The pitfalls of investigating rotational flows with the Euler equations
Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a...
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Published in | Journal of fluid mechanics Vol. 927 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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Cambridge, UK
Cambridge University Press
25.11.2021
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Abstract | Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a real flow at large Reynolds number. These facts are often overlooked and, as a result, the Euler equations are used to simulate laminar rotational flows at large Reynolds number. Based on the Fredholm alternative, an asymptotic perturbation theory is described to establish secularity conditions determined by viscosity for an inviscid solution to approximate a real viscous fluid. Four important classical inviscid solutions are investigated using the theory with the following conclusions. The Stuart cats’ eyes and Mallier–Maslowe vortices are inconsistent with any real fluid at high Reynolds number; whereas Hill's spherical vortex is confirmed to be consistent with a steady state in the spherical core region and the Lamb–Chaplygin dipole is found to be consistent with a quasi-steady state in the circular core region. These solutions have been widely used for analysing the stability of vortex flows and wakes, and their interactions with shock waves or bubbles. Serendipitously, we have revealed an original exact solution of the Navier–Stokes equations which is time dependent, has non-zero nonlinear convective terms and is restricted to a finite domain with the decay rate depending on dipole radius. |
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AbstractList | Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a real flow at large Reynolds number. These facts are often overlooked and, as a result, the Euler equations are used to simulate laminar rotational flows at large Reynolds number. Based on the Fredholm alternative, an asymptotic perturbation theory is described to establish secularity conditions determined by viscosity for an inviscid solution to approximate a real viscous fluid. Four important classical inviscid solutions are investigated using the theory with the following conclusions. The Stuart cats’ eyes and Mallier–Maslowe vortices are inconsistent with any real fluid at high Reynolds number; whereas Hill's spherical vortex is confirmed to be consistent with a steady state in the spherical core region and the Lamb–Chaplygin dipole is found to be consistent with a quasi-steady state in the circular core region. These solutions have been widely used for analysing the stability of vortex flows and wakes, and their interactions with shock waves or bubbles. Serendipitously, we have revealed an original exact solution of the Navier–Stokes equations which is time dependent, has non-zero nonlinear convective terms and is restricted to a finite domain with the decay rate depending on dipole radius. |
ArticleNumber | A42 |
Author | Smith, Warren R. Wang, Qianxi |
Author_xml | – sequence: 1 givenname: Warren R. orcidid: 0000-0002-0778-3226 surname: Smith fullname: Smith, Warren R. email: W.Smith@bham.ac.uk organization: 1School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK – sequence: 2 givenname: Qianxi orcidid: 0000-0002-0664-5913 surname: Wang fullname: Wang, Qianxi email: W.Smith@bham.ac.uk organization: 1School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK |
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Mat. Mekh. contributor: fullname: Kuzmak – volume: 399 start-page: 25 year: 1985 ident: S0022112021008053_ref29 article-title: On the development of large-sized short-scaled disturbances in boundary layers publication-title: Proc. R. Soc. Lond. A doi: 10.1098/rspa.1985.0046 contributor: fullname: Smith |
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Snippet | Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number... |
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SubjectTerms | Decay rate Dipoles Euler-Lagrange equation Eulers equations Exact solutions Fluid dynamics Fluid flow Fluid mechanics High Reynolds number JFM Papers Laminar flow Mathematical analysis Navier-Stokes equations Ordinary differential equations Perturbation theory Quasi-steady states Reynolds number Shock waves Stability Stability analysis Steady state Time dependence Viscosity Viscous fluids Vortices Wakes |
Title | The pitfalls of investigating rotational flows with the Euler equations |
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