Josephson-Anderson Relation and the Classical D’Alembert Paradox

Abstract

Generalizing prior work of P. W. Anderson and E. R. Huggins, we show that a "detailed Josephson-Anderson relation" holds for drag on a finite body held at rest in a classical incompressible fluid flowing with velocity V. The relation asserts an exact equality between the instantaneous power consumption by the drag, −F•V, and the vorticity flux across the potential mass current,Here Σij is the flux in the ith coordinate direction of the conserved jth component of vorticity and the line-integrals over are taken along streamlines of the potential flow solution u φ = ∇φ of the ideal Euler equation, carrying mass flux dJ = ρ u φ •dA. The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation (1/2) ijk Σ jk = ∂i h , where h = p+(1/2)|u| 2 is the generalized enthalpy or total pressure, extends Lighthill's theory of vorticity generation at solid walls into the interior of the flow. We use these results to explain drag on the body in terms of vortex dynamics, unifying the theories for classical fluids and for quantum superfluids. The results offer a new solution to the "D'Alembert paradox" at infinite Reynolds numbers and imply the necessary conditions for turbulent drag reduction.

Origin of enhanced skin friction at the onset of boundary-layer transition

“…This statement is consistent with our finding that the Lighthill–Panton vorticity source is the uniquely correct choice to be used as the Neumann boundary condition for the stochastic Cauchy invariant. On the other hand, Lyman's vorticity source is continuously extended into the interior of the flow by the anti-symmetric vorticity flux of Huggins (1994), and is thus related generally to pressure gradients and to energy dissipation by the Josephson–Anderson relation (Eyink 2008, 2021). The generalizations to curvilinear walls of Lighthill's wall vorticity source by Panton (1984) and by Lyman (1990) have each their own proper domains of applicability, which overlap, and one must be aware in any particular application which of the two definitions is appropriate.…”

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

“…See [42] for a mathematical treatment of Navier-Stokes solutions in such unbounded domains (and even when the solid boundary is non-smooth) and see [43] and references therein for discussion of the closely related problem of the rigid motion of the solid body B through an incompressible fluid filling the complement. We consider this particular situation because of a new mathematical approach to the d'Alembert paradox based on a Josephson-Anderson relation inspired by quantum superfluids [24], which will be the subject of a following paper [41] that builds upon our analysis here. However, our results in this paper apply with minor changes to other flows involving solid walls, including interior flows within bounding walls such as Poiseuille flows through pipes and channels.…”

“…Our next main theorem states that this spatial flux of momentum (both its components wall-parallel and wall-normal) matches onto the corresponding components of the limiting wall stress which were established in Theorem 1. Since those inviscid limits were defined as sectional distributions of the tangent and normal bundles, we must identify momentum flux (24) with similar sectional distributions. To accomplish this, we use the idea of extensions in Theorem 1 to define e.g.…”

“…for all ψ ∈ D((∂B) T , T * (∂B) T ). The righthand side is meaningful and defines a sectional distribution of the tangent bundle because of regularity (24) and linearity and continuity of Ext ∈ E T . Likewise, with Ext ∈ E N one can define Ext * (∇η h, • T + p ∇η h, ) ∈ D ((∂B) T , N (∂B) T ).…”

“…The methods of this paper can be applied to another fundamental cascade process in wall-bounded turbulence, which is the "inverse cascade" of vorticity away from the wall; e.g. see [23,24]. This topic will be discussed in detail in another work [25].…”