The integral model developed by Chin (1988) for modelling a non-buoyant turbulent jet in wave environment is improved by introducing two new parameters, i.e., the jet spreading rate c₁ and the shortening rate p_e. The parameter c₁ is used to simplify the model by explicitly describing the radial velocity and scalar profiles under the assumption of “instantaneous” Gaussian distribution. By doing so, the governing equations can be easily solved by simultaneously integrating the conservation laws of momentum and scalars across the jet cross-section. The parameter p_e is used to shorten the initial length of zone of flow establishment (ZFE), so as to more accurately account for the wave effect on the jet initial dilution near the jet nozzle. The parameters are calibrated by the particle image velocimetry (PIV)-measured data from three groups of jet experiments, i.e., the group of vertical jet towards the wave direction (vertical jet), the group of horizontal jet along the wave direction (co-wave jet) and the group of horizontal jet opposing to the wave direction (op-wave jet). The results show that both parameters are well related to the ratio of jet and wave characteristic velocities in the same group, but it is not able to be generalized among different groups. Under the same wave condition, the value of c₁ in the vertical jet is larger than that of the horizontal jets; while the value of p_e in the vertical jet is smaller than that of the horizontal jets, which indicates that the jet has a faster decay rate of centerline velocity and a wider width of jet cross-section profile in the near field when it is vertically discharged into the wave environment. With the well-calibrated parameters, the improved model can achieve a higher accuracy than the original model developed by Chin (1988).

The multiport diffuser effluent discharge facilities constructed beneath the coastal waters were simplified in the laboratory as twin buoyant jets in a wavy cross-flow environment. The near-field flow structure of twin jets was studied by series of experiments conducted in a physical wave–current flume. The particle image velocimetry (PIV) system was used to measure the velocity field of the jets in various cross-flow-only and wavy cross-flow environments. By means of flow visualization, the distinctive “effluent cloud” (EC) phenomenon was clearly observed and the jet penetration height was found to be notably increased compared with that of cross-flow-only environment at the downstream position. It was found that the wave-to-current velocity ratio R_wc is a very important parameter for effluent discharge. A new characteristic velocity u_ch and the corresponding characteristic length scale l_mb for twin buoyant jets in the wavy cross-flow environment were defined. Using curve-fitting, a new equation to estimate the effects of the jet-to-current velocity ratio (R_jc), wave-to-current velocity ratio (R_wc) and Strouhal number (St) on the jet trajectory were derived to enhance understanding the physical processes underpinning the rise and the dilution of buoyant jets, which is critical to the design of discharge facilities.

The understanding of mixing behaviors of submerged effluent discharge in buoyant jet flows within coastal dynamic environments remains incomprehensive. This study explores the mixing behaviors of buoyant jets in wavy crossflow environments by utilizing dimensional analysis and an integral model. We examine physical mechanisms underlying the linear uplift of buoyant jets and establish three semi-empirical equations pertaining to the vertical location of the cross-sectional minimum dilution, cross-sectional minimum dilution, and visible diffusion area. These equations quantify the influence of waves and buoyancy on the mixing behaviors of buoyant effluent discharge in wavy crossflow environments. The dimensional analysis demonstrates that the proposed mixing behavior equations are closely related to the buoyant characteristic length scale. We validate the reliability of these equations with that of existing ones for non-buoyant jets in wavy crossflow environments. Furthermore, this study extends the application of buoyant jet equations, evaluating their suitability by comparing them with data from cases involving buoyant jets in wavy crossflow environments. This study highlights the positive effect of wave height and buoyancy on the initial dilution of effluent discharge. Future studies can implement our semi-empirical equations on the near and far zone coupling simulations of wastewater and thermal discharges.

The mixing behavior of an oscillating jet under the influence of currents remains incomprehensive. This study uses a three-dimensional large eddy simulation model to investigate the phase-averaged and time-averaged concentration distribution of three-dimensional scalar structures in the oscillating jet under a current environment. The effects of dimensionless parameters on dilution characteristics are also analyzed. The results indicate that increasing the jet-current velocity ratio (Rjc) and the amplitude-jet velocity ratio (Raj), while decreasing the Strouhal number (St), can enhance the dilution capacity of the receiving water. To quantify the oscillatory effect of jets on the initial dilution of wastewater discharge, semiempirical equations for the cross-sectional minimum dilution (Sc) and the visual area (A25%) of the oscillating jet in a current environment are developed using the least squares method. The oscillatory nature of the jets is found to behave similarly to wave effects. Furthermore, the empirical equations for the initial dilution of oscillating jets in current environments are structurally consistent with those for non-oscillating jets in wave-current coexisting environments. This study highlights the positive impact of oscillating jets on mixing and dilution.