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.
Highlights
No general method exists to establish whether or not an exact inviscid steady state solution approximates the behaviour of a real viscous fluid at high Reynolds number
Viscous effects always represent a singular perturbation for rotational flows even in the absence of interior/boundary layers
An asymptotic technique has been introduced to establish whether or not an exact inviscid steady state solution approximates the behaviour of a real viscous fluid at high Reynolds number
Summary
We will describe the mathematical model for the Lamb–Chaplygin dipole. In order to avoid an excessive number of suffices, the same notation as in §§ 2 and 3 will be adopted for a number of the quantities; these are defined anew . The system of dimensionless equations to be studied are introduced. Two-dimensional Navier–Stokes equations for incompressible Newtonian fluids in plane polar coordinates. In which r < R is the radial coordinate, R is the radius of the circle, θ is the azimuthal coordinate, t is time, u is the radial velocity component, v is the azimuthal velocity component, p is the pressure, ω is the vorticity given by ω. The Lamb–Chaplygin dipole corresponds to C = −2U, where U is the far-field velocity of the surrounding flow
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