Inertial Momentum Dissipation for Viscosity Solutions of Euler Equations. I. Flow Around a Smooth Body

Abstract

We study the local balance of momentum for weak solutions of incompressible Euler equations obtained from the zero-viscosity limit in the presence of solid boundaries, taking as an example flow around a finite, smooth body. We show that both viscous skin friction and wall pressure exist in the inviscid limit as distributions on the body surface. We define a nonlinear spatial flux of momentum toward the wall for the Euler solution, and show that wall friction and pressure are obtained from this momentum flux in the limit of vanishing distance to the wall, for the wall-parallel and wall-normal components, respectively. We show furthermore that the skin friction describing anomalous momentum transfer to the wall will vanish if velocity and pressure are bounded in a neighborhood of the wall and if also the essential supremum of wall-normal velocity within a small distance of the wall vanishes with this distance (a precise form of the non-flow-through condition). In the latter case, all of the limiting drag arises from pressure forces acting on the body and the pressure at the body surface can be obtained as the limit approaching the wall of the interior pressure for the Euler solution. As one application of this result, we show that Lighthill's theory of vorticity generation at the wall is valid for the Euler solutions obtained in the inviscid limit. Further, in a companion work, we show that the Josephson-Anderson relation for the drag, recently derived for strong Navier-Stokes solutions, is valid for weak Euler solutions obtained in their inviscid limit.

Onsager's ‘ideal turbulence’ theory

“…I shall discuss the essential differences below. My analysis shall follow closely the work of Quan & Eyink (2022a), where momentum balance in the limit Re → ∞ is treated. It is also possible to extend this analysis to finite Reynolds numbers and some beginning steps in this direction have been taken by Eyink, Kumar & Quan (2022), following the ideas of Drivas & Eyink (2019) for energy cascade, but I consider here only the regime Re 1.…”

“…It is also possible to extend this analysis to finite Reynolds numbers and some beginning steps in this direction have been taken by Eyink, Kumar & Quan (2022), following the ideas of Drivas & Eyink (2019) for energy cascade, but I consider here only the regime Re 1. The work of Quan & Eyink (2022a) followed the same RG strategy as in prior works on the Onsager theory such as Duchon & Robert (2000), by considering the limit Re → ∞ for both the fine-grained description and the coarse-grained description with regularisation scales , h. Non-trivial consequences can be deduced simply by requiring that both descriptions lead to the same observable conclusions and exploiting the freedom to vary h, by taking h, → 0. The specific example analysed by Quan & Eyink (2022a) was external flow around a finite, smooth body B, as considered in the famous paradox of d'Alembert (1749,1768).…”

“…The work of Quan & Eyink (2022a) followed the same RG strategy as in prior works on the Onsager theory such as Duchon & Robert (2000), by considering the limit Re → ∞ for both the fine-grained description and the coarse-grained description with regularisation scales , h. Non-trivial consequences can be deduced simply by requiring that both descriptions lead to the same observable conclusions and exploiting the freedom to vary h, by taking h, → 0. The specific example analysed by Quan & Eyink (2022a) was external flow around a finite, smooth body B, as considered in the famous paradox of d'Alembert (1749,1768). As seen from the empirical results reviewed in § 4.1, this is probably the simplest flow which provides evidence for a strong dissipation anomaly.…”

“…As seen from the empirical results reviewed in § 4.1, this is probably the simplest flow which provides evidence for a strong dissipation anomaly. However, the analysis of Quan & Eyink (2022a) carries over straightforwardly to related more complex flows, such as turbulent flows through pipes with hydraulically rough (but mathematically smooth) walls. Since only mathematically smooth surfaces are considered, we are investigating the limit Re = UL/ν 1 as achieved by taking L very large (where L would be a length such as the diameter of the body or the radius of the pipe).…”

“…The analysis of Quan & Eyink (2022a) (with some pedagogical simplifications introduced by Eyink ( 2023)) begins with the fine-grained momentum balance/ Navier-Stokes equation smeared against a smooth space-time test function ϕ(x, t) :…”