itself is for the most part in a region of relatively low vorticity The high

Itself is for the most part in a region of relatively

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itself is for the most part in a region of relatively low vorticity. The high magnitudes are found near the upper boundary, along the top surface of the step, and in the “shear layer” behind the step where the vortex meets the oncoming flow. We should observe that for such a flow, except in the vicinity of the vortex, the main flow direction will be from left to right so that the velocity vectors have large u components and relatively small v components. If we now recall that in 2D the only component of the vorticity vector is ω 3 = v x u y , we can easily see why the vorticity is negative along the upper surface of the step, and positive along the upper boundary. In particular, since v is very small, we do not expect much contribution to ω 3 from v x . At the same time, along the top of the step the u component of velocity is increasing with y as it goes from a zero value on the step (due to the no-slip condition) out to the speed of the oncoming flow farther away from the step. Hence, u y > 0 holds; but this term has a minus sign in the formula for vorticity. A similar argument holds for the vorticity at the upper boundary. 3-D Shear Flow in a Box Figure 2.17 provides an image indicating some of the qualitative features of vorticity associated with the 3-D flow of Fig. 2.14(c). The colors represent magnitude of vorticity, | ω | , with blue being
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2.4. CLASSIFICATION OF FLOW PHENOMENA 35 large values and red indicating values near zero. This is a flow field of a fluid confined to a cubic box with shear induced at the top by moving the solid lid in a diagonal direction as indicated by the arrows of the figure. The main points of interest in this figure are: first, the extreme variability vortices Figure 2.17: 3-D vortical flow of fluid in a box. in a three-dimensional sense of the magnitude of vorticity throughout the flow field; second, the complicated structure of the lines indicating motion of fluid parcels; and third, particularly the vortical shape near the top of the box. It is worthwhile to recall at this point the discussion of effects of viscosity given earlier and especially how the combination of the no-slip condition and diffusion of momentum can be expected to set up such a flow field. The Potential Vortex Up to this point we have seen examples of flows that have vorticity, but no apparent vortex, and flows that have both vorticity and a vortex. We have also noted the case of uniform flow which exhibits neither vorticity nor an observable vortex. In this section we will briefly introduce the one remaining possibility: a flow containing a vortex, but for which the vorticity is identically zero. Potential flows comprise a class of idealistic flows that were once studied in great detail because in many cases it was possible to obtain exact solutions to their corresponding equation(s). They are, by definition, irrotational; but they are not necessarily trivial as is true for uniform flow. The study of such flows in modern fluid dynamics has been relegated to brief introductions, primarily
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