Combined Use of High-Resolution Numerical Schemes to Reduce Numerical Diffusion in Coupled Nonhydrostatic Hydrodynamic and Solute Transport Model
Abstract
:1. Introduction
2. Mathematical Considerations
2.1. Governing Equations
2.2. Grid and Variable Locations
2.3. Numerical Approach
2.3.1. Rans Equations
2.3.2. Solute Transport Equation
3. High-Resolution Schemes to Reduce Numerical Diffusion
3.1. Flux-Limiter
3.2. Bilinear Interpolator
- (1)
- For each time step, the particle starts at the barycenter of the j-th face at layer k, where the multistep backward Euler stream line is defined by linear interpolations to find the particle position at the end of a sub-time step (ELM step-i);
- (2)
- For each sub-time step, the particle position is analyzed in relation to the initial position (up-right, up-left, down-right, or down-left);
- (3)
- Eight node points are selected in the cell, with four points in layer k (the same layer where the particle stops) and four points in layer , depending on the particle position (up or down). The used points are always 4-face barycenters and 4 edge centers, except for the top and bottom cells, which uses 2-face barycenters, 4 edge centers, and 2 nodes;
- (4)
- The node indices are defined anticlockwise, where the first node is the initial position of the particle at time n+1;
- (5)
- Equation (31) is used to calculate the velocity component in the particle position in the sub-time step;
- (6)
- Steps (1–5) are repeated until the end of the Lagrangian trajectory.
- (1)
- For each time step, the particle starts at the horizontal-face barycenter of the i-th element at layer , where the multistep backward Euler stream line is defined by linear interpolations to find the particle position at the end of a sub-time step (ELM step-i)
- (2)
- For each sub-time step, the particle position is analyzed in relation to the initial position (up or down and in the direction of one of the four quadrants)
- (3)
- Eight node points are selected in the cell, with four points in layer (the same layer where the particle starts) and four points in layer , if the particle goes down, or , if the particle goes up. The used point always has a two-horizontal face barycenter, 4 edge centers, and 2 nodes;
- (4)
- The node indices are defined anticlockwise, where the first node is the initial position of the particle at time .
- (5)
- Equation (31) is used to calculate the velocity component in the particle position in the sub-time step.
- (6)
- Steps (1–5) are repeated until the end of the Lagrangian trajectory.
3.3. Quadratic Interpolator
- (1)
- First, 9 z-direction interpolations are carried out (black lines in Figure 3). Each vertical interpolation generates a velocity component in the horizontal plane that passes through the z-position of the particle, given by
- (2)
- Three x-direction interpolations are carried out (orange line), using the estimated velocities found in step (1), resulting in three new velocity components (white diamonds in Figure 3), given by
- (3)
- One interpolation is made to compute the y-direction displacement and find the final velocity for the particle at time
- (4)
- A new departure velocity is used to define the particle position in the next sub-time step.
- (5)
- Steps (1–4) are repeated until the end of the Lagrangian trajectory.
4. Numerical Experiments
- Standing waves in a three-dimensional closed basin: this test case verifies the capability of the model to simulate 3D linear waves, including phase and amplitude representation [6,20,48]. The motion in the basin is caused only by the initial condition of the free surface. When the roughness, viscosity and diffusivity coefficients are set equal to zero, the motion of the free surface should not lose energy. However, a wave damping is caused by the numerical diffusion of the hydrodynamic solution. We evaluated the differences in the numerical diffusion between the bilinear and quadratic interpolations, applied in ELM step-ii, as well as the numerical diffusion considering the no-advection scheme.We also compared the mass conservation of the computation domain for each time step, the cumulative mass conservation over the course of the simulation, and the mean computation time of one time-step simulation.
- Wave propagation over a submerged bar: this was an experimental model idealized by Beji and Battjes [50], and has been frequently used to validate numerical models (e.g., [4,47,48,49,51,52,53]). The experiment was used to evaluate the accuracy of representing an irregular wave pattern caused by physical changes at the bottom, by comparing the quadratic and bilinear interpolations used in ELM step-ii.
- Gravity wave test: consists of a finite-amplitude deep-water standing wave in an inviscid fluid in a nonequilibrium situation, where the baroclinic pressure makes a major contribution to promote flow. The experiment evaluated the numerical diffusion in terms of density interface expansion, analyzing the difference between the combined uses of the interpolation techniques used in ELM step-ii and different flux-limiter schemes applied in a solute transport solution. This test case also evaluated the individual effect of each interpolation technique used in the hydrodynamic solution on the solute transport solution, using different flux limiters. We also compared the mean computation time of one time-step simulation.
4.1. 3D Standing Waves in a Closed Basin
4.2. Wave Propagation over a Submerged Bar
4.3. Gravity Wave
5. Discussion
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Metrics | Quadratic | Bilinear | No-Advection |
---|---|---|---|
RMSE (mm) | 21.86 | 20.80 | 20.70 |
BIAS (mm) | 0.18 | 0.52 | 0.56 |
Error (%) | 27.22 | 25.94 | 25.79 |
KGE | 0.87 | 0.66 | 0.64 |
NSE | 0.90 | 0.91 | 0.91 |
M | Station a: x = 13.5 m | Station d: x = 17.3 m | ||
---|---|---|---|---|
Bilinear | Quadratic | Bilinear | Quadratic | |
RMSE (mm) | 3.57 | 2.17 | 5.18 | 4.29 |
BIAS (mm) | 2.01 | 0.18 | 0.87 | −0.56 |
Error (%) | 47.52 | 23.65 | 83.78 | 60.00 |
KGE | −0.90 | 0.82 | 0.23 | 0.46 |
NSE | 0.83 | 0.94 | 0.56 | 0.70 |
Station b:x= 14.5 m | Station e:x= 19.0 m | |||
RMSE (mm) | 4.79 | 4.25 | 4.97 | 3.81 |
BIAS (mm) | 0.14 | −0.64 | 2.49 | 0.76 |
Error (%) | 64.77 | 50.22 | 52.79 | 45.31 |
KGE | 0.39 | 0.35 | −6.12 | −1.18 |
NSE | 0.33 | 0.47 | 0.60 | 0.76 |
Station c:x= 15.7 m | Station f:x= 21.0 m | |||
RMSE (mm) | 4.66 | 3.51 | 4.04 | 3.67 |
BIAS (mm) | −0.19 | −1.25 | 1.34 | −0.28 |
Error (%) | 56.71 | 39.63 | 57.48 | 43.84 |
KGE | 0.65 | −1.07 | −4.92 | −0.26 |
NSE | 0.67 | 0.81 | 0.69 | 0.75 |
Flux Limiters | Fringer et al., 2005 | |||
---|---|---|---|---|
UpWind | 5.10 | 4.98 | 4.98 | 3.9 |
MUSCL | 1.66 | 1.23 | 0.48 | 0.6 |
Ultimate-Quickest | 1.40 | 0.94 | 0.46 | 0.4 |
SuperBee | 1.36 | 0.88 | 0.30 | 0.4 |
Flux Limiters | RMSE | RMSE | RMSE |
---|---|---|---|
Upwind | 2.08 | 1.97 | 1.97 |
MUSCL | 1.16 | 0.72 | 0.38 |
Ultimate-Quickest | 1.15 | 0.69 | 0.28 |
Superbee | 1.16 | 0.70 | 0.24 |
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Cunha, A.H.F.; Fragoso, C.R.; Tavares, M.H.; Cavalcanti, J.R.; Bonnet, M.-P.; Motta-Marques, D. Combined Use of High-Resolution Numerical Schemes to Reduce Numerical Diffusion in Coupled Nonhydrostatic Hydrodynamic and Solute Transport Model. Water 2019, 11, 2288. https://doi.org/10.3390/w11112288
Cunha AHF, Fragoso CR, Tavares MH, Cavalcanti JR, Bonnet M-P, Motta-Marques D. Combined Use of High-Resolution Numerical Schemes to Reduce Numerical Diffusion in Coupled Nonhydrostatic Hydrodynamic and Solute Transport Model. Water. 2019; 11(11):2288. https://doi.org/10.3390/w11112288
Chicago/Turabian StyleCunha, Augusto Hugo Farias, Carlos Ruberto Fragoso, Matheus Henrique Tavares, J. Rafael Cavalcanti, Marie-Paule Bonnet, and David Motta-Marques. 2019. "Combined Use of High-Resolution Numerical Schemes to Reduce Numerical Diffusion in Coupled Nonhydrostatic Hydrodynamic and Solute Transport Model" Water 11, no. 11: 2288. https://doi.org/10.3390/w11112288
APA StyleCunha, A. H. F., Fragoso, C. R., Tavares, M. H., Cavalcanti, J. R., Bonnet, M.-P., & Motta-Marques, D. (2019). Combined Use of High-Resolution Numerical Schemes to Reduce Numerical Diffusion in Coupled Nonhydrostatic Hydrodynamic and Solute Transport Model. Water, 11(11), 2288. https://doi.org/10.3390/w11112288