A numerical analysis of turbulent flow along an abruptly rotated cylinder

Great progress has been achieved over the past fifteen years in the computation of two-dimensional turbulent flows. The proceedings of the 1968 Stanford Conference (1) attest to the success of several methods in predicting skin friction and heat transfer coefficients, mean velocity and temperature fields, and to a lesser degree boundary layer separation. This success is due less to the fact that the physics of turbulence is well understood (it is not) than to the fact that the existent two-dimensional data obtained within pipes and on external surfaces have lent themselves to correlation. It is these correlations (particularly near-wall similarity or the law-of-the-wall) which serve as the empirical foundation of the mixing length and eddy viscosity "theories" of turbulence.

The term mathematical model may more aptly describe the mixing length/eddy viscosity approach to turbulence than the word theory, for these concepts take into account little of the basic dynamics of turbulence (its production, intensity, frequency, and dissipation). Yet these methods are significant precisely because they do predict with uncanny accuracy the gross consequences of turbulence in a number of two-dimensional flows of practical interest. Mixing length/eddy viscosity models are attractive to the engineer because these models are agreeably simplistic. That is, their formulation is algebraic and does not involve differential equations or additional turbulent transport equations. The monograph (2) of Launder and Spalding presents an excellent review and evaluation of current mathematical models of turbulence. On account of their simplicity, the mixing length/eddy viscosity models are relatively straightforward to implement and economical to use. Thus they are ideally suited for industry.

The present work is an investigation of the suitability of the eddy viscosity approach for the prediction of three-dimensional turbulent flows. The eddy viscosity formulation employed is essentially an extended two-dimensional model. Unfortunately, endeavors to correlate three-dimensional turbulent data have not been as successful as with the two-dimensional case. White (3) has neatly summarized the more significant postulations of a three-dimensional law-of-the-wall. All are patterned after the two-dimensional near-wall similarity hypothesis, and of course none can be confirmed without direct measurement of wall shear stress. No such measurements have been performed to date with the exception of the data of Pierce and Krornmenhoek (4), who did not specifically study the question of near-wall similarity in three-dimensional flows. Thus the present analysis is necessarily a simplistic one. It is based on the fact that every turbulent flow is actually three-dimensional and on the supposition that a correlation which succeeds with a two-dimensional mean velocity field may well succeed in the calculation of a three-dimensional field.