We will often in fact most of the time throughout these lectures employ a short

We will often in fact most of the time throughout

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We will often (in fact, most of the time) throughout these lectures employ a short-hand notation for partial differentiation in the form, e.g. , ∂u ∂x = u x , ∂v ∂z = v z , etc. Hence, the above formula for vorticity can be written more concisely as ω = (( w y v z ) , ( u z w x ) , ( v x u y )) T . It should be noted that no basis has been indicated in this representation (as was true in the anal- ogous representation of the velocity vector given above), and this implies that we have prescribed a vector basis—in this case ( e 1 , e 2 , e 3 ) T . We can now re-examine consequences of the definition of rotational in the context of this formula for vorticity. First, we note that any uniform flow is automatically irrotational because from Eq. (2.12) it is easily seen that all components of ω must be identically zero. But we should also observe that there is another manner in which a flow can be irrotational. Namely, if we set w y = v z u z = w x (2.16) v x = u y , then we see from Eq. (2.15) that the vorticity vector is again identically zero; but in this case we have not required velocity gradient components to be identically zero (as was true for uniform flows), so the flow field can be considerably more complex, yet still irrotational. In the following subsections we provide some specific physical examples of flows that are either rotational or irrotational. 1-D Shear Flows If we recall Figs. 2.4 and 2.14(a) we see that the only nonzero velocity component is u and that it varies only with the coordinate y . As a consequence, all velocity derivatives are identically zero except for u y . From this we see that the third component of the vorticity vector must be nonzero.
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34 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS Hence, these simple 1-D flows are rotational. It should be noted that there is no indication of swirl ( i.e. , vortices) but, on the other hand as indicated in Fig. 2.4, there is rotation of fluid elements. It is also interesting to note that the direction of vorticity in these cases is pointing out of the plane of the figures—the z direction—even though the velocity field is confined to x - y planes. Even so, this does not change our assessment of the dimensionality of the flow field: vorticity is a property of the flow, and it is pointing in the z direction; but it is changing only in the y direction (in fact, in Fig. 2.4 it is constant). 2-D Shear Flow Over a Step In this section we present a flow field that is two dimensional, and in this case it contains a very prominent vortex. This is displayed in Fig. 2.16 in which red denotes high magnitude, but negative, vorticity, and blue corresponds to positive high magnitude vorticity. The large areas of green color have nearly zero vorticity. The black lines represent paths followed by fluid elements, and the flow is from left to right. What is interesting about this particular flow is that the vortex step vortex shear layer Figure 2.16: 2-D vortex from flow over a step.
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