Applied Mathematical Model for Turbulent Boundary Layers

Abstract

We present a complete analytical model for mean velocity profile in turbulent boundary layers. The model integrates the conventional log-wake law and the Barenblatt power (scaling) law in the intermediate region so that it includes the effects of the Reynolds number. Furthermore, it includes the effects of the wall and the boundary layer edge so that the model can describe the entire turbulent boundary layer mean velocity profile analytically. The model has been confirmed with recent laboratory experimental data of Osterlund. Introduction The mean velocity profile in turbulent boundary layers is usually described by the log law or the log-wake law (Coles 1956). Barenblatt et al. (2002, 2004) challenged this conventional argument and triggered hot debates on this classical subject. They claimed that the mean flow in wall-bounded turbulent flow has a persistent dependence on the Reynolds number, and that the well known logarithmic law of the wall is invalid. They believed that a Reynolds number dependent power (scaling) law better describes velocity profiles in wall-bounded flows. In particular, zero-pressure-gradient boundary layer flows follow two sharply separated power laws (Barenblatt et al. 2002). Others (Osterlund et al. 2000, Zanoun et al. 2003, Bushmann and Gad-El-Hak 2003) defended the classical logarithmic law by doing experiments and various analyses. All the discussions are limited to the overlap layer. Our studies (Guo and Julien 2003, Guo et al. 2005) showed that both the conventional log-wake law and the Barenblatt power law have merits and weaknesses. Both laws are intermediate asymptote and are not valid near the wall and near the boundary layer edge, except that Barenblatt's two power laws (Barenblatt et al. 2002) are not smoothly connected. Nevertheless, the log-wake law is simpler, and the power law includes the effects of the Reynolds number and is more realistic. Integrating the conventional log-wake law and the Barenblatt power law and considering the effects of the wall and the upper boundary layer edge, we in this paper propose an applied mathematical model for zero-pressure-gradient turbulent boundary layers. Referring to Fig. 1, we divide a boundary layer flow into two regions: the inner and Copyright ASCE 2006 World Environmental and Water Resources Congress 2006 outer regions; or five layers: the laminar sublayer, the transition layer, the overlap layer, the wake layer, and the layer of the upper boundary layer edge. We first introduce the laws in the inner and the outer regions, respectively. We then match them to result in a complete velocity profile model that satisfies all physical requirements. 10 0 10 1 10 2 10 3 10 4 0 5 10 15 20 25 30 35 yu * /ν u/ u * laminar layer transition layer overlap layer wake layer edge layer Inner region