Josephson-Anderson Relation and the Classical D’Alembert Paradox

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.

Popular Summary

Fluid drag is very familiar, slowing down a ball tossed through the air or a diver slicing through water. But it is an old puzzle why resistance remains when bodies are large or move fast because then the fluid viscosity becomes negligible. The d’Alembert paradox is that a fluid without viscosity should exert no drag, in contradiction to experience. We have solved this puzzle.

Our inspiration comes from other physical systems where drag appears unexpectedly: quantum superfluids. These fluids have no viscosity, but above some critical speed, drag appears abruptly. In the 1960s, physicists Josephson and Anderson realized that drag in superfluids and superconductors is due to the appearance of quantized vortex lines. They showed that motion of vortices across the flow produces a drop in pressure or voltage and effective drag. Thus, drag can be eliminated by pinning the vortices, so that resistanceless flow is restored. We have derived the analogous Josephson-Anderson relation for classical fluids.

Our new result shows that the underlying mechanism of drag in classical fluids and in quantum superfluids is the same. Although vorticity or swirling motion is not quantized in water and air, drag is due to motion of vorticity across the flow. This effect can persist for large objects or high speeds, solving the old puzzle. Conversely, drag reduction, the dream of engineers, can be achieved only by lessening this cross-stream vorticity flux.

Technologies that reduce drag, such as dilute polymer additives in a fluid, can be better understood by our analysis.

Figure 1

Dissipative vortex motions in superfluid channel flow according to the Josephson-Anderson relation. Solid black dots denote quantized vortices pointing into the plane and moving across flow lines of the background superfluid velocity ${\mathbf{u}}_{\varphi }$ with a transverse vortex velocity ${\mathbf{v}}_V$, which may arise either from vortex-induced motions or from nonideal effects. When a single vortex crosses the entire channel width, the superfluid phase difference $\mathrm{\Delta }\theta$ between outflow and inflow points changes by $2\pi$, a so-called $2\pi$ phase-slip event.

Figure 2

Flow through a channel $\mathrm{\Omega }$ with inflow surface $S_{\mathrm{in}}$, outflow surface $S_{\mathrm{out}}$, and channel walls $S_w$.

Figure 3

Flow around a finite body $B$ in an unbounded region $\mathrm{\Omega }$ filled with fluid moving at a velocity $\mathbf{V}$ at far distances.

Figure 4

Streamlines of the ideal flow around a sphere.

Figure 5

Schematic of azimuthal vorticity generation on the surface of the sphere. Following the convention of Huggins [16], we use white circles to denote vorticity out of the plane (${\omega }_n>0$) and black circles to denote vorticity into the plane (${\omega }_n<0$). The mean normal (azimuthal) vorticity in the upper half plane $U$ is negative and the mean normal (antiazimuthal) vorticity in the lower half plane $L$ is positive, implying a pole-to-pole asymmetry in the pressure distribution on the surface $S$ of the sphere. The pressure drop along the surface of the sphere from $F$ to $B$ is exactly compensated by the pressure rise from $B$ to infinity along the direction of the positive streamwise axis $X$.

Figure 6

Rough illustration of the drag mechanism by generation and outward flux of vortex loops with negative azimuthal vorticity. The rings are generated by pressure drop along the surface. As the rings expand outward, they enclose greater mass flux $J$ and subtract proportionate energy from the potential flow. Although the rings initially loop around the streamwise axis $X$, they drift across in time. This implies a flux of azimuthal vorticity across $X$ and allows the pressure to recover far downstream from its drop around the sphere.