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Three-dimensional flow structure and mixing of the side thermal buoyant jet discharge in cross-flow

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Abstract

Numerical investigation of a side thermal buoyant discharge in the cross-flow is presented based on unsteady Reynolds-averaged Navier–Stokes equations closed with the realizable k-\(\upvarepsilon \) turbulence model. Emphasis is placed on the detailed three-dimensional flow evolution and scalar mixing in an incompressible turbulent environment. The present study covers the cases with different jets to cross-flow velocity ratios (R) and initial temperatures. Moreover, various flow characteristics, including vortical structures, jet trajectories, jet streamlines, and intrinsic instabilities, are examined. Mixing ability is quantified by the decay rate of scalar temperatures and velocity magnitude, the probability density function, the spatial mixing deficiencies (SMDs), power spectral density analysis, and temporal mixing deficiencies (TMDs). Comparing the simulation results with the experimental data of Abdelwahed (Surface jets and surface plumes in cross-flows. Ph.D. thesis, Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec, 1981), it shows good agreement in terms of the temperature half-width and half-thickness profiles, recirculation zone size, and jet trajectory. The numerical results of three-dimensional structures indicate a shear-layer, horseshoe, and surface roller vortices near the side-channel exit and secondary flows after the recirculation zone at the free surface of the main channel. The instantaneous temperature contours exhibit a vortex shedding phenomenon and gaps between the cross-flow and discharge jet at the shear layer. The maximum velocity magnitude location approaches the outer wall toward the main-channel downstream by increasing R. It is found that as the densimetric Froude number (\(\hbox {Fr}_{{0}})\) and R increase, the temperature dilution (S) generally decreases. The statistical analysis of TMD and SMD indicates a direct relationship between the mixing efficiency and buoyancy flux (\({F}_{{0}})\).

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Abbreviations

\({\bar{T}}\) :

Temperature (K)

\(T_{k} \) :

Reference temperature

\({T}_{\mathrm {a}}\) :

Temperature in the cross-flow inlet

\({T}_{{0}}\) :

Difference in temperature between the jet and cross-flow

T :

Time-averaged excess temperature

\({T}_{\mathrm {m}}\) :

Maximum time-averaged excess temperature

\({T}_{\mathrm {s}}\) :

Temperature scale

t :

Time

\(\alpha \) :

Probability calculated by PDF

\(\xi \) :

Statistical representation of \(\alpha \)

\({l}_{\mathrm {s}}\) :

Length scale (m)

\({t}_{\mathrm {s}}\) :

Time scale (s)

\(2\eta _{\mathrm{m}} \) :

\({T}_{\mathrm {m}}\) Half-width (m)

\(M_{0} \) :

Flow force in the side-channel inlet (\(\hbox {m}^{{4}}/\hbox {s}^{{2}})\)

\(F_{0} \) :

Buoyancy flux in the side-channel inlet (\(\hbox {m}^{{4}}/\hbox {s}^{{3}})\)

\(\Gamma _{0} \) :

Heat flux in the side-channel inlet (\(\hbox {k} \hbox {m}^{{3}}\hbox {/s}\))

\(\delta _{ij} \) :

Kronecker delta function

\(\nu _{0} \) :

Kinematic viscosity (\(\hbox {m}^{{2}}\hbox {/s}\))

\(\rho _{{0}}\) :

Density of the fluid (\(\hbox {kg/}\hbox {m}^{{3}}\))

\(\mu _{0} \) :

Molecular viscosity (kg/m s)

\(\nu _{\mathrm{t}} \) :

Turbulent viscosity (\(\hbox {m}^{{2}}\hbox {/s}\))

\({\bar{p}}\) :

Pressure

k :

Turbulent kinetic energy

\(\beta \) :

Expansion coefficient (1/K)

\(g_{i} \) :

Gravity acceleration vector in the i-direction (\(\hbox {m/}\hbox {s}^{{2}}\))

R :

Velocity ratio of the jet to the cross-flow

\(\Pr \) :

Prandtl number

\(\Pr _{\mathrm{t}}\) :

Turbulent Prandtl number

\({V}_{{0}}\) :

Streamwise velocity in the side-channel inlet (m/s)

\({A}_{{0}}\) :

Cross-sectional area in the side-channel inlet (\(\hbox {m}^{{2}}\))

d :

Side- and main-channel depth (m)

\(\hbox {Fr}_{{\mathrm{observed}}_{i}}\) :

Experimental results

\(\hbox {Fr}_{{\mathrm{predicted}_{i}}}\) :

Simulation results

U :

Streamwise velocity in the cross-flow inlet (m/s)

\({Q}_{\mathrm {o}}\) :

Flow rate in the side-channel inlet (\(\hbox {m}^{{3}}\hbox {/s}\))

\({C}_{\mathrm {1}}\), \({A}_{\mathrm {s}}\) , \({U}^{{*}}\) :

Related to the turbulence model

\({y}_{\mathrm {m }}\) :

\({x}_{\mathrm {j}}\) Coordinate of \({T}_{\mathrm {m}}\) (m)

\({\bar{u}}_{i} \), \({\bar{u}}_{j} \), \({\bar{u}}_{k}\) :

Velocity components in the i, j, and k direction (m/s)

\({x}_{\mathrm {i}}\), \({x}_{\mathrm {j}}\), \({x}_{\mathrm {k}}\) :

Cartesian coordinates in the i, j, and k direction (m)

\(\lambda _{{2}}\) :

Lambda-2 criterion

n :

Number of data

I :

Spatial index

KK :

Time-series index (s)

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  1. Department of Civil Engineering, Razi University, Kermanshah, 6714414971, Iran

    Milad Khosravi & Mitra Javan

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Khosravi, M., Javan, M. Three-dimensional flow structure and mixing of the side thermal buoyant jet discharge in cross-flow. Acta Mech 231, 3729–3753 (2020). https://doi.org/10.1007/s00707-020-02700-z

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