Experiments in Fluids

, 47:1045

Determination of skin friction in strong pressure-gradient equilibrium and near-equilibrium turbulent boundary layers

Authors

  • Shivsai Ajit Dixit
    • Department of Aerospace EngineeringIndian Institute of Science
    • Department of Aerospace EngineeringIndian Institute of Science
Research Article

DOI: 10.1007/s00348-009-0698-2

Cite this article as:
Dixit, S.A. & Ramesh, O.N. Exp Fluids (2009) 47: 1045. doi:10.1007/s00348-009-0698-2

Abstract

The conventional Clauser-chart method for determination of local skin friction in zero or weak pressure-gradient turbulent boundary layer flows fails entirely in strong pressure-gradient situations. This failure occurs due to the large departure of the mean velocity profile from the universal logarithmic law upon which the conventional Clauser-chart method is based. It is possible to extend this method, even for strong pressure-gradient situations involving equilibrium or near-equilibrium turbulent boundary layers by making use of the so-called non-universal logarithmic laws. These non-universal log laws depend on the local strength of the pressure gradient and may be regarded as perturbations of the universal log law. The present paper shows that the modified Clauser-chart method, so developed, yields quite satisfactory results in terms of estimation of local skin friction in strongly accelerated or retarded equilibrium and near-equilibrium turbulent boundary layers that are not very close to relaminarization or separation.

1 Introduction

Determination of the skin friction coefficient Cf in a turbulent boundary layer (TBL) always poses difficulties due to the sharp gradient of mean velocity in the near-wall region. This problem gets even more acute when one encounters strong pressure-gradient driven TBL flows. For a zero pressure-gradient (ZPG) turbulent boundary layer flow, the two most widely used methods for determining the skin friction are the Preston-tube method (see Patel 1965) and the Clauser-chart method (see Clauser 1954). These methods are ‘quick’ as compared to the method based on the use of streamwise momentum integral equation. Application of the momentum integral equation requires accurate measurement of several mean velocity profiles at closely spaced streamwise stations. Furthermore, the accuracy of the momentum integral method is known to degrade in adverse pressure-gradient (APG) flows (see, e.g. Bradshaw and Ferriss 1965) due to thick sidewall boundary layers that compromise the two-dimensionality of the mean flow. On the other hand, only one measurement of the ‘Preston head’ (for the Preston-tube method) or only one measured mean velocity profile (for the Clauser-chart method) is sufficient to give the local skin friction coefficient.

There have been attempts to device procedures for estimating skin friction in a variety of situations involving external as well as internal flows. Brereton (1989) has discussed the possibility of using the Clauser-chart method in the case of unsteady turbulent boundary layers. Wei et al. (2005) have shown that the Clauser-chart method can contaminate the data in such a way that the subtle Reynolds number dependence of the near-wall region does not become apparent at all. This difficulty arises because the Clauser-chart method forces the mean velocity profile to follow the universal logarithmic law, which itself is expected to exhibit a weak Reynolds number dependence (see Buschmann and Gad-el-hak 2003).

Recently, Kendall and Koochesfahani (2008) have presented an attractive proposal of using the model velocity profile proposed by Musker (1979) to estimate the skin friction in turbulent wall-bounded flows. They have shown that their approach works well even with the single data point obtained from a near-wall measurement. They have used the ZPG turbulent boundary layer data of Österlund (1999) and the fully developed turbulent pipe flow data (which they have mistakenly referred to as the favourable pressure-gradient TBL data) from the superpipe experiments of McKeon et al. (2004) to demonstrate the validity of their proposal. However, Musker (1979) has made the assumptions of linear mean velocity profile and constant total shear stress in the near-wall region while deriving the continuous closed-form expression for the mean velocity profile. Consequently, Musker’s profile itself is expected to be valid only for ZPG turbulent boundary layers since the total shear stress variation is in fact linear (not constant) for TBL flows with pressure gradients (see Townsend 1976; Patel and Head 1968, etc.).

It is well known that the mean velocity profiles depart strongly from the universal log law in case of strong FPG flows (see Patel and Head 1968; Nickels 2004; Chauhan et al.2007; Dixit and Ramesh 2008, etc.). It is also known that the log law appears to shift down and its apparent slope increases in case of strong APG turbulent boundary layer flows (see Skåre and Krogstad 1994). As a result, any method, which is based on the universal log law (directly or indirectly), is bound to yield incorrect results for skin friction in such situations. It appears that there exists no method which can yield a ‘quick’ estimate of the skin friction especially in such strong pressure-gradient TBLs, where the universal log law itself becomes invalid. Thus, it would be a distinct advance if a method could be devised that would cater to, at least equilibrium and near-equilibrium TBL flows, over a range of pressure gradients. The purpose of this paper is to present such a method which will be called as the modified Clauser-chart method (MCCM). The method is actually based on the same ideas originally put forth by Clauser (1954). However, the use of this MCCM in strong pressure-gradient situations is not straightforward for the simple reason that the magnitude of the non-dimensional pressure-gradient parameter is not known a priori. As we shall see shortly, the present MCCM in fact works quite well over a wide range of streamwise pressure gradients ranging from strong-favourable to strong-adverse. In addition, it will be shown that the range of pressure gradients covered by the present method is indeed much wider than that of, say, the Preston-tube method.

The outline of the paper is as follows. The essence of the conventional Clauser-chart method is discussed in Sect. 2, which serves as the foundation for the MCCM. Section 3 gives a brief account of the non-universal logarithmic laws in strong pressure-gradient TBL flows. These pressure-gradient-dependent log laws are the building blocks of the MCCM. The MCCM procedure is described in Sect. 4 while Sect. 5 illustrates the use of MCCM for different test cases and gives the comparison between the original (as originally given in the respective data sets) and estimated (as obtained by the MCCM) values of skin friction and other related quantities. Section 6 describes the exercises carried out to assess the effect on the MCCM of changes in the values of the Kármán constant κ0 and the intercept C0 in the standard universal log law. This is especially important in the light of recent investigations in high Reynolds number TBL flows that have shown that the asymptotic values for these constants are different from the classical belief. Section 7 gives an account of a more rational method that may be used to identify the logarithmic region in the mean velocity profile. Conclusions are presented in Sect. 8.

2 Conventional Clauser-chart method

The Clauser-chart method, in its conventional form, makes use of the fact that all fully developed ZPG turbulent boundary layers possess a logarithmic region, with ‘universal’ constants, when plotted in inner coordinates (U+ vs. y+). This is the so-called universal log law and is given by
$$ U_{ + } = {\frac{ 1}{{k_{ 0} }}}{ \ln } \left( {y_{ + } } \right) + C_{ 0} , $$
(1)
where \( U_{ + } = {U \mathord{\left/ {\vphantom {U {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }}, \)\( {\text{y}}_{ + } = {\text{y}}{{U_{\tau } } \mathord{\left/ {\vphantom {{U_{\tau } } \nu }} \right. \kern-\nulldelimiterspace} \nu } \) and κ0 and C0 are ‘universal’ constants having values 0.41 and 5.2, respectively. Here, \( U_{\tau } = \sqrt {{{\tau_{\text{w}} } \mathord{\left/ {\vphantom {{\tau_{\text{w}} } \rho }} \right. \kern-\nulldelimiterspace} \rho }} \) is the friction velocity. The choice of these values of κ0 and C0 has been made based on our own independent measurements (not shown here) in ZPG flow and this aspect is further discussed in Sect. 6. In addition, it should be noted that this pairing of constants has been used previously in the literature (see, e.g. Erm and Joubert 1991).

These values of κ0 and C0 have been somewhat controversial in view of some recent measurements in relatively high Reynolds number TBL flows. We shall discuss this issue in some detail, in the context of the MCCM in Sect. 6. However, for the present discussion, the abovementioned values of κ0 and C0 are entirely adequate.

Using the measured mean velocity profile (\( {U \mathord{\left/ {\vphantom {U {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }} \) vs. \( {y \mathord{\left/ {\vphantom {y \delta }} \right. \kern-\nulldelimiterspace} \delta } \)) in conjunction with these already-known constants κ0 and C0, the ratio \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }} \) may be calculated from Eq. 1. Here U is the free-stream velocity. There are many different but essentially equivalent ways of rearranging and solving Eq. 1, of which we prefer to use the following version for the present discussion.

The conventional Clauser-chart method proceeds in the following steps.
  1. (i)

    First, the measured mean velocity profile is plotted in \( {U \mathord{\left/ {\vphantom {U {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }} \) versus log\( \left( {{{yU_{\infty} } \mathord{\left/ {\vphantom {{yU\infty } \nu }} \right. \kern-\nulldelimiterspace} \nu }} \right) \) form. Such a plot would make the extent of the logarithmic region in the mean velocity profile apparent. Once this extent is identified by visual inspection, it is fitted with a logarithmic curve (the least-squares method). The end points of the logarithmic region are then fixed by a trial and error procedure such that the R-squared value of the fit reaches a maximum. A more rational procedure for identifying the logarithmic region in the mean velocity profile, based on the so-called log law diagnostic function Ξ, is given in Sect. 7. Therein, it is shown that the identification of the logarithmic region by visual inspection followed by a logarithmic curve-fit, as used all through this paper, is sufficiently accurate.

     
  1. (ii)

    The values of the universal constants κ0 and C0 are already known.

     
  1. (iii)
    The universal log law (Eq. 1) is rearranged as
    $$ {\frac{U}{{U_{\infty } }}}{\frac{{U{}_{\infty }}}{{U_{\tau } }}} = {\frac{ 1}{{\kappa_{0} }}}{ \ln }\left( {\frac{y}{\delta }}{\frac{{\delta U_{\infty } }}{\nu }}{\frac{{U_{\tau } }}{{U_{\infty } }}} \right) + C_{ 0} . $$
    (2)
     
  1. (iv)

    Note that Eq. 2 is applicable only in the logarithmic region of the mean velocity profile. In Eq. 2, \( {U \mathord{\left/ {\vphantom {U {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }} \) values for different values of \( {y \mathord{\left/ {\vphantom {y \delta }} \right. \kern-\nulldelimiterspace} \delta } \) (in the logarithmic region) are known. The Reynolds number \( {{\delta U_{\infty } } \mathord{\left/ {\vphantom {{\delta U_{\infty } } \nu }} \right. \kern-\nulldelimiterspace} \nu } = R_{\delta } \) is known from the measurement of the mean velocity profile and the ‘universal’ constants κ0 and C0, are also known. Thus, the only unknown in Eq. 2 is the ratio \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }}. \) Since Eq. 2 is an implicit equation in \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }}, \) the solution has to be found by an iterative procedure. This may be done essentially by carrying out a regression fit. Using the value of \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }} \) so found, the skin friction coefficient is obtained by \( C_{f} = 2\left( {{{U_{\tau } } \mathord{\left/ {\vphantom {{U_{\tau } } {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }}} \right)^{2} \). We call this step as the Cf iteration.

     

This is the essence of the conventional Clauser-chart method. Evidently, the method strongly depends upon the ‘universal’ nature of logarithmic law and hence is not suitable for situations where this universality is known to fail. Consequently, TBL flows involving strong streamwise pressure gradients are not suitable for applying the Clauser-chart method (same is true for the Preston-tube method even though the details are different). Clearly, if there exists a strong pressure-gradient counterpart of the universal log law, say a class of non-universal and pressure-gradient-dependent log laws, then extension of the Clauser-chart method should be possible even to these strong pressure-gradient situations.

In fact, such a pressure-gradient-dependence of the logarithmic region has been observed and reported in the literature, notably by Spalart and Leonard (1986), Nickels (2004) and Chauhan et al. (2007). Very recently, Dixit and Ramesh (2008) have experimentally demonstrated that there indeed exist such non-universal log laws in the case of so-called sink flow TBLs. Following section describes these non-universal log laws in some detail since they are the building blocks of the MCCM.

3 Non-universal logarithmic laws

Dixit and Ramesh (2008) have experimentally shown that in the case of so-called sink flow TBLs (FPG flows in ‘perfect’ equilibrium), the logarithmic region of the mean velocity profile shows noticeable and systematic pressure-gradient dependence. This behaviour is observed simultaneously in the inner scaling (U+ vs. y+) as well as in the defect scaling (\( U - U_{\infty } /U_{\tau } \) vs. \( y/\delta \)) over a range of FPGs. It is well known that the logarithmic relation between velocity U and wall-normal distance y may be rigorously shown to be a consequence of the overlap between the inner and outer regions of a TBL that possess different and independent scalings (see Millikan 1938). The matching argument in the overlap region may be extended to higher orders, which then yields the pressure-gradient dependence of the log laws as has been demonstrated by Dixit and Ramesh (2008).

In this paper, we shall be concerned with the non-universal or pressure-gradient-dependent log law in the inner scaling, which is given by
$$ U_{ + } = {\frac{ 1}{\kappa }}{ \ln }\left( {y_{ + } } \right) + C, $$
(3)
where, \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } = {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }(\Updelta_{p} ) \) and \( C = C(\Updelta_{p} ) \) are systematic functions of the non-dimensional pressure-gradient parameter \( \Updelta_{p} \) which is given by
$$ \Updelta_{{p}} = {\frac{\nu }{{\rho {\text{U}}_{\tau }^{ 3} }}} \, {\frac{{{\text{d}}p}}{\text{d}}x} = {\frac{ - K}{{ ({{{C}_{f} } \mathord{\left/ {\vphantom {{{\text{C}}_{\text{f}} } 2}} \right. \kern-\nulldelimiterspace} 2} )^{ 3 / 2} }}} . $$
(4)
Here, ν is the kinematic viscosity of the fluid, ρ is the density of the fluid, K is the well-known acceleration parameter given by \( K = (\nu /U_{\infty }^{2} ){\text{d}}U_{\infty } /{\text{d}}x \) and Cf is the skin friction coefficient. Since K is purely a freestream parameter, use of Δp, which contains the boundary layer information through Cf, is more relevant. In order to appreciate this pressure-gradient dependence of log laws in sink flow TBLs, we plot some data from Dixit and Ramesh (2008) in Fig. 1. Figure 1a and b, respectively, shows the variations of the slope 1/κ and the intercept C of the inner log law (Eq. 3) with Δp. Also shown is the variation of the slope as predicted by the model given by Nickels (2004) which is based on the concept of a universal critical Reynolds number for the sublayer flow.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig1_HTML.gif
Fig. 1

a Slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and b intercept C of the non-universal log law (Eq. 3) as functions of Δp, for sink flow TBL cases taken from Dixit and Ramesh (2008). Squares sink flow experiments of Dixit and Ramesh (2008); asterisk sink flow experiment (\( K = 5.39 \times 10^{ - 7} \) case) of Jones et al. (2001); continuous line polynomial fit and dashed dotted line slope variation given by the model of Nickels (2004). For other details of these data, see Dixit and Ramesh (2008)

As derived in Dixit and Ramesh (2008), Eq. 3 is strictly valid only for sink flow TBLs which are the so-called ‘perfect’ equilibrium layers (or exactly self-preserving layers according to Townsend 1976 and Rotta 1962). However, in view of the reported pressure-gradient dependence of log laws in non-sink flow cases by other researchers (see Sect. 2), it is not unnatural to expect that Eq. 3 should be applicable to ‘perfect’ equilibrium (or simply equilibrium) as well as near-equilibrium (or nearly self-preserving) TBL flows. Here, the near-equilibrium TBLs comprise the ZPG and equilibrium APG flows. As will be demonstrated later in this paper, this indeed appears to be the case. For a detailed account of exactly and nearly self-preserving layers (or equilibrium and near-equilibrium layers in the present terminology), the reader is referred to Townsend (1976). The only precaution that needs to be taken while using Eq. 3 is that the TBL should not be very close to relaminarization (for FPG layers) or separation (for APG layers) as the case may be. This ensures that the friction velocity Uτ may still be used as the velocity scale. Viewed this way, Eq. 3, which is local in nature, represents a broad class of pressure-gradient-dependent log laws for equilibrium and near-equilibrium layers, of which the universal log law (Eq. 1) is a special case. Hereafter, throughout this paper, the TBLs under consideration would be equilibrium or near-equilibrium TBLs, unless otherwise stated explicitly.

Equation 3 presents an opportunity to formulate a modification of the conventional Clauser-chart method for strong pressure-gradient TBL flows. This modified Clauser-chart method (MCCM) is described in the next section.

4 Modified Clauser-chart method (MCCM) for equilibrium and near-equilibrium TBL flows

The polynomial functions \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } = {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }\left( {\Updelta_{p} } \right) \) and C = Cp) in Eq. 3 for sink flow TBLs (shown by solid lines in Fig. 1), are given in Dixit and Ramesh (2008). However, the present work is not restricted only to the sink flow TBLs. It in fact aims at a unified procedure for estimating skin friction that is valid for equilibrium and near-equilibrium TBLs over a wide range of pressure gradients, ranging from APG to FPG. Therefore, as a first step, it is necessary to find the polynomial functions \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } = {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }\left( {\Updelta_{p} } \right) \) and \( C = C\left( {\Updelta_{p} } \right) \) in Eq. 3 for the range of pressure gradients under consideration. For this purpose, the following data sets are used.

Five experimental sink flow data sets PE1 to PE5 (here denoted by DR1 to DR5) are taken from Dixit and Ramesh (2008) and one experimental sink flow data set corresponding to \( K = 5.39 \times 10^{ - 7} \) (here denoted by JMP3) is taken from Jones et al. (2001). The data set taken from Jones et al. (2001) represents the strongest non-dimensional pressure gradient in their experiments. Herring and Norbury (1967) have presented experimental data on equilibrium FPG turbulent boundary layers for two non-dimensional pressure gradients β = −0.53 and β = −0.35 of which, the case β = −0.53 (here denoted by HN2) is taken for the present purpose (the other case β = −0.35 will be used later as a test case). Here \( \beta = (\delta^{*} /\tau_{w} ){\text{d}}p/{\text{d}}x \) is the Clauser parameter (see Clauser 1956) where τw is the wall shear stress and δ* is the displacement thickness. The abovementioned data sets represent the FPG flows. For the ZPG case, standard values of \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and C, as applicable to the universal log law (i.e. with κ0 = 0.41 and C0 = 5.2), are taken. Two APG equilibrium data sets case 5 and case 6 (here denoted as SL5 and SL6) are taken from direct numerical simulation (DNS) studies of Spalart and Leonard (1986) which are representative of the APG flows.

For each of the abovementioned data sets, the logarithmic region in the mean velocity profile was identified according to the procedure explained in Sect. 2. The inner logarithmic law (Eq. 3) was fitted to this logarithmic region using the least-squares method. The slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the inner intercept C were thus readily obtained. As a consistency check, the corresponding intercept in the outer logarithmic law was then calculated from the non-universal logarithmic skin friction law (see Dixit and Ramesh 2008 for details) and was verified by visual inspection. This procedure was applied to all data sets for the sake of uniformity. More details of this procedure can be found in Dixit and Ramesh (2008). Table 1 gives the relevant details concerning all the data sets.
Table 1

Data sets used for the polynomial functions \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } = {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }\left( {\Updelta_{p} } \right) \) and \( C = C\left( {\Updelta_{p} } \right) \)

Data set code

Cf

Δp

\( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \)

C

DR1

0.00401

−0.0086

2.349

6.960

DR2

0.00403

−0.0104

2.282

7.666

DR3

0.00416

−0.0129

2.201

8.289

DR4

0.00430

−0.0175

2.087

9.310

DR5

0.00433

−0.0288

1.999

10.603

JMP3

0.00437

−0.0053

2.379

5.403

HN2

0.00375

−0.0026

2.369

5.395

ZPG

0.0000

2.439

5.200

SL5

0.00487

0.0182

2.845

2.896

SL6

0.00456

0.0377

3.349

1.003

DR Dixit and Ramesh (2008) sink flow experiments, JMP Jones et al. (2001) sink flow experiment, HN Herring and Norbury (1967) equilibrium FPG experiment, ZPG the standard zero pressure-gradient data and SL denotes Spalart and Leonard (1986) APG equilibrium DNS

Figure 2a and b, respectively, shows the systematic variations of the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C of the inner non-universal log law (Eq. 3) with the pressure gradient parameter Δp for the cases listed in Table 1. The resulting polynomial fits, shown by solid lines in the Fig. 2a and b, are given by
$$ {\frac{1}{\kappa }} = 2.452 \, + 19.534 \, \Updelta_{p} + 113.08 \, \Updelta_{p}^{2} , $$
(5)
$$ {\text{and}}\,\,\,C = 5.3048 - 185.82 \, \Updelta_{p} + 1033.2 \, \Updelta_{p}^{2} + 25172 \, \Updelta_{p}^{3} . $$
(6)
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig2_HTML.gif
Fig. 2

a Slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and b intercept C of the non-universal log law (Eq. 3) as functions of Δp, for various cases listed in Table 1. Squares DR1 to DR5; asterisk JMP3; circle HN2; inverted triangle ZPG; triangle SL5 and SL6; continuous line polynomial fits given by Eqs. 5 and 6, respectively, and dashed line slope variation given by the model of Nickels (2004). For data set codes refer to Table 1

Also shown in Fig. 2a, is the variation of the slope as predicted by the model given by Nickels (2004) for comparison with the present polynomial fit (Eq. 5). This model is based on the concept of a universal critical Reynolds number for the sublayer flow and the range of pressure gradients for this model is \( - 0.0 2\le \Updelta_{p} \le 0.0 6 \) as given by Nickels (2004). It may be seen that the model agrees fairly well with the present polynomial fit on the FPG side (\( \Updelta_{p} < 0 \)) while the disagreement on the APG side (\( \Updelta_{p} > 0 \)) is more. In particular, the curvature of the model by Nickels (2004) and that of the present polynomial fit (Eq. 5) are totally opposite. Reasons for this apparent disagreement on the APG side are unclear at present. However, based on the facts that (a) the slope of the logarithmic variation must be positive and (b) the slope should be an increasing function of the pressure gradient parameter Δp (see also Dixit and Ramesh 2008), it may be argued that the curvature of the present polynomial fit is more plausible. Since the present purpose is to correlate the available data with a ‘working’ curve-fit, polynomials given by Eqs. 5 and 6 will be used for the rest of this paper.

Equations 5 and 6 are central to the MCCM. The main theme is that if the value of Δp for a TBL flow is known, then from Eqs. 5 and 6, one can obtain the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C of the logarithmic portion of the mean velocity profile. Once this is done, the local skin friction coefficient can be calculated from the measured mean velocity profile by making use of Eq. 3 in an exactly similar fashion as the conventional Clauser-chart method described in Sect. 2. There is, however, a subtle difficulty in this proposal. It is not possible to specify the value of Δp a priori since it depends on the skin friction coefficient Cf (see Eq. 4) which itself is an unknown. This difficulty may be overcome by using what is called here as the twofold iterative procedure as will be discussed shortly.

The MCCM proposed here uses the twofold iterative procedure and proceeds in the following steps.
  1. (i)

    The measured mean velocity profile is first plotted in \( {U \mathord{\left/ {\vphantom {U {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }} \) versus log\( \left( {{{yU_{\infty} } \mathord{\left/ {\vphantom {{yU\infty } \nu }} \right. \kern-\nulldelimiterspace} \nu }} \right) \) form. Such a plot would make the extent of the logarithmic region in the mean velocity profile apparent. Once this extent is identified by visual inspection, it is fitted with a logarithmic curve (the least-squares method). The end points of the logarithmic region are then fixed by a trial and error procedure such that the R-squared value of the fit reaches a maximum. This step is exactly the same as the one in the conventional Clauser-chart method described in Sect. 2.

     
  2. (ii)

    A guess value for Δp, close to zero (say −0.0001), is selected and then the corresponding guess values for the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C are obtained from Eqs. 5 and 6. This step should be contrasted with the corresponding step in the conventional method where the values of the universal constants κ0 and C0 are already known.

     
  3. (iii)
    The non-universal log law (Eq. 3) is rearranged as
    $$ {\frac{U}{{U_{\infty } }}}{\frac{{U{}_{\infty }}}{{U_{\tau } }}} = {\frac{ 1}{\kappa }}{ \ln }\left( {\frac{y}{\delta }}{\frac{{\delta U_{\infty } }}{\nu }}{\frac{{U_{\tau } }}{{U_{\infty } }}} \right) + C . $$
    (7)
     
  4. (iv)

    By making use of only those points which belong to the logarithmic region and the guess values for the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C, the ratio \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }} \) is evaluated from Eq. 7 by an iterative procedure (say a regression fit). The value of \( {{U_{\infty } } \mathord{\left/ {\vphantom {{U_{\infty } } {U_{\tau } }}} \right. \kern-\nulldelimiterspace} {U_{\tau } }} \) then gives the skin friction coefficient as \( C_{f} = 2\left( {{{U_{\tau } } \mathord{\left/ {\vphantom {{U_{\tau } } {U_{\infty } }}} \right. \kern-\nulldelimiterspace} {U_{\infty } }}} \right)^{2} \). We call this step as the Cf iteration. This is very similar to the one in the conventional method (see Sect. 2).

     
  5. (v)

    At this stage, it is important to note that the acceleration parameter K is a known from the given streamwise distribution of the free-stream velocity U. Using the value of Cf obtained in the previous step and the known value of K, Δp is calculated from Eq. 4. This new value of Δp is compared with the guess value used in step (ii) and if the agreement is not satisfactory, then this new value itself is taken as the next guess. We call this step as the Δp iteration. This second iteration on Δp is a key feature of the MCCM that makes it different from the conventional method.

     
  6. (vi)

    Steps (ii) to (v) are repeated till the values of Δp agree to within the desired tolerance.

     

The range of pressure gradients over which the MCCM presented here is expected to work, is evidently decided by the extreme values of the pressure gradient parameter Δp (see Table 1) used while obtaining Eqs. 5 and 6. Thus, from Table 1 and Fig. 2, the range of validity of the present MCCM can be expected to be approximately \( - 0.0 2 5\le \Updelta_{p} \le 0.0 30 \) where we have left out small regions near the extreme values of Δp that might have got affected by the end effects in curve-fitting. It is natural to compare this range with the range of Preston-tube. Patel (1965) gives the acceptable range of Δp values for the skin friction to be determined within ±6% as \( - 0.00 7< \Updelta_{p} < 0.0 1 5, \) where we have combined the ranges given separately by Patel for FPG and APG flows. It is clear that the present MCCM has much wider range (twice as large on the APG side and more than twice on the FPG side) than the Preston-tube for almost the same accuracy ±6% (accuracy to be demonstrated in the next section).

5 Illustration of the MCCM

For illustrating the MCCM, ten mean velocity profiles from different flows are chosen as test cases. Details of these profiles and their origin are discussed below.

Four profiles were selected as representative of the FPG equilibrium and near-equilibrium flows. Three of these were the sink flow profiles (designated as P1, P2 and P3) measured exclusively for the present exercise in the same experimental setup that is described in detail by Dixit and Ramesh (2008). The pressure gradients for these three profiles were ensured to be different than those for DR1 to DR5 (see Table 1). This care was taken since use has already been made of cases DR1 to DR5 for obtaining Eqs. 5 and 6 that are used in the MCCM. Profiles P1, P2 and P3 were measured using a round Pitot-tube having an outer diameter of 0.6 mm. The Pitot displacement correction suggested by MacMillan (1956) was always applied. For traversing the Pitot-tube normal to the plate, a dial-type height gauge (Mitutoyo) with a least count of 0.01 mm, was used. The friction velocity Uτ was measured for these profiles using the so-called surface hot-wire (SHW) method. More details of the SHW method, as used in this study, are given in Dixit and Ramesh (2008). One equilibrium FPG profile (β = −0.35 case, designated here as HN1) is taken from experimental studies by Herring and Norbury (1967).

Two profiles were selected as representative of the APG near-equilibrium flows. The first APG profile, was at x = 5m (designated here as SK) from the extensive experimental data of Skåre and Krogstad (1994) on equilibrium TBL flow near separation. Another equilibrium APG profile, was APG1 case (designated here as Skote1) from the DNS study by Skote et al. (1998).

It would be of interest to see how the present MCCM responds to non-equilibrium flow situations while it is actually expected to deal with the equilibrium and near-equilibrium TBL flows. To see this, following four data sets were taken into consideration. Two non-equilibrium profiles, x = 0.4m case (FPG) and x = 1m case (APG) (designated here as SW1 and SW2, respectively) were taken from the DNS study of arbitrary pressure-gradient TBLs by Spalart and Watmuff (1993). One non-equilibrium APG profile, \( x^{\prime} = 0.75 \) case (designated here as AE) was taken from the experimental study by Aubertine and Eaton (2005) and one non-equilibrium FPG profile, Case1: x = 3.11m (designated here as FW) was taken from the axisymmetric TBL experiments by Fernholz and Warnack (1998). Note that these four non-equilibrium flow cases are expected to have varying degrees of departure from equilibrium, which are primarily dictated by the rate at which a non-dimensional measure of the pressure gradient varies in the streamwise direction. The severity of the streamwise variations of pressure gradient, for these four non-equilibrium cases, increases in the order SW, AE and FW.

It is important to know how the skin friction was originally obtained (as given in corresponding references) in the selected ten test cases mentioned above. For P1, P2, P3 and FW, the skin friction was measured by the SHW technique while for AE the oil-film interferometry was used. For SK, Preston-tube was used (but the value of Δp was within the limits specified by Patel 1965) in addition to a certain profile fitting procedure. In HN1, there was no mention of the method used for skin friction measurement but here also the value of Δp was reasonably low so that Preston-tube or conventional Clauser-chart would not have been very inaccurate. Skote1, SW1 and SW2 were all DNS studies and therefore computed values of skin friction were readily available.

The MCCM described in the previous section was applied to all these profiles to obtain the skin friction coefficient Cf. As a by-product of the MCCM, the values of Δp, \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and C were also obtained. In the foregoing discussion, we shall call the values of Cf (or Uτ) and all the related parameters as (i) original if the values are as given in the original data set by the originator and (ii) estimated if the values are obtained from the MCCM.

Table 2 gives the comparison of the original and estimated values of various relevant parameters for all the ten test cases. We shall first discuss the results for equilibrium and near-equilibrium flows. In these cases (test cases P1, P2, P3, HN1, SK and Skote1), Table 2 shows that the estimated value of Cf differs from the original value by a maximum of about 4.5%. The inherent uncertainty in Cf for the SHW method and for the oil-film interferometry is about ±4% (see Fernholz et al.1996; Fernholz 2006) while that for the Preston-tube method in zero or mild pressure gradients is about ±3% (see Patel 1965). This implies that the maximum root-mean-square uncertainty in Cf for the MCCM is about ±5–6%. Thus, it is clear that the MCCM yields estimates of skin friction, for both FPG and APG (equilibrium and near-equilibrium) flows, to an accuracy which is typical of skin friction measurements. The difference in the original and estimated values of the intercept C is little higher for the APG cases. This may be attributed to the two facts (i) that there are few points on the APG side that determine the coefficients in Eqs. 5 and 6 and (ii) that the value of C for APG flows is smaller in magnitude compared to FPG flows. On the whole, the differences in the original and estimated values of the slope \( 1/\kappa \) are much smaller compared to those of the intercept C. This is consistent with the general observation that small changes in the slope of the log law cause relatively large changes in the intercept. In addition, it may be noted that the conventional Clauser-chart method gives large errors (though not shown here) in Cf (typically 15% or even larger), especially for those cases that involve stronger pressure gradients in terms of the values of Δp.
Table 2

Evaluation of the MCCM for ten test cases

Test case code

Cf

Δp

\( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \)

C

Original

 P1

0.00440

−0.0212

2.056

9.371

 P2

0.00430

−0.0144

2.134

8.532

 P3

0.00413

−0.0114

2.235

7.796

 HN1

0.00346

−0.0016

2.426

5.114

 SK

0.00055

0.0122

2.536

4.561

 Skote1

0.00467

0.0047

2.649

4.003

 SW1*

0.00487

−0.0069

2.237

7.218

 SW2*

0.00329

0.0195

3.057

1.826

 AE*

0.00185

0.0053

2.805

5.034

 FW*

0.00405

−0.0219

1.605

9.877

Estimated by the MCCM

 P1

0.00424

−0.0224

2.072

9.689

 P2

0.00436

−0.0142

2.194

8.118

 P3

0.00424

−0.0110

2.247

7.479

 HN1

0.00331

−0.0017

2.420

5.618

 SK

0.00057

0.0114

2.695

3.317

 Skote1

0.00465

0.0048

2.547

4.454

 SW1*

0.00499

−0.0066

2.327

6.580

 SW2*

0.00338

0.0187

2.846

2.412

 AE*

0.00220

0.0041

2.535

4.554

 FW*

0.00281

−0.0381

1.891

12.215

% difference of estimated value with respect to the original value

 P1

−3.6

5.8

0.8

3.4

 P2

1.3

−1.8

2.8

−4.9

 P3

2.7

−3.9

0.6

−4.1

 HN1

−4.4

7.6

−0.3

9.9

 SK

4.2

−6.7

6.3

−27.3

 Skote1

−0.5

0.8

−3.9

11.3

 SW1*

2.5

−3.7

4.0

−8.8

 SW2*

2.8

−3.9

−6.9

32.1

 AE*

18.7

−22.4

−9.6

−9.5

 FW*

−30.6

73.6

17.9

23.7

* Non-equilibrium flow. Estimated values are obtained by using Cf from the MCCM and original values are obtained by using Cf as given in the original data set

The above results can be better represented in the form of mean velocity profiles in the inner scaling where the original and the estimated values of skin friction (or the friction velocity Uτ) are both used to obtain the non-dimensional velocity profiles. Figure 3a–f shows the mean velocity profiles in inner scaling for the six equilibrium and near-equilibrium test cases viz. P1, P2, P3, HN1, SK and Skote1 (see also Table 2). Each figure contains two profiles, one plotted by using the original value of Uτ (solid symbols) and the other by using the estimated value of Uτ (hollow symbols). Also shown are the corresponding original and estimated log laws as solid and dashed lines, respectively. From the excellent agreement between the original and estimated profiles seen in these plots, it is clear that the MCCM indeed performs quite well in variety of pressure gradient situations for equilibrium and near-equilibrium TBL flows.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig3_HTML.gif
Fig. 3

Equilibrium and near-equilibrium test cases a P1, b P2, c P3, d HN1, e SK and f Skote1. Hollowsymbols indicate the estimated profile (i.e. with skin friction by the MCCM) and solid symbols indicate the original profile (i.e. with skin friction from the original data). Continuous line, original log law and dashed line, estimated log law. For test case codes, refer to Table 2

We now discuss the results for the non-equilibrium flow cases (test cases SW1, SW2, AE and FW). Interestingly, in the case of non-equilibrium flows SW1 and SW2, the MCCM seems to work quite well as can be seen in Table 2. On the other hand, it is seen to fail by a large margin for the other two non-equilibrium cases AE and FW. Figure 4a–d shows the original and estimated mean velocity profiles in inner scaling for these non-equilibrium flows, plotted in exactly similar fashion as was done for equilibrium and near-equilibrium cases. Figure 4a and b shows that for SW1 and SW2, the MCCM indeed works quite well while Fig. 4c and d shows that for AE and FW, it fails by a considerable margin. As mentioned before, the streamwise rate of change of pressure gradient is small for the SW cases and large for the FW case with the AE case somewhere in between. Thus, the degree of departure from equilibrium is expected to be the least for SW cases, maximum for the FW case and intermediate for the AE case. It thus seems plausible to speculate that the MCCM presented here for equilibrium and near-equilibrium flows is flexible enough to tolerate the degree of non-equilibrium typical of SW cases. It is, however, not clear at present, as to what the appropriate measure of the departure from equilibrium should be. From the large differences between the original and the estimated values in cases AE and FW, it is evident that a separate methodology for addressing non-equilibrium effects is required.
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Fig. 4

Non-equilibrium test cases a SW1, b SW2, c AE and d FW. Hollow symbols indicate the estimated profile (i.e. with skin friction by the MCCM) and solid symbols indicate the original profile (i.e. with skin friction from the original data). Continuous line, original log law and dashed line, estimated log law. For test case codes, refer to Table 2

The MCCM presented above may be improved by including more data points (especially on the APG side) in Fig. 2a and b that give the polynomial approximations to the pressure-gradient variations of the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C of the non-universal log law (Equation 3). The experiments carried out towards this objective should measure the skin friction coefficient by some method that does not depend, in any way, on the universal log law. One such method could be the surface hot-wire (SHW) method. The oil-film interferometry is definitely a promising choice since it does not need calibration and thus in that sense it is an absolute measurement of skin friction. Momentum integral balance is of course another choice, but has to be handled with care in APG flows.

6 Asymptotic values of κ0 and C0 in ZPG turbulent boundary layer flows and their effect on the MCCM

There have been recent advances in high Reynolds number TBL measurements with special emphasis on the measurement of skin friction by independent methods such as the oil-film interferometry (see Österlund 1999 and Österlund et al. 2000). These experiments have shed useful light on the asymptotic values of the universal constants κ0 (the Kármán constant) and C0 as well as on their possible Reynolds number dependence. Österlund et al. (2000) report the asymptotic values of κ0 and C0 to be 0.38 and 4.1. These should be contrasted with the classical values 0.41 and 5.0 (see Coles 1968) being widely used in so far. They note that the previously observed Reynolds number dependence in the value of κ0 could be due to the incorrect identification of the beginning of the logarithmic overlap region (y+ of about 50 which is traditionally used as against y+ of about 200 that they report). Nagib and Chauhan (2008) also find the asymptotic value of κ0 to be about 0.384 in the case of high Reynolds number ZPG turbulent boundary layer flow. They have processed the same data as Österlund et al. (2000) by using a composite velocity profile.

In order to study the effect of these recent asymptotic values of κ0 and C0 on the present MCCM, two exercises were carried out. In the first, the curve fits (Eqs. 5, 6) in Fig. 2a and b were reconstructed by using the recent asymptotic values κ0 = 0.38 and C0 = 4.1 for the ZPG case (Δp = 0). Consequently, Eqs. 5 and 6 underwent some changes. These new equations were then used to process the same data sets (see Table 2) using the MCCM. It was found that that the results were not very different (still within the same uncertainty band of ±5–6% in Cf) from those obtained previously with the classical values of κ0 = 0.41 and C0 = 5.2. Results of this exercise are shown below in Fig. 5 in the form of original and estimated inner velocity profiles for the six equilibrium and near-equilibrium test cases (for test case codes see Table 2). Figure 5, on comparison with Fig. 3, confirms that the MCCM estimates remain practically unaffected even after incorporating the recent asymptotic values of κ0 and C0 in place of their classical values. The MCCM is thus quite robust to changes in κ0 and C0. Incidentally, this also justifies our choice of C0 = 5.2 (based on our own measurements, not shown here, in the ZPG flow where the skin friction was obtained using Preston-tube) instead of the more-widely-used C0 = 5.0 (see Coles 1968).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig5_HTML.gif
Fig. 5

Equilibrium and near-equilibrium test cases a P1, b P2, c P3, d HN1, e SK and f Skote1. Hollow symbols indicate the estimated profile (i.e. with skin friction by the MCCM) and solid symbols indicate the original profile (i.e. with skin friction from the original data). Continuous line, original log law and dashed line, estimated log law. For test case codes, refer to Table 2. The data are reprocessed by incorporating the recent asymptotic values κ0 = 0.38 and C0 = 4.1 (see Österlund et al. 2000) in the MCCM in place of the classical values κ0 = 0.41 and C0 = 5.2

In the second exercise, two representative mean velocity profiles (the higher Reynolds number case SW981005F and the lower Reynolds number case SW981112A) were taken from Österlund’s original high Reynolds number ZPG TBL database (see Österlund 1999). This database is available online (http://www.mech.kth.se/~jens/zpg/). Both the profiles (here denoted by O1 and O2, respectively) correspond to the same x-location (x = 5.5 m). These profiles were subjected to the MCCM (Eqs. 5, 6 were used, i.e. the classical values κ0 = 0.41 and C0 = 5.2 were used) and the skin friction coefficients were obtained. Since Δp = 0 (ZPG) was known for these profiles, the Δp iteration in the MCCM routine was suppressed.

The results are shown in Table 3 where the errors in the slope \( 1/\kappa_{0} \) and the intercept C0 are seen to be large. This is, however, not surprising, as the asymptotic values of κ0 and C0 in the original data of Österlund (1999) are 0.38 and 4.1 while those used in the present MCCM are 0.41 and 5.2, respectively. What is remarkable about the data in Table 3 is that the errors in Cf are surprisingly low. Referring to Table 2, we observe that these errors are in fact the typical errors involved in any MCCM estimate. This implies that even in the case of Österlund’s data where the classical values of κ0 and C0 do not hold, the estimate of Cf by the present MCCM (that uses classical values of κ0 and C0) is quite accurate and within the acceptable limits. Considering the high Reynolds numbers involved in the data of Österlund (1999) and the wide range of pressure gradients that the present MCCM can handle, this success of the MCCM in the case of Österlund’s data is quite gratifying. Figure 6 shows the velocity profiles for these cases O1 and O2 in the usual fashion, i.e. the original and estimated format. It is clear that the estimation of skin friction by the present MCCM even in these cases is quite accurate and that the MCCM is quite robust to changes in κ0 and C0.
Table 3

Two representative high Reynolds number ZPG velocity profiles from Österlund (1999) re-evaluated using the MCCM

Test case code

Cf

Δp

\( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }_{0} \)

C0

Original

 O1

0.00229

0

2.632

4.1

 O2

0.00275

0

2.632

4.1

Estimated by the MCCM

 O1

0.0022

0

2.452

5.305

 O2

0.00263

0

2.452

5.305

% difference of estimated value with respect to the original value

 O1

−4.1

0

−6.8

29.4

 O2

−4.5

0

−6.8

29.4

Notice that relatively large errors in the slope \( 1/\kappa_{0} \) and the intercept C0 do not introduce much error in the skin friction coefficient Cf

https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig6_HTML.gif
Fig. 6

High Reynolds number ZPG test cases a O1 and b O2 from Österlund (1999) re-evaluated using the MCCM. Hollow symbols indicate the estimated profile (i.e. with skin friction by the MCCM) and solid symbols indicate the original profile (i.e. with skin friction from the original data). Continuous line, original log law and dashed line, estimated log law. For test case codes, refer to Table 3

7 Identification of the logarithmic region using the log law diagnostic function Ξ

It has been mentioned in Sect. 2 that the logarithmic region in a velocity profile in the present study was identified first by visual inspection, which was the followed by a logarithmic curve-fit to that region (the least-squares method). The final extent was decided by trial and error such that the R-squared value for the fit reaches a maximum. In this section, we describe a more rational procedure to do the same task. This is based on the so-called log law diagnostic function Ξ which will be used here in a slightly different manner than what is reported commonly in the literature (see for example Österlund 1999; Österlund et al. 2000).

The non-universal inner logarithmic law (Eq. 3) is first rewritten as
$$ {\frac{U}{{U_{\infty } }}}{\frac{{U{}_{\infty }}}{{U_{\tau } }}} = {\frac{ 1}{\kappa }}{ \ln }\left( {\frac{{yU_{\infty } }}{\nu }}{{\frac{{U_{\tau } }}{{U_{\infty } }}}} \right) + C . $$
(8)
If we call the ratio \( U_{\infty } /U_{\tau } \) (which is independent of y) as m, then Eq. 8 becomes
$$ {\frac{U}{{U_{\infty } }}} = {\frac{ 1}{m\kappa }}{ \ln }\left( {\frac{{yU_{\infty } }}{\nu }} \right) + {\frac{ 1}{m\kappa }}{ \ln }\left( {\frac{1}{m}} \right) + {\frac{C}{m}}. $$
(9)
Noting that m, κ and C are all constant with respect to y, we differentiate Eq. 9 with respect to \( yU_{\infty } /\nu \) and get
$$ \Upxi = \left( {\frac{{yU_{\infty } }}{\nu }} \right){\frac{{{\text{d}}(U/U_{\infty } )}}{{{\text{d}}(yU_{\infty } /\nu )}}} = {\frac{ 1}{m\kappa }}. $$
(10)
Here, Ξ is called as the log law diagnostic function. Thus, from the measured mean velocity data, we can identify the logarithmic region in the mean velocity profile by plotting Ξ against \( yU_{\infty } /\nu \) and looking for a range of \( yU_{\infty } /\nu \) values over which Ξ is reasonably constant.
Figures 7 and 8 show this procedure implemented for four representative test case profiles P1, P3, SK and Skote1. In order to evaluate Ξ, each mean velocity profile is fitted with a cubic spline interpolation fit consisting of closely spaced points. This spline fit is then differentiated numerically using a simple central difference formula and Ξ is evaluated using Eq. 10. Also shown in each figure is the extent of the logarithmic region that was identified by visual inspection followed by a logarithmic curve-fit (dashed line marks the beginning of this region and the dashed dotted line marks its end).
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Fig. 7

Identification of the logarithmic region from the mean velocity profile data using the log law diagnostic function Ξ. The procedure is illustrated for sink flow test cases P1 and P3. Solid lines in (a) and (c) indicate the cubic spline interpolation fits to the mean velocity profiles. Ξ constant region is the logarithmic region; dashed line, start of the logarithmic region and dashed dotted line, end of the logarithmic region as identified by visual inspection followed by a logarithmic curve-fit (the least-squares method). For test case codes see Table 2

https://static-content.springer.com/image/art%3A10.1007%2Fs00348-009-0698-2/MediaObjects/348_2009_698_Fig8_HTML.gif
Fig. 8

Identification of the logarithmic region from the mean velocity profile data using the log law diagnostic function Ξ. The procedure is illustrated for near-equilibrium APG test cases SK and Skote1. Solid lines in (a) and (c) indicate the cubic spline interpolation fits to the mean velocity profiles. Ξ constant region is the logarithmic region; dashed line start of the logarithmic region and dashed dotted line end of the logarithmic region as identified by visual inspection followed by a logarithmic curve-fit (the least-squares method). For test case codes see Table 2

In order to have an estimate of the typical uncertainty involved in using the visual + curve-fit method in lieu of the diagnostic function approach, the values of Ξ may be averaged over the region where Ξ is approximately constant. Using this averaged value of Ξ, one may then evaluate \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) using Eq. 10 in which m is known from the original value of Cf for the test case under consideration. This value of \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) may then be compared with the one that is obtained from the visual + curve-fit method. Table 4 shows such a comparison for the four test cases of Figs. 7 and 8. It may be observed that the maximum difference in the values of \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) is of the order of 3% (see also Dixit and Ramesh 2008, where differences of the order of 2% are reported for the sink flow TBL data).
Table 4

Comparison of the slope \( 1/\kappa \) of the logarithmic region of the mean velocity profile obtained by the diagnostic function approach and the visual + curve-fit method for the test cases of Figs. 7 and 8

Test case code

\( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) from Ξ

\( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) from visual + curve-fit method

% difference in \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) with respect to the value obtained from Ξ

P1

2.067

2.056

−0.5

P3

2.162

2.235

3.4

SK

2.559

2.536

−0.9

Skote1

2.713

2.649

−2.4

For test case codes see Table 2

Thus, from the considerations discussed above, it may be seen that the use of visual + curve-fit method, as is done in the present study, yields a reasonable identification procedure for identifying the logarithmic region of the mean velocity profile.

8 Conclusions

It is shown that the conventional Clauser-chart method for estimation of skin friction (which gives fairly accurate results for ZPG or mild pressure-gradient flows), originally proposed by Clauser (1954), can be modified to deal with the situations involving strong streamwise pressure gradients, provided that the equilibrium or near-equilibrium TBL under consideration is not very close to relaminarization or separation. In such cases, the overlap layer manifests itself in the form of non-universal logarithmic laws that are dependent on the local strength of the pressure gradient. Using these non-universal log laws in conjunction with the measured pressure distribution (necessary for obtaining the acceleration parameter K) and a measured mean velocity profile, it is possible to obtain the local skin friction coefficient to an accuracy, which is typical of skin friction measurements. This modified Clauser-chart method (MCCM) employs a twofold iterative procedure (one iteration on Cf and the other on Δp) in contrast to the conventional method that involves only one iteration (on Cf alone). As a by-product of this MCCM, one obtains the local pressure gradient parameter Δp and the slope \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and the intercept C of the non-universal log law for that profile. It is also demonstrated that the MCCM is quite robust to the changes in the universal values of the Kármán constant κ0 and the intercept C0 for the ZPG turbulent boundary layer.

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© Springer-Verlag 2009