Version 1
: Received: 30 October 2021 / Approved: 1 November 2021 / Online: 1 November 2021 (11:28:02 CET)
Version 2
: Received: 15 November 2021 / Approved: 15 November 2021 / Online: 15 November 2021 (13:33:20 CET)
Version 3
: Received: 5 January 2022 / Approved: 6 January 2022 / Online: 6 January 2022 (11:26:17 CET)
Version 4
: Received: 7 June 2022 / Approved: 8 June 2022 / Online: 8 June 2022 (12:27:48 CEST)
How to cite:
Sun, B. Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface. Preprints2021, 2021110008. https://doi.org/10.20944/preprints202111.0008.v4
Sun, B. Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface. Preprints 2021, 2021110008. https://doi.org/10.20944/preprints202111.0008.v4
Sun, B. Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface. Preprints2021, 2021110008. https://doi.org/10.20944/preprints202111.0008.v4
APA Style
Sun, B. (2022). Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface. Preprints. https://doi.org/10.20944/preprints202111.0008.v4
Chicago/Turabian Style
Sun, B. 2022 "Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface" Preprints. https://doi.org/10.20944/preprints202111.0008.v4
Abstract
A century-old scientific conundrum is solved in this paper. The Prandtl mixing length modelled plane boundary turbulent flow is described by: $\frac{du^+}{d{y^+} }+\kappa^2{y^+} ^2\left( \frac{du^+}{d{y^+} }\right)^2=1$, together with boundary condition $ {y^+} =0:\, u^+=0$. Only an approximate solution to this nonlinear ordinary differential equation (ODE) has been sought so far, however, the exact solution to this ODE has not been obtained. By introducing a transformation, $2\kappa y^+=\sinh \xi$, I successfully find the exact solution of the ODE as follows: $u^+=\frac{1}{\kappa}\ln(2\kappa {y^+} +\sqrt{1+4\kappa^2{y^+} ^2})-\frac{2{y^+} }{1+\sqrt{1+4\kappa^2{y^+} ^2}}$.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Bohua Sun
Commenter's Conflict of Interests: Author