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The diagnostic plot — a litmus test for wall bounded turbulence data
European Journal of Mechanics B/Fluids 29 (2010) 403–406
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
The diagnostic plot — a litmus test for wall bounded turbulence data P. Henrik Alfredsson ∗ , Ramis Örlü Linné Flow Centre, KTH Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
article
info
Article history: Received 23 February 2010 Received in revised form 30 June 2010 Accepted 20 July 2010 Available online 27 July 2010 Keywords: Wall bounded turbulence Mean velocity distribution Near wall behaviour
abstract A diagnostic plot is suggested that can be used to judge wall bounded turbulence data of the mean and the rms of the streamwise velocity in terms of reliability both near the wall, around the maximum in the rms as well as in the outer region. The important feature of the diagnostic plot is that neither the wall position nor the friction velocity needs to be known, since it shows the rms value as a function of the streamwise mean velocity, both normalized with the free stream velocity. One must remember, however, that passing the test is a necessary, but not sufficient condition to prove good data quality. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction During the last decade there has been a renewed interest in how averaged quantities in wall bounded turbulence vary in the direction normal to the wall, especially with regard to the mean and rms distributions of the streamwise velocity (see e.g. [1–3]). Comparing data from different facilities and/or different techniques is, however, often inconclusive since the inaccuracies of the measured quantities may be larger than the trends one wants to investigate. In recent conference contributions [4,5] the so called diagnostic plot was introduced as a tool to clarify whether boundary layer data are corrupted by measurement problems. In this paper we expand on this method to show how the behaviour of this plot can be explained from what we already know about the scaling of boundary layer data. We also show that it works equally well for turbulent channel flow data. 2. Motivation For studies in wall bounded turbulent flows most measurements of the streamwise mean velocity (U) and the rms of the fluctuating streamwise velocity (u0 ) distributions are made with hot-wire anemometry using single wire probes. The technique has itself several limitations, such as frequency response, spatial resolution, wall interference and heat conduction to the wall, calibration at low velocities, determination of the wire position relative to the wall etc. Most of them can be mastered by an experienced experimentalist under well controlled conditions.
∗
Corresponding author. E-mail address: [email protected] (P.H. Alfredsson).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.07.006
An important quantity is the wall shear stress (τ ) or rather the friction velocity (uτ ). Since uτ scales the mean velocity and also determines the viscous length scale `∗ = ν/uτ , and thereby the coordinate y+ , i.e. the normal coordinate y normalized with `∗ , the comparison and scaling between different measurements is utterly dependent on the accuracy of uτ itself. For smooth-wall boundary layers, oil–film interferometry can now be used to determine uτ with an accuracy of around 1% [6,7], but it is neither standard practice in most laboratories, nor simple enough to do in a short time for diagnostic purposes (see also Ref. [8], which gives an accuracy of ±1.5%). Let us also one more time point out that the determination of the probe position relative to the wall is crucial when plotting data in the standard inner scaling. This may at first look like a trivial problem but to determine the position with an accuracy of better than one viscous length scale is not easy if it has to be done from outside the wind-tunnel test section [9]. A possibility is to fit the mean velocity profile to some pre-described distribution for the near-wall behaviour. However in that case it is necessary that the measurements of the mean velocity are accurate in themselves. The diagnostic plot can be used to check whether the velocity data measured close to the wall conform to expected physical characteristics or if they are affected by spatial averaging or other near-wall effects. The diagnostic plot allows the experimentalist to do this without having to determine either the position of the wall or the friction velocity. However, as will be shown here it also allows a characterization of the flow, not only close to the wall, but also in the rest of the boundary layer. It may be pointed out that the diagnostic plot can also be useful to check the behaviour of numerical simulations, especially for LES/DES where the skin friction may not be accurately known.
404
P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
a
b
c
d
Fig. 1. Profiles of the mean streamwise velocity and rms streamwise velocity in inner (a, b) and outer (c, d) scaling, respectively. U + = y+ and U + = κ −1 ln y+ + B, with κ = 0.384 and B = 4.17 [13] are given as dashed lines. : Reθ = 2540, : Reθ = 8100, 4: Reθ = 18 700, —: DNS at Reθ = 2510.
3. The basic data in standard form
4. The diagnostic plot
In this paper we will make use of three different velocity profile measurements carried out in the MTL wind tunnel at KTH [10,11] as well as a DNS reported by Schlatter et al. [12]. Typical plots of U and u0 are shown in Fig. 1(a) and (b) for the three different Reθ , viz. 2540, 8100 and 18 700 (Reθ is the Reynolds number based on the momentum-loss thickness and free-stream velocity and the first is taken at x = 1.62 m and the other two at x = 3.62 m from the leading edge of the plate). Both U and u0 are normalized with the friction velocity uτ (based on oil–film interferometry) and plotted as a function of y+ . The probe lengths for the three cases are, 15, 26 and 60 `∗ , respectively, which would give fairly good resolution for the smallest Reθ but rather large spatial averaging effects in the near-wall region for the highest Reθ [11]. The DNS data plotted in the figure have a Reθ of 2510, i.e. approximately the same as the lowest experimental case, and it has been shown in Ref. [12] that the experiments and DNS show a very good agreement, with respect to the shape factor, skin friction coefficient, as well as the mean and rms distribution of the streamwise velocity normal to the wall. In Fig. 1(c) and (d) we show the data plotted against the outer lengthscale, i.e. as function of y/∆ where ∆ = δ∗ U∞ /uτ and δ∗ and U∞ are the displacement thickness and free stream velocity, respectively. The mean flow data are plotted in the standard wake formulation (W + = (U∞ − U )/uτ ). As expected the mean flow data overlap nicely from the edge of the boundary layer all the way through the logarithmic region, whereas the u0 -distributions have a much smaller range of overlap.
In the diagnostic plot u0 is plotted as function of U, where both quantities are normalized by U∞ , thereby avoiding any uncertainties in both the wall position and uτ . For fully developed turbulent boundary layers that are plotted in this way, distributions of U and u0 that are taken at the same Re should fall on top of each other if accurately measured (or simulated). The values of U /U∞ are between 0 and 1, and Table 1 gives approximate values for the different flow regions typically associated with turbulent boundary layer flows (for the lowest and highest Reynolds number of the present experiments). 4.1. Near-wall region Close to the wall (y+ < 10) we expect that the distribution should be nearly self-similar and independent of Re. The relative level of the rms of the wall shear-stress fluctuations, τ 0 /τ , is related to u0 /U as (see e.g. Ref. [14])
τ0 u0 = lim . y→0 U τ
(1)
Experimentally the value of τ 0 /τ was given as 0.39 in Ref. [14] at a Reθ of about 3000, however Ref. [2] indicates that it may increase slightly with Re. For the boundary layer DNS in [15,16] the value increases from 0.403 to 0.429 when Reθ increases from 670 to 4300. Since U varies linearly with y near the wall, Eq. (1) shows that u0 also varies linearly with y with a constant slope (e.g. u0 = 0.40U). Simulations and experiments show that the distribution is almost
P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
405
Table 1 Different regions in the diagnostic plot. Reθ = 2540 0 < y+ < 5 5 < y+ < 50 50 < y+ , y/δ < 0.15 0.15 < y/δ < 1
Viscous sublayer Buffer region Logarithmic region Wake region
0.00 0.20 0.60 0.71
Reθ = 18 700
< U /U∞ < U /U∞ < U /U∞ < U /U∞
< 0.20 < 0.60 < 0.71 < 1.00
0.00 0.16 0.50 0.75
< U /U∞ < U /U∞ < U /U∞ < U /U∞
< 0.16 < 0.50 < 0.75 < 1.00
The review in Ref. [3] of available data indicates that the maximum is located at a constant y+ independent of Re. This value is close to + y+ = 15 and corresponds to a value of Umax ≈ 10.6. Using these relations we can find a direct relation between the maximum value of u0 and the mean velocity U at the maximum, which turns out to be a straight line in the diagnostic plot, namely u0
max
Fig. 2. The data from Fig. 1 shown in the diagnostic plot. The dashed line corresponds to the maximum in u0 obtained from the correlation in Ref. [2] and assuming that the maximum is located at y+ = 15, and the filled symbols to Reynolds numbers of the present experiments. The dash–dotted line indicates the tangent to the near-wall data and has a slope of 0.40. The measured points below this line for U /U∞ < 0.15 are clearly affected by the wall.
linear up to y+ = 5 (corresponding to U /U∞ = 0.2 for Reθ = 2540) and thereafter the slope decreases [3]. As can be seen in Fig. 2 the points closest to the wall (i.e. points within the viscous sublayer for the lowest Re) deviate below the straight line indicating that the measured u0 values are too small and/or the U values are too high. Hence the diagnostic plot indicates clearly those measurement points for which the measured values show a wall interference effect, which is not directly evident in Fig. 1(a) and (b). It should be noted that, although the slope increases weakly with Re and decreases with increasing spatial resolution effects, the important point here is that suspicious measurement points, due to wall interference, can be identified through their deviation from the tangent to the data points in the diagnostic plot.
max
u
0
uτ
= 1.86 + 0.28 log(Reθ )
(2)
u0
=
U∞
uτ U∞
[1.86 + 0.28 log(Reθ )].
(3)
2
=
uτ
2
U∞
= [κ
−1
ln(Reθ ) + C ]
−2
U∞
U∞ − U uτ
(6)
=f
y ∆
,
(7)
whereas the rms level can be written
=g
y ∆
.
(8)
These can be combined to give u0
U∞
+ = Umax
uτ U∞
.
y
= 1−
=f
U
g (y/∆)
(9)
f (y/∆)
U∞
(4)
(5)
−1
U∞ − U
=F
uτ
U U∞
;
U∞ uτ
.
(10)
Finally we can write u0 U∞
where κ = 0.384 and C = 4.127 [17]. The velocity Umax at the maximum of u0 can simply be written as Umax
,
Also in the outer region the distributions at different Reynolds numbers previously shown in Fig. 1 nicely overlap in the diagnostic plot. From the accepted standard scaling behaviour of the mean and fluctuating streamwise velocity in the outer region the mean velocity can be written in defect form
∆
This relation can be used together with a skin friction formula to find how the maximum should vary in the diagnostic plot. Here we use the Coles–Fernholz skin friction relation which can be written Cf
U∞
and both g and f decrease with increasing y/∆. We may express y/∆ in terms of U /U∞ using Eq. (7) giving
or max
U∞
U
4.3. The outer region
uτ
Marusic and Kunkel [2] give an expression for how the maximum of u0 increases with Reθ
= C1 + C2
where C1 = 0.047 and C2 = 0.157. As can be seen in Fig. 2 the maximum value decreases with increasing Re in this scaling. In this figure we have also put symbols where we would, based on Eq. (6), expect the present Reynolds number data to have their maxima. As can be seen the maxima show up at the correct U /U∞ although the differences between the estimated and measured values at the two highest Re are an effect mainly due to spatial averaging (see e.g. [11,18,19]). Hence, the location in the diagnostic plot of the maximum in u0 can be used as an indication on the accuracy of the measurements in the buffer region. It should be noted that in Ref. [19] a different relation for the maximum value of u0 was given, however, there are only small differences compared to Eq. (2) for the present Reynolds number range.
u0
4.2. The maximum of u0
=G
U U∞
;
U∞ uτ
,
(11)
which indicates that in the outer region the u0 /U∞ versus U /U∞ distribution has a Reynolds number dependence, expressed here as U∞ /uτ . The employed experimental and numerical data within the Reynolds number range considered here indicate, however, that this Reynolds number dependence is small. This is also the case for the data shown in Fig. 2 (remember that the data shown span
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P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
stream velocity from a few measurement points in the outer region of u0 and U, which may be helpful when studying atmospheric boundary layers. Acknowledgements
8
The authors thank their co-authors in Ref. [4], Dr. Philipp Schlatter for access to the DNS data as well as participants of the NORDITA Programme on Turbulent boundary layers and Turbulent Combustion in 2010 for valuable discussions. We also want to thank the authors of Refs. [20–23] for making their data available on the web, as well as the referees for their constructive criticisms. This work was partly sponsored by the Swedish Research Council. References
Fig. 3. DNS data for channel flows with Reτ = 110, 150, 180, 395, 550, 640, 950, 1020, 2000. The three lowest Reτ are shown as dashed lines, and the maxima of u0 /U∞ for all data sets decrease as Reτ increase. The dash–dotted lines correspond to u0 /U = 0.32, 0.36 and 0.40, respectively.
over almost one decade in Reynolds number) and it is interesting to see that in the diagnostic plot the different Re data collapse also in the logarithmic region. For this part of the boundary layer the diagnostic plot gives a good indication whether the boundary layer is in a natural state. 5. Conclusions and outlook We have shown here how the behaviour of the distribution of u0 in the diagnostic plot can be obtained from what we already know about its scaling in the boundary layer. We have demonstrated that the diagnostic plot may detect measured data in the near-wall region that may suffer from various problems. For the Reynolds number range considered here, the distribution in the outer region appears to be self-similar, and the behaviour of the near-wall peak of u0 /U∞ has been given. All these features of the diagnostic plot make it possible to use it as a ‘‘litmus’’ test on the quality of measured data, without the need to determine the friction velocity or the position of the wall. One must remember, however, that passing the test is a necessary, but not sufficient condition to prove good data quality. We have also tested the diagnostic plot on DNS data for turbulent channel flow available in the literature [20–23]. For Reynolds numbers in the range 395–2000 (based on friction velocity and half channel height) the data collapse nicely (see Fig. 3) both in the inner and outer regions (although with a different shape in the outer region compared to boundary layers since the turbulence intensity does not go to zero on the centreline). For the three lowest Reynolds numbers it is seen that they deviate significantly in the outer region (i.e. in the centre of the channel) and this indicates that the flow for these Reynolds numbers has not reached a self-similar state. An analogous behaviour of the diagnostic plot should probably also be expected in pipe flows. The diagnostic plot should therefore be useful for anybody doing measurements in wall bounded flows. The diagnostic plot also has a potential to be used to evaluate the accuracy of LES/DES since for such simulations the skin friction cannot be directly determined from near-wall data. For experiments in rough wall boundary layers both the wall distance and skin friction are usually not known accurately and also here the diagnostic plot may prove useful. Some preliminary tests show that the diagnostic plot may also be useful in evaluating pressure gradient boundary layers. Another possibility that has not yet been explored is that if the Reynolds number variation is small in the outer region the diagnostic plot may be used to estimate the boundary layer free
[1] P.A. Monkewitz, K.A. Chauhan, H.M. Nagib, Self-consistent high-Reynoldsnumber asymptotics for zero-pressure-gradient turbulent boundary layers, Phys. Fluids 19 (2007) 115101. [2] I. Marusic, G.J. Kunkel, Streamwise turbulence intensity formulation for flatplate boundary layers, Phys. Fluids 15 (2003) 2461–2464. [3] M.H. Buschmann, T. Indinger, M. Gad-el-Hak, Near-wall behavior of turbulent wall-bounded flows, Int. J. Heat Fluid Flow 30 (2009) 993–1006. [4] P.H. Alfredsson, R. Örlü, T. Kurian, J.H.M. Fransson, A. Segalini, J.-D. Rüedi, A. Talamelli, The diagnostic plot—a new way to appraise turbulent boundarylayer data, in: B. Eckhardt (Ed.), Advances in Turbulence XII, 2010, pp. 609–612. [5] P.H. Alfredsson, R. Örlü, The diagnostic plot—a new way to appraise turbulent boundary layer data, Bull. Amer. Phys. Soc. 54 (2009) 82. [6] J.-D. Rüedi, R. Duncan, S. Imayama, K. Chauhan, ICET Team, Accurate and independent measurements of wall-shear stress in turbulent flows, Bull. Amer. Phys. Soc. 54 (2009) 20. [7] I. Marusic, B.J. McKeon, P.A. Monkewitz, H.M. Nagib, A.J. Smits, K.R. Sreenivasan, Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues, Phys. Fluids 22 (2010) 065103. [8] G. Pailhas, P. Barricau, Y. Touvet, L. Perret, Friction measurement in zero and adverse pressure gradient boundary layer using oil droplet interferometric method, Exp. Fluids 47 (2009) 195–207. [9] R. Örlü, J.H.M. Fransson, P.H. Alfredsson, On near wall measurements of wall bounded flows—the necessity of an accurate determination of the wall position, Prog. Aerosp. Sci. (2010) (in press). http://dx.doi.org/10.1016/j.paerosci.2010.04.002. [10] R. Örlü, Experimental studies in jet flows and zero pressure-gradient turbulent boundary layers, Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden, 2009. http://www.mech.kth.se/thesis/2009/phd/phd_2009_ramis_orlu.pdf. [11] R. Örlü, P.H. Alfredsson, On spatial resolution issues related to time-averaged quantities using hot-wire anemometry, Exp. Fluids 49 (2010) 101–110. [12] P. Schlatter, R. Örlü, Q. Li, G. Brethouwer, J.H.M. Fransson, A.V. Johansson, P.H. Alfredsson, D.S. Henningson, Turbulent boundary layers up to Reθ = 2500 studied through simulation and experiment, Phys. Fluids 21 (2009) 051702. [13] K.A. Chauhan, P.A. Monkewitz, H.M. Nagib, Criteria for assessing experiments in zero pressure gradient boundary layers, Fluid Dynam. Res. 41 (2009) 021404. [14] P.H. Alfredsson, A.V. Johansson, J.H. Haritonidis, H. Eckelmann, On the fluctuating wall shear stress and velocity field in the viscous sublayer, Phys. Fluids 31 (1988) 1026–1033. [15] P. Schlatter, Q. Li, G. Brethouwer, A.V. Johansson, D.S. Henningson, Simulations of spatially evolving turbulent boundary layers up to Reθ = 4300, Int. J. Heat Mass Transfer 31 (2010) 251–261. [16] P. Schlatter, R. Örlü, Assessment of direct numerical simulation data of turbulent boundary layers, J. Fluid Mech. (2010) (in press). http://dx.doi.org/10.1017/S0022112010003113. [17] H.M. Nagib, K.A. Chauhan, P.A. Monkewitz, Approach to an asymptotic state for zero pressure gradient turbulent boundary layers, Phil. Trans. R. Soc. A 365 (2007) 755–770. [18] A.V. Johansson, P.H. Alfredsson, Effects of imperfect spatial resolution on measurements of wall bounded turbulent shear flows, J. Fluid Mech. 137 (1983) 409–421. [19] N. Hutchins, T.B. Nickels, I. Marusic, M.S. Chong, Hot-wire spatial resolution issues in wall-bounded turbulence, J. Fluid Mech. 635 (2009) 103–136. [20] T. Tsukahara, Y. Seki, H. Kawamura, D. Tochio, DNS of turbulent channel flow at very low Reynolds numbers, in: Proc. of the Forth Int. Symp. on Turbulence and Shear Flow Phenomena, Williamsburg, USA, 2005, pp. 935–940. [21] H. Abe, H. Kawamura, Y. Matsuo, Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence, ASME J. Fluids Eng. 123 (2001) 382–393. [22] H. Abe, H. Kawamura, Y. Matsuo, Surface heat-flux fluctuations in a turbulent channel flow up to Reτ = 1020 with Pr = 0.025 and 0.71, Int. J. Heat Fluid Flow 25 (2004) 404–419. [23] J.C. del Alamo, J. Jimenez, P. Zandonade, R.D. Moser, Scaling of the energy spectra of turbulent channels, J. Fluid Mech. 500 (2004) 135–144.
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
The diagnostic plot — a litmus test for wall bounded turbulence data P. Henrik Alfredsson ∗ , Ramis Örlü Linné Flow Centre, KTH Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
article
info
Article history: Received 23 February 2010 Received in revised form 30 June 2010 Accepted 20 July 2010 Available online 27 July 2010 Keywords: Wall bounded turbulence Mean velocity distribution Near wall behaviour
abstract A diagnostic plot is suggested that can be used to judge wall bounded turbulence data of the mean and the rms of the streamwise velocity in terms of reliability both near the wall, around the maximum in the rms as well as in the outer region. The important feature of the diagnostic plot is that neither the wall position nor the friction velocity needs to be known, since it shows the rms value as a function of the streamwise mean velocity, both normalized with the free stream velocity. One must remember, however, that passing the test is a necessary, but not sufficient condition to prove good data quality. © 2010 Elsevier Masson SAS. All rights reserved.
1. Introduction During the last decade there has been a renewed interest in how averaged quantities in wall bounded turbulence vary in the direction normal to the wall, especially with regard to the mean and rms distributions of the streamwise velocity (see e.g. [1–3]). Comparing data from different facilities and/or different techniques is, however, often inconclusive since the inaccuracies of the measured quantities may be larger than the trends one wants to investigate. In recent conference contributions [4,5] the so called diagnostic plot was introduced as a tool to clarify whether boundary layer data are corrupted by measurement problems. In this paper we expand on this method to show how the behaviour of this plot can be explained from what we already know about the scaling of boundary layer data. We also show that it works equally well for turbulent channel flow data. 2. Motivation For studies in wall bounded turbulent flows most measurements of the streamwise mean velocity (U) and the rms of the fluctuating streamwise velocity (u0 ) distributions are made with hot-wire anemometry using single wire probes. The technique has itself several limitations, such as frequency response, spatial resolution, wall interference and heat conduction to the wall, calibration at low velocities, determination of the wire position relative to the wall etc. Most of them can be mastered by an experienced experimentalist under well controlled conditions.
∗
Corresponding author. E-mail address: [email protected] (P.H. Alfredsson).
0997-7546/$ – see front matter © 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2010.07.006
An important quantity is the wall shear stress (τ ) or rather the friction velocity (uτ ). Since uτ scales the mean velocity and also determines the viscous length scale `∗ = ν/uτ , and thereby the coordinate y+ , i.e. the normal coordinate y normalized with `∗ , the comparison and scaling between different measurements is utterly dependent on the accuracy of uτ itself. For smooth-wall boundary layers, oil–film interferometry can now be used to determine uτ with an accuracy of around 1% [6,7], but it is neither standard practice in most laboratories, nor simple enough to do in a short time for diagnostic purposes (see also Ref. [8], which gives an accuracy of ±1.5%). Let us also one more time point out that the determination of the probe position relative to the wall is crucial when plotting data in the standard inner scaling. This may at first look like a trivial problem but to determine the position with an accuracy of better than one viscous length scale is not easy if it has to be done from outside the wind-tunnel test section [9]. A possibility is to fit the mean velocity profile to some pre-described distribution for the near-wall behaviour. However in that case it is necessary that the measurements of the mean velocity are accurate in themselves. The diagnostic plot can be used to check whether the velocity data measured close to the wall conform to expected physical characteristics or if they are affected by spatial averaging or other near-wall effects. The diagnostic plot allows the experimentalist to do this without having to determine either the position of the wall or the friction velocity. However, as will be shown here it also allows a characterization of the flow, not only close to the wall, but also in the rest of the boundary layer. It may be pointed out that the diagnostic plot can also be useful to check the behaviour of numerical simulations, especially for LES/DES where the skin friction may not be accurately known.
404
P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
a
b
c
d
Fig. 1. Profiles of the mean streamwise velocity and rms streamwise velocity in inner (a, b) and outer (c, d) scaling, respectively. U + = y+ and U + = κ −1 ln y+ + B, with κ = 0.384 and B = 4.17 [13] are given as dashed lines. : Reθ = 2540, : Reθ = 8100, 4: Reθ = 18 700, —: DNS at Reθ = 2510.
3. The basic data in standard form
4. The diagnostic plot
In this paper we will make use of three different velocity profile measurements carried out in the MTL wind tunnel at KTH [10,11] as well as a DNS reported by Schlatter et al. [12]. Typical plots of U and u0 are shown in Fig. 1(a) and (b) for the three different Reθ , viz. 2540, 8100 and 18 700 (Reθ is the Reynolds number based on the momentum-loss thickness and free-stream velocity and the first is taken at x = 1.62 m and the other two at x = 3.62 m from the leading edge of the plate). Both U and u0 are normalized with the friction velocity uτ (based on oil–film interferometry) and plotted as a function of y+ . The probe lengths for the three cases are, 15, 26 and 60 `∗ , respectively, which would give fairly good resolution for the smallest Reθ but rather large spatial averaging effects in the near-wall region for the highest Reθ [11]. The DNS data plotted in the figure have a Reθ of 2510, i.e. approximately the same as the lowest experimental case, and it has been shown in Ref. [12] that the experiments and DNS show a very good agreement, with respect to the shape factor, skin friction coefficient, as well as the mean and rms distribution of the streamwise velocity normal to the wall. In Fig. 1(c) and (d) we show the data plotted against the outer lengthscale, i.e. as function of y/∆ where ∆ = δ∗ U∞ /uτ and δ∗ and U∞ are the displacement thickness and free stream velocity, respectively. The mean flow data are plotted in the standard wake formulation (W + = (U∞ − U )/uτ ). As expected the mean flow data overlap nicely from the edge of the boundary layer all the way through the logarithmic region, whereas the u0 -distributions have a much smaller range of overlap.
In the diagnostic plot u0 is plotted as function of U, where both quantities are normalized by U∞ , thereby avoiding any uncertainties in both the wall position and uτ . For fully developed turbulent boundary layers that are plotted in this way, distributions of U and u0 that are taken at the same Re should fall on top of each other if accurately measured (or simulated). The values of U /U∞ are between 0 and 1, and Table 1 gives approximate values for the different flow regions typically associated with turbulent boundary layer flows (for the lowest and highest Reynolds number of the present experiments). 4.1. Near-wall region Close to the wall (y+ < 10) we expect that the distribution should be nearly self-similar and independent of Re. The relative level of the rms of the wall shear-stress fluctuations, τ 0 /τ , is related to u0 /U as (see e.g. Ref. [14])
τ0 u0 = lim . y→0 U τ
(1)
Experimentally the value of τ 0 /τ was given as 0.39 in Ref. [14] at a Reθ of about 3000, however Ref. [2] indicates that it may increase slightly with Re. For the boundary layer DNS in [15,16] the value increases from 0.403 to 0.429 when Reθ increases from 670 to 4300. Since U varies linearly with y near the wall, Eq. (1) shows that u0 also varies linearly with y with a constant slope (e.g. u0 = 0.40U). Simulations and experiments show that the distribution is almost
P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
405
Table 1 Different regions in the diagnostic plot. Reθ = 2540 0 < y+ < 5 5 < y+ < 50 50 < y+ , y/δ < 0.15 0.15 < y/δ < 1
Viscous sublayer Buffer region Logarithmic region Wake region
0.00 0.20 0.60 0.71
Reθ = 18 700
< U /U∞ < U /U∞ < U /U∞ < U /U∞
< 0.20 < 0.60 < 0.71 < 1.00
0.00 0.16 0.50 0.75
< U /U∞ < U /U∞ < U /U∞ < U /U∞
< 0.16 < 0.50 < 0.75 < 1.00
The review in Ref. [3] of available data indicates that the maximum is located at a constant y+ independent of Re. This value is close to + y+ = 15 and corresponds to a value of Umax ≈ 10.6. Using these relations we can find a direct relation between the maximum value of u0 and the mean velocity U at the maximum, which turns out to be a straight line in the diagnostic plot, namely u0
max
Fig. 2. The data from Fig. 1 shown in the diagnostic plot. The dashed line corresponds to the maximum in u0 obtained from the correlation in Ref. [2] and assuming that the maximum is located at y+ = 15, and the filled symbols to Reynolds numbers of the present experiments. The dash–dotted line indicates the tangent to the near-wall data and has a slope of 0.40. The measured points below this line for U /U∞ < 0.15 are clearly affected by the wall.
linear up to y+ = 5 (corresponding to U /U∞ = 0.2 for Reθ = 2540) and thereafter the slope decreases [3]. As can be seen in Fig. 2 the points closest to the wall (i.e. points within the viscous sublayer for the lowest Re) deviate below the straight line indicating that the measured u0 values are too small and/or the U values are too high. Hence the diagnostic plot indicates clearly those measurement points for which the measured values show a wall interference effect, which is not directly evident in Fig. 1(a) and (b). It should be noted that, although the slope increases weakly with Re and decreases with increasing spatial resolution effects, the important point here is that suspicious measurement points, due to wall interference, can be identified through their deviation from the tangent to the data points in the diagnostic plot.
max
u
0
uτ
= 1.86 + 0.28 log(Reθ )
(2)
u0
=
U∞
uτ U∞
[1.86 + 0.28 log(Reθ )].
(3)
2
=
uτ
2
U∞
= [κ
−1
ln(Reθ ) + C ]
−2
U∞
U∞ − U uτ
(6)
=f
y ∆
,
(7)
whereas the rms level can be written
=g
y ∆
.
(8)
These can be combined to give u0
U∞
+ = Umax
uτ U∞
.
y
= 1−
=f
U
g (y/∆)
(9)
f (y/∆)
U∞
(4)
(5)
−1
U∞ − U
=F
uτ
U U∞
;
U∞ uτ
.
(10)
Finally we can write u0 U∞
where κ = 0.384 and C = 4.127 [17]. The velocity Umax at the maximum of u0 can simply be written as Umax
,
Also in the outer region the distributions at different Reynolds numbers previously shown in Fig. 1 nicely overlap in the diagnostic plot. From the accepted standard scaling behaviour of the mean and fluctuating streamwise velocity in the outer region the mean velocity can be written in defect form
∆
This relation can be used together with a skin friction formula to find how the maximum should vary in the diagnostic plot. Here we use the Coles–Fernholz skin friction relation which can be written Cf
U∞
and both g and f decrease with increasing y/∆. We may express y/∆ in terms of U /U∞ using Eq. (7) giving
or max
U∞
U
4.3. The outer region
uτ
Marusic and Kunkel [2] give an expression for how the maximum of u0 increases with Reθ
= C1 + C2
where C1 = 0.047 and C2 = 0.157. As can be seen in Fig. 2 the maximum value decreases with increasing Re in this scaling. In this figure we have also put symbols where we would, based on Eq. (6), expect the present Reynolds number data to have their maxima. As can be seen the maxima show up at the correct U /U∞ although the differences between the estimated and measured values at the two highest Re are an effect mainly due to spatial averaging (see e.g. [11,18,19]). Hence, the location in the diagnostic plot of the maximum in u0 can be used as an indication on the accuracy of the measurements in the buffer region. It should be noted that in Ref. [19] a different relation for the maximum value of u0 was given, however, there are only small differences compared to Eq. (2) for the present Reynolds number range.
u0
4.2. The maximum of u0
=G
U U∞
;
U∞ uτ
,
(11)
which indicates that in the outer region the u0 /U∞ versus U /U∞ distribution has a Reynolds number dependence, expressed here as U∞ /uτ . The employed experimental and numerical data within the Reynolds number range considered here indicate, however, that this Reynolds number dependence is small. This is also the case for the data shown in Fig. 2 (remember that the data shown span
406
P.H. Alfredsson, R. Örlü / European Journal of Mechanics B/Fluids 29 (2010) 403–406
stream velocity from a few measurement points in the outer region of u0 and U, which may be helpful when studying atmospheric boundary layers. Acknowledgements
8
The authors thank their co-authors in Ref. [4], Dr. Philipp Schlatter for access to the DNS data as well as participants of the NORDITA Programme on Turbulent boundary layers and Turbulent Combustion in 2010 for valuable discussions. We also want to thank the authors of Refs. [20–23] for making their data available on the web, as well as the referees for their constructive criticisms. This work was partly sponsored by the Swedish Research Council. References
Fig. 3. DNS data for channel flows with Reτ = 110, 150, 180, 395, 550, 640, 950, 1020, 2000. The three lowest Reτ are shown as dashed lines, and the maxima of u0 /U∞ for all data sets decrease as Reτ increase. The dash–dotted lines correspond to u0 /U = 0.32, 0.36 and 0.40, respectively.
over almost one decade in Reynolds number) and it is interesting to see that in the diagnostic plot the different Re data collapse also in the logarithmic region. For this part of the boundary layer the diagnostic plot gives a good indication whether the boundary layer is in a natural state. 5. Conclusions and outlook We have shown here how the behaviour of the distribution of u0 in the diagnostic plot can be obtained from what we already know about its scaling in the boundary layer. We have demonstrated that the diagnostic plot may detect measured data in the near-wall region that may suffer from various problems. For the Reynolds number range considered here, the distribution in the outer region appears to be self-similar, and the behaviour of the near-wall peak of u0 /U∞ has been given. All these features of the diagnostic plot make it possible to use it as a ‘‘litmus’’ test on the quality of measured data, without the need to determine the friction velocity or the position of the wall. One must remember, however, that passing the test is a necessary, but not sufficient condition to prove good data quality. We have also tested the diagnostic plot on DNS data for turbulent channel flow available in the literature [20–23]. For Reynolds numbers in the range 395–2000 (based on friction velocity and half channel height) the data collapse nicely (see Fig. 3) both in the inner and outer regions (although with a different shape in the outer region compared to boundary layers since the turbulence intensity does not go to zero on the centreline). For the three lowest Reynolds numbers it is seen that they deviate significantly in the outer region (i.e. in the centre of the channel) and this indicates that the flow for these Reynolds numbers has not reached a self-similar state. An analogous behaviour of the diagnostic plot should probably also be expected in pipe flows. The diagnostic plot should therefore be useful for anybody doing measurements in wall bounded flows. The diagnostic plot also has a potential to be used to evaluate the accuracy of LES/DES since for such simulations the skin friction cannot be directly determined from near-wall data. For experiments in rough wall boundary layers both the wall distance and skin friction are usually not known accurately and also here the diagnostic plot may prove useful. Some preliminary tests show that the diagnostic plot may also be useful in evaluating pressure gradient boundary layers. Another possibility that has not yet been explored is that if the Reynolds number variation is small in the outer region the diagnostic plot may be used to estimate the boundary layer free
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