the oncoming flow effectively narrowing the flow passage giving rise to the

# The oncoming flow effectively narrowing the flow

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the oncoming flow, effectively narrowing the flow passage (giving rise to the terminology vena contracta ), increasing the flow speed, and thus decreasing the pressure—a head loss. It is also interesting to note that the extent of effective contraction of passage diameter can be greatly reduced simply by rounding the corners of the inlet, rather than using sharp edges. This can be
4.5. PIPE FLOW 149 recirculation zones vena contracta Figure 4.21: Flow through sharp-edged inlet. seen in Table 4.2 which contains experimental data for the value of the loss coefficient K as a function of radius of curvature of the inlet edge normalized by the downstream diameter. Table 4.2 Loss coefficient for different inlet radii of curvature D R / D K 0.0 0.5 0.02 0.28 0.06 0.15 0.15 0.04 R As the accompanying sketch implies, the amount of flow separation is decreased as the radius of curvature of the corner increases, resulting in significant decrease in the extent of vena contracta and associated effective flow blockage. Contracting pipes . Another often-encountered flow situation that can result in significant head loss is that of a contracting pipe or duct. A general depiction of such a configuration is presented in Fig. 4.22. It is clear that multiple recirculation zones are
150 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS D 1 recirculation zones D 2 θ Figure 4.22: Flow in contracting pipe. Table 4.3 Loss coefficients for contracting pipes D /D 2 1 K θ = 60 ° θ = 180 ° 0.2 0.08 0.49 0.4 0.07 0.42 0.6 0.06 0.32 0.8 0.05 0.18 present. In the case of circular pipes these wrap around the entire inner surface of the pipe, while for more general geometries they can take on very complicated shapes. In any case, they result in some effective blockage of the pipe or duct, and consequently additional head loss. It is reasonable to expect that as the angle θ becomes small, or as D 2 D 1 , the size of the separated regions should decrease, and the loss factor will correspondingly be reduced. This is reflected to some extent in Table 4.3. In particular, we see for the case of θ = 60 that the value of K decreases as D 2 D 1 , although in this rather mild situation no value of K is excessive. The case of abrupt contraction ( θ = 180 ) shows a much more dramatic effect as pipe diameters approach the same value. Namely, for the smallest downstream pipe (greatest amount of contrac- tion), the value of K is 0.49; but this decreases to 0.18 when the pipe diameter ratio is 0.8. It is of interest to note that this case is essentially the same as that of a sharp-edged inlet, discussed above, with D 1 → ∞ . Furthermore, there is a reasonably-accurate empirical formula that represents this case of a rapidly contracting pipe: K 1 2 bracketleftBigg 1 parenleftbigg D 2 D 1 parenrightbigg 2 bracketrightBigg . (4.74) Rapidly-expanding pipes . The reader should recall that we earlier treated this example using the control-volume momentum equation, and then repeated the analysis using Bernoulli’s equation to find that the results were considerably different. Figure 4.8 provides a detailed schematic; here we will repeat a few essential features in Fig. 4.23. We discussed details of the physics of this flow
4.5. PIPE FLOW 151 D 1 D 2 Figure 4.23: Flow in expanding pipe.

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