Generalizing the 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 $\mathbf{V}$. The relation asserts an exact equality between the instantaneous power consumption by the drag $-\mathbf{F}·\mathbf{V}$ and the vorticity flux across the potential mass current $-(1/2)\int dJ\int {\epsilon }_{ijk}{\mathrm{\Sigma }}_{ij}d{\ell }_k$. Here, ${\mathrm{\Sigma }}_{ij}$ is the flux in the $i$th coordinate direction of the conserved $j$th component of vorticity, and the line integrals over $\mathbf{\ell }$ are taken along streamlines of the potential-flow solution ${\mathbf{u}}_{\varphi }=\mathbf{\nabla }\varphi$ of the ideal Euler equation, carrying mass flux $dJ=\rho {\mathbf{u}}_{\varphi }·d\mathbf{A}$. Drag and dissipation are thus associated with the motion of vorticity relative to this background ideal potential flow solving Euler’s equation. The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation $(1/2){\epsilon }_{ijk}⟨{\mathrm{\Sigma }}_{jk}⟩={\partial }_i⟨h⟩$, where $h=p+(1/2)|\mathbf{u}{|}^2$ 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, provide an explanation for a long-standing puzzle about the experimental conditions required for anomalous turbulent energy dissipation, and imply the necessary and sufficient conditions for turbulent drag reduction.

中文翻译：

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约瑟夫森-安德森关系和古典达朗贝尔悖论

概括 PW Anderson 和 ER Huggins 的先前工作，我们表明“详细的约瑟夫森 - 安德森关系”适用于在经典不可压缩流体中保持静止的有限物体的阻力 $\mathbf{伏}$. 该关系断言阻力的瞬时功耗之间完全相等$-\mathbf{F}·\mathbf{伏}$ 和穿过潜在质量流的涡流 $-(1/2)\int dJ\int {\epsilon }_{\mathrm{一世}j克}{\mathrm{\Sigma }}_{\mathrm{一世}j}d{\ell }_{克}$. 这里，${\mathrm{\Sigma }}_{\mathrm{一世}j}$ 是通量 $\mathrm{一世}$守恒的第 th 坐标方向 $j$涡量的第 th 分量，以及线积分 $\mathbf{\ell }$ 沿着势流解决方案的流线 ${\mathbf{你}}_{\phi }=\mathbf{\nabla }\phi$ 携带质量通量的理想欧拉方程 $dJ=\rho {\mathbf{你}}_{\phi }·d\mathbf{一种}$. 因此，阻力和耗散与求解欧拉方程的这个背景理想势流相关的涡量运动有关。结果概括了 MJ Lighthill 关于流过物体的理论，特别是稳态关系$(1/2){\epsilon }_{\mathrm{一世}j克}⟨{\mathrm{\Sigma }}_{j克}⟩={\partial }_{\mathrm{一世}}⟨H⟩$， 在哪里 $H=磷+(1/2)|\mathbf{你}{|}^2$是广义焓或总压力，将莱特希尔在固体壁上产生涡量的理论扩展到流动的内部。我们使用这些结果从涡流动力学的角度解释对身体的阻力，统一经典流体和量子超流体的理论。结果为无限雷诺数下的“d'Alembert悖论”提供了新的解决方案，为关于异常湍流能量耗散所需实验条件的长期难题提供了解释，并暗示了湍流减阻的充分必要条件.