Determination of skin friction in strong pressure-gradient equilibrium and near-equilibrium turbulent boundary layers
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DOI: 10.1007/s00348-009-0698-2
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- Dixit, S.A. & Ramesh, O.N. Exp Fluids (2009) 47: 1045. doi:10.1007/s00348-009-0698-2
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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 C_{f} 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 κ_{0} and the intercept C_{0} 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
These values of κ_{0} and C_{0} 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 κ_{0} and C_{0} 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 κ_{0} and C_{0}, 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.
- (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.
- (ii)
The values of the universal constants κ_{0} and C_{0} are already known.
- (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)
- (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 κ_{0} and C_{0}, 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 C_{f} 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).
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 U_{τ} 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 = C(Δ_{p}) 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 τ_{w} 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 κ_{0} = 0.41 and C_{0} = 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.
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 | C_{f} | Δ_{p} | \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) | 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 |
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 Δ_{p} (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 Δ_{p} 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 Δ_{p} a priori since it depends on the skin friction coefficient C_{f} (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.
- (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.
- (ii)
A guess value for Δ_{p}, 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 κ_{0} and C_{0} are already known.
- (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)
- (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 C_{f} iteration. This is very similar to the one in the conventional method (see Sect. 2).
- (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 C_{f} obtained in the previous step and the known value of K, Δ_{p} is calculated from Eq. 4. This new value of Δ_{p} 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 Δ_{p} iteration. This second iteration on Δ_{p} is a key feature of the MCCM that makes it different from the conventional method.
- (vi)
Steps (ii) to (v) are repeated till the values of Δ_{p} 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 Δ_{p} (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 Δ_{p} 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 Δ_{p} 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 U_{τ} 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 Δ_{p} 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 Δ_{p} 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 C_{f}. As a by-product of the MCCM, the values of Δ_{p}, \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) and C were also obtained. In the foregoing discussion, we shall call the values of C_{f} (or U_{τ}) 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.
Evaluation of the MCCM for ten test cases
Test case code | C_{f} | Δ_{p} | \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa } \) | 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 |
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 κ_{0} and C_{0} 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 κ_{0} (the Kármán constant) and C_{0} as well as on their possible Reynolds number dependence. Österlund et al. (2000) report the asymptotic values of κ_{0} and C_{0} 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 κ_{0} 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 κ_{0} 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 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 κ_{0} = 0.41 and C_{0} = 5.2 were used) and the skin friction coefficients were obtained. Since Δ_{p} = 0 (ZPG) was known for these profiles, the Δ_{p} iteration in the MCCM routine was suppressed.
Two representative high Reynolds number ZPG velocity profiles from Österlund (1999) re-evaluated using the MCCM
Test case code | C_{f} | Δ_{p} | \( {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa }_{0} \) | C_{0} |
---|---|---|---|---|
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 |
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).
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 |
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 C_{f} and the other on Δ_{p}) in contrast to the conventional method that involves only one iteration (on C_{f} alone). As a by-product of this MCCM, one obtains the local pressure gradient parameter Δ_{p} 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 κ_{0} and the intercept C_{0} for the ZPG turbulent boundary layer.