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Research Papers: Fundamental Issues and Canonical Flows

# Mean Velocity, Reynolds Shear Stress, and Fluctuations of Velocity and Pressure Due to Log Laws in a Turbulent Boundary Layer and Origin Offset by Prandtl Transposition Theorem

[+] Author and Article Information
Noor Afzal

12080 Kirkbrook Drive,
Saratoga, CA 95070;
Embassy Hotel,
Rasal Ganj,
Aligarh 202001, India

Abu Seena

Samsung C&T,
Tower 2, 145, Pangyoyeok-ro, Bundang-gu,
Seongnam-Si 13530, Gyeonggi-do,
South Korea

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received October 1, 2015; final manuscript received February 6, 2018; published online March 16, 2018. Assoc. Editor: Elias Balaras.

J. Fluids Eng 140(7), 071204 (Mar 16, 2018) (23 pages) Paper No: FE-15-1708; doi: 10.1115/1.4039259 History: Received October 01, 2015; Revised February 06, 2018

## Abstract

The maxima of Reynolds shear stress and turbulent burst mean period time are crucial points in the intermediate region (termed as mesolayer) for large Reynolds numbers. The three layers (inner, meso, and outer) in a turbulent boundary layer have been analyzed from open equations of turbulent motion, independent of any closure model like eddy viscosity or mixing length, etc. Little above (or below not considered here) the critical point, the matching of mesolayer predicts the log law velocity, peak of Reynolds shear stress domain, and turbulent burst time period. The instantaneous velocity vector after subtraction of mean velocity vector yields the velocity fluctuation vector, also governed by log law. The static pressure fluctuation $p′$ also predicts log laws in the inner, outer, and mesolayer. The relationship between $u′/Ue$ with u/Ue from structure of turbulent boundary layer is presented in inner, meso, and outer layers. The turbulent bursting time period has been shown to scale with the mesolayer time scale; and Taylor micro time scale; both have been shown to be equivalent in the mesolayer. The shape factor in a turbulent boundary layer shows linear behavior with nondimensional mesolayer length scale. It is shown that the Prandtl transposition (PT) theorem connects the velocity of normal coordinate y with s offset to y + a, then the turbulent velocity profile vector and pressure fluctuation log laws are altered; but skin friction log law, based on outer velocity Ue, remains independent of a the offset of origin. But if skin friction log law is based on bulk average velocity Ub, then skin friction log law depends on a, the offset of origin. These predictions are supported by experimental and direct numerical simulation (DNS) data.

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## Figures

Fig. 3

The turbulent boundary layer DNS data from group of Wu [37] in 2010: Reynolds shear stress distribution (a) τ+ versus ζ in the mesolayer, (b) self-similar (τ+−1)Rτ1/2 versus ζ in the mesolayer; predictions in – – – outer mesolayer and —– mesolayer domain. (c) Reynolds shear stress gradient ζ∂τ+∂ζ versus ζ in the mesolayer variables and (d) Reynolds shear stress gradient y+∂τ+∂y+ versus y+ in wall variables. The ratio of mean velocity gradient to stress gradient (e) in the mesolayer variables S=(∂2u+/∂ζ2)/(∂τ+/∂ζ) versus ζ and (f) in wall variables S=(∂2u+/∂y+2)/(∂τ+/∂y+) versus y+.

Fig. 4

The turbulent boundary layer DNS data from group of Jimenez et al. [34,35] during 2010 and 2013: (a) the streamwise fractional velocity defect u+ − Um+ and velocity fluctuations u′+ with mesolayer variable ζ. (c) The first derivative ζ∂u+/ in the mesolayer variable ζ, (b) y+∂u+/∂y+ in wall variable y+, and (d) the velocity derivative (y+/u+)∂u+/∂y+ for power law in wall variable y+. The second derivative of velocity (e) M2 = ζ22u+/∂ζ2 with ζ in the mesolayer variables and (f) M2=y+2∂2u+/∂y+2 with y+ in wall variables.

Fig. 6

The turbulent boundary layer DNS data of Wu [37] during 2010: (a) the streamwise fractional velocity defect u+ − Um+ in the mesolayer variable ζ. (b) The first derivative ζ∂u+/ in the mesolayer variable ζ, (c) y+∂u+/∂y+ in wall variable y+, and (d) the velocity derivative (y+/u+)∂u+/∂y+ for power law in wall variable y+. The second derivative of velocity (e) M2=ζ2∂2u+/∂ζ2 with ζ in the mesolayer variables and (f) M2=y+2∂2u+/∂y+2 with y+ in wall variables.

Fig. 7

The turbulent boundary layer DNS data of Wu [37] during 2010: mean velocity u+ in streamwise direction in mesolayer variable ζ

Fig. 1

The turbulent boundary layer DNS data from Jimenez et al. [34,35] during 2010 and 2013: Reynolds shear stress distribution (a) τ+ versus ζ in the mesolayer, (b) self-similar (τ+−1)Rτ1/2 versus ζ in the mesolayer; predictions in – – – outer mesolayer and —– mesolayer domain. (c) Reynolds shear stress gradient ζ∂τ+∂ζ versus ζ in the mesolayer variables and (d) Reynolds shear stress gradient y+∂τ+∂y+ versus y+ in wall variables. The ratio of mean velocity gradient to stress gradient (e) in the mesolayer variables S=(∂2u+/∂ζ2)/(∂τ+/∂ζ) versus ζ and (f) in wall variables S=(∂2u+/∂y+2)/(∂τ+/∂y+) versus y+.

Fig. 5

The turbulent boundary layer DNS data of Schlatter and Orlu [36] during 2010: (a) the streamwise fractional velocity defect u+ − Um+ and velocity fluctuations u′+ with mesolayer variable ζ. (c) The first derivative ζ∂u+/ in the mesolayer variable ζ, (b) y+∂u+/∂y+ in wall variable y+, and (d) the velocity derivative (y+/u+)∂u+/∂y+ for power law in wall variable y+. The second derivative of velocity (e) M2=ζ2∂2u+/∂ζ2 with ζ in the mesolayer variables and (f) M2=y+2∂2u+/∂y+2 with y+ in wall variables.

Fig. 8

The turbulent boundary layer DNS data of Jimenez et al. [34,35] during 2010 and 2013: (a) ⋯⋯u′/Ue=0.171u/Ue+0.016, marked 1, is the mesolayer (intermediate layer); – – – u′/Ue=0.29u/Ue−0.26(u/Ue)2, marked 2, is the outer layer; and —- u′/Ue=0.4u/Ue, marked 3, is lower part of inner wall layer. (b) Velocity fluctuation u′/u with mean velocity distribution u/Ue in outer layer ⋯⋯u′/u=0.28−0.26u/Ue.

Fig. 9

The turbulent boundary layer DNS data of Schlatter and Orlu [36] during 2010: (a) ⋯⋯u′/Ue=0.171u/Ue+0.016, marked 1, is the mesolayer (intermediate layer); – – – u′/Ue=0.29u/Ue−0.26(u/Ue)2, marked 2, is the outer layer; and —- u′/Ue=0.4u/Ue, marked 3, is lower part of inner wall layer. (b) Velocity fluctuations u′/u with mean velocity distribution u/Ue in outer layer ⋯⋯u′/u=0.28−0.26u/Ue.

Fig. 2

The turbulent boundary layer DNS data of Schlatter and Orlu [36] during 2010: Reynolds shear stress distribution (a) τ+ versus ζ in the mesolayer, (b) self-similar (τ+−1)Rτ1/2 versus ζ in the mesolayer; predictions in – – – outer mesolayer and —– mesolayer domain. (c) Reynolds shear stress gradient ζ∂τ+∂ζ versus ζ in the mesolayer variables and (d) Reynolds shear stress gradient y+∂τ+∂y+ versus y+ in wall variables. The ratio of mean velocity gradient to stress gradient (e) in the mesolayer variables S=(∂2u+/∂ζ2)/(∂τ+/∂ζ) versus ζ and (f) in wall variables S=(∂2u+/∂y+2)/(∂τ+/∂y+) versus y+.

Fig. 12

Wall pressure fluctuation p′w+ and friction Reynolds number Rτ from DNS data of Jimenez et al. [34,35] during 2010and 2013 and Schlatter and Orlu [36] during 2010 in turbulent boundary layer. Our proposed prediction —– p′w+=−1.06 log Rτ+2.2.

Fig. 13

The Taylor microlength scale in mesolayer λ/δm versus ζ for various values of Rτ from DNS data of Segalini et al. [43] during 2011 in turbulent boundary layer. ——- log region λ/δm=1.42  log10ζ+0.7.

Fig. 15

Comparison of Reynolds shear stress maxima in turbulent boundary layer over smooth surface from DNS data of Schlatter and Orlu, Jimenez et al., Wu, Wu and Moin, Lee and Sung, and Spalart. (a) Maxima of Reynolds shear stress location in mesolayer variable ζm versus Rτ, (b) maxima of Reynolds shear stress location in wall variable y+m versus Rτ, and (c) magnitude of maximum of Reynolds shear stress τm versus Rτ.

Fig. 14

Velocity ratio Up/Uc at the peak of Reynolds shear stress with Rτ in mesolayer from DNS data of Schlatter and Orlu [36] during 2010 and Wu [37] during 2010 in turbulent boundary layer

Fig. 10

The turbulent boundary layer DNS data of Jimenez et al. [34,35] during 2010 and 2013: pressure fluctuation profiles in (a) inner variables p′+−p′w+ and distance variable y+, (b) mesolayer variables p′+−p′w+/2 and distance variable ζ=y+Y, (c) outer variables p′ and distance variable Y, and (d) the functional derivate ζdp′+/dζ against ζ in mesolayer

Fig. 11

The turbulent boundary layer DNS data of Schlatter and Orlu [36] during 2010: pressure fluctuation profiles in (a) inner variables p′+−p′w+ and distance variable y+, (b) mesolayer variables p′+−p′w+/2 and distance variable ζ=y+Y, (c) outer variables p′ and distance variable Y, and (d) the functional derivate ζdp′+/dζ against ζ, in mesolayer

Fig. 19

The scaling of turbulent bursting time scale, in the mesolayer as Taylor micro time scale λt in turbulent boundary layer: (a) bursting period Tb/λt versus ζ(=y+Rτ−1/2), — Tb/λt ∼ 18 prediction in mesolayer overlap region. (b) Burst time mode Tbm/λt versus ζ(=y+Rτ−1/2), — Tbm/λt = 4.5 prediction in mesolayer overlap region; (c) burst time mode △Tb/λt versus ζ(=y+Rτ−1/2), – △Tb/λt ∼ 5 prediction in mesolayer overlap region; and (d) burst time mode △Tbm/λt versus ζ(=y+Rτ−1/2), – △Tbm/λt ∼ 1 prediction in mesolayer overlap region. Data is from Metzger et al. [47].

Fig. 20

Our prediction of shape factor H in linear mesolayer variable Rτ−1/2 shows universal relation from extensive experimental and DNS data. (a) The prediction (94) of shape factor H=1.25+4.6 Rτ−1/2 supports data for Rτ−1/2≤0.082. (b) The shape factor H data in conventional variables Rτ from relation (94) hold for Rτ ≥ 200.

Fig. 21

Our prediction of shape factor H in linear mesolayer variable Rθ−1/2 shows universal relation supported by experimental and DNS data. (a) The prediction (95) of shape factor H=1.28+5.7 Rθ−1/2 supports data for Rθ−1/2≤0.04. (b) The shape factor H data in conventional variables Rθ with present relation (95) holds for Rθ ≥ 500, and earlier prediction of Monkewitz et al. [30] during 2007 from six terms of series (96) in lnnRθ.

Fig. 16

Comparison of time period of turbulent bursts scaling in the mesolayer Tm=Tb(uτ3/νδ)1/2 versus Rτ due to Afzal [9] in 1984 with experimental data from various sources in turbulent boundary layer over a smooth wall

Fig. 17

The occurrence of VITA events in turbulent flow over a fully smooth surface as a function of the averaging time: n* = ntm denotes the frequency normalized with the inverse of tm and T* = T/tm the averaging time window normalized by tm=(νδ/uτ3)1/2 due to Afzal [9] during 1984: data of Osterlund et al. [46] during 2003 in turbulent boundary layer

Fig. 18

The scaling of turbulent bursting in the mesolayer ofAfzal [9] during 1984 in turbulent boundary layer: (a) burstingperiod Tm(ζ)=(uτ3/νδ)1/2Tb(y+) versus ζ=y+Rτ−1/2, prediction Tm = 2 in the mesolayer domain. (b) TmM(ζ)=(uτ3/νδ)1/2Tmb(y+) versus ζ=y+Rτ−1/2, (c) burst time mode △Tm(ζ)=(uτ3/νδ)1/2△Tb(y+) versus ζ=y+Rτ−1/2, prediction △Tm = 0.55 in the mesolayer domain 1.5 < ζ < 8, and (d) burst time mode △TmM(ζ)=(uτ3/νδ)1/2△Tmb(y+) versus ζ=y+Rτ−1/2. Data is from Metzger et al. [47].

Fig. 22

Our prediction of shape factor H in linear mesolayer variable Rδ*−1/2 shows universal relation supported by experimental and DNS data. (a) The prediction (97) of shape factor H=1.28+5 Rδ*−1/2 supports data for Rδ*−1/2≤0.032. (b) The shape factor H data in conventional variables Rδ* by present relation (97) hold for Rδ*≥800, and Monkewitz et al. [30] during 2007 the six terms of series (98) in lnnRδ*.

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