# Improvement of Non-Hydrostatic Hydrodynamic Solution Using a Novel Free-Surface Boundary Condition

^{1}

^{2}

^{*}

## 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 ($\tilde{\eta}$) through the preconditioned conjugate gradient iterations until the residual norm is smaller than a given tolerance ${\u03f5}_{q}$.
- Numeric solution of provisional velocity field ($\tilde{u}$ and $\tilde{w}$).
- Solution of non-hydrostatic pressure (q) through the preconditioned conjugate gradient iterations until the residual norm is smaller than a given tolerance ${\u03f5}_{q}$.
- 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., $k=m$ to $M-1$), the Equation (17) is applied using its original form;
- (ii)
- For the top layer ($k=M$), Equation (17) is adapted to take into account the influence of FSFS height ($\mathsf{\Delta}{z}_{FSFS}$) in $\mathsf{\Delta}{z}_{i,M+\frac{1}{2}}^{n}$ and $\mathsf{\Delta}{z}_{j\left(i,l\right),k}^{n}$:$$\begin{array}{c}g{\theta}^{2}\mathsf{\Delta}{t}^{2}\left[{\displaystyle \sum _{l=1}^{{S}_{i}}}{s}_{i,L}{\lambda}_{j\left(i,l\right)}\left(\mathsf{\Delta}{z}_{j\left(i,l\right),M}^{n}-\mathsf{\Delta}{z}_{FSFS}\right)\frac{{\tilde{q}}_{i\left[j\left(i,l\right),1\right],M}^{n+1}-{\tilde{q}}_{i\left[j\left(i,l\right),2\right],M}^{n+1}}{{\delta}_{j\left(i,l\right)},M}\right]+\hfill \\ \left[{V}_{i}\left(\frac{{\tilde{q}}_{i,M}^{n+1}-{\tilde{q}}_{i,M+1}^{n+1}}{\mathsf{\Delta}{z}_{i,M+\frac{1}{2}}^{n}}-\frac{{\tilde{q}}_{i,M-1}^{n+1}-{\tilde{q}}_{i,M}^{n+1}}{\mathsf{\Delta}{z}_{i,M-\frac{1}{2}}^{n}}\right)\right]=g\theta \mathsf{\Delta}t{V}_{i}\left({\tilde{w}}_{i,M-\frac{1}{2}}^{n+1}-{\tilde{w}}_{i,M+\frac{1}{2}}^{n+1}\right)-\hfill \\ g\theta \mathsf{\Delta}t{\displaystyle \sum _{l=1}^{{S}_{i}}}{s}_{i,L}{\lambda}_{j\left(i,l\right)}\left(\mathsf{\Delta}{z}_{j\left(i,l\right),M}^{n}-\mathsf{\Delta}{z}_{FSFS}\right){\tilde{u}}_{j\left(i,l\right),M}^{n+1},\phantom{\rule{1.em}{0ex}}k=M,\hfill \end{array}$$$${z}_{i,M+\frac{1}{2}}^{n}=\frac{1}{2}[(\mathsf{\Delta}{z}_{i,M}^{n}-\mathsf{\Delta}{z}_{FSFS})+\mathsf{\Delta}{z}_{FSFS}]=\frac{1}{2}(\mathsf{\Delta}{z}_{i,M}^{n})$$$$\mathsf{\Delta}{z}_{i,M-\frac{1}{2}}^{n}=\frac{1}{2}[(\mathsf{\Delta}{z}_{i,M}^{n}-\mathsf{\Delta}{z}_{FSFS})+\mathsf{\Delta}{z}_{i,M-1}^{n}];\phantom{\rule{4pt}{0ex}}$$
- (iii)
- For the FSFS layer ($k=M+1$), Equation (18) is adapted to take into account the $FSFS$ height and the velocity field in the layer M. Preliminary simulations showed that making $\mathsf{\Delta}{z}_{FSFS}=0$ 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

**${\mathsf{\Phi}}_{\epsilon}$**) increases over time step, and it becomes bigger with the reduction of the number of layers (Table 1). For simulations CBA approach, these errors are higher; for instance, the 20-layer CBA simulation had comparable results with the 8-layers FSFS simulation (Figure 3). Furthermore, when phase error is critical, a decrease in the cumulative free surface elevation error in the low vertical resolution results occurs (less than ten) due to an increase in wave periods. Due to this, results with fewer layers appeared better than those with more layers, as can be seen by comparing FSFS-5L with FSFS-10L results (Figure 3). These effects complicate the direct comparison between methods; however, the cumulative residual free surface elevation error (Figure 4) can clarify the matter. Due to this, the calculated metrics (Table 2 and Table 3) was only comparable between 0 and 10 s of simulations (Figure 4). This effect can be better verified in CBA accumulated residual series when the graphic changes the slope, which occurs sooner as the number of layers reduced (Figure 4).

#### 3.2. Wave Propagation over a Submerged Bar

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Valipour, R.; Bouffard, D.; Boegman, L.; Rao, Y.R. Near-inertial waves in Lake Erie. Limnol. Oceanogr.
**2015**, 60, 1522–1535. [Google Scholar] [CrossRef] - De Brito, A.N., Jr.; Fragoso, C.R., Jr.; Larson, M. Tidal exchange in a choked coastal lagoon: A study of Mundaú Lagoon in northeastern Brazil. Reg. Stud. Mar. Sci.
**2018**, 17, 133–142. [Google Scholar] [CrossRef] - Vilas, M.P.; Marti, C.L.; Adams, M.P.; Oldham, C.E.; Hipsey, M.R. Invasive macrophytes control the spatial and temporal patterns of temperature and dissolved oxygen in a shallow lake: A proposed feedback mechanism of macrophyte loss. Front. Plant Sci.
**2017**, 8, 2097. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Soulignac, F.; Vinçon-Leite, B.; Lemaire, B.J.; Martins, J.R.S.; Bonhomme, C.; Dubois, P.; Mezemate, Y.; Tchiguirinskaia, I.; Schertzer, D.; Tassin, B. Performance Assessment of a 3D Hydrodynamic Model Using High Temporal Resolution Measurements in a Shallow Urban Lake. Environ. Model. Assess.
**2017**, 22, 309–322. [Google Scholar] [CrossRef] - Munar, A.M.; Cavalcanti, J.R.; Bravo, J.M.; Fan, F.M.; da Motta-Marques, D.; Fragoso, C.R., Jr. Coupling large-scale hydrological and hydrodynamic modeling: Toward a better comprehension of watershed-shallow lake processes. J. Hydrol.
**2018**, 564, 424–441. [Google Scholar] [CrossRef] - Zhang, J.; Liang, D.; Liu, H. A hybrid hydrostatic and non-hydrostatic numerical model for shallow flow simulations. Estuar. Coast. Shelf Sci.
**2018**, 205, 21–29. [Google Scholar] [CrossRef] - Stelling, G.; Zijlema, M. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation. Int. J. Numer. Methods Fluids
**2003**, 43, 1–23. [Google Scholar] [CrossRef] - Yuan, H.; Wu, C.H. An implicit three-dimensional fully non-hydrostatic model for free-surface flows. Int. J. Numer. Methods Fluids
**2004**, 46, 709–733. [Google Scholar] [CrossRef] - Zijlema, M.; Stelling, G.S. Further experiences with computing non-hydrostatic free-surface flows involving water waves. Int. J. Numer. Methods Fluids
**2005**, 48, 169–197. [Google Scholar] [CrossRef] - Lv, B. A higher-efficient three-dimensional numerical model for small amplitude free surface flows. China Ocean Eng.
**2014**, 28, 617–628. [Google Scholar] [CrossRef] - Liu, X.; Ma, D.G.; Zhang, Q.H. A higher-efficient non-hydrostatic finite volume model for strong three-dimensional free surface flows and sediment transport. China Ocean Eng.
**2017**, 31, 736–746. [Google Scholar] [CrossRef] - Escalante, C.; Fernández-Nieto, E.; de Luna, T.M.; Castro, M. An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves. J. Sci. Comput.
**2019**, 79, 273–320. [Google Scholar] [CrossRef] - Wadzuk, B.M.; Hodges, B.R. Hydrostatic and Non-Hydrostatic Internal Wave Models; Technical report; Center for Research in Water Resources, University of Texas at Austin: Austin, TX, USA, 2004. [Google Scholar]
- Bohacek, J.; Kharicha, A.; Ludwig, A.; Wu, M.; Karimi-Sibaki, E.; Paar, A.; Brandner, M.; Elizondo, L.; Trickl, T. A (non-) hydrostatic free-surface numerical model for two-layer flows. Appl. Math. Comput.
**2018**, 319, 301–317. [Google Scholar] [CrossRef] - Escalante, C.; de Luna, T.M.; Castro, M. Non-hydrostatic pressure shallow flows: GPU implementation using finite volume and finite difference scheme. Appl. Math. Comput.
**2018**, 338, 631–659. [Google Scholar] [CrossRef] [Green Version] - Kanarska, Y.; Maderich, V. A non-hydrostatic numerical model for calculating free-surface stratified flows. Ocean Dyn.
**2003**, 53, 176–185. [Google Scholar] [CrossRef] - Wang, K.; Jin, S.; Liu, G. Numerical modelling of free-surface flows with bottom and surface-layer pressure treatment. J. Hydrodyn.
**2009**, 21, 352–359. [Google Scholar] [CrossRef] - Casulli, V. A semi-implicit finite difference method for non-hydrostatic, free-surface flows. Int. J. Numer. Methods Fluids
**1999**, 30, 425–440. [Google Scholar] [CrossRef] - Namin, M.; Lin, B.; Falconer, R. An implicit numerical algorithm for solving non-hydrostatic free-surface flow problems. Int. J. Numer. Methods Fluids
**2001**, 35, 341–356. [Google Scholar] [CrossRef] - Monteiro, L.R.; Schettini, E.B.C. Comparação entre a aproximação hidrostática e a não-hidrostática na simulação numérica de escoamentos com superfície livre. Rev. Bras. De Recur. Hídricos
**2015**, 20, 1051–1062. [Google Scholar] - Bergh, J.; Berntsen, J. The surface boundary condition in nonhydrostatic ocean models. Ocean Dyn.
**2010**, 60, 301–315. [Google Scholar] [CrossRef] - Jankowski, J.A. A Non-Hydrostatic Model for Free Surface Flows. Ph.D. Thesis, Inst. für Strömungsmechanik und Elektronisches Rechnen im Bauwesen, University of Hannover, Hannover, Germany, 1999. [Google Scholar]
- Casulli, V.; Lang, G. Mathematical Model UnTRIM, Validation Document 1.0; The Federal Waterways Engineering and Research Institute (BAW): Hamburg, Germany, 2004. [Google Scholar]
- Cui, H.; Pietrzak, J.; Stelling, G. Improved efficiency of a non-hydrostatic, unstructured grid, finite volume model. Ocean Model.
**2012**, 54, 55–67. [Google Scholar] [CrossRef] - Lu, X.; Dong, B.; Mao, B.; Zhang, X. A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation. Ocean Model.
**2015**, 96, 187–202. [Google Scholar] [CrossRef] - Dingemans, M. Comparison of computations with Boussinesq-like models and laboratory measurements. In Memo in Framework of MAST Project (G8-M), Delft Hydraulics Memo H1684. 12; Deltares: Delft, The Netherlands, 1994. [Google Scholar]
- Fringer, O.; Armfield, S.; Street, R. Reducing numerical diffusion in interfacial gravity wave simulations. Int. J. Numer. Methods Fluids
**2005**, 49, 301–329. [Google Scholar] [CrossRef] - Yin, J.; Sun, J.W.; Wang, X.G.; Yu, Y.H.; Sun, Z.C. A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model. China Ocean Eng.
**2017**, 31, 261–271. [Google Scholar] [CrossRef] - Casulli, V. Semi-implicit finite difference methods for the two-dimensional shallow water equations. J. Comput. Phys.
**1990**, 86, 56–74. [Google Scholar] [CrossRef] - Casulli, V.; Cheng, R.T. Semi-implicit finite difference methods for three-dimensional shallow water flow. Int. J. Numer. Methods Fluids
**1992**, 15, 629–648. [Google Scholar] [CrossRef] - Hodges, B.R.; Imberger, J.; Saggio, A.; Winters, K.B. Modeling basin-scale internal waves in a stratified lake. Limnol. Oceanogr.
**2000**, 45, 1603–1620. [Google Scholar] [CrossRef] [Green Version] - Beji, S.; Battjes, J. Experimental investigation of wave propagation over a bar. Coast. Eng.
**1993**, 19, 151–162. [Google Scholar] [CrossRef] - Beji, S.; Battjes, J. Numerical simulation of nonlinear wave propagation over a bar. Coast. Eng.
**1994**, 23, 1–16. [Google Scholar] [CrossRef] - Park, J.C.; Kim, M.H.; Miyata, H. Fully non-linear free-surface simulations by a 3D viscous numerical wave tank. Int. J. Numer. Methods Fluids
**1999**, 29, 685–703. [Google Scholar] [CrossRef] - Walters, R.A. A semi-implicit finite element model for non-hydrostatic (dispersive) surface waves. Int. J. Numer. Methods Fluids
**2005**, 49, 721–737. [Google Scholar] [CrossRef] - Chen, X. A fully hydrodynamic model for three-dimensional, free-surface flows. Int. J. Numer. Methods Fluids
**2003**, 42, 929–952. [Google Scholar] [CrossRef]

**Figure 1.**Model representation of the grid. (Source: Casulli and Lang [23]).

**Figure 2.**The initial free-surface profile for a linear 3D standing wave oscillation in a closed basin. (Source: Yuan and Wu [8]).

**Figure 3.**Free surface elevation at $x=y=0.25$ m for 30 s of simulation: (

**top**) Comparing analytic solution with simulated solution for 20 to 5 layers scenario with fictional sublayer at the free-surface ($FSFS$) condition (left side); (

**middle**) Comparing analytic solution with simulated solution for 20 to 5 layers scenario with Classic Boundary Condition Approach ($CBA$) condition (right side); and (

**bottom**) Comparing both methods thought the 20 layer scenario and 8 layers scenario (at bottom).

**Figure 4.**Free surface elevation accumulated residuals series for the FSFS approach (

**left**) and CBA approach (

**right**), at x = y = 0.25 m for 30 s of simulation, comparing different layers scenarios.

**Figure 5.**Scheme of experimental bottom geometry and location of wave level gauges. (Source: Beji and Battjes [32]).

**Figure 6.**Comparisons between experimental (circles) and computed data with 20-layers (solid black), 16-layers (dashed gray), 10-layers (solid red), and 5-layers (dashed blue), at 6 different level gauges.

**Table 1.**Computational cost, Phases Error, and the number of wave periods between different methods and scenarios. The model was implemented with Fortran and simulated in a machine using an Intel R Xenon R CPU-E5-1620 3.7 GHz computer with 32 GB of RAM.

N°L | FSFS | CBA | ||||
---|---|---|---|---|---|---|

$\mathsf{\Delta}\mathit{t}\left(\mathit{s}\right)$ | N° T | ${\mathbf{\Phi}}_{\mathit{\epsilon}}$ | $\mathsf{\Delta}\mathit{t}\left(\mathit{s}\right)$ | N° T | ${\mathbf{\Phi}}_{\mathit{\epsilon}}$ | |

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 |

**Table 2.**Metrics between the analytic and simulated results from the FSFS method for each scenario for the first 10 s of simulation.

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 |

**Table 3.**Metrics between the analytic and simulated results from the CBA method for each scenario for the first 10 s of simulation.

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 |

**Table 4.**Nemenyi posthoc test comparing the FSFS residue series of the simulation with 20 to 5 vertical layers to identify the significative statistical difference between results.

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 |

**Table 5.**Statistics metrics between simulated and experimental results for the six stations for each used layer scenario with the FSFS method.

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 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Cunha, 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