set" of starting values to be used in conjunction with the stream function to arrive at the required solution for the temperature profile.

The results obtained from the research indicate that the major effect of curvature on the mixing zone in the free jet boundary is alteration of the thermal boundary layer thickness with increasing value of the curvature parameter. This effect was seen to exist both in the presence of, and in the absence of thermal radiation. Calculations were performed for differing values of the conduction-to-radiation parameter, N, which defines the relative degree of radiative versus conductive contribution to the thermal boundary layer. For values of N greater than l, which coincides with the lessening influence of radiative transfer, the thermal boundary layer was found to grow with increases in the curvature parameter. On the other hand, however, the dominant influence of radiative transfer represented by values of N less than l showed a decrease in the thickness of the thermal boundary layer for increasing curvature. Coincident with the change in boundary layer thickness for this situation, a decrease in the value of the temperature along the major streamline accompanied the increase in curvature. These two phenomena, as related to optically thick radiation, can be explained through the fact that optically thick radiation restricts thermal transfers among molecules in the fluid to neighbors in close proximity of one another. Others had established before this study that the velocity profile within the outer confines of the boundary layer changed significantly with curvature. The resultant effect on the motion of the molecules in the flow field in this region would obviously have an effect on the capability for radiative transfers within the flow.

The linearizing technique which was utilized for the first order approximation to the solution of the energy equation provides, at best, some insight into the general behavior of the temperature profile within the thermal boundary layer. The roughness of this approximation is due primarily to the assumption of very small temperature differences across the mixing zone. The accuracy of the results is also affected by the error inherent in adoption of the optically thick approximation for radiative heat flux across the thermal boundary layer. Future studies should concentrate on development of techniques which allow more precise analyses to be conducted. A good point of departure would be one where the nonlinear form of the energy equation is used to more adequately predict the temperature profile in the boundary layer. More exact expressions for the radiative heat flux may also be employed in an effort to gain a better perspective into the problem of completely specifying thermal behavior.