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EAS4101_S11_HW5S - 6 Given A circular...

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Unformatted text preview: 6. Given: A circular cylinder. Find:Describe the flow field over the circular cylinder as the Reynolds number based on diameter increases from 0.1 to 200,000, discussing the physics of how viscous effects dominate the flow. Please sketch out various flow regimes and discuss the structure of the cylinder wake as a function of Reynolds number. Note: I expect a strong effort of 35 pages of explanation here. SOLUTION: Beginning at a very low Reynolds number, Re=0.1, the inertial forces are negligible and the flow pattern is dominated by the viscous forces. There is also a fore ­aft flow field symmetry. This is known as creeping flow. Hence to non ­dimensionalize the momentum equation, a viscous pressure scale must be used in place of the usual inertial viscous pressure scale momentum equation . This results in a dimensionless • And neglecting the left term because the Reynolds number is very small, we arrive at the Stokes flow assumption: • Completely neglecting inertial terms implies that the forces on the cylinder are a function of only the freestream velocity, the viscosity, and the body length scale. For two dimensional flow, this results in • Physically this cannot be, otherwise the size of the body would play no role in the drag force, for example. This is known as Stokes’ paradox – though the inertial forces may be approximated as negligible for creeping flow, physically they are always important. Ironically, the coefficient of drag over the cylinder is highest for this type of flow, with . From Stokes’ solution for an immersed sphere in creeping flow, it can be shown that creeping flow streamlines and velocities are independent of viscosity. Furthermore, the effect of the immersed body is extreme; for example, at 10 diameters away from the body, the flow is still about 10 percent below freestream values. Figure 1. Creeping flow over a cylinder (from Van Dyke). As the Reynolds number increases, the vertical symmetry of the flow pattern breaks down, and by Re=4, the flow separates and forms a pair of counter ­rotating vortex sheets which remain attached over the rear of the cylinder, as shown in Figure 2. The flow field at this stage is still steady. Figure 2. Stationary eddies in wake behind cylinder (from Van Dyke). By a Reynolds number of 40, the wake behind the cylinder becomes unstable and discrete vortices begin to roll up and are alternately shed from the surface of the cylinder, forming what is known as the von Karman vortex street, as shown in Figure 3. The velocity of these shed vortices is slightly less than the freestream velocity. At this point, the Strouhal number becomes an important dimensionless parameter, • which is a measure of the ratio of the unsteady inertial forces to the steady inertial forces. The Strouhal number remains roughly constant at 0.2 for a wide range of Reynolds numbers. Figure 3. Von Karman vortex street (from Van Dyke). When the Reynolds number reaches ~100, the eddies which had remained attached themselves begin to periodically be shed from the cylinder. As this occurs, the lift and drag force locations begin to oscillate as flow is alternately attached and then separated over one half of the cylinder. The results of this periodic oscillation can be catastrophic in structures that have a natural vibration mode which corresponds to the frequency of oscillation, such as the infamous Tacoma Narrows bridge disaster of 1940. When the Reynolds number reaches about 200, the vortex sheet downstream from the cylinder transitions to turbulence. This turbulence occurs in the shed vortices as the Reynolds number continues to increase to about 400. Within this range, 200<Re<400, the Strouhal number becomes erratic; after which it stabilizes again. Around this range, the drag coefficient has also decreased to about 1 and remains roughly constant. At this point, the pressure forces are dominating the drag and hence we can say we are in the “high Reynolds number” range, where inertial forces dominate over viscous forces except for in the thin boundary layer, where viscous forces and vorticity are present. The drag suddenly drops again at a Reynolds number of 30,000, because the laminar boundary layer has transitioned to turbulent, and remains attached over a larger portion of the cylinder. This is due to the fact that a turbulent boundary layer velocity profile is “fuller” than a laminar one – the random fluctuations characteristic of a TBL allow a greater diffusion of momentum away from the free stream. A laminar vs. turbulent boundary layer flow over a sphere is shown in Figures 4 and 5. The flow clearly separates sooner for the laminar boundary layer case than for the turbulent boundary layer case. Hence, pressure drag is higher on the sphere with the laminar boundary layer. A similar phenomenon occurs for the two dimensional cylinder. Figure 4. Separation over a sphere – laminar boundary layer (from Van Dyke). Figure 5. Separation over a sphere – turbulent boundary layer (from Van Dyke). Once the Reynolds number reaches around 300,000, the final flow pattern of flow becoming turbulent over the front half of the cylinder is observed, and the drag steadily increases from there. ...
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