## Abstract

Flat plate turbulent boundary layers under zero pressure gradient at high Reynolds numbers are studied to reveal appropriate scale relations and asymptotic behaviour. Careful examination of the skin-friction coefficient results confirms the necessity for direct and independent measurement of wall shear stress. We find that many of the previously proposed empirical relations accurately describe the local *C*_{f} behaviour when modified and underpinned by the same experimental data. The variation of the integral parameter, *H*, shows consistent agreement between the experimental data and the relation from classical theory. In accordance with the classical theory, the ratio of *Δ* and *δ* asymptotes to a constant. Then, the usefulness of the ratio of appropriately defined mean and turbulent time-scales to define and diagnose equilibrium flow is established. Next, the description of mean velocity profiles is revisited, and the validity of the logarithmic law is re-established using both the mean velocity profile and its diagnostic function. The wake parameter, *Π*, is shown to reach an asymptotic value at the highest available experimental Reynolds numbers if correct values of logarithmic-law constants and an appropriate skin-friction estimate are used. The paper closes with a discussion of the Reynolds number trends of the outer velocity defect which are important to establish a consistent similarity theory and appropriate scaling.

## 1. Introduction

Turbulent boundary layers (TBLs) are among the most intriguing and important turbulent flows which have so far resisted satisfactory physical and mathematical modelling. From the theoretical standpoint, the question of the asymptotic behaviour in the limit of infinite Reynolds number arises naturally when treating the boundary layer as a singular perturbation problem. This latter framework has considerably clarified the problem, but not the solution. Since the Reynolds averaged equations are not closed, even the functional form of the inner (near-wall) and the outer asymptotic expansions is still being debated. Hence, in this paper, we start from the recent extensive experimental data over a wide range of Reynolds numbers and attempt to extract from them new information, and thereby, understand the behaviour of zero pressure gradient (ZPG) TBLs at high Reynolds numbers. All along, we will be concerned with the question of whether the trends extracted from the data already represent the true asymptotic behaviour or are just engineering approximations valid over a limited range of Reynolds numbers. We note already here that this endeavour is more often than not hampered by the requirement of an experimental accuracy which will never be attained.

Recently, advances in measurement techniques, equipment and facilities have enabled experiments at high Reynolds numbers (Hites 1997; Österlund 1999; DeGraff & Eaton 2000; Knobloch & Fernholz 2002; Nagib *et al*. 2004*b*) which will be used, complemented by data from some classical experiments (Schultz-Grunow 1940; Smith & Walker 1958; Karlsson 1980; and data found in Coles 1968). This study was primarily motivated by the experiments of Nagib *et al*. (2004*b*) in the National Diagnostic Facility (NDF) at IIT and those of Österlund (1999) at KTH. The range of Reynolds number *Re*_{θ} based on momentum thickness for these experiments is between 2500 and 70 000 for ZPG. The classical theory of logarithmic similarity in the overlap region will be used and extended to explain the behaviour of various parameters and trends seen in experiments.

## 2. Skin-friction and integral momentum equation

The Kármán integral momentum equation obtained from the Reynolds-averaged *x*-momentum equation for ZPG TBLs is given as(2.1)The skin-friction coefficient, *C*_{f}, can be found by direct measurement of *u*_{τ} or by calculating d*θ*/d*x* from the mean velocity. The estimation of d*θ*/d*x* requires accurate measurement of velocity throughout the boundary layer especially near the wall, and the streamwise derivative of *θ* is difficult to obtain. Traditionally, except for a few experiments where wall shear stress was measured directly, most experimentalists estimate *u*_{τ} from a best-fit to the well-known logarithmic law for the mean velocity in inner scale, using typically 0.41 and 5.0 for the Kármán constant *κ* and additive constant *B*, respectively. However, with the relatively new measurement technique of oil-film interferometry, accurate and independent wall shear stress measurements over a wide range of *Re* have been documented by Österlund (1999) and Nagib *et al*. (2004*a*), leading to a modification of these traditionally used values. Oil-film interferometry is currently the most reliable method for accurate and direct measurement of mean skin friction (within 1.5%) and it can also measure its direction (Ruedi *et al*. 2003). In this section, we use the direct wall shear stress measurements of the experiments at IIT and KTH to study the development of *C*_{f} and evaluate the quality of various relations which have been proposed over the years.

Figure 1 shows the decrease of *C*_{f} with *Re*_{θ}. The good collapse of diverse data, independent of how the flow was established and how the Reynolds number was reached (with a long fetch or a high free-stream velocity), indicates that *C*_{f} exhibits self-similarity. Along with the experiments, the predicted skin-friction coefficient using some of the empirical relations found in Schlichting (1979), George & Castillo (1997) and White (2006) is also plotted. These relations and their functional forms are listed in table 1. In the top part of the figure, various relations are plotted using the original coefficient values given by the respective researchers. It is clearly seen that none of the original relations are in good agreement with the recent experimental data from KTH and NDF. This is primarily due to the use of experimental data over a small Reynolds number range to establish the coefficients required for the various forms of proposed relations.

In the following, we will use the Coles–Fernholz relation for further theoretical developments. Even though there is no proof that this relation is the most appropriate, it has relatively solid foundations in Clauser theory (1954), which uses *u*_{τ}/*ν* as the inner length-scale, *Δ*=*U*_{∞}*δ*^{*}/*u*_{τ} as the outer length-scale and *u*_{τ} as the velocity scale for both the layers. As shown by Fernholz & Finley (1996; see also Wilcox 1995), the Coles–Fernholz 1 relation involving is obtained by patching or matching of the inner and outer expressions for the mean velocity profile in the overlap region where it is assumed to be given by the ‘log-law’. Here, we choose to start with the Coles–Fernholz 2 relation, which amounts to replacing *Δ* by *Λ*=*U*_{∞}*θ*/*u*_{τ} in the Clauser theory. Some reasons are given below and elaborated on in a forthcoming paper by Monkewitz & Nagib (submitted). Therefore, we start with the relation(2.2)where dots signify that equation (2.2) is considered as a truncated version of the full *leading order* of an asymptotic series. This is in keeping with Crighton & Leppington (1973) who showed that in the presence of logarithms, terms of the form *Re*^{−m}ln^{n}(*Re*) must be treated as order *m* terms for any *n*. In other words, it is not consistent to add a term to equation (2.2) before one is sure that the leading order is complete (see Monkewitz & Nagib submitted).

Using the definition of shape factor *H*=*δ*^{*}/*θ*, one can express the above equation in terms of as(2.3)When considering that *H*→1 in the limit of infinite Reynolds number (see §3), it follows immediately from equation (2.3) that *C* must be asymptotically equal to *C*^{*}, meaning that the truncated equations (2.2) and (2.3) are asymptotically equivalent. This justifies the choice of equation (2.2) as starting point of our analysis which is thereby simplified. The same choice of the Coles–Fernholz 2 relation has been made by Österlund (1999) to fit the KTH oil-film data with the modified values of 0.38 and 4.1 for *κ* and *C*, respectively. One has to note, though, that the mean velocity defect in the outer region shows, within experimental accuracy, *Re*-independent scaling with *y*/*Δ*, as shown in §5. This appears to be an argument against our choice of starting point, but this discussion goes beyond the scope of the present paper and is covered by Monkewitz & Nagib (submitted).

As seen in table 1, using the collective NDF and KTH oil-film data, the best fits for *C* and *C*^{*} are 4.127 and 3.354, respectively. This clearly demonstrates that additional terms are needed in equations (2.2) and (2.3) if one wants to satisfy the asymptotic requirement of *C*=*C*^{*} *and* obtain a good fit of *C*_{f} in the Reynolds number range of the KTH and NDF data. It should be noted though that an extremely high experimental accuracy of the order of 0.1% is required to reliably determine such additional terms (see attempt made by Monkewitz & Nagib). Many other relations describing the *Re* dependence of the skin friction for ZPG TBLs are listed in table 1. All of them are based on some integral form of the boundary layer equations of Prandtl, complemented by empirical data fits. We observe that two of them, the laws of Kármán–Schoenherr and of White (2006), are closely related to the Coles–Fernholz relations in that, when expanded in inverse powers of ln(*Re*), the largest term is proportional to ln^{−2}(*Re*). In order to compare all the relations on the same graph as a function of *Re*_{θ}, one needs *H* (see §3) and a relation between *Re*_{x} and *Re*_{θ} (the reader is reminded that *x* is measured from the leading edge). Such a relation can be obtained by introducing equation (2.2) into the Kármán integral momentum equation(2.4)where *x*′ denotes the streamwise distance from a corrected origin which accounts for the length of the initial laminar boundary layer and/or the location of the virtual origin. In laboratory experiments, TBLs are typically generated by artificially tripping an initially laminar boundary layer, but it is not practical to relate the distance of the trip from the leading edge to the origin of *x*′. The inverse of equation (2.4) can now be readily integrated to yield(2.5)To finally relate to *Re*_{x}, an empirical relation is sought which provides a good fit of equation (2.5) to the combined data of KTH and NDF. It is given by(2.6)This relation is compared to various data in figure 3 which clearly shows that equation (2.6) with equation (2.5) is quite accurate in describing also the experiments other than the KTH and NDF measurements. However, nothing can be said about the performance of relation (2.6) outside the Reynolds number range where data are available. Alternatively, one can use a simple power-law relation between *Re*_{x} and *Re*_{θ}, also shown in figure 3, again over a limited *Re* range.

To investigate the reasons for the discrepancies between the different relations of table 1 in the top part of figure 1, they were modified by changing only one empirical coefficient to obtain a least-square fit with the high Reynolds number data of KTH and NDF. The coefficients that are modified are shown in bold font in the respective equations and their modified values are shown in the last column of table 1. From the plot in the bottom part of figure 1, we find that *all* the modified relations in table 1 fit the data within experimental accuracy, while the slightly modified version of the nearly a century old Prandtl–Kármán skin-friction relation appears to be exceptionally good. The collapse of all profiles is remarkable near *Re*_{θ}≈20 000, around which data from several of the previous experiments were made. The conclusion is that many of the commonly used skin-friction relations for ZPG TBLs provide correct predictions for all laboratory Reynolds numbers if they are underpinned by the *same* accurate measurements. Therefore, as concluded recently by Nagib *et al*. (2004*a*), the choice between the various relations, including the choice between log-law- and power-law-based relations, can only be resolved by theoretical arguments which are consistent with the behaviour of velocity profiles over a wide range of Reynolds numbers.

These observations resolve the problem of estimating *C*_{f} for most Reynolds numbers of practical interest. We find that both *C*_{f}(*Re*_{θ}) and are equivalent when used at finite Reynolds numbers, with appropriate but different *C* and *C*^{*}, as the differences between the predicted *u*_{τ} are less than the experimental accuracy. Therefore, the choice of a *C*_{f} relation for numerical estimates depends generally on convenience. For most of the experimental data available, either the near-wall velocity measurements are missing or they contain larger errors. Hence, the determination of *δ*^{*} is relatively more accurate than *θ* which leads us to use to estimate *u*_{τ} in some of the following analysis. However, the above considerations do not answer the question of the asymptotic behaviour of *C*_{f}. To illustrate this, the *C*_{f} predictions by the various modified relations are extended in figure 2 to *Re*_{θ}=10^{20}. It is not surprising that the differences between the various predictions increase steadily: while they are less than 0.5% at *Re*_{θ}=20 000, they increase to over 40% beyond an *Re*_{θ} of 10^{16}.

## 3. Development of integral parameters

In wall-bounded flows, the exact equation for *H*=*δ*^{*}/*θ* can be derived using definitions of *δ*^{*} and *θ* (Tennekes & Lumley 1972; Nagib *et al*. 2005) and given as(3.1)The variation of *H* can now be described by two parameters: and *C*′. Note that this expression for *H* is exact. Mean velocity profiles from experiments show that *C*′ asymptotes to a constant value at high Reynolds numbers in ZPG TBLs. This is consistent with the finding in §5 that the outer () scales with (*y*/*Δ*) except near the wall. Therefore, the wall region contributes an *O*(1/*Re*_{θ}) correction to *C*′ and we find that(3.2)equations (3.1) and (3.2) now express *H* as a function of *Re*_{θ} only. The shape factor for experiments in ZPG TBLs is plotted against *Re*_{θ} in figure 4. The data included represent more than 400 data points covering a wide range of Reynolds number and clearly show that *H* decreases with *Re*_{θ} and does not level off at the traditional value of about 1.3. The relation (3.1), with *C*′ given by equation (3.2), is also included in figure 4 and found to fit the data very well. The dotted lines represent ±3% deviation from equation (3.1) with *C*′=7.135, enclosing 97% of data plotted. The analytical curve also agrees well with the high *Re* atmospheric boundary layer data of Priyadarshana & Klewicki (2004) and the low *Re* DNS data of Spalart (1988), except for the lowest Reynolds number. It is gratifying to note that both of these are very different from all the other laboratory data included in the figure. Since in the limit of infinite Reynolds number, , as shown in the previous section, and the shape factor, *H*, must approach unity. The trend of experimental data and their agreement with equation (3.1) confirms this prediction. However, *H* approaches this limit extremely slowly as shown in the inset of figure 4. The experimental verification of this limit is impossible and even getting within 10% of the asymptotic limit of *H* will likely remain forever beyond the reach of experimental facilities. As discussed in Nagib *et al*. (2005), it is not informative to simply associate *H*→1 with *θ*=*δ*^{*}, since in this limit *θ* and *δ*^{*} are either infinite or zero, depending on how the Reynolds number has been increased. If *Re*→∞ is achieved by increasing *U*_{∞} at a fixed *x*, the limit of (*δ*^{*}−*θ*)/*θ*=*H*−1 is zero. Using L'Hopital's rule, one has , which tells us that for increasing *U*_{∞}, the difference (*δ*^{*}−*θ*) goes to zero faster than *θ*. On the other hand, if *Re*→∞ is achieved by increasing *x* with a fixed *U*_{∞}, one finds by an analogous argument that (*δ*^{*}−*θ*) goes to infinity slower than *θ*.

Another important parameter *D* is the ratio of ‘Clauser’ *Δ* to the local boundary layer thickness *δ*, which is a constant according to the classical theory. In experiments, the theoretical boundary layer thickness *δ* is undefined. In this paper, *δ*_{99} will be used based on the location where the mean velocity is 99% of the free-stream velocity. Figure 5 shows this ratio, denoted henceforth by *D*_{99}, for various experimental data. It is seen that *Δ*/*δ*_{99} can be considered a constant near 4.5 for the KTH and the NDF data. The high *Re* data of Knobloch & Fernholz (2002) also confirm this. However, at low *Re*, the experimental ratio is below 4.5 and gradually increases with Reynolds number towards this value. This behaviour is attributed to the flow being not fully developed1 at low Reynolds numbers, where the mean flow profiles are not expected to be self-similar. It should be noted though that the data of Smith & Walker (1958) and Karlsson (1980) level off at a value of *Δ*/*δ*_{99} somewhat below the NDF data. Since *D* is nearly proportional to *D*_{99} over the practical range of Reynolds numbers, the finding that *Δ*/*δ*_{99} asymptotes to a constant or ‘near constant’ is a further confirmation of Clauser's theory and plays an important role in the dynamics, which is the subject of §4. An expression for *D*_{99} can be constructed using equations (5.2) and (2.2) to yield(3.3)With *Π*_{99} a constant2 equal to 0.55, as shown later in §5, the expression (3.3) is also included in figure 5. It is seen to have a slightly upward trend over this Reynolds number range, which matches the trend of the KTH and NDF data very well. We note, however, that in the limit of infinite Reynolds number , which is not consistent with the classical theory. If the boundary layer thickness *δ* is used instead, the term involving (*HRe*_{θ}) in equation (3.3) disappears, and we find that *D* is nearly a constant equal to 3.5 for high *Re* boundary layers, as demonstrated by Chauhan & Nagib (in press *a*). To calculate *D*, the value of *δ* is obtained by fitting a composite profile. However, such an approach requires an analytical form for the composite or outer profile, which goes beyond the scope of this paper.

## 4. Ratio of turbulent and mean time-scales

From the physical characteristics of turbulence, it is well known that the time-scale of the large-scale turbulence is of the same order as the convective time-scale of the mean flow (Tennekes & Lumley 1972). Such an order of magnitude relation will now be considered as a *criterion* for a ‘well-behaved’ interaction between the mean flow and the turbulence. In TBLs, an order of magnitude estimate of turbulent time-scale can be obtained with the length-scale *δ* or *δ*_{99} and the velocity scale *u*_{τ}. For the mean flow, the convective time-scale can be based on the overall development length of the boundary layer, estimated as *L*≈*x* and the free-stream velocity, *U*_{∞}. The behaviour of *δ*_{99}/*u*_{τ} with *x*/*U*_{∞} for ZPG TBLs was recently presented by us in Nagib *et al*. (2005), where it was demonstrated that the two time-scales are very well correlated in a large number of experiments, regardless of flow history or initial conditions. In addition, it was noted that the ratio *Ω* of the turbulent time-scale *δ*_{99}/*u*_{τ} and the mean flow time-scale *x*/*U*_{∞} was very close in value to *κ*. Additional theoretical developments and the analysis of more experimental data have further clarified this. The non-dimensional ratio *Ω* can be written as(4.1)Figure 6 shows the variation of *Ω* for various experiments in ZPG TBLs. The consistent behaviour within individual datasets and the remarkable agreement between datasets obtained with different measuring techniques in different experimental facilities clearly point to the significance of *Ω*. The physical interaction/balance of mean flow with the turbulence which has been invoked in theory is experimentally demonstrated by the parameter *Ω*. This agreement is even better in both magnitude and trend when only the KTH, NDF and Smith & Walker (1958) data are considered. From the agreement in figure 6 between data from different experiments with different initial flow conditions, it is concluded that in fully developed ZPG TBLs at sufficiently high Reynolds numbers, the streamwise development of the mean flow and the behaviour of the large-scale turbulence are independent of flow history or ‘initial conditions’.

Based on the results shown in the previous sections, an explicit expression for *Ω* as a function of *Re*_{θ} can be developed. The expressions for *H*, *Re*_{x}/*Re*_{θ} and in equation (4.1) are taken from equations (2.2), (2.5), (2.6) and (3.1), while *Δ*/*δ*_{99} is taken to be a constant3 equal to 4.5 to yield(4.2)where is defined in equation (2.5). It can be seen in figure 6 that the theoretical expression (4.2) is in very good agreement with the experimental data which confirms the validity of the approximations made. The scatter of various experiments is primarily due to two reasons: the lack of reliable information regarding the origin of the boundary layer and inaccuracies in the estimates of skin friction. However, all the experiments show a decreasing trend of *Ω* with *Re*. The asymptotic value of *Ω* based on equation (4.2) is 0.222 at infinite *Re*. Considering the limit *Re*_{θ}→∞, we note that . This yields the asymptotic behaviour of *Ω* and *C*_{f}, given by equations (4.2) and (2.1), respectively, as(4.3)

The time-scale ratio *Ω* can be interpreted as the ratio of *local time-scale* of turbulence to the *age* of turbulence. The changing mean flow and the resulting change of wall shear stress induces a change in the structure of turbulence and hence *δ*_{99}/*u*_{τ} increases. But as we see in the figure, *Ω* decreases with increasing *Re* implying that at finite *Re* a lag exists between the two time-scales, i.e. *x*/*U*_{∞} increases at a faster rate than *δ*_{99}/*u*_{τ}. The fact that at lower Reynolds numbers *Ω* is higher implies that turbulence is taking a longer time to come to equilibrium with the mean flow. This relative lag decreases as the flow develops and at infinite *Re*, *δ*_{99}/*u*_{τ} and *x*/*U*_{∞} become so large that the lag becomes insignificant and *Ω* asymptotes to a constant. There is also a practical side to the above development: given that *U*_{∞} and *x* are often known, all the parameters needed to describe the boundary layer can be obtained by starting with *Re*_{x} and using equations (2.2), (2.5), (2.6), (3.1), (3.3) and (4.1). The consistency of this approach is confirmed by the agreement of respective curves with experiments in figures 1, 3–6. In addition, equation (4.3) reveals an important conclusion about power relations often used to represent the streamwise development of boundary layer thicknesses. Such power-like relations, obtained empirically from fits to experimental data, can only be an approximation over a limited *Re* range, and are highly misleading when used as the functional form *in lieu* of equation (2.6) at all *Re*_{θ} or even over any extended range of *Re*. Considering the good agreement of curves obtained from the Coles–Fernholz relation with experiments in figures 1 and 3, one can use a local power fit for *Re*_{θ} versus *Re*_{x} to find the exponent of the power-like local dependence. Similarly, from equations (3.1) and (3.3), the exponents for and *Re*_{δ} can also be found. Using such an approach, one finds that the exponent increases with increasing *Re* towards a limit of 1, which is also concluded from equation (4.3). At infinite *Re*, the exponent for *θ* becomes nearly equal to that for *δ*^{*}, consistent with the behaviour of *H*. On the other hand, the exponent for *δ* is always higher than both, implying that the physical boundary layer thickness grows faster than the integral thicknesses, and that the ratios *δ*/*θ* and *δ*/*δ*^{*} will tend to infinity.

## 5. Description of mean velocity, *U*

Figure 7 shows the mean velocity profile in inner scaling for selected datasets of NDF and KTH. The profiles plotted cover a wide range of Reynolds numbers and have been obtained at different *x* locations and for different free-stream velocities. We observe that all the profiles collapse near the wall and in the overlap region, independent of how the Reynolds number was achieved. The classical logarithmic law,(5.1)is clearly exhibited by the profiles for *y*^{+}≥200, and is also demonstrated with a constant log-law diagnostic function *Ξ*=*y*^{+}(d*U*^{+}/d*y*^{+}), which equals 1/*κ* in the plot on the top. The collapse displayed in the mean velocity profiles can also be seen in the more sensitive diagnostic function. We find that the diagnostic function remains constant in the overlap region from *y*^{+}≥200 to *y*/*δ*=0.15 (Österlund 1999). Also shown in the figure is *U*^{+} and *Ξ* for channel flow DNS performed by Hoyas & Jiménez (2006) at *Re*_{τ}=2000. The lack of a well-developed logarithmic overlap region points to the fact that all available DNS of wall-bounded flows remain far from representing high *Re* conditions. We include the DNS data here to also point out the large difference between the magnitude of the outer component of the mean velocity profile between boundary layers and the channel flow. Based on the observed logarithmic behaviour in the overlap region of experimental boundary layers, the mean velocity profile in the outer region can be written as(5.2)The wake parameter *Π* is the coefficient of a universal profile fitted to the deviation of the mean velocity from the logarithmic law (Coles 1968). By definition, at *y*=*δ*. In its original form, the wake function was modelled as the square of a sine function. Later, different wake functions in the form of cubic or quartic polynomials were adopted. In the conventional approach of Coles, the outer part of a measured velocity profile is least-square fitted by equation (5.2) to determine *Π*, *δ* and *u*_{τ}. However, the available wake functions in the literature have different limitations, e.g. a finite slope at the boundary edge for the Coles sine function. For this reason, if one tries to fit experimental data of *U* versus *y* with the same inner layer function but different wake functions, one obtains different values for *Π*, *δ* and *u*_{τ}. These values and their accuracy will depend on the type of wake function fitted and the number of data points available for the fit (Chauhan & Nagib in press *b*).

To avoid any ambiguity, we find *Π* here in an indirect way as suggested by Nagib *et al*. (2005), which is based on the property . This leads to a relation for *Π*_{99}, which is obtained using *δ*_{99} that can be written as(5.3)Note that near the edge of the boundary layer, the wall-normal velocity gradient is very small, and hence the reasonable approximation, , has been used in equation (5.3). The wake parameter determined with equation (5.3) for various experiments is plotted in figure 8 against *Re*_{θ}. The values of *κ* and *B* used are 0.384 and 4.173, respectively, and the Coles–Fernholz relation (2.3) is used to determine *u*_{τ}. As demonstrated in the figure, various experiments agree remarkably well in both magnitude and trend of *Π*_{99}. For *Re*_{θ}>6000, the experiments indicate that *Π*_{99} remains constant near 0.55, which is also the value suggested in the classical theory and by Coles. It should be noted again that values of *Π*_{99} are obtained only from and *δ*_{99}, which are determined from direct measurements and not dependent on any assumptions or theory. However, the observed agreement is only achieved with the recently established values of *κ* and *C* or *C*^{*} in the modified Coles–Fernholz relation based on the oil-film data of KTH and NDF, as well as *B*, and again *κ* from the mean velocity profiles. An interesting trend that has been speculated on by several researchers is the decline in the values of *Π* after it reaches the ‘asymptotic’ level, as seen in data of Smith & Walker (1958) and DeGraff & Eaton (2000). This behaviour has also been noted privately by Professor Don Coles in some other datasets. We believe that such behaviour is only a manifestation or symptom of underdeveloped or anaemic boundary layers that were set-up to achieve high Reynolds numbers with inadequate development fetch (Chauhan & Nagib in press *a*). As the structure of mean turbulence in the boundary layer is dependent on the structure of the mean flow, the behaviour of mean flow parameters observed in figures 6 and 8 should be reflected in profiles of turbulence intensity.

The classical TBL theory has two length-scales, *ν*/*u*_{τ} and *δ*, but only one velocity scale, *u*_{τ}. The appropriateness of *u*_{τ} as the velocity scale for the velocity defect in the outer part of the TBL has nevertheless been questioned by many researchers. Two new outer velocity scalings have been proposed in the past decade by George & Castillo (1997) and Zagarola & Smits (1998). George & Castillo proposed power law behaviour in the overlap region, with the argument that *U*_{∞} is the proper outer velocity scale. On the other hand, Zagarola & Smits proposed a scaling with *U*_{∞}*δ*^{*}/*δ*, based on the superpipe data.4 Instead of discussing these scales, we look at the appropriateness of *u*_{τ} as the velocity scale based on some fundamental definitions in boundary layers. The displacement thickness, *δ*^{*}, can be expressed in terms of the normalized velocity defect, *W*^{+}, as(5.4)The above equation directly translates into an integral constraint for the mean velocity defect given as(5.5)The mean velocity defect , by definition, must satisfy the above integral constraint for all boundary layers and at all Reynolds numbers. This makes the choice of *u*_{τ} as velocity scale and *Δ* as length-scale promising. Even though the area under the profiles of does not change with Reynolds number, nothing can be said about its shape yet. Similarly, the definition of *θ* is expressed in terms of *W*^{+} to obtain(5.6)Here, is defined similar to *Δ* (Nagib *et al*. 2005). Equation (5.6) outlines another integral constraint for *W*^{+}(*y*/*Λ*) which includes the development of the boundary layer through the shape factor *H*. The two functions, *W*^{+}(*y*/*Λ*) and *W*^{+}(*y*/*Δ*), are plotted in figure 9 for profiles from NDF, KTH and Knobloch & Fernholz (2002). The plots on the left of figure 9 with linear scale give a good impression of the difference between *W*^{+}(*y*/*Δ*) and *W*^{+}(*y*/*Λ*), and their associated integral constraints equations (5.5) and (5.6), respectively. As the integral under *W*^{+}(*y*/*Λ*) has to approach unity for *Re*→∞, one can appreciate how much this curve has to shift between *Re*_{θ}=10^{5} and, say, 10^{60}, where *H* is finally getting close to unity. In the log-linear plot, we see that it is the outer part of *W*^{+}(*y*/*Λ*) which has to move towards smaller *y*/*Λ* in order to reduce the area under the curve. This slow trend can already be detected over the limited range of *Re*_{θ} between 2500 and 115 000 covered by the experimental data. On the other hand, the profiles of *W*^{+}(*y*/*Δ*) show no visible trend over the *Re* range of the measurements. This suggests that the mean velocity defect scales with *u*_{τ} as velocity scale and *Δ* as length-scale consistent with equation (5.5). However, it should be noted that the intercept of *W*^{+} at *y*/*Δ*=0 is , which increases with Reynolds number. Therefore, profiles of *W*^{+}(*y*/*Δ*) should change their shape to satisfy the integral constraint (5.5) at all Reynolds numbers. This could be achieved in two ways. First, the profiles need to change only near the wall with the intercept moving up and the viscous sublayer becoming narrower to keep the area constant as *Re* increases. This would result in *Re*-independent self-similarity of *W*^{+}(*y*/*Δ*) in both overlap and outer regions, similar to the one in figure 9. To validate this only from the mean velocity profiles, one would need accurate measurements very near the wall which are difficult and hence not widely available, especially at high *Re*. A plausible alternative would be a profile which not only changes near the wall but also changes in the overlap and/or outer region. However, such Reynolds number changes are extremely slow and again require very high measurement accuracies. The mean velocity data from experiments available to date are not sufficient to address the issues discussed here. Therefore, verification of a proposed similarity theory for the mean velocity cannot be based on equations (5.5) and (5.6) alone, but also needs to take into account the experimental behaviour of the various parameters discussed in §§2–5.

## 6. Conclusions

Armed with the recent experimental results in higher Reynolds number ZPG TBLs, we have revisited and carefully examined the classical theory and the trends of various critical mean-flow parameters as the Reynolds number grows beyond any experimentally documented values. Starting with the cornerstone of the recent experiments, namely the accurate and the independent measurements of the wall shear stress, we systematically clarify most apparent inconsistencies among experiments, empirical correlations and theory. We find that the trends in the data are very well reproduced by relations emerging from the classical theory and a log-law based on *κ* and *B* equal to 0.384 and 4.173, respectively. With this new foundation, we find the agreement between many of the classical experiments and the recent data from the NDF and the KTH to be most gratifying.

## Footnotes

One contribution of 14 to a Theme Issue ‘Scaling and structure in high Reynolds number wall-bounded flows’.

↵The term ‘fully developed’ refers to the independence of normalized mean flow from how the flow was established.

↵The notation of

*Π*_{99}is used to indicate that the wake parameter is obtained using*δ*_{99}and not the physical boundary layer thickness,*δ*.↵Since the expression for

*Ω*will be compared to the KTH and NDF data, we use the average value of*D*_{99}for these datasets and not the probably higher asymptotic limit.↵As noted in §3, the ratio (

*U**)/(_{∞}δ*u*_{τ}*δ*) asymptotes to a constant at high Reynolds number. Hence, the Zagarola-Smits velocity scale is simply a multiple of*u*_{τ}in ZPG TBL.- © 2007 The Royal Society