due to the effectiveness of CFD in calculating general flow fields for which

Due to the effectiveness of cfd in calculating

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due to the effectiveness of CFD in calculating general flow fields for which few assumptions need be made; but they once formed the basis of most incompressible aerodynamics analyses. We will not provide a detailed description of the potential flow we are considering here because it will not be of
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36 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS particular use in our later studies. On the other hand, it does provide a final example associated with vorticity—actually, lack thereof in this case—of which we should be aware. Figure 2.18 provides a sketch of the potential vortex flow field. It is important to examine the orientation of the fluid elements shown in this figure and compare this with what would occur velocity profile parcel fluid θ r Figure 2.18: Potential Vortex. in “rigid-body rotation” of the same fluid. Potential vortex flow is one dimensional, depending only on the r coordinate when represented in polar coordinates. Its radial velocity component u is identically zero, as is the z (into the paper) component; its azimuthal component is given by v = K r , where K is a constant. This can be shown to satisfy ∇× U = 0 in cylindrical polar coordinates, and thus the flow is irrotational. It should be noted that the motion of any particular fluid element does not correspond to rigid-body rotation—which produces non-zero vorticity. In fact, we see from the figure that as the fluid element moves in the θ direction it maintains the same face perpendicular to the radial direction (as would happen for rigid-body rotation); but it is distorted in such a way that originally perpendicular faces are distorted in opposite directions, leaving the net rotation of the fluid element at zero. Thus, we see that (net) rotation of fluid elements is required for non-zero vorticity. Indeed, it is possible to provide a completely physically-based derivation of fluid element rotation, the outcome of which shows that vorticity, as defined above, is just twice the rotation (which is zero for the potential vortex). 2.4.5 Viscous and inviscid flows It is a physical fact that all fluids possess the property of viscosity which we have already treated in some detail. But in some flow situations it turns out that the forces on fluid elements that arise from viscosity are small compared with other forces. Understanding such cases will be easier after we have before us the complete equations of fluid motion from which we will be able to identify
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2.4. CLASSIFICATION OF FLOW PHENOMENA 37 appropriate terms and estimate their sizes. But for now it is sufficient to consider a case in which viscosity is small (such as a gas flow at low temperature); hence, the shear stress will be reasonably small (recall Eq. (2.2)) and, in turn, the corresponding shear forces will be small. Assume further that pressure forces are large by comparison with the shear forces. In this situation it might be appropriate to treat the flow as inviscid and ignore the effects of viscosity. On the other hand, in
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