Flow property and self-similarity in steady hydraulic jumps

Abstract

The flow structure in a steady hydraulic jump in both the non-aerated and aerated regions was measured using the image-based particle image velocimetry and bubble image velocimetry techniques, respectively. Three highly aerated steady jumps with Froude numbers varying from 4.51 to 5.35 were tested, and a weak jump with a Froude number of 2.43 was generated for comparison. Mean velocities and turbulence statistics were obtained by ensemble averaging the repeated velocity measurements. Based on the mean velocities, the flow structure in the steady jumps was classified into four regions to distinguish their distinct flow behaviors; they are the potential core region, the boundary layer region, the mixing layer region, and the recirculation region. The flow structure in the weak jump features only three regions without the recirculation region. In addition, spatial variations of mean velocities, turbulence intensity, and Reynolds stresses were also presented. It was observed that the maximum horizontal bubble velocity and maximum horizontal water velocity occur at the same location in the overlapping regions of potential core and mixing layer. The ratio between the maximum horizontal bubble velocity and maximum horizontal water velocity is between 0.6 and 0.8, depending on the Froude number. Examining the mean horizontal bubble velocities in the mixing layer, a similarity profile was revealed with representative mixing layer thickness as the characteristic length scale and the difference between the maximum positive and maximum negative velocities as the characteristic velocity scale. It was also found that the mean horizontal water velocities in the near-wall region are self-similar and behave like a wall jet. Further analyzing autocorrelation functions and energy spectra of the water and bubble velocity fluctuations found that the energy spectra in the water region follow the −5/3 slope, whereas the spectra in the bubble region follow a −2/5 slope. In addition, the integral length scale of bubbles is one order of magnitude shorter than that of water.

Appendix: Turbulence statistics of weak jump with F r  = 2.43

In addition to the three steady hydraulic jumps (F _r  = 4.51, 5.00, and 5.35) investigated in the present study, a weak jump with a Froude number of F _r  = 2.43 was also measured using the same techniques and analyzed using the same procedure. The turbulence statistics of the weak jump are plotted in Fig. 24. By comparing with the turbulence statistics of steady jumps in Fig. 15, the high turbulence region in the weak jump overlaps with the region of high aeration (near free surface and behind the toe) as shown in Fig. 24. On the contrary, the high turbulence region occurs coincidently with the maximum positive velocity region around the lower bound of the mixing layer in the steady jumps as shown in Fig. 15. The distributions of Reynolds stresses between the weak jump and the steady jumps are also very different.

Fig. 24

Turbulence statistics of the weak jump with F _r  = 2.43. a $U_{\text{rms}} = \sqrt {\overline{{u^{\prime 2} }} } /u_{1}$; b $V_{\text{rms}} = \sqrt {\overline{{v^{\prime 2} }} } /u_{1}$; c in-plane turbulence intensity $I = \sqrt {\overline{{u^{\prime 2} }} + \overline{{v^{\prime 2} }} } /u_{1}$; d Reynolds stresses $\Uplambda = - \overline{{u^{\prime } v^{\prime } }} /u_{1}^{2}$