The pitfalls of investigating rotational flows with the Euler equations

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.

A theoretical model for the growth of spherical bubbles by rectified diffusion

“…(3) This model, together with our recent research on strongly nonlinear analysis (Smith & Wang 2017, 2018, 2021, is the basis of our objective to simulate bubble growth over many millions of cycles of oscillation.…”

“…This new system of equations is important for the following reasons. These ordinary differential equations are straightforward to integrate numerically, whereas the well-established mathematical model has proved to be intractable. The spatial extent of the diffusion boundary layer in the liquid adjacent to the bubble interface is sufficiently small (), that the model applies to bubbles even in a dense cloud. This model, together with our recent research on strongly nonlinear analysis (Smith & Wang 2017, 2018, 2021), is the basis of our objective to simulate bubble growth over many millions of cycles of oscillation. …”

The role of surface tension on the growth of bubbles by rectified diffusion

“…The Peclet number typically has a value which is greater than one hundred thousand. Therefore the well-established physical model may be simplified using the latest techniques in perturbation theory [33] . This asymptotic analysis was undertaken in our previous article [9] .…”

A tractable mathematical model for rectified diffusion

“…liquid drops damped by viscosity (Smith 2010), the finite-amplitude travelling waves in two-dimensional plane Poiseuille flow (Smith & Wissink 2015), the finite-amplitude travelling and standing waves in two-dimensional Kolmogorov flow (Smith & Wissink 2018) and steady states and quasi-steady states in vortex dynamics (Smith & Wang 2021). In rectified diffusion, the time scales in the mass transport boundary layer adjacent to the bubble may only be systematically analysed with these techniques.…”

“…The analysis of the basic model for rectified diffusion (and more sophisticated models such as the inclusion of thermal effects) has been made possible by the latest developments in singular perturbation theory. In the last few years, these state-of-the-art techniques have led to new results in the strongly nonlinear analysis of the oscillations of spherical bubbles damped by viscosity (Smith & Wang 2017), the oscillations of spherical bubbles damped by acoustic radiation (Smith & Wang 2018), the axisymmetric oscillations of liquid drops damped by viscosity (Smith 2010), the finite-amplitude travelling waves in two-dimensional plane Poiseuille flow (Smith & Wissink 2015), the finite-amplitude travelling and standing waves in two-dimensional Kolmogorov flow (Smith & Wissink 2018) and steady states and quasi-steady states in vortex dynamics (Smith & Wang 2021). In rectified diffusion, the time scales in the mass transport boundary layer adjacent to the bubble may only be systematically analysed with these techniques.…”