3. Turbulent Flow Structure
The velocity data acquired using the LDV can be analyzed for turbulence in the wake. In order to estimate the turbulence intensity, velocity deviations are computed as
where u is a velocity vector and ub is a base velocity. As the bore passes the obstacle, there are large changes in the flow structure. These changes represent gradually varying base flow but not consider turbulence. Therefore defining ub as the time averaged velocity is not applicable, rather, a type of moving average is needed for this transient flow. However calculating a moving average, u, using
where t is time and Δt represents the length of the time interval the velocity is averaged over, does not solve the problem, as the changes in the flow are very abrupt at the time of bore impact. A single averaging interval Δt provides too much averaging in some portions of the velocity time history and does not provide enough averaging in others. A different definition of a base velocity ub is needed. First the time of bore arrival, t0, needs to be identified from the velocity time history. Before that the water is still so . After the bore arrived, the base velocity is calculated using repeated moving average calculations.
where i is the step number in the repetition and u0 = u. Four repetitions are used so ub = u4. The value of the half-interval Δt is set to 0.1 s. Figure 3.1 shows examples of these calculations at different locations in the wake of the circular column. The broken vertical lines indicate the interval used in the calculations for turbulent intensity to be shown in Table 2.
Figure 3.1. Velocity data and the base velocity in the wake of the circular column for h1 = 250 mm: a) x = 350 mm, y = 0 mm, and z = 15 mm; b) x = 470 mm, y = 210 mm, and z = 35 mm.
The turbulence intensity is then calculated as the root-mean-square average of the velocity deviations or
where the averaging interval is 20 times that used for the base velocity calculations or 20 Δt = 2 sec.
The LDV velocity data for u and v are available at 24 points on a three-dimensional rectangular grid at x = 210 mm, 350 mm and 470 mm, y = 0 mm, 70 mm, 140 mm and 210 mm, and z = 15 mm and 35 mm – the horizontal and vertical coordinates of the measurement points are depicted in Fig. 3.2.
Figure 3.2. Locations of LDV measurements in the wake of the circular column: a) the horizontal coordinates, b) the vertical locations.
Figure 3.3 shows results of the root-mean-square calculations starting 0.5 s after the bore arrival. Away from the centerline there is a clear decrease in turbulence intensity in time, in both the x and y-directions. The magnitudes of u’rms and v’rms are about the same away from the centerline, while v’rms > u’rms near the centerline.
Figure 3.3. Root-mean-square average of velocity deviations in the wake of the circular column for h1 = 250 mm: a) x = 350 mm, y = 0 mm, and z = 15 mm; b) x = 470 mm, y = 210 mm, and z = 35 mm
Using root mean square values over the interval described above, we compute the relative strength , where , and the results are listed in Table 2 and also plotted in Figs. 3.4 and 3.5. Generally the velocity fluctuation in the x-direction is greater at the lower elevation, but that does not hold for the y-direction. There the turbulence is greater at the lower elevation on the centerline, but it varies outside of it. On the centerline the turbulence intensity in the y-direction is considerably higher than that in the x-direction, while the magnitude is similar in either direction away from the centerline. Interestingly, these observations are true at the different downstream locations as shown in Figs. 3.4 and 3.5.
Recommendation: Our LDV and DPIV data are available for validation of turbulence models. Free-surface wake turbulence is a very challenging problem.
Table 2. Velocity deviation as a ratio of the mean velocity in the wake of the circular column for h1 = 250 mm.
Figure 3.4. The flow-wise variations in the turbulence intensities.
Figure 3.5. The span-wise variations in the turbulence intensities.