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# Mean velocity, Reynolds shear stress, 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

Embassy Hotel, Rasal Ganj, Aligarh, 202001 India, and 12080 Kirkbrook Dr, Saratoga, California CA 95070 USA
prof_noor_afzal@yahoo.com

Abu Seena

Samsung C&T, Tower 2, 145, Pangyoyeok-ro, Bundang-gu, Seongnam-Si, Gyeonggi-do, 13530 Republic of Korea
seena.abu@samsung.com

1Corresponding author.

ASME 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, 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 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, yield 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 $u'/U_e$ with $u/U_e$ from structure of turbulent boundary layer is presented in inner, meso and outer layers. The shape factor in a turbulent boundary layer shows linear behavior with non-dimensional mesolayer length scale. It is shown that the Prandtl transposition theorem, connects the velocity of normal co-ordinate $y$ with offset a 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 $U_e$, remains independent of $a$ the offset of origin. But if skin friction log law is based on bulk average velocity $U_b$ then skin friction log law depends on $a$, the offset of origin. These predictions are supported by experimental and DNS data.

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