Turbulence characteristics of the flow resulting from the hydrodynamic interaction of multiple counter flow jets in expanding channels

Abstract

Various methods exist to stabilize asymmetric hydraulic jumps occurring in suddenly expanding channels. In this study, the effects of the interaction of multiple submerged counter flow jets are investigated experimentally in terms of the hydrodynamic characteristics of the flow resulting in the tailwater channel. To this aim, three-dimensional flow velocities downstream of the jet system were measured for different configurations of the device. The stability in the performance of the dissipation system under variable tailwater conditions was analyzed in terms of flow velocity distribution, turbulent kinetic energy, turbulence intensity and Reynolds shear stresses. The results reveal the good performance of the investigated system in preventing the phenomena of flow instability associated with suddenly expanding channels.

Appendix

1.1 Selection of the appropriate duration of velocity measurements

In a first phase of the experimental campaign, a number of preliminary runs lasting more than 30 s were performed to determine the appropriate duration of the measurements which ensured the convergence of velocity data.

In particular, in order to select the most efficient length of the time window, a sensitivity analysis was carried out on a time series of 75 s. To analyze the convergence of the second order turbulence statistics, the standard deviation ${(\sigma }_{v})$ of the signal (strongly correlated with the majority of the considered turbulence indicators) was calculated according to Equation A1 for all the possible subsamples on time windows of 5, 10, 15, 20, 30, 40 and 50 s in the series:

$${\sigma }_{vi}=\sqrt{\frac{{\sum }_{i=1}^{N}{\left({\mathrm{v}}_{i}-\overline{v }\right)}^{2}}{N-1}},$$

(A1)

where v_i is the ith velocity sample, $\overline{v }$ is the average velocity and N is the number of values sampled in that time window.

This allowed to compute the standard deviation of the corresponding ${\sigma }_{vi}$, as follows:

$${\sigma }_{\sigma }=\sqrt{\frac{{\sum }_{i=1}^{{N}_{w}}{\left({\upsigma }_{vi}-\tilde{\sigma }\right)}^{2}}{{N}_{w}-1}},$$

(A2)

where ${\sigma }_{vi}$ is the standard deviation computed for the ith subsample, $\tilde{\sigma }$ is the average value of the ${\sigma }_{vi}$ and N_w is the number of subsamples with the given duration.

An example of the typical observed result is shown in Fig. 

Fig. 10

Results of the analysis supporting the selection of the duration of velocity measurements

10, which reports ${\sigma }_{\sigma }$ values observed for different sampling durations of an experimental time series. Such results, indicating no significant reduction of the variance for higher durations of the signal, suggested 30 s as the most efficient sampling time to be considered for the experiments.

1.2 Selection of the spatial resolution of velocity measurements in the transversal direction

To support the selection of the spatial resolution of the measurement in the transversal direction, a sensitivity analysis was performed by comparing the point velocities detected with two different measurement spacings (0.05 and 0.1 m). As an example (for Configuration 2, under maximum tailwater condition (h_S), cross section x = 1 m, height z = 0.02 m), Fig. 

Fig. 11

Average velocity values (u, v and w) observed by adopting a 0.05 m measurement grid, compared to the ones obtained by interpolating the data at the resolution of 0.1 m. The differences [%] shown in the bottom panels are calculated as follows: 100·(V_{interp_10} – V_{meas_5})/V_{meas_5}, where V_{meas_5} is the generic velocity component measured at a resolution of 0.05 m and V_{interp_10} is the value obtained in the same point by interpolating the data from the measurements at a resolution of 0.10 m

11 shows the average velocity values (u, v and w) measured by adopting a resolution of 0.05 m and the ones obtained by interpolating the 0.10 m resolution data. The results indicated relative differences up to about 5%, which were considered acceptable for the main aim of the study, i.e., on gaining more insights on the dissipative performance of the tested device.