# Turbulence characteristics of 45° inclined dense jets

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## Abstract

In the present study, we performed an extensive laboratory investigation to quantify the turbulence characteristics of 45° inclined dense jets using Particle Image Velocimetry (PIV) over a wide range of Densimetric Froude Number. The objective was to provide benchmark data to guide high resolution turbulence numerical simulations for dense jets in the future. The PIV measurements were sampled at a relatively high frequency of 50 Hz, which enabled the analysis of second order turbulence statistics as well as the turbulence kinetic energy spectrum (including the inertial subrange) along the curvilinear jet trajectory, which has hitherto not been reported. The measurements showed that the spectral profile was flat near the discharge port with the potential core, since the Kelvin–Helmholtz shear-induced turbulence at the jet boundaries had not fully penetrated to the core. The spectral profile then evolved along the trajectory with progressive steepening towards the higher frequencies, and a fully-developed profile appeared beyond the terminal rise with a clearly identifiable inertial subrange for the energy cascade. In parallel, we also performed numerical simulations using the Large Eddy Simulations (LES) approach with the Dynamic Smagorinsky sub-grid model for the specific discharge conditions as in the experiments. The LES approach followed that of Zhang et al. (Environ Fluid Mech 16(1):101–121, 2016, J Hydro Environ Res 15:54–66, 2017) using GCI as the grid convergence criteria. The comparison showed that the time-averaged first order mixing characteristics of the inclined dense jet can be simulated reasonably well comparing to the experimental data. In terms of the turbulence kinetic energy spectrum, the low frequencies of the production range were also well captured by the simulations. However, the simulated transitional spectra towards the inertial subrange decayed substantially faster than the experiments. The discrepancy was attributed to the fact that the grid resolution was not sufficiently fine in the simulations (which were constrained by the available computational resources and time), such that stratified effects remained present inside the sub-grids producing additional turbulence energy that were not represented by the Dynamic Smagorinsky model. Thus, the numerical investigation showed that further improvement in sub-grid models that can incorporate the stratified effects would be desirable in the future for engineering simulations.

## Keywords

Inclined dense jet PIV Large Eddy Simulations (LES) Turbulence kinetic energy (TKE) Turbulence kinetic energy spectrum## List of symbols

*b*_{e}Velocity 1/e width

*C*_{0}Initial concentration

*C*_{m}Centerline concentration

*C*_{s}Smagorinsky constant

*D*Nozzle diameter

*Fr*Densimetric Froude number

*g*Gravitational acceleration

*g′*Reduced gravitational acceleration

*h*Port height

*H*Distance from the port to the water surface

*L*Distance from the port to the back and front vertical boundary

*L*_{M}Jet characteristic length scale

*L*_{ij}Resolved turbulent stress

*M*_{ij}Anisotropic part of the turbulent stress

*p*Pressure

*Q*_{j}SGS scalar flux or turbulent scalar flux

*r*Radial distance

*Re*Reynolds number

*s*Stream-wise distance from the nozzle

- \(\tilde{S}\)
Local strain rate

*S*_{Ct}Turbulent Schmidt number

- \(\widetilde{S}_{ij}\)
Rate of strain tensor for the resolved scale

*t*Time

*t*_{s}Run time of a simulation

*U*Stream-wise velocity

*U′*Fluctuation of stream-wise velocity

*U*_{0}Discharge velocity

*U*_{m}Jet velocity along the centerline

*U′*_{rms}Root mean square stream-wise velocity fluctuation

*u*_{i},*u*_{j}Velocity in

*i*,*j*direction, respectively*V*Radial velocity

*V′*Fluctuation of radial velocity

*V′*_{rms}Root mean square radial velocity fluctuation

*W*Distance from the nozzle to the left and right vertical boundary

*x*,*y*,*z*Cartesian Coordinates in the horizontal, lateral and vertical direction, respectively

*x*_{m}Horizontal location of centerline peak

- ∆
LES filter width

*ρ*Fluid density

*ρ*_{a}Ambient density

*ρ*_{b}Effluent density

*μ*Fluid viscosity

*μ*_{t}SGS eddy viscosity or turbulent eddy viscosity

*ϕ*Scalar concentration

*ε*Dissipation rate

- \(\tau_{ij}\)
SGS Reynolds stresses or Reynolds stresses

*τ*_{kk}Isotropic part of SGS stress

- Γ
Scalar diffusivity

- Γ
_{t} Turbulent dispersivity

## Notes

### Acknowledgements

The authors would like to thank the Nanyang Environment and Water Research Institute and the Interdisciplinary Graduate School at Nanyang Technological University, for the award of research scholarship to the first author.

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