Research Article A Simple Model for Turbulent Boundary Layer Momentum Transfer on a Flat Plate A simple model is presented for turbulent momentum transfer on a flat plate. The proposed model is based on some polynomial velocity profiles in a laminar sublayer as well as in a fully developed boundary layer and two integral boundary layer equations. The model could be used for the calculation of boundary layer thickness, velocity profile and skin friction factor on the flat plate. The calculated results are in very good agreement with other proposed empirical correlations. Keywords: Flat plate, Fluids, Momentum transfer, Velocity distribution Received: December 30, 2009; revised: March 03, 2010; accepted: March 08, 2010 DOI: 10.1002/ceat.200900634 1 Introduction Most flows that occur in practical applications are turbulent. This term denotes a motion in which an irregular fluctuation (mixing or eddying motion) is superimposed on the main stream. For the case of a turbulent boundary layer, the Navier- Stokes equations, which govern the motion of a Newtonian viscous fluid, were formulated well over a century ago. The most straightforward method of attacking any fluid dynamics problem is to solve these equations for the appropriate bound- ary conditions. Analytical solutions are few and trivial and, even with today’s supercomputers, a numerically exact solu- tion of the complete equations for the three-dimensional, time-dependent motion of turbulent flow is prohibitively ex- pensive, except for basic research studies in simple configura- tions at low Reynolds numbers. Therefore, the straightforward approach is still impracticable for engineering purposes. Con- sidering the successes of the pre-computer age, one might ask whether it is necessary to gain a greater understanding of fluid dynamics and develop new computational techniques, with their associated efforts and costs. Textbooks on fluid dynamics reveal two approaches to understanding fluid dynamics pro- cesses. The first approach is to derive useful correlations through a progression from demonstrative experiments to de- tailed experimental investigations which yield additional un- derstanding and subsequent improvement. The second ap- proach is to solve simplified versions of fluid dynamics equations for the conservation of mass, momentum and ener- gy for comparatively simple boundary conditions. There is great advantage in combining both approaches when address- ing complex fluid dynamics problems, but interaction between these two approaches has been limited until recently by the narrow range of useful solutions that could be obtained by an- alytical methods or simple numerical computations. It is therefore evident that any method for increasing the accuracy of computational methods by solving more complete forms of the conservation equations than has been possible up to now will be welcome. A turbulent boundary layer on a flat plate with ∂u/∂x = 0 is an equilibrium boundary layer. It has been and still is used ex- tensively for the purpose of turbulence research. It is also an important engineering approximation for many problems. For example, the boundary layer on an airline fuselage in cruise conditions is largely similar to the flat-plate boundary layer. There is still a debate in the literature on the proper scaling of the turbulent boundary layer along a flat plate with or without a streamwise pressure gradient. Mainly based on physical argu- ments, the classical scaling for the turbulent boundary layer in a zero-pressure gradient was derived by different scientists, in- cluding Von Kärmän [1], Millikan [2], Rotta [3], Clauser [4], and Coles [5]. To solve the turbulent boundary layer equations, a turbulent model is needed to represent the Reynolds shear stress, i.e. u’v’ in the boundary layer equation. The following four turbulence models were considered in this case: – the algebraic model of Cebeci and Smith [6], – the two-equation low-Reynolds number k-e model of Laun- der and Sharma [7], – the two-equation low-Reynolds number k-x model of Wil- cox [8], – and the differential Reynolds stress model (DRSM) of Han- jalic ´ et al. [9] (see also Jakirlic ´ et al. [10]). For the precise formulation of the different models, the readers are referred to the original papers or to Henkes [11]. The algebraic model and the two-equation models approxi- mate the turbulence through a turbulent viscosity, which is de- fined as u′v ′  v t ∂u=∂y. The algebraic model couples v t to Chem. Eng. Technol. 2010, 33, No. 6, 867–877 © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.cet-journal.com Mohammad Hasan Khademi 1 Ali Zeinolabedini Hezave 1 Dariush Mowla 1 Mansour Taheri 1 1 School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran. – Correspondence: Prof. D. Mowla (dmowla@shirazu.ac.ir), School of Chemical and Petroleum Engineering, Shiraz University, Shiraz 71345, Iran. Fluids 867