New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Characteristic length scale of the intermediate structure in zeropressuregradient boundary layer flow

Contributed by G. I. Barenblatt
Abstract
In a turbulent boundary layer over a smooth flat plate with zero pressure gradient, the intermediate structure between the viscous sublayer and the free stream consists of two layers: one adjacent to the viscous sublayer and one adjacent to the free stream. When the level of turbulence in the free stream is low, the boundary between the two layers is sharp, and both have a selfsimilar structure described by Reynoldsnumberdependent scaling (power) laws. This structure introduces two length scales: one—the wallregion thickness—determined by the sharp boundary between the two intermediate layers and the second determined by the condition that the velocity distribution in the first intermediate layer be the one common to all wallbounded flows and in particular coincide with the scaling law previously determined for pipe flows. Using recent experimental data, we determine both these length scales and show that they are close. Our results disagree with the classical model of the “wake region.”
Turbulent boundary layer flow over a smooth flat plate outside a close vicinity of the plate tip contains two unambiguous elements: the viscous sublayer adjacent to the plate, where the velocity gradient is large and the viscous stress is comparable with the Reynolds stress, and the statistically uniform free stream.
According to classical theory (1), the region intermediate between these two consists of two layers with different properties. The first, adjacent to the viscous sublayer, is a universal, Reynoldsnumberindependent logarithmic layer. In the second, the “wake region,” there is a smooth transition from the universal logarithmic layer to the free stream.
Our analysis of all available experiments (2−4) contradicts this classical theory. Indeed, in the clearcut case of a smooth plate and low free stream turbulence, the intermediate structure does consist of two layers. However, the boundary between them is sharp. Most important, both layers are selfsimilar, substantially Reynoldsnumberdependent, and described by different scaling laws. It is interesting to note (see the details below) that the same configuration of two selfsimilar layers with a sharp interface between them can be seen in all runs used in ref. 1 for the illustration of the wake region model.
We found it possible (2−4) to introduce a characteristic length scale Λ such that the average velocity distribution in the first intermediate layer coincides with Reynoldsnumberdependent scaling law obtained previously for pipe flows, when the Reynolds number is chosen as UΛ/ν, where U is the free stream velocity and ν is the fluid's kinematic viscosity. The sharp boundary between the selfsimilar intermediate layers also defines a length scale λ. We show, by analysis of experimental data, that these two length scales λ and Λ are close.
Background.
In a previous paper (2), we noted that when the turbulence level in the free stream is small, the intermediate structure between the viscous sublayer and the free stream consists of two selfsimilar layers: one adjacent to the viscous sublayer where the average velocity profile is described by the scaling law 1 and one adjacent to the free stream where 2 Here, 3 u is the average velocity; τ is the shear stress at the wall; ρ and ν are the fluid density and kinematic viscosity; and A, B, α, and β are Reynoldsnumberdependent constants.
Our processing of all experimental data available in the literature (3, 4) confirmed these observations and showed that it is always possible to find a length scale Λ such that, setting Re = UΛ/ν, we can represent the scaling law of Eq. 1 in the form 4 obtained by us earlier for pipe flows (see, e.g., ref. 5). This result suggests that the structure of wall regions in all wallbounded shear flows at large Reynolds numbers is identical, if the length scale and velocity scale are selected properly. The natural question is, however, what is the physical meaning of this length scale Λ in boundary layer flow? This question is of substantial importance and should be clarified for proper understanding of the identity of scaling laws for different wallbounded shear flows.
We note that the intermediate structure has another characteristic length scale λ—the wallregion thickness—determined by the sharp intersection η = η_{*} of the two velocity distribution laws φ = Aη^{α} and φ = Bη^{β} valid in the different layers. We have 5 such that 6 and the wallregion thickness λ is determined by the relation 7 On the other hand, the characteristic length scale Λ is determined by the relation 8 such that the ratio of these two scales is 9
Analysis of Experimental Data.
We analyzed the recent data of J. M. Österlund presented on the internet (www.mesh.kth.se/∼jens/zpg/). The data seem to us to be reliable, however much we disagree with the processing and interpretation in the paper by Österlund et al. (ref. 6; see also ref. 4). All 70 runs presented on the internet give the characteristic brokenline average velocity distribution in lg η, lg φ coordinates (see the examples in Fig. 1; all of the other cases are similar), such that the possibility of determining A, α, B, β, and η_{*} accurately from these experimental data is unquestionable. These results are presented in Table 1 for all of Österlund's experiments where Re_{θ} = Uθ/ν > 10,000. Here, θ is the momentum thickness; the runs in the Österlund's experimental data are labeled by Re_{θ}.
The effective Reynolds number Re was obtained (4) by the formula 10 where lnRe_{1} and lnRe_{2} are the solutions of the equations 11 and the values of A and α were obtained by standard statistical processing of Österlund's data. For Re_{θ} > 10,000, the difference δ between lnRe_{1} and lnRe_{2} does not exceed 3%, such that they coincide within experimental accuracy.
According to Eqs. 8 and 9, 12 The data for u_{*} and U are presented by Österlund on the internet for each run. In Fig. 2, we present the values of lg(Λ/λ) for all runs. The mean value of lg(Λ/λ) is approximately 0.2, such that the characteristic length scale Λ is about 1.6 times the thickness of the wall region.
If we take into account that Λ is calculated from the value of Re and that lnRe, not Re itself, has been determined from experiment, the ratios Λ/λ as shown in Fig. 2 are close to 1.5.
We processed in refs. 3 and 4 the data of 90 zeropressuregradient boundary layer experiments at low free stream turbulence performed by different authors during the last 25 years—all of the experiments of this kind available to us. Without exception, all runs revealed identical configurations of the intermediate structure in the boundary layer: two adjacent selfsimilar layers separated by a sharp interface.
According to the classical model (1), the intermediate structure consists of the (universal) logarithmic layer and a nonselfsimilar wake region smoothly matching the logarithmic layer. It was natural to process also the very data used in ref. 1 for the justification of the wake region model with the general procedure we used on the other data. The data presented in figure 21 of ref. 1 were scanned and replotted in lg η, lg φ coordinates as was done for all experimental data processed in refs. 3 and 4. Processing revealed the same brokenline structure, i.e., two adjacent selfsimilar layers (see Table 2 and Fig. 3, where a typical example is presented). The difference between lnRe_{1} and lnRe_{2} determined from the wall layer data is small: this result shows that the procedure is adequate. We conjecture that the values of β are larger than those in newer experiments because of a nonzero pressure gradient in all these runs (see ref. 1). The results of our processing fail to confirm the wake region model proposed in ref. 1.
Conclusions.
We have shown that one can find a length scale Λ, such that, if the Reynolds number Re in a zeropressuregradient boundary layer flow is defined by Re = UΛ/ν, where U is the free stream velocity and ν is the kinematic viscosity, then the scaling law for the selfsimilar region adjacent to the viscous sublayer coincides with the scaling law for turbulence pipe flow. Using the recent experimental data of Österlund (www.mesh.kth.se/∼jens/zpg/), we confirmed this fact and reached the important conclusion that Λ is about 1.6 times the wallregion thickness. Our results are in disagreement with the classical model of the wake region in the boundary layer (1).
Acknowledgments
This work was supported in part by the Applied Mathematics subprogram of the U.S. Department of Energy under Contract DEAC0376SF00098 and in part by National Science Foundation Grants DMS 9416431 and DMS 9732710.
 Accepted February 10, 2000.
 Copyright © 2000, The National Academy of Sciences
References
 ↵
 ↵
 Barenblatt G I,
 Chorin A J,
 Hald O,
 Prostokishin V M
 ↵
 ↵
 Barenblatt G I,
 Chorin A J,
 Prostokishin V M
 ↵
 Barenblatt G I,
 Chorin A J,
 Prostokishin V M
 ↵
Citation Manager Formats
More Articles of This Classification
Physical Sciences
Related Content
 No related articles found.