The limit of zero viscosity
We would like to point out that none of the flows we have described are anything like the potential flow solution we found in the preceding chapter. This is, at first sight, quite surprising. After all, $\ReynoldsR$ is proportional to $1/\eta$. So $\eta$ going to zero is equivalent to $\ReynoldsR$ going to infinity. And if we take the limit of large $\ReynoldsR$ in Eq. (41.23), we get rid of the right-hand side and get just the equations of the last chapter. Yet, you would find it hard to believe that the highly turbulent flow at $\ReynoldsR=10^7$ was approaching the smooth flow computed from the equations of “dry” water. How can it be that as we approach $\ReynoldsR=\infty$, the flow described by Eq. (41.23) gives a completely different solution from the one we obtained taking $\eta=0$ to start out with? The answer is very interesting. Note that the right-hand term of Eq. (41.23) has $1/\ReynoldsR$ times a second derivative. It is a higher derivative than any other derivative in the equation. What happens is that although the coefficient $1/\ReynoldsR$ is small, there are very rapid variations of $\FLPOmega$ in the space near the surface. These rapid variations compensate for the small coefficient, and the product does not go to zero with increasing $\ReynoldsR$. The solutions do not approach the limiting case as the coefficient of $\nabla^2\FLPOmega$ goes to zero. You may be wondering, “What is the fine-grain turbulence and how does it maintain itself? How can the vorticity which is made somewhere at the edge of the cylinder generate so much noise in the background?” The answer is again interesting. Vorticity has a tendency to amplify itself. If we forget for a moment about the diffusion of vorticity which causes a loss, the laws of flow say (as we have seen) that the vortex lines are carried along with the fluid, at the velocity $\FLPv$. We can imagine a certain number of lines of $\FLPOmega$ which are being distorted and twisted by the complicated flow pattern of $\FLPv$. This pulls the lines closer together and mixes them all up. Lines that were simple before will get knotted and pulled close together. They will be longer and tighter together. The strength of the vorticity will increase and its irregularities—the pluses and minuses—will, in general, increase. So the magnitude of vorticity in three dimensions increases as we twist the fluid about. You might well ask, “When is the potential flow a satisfactory theory at all?” In the first place, it is satisfactory outside the turbulent region where the vorticity has not entered appreciably by diffusion. By making special streamlined bodies, we can keep the turbulent region as small as possible; the flow around airplane wings—which are carefully designed—is almost entirely true potential flow.