Question 4.SP.2: Figure 4.12 represents flow toward and around a bridge pier ......
Figure 4.12 represents flow toward and around a bridge pier where 5 ft and U_0=10 \mathrm{fps}. (a) Make a plot of the velocity along the flow centerline to the left of the solid, and along the boundary of the solid. (b) By what percentage does the maximum velocity along the boundary exceed the uniform velocity? (c) How far from the stagnation point does a velocity of 7.5 fps occur?
{ }^7 This surface shape is the boundary between the given flow field and that issuing from a source of strength Q=bdU0 located at S, where d is the source length perpendicular to the figure (see Prob. 14.14).
Eq. (4.19): \begin{gathered}V \Delta n=\text { const. }=U_0 \Delta n_0 \end{gathered}
So V=\left(\Delta n_0 / \Delta n\right) 10 \mathrm{fps}
Use b = 5 ft to scale 1 ft distances along the centerline and around the boundary of the solid. On Fig. 4.12 measure the net “square” sizes, in both the flow (\Delta L) and peroendicular (\Delta W) directions, using three or four squares where appropriate and taking the average. Calculate \Delta n and Vas shown in the table:
(b) V_{\max } / U_{0}=13.3 / 10.0=1.33
Therefore V_{\max } is 33 % greater than U_{0}
(c) From the plot shown, a velocity of 7.5 \mathrm{fps} occurs at about -1.9 \mathrm{ft} and +0.4 \mathrm{ft} from the stagnation point
Note: If the flow net had been constructed perfectly, the respective average \Delta L and \Delta W values would have been identical. Even so, the results obtained here are quite accurate, because the respective values of \Delta L and \Delta W were averaged. This problem can also be solved analytically using principles of hydrodynamics, which yield V_{\max } / U_{0}=1.260 (see, e.g., Prob. 14.14).
V \Delta n=\text { constant } (4.19)
| Distance from stagnation pt, ft | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \text { Average } \Delta L, \mathrm{~mm} | 0.98 | 1.02 | 1.06 | 1.06 | 1.17 | 1.25 | – | 0.88 | 0.7 | 0.67 | 0.66 | 0.73 | 0.75 | 0.78 | 0.84 |
| \text { Average } \Delta W, \mathrm{~mm} | 0.88 | 0.91 | 0.94 | 1.00 | 1.30 | 1.80 | – | 0.95 | 0.8 | 0.74 | 0.75 | 0.76 | 0.79 | 0.9 | 0.93 |
| \begin{aligned}& \Delta n=\frac{1}{2}(\Delta L+\Delta W) \\& \mathrm{mm}\end{aligned} | 0.93 | 0.97 | 1.00 | 1.03 | 1.24 | 1.53 | – | 0.92 | 0.75 | 0.71 | 0.71 | 0.74 | 0.77 | 0.84 | 0.88 |
| \Delta n_0 / \Delta n=0.93 / \Delta n | 1.00 | 0.96 | 0.93 | 0.90 | 0.75 | 0.61 | – | 1.02 | 1.24 | 1.32 | 1.32 | 1.25 | 1.21 | 1.11 | 1.05 |
| V=10\left(\Delta n_0 / \Delta n\right), \mathrm{fps} | 10.0 | 9.6 | 9.3 | 9.0 | 7.5 | 6.1 | 0 | 10.2 | 12.4 | 13.2 | 13.2 | 12.5 | 12.1 | 11.1 | 10.5 |
