Improvement of Non-Hydrostatic Hydrodynamic Solution Using a Novel Free-Surface Boundary Condition
Abstract
:1. Introduction
2. Methods
2.1. Governant Equations
2.2. Grid and Variables Locations
2.3. Numerical Approximation
- Definition of initial parameters, initial conditions, and boundary conditions.
- Solution of convective terms using the Eulerian–Lagrangian Method.
- Determination of the provisional free-surface elevation () through the preconditioned conjugate gradient iterations until the residual norm is smaller than a given tolerance .
- Numeric solution of provisional velocity field ( and ).
- Solution of non-hydrostatic pressure (q) through the preconditioned conjugate gradient iterations until the residual norm is smaller than a given tolerance .
- Numeric correction of velocity field and free surface elevation.
2.3.1. Non-Hydrostatic Pressure Discretization
2.3.2. Free-Surface Boundary Condition Treatment
- (i)
- For bottom and middle layers (i.e., to ), the Equation (17) is applied using its original form;
- (ii)
- For the top layer (), Equation (17) is adapted to take into account the influence of FSFS height () in and :
- (iii)
- For the FSFS layer (), Equation (18) is adapted to take into account the height and the velocity field in the layer M. Preliminary simulations showed that making a stable solution is achieved for any vertical discretization:
2.4. Numerical Experiments
- a
- Standing waves in a three-dimensional closed basin: This test was widely applied in the literature to verify the model’s ability to simulate 3D linear waves comparing the analytic solution with the numerical solution in regard to phase and amplitude representation [8,9,11,20]. We evaluated the model capability in calculating the wave celerity and frequency wave dispersion with the Classic Boundary Approach (hereafter named CBA) and with the proposed FSFS boundary condition. We used six different vertical resolutions (20~5 layers), as most of the previous studies do, and since previous analyses showed that more than 20 layers do not have substantial improvement over 20 layers non-hydrostatic solution. We compare the free surface elevation cumulative phase error, the mean one time-step computational cost, the number of wave periods, and the relation with the free surface vertical velocity after 30 s of simulation in comparison with the analytical solution. We also compared the free surface elevation results with some metrics (RMSE, BIAS, Volume Error, KGE, NSE). At last, we statistically tested the residual series (the difference between analytic and simulated results) with the non-parametric Kruskal–Wallis test followed by a post-hoc Nemenyi to identify significant differences concerning the analytic results. The mean time of one time-step simulation was computed using an Intel® Xenon® CPU-E5-1620 3.7 GHz computer with 32 GB of RAM in a Fortran-based numerical model.
- b
- The wave propagation over a submerged bar: This test case was an experimental model idealized by [32], and was frequently used to validate numerical models (e.g., [7,8,24,26,28,33]). This experiment was used to evaluate accuracy to represent a non-linear wave pattern due to physical changes at the bottom, by comparing free surface elevation between the FSFS approach with a different vertical resolution between simulated and experimental results. To evaluate the model’s performance was used a few metrics (RMSE, BIAS, Volume Error, KGE, NSE) and the statistical test non-parametric Kruskal–Wallis test followed by a post-hoc Nemenyi test applied at residuals series.
3. Results
3.1. 3D Standing Waves in a Closed Basin
3.2. Wave Propagation over a Submerged Bar
4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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N°L | FSFS | CBA | ||||
---|---|---|---|---|---|---|
N° T | N° T | |||||
20-Layers | 2.8 | 10 | 0.3 | 2.62 | 10 | 1.6 |
16-Layers | 2.21 | 10 | 0.4 | 2.14 | 10 | 2 |
13-Layers | 1.68 | 10 | 0.6 | 1.67 | 10 | 2.4 |
10-Layers | 1.54 | 10 | 1 | 1.45 | 11 | 3.1 |
8-Layers | 1.51 | 10 | 1.5 | 1.34 | 11 | 3.8 |
5-Layers | 1.22 | 11 | 2.1 | 1.08 | 12 | 4.6 |
Metrics | FSFS-20L | FSFS-16L | FSFS-13L | FSFS-10L | FSFS-8L | FSFS-5L |
---|---|---|---|---|---|---|
RMSE (mm) | 7.40 | 11.52 | 18.29 | 30.53 | 45.30 | 89.20 |
BIAS (mm) | 0.32 | 0.32 | −0.77 | −2.11 | −3.54 | −7.61 |
Error (%) | 9.25 | 14.28 | 22.55 | 37.79 | 56.60 | 114.13 |
KGE | 0.93 | 0.93 | 0.82 | 0.52 | 0.18 | −0.88 |
NSE | 0.99 | 0.97 | 0.93 | 0.81 | 0.57 | −0.66 |
Metrics | CBA-20L | CBA-16L | CBA-13L | CBA-10L | CBA-8L | CBA-5L |
---|---|---|---|---|---|---|
RMSE (mm) | 47.06 | 57.68 | 68.75 | 83.48 | 95.63 | 107.80 |
BIAS (mm) | −4.35 | −5.65 | −6.88 | −7.96 | −7.57 | −1.06 |
Error (%) | 58.94 | 72.70 | 87.10 | 106.42 | 123.39 | 145.02 |
KGE | 0.004 | −0.30 | −0.61 | −0.91 | −0.93 | −0.20 |
NSE | 0.54 | 0.31 | 0.017 | −0.45 | −0.90 | −1.42 |
N°-L | FSFS-20L | FSFS-16L | FSFS-13L | FSFS-10L | FSFS-8L |
---|---|---|---|---|---|
FSFS-16L | 0.97833 | - | - | - | - |
FSFS-13L | 0.24554 | 0.69401 | - | - | - |
FSFS-10L | <0.05 | <0.05 | 0.05269 | - | - |
FSFS-8L | <0.05 | <0.05 | <0.05 | <0.05 - | |
FSFS-5L | <0.05 | <0.05 | <0.05 | 0.70396 | 0.5515 |
Station a: x = 13.5 m | Station d: x = 17.3 m | |||||||
---|---|---|---|---|---|---|---|---|
Metrics | 20L | 16L | 10L | 5L | 20L | 16L | 10L | 5L |
RMSE (mm) | 2.67 | 2.31 | 3.40 | 5.23 | 4.12 | 4.08 | 5.57 | 8.05 |
BIAS (mm) | −0.67 | −0.70 | −1.26 | −2.70 | −0.60 | −0.44 | −1.34 | −3.14 |
Error (%) | 26.47 | 25.37 | 34.07 | 53.94 | 59.29 | 57.97 | 79.03 | 116.84 |
KGE | −0.95 | −1.06 | −2.69 | −6.92 | −4.57 | −3.06 | −11.30 | −27.85 |
NSE | 0.92 | 0.94 | 0.87 | 0.69 | 0.76 | 0.76 | 0.56 | 0.08 |
Station b: x = 14.5 m | Station e: x = 19.0 m | |||||||
RMSE (mm) | 3.68 | 4.21 | 4.18 | 6.74 | 4.30 | 4.23 | 5.69 | 7.22 |
BIAS (mm) | −0.80 | −0.73 | −1.23 | −2.58 | −0.65 | −0.50 | −1.39 | −3.28 |
Error (%) | 37.73 | 43.83 | 44.30 | 76.27 | 48.15 | 47.35 | 62.91 | 75.29 |
KGE | −0.83 | −0.68 | −1.80 | −4.88 | −54.45 | −41.24 | −117.00 | −277.78 |
NSE | 0.82 | 0.77 | 0.77 | 0.41 | 0.72 | 0.72 | 0.50 | 0.20 |
Station c: x = 15.7 m | Station f: x = 21.0 m | |||||||
RMSE (mm) | 3.91 | 3.64 | 5.45 | 7.64 | 4.57 | 4.59 | 5.61 | 7.09 |
BIAS (mm) | −0.63 | −0.48 | −1.37 | −3.32 | −0.65 | −0.48 | −1.38 | −3.25 |
Error (%) | 44.46 | 40.98 | 62.63 | 88.06 | 56.63 | 56.12 | 67.62 | 76.43 |
KGE | −1.13 | −0.63 | −3.58 | −10.09 | −9.80 | −7.09 | −22.02 | −53.11 |
NSE | 0.79 | 0.82 | 0.60 | 0.21 | 0.69 | 0.68 | 0.53 | 0.24 |
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Cunha, A.H.F.; Fragoso, C.R., Jr.; Chalegre, C.L.B.; Motta-Marques, D. Improvement of Non-Hydrostatic Hydrodynamic Solution Using a Novel Free-Surface Boundary Condition. Water 2020, 12, 1271. https://doi.org/10.3390/w12051271
Cunha AHF, Fragoso CR Jr., Chalegre CLB, Motta-Marques D. Improvement of Non-Hydrostatic Hydrodynamic Solution Using a Novel Free-Surface Boundary Condition. Water. 2020; 12(5):1271. https://doi.org/10.3390/w12051271
Chicago/Turabian StyleCunha, Augusto Hugo Farias, Carlos Ruberto Fragoso, Jr., Cayo Lopes Bezerra Chalegre, and David Motta-Marques. 2020. "Improvement of Non-Hydrostatic Hydrodynamic Solution Using a Novel Free-Surface Boundary Condition" Water 12, no. 5: 1271. https://doi.org/10.3390/w12051271