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

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A Method to Determine Wall Shear Stress from Mean Profiles in Turbulent Boundary Layers Noname manuscript No. (will be inserted by the editor) A method to determine wall shear stress from mean profiles in turbulent boundary layers Praveen Kumar · Krishnan Mahesh Received: date / Accepted: date Abstract The direct measurement of wall shear stress and hydrodynamic applications. Under most practical in turbulent boundary layers (TBL) is challenging, there- conditions, the boundary layer is turbulent, and is com- fore requiring it to be indirectly determined from mean monly described using first- and second-order flow field profile measurements. Most popular methods assume statistics (Pope, 2001). the mean streamwise velocity to satisfy either a loga- In turbulent boundary layers, the skin-friction is rithmic law in the inner layer or a composite velocity represented by the friction velocity defined as uτ = profile with many tuned constants for the entire TBL, τw/ρ where τw is the mean wall-stress and ρ is den- and require reliable data from the noise-prone inner sity of the fluid. u is a very important velocity scale in p τ layer. A simple method is proposed to determine the TBL due to its common use in most scaling laws. Un- wall shear stress in zero pressure gradient TBL from fortunately, the direct measurement of uτ is often not measured mean profiles, without requiring noise-prone feasible, requiring indirect methods to determine it. Al- near-wall data. The method requires a single point mea- though a number of indirect techniques are available for surement of mean streamwise velocity and mean shear determining uτ , none are universally accepted. stress in the outer layer, preferably between 20 to 50 % An indirect technique can be formulated if the be- of the TBL, and an estimate of boundary layer thick- havior of the mean velocity profile is known in viscous ness and shape factor. The friction velocities obtained units. Several so called ‘law of the wall’ have been pro- using the proposed method agree with reference values, posed for the inner layer (typically y/δ < 0.15) that to within 3% over a range of Reynolds number. relate the inner scaled mean streamwise velocity (U +) to wall normal coordinate (y+), where ‘+’ denotes nor- Keywords Turbulent boundary layer Wall shear · malization with uτ and ν. Table 1 list some of the pop- stress ular formulae for the law of the wall. The readers are referred to Zhang et al. (2021) for a more extensive list. Boundary layer researchers commonly use the Clauser 1 Introduction chart method (Fernholz and Finley, 1996), which as- sumes that the mean velocity satisfies the universal log- Flow over a solid surface or wall produces a thin region arithmic law arXiv:2106.03981v1 [physics.flu-dyn] 7 Jun 2021 of high shear close to the wall due to no-slip conditions 1 U + = lny+ + B (1) imposed at surface by viscosity. This thin near-wall κ ‘boundary layer’ has been extensively studied since the over the range y+ > 30 and y/δ 0.15, where κ 0.4 ≤ ≈ seminal work of Prandtl (Schlichting, 1968). The high and B 5. The exact values of the constants κ and ≈ shear induces large tangential stresses at the wall, which B, their universality, and the exact region of validity leads to drag and energy expenditure in aerodynamic of Eq. (1) have been widely debated in the literature Praveen Kumar · Krishnan Mahesh (Smits et al., 2011). Table 2 shows some of the popular Department of Aerospace Engineering and Mechanics, values of the constants for zero pressure gradient TBL University of Minnesota, Minneapolis, Minnesota 55455, USA and region of validity. Using the Clauser chart method, E-mail: [email protected] uτ is determined to be the value that best fits the mea- 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-L´opez 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 uni- versal 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 ve- locity and Reynolds shear stress profiles.
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