1.
Validation test of a solitary
wave over an erodible
sloped beach for XBeach in
Non-Hydrostatic mode
Giulia Mancini, Riccardo Briganti, Nicholas Dodd
University of Nottingham, UK
Fangfang Zhu
University of Nottingham Ningbo, China
Environmental Fluid Mechanics and Geoprocesses
Research Group

2.
• Introduction and Motivation
• Aim and Objectives
• Methodology
• Results
• Conclusions

3.
Introduction and Motivation
The non-hydrostatic version of XBeach model allows to simulate the propagation of
individual waves in the nearshore region up to the shoreline and bed changes
However, modelling of the
intra-swash sediment transport
and, in turn, morphodynamic evolution
of beaches needs further investigation
x=44.5m x=75.86m
Simulated and measured velocity and suspended sediment
concentration in the swash zone
Bichromatic broad-banded
wave condition

4.
Aim and Objectives
Aim: Improve the modelling of intra-swash processes in wave resolving models
• Simulation of a solitary wave
over an erodible beach
Single swash event without the
complexity of swash-swash
interaction present in wave groups
• Comparison with the high accuracy
numerical solution obtained by
Zhu and Dodd (2015)
Analytical solution used as a
benchmark for assessing XBeach
response
Objectives

5.
Methodology
Zhu and Dodd (2015): XBeach in Non-hydrostatic mode
(“XBeachX BETA” release):
• 1D - Non Linear Shallow Water Equations
including bed shear stress
•
𝜕𝑧 𝑏
𝜕𝑡
+ ξ
𝜕𝑞 𝑏
𝜕𝑥
= ξ(𝐷 − 𝐸)
•
𝜕ℎ𝐶
𝜕𝑡
+
𝜕ℎ𝑢𝐶
𝜕𝑥
= 𝑚 𝑒
𝑢2
𝑢0
− 𝑤𝑠C = E − D
(suitable for swash zone flow, but unlike
XBeach frequency dispersion is not modelled)
+
(Pritchard and Hogg, 2005)
• 1D - Non Linear Shallow Water Equations +
Non-hydrostatic pressure term
(comparable to SWASH one single layer version)
Governing equations:
•
𝜕ℎ𝐶
𝜕𝑡
+
𝜕ℎ𝑢𝐶
𝜕𝑥
+ 𝑑𝑖𝑓𝑓 =
ℎ𝐶 𝑒𝑞−ℎ𝐶
𝑇𝑠
= 𝐸 − 𝐷
•
𝜕 𝑧 𝑏
𝜕𝑡
+ ξ
𝜕𝑞 𝑏
𝜕𝑥
= ξ(D − E)
+
(Galappatti and Vreugdenhil, 1985)
Governing equations: (solved by means of
method of characteristics)

6.
Methodology
- Bed load (Van Thiel – Van Rijn)
Equilibrium concentration:
- Suspended load (Van Thiel – Van Rijn)
As all waves are resolved in
extended NLSWE, in the non-
hydrostatic solver all intra-wave
velocities are currently included
in “u” term (thus, urms is null)
𝜕ℎ𝐶
𝜕𝑡
+
𝜕ℎ𝑢𝐶
𝜕𝑥
+ 𝑑𝑖𝑓𝑓 =
ℎ𝐶𝑒𝑞 − ℎ𝐶
𝑇𝑠
= E − D
𝐶𝑒𝑞 = max min 𝐶𝑒𝑞,𝑏,
1
2
𝐶 𝑚𝑎𝑥 + min 𝐶𝑒𝑞,𝑠,
1
2
𝐶 𝑚𝑎𝑥 , 0
𝐶𝑒𝑞,𝑏 =
𝐴 𝑠𝑏
ℎ
𝑢2 + 0.64𝑢 𝑟𝑚𝑠
2
− 𝑈𝑐𝑟
1.5
𝐶𝑒𝑞,𝑠 =
𝐴 𝑠𝑠
ℎ
𝑢2 + 0.64𝑢 𝑟𝑚𝑠
2
− 𝑈𝑐𝑟
2.4
XBeach in Non-hydrostatic mode: sediment transport formulation implemented were originally
validated only for the wave-averaged framework

7.
Methodology
Zhu and Dodd (2015):
• Initial condition along the x-domain (t=0s, x)
• Solitary wave simulated by means of 1st order
theory of cnoidal waves by Mei (1990)
XBeach:
• Boundary condition located at x=-20 m
• Time series of u(t) and η(t) provided by
Zhu and Dodd (2015)

8.
Results
• Initial shoreline position located at x = 5m,
where the bore collapses
• In the uprush sediment is entrained in the
water column and transported onshore
• In the backwash sediment is entrained in
the water column and moved offshore
• Bed step formation due to collision of
backwash bore and still water mass

9.
Results
Hydrodynamics:
𝑡/ ℎ0/𝑔 𝑡/ ℎ0/𝑔
𝑢/𝑔ℎ0𝑢/𝑔ℎ0𝑢/𝑔ℎ0

10.
Results
Bed step
Bed step
Erosion
Erosion
Bed changes:
0
𝑡/ℎ0/𝑔
Deposition
Bed step
Erosion
Deposition

11.
Results
•
𝜕ℎ𝐶
𝜕𝑡
+
𝜕ℎ𝑢𝐶
𝜕𝑥
= 𝐸 − 𝐷
•
𝜕 𝑧 𝑏
𝜕𝑡
+ ξ
𝜕𝑞 𝑏
𝜕𝑥
= −ξ(E − D)
Sediment transport equation:
Exner equation:
x=6m
x=12m
Source term
Source term

12.
Results
x = 6m
XBeachZhu and Dodd (2015)
𝑡/ ℎ0/𝑔 𝑡/ ℎ0/𝑔
Flow reversal
• The advection term dominates on
the source term
• The source term has the same order
of magnitude as the advection term

13.
Results
x = 6m
XBeachZhu and Dodd (2015)
𝑡/ ℎ0/𝑔 𝑡/ ℎ0/𝑔
0
𝑡/ℎ0/𝑔
Exner equation

14.
Results
x = 12m
XBeachZhu and Dodd (2015)
𝑡/ ℎ0/𝑔 𝑡/ ℎ0/𝑔
• The advection term dominates on
the source term
• The source term has the same order
of magnitude as the advection term
Flow reversal

15.
Results
x = 12m
XBeachZhu and Dodd (2015)
𝑡/ ℎ0/𝑔 𝑡/ ℎ0/𝑔
0
𝑡/ℎ0/𝑔
Exner equation

16.
Conclusions
Ongoing work: inclusion of further sediment
transport formulation (Pritchard and Hogg, 2005)
and testing of multiple events
• Despite the different governing equations, results highlight a good agreement between
the two models in terms of hydrodynamics response
0
• The main qualitative difference is noticed in the uprush suspended sediment transport
and morphodynamic response
Sediment transport formulation
implemented were originally validated
only for the wave-averaged framework
𝑡/ ℎ0/𝑔
𝑡/ℎ0/𝑔
Flow reversal

17.
Thank you for your attention!
Giulia Mancini
Email: evxgm7@nottingham.ac.uk
References:
XBeach model:
• D. Roelvink, et al. XBeach model description and manual. Unesco-IHE Institute for Water Education, Deltares and
Delft University of Technology, 2010
• P. Smit, G. Stelling, J.A. Roelvink , J. Van Thiel de Vries, R. McCall, A. van Dongeren, C. Zwinkels, R. Jacobs.
XBeach: Non-hydrostatic model: Validation, verification and model description. Delft University of Technology,
2010
CoSSedM-HYDRALAB IV project:
• J.M. Alsina, E.M. Padilla, I. Cáceres. Sediment transport and beach profile evolution induced by bi-chromatic wave
groups with different group periods. Coastal Engineering 114: 325-340, 2016
• G. Ruffini, R. Briganti, J. Alsina, M. Brocchini, N. Dodd, R. McCall. Manuscript under review
Zhu and Dodd (2015) model:
• F. Zhu and N. Dodd. The morphodynamics of a swash event on an erodible beach. Journal of Fluid Mechanics,
762:110–140, 2015
• D. Pritchard and A. J. Hogg. On the transport of suspended sediment by a swash event on a plane beach. Coastal
Engineering, 52(1):1–23, 2005