Matthias H. Buschmann

Debate Concerning the Mean-Velocity Profile of a Turbulent Boundary Layer

There has been considerable controversy during the past few years concerning the validity of the classical log law that describes the overlap region of the mean-velocity profile in the canonical turbulent boundary layer. Alternative power laws have been proposed by Barenblatt, Chorin, George, and Castillo, to name just a few. Advocates of either law typically have used selected data sets to foster their claims. The experimental and direct numerical simulation data sets from six independent groups are analyzed. For the range of momentum-thickness Reynolds numbers of 5£ 10 2 ‐2:732£ 10 4 , the best-fit values are determined for the “constants” appearing in either law. Our strategy involves calculating the fractional difference between the measured/computed mean velocity and that calculated using either of the two respective laws. This fractional difference is bracketed in the region § §0:5%, so that an accurate, objective measure of the boundary and extent of either law is determined. It is found that, although the extent of the power-law region in outer variables is nearly constant over a wide range of Reynolds numbers, the log-region extent increases monotonically with Reynolds number. The log law and the power law do not cover the same portion of the velocity profile. A very small zone directly above the buffer layer is not represented by the power law. On the other hand, the inner region of the wake zone is covered by it. In the region where both laws show comparable fractional differences, the mean and variance were calculated. From both measures, it is concluded that the examined data do not indicate any statistically significant preference toward either law. I. The Opening Arguments T HE Reynolds numbers encountered in many practical situations are typically several orders of magnitude higher than those studied computationally or even experimentally. High-Reynoldsnumber research facilities are expensive to build and operate, and the few existing are heavily scheduled with mostly developmental work. For traditional wind tunnels, additional complications are introduced at high speeds due to compressibility effects and probe-resolution limitations near walls. Likewise, full computational simulation of high-Reynolds-number flows is beyond the reach of current capabilities. Understanding of turbulence and modeling will, therefore, continue to play a vital role in the computation of high Reynolds number practical flows using the Reynolds averaged Navier‐Stokes equations. Because the existing knowledge base, accumulated mostly through physical as well as numerical experiments, is skewed toward the low Reynolds numbers, the key question in modeling high-Reynolds-number flows is what the Reynolds number effects are on the mean and statistical turbulence quantities. One of the fundamental tenets of boundary-layer research is the idea that, for a given geometry, any statistical turbulence quantity (mean, rms, Reynolds stress, etc.) measured at different facilities and at different Reynolds numbers will collapse to a single universal profile when nondimensionalized using the proper length and velocity scales. (Different scales are used near the wall and away from it.) This is termed self-similarity or self-preservation and allows convenient extrapolation from the low-Reynolds-number laboratory experiments to the much higher-Reynolds-number situations encountered in typical field applications. The universal log profile

Generalized logarithmic law and its consequences

There has been considerable controversy during the past few years concerning the validity of the universal logarithmic law that describes the mean velocity profile in the overlap region of a turbulent wall-bounded flow. Alternative Reynolds-number-dependent power laws have been advanced. We propose herein an extension of the classical two-layer approach to higher-order terms involving the Karman number and the dimensionless wall-normal coordinate. The inner and outer regions of the boundary layer are described using Poincare expansions, and asymptotic matching is applied in the overlap zone. Because of the specific sequence of gauge functions chosen, the resulting profile depends explicitly on powers of the reciprocal of the Karman number. The generalized law does not exhibit a pure logarithmic region for large but finite Reynolds numbers. On the other hand, the limiting function of all individual Reynolds-number-dependent profiles described by the generalized law shows a logarithmic behavior. As compared to either the simple log or power law, the proposed generalized law provides a superior fit to existing high-fidelity data

Evidence of Nonlogarithmic Behavior of Turbulent Channel and Pipe Flow

Employing data of peak position and peak values of Reynolds shear stress provided by canonical channel flow direct numerical simulations and pipe and channel flow experiments, three recent theories predicting the mean-velocity profile of wall-bounded turbulent flows are compared. Based on that analysis it is verified that the mean-velocity profile of such internal flows is logarithmic only to leading order and that the higher-order terms are proportional to reciprocal powers of the Karman number. This finding clearly indicates that higher-order effects with respect to the Karman number and the wall-normal coordinate must be considered. Furthermore, our results show that even the statistics of the flow region closest to the wall are affected by flow phenomena that scale with outer variables.

Reynolds-Number-Dependent Scaling Law for Turbulent Boundary Layers

Based on an extension of the two-layer approach a compact function for the mean velocity profile of a turbulent boundary layer is presented. The profile shows an explicit dependence on the Karman number. It is applied succesfully to profiles over a large Reynolds number range.

Turbulent boundary layers: reality and myth

In this paper, we discuss several recent theoretical approaches to wall-bounded flows such as turbulent boundary layers and pipe flows. Specifically, the power law theories by Barenblatt and George and a higher-order approach proposed by Buschmann and Gad-el-Hak are discussed. By employing probably the best data sets currently available worldwide, we uncover the qualities of these approaches. The outcome shows that the mean-velocity profile of turbulent wall-bounded flows is much more complex than the classical logarithmic law or a simple power law. A Reynolds number dependence of the mean profile persisting for arbitrarily high but finite Reynolds numbers seems to be highly likely. Both experimental and theoretical efforts are needed to solve the currently open questions.

Effects of Outer Scales on the Peaks of Near-Wall Reynolds Stresses

The classical view of wall-bounded turbulence suggests that the near-wall region should be scaled with characteristic scales that are closely related to that region. For the last decade, however, alternative concepts considering the in uence of outer scales were proposed. Herein, we show that the peak values of the Reynolds stresses in di erent geometries (e.g., zero-pressure-gradient boundary layers, and pipe and channel ows) collapse in single Reynolds-number-independent curve when scaled with an alternative mixed scaling based on u 3=2 u 1=2 e .