Notes

Chapter 8: Implications for Everyday Systems

Section 4: Fluid Flow


Flows past objects

By far the most experimental data has been collected for flows past cylinders. The few comparisons that have been done indicate that most results are extremely similar for plates and other non-streamlined or "bluff" objects. For spheres at infinitesimal Reynolds numbers a fairly simple exact analytical solution to the Navier-Stokes equations was found by George Stokes in 1851, giving a drag coefficient of 6 π/R. For a cylinder, there are difficulties with boundary conditions at infinity, but the drag coefficient was nevertheless calculated by William Oseen in 1915 to be 8 Pi/(R (1/2+Log[8/R]-EulerGamma)). At infinitesimal Reynolds number the flow around a symmetrical object is always symmetrical. As the Reynolds number increases, it becomes progressively more asymmetrical, and at R 6 for a cylinder, closed eddies begin to appear behind the object. The length of the region associated with these eddies is found to grow almost perfectly linearly with Reynolds number. At R 30 - 40 for a cylinder, oscillations are often seen in the eddies, and at R 46 - 49, a vortex street forms. Increasingly accurate numerical calculations based on direct approximations to the Navier-Stokes equations have been done in the regime of attached eddies since the 1930s. For a vortex street no analytical solution has ever been found, and indeed it is only recently that the general paths of fluid elements have even been accurately deduced. A simple model due to Theodore von Kármán from 1911 predicts a relative spacing of π/Log[1+Sqrt[2]] between vortices, and bifurcation theory analyses have provided some justification for some such result. Over the range 50\[LessTilde] R \[LessTilde]150 vortices are found to be generated at a cylinder with almost perfect periodicity at a dimensionless frequency (Strouhal number) that increases smoothly from about 0.12 to 0.19. But even though successive vortices are formed at fixed intervals, irregularities can develop as the array of vortices goes downstream, and such irregularities seem to occur at lower Reynolds numbers for flows past plates than cylinders. Some direct calculations of interactions between vortices have been done in the context of the Navier-Stokes equations, but the cellular automaton approach of page 378 seems to provide essentially the first reliable global results. In both calculations and experiments, there is often sensitivity to details of whatever boundary conditions are imposed on the fluid, even if they are far from the object. Results can also be affected by the history of the flow. In general, the early way the flow develops over time typically mirrors quite precisely the long-time behavior seen at successively greater Reynolds numbers. In experiments, the process of vortex generation at a cylinder first becomes irregular somewhere between R=140 and R=194. After this surprisingly few qualitative changes are seen even up to Reynolds numbers as high as 100,000. There is overall periodicity much like in a vortex street, but the detailed motion of the fluid is increasingly random. Typically the scale of the smallest eddies gets smaller in rough correspondence with the R^-(3/4) prediction of Kolmogorov's general arguments about turbulence. In flow past a cylinder, there are various quite sudden changes in the periodicity, apparently associated with 3D phenomena in which the flow is not uniform along the axis of the cylinder. The drag coefficient remains almost constant at a value around 1 until R3×10^5, at which point it drops precipitously for a while. This phenomenon is associated with details of flow close to the cylinder. At lower Reynolds numbers, the flow is still laminar when it first comes around the cylinder; but there is a transition to turbulence in this boundary layer after which the fluid can in effect slide more easily around the cylinder. When the speed of the flow passes the speed of sound in the fluid, shocks appear. Usually they form simple geometrical patterns (see below), and have the effect of forcing the turbulent wake behind the cylinder to become narrower.


From Stephen Wolfram: A New Kind of Science [citation]