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Figure 4-7. Normalized terminal height rise for different experimental conditions for different wave periods (T1=Yellow, T2= Blue, T3=Gray, without wave=Red), jet angle is (a) 45 (b) 60 environment, for a constant wave height and different wave periods, which are indicated in Moreover, in the presence of only the current (symbols with red color), the terminal rise height

Figure 4 -7. Normalized terminal height rise for different experimental conditions for different wave periods (T1=Yellow, T2= Blue, T3=Gray, without wave=Red), jet angle is (a) 45 (b) 60 environment, for a constant wave height and different wave periods, which are indicated in Moreover, in the presence of only the current (symbols with red color), the terminal rise height

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causes variations in the third (vertical) direction. (Vreugdenhil, 1994). Moreover, density stratification due to differences in temperature or salinity
causes variations in the third (vertical) direction. (Vreugdenhil, 1994). Moreover, density stratification due to differences in temperature or salinity
Figure 1-3 Dispersion of the effluent from a negatively buoyant jet
Figure 1-3 Dispersion of the effluent from a negatively buoyant jet
waves on jets. Dilution was determined by measuring the conductivity where the plume surfaces
waves on jets. Dilution was determined by measuring the conductivity where the plume surfaces
Figure 1-6 Fluid element for conservation law equal to the rate at which mass leaves the system plus the accumulation of mass within the
Figure 1-6 Fluid element for conservation law equal to the rate at which mass leaves the system plus the accumulation of mass within the
in which, vectors U, F, G, and Hf are defined as nd v are the depth-averaged velocities in x- and y-direction, respectively, p is the average density
in which, vectors U, F, G, and Hf are defined as nd v are the depth-averaged velocities in x- and y-direction, respectively, p is the average density
Figure 2-1 Triangular cell used in the proposed model Kurganov and Petrova to variable density shallow water equations written as
Figure 2-1 Triangular cell used in the proposed model Kurganov and Petrova to variable density shallow water equations written as
Figure 2-2 Scheme used for calculating non-limited derivatives
Figure 2-2 Scheme used for calculating non-limited derivatives
const. ). Therefore, the source term should satisfy the following two conditions: To derive the desired well-balanced quadrature for the integrals in (2.12), we first apply the zero, when experiencing the so-called “lake at rest” situation (h + B = const.,u=v=0,p =
const. ). Therefore, the source term should satisfy the following two conditions: To derive the desired well-balanced quadrature for the integrals in (2.12), we first apply the zero, when experiencing the so-called “lake at rest” situation (h + B = const.,u=v=0,p =
Figure 2-3 Notations used in calculating the spatial derivatives
Figure 2-3 Notations used in calculating the spatial derivatives
Various aspects of the numerical method explained above are further tested and verified in this section. The accuracy of the method is tested for several benchmarks and compared to the available solutions in the literature. Also in order to justify and highlight the well-balanced property of the solution, the results of a well-balanced and non-well-balanced solutions for small perturbations are
Various aspects of the numerical method explained above are further tested and verified in this section. The accuracy of the method is tested for several benchmarks and compared to the available solutions in the literature. Also in order to justify and highlight the well-balanced property of the solution, the results of a well-balanced and non-well-balanced solutions for small perturbations are
Figure 2-4(a) Grid, and (b) bed topography used for benchmark 1 topography of the hump is shown in Figure 2.4(b). The simulation time-step was set to 0.001 sec. The computational domain is [0,2] x [0,1], with a
Figure 2-4(a) Grid, and (b) bed topography used for benchmark 1 topography of the hump is shown in Figure 2.4(b). The simulation time-step was set to 0.001 sec. The computational domain is [0,2] x [0,1], with a
Figure 2-5 Water surface level for benchmark 1, at t=0.07 s: (a) 3-D view, (b) top plot
Figure 2-5 Water surface level for benchmark 1, at t=0.07 s: (a) 3-D view, (b) top plot
Figure 2-6(a) Water surface elevation, and (b) water velocity vectors at 7.3 sec after the start of the analysis solution in Brice et al. (2010). Also, the velocity vectors in the domain are shown in Figure 2.6(b). 2.3.2.2. Variable density case
Figure 2-6(a) Water surface elevation, and (b) water velocity vectors at 7.3 sec after the start of the analysis solution in Brice et al. (2010). Also, the velocity vectors in the domain are shown in Figure 2.6(b). 2.3.2.2. Variable density case
Figure 2-7(a) Water surface elevation, and (b) density distribution 5.2 sec after the dam break initiation 2.3.3. Variable density circular dam break f the fluids is presented: the results demonstrate that the smooth solution and the symmetry are
Figure 2-7(a) Water surface elevation, and (b) density distribution 5.2 sec after the dam break initiation 2.3.3. Variable density circular dam break f the fluids is presented: the results demonstrate that the smooth solution and the symmetry are
Figure 2-9 Variable density circular dam break, Predicted free surface elevation at t=0.55(a)pin R= 1000 kg/m’, (b) pin R= 200 kg/m?
Figure 2-9 Variable density circular dam break, Predicted free surface elevation at t=0.55(a)pin R= 1000 kg/m’, (b) pin R= 200 kg/m?
Figure 2-10 Variable density circular dam break, Predicted depth-averaged velocity in x direction at t=0.55. (a)pin R= 1000 kg/m’, (b) pin R= 200 kg/m?
Figure 2-10 Variable density circular dam break, Predicted depth-averaged velocity in x direction at t=0.55. (a)pin R= 1000 kg/m’, (b) pin R= 200 kg/m?
Figure 2-11 Dry bed test case (a) initial depth profile, (b) velocity distribution at t= 4 (s), and (c) depth profile at t= 4 (s) in the velocity profile. The present results are very similar to the reference solution (Toro 2001)
Figure 2-11 Dry bed test case (a) initial depth profile, (b) velocity distribution at t= 4 (s), and (c) depth profile at t= 4 (s) in the velocity profile. The present results are very similar to the reference solution (Toro 2001)
Figure 2-12(a) Topology, and (b) density perturbation in benchmark 5 into 4x200x200 triangles and the time step is At = 0.001 sec. The bed topography is given as (2.34) advantage of using a well-balanced scheme. The computational domain is [0, 2] x [0, 1] discretized
Figure 2-12(a) Topology, and (b) density perturbation in benchmark 5 into 4x200x200 triangles and the time step is At = 0.001 sec. The bed topography is given as (2.34) advantage of using a well-balanced scheme. The computational domain is [0, 2] x [0, 1] discretized
Figure 2-13 Time history of the water surface elevation computed using the well-balanced (left column) and non- well-balanced (right column) central upwind scheme.
Figure 2-13 Time history of the water surface elevation computed using the well-balanced (left column) and non- well-balanced (right column) central upwind scheme.
Figure 2-14(a) Plan view, and (b) bed topography in benchmark 6
Figure 2-14(a) Plan view, and (b) bed topography in benchmark 6
Figure 2-15 Time history of the water elevation in the solution computed by the well-balanced (left column) and non-well-balanced (right column) central upwind scheme in benchmark 6
Figure 2-15 Time history of the water elevation in the solution computed by the well-balanced (left column) and non-well-balanced (right column) central upwind scheme in benchmark 6
where p is the pressure, u; and u, are the velocities in the x- and y-directions, respectively, p is equations for the Cartesian coordinates for the LES method are as follows (Pope, 2002): the fluid density, g is the gravitational acceleration, 1 is the fluid viscosity, @ is the scal: concentration, I" is the scalar diffusivity, t,; are the SGS Reynold stresses, and Q; is the SG
where p is the pressure, u; and u, are the velocities in the x- and y-directions, respectively, p is equations for the Cartesian coordinates for the LES method are as follows (Pope, 2002): the fluid density, g is the gravitational acceleration, 1 is the fluid viscosity, @ is the scal: concentration, I" is the scalar diffusivity, t,; are the SGS Reynold stresses, and Q; is the SG
Figure 3-1 Hydraulic flume layout and computational domain used in LES model of flow over a dune
Figure 3-1 Hydraulic flume layout and computational domain used in LES model of flow over a dune
instabilities like Kelvin-Helmholtz or barotropic seiches. Figure 3-2 The unstructured mesh of the domain of the LES model of flow over a dune performed around the dune to show the capability of the model in reproducing shear interfacial
instabilities like Kelvin-Helmholtz or barotropic seiches. Figure 3-2 The unstructured mesh of the domain of the LES model of flow over a dune performed around the dune to show the capability of the model in reproducing shear interfacial
Figure 3-3 (a) Using different filters for PIV and LIF cameras; (b) PIV, LIF and raw images of the experiment 3.4.Results and discussion experiments were used in this study to verify the numerical simulation. In this section, the results of the numerical simulation with LES and the experimentally-collected
Figure 3-3 (a) Using different filters for PIV and LIF cameras; (b) PIV, LIF and raw images of the experiment 3.4.Results and discussion experiments were used in this study to verify the numerical simulation. In this section, the results of the numerical simulation with LES and the experimentally-collected
Figure 3-4 Time-averaged mean velocity profile of the flow over the dune: (a) experimental results; (b) LES numerical model prediction (red line is the shear layer position) dune’s summit, it accelerates and continues downstream.
Figure 3-4 Time-averaged mean velocity profile of the flow over the dune: (a) experimental results; (b) LES numerical model prediction (red line is the shear layer position) dune’s summit, it accelerates and continues downstream.
Figure 3-5 LES predicted vs. observed values of the velocity for three different flow regions: (a) for -100<x<0 mm; (b) for 0<x<50 mm; and (c) for 50<x<100 mm
Figure 3-5 LES predicted vs. observed values of the velocity for three different flow regions: (a) for -100<x<0 mm; (b) for 0<x<50 mm; and (c) for 50<x<100 mm
Figure 3-6 Comparison of the LES modelled and measured velocity shear interface positions
Figure 3-6 Comparison of the LES modelled and measured velocity shear interface positions
Figure 3-7 Comparison of the LES modelled and measured vertical velocity profile:
Figure 3-7 Comparison of the LES modelled and measured vertical velocity profile:
the slope. This pattern was also observed by Fouli and Zhu (2011). Figure 3.8 also shows that
the slope. This pattern was also observed by Fouli and Zhu (2011). Figure 3.8 also shows that
Figure 3-8 Series of instantaneous images of the: (a) measured and (b) LES modeled (c) RANS modeled two-layer flows. The time lag between the images is 0.1 s.
Figure 3-8 Series of instantaneous images of the: (a) measured and (b) LES modeled (c) RANS modeled two-layer flows. The time lag between the images is 0.1 s.
amplitude over time because of frictional losses. In this study, in both the numerical and the experimental graphs, the net flow rate oscillates around
amplitude over time because of frictional losses. In this study, in both the numerical and the experimental graphs, the net flow rate oscillates around
modeling small-scale turbulent features of a flow field. Figure 3-10 Time-averaged thickness of the shear layer obtained from the numerical model results and data measured from experimental tests
modeling small-scale turbulent features of a flow field. Figure 3-10 Time-averaged thickness of the shear layer obtained from the numerical model results and data measured from experimental tests
Figure 3-11 Time-averaged Bulk Richardson number obtained from the numerical model results and data measured from experimental tests
Figure 3-11 Time-averaged Bulk Richardson number obtained from the numerical model results and data measured from experimental tests
Figure 3-12 Time-averaged entrainment rates obtained from the numerical model results and the experimental tests downstream, the flow entrainments start to decrease.
Figure 3-12 Time-averaged entrainment rates obtained from the numerical model results and the experimental tests downstream, the flow entrainments start to decrease.
Figure 4-1. Experimental set up for PIV-LIF study of buoyant plume under the combined effects of waves and current effluent and the ambient water was 2.136. The laser set up used for this experiment is shown in Figure 4.1. The laser source is placed below
Figure 4-1. Experimental set up for PIV-LIF study of buoyant plume under the combined effects of waves and current effluent and the ambient water was 2.136. The laser set up used for this experiment is shown in Figure 4.1. The laser source is placed below
Figure 4-2. LIF images of mean concentration field of an INBJ with Fr=34 released with an angle of 45 to the horizontal in receiving environment with wave with T=0.9 (s) and H=1.9 cm
Figure 4-2. LIF images of mean concentration field of an INBJ with Fr=34 released with an angle of 45 to the horizontal in receiving environment with wave with T=0.9 (s) and H=1.9 cm
Figure 4-3. Jet centerline superimposed onto the contours of the velocity field (W-01) are removed and gaps were filled using interpolation.
Figure 4-3. Jet centerline superimposed onto the contours of the velocity field (W-01) are removed and gaps were filled using interpolation.
Figure 4-5. Characteristics of a single inclined dense jet in stagnant water Table 4-2. Comparison of coefficients for stagnant ambient for deep water condition
Figure 4-5. Characteristics of a single inclined dense jet in stagnant water Table 4-2. Comparison of coefficients for stagnant ambient for deep water condition
Figure 4-6. Dimensionless relationship between the centerline trajectory and the horizontal distance in MDFF region for the jet in (a) and (c) waves only for 45 and 60 (b) and (d) waves and current for 45 and 60. numbers are normalized by the length scales, Jy and Ja, corresponding to each experimental As shown in Figures 4.6a and 4.6c, the relationship between the jet centerline trajectory and th Centerline trajectories of the jets versus horizontal distance in the presence of wave only and for
Figure 4-6. Dimensionless relationship between the centerline trajectory and the horizontal distance in MDFF region for the jet in (a) and (c) waves only for 45 and 60 (b) and (d) waves and current for 45 and 60. numbers are normalized by the length scales, Jy and Ja, corresponding to each experimental As shown in Figures 4.6a and 4.6c, the relationship between the jet centerline trajectory and th Centerline trajectories of the jets versus horizontal distance in the presence of wave only and for
to the wave-only and the wave-current conditions, respectively. respectively. The terminal rise heights are normalized by the length scales z, and za corresponding periods, in wave only environment, can be approximated by the following relationship: Terminal rise height of the jet with an angle of 45'versus various wave periods for the wave-only
to the wave-only and the wave-current conditions, respectively. respectively. The terminal rise heights are normalized by the length scales z, and za corresponding periods, in wave only environment, can be approximated by the following relationship: Terminal rise height of the jet with an angle of 45'versus various wave periods for the wave-only
Figure 4-9. Normalized centerline peak in different experimental conditions for different (T1=Yellow, T2= Blue, T3=Gray, without waves=Red), jet angle is (a) 45 (b) 60 similar behavior, the results are only presented for the jet with angle of 45.
Figure 4-9. Normalized centerline peak in different experimental conditions for different (T1=Yellow, T2= Blue, T3=Gray, without waves=Red), jet angle is (a) 45 (b) 60 similar behavior, the results are only presented for the jet with angle of 45.
Figure 4-10. Comparison of mean concentration field of the jet in different wave and current conditions
Figure 4-10. Comparison of mean concentration field of the jet in different wave and current conditions
Figure 4-4-21. Minimum dilution at centerline peak as a function of wave period for different environment condition
Figure 4-4-21. Minimum dilution at centerline peak as a function of wave period for different environment condition
Figure 4-14-3. Minimum dilution at return point as a function of wave period is presented in Figure 4.12 for wave-only tests.
Figure 4-14-3. Minimum dilution at return point as a function of wave period is presented in Figure 4.12 for wave-only tests.
Figure 4-4-43. Cross-sectional velocity distribution at various locations (W-01) downstream of the nozzle Gaussian distribution held up to 19D.
Figure 4-4-43. Cross-sectional velocity distribution at various locations (W-01) downstream of the nozzle Gaussian distribution held up to 19D.
Figure 4-14. Normalized velocity profiles at various downstream cross-sections (W-01) 4.4. Conclusions
Figure 4-14. Normalized velocity profiles at various downstream cross-sections (W-01) 4.4. Conclusions
Figure 5-1 Experimental setup for PIV-LIF study of buoyant plumes under the combined effects of waves and current (Khorshid et al., 2019)
Figure 5-1 Experimental setup for PIV-LIF study of buoyant plumes under the combined effects of waves and current (Khorshid et al., 2019)
Figure 0-2. Computational domain: (a) refined unstructured mesh system; (b) mesh refinement around the nozzle (zoomed in
Figure 0-2. Computational domain: (a) refined unstructured mesh system; (b) mesh refinement around the nozzle (zoomed in
where u, v, and w are the mean velocity components in the x, y, and z directions, respectively, fis the turbulent kinematic viscosity and v is the kinematic viscosity, g is the gravity acceleration, p
where u, v, and w are the mean velocity components in the x, y, and z directions, respectively, fis the turbulent kinematic viscosity and v is the kinematic viscosity, g is the gravity acceleration, p
Figure 5-3. Water and air phases in simulated case 5.2.3. Turbulence models:
Figure 5-3. Water and air phases in simulated case 5.2.3. Turbulence models:
Table 5-1. Numerical cases - characteristic presented in the following sections. The characteristics of the three cases are summarized below. The inclined negatively buoyant jets are characterized by the jet discharge volume flux (Qo) and (totalling 12 simulations: 3 cases x 4 turbulence models), and the comparative results are the kinematic momentum flux (Mo), as given below (Lai and Lee, 2012):
Table 5-1. Numerical cases - characteristic presented in the following sections. The characteristics of the three cases are summarized below. The inclined negatively buoyant jets are characterized by the jet discharge volume flux (Qo) and (totalling 12 simulations: 3 cases x 4 turbulence models), and the comparative results are the kinematic momentum flux (Mo), as given below (Lai and Lee, 2012):
Figure 0-5. Concentration map of an INBJ in a receiving environment with waves of T=0.9 (s) and H=1.9 cm (W-01); results of LRR simulation outlet.
Figure 0-5. Concentration map of an INBJ in a receiving environment with waves of T=0.9 (s) and H=1.9 cm (W-01); results of LRR simulation outlet.
Table 5-2. Error coefficient of calculated the centerline trajectory using various turbulence models In Table 5.3-5.5, using dimensional analysis, the various geometrical and mixing quantities are
Table 5-2. Error coefficient of calculated the centerline trajectory using various turbulence models In Table 5.3-5.5, using dimensional analysis, the various geometrical and mixing quantities are
Table 5-3. Comparison of numerical and experimental coefficients for wave period 0.9 (s) Table 5-4. Comparison of numerical and experimental coefficients for wave period 1.1 (s)
Table 5-3. Comparison of numerical and experimental coefficients for wave period 0.9 (s) Table 5-4. Comparison of numerical and experimental coefficients for wave period 1.1 (s)
Table 5-5. Comparison of numerical and experimental coefficients for wave period 1.3 (s)
Table 5-5. Comparison of numerical and experimental coefficients for wave period 1.3 (s)
Figure 5-7. Dimensionless relationship between the centerline trajectory and the horizontal distance in the JDR region
Figure 5-7. Dimensionless relationship between the centerline trajectory and the horizontal distance in the JDR region
Table 5-6. Error coefficient of calculated the centerline trajectory using various turbulence models in the JDR regior 5.3.3. Terminal Rise Height
Table 5-6. Error coefficient of calculated the centerline trajectory using various turbulence models in the JDR regior 5.3.3. Terminal Rise Height
Table 5-7. Error coefficient of the calculated terminal rise height for various turbulence models Table 5-8. Error coefficient of the calculated terminal rise location for various turbulence models data comparing to the RNG and the non-linear k-e.
Table 5-7. Error coefficient of the calculated terminal rise height for various turbulence models Table 5-8. Error coefficient of the calculated terminal rise location for various turbulence models data comparing to the RNG and the non-linear k-e.
Figure 5-8. Normalized terminal rise height (circle symbol is associated to W-01, diamond symbols associated to W-02, and triangular symbols are associated to W-03)
Figure 5-8. Normalized terminal rise height (circle symbol is associated to W-01, diamond symbols associated to W-02, and triangular symbols are associated to W-03)
Figure 0-9. Dimensionless relationship between the terminal rise height and the wave periods for the jet
Figure 0-9. Dimensionless relationship between the terminal rise height and the wave periods for the jet
Table 5-9. E,, and E,, values for four turbulence models and comparison with values from experimental results results.
Table 5-9. E,, and E,, values for four turbulence models and comparison with values from experimental results results.
Figure 5-5-20. Minimum dilution at centerline peak
Figure 5-5-20. Minimum dilution at centerline peak
Table 5-10. Error coefficient of calculated minimum dilution at centerline peak for the various turbulence model
Table 5-10. Error coefficient of calculated minimum dilution at centerline peak for the various turbulence model
Figure 0-11. Minimum dilution at the return point as a function of wave period are in better agreement with the experimental data than the other turbulence models.
Figure 0-11. Minimum dilution at the return point as a function of wave period are in better agreement with the experimental data than the other turbulence models.
Table 5-11. Ey» and E,- values for the four turbulence models and comparison with values from experimental results
Table 5-11. Ey» and E,- values for the four turbulence models and comparison with values from experimental results
Table 5-12. Error coefficient of calculated minimum dilution at the return point for the various turbulence model; 5.3.5. Cross-sectional Velocity Profile with the experimental results, with the LRR being slightly better than the realizable k-e model.
Table 5-12. Error coefficient of calculated minimum dilution at the return point for the various turbulence model; 5.3.5. Cross-sectional Velocity Profile with the experimental results, with the LRR being slightly better than the realizable k-e model.
Table 5-13. Error coefficient of calculated normalized velocity for the upper half of the three cross-sections for the various turbulence models 5.4. Discussion and Conclusions
Table 5-13. Error coefficient of calculated normalized velocity for the upper half of the three cross-sections for the various turbulence models 5.4. Discussion and Conclusions
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