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
Structure of the zeropressuregradient turbulent boundary?layer
Abstract
A processing of recent experimental data by Nagib and Hites [Nagib, H. & Hites, M. (1995) AIAA paper 950786, Reno, NV) shows that the flow in a zeropressuregradient turbulent boundary layer, outside the viscous sublayer, consists of two selfsimilar regions, each described by a scaling law. The results concerning the Reynoldsnumber dependence of the coefficients of the wallregion scaling law are consistent with our previous results concerning pipe flow, if the proper definition of the boundary layer Reynolds number (or boundary layer thickness) is used.
The currently dominant engineering theory of the zeropressuregradient turbulent boundary layer was proposed by Coles (1). An exposition, closely following the original work, can be found in Monin and Yaglom (2) and a discussion can be found in the instructive paper by Fernholz and Finley (3). We do not reproduce this theory here, noting only that beside some invariance assumptions, common to semiempirical theories of turbulence, Coles’ theory introduces additional parameters and approximations, convenient for engineering calculations but without a direct physical meaning. An instructive survey of the general properties of turbulent boundary layers can be found in Sreenivasan (4).
In the present paper we start by a very simple processing of recent experimental data of Nagib and Hites (5, 6). Our study indicates that the flow outside of the viscous sublayer consists basically of two selfsimilar regions: the inner region (wall region) and an outer region. In both regions the mean velocity distribution can be very accurately described by scaling (power) laws, different for the inner and the outer regions. The boundaries between the inner and outer regions and the boundary between the outer region of the boundary layer and the free stream flow are rather sharp. We show further that the scaling law for the inner region is almost identical to the scaling law proposed for pipes (7, 8). However, to reveal this identity a redefinition of the Reynolds number for the boundary layer was needed.
Processing of the Experimental Data
The experimental data of Nagib and Hites (5, 6) as well as earlier data of Naguib (6), are presented in Fig. 1. (We are most grateful to H. Nagib and M. Hites who supplied us with tables.) To reveal the scaling laws, we simply presented their results in doublelogarithmic rather than the semilogarithmic coordinates which are commonly used for processing such data. The results are presented in Fig. 2 a–h. The instructive common feature of these figures is that outside the viscous sublayer the velocity distribution in the flow is represented by a broken line—a combination of two different scaling (power) laws separated by a sharp boundary.
The parameters of the scaling laws are presented in Table 1: the scaling law in the inner region is assumed to have the form θ = Aη^{α}; in the outer region the assumption is θ = Bη^{β}. Here the standard notations are used: 1 where u is the mean velocity; u_{?} is , the dynamic or friction velocity; τ is the shear stress at the wall; ν is the kinematic viscosity; ρ is the density of the fluid; θ is the momentum displacement thickness; and U is the freestream velocity.
We see that the slope α and the coefficient of the inner scaling law A are slightly Redependent. For the outer scaling law the Re dependence, if it exists, is weaker. The power β is close to 0.2 = 1/5. The evidence therefore shows that the boundary layer between the viscous sublayer and the free stream consists of two different selfsimilar regions.
A Comparison with the Wall Law of the Flow in the Pipes and an Effective Reynolds Number for the Turbulent Boundary Layer
To interpret our result we turn to the scaling for the intermediate region of pipe flow—the region between the viscous sublayer and close vicinity of the axis (7, 8). It has the form 2 where ū is the average velocity (total discharge divided by the crosssection’s area) and d is the diameter of the pipe.
Intuitively it is clear that at moderate values of η the scaling law (2) and the scaling law for the inner region of the turbulent boundary layer should coincide. The problem is to establish a correspondence between the welldefined Re of pipe flow and the illdefined quantity Re_{θ}.
If such a correspondence does exist, then with a redefined Reynolds number of the boundary layer Re the scaling law (2) should be valid for the boundary layer. Therefore two Reynolds numbers for zeropressuregradient turbulent boundary layer were introduced, Re_{BL}^{(1)} and Re_{BL}^{(2)}, obtained by processing experimental data in the following way: 3 The question is whether ln Re_{BL}^{(1)} and ln Re_{BL}^{(2)} are close; the results are presented in Table 2.
As we can see, the logarithms of Re_{BL}^{(1)} and Re_{BL}^{(2)} are close. (Only the logarithms should be compared because the small parameter of the theory is 1/ln Re.) Therefore, we introduce the effective Reynolds number Re for the turbulent boundary layer by the formula 4 as a basic Reynolds number. The ratio Re_{θ}/Re, i.e., the ratio of the momentum thickness θ of the boundary layer to the effective length scale, is of primary interest. Table 2 and Fig. 3 suggest that basically this ratio is a constant, approximately equal to 1/3. The most important point is the very existence of the effective length scale. Another point of interest is the ratio of the free stream velocity U to the friction velocity u_{?}. Table 2 and Fig. 4 show that basically U/u_{?} is a linear function of ln Re, as is the ratio ū/u_{?} in pipe flow. We should mention that in the zeropressuregradient turbulent boundary layer problem there is an uncontrollable parameter: the level of turbulence in the outer flow. The scatter in the values of B might be due to the influence of this parameter.
Conclusions
We have shown that the structure of the zeropressuregradient boundary layer consists of two selfsimilar flows having different scaling laws. Both laws reveal incomplete similarity in a basic parameter. The introduction of the effective Reynolds number of the boundary layer allowed us to establish a correspondence between the scaling law in the inner part of the boundary layer and the scaling wall law in a pipe. The experiment of Nagib and Hites, as well as earlier experiments of Naguib, suggest that a properly defined effective Reynolds number for boundary layer flow gives an appropriate characterization of the flow regime. The ratio of the effective length scale to the momentum thickness of the boundary layer seems to be a constant, approximately equal to 3. The ratio of the free stream velocity to the friction velocity is a linear function of the logarithm of the effective Reynolds number. The scaling relationship for the second regime can also be represented in the form 5 if we prefer the external velocity U to be the basic variable in the second regime. The values of B′ are presented in Table 1. The scatter remains practically the same, and may support our opinion that it is due to the uncontrolled parameter.
It seems of interest to verify all these conclusions using other experimental data.
Acknowledgments
We acknowledge with gratitude the cooperation of Prof. H. M. Nagib, Dr. M. Hites, and Prof. C. Wark. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DEAC0376SF00098, and in part by the National Science Foundation Grants DMS9414631 and DMS8919074.
Footnotes

A. J. Chorin
 Accepted May 19, 1997.
 Copyright ? 1997, The National Academy of Sciences of the USA
References
 ?
 ?
 Monin A S,
 Yaglom A M
 ?
 ?
 GadelHak M
 Sreenivasan K R
 ?
 Hites M
 ?
Nagib, H. & Hites, M. (1995) AIAA paper 950786, Reno, NV.
 ?
 Barenblatt G I
 ?
 Barenblatt G I,
 Chorin A J,
 Prostokishin V M