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Version 4
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Closed Form Solution of Plane-Parallel Turbulent Flow Along an Unbounded Plane Surface
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)
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. 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. Preprints 2021, 2021110008. 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: $u^+=\frac{1}{\kappa}\ln(2\kappa {y^+} +\sqrt{1+4\kappa^2{y^+} ^2})-\frac{2{y^+} }{1+\sqrt{1+4\kappa^2{y^+} ^2}}$.
Keywords
turbulent flow; Prandtl mixing length; Reynolds number; boundary layer
Subject
Physical Sciences, Fluids and Plasmas Physics
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.
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Commenter: Bohua Sun
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