A Method to Determine Wall Shear Stress from Mean Profiles in Turbulent Boundary Layers

arXiv:2106.03981v1 [physics.flu-dyn] 7 Jun 2021 ed oda n nryepniuei aerodynamic in expenditure energy and drag which to wall, the high leads at The stresses tangential 1968). large (Schlichting, induces shear near-wall Prandtl the of thin since work studied This seminal extensively been viscosity. has conditions by layer’ no-slip ‘boundary to surface due wall at region the thin imposed to a close produces shear wall high or of surface solid a over Flow Introduction 1 stress owti %oe ag fRyod number. Reynolds Keywords of range a over values, 3% reference within with obtained to agree velocities thick- method friction layer proposed the The boundary using factor. of shape estimate % and an 50 ness and to 20 TBL, between the preferably of layer, shear outer mean the and in velocity stress streamwise mea- mean point of single surement a from requires method TBL noise-prone The data. gradient requiring near-wall without the pressure profiles, determine zero mean to in measured inner proposed stress noise-prone is shear the method wall simple from A TBL, data layer. entire velocity reliable the composite require for a constants and or tuned layer loga- many with inner a profile the either in satisfy law to rithmic velocity assume streamwise methods mean popular the mean Most from determined measurements. indirectly profile be to it requiring there- challenging, fore is (TBL) layers boundary turbulent in Abstract date Accepted: / date Received: in profiles Kumar mean Praveen from stress layers shear wall boundary determine turbulent to method A rve Kumar Praveen -al [email protected] US 55455, E-mail: Minnesota Minneapolis, Mechanics, Minnesota, and of University Engineering Aerospace of Department wl eisre yteeditor) the by inserted be (will No. manuscript Noname h ietmaueeto alserstress shear wall of measurement direct The ubln onaylayer boundary Turbulent · rsnnMahesh Krishnan · rsnnMahesh Krishnan · alshear Wall A B u oiscmo s nms cln as Un- laws. scaling of most measurement in direct use the common fortunately, its to due TBL iyo h fluid. the of sity and vrterange the over ersne ytefito eoiydfie as defined velocity friction the by represented field 2001). flow (Pope, second-order statistics and first- com- using is described and turbulent, monly is layer practical boundary most the conditions, Under applications. hydrodynamic and n eino aiiy sn h lue hr method, chart Clauser TBL the u gradient Using validity. pressure of popular zero region the for literature and of constants the some the shows in of 2 Table debated values 2011). widely al., been et have (Smits (1) Eq. of B oe o h ne ae (typically pro- layer been inner have wall’ the the viscous for of in posed ‘law known called is so profile Several velocity units. mean the of havior huhanme fidrc ehiusaeaalbefor available are determining techniques indirect Al- of number it. a determine though to methods indirect requiring feasible, aiainwith malization eaeteinrsae ensraws eoiy( velocity streamwise mean scaled inner the relate p U log- as- universal law the which satisfies arithmic velocity 1996), mean the Finley, that and sumes (Fernholz method list. are extensive chart more readers a The for (2021) wall. al. the et of Zhang law to referred the for formulae ular owl omlcodnt ( coordinate normal wall to τ + hi nvraiy n h xc eino validity of region exact the and universality, their , τ sdtrie ob h au htbs t h mea- the fits best that value the be to determined is w ntruetbudr aes h knfito is skin-friction the layers, boundary turbulent In nidrc ehiu a efruae ftebe- the if formulated be can technique indirect An onaylyrrsacescmol s h Clauser the use commonly researchers layer Boundary = B /ρ κ 1 ≈ where ln y .Teeatvle fteconstants the of values exact The 5. + u + τ τ y oeaeuieslyaccepted. universally are none , w B u u + τ τ stema alsrs and wall-stress mean the is > and savr motn eoiysaein scale velocity important very a is 0and 30 ν al itsm ftepop- the of some list 1 Table . y + y/δ ,wee‘’dntsnor- denotes ‘+’ where ), ≤ 0 . 5 where 15, / < y/δ u τ sotnnot often is 0 . ρ 5 that 15) κ sden- is κ u ≈ U τ and 0 (1) + . = 4 ) 2 Praveen Kumar, Krishnan Mahesh

Table 1 Popular formulae for the law of the wall.

Authors Formulae Region where valid Prandtl (1910) U + = y+ 0 ≤ y+ ≤ 11.5 Taylor (1916) U + = 2.5lny+ + 5.5 y+ ≥ 11.5 von K´arm´an (1939) U + = y+ 0 ≤ y+ < 5 U + = 5lny+ - 3.05 5 ≤ y+ < 30 U + = 2.5lny+ + 5.5 y+ ≥ 30 Rannie (1956) U + = 1.454tanh(0.0688y+) 0 ≤ y+ < 27.5 U + = 2.5lny+ + 5.5 y+ ≥ 27.5 Werner and Wengle (1993) U + = y+ 0 ≤ y+ < 11.81 U + = 8.3(y+)1/7 y+ ≥ 11.81 + + Reichardt (1951) U + = 2.5ln(1 + 0.4y+) + 7.8(1 − e−y /11 − (y+/11)e−0.33y ) y+ ≥ 0 + 2 + 3 + 4 Spalding (1961) y+ = U + + 0.1108 e0.4κU + − 1 − κU + − (κU ) − (κU ) − (κU ) y+ ≥ 0 2 6 24 + + + + Monkewitz et al. (2007) Uinner = Uinner,23 + Uinner,25, where y ≥ 0 + +2 + Uinner,23 = 0.68285472 ln(y + 4.7673096y + 9545.9963) +1.2408249 arctan(0.010238083y+ + 0.024404056) +1.2384572 ln(y+ + 95.232690) − 11.930683 + +2 + Uinner,25 = −0.50435126 ln(y − 7.8796955y + 78.389178) +4.7413546 arctan(0.12612158y+ − 0.49689982) −2.7768771 ln(y+2 + 16.209175y+ + 933.16587) +0.37625729 arctan(0.033952353y+ + 0.27516982) +6.5624567 ln(y+ + 13.670520) + 6.1128254

Table 2 Values of the log law constants in TBL.

Authors Formulae Region where valid + Coles (1956) κ = 0.41, B = 5 50 < y < 0.2Reτ + Osterlund¨ (1999) κ = 0.38, B = 4.1 50 ≤ y ≤ 0.15Reτ + Nagib et al. (2007) κ = 0.384, B = 4.173 50 ≤ y ≤ 0.15Reτ 1/2 + Marusic et al. (2013) κ = 0.39, B = 4.3 3Reτ < y < 0.15Reτ + Monkewitz et al. (2007) κ = 0.384, B = 4.17 50 ≤ y ≤ 0.15Reτ + Monkewitz (2017) κ = 0.384, B = 4.22 50 ≤ y ≤ 0.15Reτ

sured mean streamwise velocity profile to Eq. (1) in A more recent approach to determine uτ (e.g. Samie et al., its range of validity. For example, De Graaff and Eaton 2018) is to fit the entire mean streamwise velocity pro- (2000) obtained uτ using the Clauser chart method by file to the composite profile of Chauhan et al. (2009): fitting the mean velocity data to Eq. (1) in the region y+ > 50 and y/δ < 0.2 with κ =0.41 and B = 5. 2Π U + = U + + W (η), (2) Several researchers have recognized and documented composite inner κ the limitations of the Clauser chart method. George and Castillo (1997) showed clear discrepancies between mean ve- where, locity profiles scaled using direct measurements of uτ and approximations using the Clauser chart method. Wei et al. (2005) explicitly illustrated how the Clauser 1 y+ a U + = ln − + chart method can mask subtle Reynolds-number-dependent inner κ a behavior, causing significant error in determining u . − τ R2 a (y+ α)2 + β2 Several past works have attempted to improve the Clauser (4α + a)ln − a(4α a) − R y+ a chart method by optimizing the constants of Eq. 1. − p − α y+ α α Rodr´ıguez-López et al. (2015) analyzed several such past + (4α +5a) arctan − + arctan , β β β efforts and proposed a better method by optimizing the various parameters appearing in the mean velocity pro- (3) file of the entire TBL. However, as mentioned by the authors themselves, their method was sensitive to the assumed universal velocity profile, and the definition of with α = ( 1/κ a)/2, β = √ 2aα α2, and R = − − − − the error which was used for the optimization. α2 + β2. The chosen values of κ = 0.384 and a = p A method to determine wall shear stress from mean profiles in turbulent boundary layers 3

10.3061. The wake function W is given by a universal W . However, this makes the method sensi- − tive to the choice of W and the constants in the law of W (η)= the wall. It is clear that most of the issues in determining 4 5 6 7 1 exp( (1/4)(5a2+6a3+7a4)η +a2η +a3η +a4η ) u from profile data stem from the presumption of a − − τ universal mean velocity profile that is valid in the in- 1 1 exp( (a2 +2a3 +3a4)/4) 1 lnη , ner layer or throughout the TBL. Therefore, a method − − − 2Π is proposed in this paper to determine uτ from mean (4) profile measurements without any assumption of a universal mean velocity profile, thereby avoiding all issues where a2 = 132.8410, a3 = 166.2041, a4 = 71.9114, − associated with uncertainties in the constants of the as- the wake parameter Π =0.45 and η = y/δ. Chauhan et al. sumed profile. The method utilizes profiles of the mean (2009) have also shown that the accuracy of the ap- velocity and the mean shear stress without requiring proach to be within 2% of those determined by direct ± any near-wall (y < 0.2δ) data. The method is based on measurements. Although the method appears attrac- integral analysis of the streamwise momentum equation tive, it still requires accurate near-wall measurements but does not require profiles at multiple streamwise lo- to determine uτ reliably. cations, wall-normal integration of data or any iterative Several recent studies (e.g. Fukagata et al., 2002; procedure to determine u accurately. Mehdi and White, 2011; Mehdi et al., 2014) have at- τ The paper is organized as follows. The proposed tempted to relate uτ to measurable profile quantities method is described in Section 2. Section 3 describes using the governing equations of the mean flow. These the databases used in the present work. The perfor- methods often require additional measurements and as- mance of the method is discussed in Section 4. Section sumptions to account for the unknown quantities. For 5 concludes the paper. example, Mehdi and White (2011) and Mehdi et al. (2014) obtained uτ using the measured mean streamwise velocity and Reynolds shear stress profiles. However, their 2 The proposed method method was sensitive to noisy or missing near-wall data. They overcome this limitation by assuming a shape for The boundary layer approximations for the time-averaged the total shear stress profile and by smoothing the mea- Navier–Stokes equations in Cartesian coordinates yield sured data. ∂U ∂V + =0, (5) Volino and Schultz (2018) used integral analysis of ∂x ∂y the mean streamwise momentum equation to obtain ∂U ∂U 1 dP ∂T a relation between mean velocity and Reynolds shear U + V = + , (6) ∂x ∂y −ρ dx ∂y stress profiles, that was subsequently used to determine where U and V are the streamwise (x) and the wall- uτ from measured profiles. The utility of the method was shown through application to experimental data normal (y) components of mean velocity vector, ρ is including zero pressure gradient cases from smooth and the fluid density, P is the mean pressure and T is the rough walls, and smooth wall cases with favorable and mean total stress, i.e., the sum of the viscous stress (ν∂U/∂y) and the Reynolds shear stress ( u′v′). ν is adverse pressure gradients. Although their method is − generally applicable, it has two main issues: (i) the for- the kinematic viscosity. Using Eqs. (5) and (6), and mulation attempts to minimize dependence on stream- neglecting the pressure gradient term, it can be shown wise gradients, but some dependence remains, making that ∂T ∂V ∂V data from two or more streamwise locations necessary = Ue + (Ue U) . (7) and the process of determining uτ iterative, and (ii) ∂y − ∂y − ∂y since the method requires numerical integration of the Integrating Eq. (7) from a generic y to y = δ and nor- profiles in the wall-normal direction, the accuracy dete- malizing in viscous units yield, riorates if near-wall data is omitted. In order to improve δ+ + + + + V + + ∂V + the accuracy in such cases, they either used the trape- T = Ue Ve 1 (Ue U ) + dy .(8) − V − + − ∂y zoidal rule or an assumed velocity profile to account for e Zy the missing contribution. The requirement to have data Kumar and Mahesh (2021) used available TBL data to at multiple streamwise locations to obtain streamwise model the second term in the rhs of Eq. (8) to obtain gradients was mitigated by Womack et al. (2019) by a modeled total stress, essentially using a skin-friction law and connecting the V y T + = H 1 + (H 1) 1 (9) streamwise gradient to the wall-normal gradient using − V − δ − e 4 Praveen Kumar, Krishnan Mahesh

Table 3 TBL datasets considered in this paper.

Case Reθ Reτ H Database S1000 1006 359 1.4499 DNS (Schlatter and Orlü,¨ 2010) S2000 2001 671 1.4135 DNS (Schlatter and Orlü,¨ 2010) S3030 3032 974 1.3977 DNS (Schlatter and Orlü,¨ 2010) S4060 4061 1272 1.3870 DNS (Schlatter and Orlü,¨ 2010) S5000 5000 1571 1.3730 DNS (Sillero et al., 2013) S6000 6000 1848 1.3669 DNS (Sillero et al., 2013) S6500 6500 1989 1.3633 DNS (Sillero et al., 2013) E6920 6919 1951 1.3741 Expt (Morrill-Winter et al., 2015) E9830 9826 2622 1.3661 Expt (Morrill-Winter et al., 2015) E11200 11200 2928 1.3366 Expt (Morrill-Winter et al., 2015) E15120 15116 3770 1.3350 Expt (Morrill-Winter et al., 2015) E15470 15469 3844 1.3476 Expt (Morrill-Winter et al., 2015) E24140 24135 5593. 1.3264 Expt (Morrill-Winter et al., 2015) E26650 26647 6080 1.2938 Expt (Morrill-Winter et al., 2015) E36320 36322 7894 1.2774 Expt (Morrill-Winter et al., 2015) E21630 21632 5100 1.3 Expt. (Baidya et al., 2017) E51520 51524 10600 1.28 Expt. (Baidya et al., 2017)

+ + where, the relation Ue Ve = H (Wei and Klewicki, 0.05 2016; Kumar and Mahesh, 2018) is used. The mean shear stress model was validated for a range of Re using past simulations and experiments. 0.045 Since, most past work do not report the V profile, u /U Kumar and Mahesh (2021) also provided a compact model τ e 0.04 for V/Ve as function of y/δ, i.e., 3 V y y = tanh a + b , (10) 0.035 V δ δ e where the model constants a = 0.5055 and b = 1.156. 0.03 Now, Eq. (9) can be rearranged to obtain 0 0.2 0.4 0.6 0.8 1 T uτ = . (11) 0.05 H(1 V/Ve) + (H 1)(y/δ 1) s − − − Figure 1 shows the rhs of Eq. (11) for cases S1000, 0.045 S2000, S3030 and S4060 (i.e. Reθ = 1006 to 4061, see Table 3) using actual V/Ve (Figure 1 (a)) and mod- uτ /Ue eled V/Ve (Figure 1 (b)) using Eq. (10). The true val- 0.04 ues of uτ are also shown for comparison. Each symbol represents u predicted using data point at only that τ 0.035 location in the boundary layer. It is clear that Eq. (11) predicts utau accurately for any data point in the range y/δ < 0.6. It is also evident that as Re increases, the 0.03 0 0.2 0.4 0.6 0.8 1 difference between the uτ obtained using actual V/Ve y/δ and modeled V/Ve becomes increasingly smaller. Since V is often not available, all subsequent results in this Fig. 1 Profiles of rhs of Eq. (11) (symbols) are compared paper use modeled V (Eq. (10)). to true uτ (lines) for cases S1000, S2000, S3030, and S4060 The proposed method determines uτ from Eq. (11). (from top to bottom in that order), using actual V/Ve (a) In principle, only one measurement location anywhere and modeled V/Ve (b) using Eq. (10). in the range y/δ < 0.6 is sufficient to obtain uτ as observed in figure 1. However, if a coarse profile measurement is available, uτ value is the horizontal line which best fits the rhs of Eq. (11) data in the range 0.2 < y/δ < 0.5. The region y/δ < 0.2 is deliberately A method to determine wall shear stress from mean profiles in turbulent boundary layers 5 avoided since this region is prone to noise in experi- 0.07 ments. 0.06

0.06 0.04 3 TBL databases uτ /Ue 0.02 0 0.5 1 Table 3 lists the relevant details of all the direct nu- 0.05 merical simulation (DNS) and experimental databases used in this paper. Note that the correlation Reτ = 0.04 1.13 Re0.843 proposed by Schlatter and Orl¨u(2010)¨ × θ is used to obtain Reθ from the reported values of Reτ in the experiments of Morrill-Winter et al. (2015) and 0.03 0.2 0.3 0.4 0.5 0.6 Baidya et al. (2017). The readers are referred to the pa- pers cited in the table for numerical and experimental 0.07 0.06 details. Note that throughout the paper, ‘true uτ ’ is the value reported in the reference paper. 0.06 0.04

uτ /Ue 0.02 4 Results 0 0.5 1 0.05

Figure 2 shows the predicted uτ for TBL cases ranging from Reθ = 1006 to 4061 using the DNS data of 0.04 Schlatter and Orlü(2010).¨ The black lines show the bounds of 3 % deviation from the true values. Fig- ± 0.03 ure 3 shows similar results for TBL cases ranging from 0.2 0.3 0.4 0.5 0.6 Reθ = 5000 to 6500 using the DNS data of Sillero et al. (2013). In all these cases, the proposed method accu- 0.07 0.06 rately predicts uτ using the DNS data in the range 0.2 < y/δ < 0.6. 0.06 0.04 Next, the proposed method is used to predict uτ using the experimental database of Morrill-Winter et al. uτ /Ue 0.02 0 0.5 1 (2015) and Baidya et al. (2017). Note that both these 0.05 experimental campaigns used the composite fit method to determine uτ . Figures 4 and 5 show the predicted uτ compared for the experimental data of Morrill-Winter et al. 0.04 (2015). Overall, the proposed method agrees to the data obtained from the composite fit approach to within 0.03 3% in the region 0.2 < y/δ < 0.5 for all the cases 0.2 0.3 0.4 0.5 0.6 except E36320, where it is within 3% in the region 0.07 0.2 < y/δ < 0.35. 0.06 Figure 6 shows the predicted uτ compared to the experimental data of Baidya et al. (2017). Note that the 0.06 0.04 contribution of viscous stress to the total stress is ig- uτ /Ue 0.02 nored for these data sets. Overall, the proposed method 0 0.5 1 agrees with the composite fit approach to within 3% in 0.05 the region 0.2 < y/δ < 0.35. An important point to note is that the proposed 0.04 method requires the knowledge of H in addition to δ. From the definition of δ∗ and θ, it is clear that the near- wall region contributes more to the former. Obtaining 0.03 0.2 0.3 0.4 0.5 0.6 H from the measured data in the absence of near-wall y/δ resolution can be challenging. It is tempting to assume the one-seventh power law for near-wall data to account Fig. 2 Predicted uτ is compared to true uτ (red line) for for the contribution of missing data to the overall H. cases S1000 (a), S2000 (b), S3030 (c), and S4060 (d). Note that the black lines show ±3 % of of the red line. 6 Praveen Kumar, Krishnan Mahesh

0.07 0.06 0.06 0.06

0.06 0.04 0.05 0.04 uτ /Ue 0.02 uτ /Ue 0.02 0 0.5 1 0 0.5 1 0.05 0.04

0.04 0.03

0.03 0.02 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6

0.07 0.06 0.06 0.06

0.06 0.04 0.05 0.04 uτ /Ue 0.02 uτ /Ue 0.02 0 0.5 1 0 0.5 1 0.05 0.04

0.04 0.03

0.03 0.02 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6

0.07 0.06 0.06 0.06

0.06 0.04 0.05 0.04 uτ /Ue 0.02 uτ /Ue 0.02 0 0.5 1 0 0.5 1 0.05 0.04

0.04 0.03

0.03 0.02 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 y/δ 0.06 0.06 Fig. 3 Predicted uτ is compared to true uτ (line) for cases S5000 (a), S6000 (b), and S6500 (c). Note that the black lines 0.04 show ±3 % of the red line. 0.05

uτ /Ue 0.02 0 0.5 1 However, the goal here is to avoid using the inner layer 0.04 altogether. Hence, the sensitivity of the obtained uτ to H is assessed using error analysis. It can be shown that 0.03 the relative error in the friction velocity (ǫuτ ) is related to the relative error in the shape factor (ǫH ) as 0.02 0.5H(f η) 0.2 0.3 0.4 0.5 0.6 ǫu = − ǫH . (12) τ H(1 f) + (H 1)(η 1) y/δ − − − Figure 7 shows the term in parenthesis on the rhs of Fig. 4 Predicted uτ is compared to uτ obtained from the Eq. (12) for S1000 and S4000 cases; note that it is of composite ﬁt (red line) for cases E6920 (a), E9830 (b), E11200 (c), and E15120 (d). Note that the black lines show ±3 % of the red line. A method to determine wall shear stress from mean proﬁles in turbulent boundary layers 7

0.06 0.06 0.05

0.04 0.05 0.045

uτ /Ue 0.02 0 0.5 1 0.04 0.04 uτ /Ue 0.035 0.03 0.03

0.02 0.025 0.2 0.3 0.4 0.5 0.6 0.02 0.06 0.06 0.2 0.3 0.4 0.5 0.6 y/δ 0.05 0.04 0.05 uτ /Ue 0.02 0 0.5 1 0.04 0.045

0.03 0.04 uτ /Ue 0.035 0.02 0.2 0.3 0.4 0.5 0.6 0.03 0.06 0.06 0.025

0.04 0.02 0.05 0.2 0.3 0.4 0.5 0.6

uτ /Ue 0.02 y/δ 0 0.5 1 0.04

Fig. 6 Predicted uτ is compared to true uτ (line) for cases E21630 (a) and E51520 (b). Note that the error bar shows ±3 0.03 % of the red line.

0.02 0.2 0.3 0.4 0.5 0.6

0.06 0.06 the order 10−1. Hence, it can be concluded that the predicted uτ is relatively insensitive to the error in H. 0.04 0.05 Instead of obtaining H from measurements, one can use H = 1.36 at high Re (say Reθ > 6500) without uτ /Ue 0.02 0 0.5 1 0.04 compromising accuracy of the predicted uτ .

0.03 Consider an experimental campaign where there is a coarse field measurement from which δ can be obtained. Note that all it takes to determine uτ from Eq. (11) is 0.02 to have a probe at any location below 0.2 < y/δ < 0.2 0.3 0.4 0.5 0.6 0.5 to measure U and T u′v′. The flexibility and y/δ ≈ − limited data requirement of the proposed method makes it powerful, especially when data from the entire inner Fig. 5 Predicted uτ is compared to uτ obtained from the composite fit (red line) for cases E15470 (a), E24140 (b), layer is unreliable or missing. E26650 (c), and E36320 (d). Note that the black lines show ±3 % of the red line. 8 Praveen Kumar, Krishnan Mahesh

0 Chauhan KA, Monkewitz PA, Nagib HM (2009) Cri- teria for assessing experiments in zero pressure gradient boundary layers. Fluid Dynamics Research -0.05 41(2):021404 Coles D (1956) The law of the wake in the turbulent boundary layer. Journal of Fluid Mechanics 1(2):191– -0.1 226 De Graaﬀ DB, Eaton JK (2000) Reynolds-number scal- -0.15 ing of the ﬂat-plate turbulent boundary layer. Jour- nal of Fluid Mechanics 422:319–346 Fernholz HH, Finley PJ (1996) The incompressible -0.2 0.2 0.3 0.4 0.5 0.6 zero-pressure-gradient turbulent boundary layer: an y/δ assessment of the data. Progress in Aerospace Sci- ences 32(4):245–311

Fig. 7 Sensitivity of the predicted uτ to the error in H is Fukagata K, Iwamoto K, Kasagi N (2002) Contribu- shown by plotting the term in brackets in (12) for S1000 tion of reynolds stress distribution to the skin friction (blue) and S4000 (red). in wall-bounded flows. Physics of Fluids 14(11):L73– L76 5 Conclusions George WK, Castillo L (1997) Zero-pressure-gradient turbulent boundary layer. Applied Mech Review A novel method to determine uτ from measured pro- 50(11):689–729 file data is proposed. The method is based on integral von Kármán T (1939) The analogy between fluid fric- analysis of the governing equations of the mean flow tion and heat transfer. Transactions of the American and like all other such methods, requires shear stress Society of Mechanical Engineers 61:705–710 profile in addition to mean streamwise velocity profile Kumar P, Mahesh K (2018) Analysis of axisymmet- to determine uτ . However, unlike all the other meth- ric boundary layers. Journal of Fluid Mechanics ods, the proposed method does not require any near- 849:927–941 wall (y/δ < 0.2) data, thereby avoiding the noise-prone Kumar P, Mahesh K (2021) Simple model for mean region of the profile altogether. The proposed method stress in turbulent boundary layers. Physical Review is shown to predict uτ accurately for a range of Re and Fluids 6(2):024603 is relatively insensitive to measurement errors. There- Marusic I, Monty JP, Hultmark M, Smits AJ (2013) On fore, the proposed method provides a simple alternative the logarithmic region in wall turbulence. Journal of to current popular methods to indirectly determine uτ Fluid Mechanics 716 within an acceptable uncertainty by just using few data Mehdi F, White CM (2011) Integral form of the skin points in the entire boundary layer. friction coefficient suitable for experimental data. Ex- periments in Fluids 50(1):43–51 Acknowledgements This work is supported by the United Mehdi F, Johansson TG, White CM, Naughton JW States Office of Naval Research (ONR) under ONR Grant (2014) On determining wall shear stress in spatially N00014-20-1-2717 with Dr. Peter Chang as technical moni- developing two-dimensional wall-bounded flows. Ex- tor. The authors thank Prof. J. Klewicki for providing the experimental data published in Morrill-Winter et al. (2015). periments in Fluids 55(1):1–9 Monkewitz PA (2017) Revisiting the quest for a universal log-law and the role of pressure gradient in Conflict of interest “canonical” wall-bounded turbulent flows. Physical Review Fluids 2(9):094602 The authors declare that they have no conflict of inter- Monkewitz PA, Chauhan KA, Nagib HM (2007) est. Self-consistent high-reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Physics of Fluids 19(11):115101 References Morrill-Winter C, Klewicki J, Baidya R, Marusic I (2015) Temporally optimized spanwise vorticity sen- Baidya R, Philip J, Hutchins N, Monty JP, Maru- sor measurements in turbulent boundary layers. Ex- sic I (2017) Distance-from-the-wall scaling of turbu- periments in Fluids 56(12):1–14 lent motions in wall-bounded flows. Physics of Fluids 29(2):020712 A method to determine wall shear stress from mean profiles in turbulent boundary layers 9

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