Flow past a circular cylinder
Let’s go back to the problem of low-speed (nearly incompressible) flow over the cylinder. We will give a qualitative description of the flow of a real fluid. There are many things we might want to know about such a flow—for instance, what is the drag force on the cylinder? The drag force on a cylinder is plotted in Fig. 41–4 as a function of $\ReynoldsR$—which is proportional to the air speed $V$ if everything else is held fixed. What is actually plotted is the so-called drag coefficient $C_D$, which is a dimensionless number equal to the force divided by $\tfrac{1}{2}\rho V^2Dl$, where $D$ is the diameter, $l$ is the length of the cylinder, and $\rho$ is the density of the liquid: \begin{equation*} C_D=\frac{F}{\tfrac{1}{2}\rho V^2Dl}. \end{equation*}  The coefficient of drag varies in a rather complicated way, giving us a pre-hint that something rather interesting and complicated is happening in the flow. We will now describe the nature of flow for the different ranges of the Reynolds number. First, when the Reynolds number is very small, the flow is quite steady; that is, the velocity is constant at any place, and the flow goes around the cylinder. The actual distribution of the flow lines is, however, not like it is in potential flow. They are solutions of a somewhat different equation. When the velocity is very low or, what is equivalent, when the viscosity is very high so the stuff is like honey, then the inertial terms are negligible and the flow is described by the equation \begin{equation*} \nabla^2\FLPOmega=\FLPzero. \end{equation*}  This equation was first solved by Stokes. He also solved the same problem for a sphere. If you have a small sphere moving under such conditions of low Reynolds number, the force needed to drag it is equal to $6\pi\eta aV$, where $a$ is the radius of the sphere and $V$ is its velocity. This is a very useful formula because it tells the speed at which tiny grains of dirt (or other particles which can be approximated as spheres) move through a fluid under a given force—as, for instance, in a centrifuge, or in sedimentation, or diffusion. In the low Reynolds number region—for $\ReynoldsR$ less than $1$—the lines of $\FLPv$ around a cylinder are as drawn in Fig. 41–5. If we now increase the fluid speed to get a Reynolds number somewhat greater than $1$, we find that the flow is different. There is a circulation behind the sphere, as shown in Fig. 41–6(b). It is still an open question as to whether there is always a circulation there even at the smallest Reynolds number or whether things suddenly change at a certain Reynolds number. It used to be thought that the circulation grew continuously. But it is now thought that it appears suddenly, and it is certain that the circulation increases with $\ReynoldsR$. In any case, there is a different character to the flow for $\ReynoldsR$ in the region from about $10$ to $30$. There is a pair of vortices behind the cylinder. The flow changes again by the time we get to a number of $40$ or so. There is suddenly a complete change in the character of the motion. What happens is that one of the vortices behind the cylinder gets so long that it breaks off and travels downstream with the fluid. Then the fluid curls around behind the cylinder and makes a new vortex. The vortices peel off alternately on each side, so an instantaneous view of the flow looks roughly as sketched in Fig. 41–6(c). The stream of vortices is called a “Kármán vortex street.”  They always appear for $\ReynoldsR>40$. We show a photograph of such a flow in Fig. 41–7. The difference between the two flows in Fig. 41–6(c) and 41–6(b) or 41–6(a) is almost a complete difference in regime. In Fig. 41–6(a) or (b), the velocity is constant, whereas in Fig. 41–6(c), the velocity at any point varies with time. There is no steady solution above $\ReynoldsR=40$—which we have marked on Fig. 41–4 by a dashed line. For these higher Reynolds numbers, the flow varies with time but in a regular, cyclic fashion. We can get a physical idea of how these vortices are produced. We know that the fluid velocity must be zero at the surface of the cylinder and that it also increases rapidly away from that surface. Vorticity is created by this large local variation in fluid velocity. Now when the main stream velocity is low enough, there is sufficient time for this vorticity to diffuse out of the thin region near the solid surface where it is produced and to grow into a large region of vorticity. This physical picture should help to prepare us for the next change in the nature of the flow as the main stream velocity, or $\ReynoldsR$, is increased still more. As the velocity gets higher and higher, there is less and less time for the vorticity to diffuse into a larger region of fluid. By the time we reach a Reynolds number of several hundred, the vorticity begins to fill in a thin band, as shown in Fig. 41–6(d). In this layer the flow is chaotic and irregular. The region is called the boundary layer and this irregular flow region works its way farther and farther upstream as $\ReynoldsR$ is increased. In the turbulent region, the velocities are very irregular and “noisy”; also the flow is no longer two-dimensional but twists and turns in all three dimensions. There is still a regular alternating motion superimposed on the turbulent one. As the Reynolds number is increased further, the turbulent region works its way forward until it reaches the point where the flow lines leave the cylinder—for flows somewhat above $\ReynoldsR=10^5$. The flow is as shown in Fig. 41–6(e), and we have what is called a “turbulent boundary layer.” Also, there is a drastic change in the drag force; it drops by a large factor, as shown in Fig. 41–4. In this speed region, the drag force actually decreases with increasing speed. There seems to be little evidence of periodicity. What happens for still larger Reynolds numbers? As we increase the speed further, the wake increases in size again and the drag increases. The latest experiments—which go up to $\ReynoldsR=10^7$ or so—indicate that a new periodicity appears in the wake, either because the whole wake is oscillating back and forth in a gross motion or because some new kind of vortex is occurring together with an irregular noisy motion. The details are as yet not entirely clear, and are still being studied experimentally.