Mean Velocity of Fully Developed Turbulent Pipe Flows

Abstract

The mean flow profile of a turbulent flow in a smooth channel or a pipe is a classical topic in fluid mechanics [1–4]. For fully developed turbulent pipe flows, von Karman and Prandtl’s pioneering analyses indicate two dominant regions: the near-wall laminar sublayer and the turbulent region away from the wall. The two regions are connected by a small intermediate buffer zone. In both regions, u , the mean velocity scaled with the friction speed u , is a function of y , the distance from the wall scaled with the ratio of kinematic viscosity to friction speed =u . This relationship u f y is known as the classical “law-of-the-wall.” The Reynolds-averaged Navier–Stokes equation describes the mean axial speed of a fully developed turbulent flow and involves the viscous stresses and the turbulent Reynolds stresses. This equation is unclosed because there is no physical rule relating between the turbulent stresses and the mean axial speed. Yet, experiments show that the magnitudes of the stresses differ vastly for each region. Very near the wall, in the viscous sublayer where y 70 is characterized by a negligible viscous stress and a dominant Reynolds stress. Modeling the Reynolds stress in this region by the linear mixing-length approach of Prandtl [3] results in the “logarithmic velocity distribution law.” The logarithmic relationship was also derived from a dimensional analysis requiring the continuity of the velocity gradient in the buffer zone when the Reynolds number is sufficiently high and using u as the speed scale in both the near-wall and turbulent region [4]. In the buffer zone, the Reynolds and viscous stresses are of the same magnitude. Von Karman suggested that in the buffer zone u is also related to the logarithm of y . In both the buffer and turbulent zones, the logarithmic functions include two constants that may be determined experimentally. This semi-empirical approach showed a nice agreement with measurements [2]. It also resulted in “Prandtl’s universal law of friction for smooth pipes”which relates between the friction factor and the Reynolds number and agrees with data. Note that van Driest [5] proposed a damping function for the turbulent stress in the viscous sublayer and the buffer zone which resulted in a good prediction of the mean velocity in these regions. In recent years, new approaches raised controversy about the scales of both distance and speed in the outer or core region for pipe flows. The pipe radius has been proposed as an additional dominant length scale. Arguments have been made for the velocity scale to be either the maximum (centerline) speed [6] or the velocity deficit [7]. The functional form of the scaling law in the overlap region is a subject of even greater controversy. It has been suggested that a power law rather than a log law governs the relationship between velocity and distance from the wall [6–8]. Note, however, that Wosnik et al. [6], Zagarola and Smits [7], and Barenblatt [8] power laws are derived from different approaches. For a thorough review of the scales in turbulent pipe flows, see Pope [9]. Wosnik et al. [6] proposed a more detailed description of the pipe flow structure, see also Pope, [9] chapter 7. They divided the flow into the main “viscous sublayer,” “overlap,” and “outer” regions. The near-wall region, where 0 200; 000. They also found that the logarithmic law region holds for a larger range, 600< y < 0:12K. Another set of recent pipe flow experiments over a wide range of Reynolds numbers (10< Re < 1; 000; 000) was conducted by Received 10 December 2004; revision received 18 April 2006; accepted for publication 19May 2006. Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code $10.00 in correspondence with the CCC. AIAA JOURNAL Vol. 44, No. 11, November 2006