The LOG-EXP formula of the wall 1 A single formula for the law of the wall and its application to wall-modelled 2 large-eddy simulation 1, 2 1, 3 1, 3, a) 3 Fengshun Zhang( Îz), Zhideng Zhou( h志{), Xiaolei Yang( hS雷), 2 4 and Huan Zhang( ") 1) 5 The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, 6 Chinese Academy of Sciences, Beijing 100190, China 2) 7 Department of Mechanics and Engineering Science, Lanzhou University, 8 Lanzhou 730000, China 3) 9 School of Engineering Sciences, University of Chinese Academy of Sciences, 10 Beijing 100049, China 11 (Dated: 20 April 2021) 12 In this work, we propose a single formula for the law of the wall, which is dubbed as 13 the logarithmic-exponential (LOG-EXP) formula, for predicting the mean velocity profile 14 in different regions near the wall. And then a feedforward neural network (FNN), whose 15 inputs and training data are based on this new formula, is trained for the wall-modelled 16 large-eddy simulation (WMLES) of turbulent channel flows. The direct numerical simula- 17 tion (DNS) data of turbulent channel flows is used to evaluate the performance of both the 18 formula and the FNN. Compared with the Werner-Wengle (WW) model for the WMLES, 19 a better performance of the FNN for the WMLES is observed for predicting the Reynolds 20 stresses. arXiv:2104.09054v1 [physics.flu-dyn] 19 Apr 2021 a)Electronic mail: [email protected] 1 21 I. INTRODUCTION 1,2 22 The law of the wall is one of the cornerstones in wall-bounded turbulent flows . Different 23 formulae have been proposed in the literature for the law of the wall. Roughly, they can be di- 24 vided into two groups, i.e., the piecewise formulae and the single formulae. In the early year, the 25 piecewise function was developed to describe the dynamics in different near-wall regions of the 26 inner layer (e.g., for turbulent channel flows, it is located in the range of 0 ≤ y ≤ 0:1d, where y 3–10 27 denotes the wall-normal direction and d is the half-height of the channel) , which includes the 1 28 viscous sublayer, the buffer layer and the logarithmic layer . The most widely used law of the wall 29 describes the viscous sublayer using the linear profile, i.e., U+ = y+; (1) + + p 30 where U = U=ut , y = y=dv with the friction velocity ut = tw=r (where tw is the wall shear 31 stress and r is the fluid density) and the viscous scale dv = n=ut (n is the kinematic viscosity of 32 the fluid), and the logarithmic layer using the logarithmic law, i.e., 1 U+ = lny+ + 5:0; (2) k 33 where k = 0:4 is the Kármán constant. A list of different piecewise formulae proposed in the 34 literature for the law of the wall is shown in Table I. There are two disadvantages of the piecewise 35 formulae: 1) the velocity in the buffer layer is not accurately described; 2) the velocity derivative 36 is discontinuous because of the piecewise nature of these formulae. To resolve these two issues, 37 different types of the law of the wall based on single formula have been proposed in the litera- 11–16 17,18 38 ture, which are often of two forms, i.e., the analytical form and the numerical form . The 12,14–16 39 analytical forms are usually complicated or do not have a sound prediction , while the nu- 17,18 40 merical forms have a better prediction of the velocity profile , but do not have a clear physical 41 meaning or difficult to use in practice. Different single formulae for the law of the wall proposed 4243 in the literature are shown in Table II and Table III. 44 In this work, we propose a new single formula named as the logarithmic-exponential (LOG- 45 EXP) formula shown as follows: + + + + 1 + − y − y U y = ln 1 + ky + A 1 − e B +C 1 − e D ; (3) k 46 where A = 11:630, B = 7:194, C = −4:472 and D = 2:766, validate the proposed formula us- 28 47 ing direct numerical simulation (DNS) data , the experimental data from other canonical wall 2 Authors Formulae Prandtl3 U+ = y+ for 0 ≤ y+ ≤ 11:5 Taylor4 U+ = 2:5ln(y+) + 5:5 for 11:5 ≤ y+ 8 > + + + > U = y , for 0 ≤ y < 5 <> Von Karman 5 U+ = 5ln(y+) − 3:05 for 5 ≤ y+ < 30 > > :> U+ = 2:5ln(y+) + 5:5 for 30 ≤ y+ 8 + h + + i−1 > + R y 2 + + −n2U y + + < U = 0 1 + n U y 1 − e dy , where n = 0:124 for 0 ≤ y < 26 Deissler 6 :> U+ = 2:78ln(y+) + 3:8, for 26 ≤ y+ 8 < U+ = 1:454tanh(0:0688y+) for 0 ≤ y+ < 27:5 Rannie 7 : U+ = 2:5ln(y+) + 5:5 for 27:5 ≤ y+ 8 + + + > U = y for 0 ≤ y < 5 > > <> U+ = Aln(y+) + B, Breuer & Rodi 8 − + > where A = k 1 ln(30E) − 5 =ln(6), B = 5 − Aln(5) for 5 ≤ y < 30 > > :> U+ = k−1 ln(Ey+), where E = 9:8 for 30 ≤ y+ 8 < U+ = y+ for 0 ≤ y+ < 11:81 Werner & Wengle 9 : U+ = A(y+)B , where A = 8:3;B = 1=7 for 11:81 ≤ y+ 8 > U+ = y+ for 0 ≤ y+ < y+ > C1 <> 10 + + B1 + 1=(1−B1) + + + Inagaki et al. U = A1 (y ) , where A1 = 2:7;B1 = 1=2;yC = A1 for yC ≤ y < yC > 1 1 2 > B 1=(B −B ) > U+ = A (y+) 2 , where A = 8:6;B = 1=7;y+ = (A =A ) 1 2 for y+ ≤ y+ : 2 2 2 C2 2 1 C2 TABLE I. Formulae for the law of the wall: piecewise function. 29–31 48 bounded flows and the classic law of the wall (i.e., Eq. (1) and Eq. (2)), and apply the new 49 single formula to wall-modelled large-eddy simulation (WMLES) via a feedforward neural net- 50 work (FNN) model for explicitly computing the wall shear stress using the wall-normal distance 51 and streamwise velocity. 52 The rest of this paper is organized as follows: in section II, the derivation process of the LOG- 53 EXP formula is presented; the proposed formula is validated using the DNS data of turbulent 54 channel flows and the experimental data of other canonical wall bounded flows in section III; then 55 it is applied WMLES via a feedforward neural network in section V; at last, conclusions are drawn 3 Authors Formulae + 11 y y+ + Reichardt + + − 11 −0:33y U = 2:5ln(1 + 0:4y ) + 7:8 1 − e − 11 e + + 2 + 3 + 4 Spalding12 + + −A kU + (kU ) (kU ) (kU ) f (U ) = U + e e − 1 − kU − 2! − 3! − 4! Rasmussen13 y+ = f (U+), where f (U+) = U+ + e−A 2cosh(kU+) − (kU+)2 − 2 ;A = 2:2;k = 0:4 h + i + 9:6 U+ = : −1 2y −8:15 + (y +10:6) − : + 5 424tan 16:7 log10 2 2 3 52 Musker14 (y+ −8:15y++86) n h y 2 y 3i h y 2 y io 2:44 P 6 d − 4 d + d 1 − d , P = 0:55 + kg P; y + y 1 y 2 1 y 3 y e ( d ) = f (U );g P; = (1 + 6P) − (1 + 4P) Dean15 d k d k d where f (U+)is given by Spalding or Rasmussen’s expressions listed above + + + Uinner = Uinner, 23 +Uinner, 25 + +2 + Uinner, 23 = 0:68285472ln y + 4:7673096y + 9545:9963 + 1:2408249arctan(0:010238083y+ + 0:024404056)+ Monkewitz, 1:2384572ln(y+ + 95:232690) − 11:930683 + +2 + Chauhan Uinner;25 = −0:50435126ln y − 7:8796955y + 78:389178 + and Nagib 16 4:7413546arctan(0:12612158y+ − 0:49689982) −2:7768771ln y+2 + 16:209175y+ + 933:16587 + 0:37625729arctan(0:033952353y+ + 0:27516982)+ 6:5624567ln(y+ + 13:670520) + 6:1128254 " #−1 r + 2 y+ 2 y 17 + R + − 26 + van Driest U = 0 2 1 + 1 + 0:64y 1 − e dy ¶P 3=2 + + sign( )(1−a) y +sign(tw)a ¶U = ¶x ¶y+ 1+ nt Duprat et al.18 n h ib + 3 2 nt + + 3=2 −y =(1+Aa ) n = ky a + y (1 − a) 1 − e , where b = 0:78;A = 17;a = 1 1=2 + + R y+ 1 1 2 s U (y ) = − 2 + 2 1 + 4l(s) 1 − ds 0 2l(s) 2l(s) Rt 19 + m Cantwell ky+ 1−e−(y =a) (y+) = , where k = 0:4092;a = 20:095;m = 1:621;b = 0:3195;n = 1:619: l + n1=n 1+ y bRt q q + 1 +2 1 + +2 + U = + 1 − 1 + 4Im + ln 2I + 1 + 4Im + d ; 2kIm k m l Rotta20 + + where Im = k(y − dl), Im = ut Im=n and dl = ut dl=n + 21,22 + dl = 5:0 and experimental data suggests dl = 7:0: TABLE II. Formulae for the law of the wall: single formula. 4 Authors Formulae h + + i + + + + 1 + + + + 2 3 + + + + 3 −3y =yc U = yc 1 − 1 + 2(y =yc ) + 2 (3 − px yc )(y =yc ) − 2 px yC (y =yc ) e + p 4 8 + + + + 6 5(h +h ) 23 1+px yc 1+(0:6(y =y )) − Nickels ln c + b 1 − e 1+5h3 ;where h = y=d; p+ = (n=rU3)=(dp=dx); 6ko 1+h6 x t + +3 +2 2 UT yc p px yc + yc − Rc = 0;Rc = n and UT = t(y = yc)=r 2 U+ = 1 y+ − a + 2y+2 2 = f +; = nm ;a = =h+; l arctanl 2l 2 ln 1 l , where l a a 2 1 24 Haritonidis n is the number of ejections of equal strength, f = 1=Dtb is the bursting frequency, −1 −1 m = kn (Dte=dtb) and Dte is the duration of the ejections.