Mesolayer theory for turbulent flows

Abstract

ONG and Chen,1 Afzal,2'3 and Afzal and Bush4 have reported the intermediate layer theory for turbulent shear flows. The velocity in the intermediate (meso) layer, as postulated by Long and Chen,1 is of the order of the friction velocity. The inner and intermediate layers in the mesolayer theory1 were regarded as a composite expansion that was matched with the outer defect layer. Their theory showed that for a boundary layer the logarithmic region was absent and that for a pipe flow the effect of the mesolayer was weak, although it did tend to modify the classical logarithmic behavior. In contrast to Long and Chen, Afzal2'3 has shown that the velocity in the intermediate layer is of the order of unity. Afzal's theory2'3 dealt separately with three (inner, intermediate, and outer) layers that were matched in the two overlap domains to show the possibility of two logarithmic regions rather than the one proposed by classical theory. Afzal2'3 and Afzal and Bush4 have shown that, in terms of classical theory, there is a distinct intermediate limit between the outer defect layer and the inner wall layer. This (new) three-layer theory modifies and/or extends the (old) two-layer theory. The implications of turbulent modeling by eddy viscosity and mixing length have been studied by Afzal and Bush.4 The aim of this Note is to modify Long and Chen's mesolayer theory in the light of Afzal's proposition2'3 that the velocity in the intermediate layer is of the order of C/ c, where 0 ( U c) >0 ( UT ). On the basis of the above stated proposition, a composite expansion for the inner and intermediate layer is formulated. The composite expansion is matched with the outer defect layer to show that the classical logarithmic region always exists, provided Uc is judiciously selected. It is further shown that the intermediate layer, when analyzed in terms of the two-layer classical theory, plays an insignificant role as far as the lowest-order results are concerned. However, this intermediate layer plays a significant role in the first-order theory where a three-layer picture of the flow is required.