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Volume 13, Issue 3
Singularity of Navier-Stokes Equations Leading to Turbulence

Hua-Shu Dou

Adv. Appl. Math. Mech., 13 (2021), pp. 527-553.

Published online: 2020-12

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  • Abstract

Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in pressure driven flows, velocity discontinuity occurs at this point. Meanwhile, pressure pulse is produced at the discontinuity due to conservation of the total mechanical energy. This discontinuity makes the Navier-Stokes equations be singular and causes the flow to become indefinite. The analytical results show that the singularity of the Navier-Stokes equations is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence. Since the velocity is not differentiable at the singularity, there exist no smooth and physically reasonable solutions of Navier-Stokes equations at high Reynolds number (beyond laminar flow). The negative spike of velocity and the pulse of pressure due to discontinuity have obtained agreement with experiments and simulations in literature qualitatively.

  • Keywords

Navier-Stokes equations, singularity, discontinuity, total mechanical energy, turbulence.

  • AMS Subject Headings

76D03, 76D05, 76E05, 76F06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-13-527, author = {Dou , Hua-Shu}, title = {Singularity of Navier-Stokes Equations Leading to Turbulence}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {13}, number = {3}, pages = {527--553}, abstract = {

Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in pressure driven flows, velocity discontinuity occurs at this point. Meanwhile, pressure pulse is produced at the discontinuity due to conservation of the total mechanical energy. This discontinuity makes the Navier-Stokes equations be singular and causes the flow to become indefinite. The analytical results show that the singularity of the Navier-Stokes equations is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence. Since the velocity is not differentiable at the singularity, there exist no smooth and physically reasonable solutions of Navier-Stokes equations at high Reynolds number (beyond laminar flow). The negative spike of velocity and the pulse of pressure due to discontinuity have obtained agreement with experiments and simulations in literature qualitatively.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0063}, url = {http://global-sci.org/intro/article_detail/aamm/18496.html} }
TY - JOUR T1 - Singularity of Navier-Stokes Equations Leading to Turbulence AU - Dou , Hua-Shu JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 527 EP - 553 PY - 2020 DA - 2020/12 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0063 UR - https://global-sci.org/intro/article_detail/aamm/18496.html KW - Navier-Stokes equations, singularity, discontinuity, total mechanical energy, turbulence. AB -

Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in pressure driven flows, velocity discontinuity occurs at this point. Meanwhile, pressure pulse is produced at the discontinuity due to conservation of the total mechanical energy. This discontinuity makes the Navier-Stokes equations be singular and causes the flow to become indefinite. The analytical results show that the singularity of the Navier-Stokes equations is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence. Since the velocity is not differentiable at the singularity, there exist no smooth and physically reasonable solutions of Navier-Stokes equations at high Reynolds number (beyond laminar flow). The negative spike of velocity and the pulse of pressure due to discontinuity have obtained agreement with experiments and simulations in literature qualitatively.

Hua-Shu Dou. (1970). Singularity of Navier-Stokes Equations Leading to Turbulence. Advances in Applied Mathematics and Mechanics. 13 (3). 527-553. doi:10.4208/aamm.OA-2020-0063
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