A theoretical study of the transference of heat and momentum across turbulent incompressible boundary layers

Abstract

A survey and evaluation of some models of turbulence for isothermal turbulent flows is made. Models such as mixing length, one-equation, two-equations and three-equations are solved with the aid of a high speed computer for annular turbulent flows. The results are compared with each other and with experiment and the significance is discussed. The three equation model (three transport equations plus the mean velocity equation) emerges as the most accurate and capable of the widest application: one set of constants only is sufficient to solve a number of turbulent flows. Also, this model does not require the prescription of any arbitrary length scale. A study of the effect of varying the constants in the three-equation model shows that the velocity and shear stress profiles are insensitive to the variation of the constants. A variation of up to 50%, in the value of the constants produces, at most, less than 2% variation in the velocity and shear stress profiles. Only the turbulence energy distribution shows some sensitivity. The position of maximum velocity for smooth annuli with different radius ratios, as well as friction factors for a number of wall conditions are calculated with the three-equation model. The comparison between predictions and experimental data shows a fairly good agreement. Starting from this three-equation model, an extended model, capable of predicting turbulent, two-dimensional, incompressible thermal boundary layers is developed. Three more equations are incorporated in the isothermal model, namely, (1) mean temperature equation (T), (2) convective heat flux equation (uyT') and (3) equation for the intensity of temperature fluctuation (1/2T'2). Appropriate approximations are introduced and the new model of parabolic differential equations is solved simultaneously with the equations for the isothermal flow. The new five-equations model (five transport equations plus mean velocity and mean temperature equations) is applied to a number of real flows, with and without the presence of walls. Both rough and smooth walls are considered. Generally, good agreement is obtained when predicted results are compared with the available experimental data.