Skin Friction and the Inner Flow in Pressure Gradient Turbulent Boundary Layers

Abstract

This investigation will look at multiple methods to determine the wall shear stress for several pressure gradient turbulent boundary layer flows, particularly favorable pressure gradient and zero pressure gradient. These methods include using the slope at the wall, the integrated bounary layer equation, momentum integral equation and the Clauser method. In order to perform this study, 2D Laser Doppler Anemometry, (LDA), measurements of the velocity field near the wall for various streamwise positions have been carried out at the Chalmers L2 wind-tunnel. With the resulting wall shear stress calculations, the effects of pressure gradient and upstream conditions will be investigated on the inner region of the velocity profiles and Reynolds stresses. As will be seen, the integrated boundary layer equation is the most accurate technique to determine the wall shear stress when direct measurements are not available. In addition, the velocity profiles show a mild effect of the pressure gradient. The Reynolds stresses show a large effect of the pressure gradient in inner variables, but not below, y and components changes significantly due to the external pressure gradient, damping them as much as 40%, though the streamwise component exhibits an insignificant amount of change. Introduction The effects of Reynolds number and pressure gradient on the skin friction coefficient, Cf , and the velocity field have long been debated as well as the accuracy ∗MS, Rensselaer Polytechnic Institute, Department of Mechanical, Aeronautical and Nuclear Engineering, Troy, NY 12180 †PhD, Rensselaer Polytechnic Institute, Department of Mechanical, Aeronautical and Nuclear Engineering, Troy, NY 12180 ‡Post-doctoral fellow, The Johns Hopkins University, Department of Mechanical Engineering, Baltimore, MD §Associate Professor, Chalmers Institute of Technology, Gothenburg, Sweden ¶Associate Professor, Rensselaer Polytechnic Institute, Department of Mechanical, Aeronautical and Nuclear Engineering, Troy, NY 12180, also Research Professor at University of Puerto Rico-, Mayaguez, Mayaguez, P.R., Department of Mechanical Eng. Copyright c © 2006 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royaltyfree license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright owner. of various techniques used to determine this quantity. Some of the more common methods to obtain the skin friction include direct measurements such as oil-film interferometry, force balance and the newer MEMS devices. Common velocity based methods include computing the slope at the wall and evaluating the momentum equation. Various in-direct measurements such as the Clauser method and Preston total head tubes are also used. Many of the theories and methods have been proven valid for the ZPG boundary layer, in various ranges of Reynolds number. However, these theories become more complex in the presence of an external pressure gradient. Accurate values of Cf are of particular importance due to its direct relationship to u∗, the friction velocity. Through similarity analysis, George and Castillo (1997) found that the friction velocity is the correct velocity scale for the inner region of the boundary layer. Also, the inner similarity length scale involves u∗ and the Reynolds stress inner similarity scale is u∗. Accurate values of the friction velocity are needed to view profiles in inner similarity variables. Also, obtaining accurate values of Cf will enable better predictions of the friction drag for airplanes, submarines, and ships. With knowledge of the behavior of the skin friction coefficient under a variety of flow conditions, different ideas can be implemented for drag reduction studies. To aid in determining accurate values of the skin friction, this investigation will evaluate different techniques in terms of their accuracy and limitations. Also, the effects of an external pressure gradient on the skin friction, as well as the scaled velocity profiles and Reynolds stresses in inner variables will be examined. Experimental Setup This analytical study will look at four methods to calculate the wall shear stress for different pressure gradient turbulent boundary layer flows. Specifically, the investigation will use the ZPG data of Castillo/Johansson (CJ) (2002) as well as the developing FPG flow of Cal et al. (2005). Both of these experiments were performed in the Chalmers University of Technology L2 wind-tunnel, where upstream conditions such as wind-tunnel speed and trip wire size and location could be controlled. The wind-tunnel has a test section that is 3 m long, 1.8 m wide and 1.3 m high and is a closed loop design. In each case, the flat plate was installed vertically in the wind-tunnel and various wind-tunnel speeds were