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Advances in Water Resources 123 (2019) 173–188

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Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

Numerical simulation of floods from multiple sources using an adaptive anisotropic unstructured mesh method

R. Hu a , b , F. Fang a , b , ∗ , P. Salinas a , b , C.C. Pain a , b , N.D. Sto.Domingo c , O. Mark c

a Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, UK b Novel Flood Modelling and Simulation Group, Department of Earth Science and Engineering, Imperial College London, UK c DHI, Hørsholm, Denmark

a r t i c l e i n f o

Keywords:

Anisotropic dynamic mesh optimization Double control-volume finite element method DCV-FEM Joint flood

a b s t r a c t

The coincidence of two or more extreme events (precipitation and storm surge, for example) may lead to severe floods in coastal cities. It is important to develop powerful numerical tools for improved flooding predictions (especially over a wide range of spatial scales - metres to many kilometres) and assessment of joint influence of extreme events. Various numerical models have been developed to perform high-resolution flood simulations in urban areas. However, the use of high-resolution meshes across the whole computational domain may lead to a high computational burden. More recently, an adaptive isotropic unstructured mesh technique has been first introduced to urban flooding simulations and applied to a simple flooding event observed as a result of flow exceeding the capacity of the culvert during the period of prolonged or heavy rainfall. Over existing adaptive mesh refinement methods (AMR, locally nested static mesh methods), this adaptive unstructured mesh technique can dynamically modify (both, coarsening and refining the mesh) and adapt the mesh to achieve a desired precision, thus better capturing transient and complex flow dynamics as the flow evolves.

In this work, the above adaptive mesh flooding model based on 2D shallow water equations (named as Floodity) has been further developed by introducing (1) an anisotropic dynamic mesh optimization technique (anisotropic-DMO); (2) multiple flooding sources (extreme rainfall and sea-level events); and (3) a unique combination of anisotropic-DMO and high-resolution Digital Terrain Model (DTM) data. It has been applied to a densely urbanized area within Greve, Denmark. Results from MIKE 21 FM are utilized to validate our model. To assess uncertainties in model predictions, sensitivity of flooding results to extreme sea levels, rainfall and mesh resolution has been undertaken. The use of anisotropic-DMO enables us to capture high resolution topographic features (buildings, rivers and streets) only where and when is needed, thus providing improved accurate flooding prediction while reducing the computational cost. It also allows us to better capture the evolving flow features (wetting-drying fronts).

1

u t i f ( e i p c r h

T d e 2 m m c t e ( d M

hRA0

. Introduction

Over the last two decades, the risk of urban flooding in heavily pop-lated coastal regions has increased and is expected to increase fur-her, mainly due to the urbanization and climate change. Urban flood-ng in coastal regions could be caused by a single source (heavy rain-all, high sea levels or storms), or several sources acting in combination Dawson et al., 2008 ). Due to this increasing high risk, the combinedffect and the joint probability of multiple extreme floods are gainingmportance in flooding simulations ( Lian et al., 2013 ). Improving theredictive capabilities in such cases is critical for populated areas, espe-ially cities. It is therefore important to develop an efficient and accu-ate numerical model for studying floods caused by several concurrent

azards in coastal cities.

∗ Corresponding author at: Applied Modelling and Computation Group, DepartmenE-mail address: [email protected] (F. Fang).

r

ttps://doi.org/10.1016/j.advwatres.2018.11.011 eceived 14 February 2018; Received in revised form 22 August 2018; Accepted 24 Nvailable online 29 November 2018 309-1708/© 2018 The Authors. Published by Elsevier Ltd. This is an open access ar

t of Earth Science and Engineering, Imperial College London, UK.

An overview of flood inundation models has been given byeng et al. (2017) . In the past, various numerical models have beeneveloped to simulate flood inundation ( Chen et al., 2012; van Dijkt al., 2014; Hunter et al., 2008; Soares-Frazão et al., 2008; Son et al.,016 ). These models are classified into three categories: (1) empiricalethods such as measurements ( O’Connor and Costa, 2004 ) and re-ote sensing ( Smith, 1997 ); (2) hydrodynamic models; and (3) con-

eptual models for large floodplains ( Teng et al., 2015 ) and probabilis-ic flood risk assessment ( Apel et al., 2004 ). The hydrodynamic mod-ls include one-dimensional (1D) ( Huber et al., 1988 ), two-dimensional2D) ( DHI, 2009; Mignot et al., 2006; Xia et al., 2017 ) and three-imensional (3D) models ( Prakash et al., 2014; Zhang et al., 2016 ).IKE Flood ( DHI, 2014 ) has the capability of simulating combined

iver, sewer and floodplain modelling with high resolution and relia-

ovember 2018

ticle under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

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R. Hu, F. Fang and P. Salinas et al. Advances in Water Resources 123 (2019) 173–188

Fig. 1. (a) Situation of study area in Greve, Municipality of Denmark. (b) DTM with buildings of study area (resolution of 1.6 m ×1.6 m) - generated by the GIS (Geographical Information System) software, see ArcGIS (2010) .

b o T s b 2 c r h e e ( p N U d c B f

F s ( fi C i v w l fl s a l F (

o b b

w D o t S t p

a s G f t w S

2

( a t d e M i o f o

t a t s c t t t o

ility ( Sto. Domingo et al., 2010 ). Recently various efforts to computeverland flows by solving the shallow-water equations have been made.hese studies have simulated overland flows under extreme and un-teady rainfall conditions and spatially constant infiltration rates haveeen taken into account ( Bellos and Tsakiris, 2016; Nguyen et al.,016; Singh et al., 2014 ). However, due to the complexity and un-ertainty of flood modelling, efficient simulation of flooding at high-esolution terrain remains a significant challenge in hydrologic andydraulic studies. For efficient and accurate flood inundation mod-lling, numerous methods have been developed, such as grid coars-ning methods ( Hartnack et al., 2009 ), cellular automata approach Dottori and Todini, 2011 ), and speeding-up strategies such as parallelrocessing ( Hu and Song, 2018; Sanders et al., 2010; Zhang et al., 2014 ).guyen et al. (2016) have proposed a coupled model called HiResFlood-CI, which combines the hydrological model HL-RDHM and the hydro-ynamic BreZo model while ensuring a bare minimum computationalost. HiResFlood-UCI uses HL-RDHM as a rainfall-runoff generator andreZo as the hydrological routing scheme. This model has been success-ully applied to a catchment of the Illinois river in USA.

More recently, a new 2D double control-volume finite element (DCV-EM) flood model has been developed together with the adaptive un-tructured mesh technology and validated by a simple flooding case Hu et al., 2018 ). In comparison to adaptive mesh refinement (AMR) (ane structured mesh nested within a coarse mesh) technique ( Berger andolella, 1989 ), the DMO technique is able to adapt the mesh optimally

n time and space in response to the evolving flow features, thus pro-iding sufficient mesh resolution where and when it is required. In thisork, we have further developed this adaptive unstructured mesh shal-

ow water model with anisotropic considerations for modelling urbanoods from multiple sources (rainfall and storm surge). The implicit 𝜃-cheme has been adopted for solving the shallow water equations andpplications to urban floods caused by multiple sources (rains and seaevels). In the DCV-FEM scheme, the velocity components are discretisedE-wise, while the pressure/free surface height is discretised CV-wise Salinas et al., 2017 ).

In this paper, a DCV-FEM adaptive mesh urban flooding model basedn 2D shallow water equations named as Floodity ( Hu et al., 2018 ), haseen further developed for modelling the concurrent flooding and haseen successfully applied to a 2.2 km ×1.7 km densely urbanized area

174

ithin Greve, Denmark. This is the first time to apply the anisotropic-MO method to simulate urban floods caused by multiple sources basedn high resolution Digital Terrain Model (DTM) data. Model valida-ion has been performed in comparison with results from MIKE 21 FM.ensitivity of the extreme sea levels, rainfall and mesh resolution tohe flood volume has been explored to assess uncertainties in modelredictions.

The structure of this paper is as follows: The governing equations andnisotropic-DMO method are introduced in Section 2 . Section 3 demon-trates the application of Floodity to simulate flood events withinreve, Denmark. It describes the study site and geospatial data used

or modelling, and specifies boundary conditions and parameter set-ings. Section 4 presents and discusses the Floodity results by comparingith results from MIKE 21 FM. Finally, some conclusions are given inection 5 .

. Methodology

In this work, we have adopted the element pair P 1 DG- P 1 CV Salinas et al., 2017 ) (a modification of linear discontinuous velocitynd continuous pressure representations) for 2D shallow water simula-ions. In DCV-FEM scheme used here, the pressure (free surface height) isiscretized CV-wise rather than FE-wise in the classic CV-FEM ( Forsytht al., 1989; Geiger et al., 2004; Gomes et al., 2017; Jackson et al., 2015;atthai et al., 2007 ). The DCV-FEM provides significant improvements

n the quality of the pressure matrix that can be solved efficiently evenn highly anisotropic elements. We also use flux limiting of the free sur-ace height based on the Normalized Variable Diagram (NVD) approachf Leonard (1988) .

The Discontinuous Galerkin (DG) method is used for the discretiza-ion of the momentum equation. The DG approach is very powerfuls it has a natural dissipation associated with it. To robustly stabilizehe shallow water momentum equation, a non-linear Petrov–Galerkincheme ( Farrell and Maddison, 2011 ) is used. It is a mathematicallyonsistent residual scheme and converges to the governing equations ashe mesh and time step size are refined. Due to a diffusion term propor-ional introduced to the residual of the momentum equation, it provideshe robustness needed when there are sharp changes in velocity (usuallyccurs near wetting and drying fronts).


Fig. 2. (a) Forecast of water levels issued at 12:00 on 29th November 2015. (b) Correlation coefficient at function of forecast time.

t c a l i t

2

2

t

A non-linear 𝜃 method is used for the time discretization in whichhe value of 𝜃 (between 0.5 and 1) is adjusted in space and time. 𝜃 isalculated at each CV face based on the satisfaction of a Total Vari-tional Diminishing (TVD) criterion ( Pavlidis et al., 2016 ). The non-inear iteration is based on the fixed-point iteration method describedn Salinas et al. (2017) . This is important as wetting and drying (due tohe non-linear drag and inertia) is highly non-linear.

175

.1. Governing equations

.1.1. Momentum and continuity equations

For depth averaged velocity u in non-conservative form, the momen-um equation or shallow water is written as:

𝜕𝐮 + 𝐮 ⋅ ∇ 𝐮 + 𝐶 𝑓 𝐮 − 𝜇∇ 2 𝐮 = −∇ 𝑝 + 𝐬 𝐛 , (1)

𝜕𝑡


Fig. 3. (a) Initial water depth in a scenario of 2100 upper extreme water level. (b) Future extreme seawater levels by 2100 (considering the worst climate change scenario for 100-yr projection), see Berbel Roman (2014) . (c) Extreme 24 h design rainfall for 2-yr, 10-yr and 100-yr return period (considering the effects of climate change), see Berbel Roman (2014) .

a

w t t f

𝑝

a

ℎ

w d h t

2

S

𝐧

i t l

𝐶

a c t o w d v w w

t ( t T t d c

2

s c p t

v t

nd the continuity equation is written as:

𝜕ℎ

𝜕𝑡 + ∇ ⋅ ( ℎ 𝐮 ) = 𝑠 ℎ , (2)

here h is the water depth, C f is the volumetric drag coefficient, 𝜇 ishe dynamic viscosity, p is the depth averaged pressure, s b is the sourceerm of velocity (unit: m s −2 ), s h is the source term of mass (from rainfallor example, unit: m s −1 ), and p is calculated:

= 𝑔( ℎ + 𝑑) , (3)

nd thus

= 𝑝 𝑔 − 𝑑, (4)

here g is the gravitational acceleration, and d is bathymetry (the heighteviation from a horizontal and flat plain). The pressure and free surfaceeight are defined CV-wise, contrary to the classical CV-FEM, wherehey are FE-wise.

.1.2. Drag coefficient

A commonly used bottom stress parameterization is the Manning-trickler formulation ( Keulegan, 1938; Rouse, 1961 ):

⋅ 𝜈∇ 𝐮 = 𝑛 2 𝑚 𝑔 |𝐮 |𝐮 ℎ

1 3

, on Γ𝑏𝑜𝑡𝑡𝑜𝑚 , (5)

n which n is the unit normal to the bottom surface Γbottom normal, 𝜈 ishe kinematic viscosity, and n m is the Manning coefficient. The formu-ation for the volumetric drag coefficient C f is:

𝑓 = 𝑛 2 𝑚 𝑔 |𝐮 |

max { ℎ, ℎ } 1 3

. (6)

min e

176

Wetting and drying: Here we use the thin film wetting and dryinglgorithms, which specify a minimum threshold depth that defines theategories of wet or dry in the model. Importantly, we have introducedhe flux limiter to ensure the positive water depth and avoid the physicalscillation. ℎ min = 0 . 01 mm was finally chosen as considering a layer ofater of 0.01 mm in the dry areas is physically plausible, while also re-ucing the non-linear behaviour which may be introduced by a smalleralue (e.g. 0.001 mm). In order to enable the CVs to wet and dry freely,e perform an element wise average of the node-wise depths of waterithin each element. We then use this average in Eq. (6) .

Representation of buildings: The most common methods for simulatinghe water flow among structures are: (1) blocking-out of the solid area;2) local elevation rise of the solid area; (3) local increase of roughness ofhe solid area either via the Manning coefficient increase ( Bellos, 2012 ).he method used here is similar to the local elevation rise. The eleva-ion and shape of buildings are embedded in the realistic topographyata. With the use of anisotropic-DMO, the details of buildings can beaptured as the water floods around them.

.1.3. Boundary conditions for the joint flooding events

The boundary conditions need to be set for the water depth h , pres-ure p or velocity u . We can define either pressure or velocity boundaryonditions but not both ( Gresho and Sani, 1998 ). Here we specify theressure p boundary conditions and then the velocity is calculated fromhe pressure gradient.

h boundary conditions: For the joint flooding events involved with plu-ial and coastal flooding, the water depth h boundary conditions alonghe coastline include: (1) time series of extreme water level heights estimated from storm-surge models or the historical data observed; (2)


Fig. 4. Water depths obtained from a MIKE 21 FM model updated from Berbel Roman (2014) (left column), a Floodity model with a mesh resolution 10 m (middle column) and results showing the corresponding mesh (right column) based on bathymetric data without buildings in a scenario of 2100 upper extreme sea-level event at time level 𝑡 = 2 ℎ (first row), 8 h (middle row), and 18 h (bottom row).

b d

p v a i u

s m s o

r t p r o i o

2

f i

athymetry (terrain elevation) d along the coastline. Thus, the waterepth h is calculated from ℎ = 𝑒 − 𝑑.

p boundary conditions: Once the h boundary conditions are set, theressure p boundary conditions are specified through Eq. (3) , and theelocity is calculated from the pressure gradient ( Gresho and Sani, 1998 )nd evolve to point into or out of the domain (depending on the dynam-cs). The boundaries (except the coastline) over the domain can be setp as closed or open based on actual situation.

Rainfall as source term: The rainfall needs to be considered as theource term s h in Eq. (2) . Generally, given a large domain (a large catch-ent), where the characteristics of the subcatchments are found to be

ignificantly different from each other, the effect of spatial distributionf rainfall is considered. This mean that different subcatchments may

177

eceive different amounts of rainfall, namely different rainfall intensityime series are induced according to the region. However, since the com-utation domain in this study is relatively small, we assume that theainfall is uniform across the domain. Due to the fact that the effectf rainfall is relatively small compared with that of incoming waves,nfiltration is not considered here. An average depth of runoff is thenbtained, which contributes to the calculation of the water depth h .

.2. Anisotropic dynamic mesh optimization technique (anisotropic-DMO)

The anisotropic-DMO has the advantage of capturing details of sur-ace and local flows (wetting-drying front) during the process of flood-ng modelling. The use of anisotropic-DMO can efficiently provide a high


Fig. 5. Flood volume obtained from a MIKE 21 FM model and a Floodity model with a mesh resolution 10 m based on bathymetric data without buildings in a scenario of 2100 upper extreme sea-level event.

m i o a f

p T m o a

p

𝑀

w t (

𝐻

e

i 𝑀

P

m k m a

𝑀

w 𝑀

m

n r (

𝜆

where

F

1t

esh resolution where and when it is needed ( Hiester et al., 2011 ). Thats, finer meshes are placed only in specific regions where the variationsf flow variable solutions are relatively large (e.g. flow around buildingsnd along the flooding paths), while coarser meshes are used in areas farrom these regions, where inundation has not yet occurred for example.

Here, the anisotropic-DMO technique relies on the derivation of ap-ropriate error measures, which dictate how the mesh is to be modified.he required error measure is defined in the form of a metric tensor. Theetric is derived from a solution field variable and an error norm based

n the interpolation error ( Pain et al., 2001 ) defined to make sure that desired element length can be obtained while having a required inter-

ig. 6. Flood depth time series at detector locations P1, P2, P3 and P4 (see Fig. 1 (b0 m, and 5 m respectively, based on bathymetric data without buildings in a scenaro color in this figure, the reader is referred to the web version of this article.)

178

olation error. Thus, the metric 𝑀 is calculated from

̂= �̂�𝜖|𝐻|, (7)

here ̂𝛾 is a scalar constant, and ̂𝛾 = 1 is used here, 𝜖 is a required in-erpolation error, and H is the Hessian matrix for a specified field 𝜓( Ω)here, the field of water depth):

=

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 𝜕 2 𝜓

𝜕𝑥 2 𝜕 2 𝜓

𝜕 𝑥𝜕 𝑦

𝜕 2 𝜓

𝜕 𝑦𝜕 𝑥

𝜕 2 𝜓

𝜕𝑦 2

⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . (8)

The desired edge length, h i , in the direction of the i th eigenvector i , of the metric 𝑀 , is defined as ℎ 𝑖 = 1∕

√𝜆𝑖 , i.e., anisotropic as well as

nhomogeneous resolution results from a mesh that respects the metric ̂, where 𝜆i is the eigenvalue associated with e i ( Piggott et al., 2009;

ower et al., 2006 ). It is advantageous to modify the metric to impose the maximum and

inimum element sizes on the mesh, especially for problems that havenown high-aspect ratio dynamics or domains. To impose these maxi-um and minimum constraints directionally, ̂𝑀 is modified and defined

s

= 𝑉 𝑇 ∧ 𝑉 , (9)

here V is a rotation matrix that includes the eigenvectors of the metric ̂calculated from Eq. (7) and general directions for the maximum andinimum edge length can be introduced through the use of V .

To bound the aspect ratio of elements in physical space, there is aeed to limit the ratio of eigenvalues. For example, to limit the aspectatio of elements to be below a , the eigenvalues are modified as follows Pain et al., 2001 ):

�̃� = max {

𝜆′𝑗 , 1 𝑎 2

max 𝑗=1 , 2 , 3

𝜆′𝑗

} ∀𝑗 ∈ {1 , 2 , 3} , (10)

)) simulated using MIKE 21 FM and Floodity with a mesh resolution of 20 m, io of 2100 upper extreme sea-level event. (For interpretation of the references


Fig. 7. Water depths at time level 𝑡 = 15 h obtained from a MIKE 21 FM model (first row), a Floodity model with a mesh resolution 10 m (middle row) and results showing the corresponding mesh (bottom row) based on bathymetric data without buildings in scenarios of an individual 2100 upper extreme sea-level event (left column) and a joint event with 100-yr return period rainfall (right column).

𝜆

i m

t

𝑆

t

′𝑗 = min

{ 1

ℎ 2 min

, max

{ |𝜆𝑗 |, 1 ℎ 2 max

} } ∀𝑗 ∈ {1 , 2 , 3} . (11)

n which a is the maximum aspect ratio. h min and h max are the mini-um and maximum element sizes. A uniform isotropic element can be

179

ransformed to an adapted anisotropic element under the transformation

= 𝑉 √| ∧ |−1 in the physical domain, achieving the desired interpola-

ion error everywhere.


Fig. 8. Flood depth time series at detector locations P1, P2, P3 and P4 (see Fig. 1 (b)) simulated using MIKE 21 FM and Floodity with a mesh resolution of 20 m, 10 m, and 5 m respectively, based on bathymetric data without buildings in a scenario of 2-yr return period rainfall and 2100 upper extreme sea-level event.

Fig. 9. Flood velocity time series at detector locations P1, P2, P3 and P4 (see Fig. 1 (b)) simulated using MIKE 21 FM and Floodity with a mesh resolution of 10 m based on bathymetric data without buildings in a scenario of 2-yr return period rainfall and 2100 upper extreme sea-level event.

i n d t

3

3

The Galerkin interpolation technique ( Farrell and Maddison, 2011 )s used for interpolating the solutions from the previous mesh onto theewly adapted mesh. The overhead and extra computational cost intro-uced during the adaptive mesh procedure is very low, around 10% ofhe total CPU time ( Hu et al., 2018 ).

t a

180

. Model application

.1. Descriptions of study site and data

To assess the performance of anisotropic-DMO in flooding modelling,he new flooding model has been applied to an urban area located within 2.2 km ×1.7 km densely urbanized area within Greve, Denmark.


Fig. 10. Water depths at time level 𝑡 = 15 h obtained from a MIKE 21 FM model (first row), a Floodity model with a mesh resolution 5 m (middle row) and results showing the corresponding mesh (bottom row) based on bathymetric data with buildings in scenarios of an individual 2100 upper extreme sea-level event (left column) and a joint event with 100-yr return period rainfall (right column).

181


Fig. 11. Details of the areas marked with rectangles ( Fig. 10 ) obtained from MIKE 21 FM modelling (left column), a Floodity model with a mesh resolution 10 m (middle column) and 5 m (right column). The bottom shows more details of the anisotropic unstructured meshes.

w d o e e o w o l i e i d i t g r b

3

c t

T a h t c s fi l

w a b c i a e S

m m a

i m ( w

The study area is located in the northeastern part of Greve, Denmark,hich covers part of the coastal area (see Fig. 1 (a)). Historical extraor-inary flood events which were caused by a series of rain events haveccurred in Greve ( Kommune, 2007 ). In addition to extreme rainfallvents, Greve is also vulnerable to flood induced by extreme sea-levelvents along its coast. For example, the most extreme historical floodccurred on 13th October 1760 with a maximum water level of 3.7 m,as caused by a very serious storm surge ( Madsen, 2008 ). Recurrencef any of these flood events would cause numerous damages (e.g. loss ofife, direct damages to roads, railways and buildings, indirect damagesncluding loss of income, clean-up cost, turnover loss, cost of illnesses,tc). As a consequence, it is important to develop flooding models tomprove the accuracy of flood predictions. In the Greve case study, theigital elevation data provided was quality assured and buildings werencorporated into the DTM (Digital Terrain Model) data with a resolu-ion of 1.6 m ( Fig. 1 (b)), which was detailed enough to describe topo-raphic features (buildings, rivers and streets). Data of different extremeainfall events as well as extreme sea-level events have been used as theoundary conditions and sources.

.2. Extreme sea-level and rainfall events

Due to the impact of climate change in the next 100 years, futurelimate change conditions should be taken into account to estimatehe future extreme sea-level and rainfall events ( Berbel Roman, 2014 ).

182

o forecast storm surges, a hydrodynamic model has been calibratedgainst historical events during 2000–2015. The capability of this builtydrodynamic model to forecast storm surges was evaluated against his-orical storm event during 2010–2017. An example of a real time fore-ast issued during the storm “Gorm ” on 29th November 2015 can beeen in Fig. 2 (a). From Fig. 2 (b) it can be seen how the correlation coef-cient between observed and forecasted water levels goes down as the

ead time increased. The 3D hydrodynamic model was built using MIKE 3 FM, a soft-

are tool for modelling unsteady three-dimensional flows, which uses flexible mesh calculation grid taking into account density variations,athymetry and external forcing, such as meteorology, tidal elevations,urrents and other hydrographic conditions ( DHI, 2016 ). Meteorolog-cal forcing for the model is obtained from a WRF (Weather Researchnd Forecasting) limited-area numerical weather prediction model cov-ring Northern Europe with a resolution of 0.1 degrees, which is run bytormGeo in Norway. More information about the calibration, and theodel in general, are given in DHI (2011) . A description of the aboveethod for estimating storm surge event time series for climate change

nalysis is described in Berbel Roman (2014) . Fig. 3 (b) shows the extreme sea water levels which were used as

nput boundary conditions along the coastline. According to Berbel Ro-an (2014) , to estimate the expected changes in sea surges in future

up to year 2100), hydrodynamic simulations were carried out, whichere driven by the wind and atmospheric pressure results from three


Fig. 12. Buildings gradually become visible as the flood water spreads west and northward. The left column shows the plane view of surface topography. The right column shows the corresponding mesh.

r i t f a s t

t ‘ s s f a

egional climate models. A general observed extreme event pattern wasdentified based on the past observed extreme sea-level events. The fu-ure extreme water level event time series ( Fig. 3 (b)) were then obtainedrom the general observed pattern by scaling to a given return periodnd adding estimates of mean sea level rise and change in storm surgeignal. The water level calculated with statistics projections of 100-yr re-urn period considering climate change under a present scenario iden-

183

ified as ‘current’ and a future scenario identified as ‘2100 mean’ and2100 upper’ depending of the climate factor considered. These extremeea-level events (current, 2100 mean and 2100 upper) lasted 24 h. Aseen in Fig. 3 (b), for a return period of 100 years, the maximum valueor the extreme water levels is 1.52 m (current), 2.25 m (2100 mean)nd 3.08 m (2100 upper).


Fig. 13. Flood depth time series at detector locations P1, P2, P3 and P4 (see Fig. 1 (b)) simulated using MIKE 21 FM and Floodity with a mesh resolution of 20 m, 10 m, and 5 m respectively, based on bathymetric data with/without buildings in a scenario of 100-yr return period rainfall and 2100 upper extreme sea-level event.

Fig. 14. Sensitivity of flood volume results to forcing inputs in scenarios: (a) individual extreme sea-level events (shown in Fig. 3 (b)); (b) both the individual 2100 upper sea-level flood event and the joint event with 100-yr return period rainfall; and (c) mesh resolutions of 20 m, 10 m, and 5 m. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

q 8 t r d m s t

3

r h u

Table 1

Adapting mesh schemes for the Floodity simulations.

Minimum element size Maximum element size Adaptive time step Δt ( s )

20 200 [5,10] 10 200 [3,10] 5 200 [1,10]

r t ( n s a t

Extreme precipitation data in this case study is obtained by a fre-uency analysis, where the rainfall data (over more than 10 years) from3 stations in Denmark is used. By running a regional statistical ex-reme model, the intensities of rainfall for the 2-yr, 10-yr and 100-yreturn period have been obtained, then multiplied by a factor of 1.1–1.5ue to the impact of climate change in the next 100 years ( Berbel Ro-an, 2014 ). The designed intensities are shown in Fig. 3 (c). It is as-

umed that for a given rainfall event, a uniform rainfall is falling overhe whole area of the domain (2.2 km ×1.7 km).

.3. Model applications

A series of model simulations using anisotropic-DMO have been car-ied out to assess the performance of the new flooding model developedere. The extreme seawater level event ( Fig. 3 (b)) which lasts 24 h issed as input boundary condition on the sea boundary ( Fig. 1 (b)), the

184

emaining boundaries are set up as closed (no flow). In these simula-ions, sea water enters the densely urbanized area from the sea boundary Fig. 1 (b)) and the extreme rainfall events ( Fig. 3 (c)) take place simulta-eously. Fig. 3 (a) shows the initial water depth within the domain in acenario of 2100 upper extreme water level. The adapting mesh schemesre listed in Table 1 . The mesh is adapted to ensure an absolute error inhe water depth field of 0.01 m and the aspect ratio is 100.


Table 2

Node and element number of the adaptive unstructured mesh with mesh resolution 20 m, 10 m, and 5 m for Floodity modelling.

Meshing Type Time Level (min) Node #(20 m) Element #(20 m) Node #(10 m) Element #(10 m) Node #(5 m) Element #(5 m)

Fixed mesh 0–120 13,033 25,666 51,367 101,939 204,980 408373 0 13,033 25,666 51,367 101,939 204,980 408373 5 3120 6095 8166 16,157 19,825 39446

Adaptive mesh 200 3395 6647 9064 17,952 21,797 43390 600 4904 9620 12,897 25,548 29,289 58275 1200 4666 9156 12,520 24,806 28,235 56176 1440 4581 8990 12,271 24,315 27,485 54686

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. Results and discussion

.1. Individual flooding events

.1.1. Flooding map

Fig. 4 shows the flood propagation process over the urban area in scenario of an individual 2100 upper extreme sea-level event. It cane observed that in most of the inundation area, the solutions of waterepths obtained from Floodity are in good agreement with those fromIKE 21 FM. The mesh is optimally adapted according to the evolvingow features in time and space, thus providing sufficient mesh resolu-ion where and when it is required (right panel in Fig. 4 ). For exam-le, the fine mesh is located along the flood propagation path while theoarser mesh is used in the areas where inundation has not occurredet. To further estimate and compare the flood extent, the flood volumeuring the flood propagation process is calculated ( Fig. 5 ). It is clearlyhat flood volume obtained from Floodity is a little higher than that fromIKE 21 FM during the whole period, whilst the general trends of both

re consistent.

.1.2. Comparison with MIKE 21 FM results at detector locations

To further evaluate the performance of Floodity using different meshesolutions, the time series of water depth at detector locations P1, P2,3 and P4 are plotted in Fig. 6 , in comparison to those from MIKE 21M. In Fig. 6 , the blue, green and black lines represent the time series ofater depth predicted by the new anisotropic unstructured mesh flood-

ng model Floodity with a minimum mesh size of 20 m, 10 m, and 5 mespectively. It can be seen that a good agreement is achieved betweenhe results from both the fixed and adaptive mesh simulations (exceptor that with a minimum mesh size of 20 m) at detectors P3 and P4 dur-ng the flooding propagation period [6.7, 24] h. However, MIKE 21 FMredicts an earlier flood arrival time at P3 and P4 than that predictedy the Floodity simulations. The results of water depth at detectors P1nd P2 obtained by both the MIKE 21 FM and Floodity simulations areery close to each other when almost the same mesh resolution (10 m)s used over the inundated regions. The detectors P1 and P2 are locatedithin a narrow open channel, where a high resolution mesh (a mesh

ize smaller than 10 m at least) is required to represent it. We can seehat the Floodity simulation with a minimum mesh size of 5 m predicts deeper water depth at P1 and P2 than that in the simulations with thearge mesh size (10 m and 20 m). It proves that more detailed solutionsan be obtained in the local areas when using an adaptive mesh insteadf a fixed mesh. We also note that using anisotropic-DMO, Floodity isapable of providing reasonable results even with use of a coarser mesh20 m).

For comparison purposes, MIKE 21 FM results from the model up-ated from Berbel Roman (2014) are used as a reference solution inhis study. The original model domain in Berbel Roman (2014) has dimension of 2.3 km ×7.5 km, where the northeastern region is se-ected as our computational domain ( Fig. 3 (a)). Berbel Roman (2014) di-ided the domain in 9 sub-regions with the element size in range of10,100] m , where large elements were used to represent the surround-ngs of harbours, the coastline and the train tracks were representedith smaller elements. Thus, a flexible mesh with the element size in

185

ange of [10,100] m is used in the MIKE 21 FM simulations while thedaptive meshes with a minimum mesh size of 20 m, 10 m, and 5 m aresed in Floodity. A comparison of water depth results using the adaptiveFloodity) and fixed (MIKE 21 FM) unstructured mesh has been carriedut.

.2. Joint flooding events

The newly developed adaptive mesh flooding model is further usedor simulating floods under the combined impacts of the events (extremeainfall and sea level). Fig. 7 shows the results of water depth from MIKE1 FM and Floodity with a minimum adaptive mesh size of 10 m at timeevels 𝑡 = 15 h . It presents the flood propagation process over the urbanrea in scenarios of an individual 2100 upper extreme sea-level eventnd a joint event with 100-yr return period rainfall respectively. Dueo the effect of rainfall, the joint event has larger flood areas than thendividual event, as shown in the areas marked with rectangles in Fig. 7 .

Figs. 8 and 9 show the time series of water depth at four detector lo-ations predicted by these models respectively. The total rainfall amountn 24 h is 28.1 mm, 45.7 mm and 82.1 mm within the 2.2 km ×1.7 kmtudy area for the 2-yr, 10-yr, 100-yr return period respectively. Thus,n comparison to the extreme sea level, the rainfalls have a relativelymaller impact on the inundation extent. Again, it can be observed thatnder the scenarios of joint flooding events, the water depths and ve-ocity obtained from Floodity using the minimum mesh size of 10 m aren good agreement with those from MIKE 21 FM simulations.

.3. Impact of buildings on flooding simulations

The use of anisotropic-DMO in flooding modelling can better capturehe evolving flow features and topographic features (buildings, riversnd streets), thus providing improved accurate flooding prediction. Tourther demonstrate the capability of the flooding model, Floodity haseen applied to the joint flooding events with the bathymetric datancluding buildings, which were represented as impervious obstacleslocking the flow path.

Fig. 10 presents the flooding map over the urban area at time level = 15 h in scenarios of an individual and joint flooding event respec-ively. It can be observed that Floodity results with a mesh resolution m have a larger inundation extent than MIKE 21 FM results and presentore details of topographic features, including buildings and channels.he details of roads, buildings and channels can be observed clearlyith an increased mesh resolution around them. Fig. 11 provides theetails of the areas marked with rectangles in Fig. 10 . It is noted thathe information of roads and channels has been lost in the MIKE 21 FMimulations. However, with the use of anisotropic-DMO, Floodity canapture the details of topographic features even with almost the sameesh resolution (10 m) as that used in the MIKE simulation. The bottom

n Fig. 11 shows more details of the anisotropic unstructured meshes,here the adapted anisotropic elements are placed under the limitationf the aspect ratio of elements.

In these simulations, the topographical data (digital elevation data)s available with a high resolution of 1.6 m. The availability of high-esolution topographical data is important for the accurate numerical


Fig. 15. The numbers of nodes and elements used in Floodity modelling for mesh resolution 20 m, 10 m, and 5 m during the simulation period [0, 1440] min.

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imulation of urban food inundation. However, high-resolution topo-raphical data requires a high computational effort, thus, resulting in computationally demanding flood modelling. Using anisotropic-DMO,he topographical data over the domain is obtained by interpolating theigh resolution (1.6 m) data onto the adapted mesh at each time level.herefore, the high-resolution topographical data is only used in theooded region while the low-resolution data is used in the rest of theomain, thus reducing the computational cost. In addition, the detailsf buildings can be represented accurately as the flooding water spreadscross the domain (see Figs. 10 and 12 ).

Fig. 13 shows the time series of water depth at four detector loca-ions simulated using MIKE 21 FM and Floodity with three differenteshes in a scenario of a joint flooding event from extreme sea level

nd rainfall for a 100-yr return period. It can be observed that due tohe impact of buildings, the water depths obtained from Floodity have slight difference from that of MIKE 21 FM. A deeper water depth at1 and P2 locations is predicted when using the 5 m resolution adaptiveesh than that using the fixed mesh (MIKE 21 FM), similarly as which

s shown in Fig. 6 . This is due to the fact that the detectors P1 and P2re located at the channel, thus having a larger water depth. Again thisroves that the adaptive mesh flooding model can provide relatively ac-urate predictions. Also note that there is an arrival time lag at detectorsf the impact of buildings is considered in the flooding simulations. As result, the arrival time at P3 and P4 has nearly 140 min’ time lag inloodity with a mesh resolution of 5 m results, in comparison to MIKE1 FM results.

.4. Sensitivity to the forcing inputs and mesh resolution

Extreme joint flooding is the product of a wide range of interactingrocesses. Here the uncertainties from the forcing inputs are the extremeea levels and rainfall. In addition, the mesh resolution is one of criticalarameters in flooding modelling. In this section, sensitivity analysis ofood volume over inundated areas to extreme sea levels, rainfall and

186

esh resolution has been explored and shown in Fig. 3 (a), (b) and (c),espectively.

The impact of the incoming sea levels on flooding results has beennvestigated and the corresponding results in Fig. 14 (a). There are threecenarios of individual extreme sea-level events shown in Fig. 3 (b),here the maximum value for the extreme incoming water levels is.52 m (current), 2.25 m (2100 mean) and 3.08 m (2100 upper) peak-ng at 𝑡 = 9 . 5 h . In Fig. 14 (a), we can see that the flood volume becomearge with an increased incoming wave level. The flood volume peakspproximately at 𝑡 = 9 . 5 h when the incoming wave is reaching its ex-reme. There is a slight time lag in the peak of flood volumes in thecenario of the current extreme sea-level event.

Further investigation of the effect of rainfall on flood volume haseen undertaken in the scenario of the joint flood event. A comparisonf flood volumes between the scenarios of the individual and joint floodvents is provided in Fig. 14 (b). The solid line is the flood volume duringhe individual extreme sea-level event (2100 upper) while the dashedine represents the joint flood event with the 100-yr return period rain-all event during the rainfall period 𝑡 = 13 − 16 h . The influence of rain-all is reflected by the divergence of the flood volume during rainfall.he largest difference in flood volume is 1.10 ×10 5 m 3 at 𝑡 = 14 . 08 h .

Fig. 14 (c) presents the flood volume results from the simulations us-ng different mesh resolutions. It is found that the peak values of floodolume results are very close in all simulations, while the peak time dif-ers greatly when using the mesh resolutions of 20 m (dotted blue line)nd 10 m (dotted green line) as well as 5 m (solid black line). The reasonor this is that the blocking effect of buildings cannot be represented inood modelling with use of coarse mesh resolutions (20 m, here) due tohe failure of capturing the details of buildings.

In general, the flood volume results are sensitive to both incomingea levels and rainfall. However, in joint flood events, the effect of rain-all is relatively small in comparison to extreme incoming waves. Inooding modelling the mesh resolution is the key to capture the detailsf complex topography, for example, buildings and channels. One can


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ee the blocking effect of buildings only when the buildings are capturedith high mesh resolutions.

.5. Performance of floodity modelling

As seen in Fig. 15 and Table 2 , an unstructured mesh with 20 m res-lution generated by Gmsh ( Geuzaine and Remacle, 2009 ) consists of3,033 nodes and 25,666 unstructured triangle elements, while 10 mesolution consists of 51,367 nodes and 101,939 elements, and 5 mesolution mesh contains 204,980 nodes and 408,373 elements. Theseeshes are used during the whole simulation period for fixed unstruc-

ured mesh modelling, and used as the initial mesh for the adaptiveesh simulations. After first adapting the mesh, the numbers of nodes

nd elements used for adaptive mesh of 20 m, 10 m, and 5 m resolu-ion are reduced by 74%, 82% and 88% respectively, then gradually in-reased during 𝑡 = [1, 600] min and decreased during 𝑡 = [600, 1440]in as the flooding water retreats. Above all, Floodity is less computa-

ionally expensive than fixed unstructured mesh modelling. The use ofdaptive unstructured meshes improves the computational efficiency.o further reduce the computational cost, various numerical techniquesan be adopted in the flood model Floodity, for example, parallel com-uting using MPI.

. Conclusions

Realising the importance of flood coincidence risk assessments, weave further developed the adaptive unstructured mesh flooding modelFloodity, Hu et al., 2018 ) for the joint urban flood events caused byultiple sources (extreme rainfall and sea-level events) and successfully

pplied to Greve in Denmark. By introducing the anisotropic-DMO tech-ique, the features of flooding flows (local flows around the buildingsr the wetting and drying front, for example) are able to be better cap-ured while reducing computational cost without sacrificing accuracyf flooding simulations. With a unique combination of anisotropic-DMOnd high-resolution Digital Terrain Model (DTM) data, the complex ur-an topography can be accurately represented when/where needed byncreasing the mesh resolution (around the buildings, for example) dy-amically when the flooding water spreads over the urban area. Thisew Floodity model has been applied to several flooding scenarios thatappened in Greve, Denmark, where the flood is induced by differentombinations of extreme incoming sea levels and rainfall. A compari-on between Floodity and MIKE 21 FM results has been undertaken. Itas been found that Floodity is able to provide relatively accurate re-ults while the computational cost is reduced by 20–88% in comparisono fixed mesh models. To assess uncertainties in model predictions, theensitivity of flood volumes to extreme sea levels and rainfalls has beenxplored. In joint flood events, we found that the flood volume over thenundated area is more sensitive to sea levels than rainfall. Extreme sea-evel events with the higher peak water levels induce higher peak floodolume while the impact of rainfall is relatively small. The sensitivity ofood results to the mesh resolution is also investigated. In flood mod-lling, the blocking effect of buildings on the peak time of flood volumesan be seen only when using high resolution meshes and Digital Terrainodel data.

Flood modelling is a complex and parametric problem. The inputncertainty is one of the main sources of uncertainty. In this paper, weainly focused on the simulation of flooding from multiple sources. Weave done some basic sensitivity analysis. Given its complexity, in fu-ure, we will further carry out uncertainty analysis using advanced nu-erical techniques, for example, the adjoint sensitivity and uncertainty

nalysis ( Cacuci et al., 2003 ). In this work, the effect of rainfall is rel-tively small compared to incoming waves. So infiltration is not takennto account here, namely all amount of rainfall water becomes pondedater on ground surface. We will further introduce infiltration rate in

uture work. Due to the lack of optimization of codes, the CPU time

187

s not demonstrated here. Instead, we have demonstrated the compu-ational cost is significantly reduced by the decrease of the number ofodes used while the accuracy remains the same or better than that inxed mesh modelling. In future work we will focus on the optimizationf codes (data structures).

cknowledgements

The Authors acknowledge the support of: the NSFC grant 11502241 ,unding from the European Union Seventh Framework ProgrammeFP7/20072013) under grant agreement No. 603663 for the researchroject PEARL (Preparing for Extreme and Rare events in coastalegions), the EPSRC MEMPHIS multi-phase flow programme grant EP/K003976/1 ) and EPSRC (MAGIC) ( EP/N010221/1 ). Funding foralinas from EPSRC (Smart-GeoWells grant EP/R005761/1 ) is gratefullycknowledged. We would like to thanks Dr. Linmei Nie’s valuable ad-ice. The data used are listed in the references, tables and figures. Weould like to thank the four reviewers for their in-depth comments that

ontributed to improving the presentation of our paper.

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Numerical simulation of floods from multiple sources using an adaptive anisotropic unstructured mesh method1 Introduction2 Methodology2.1 Governing equations2.1.1 Momentum and continuity equations2.1.2 Drag coefficient2.1.3 Boundary conditions for the joint flooding events

2.2 Anisotropic dynamic mesh optimization technique (anisotropic-DMO)

3 Model application3.1 Descriptions of study site and data3.2 Extreme sea-level and rainfall events3.3 Model applications

4 Results and discussion4.1 Individual flooding events4.1.1 Flooding map4.1.2 Comparison with MIKE 21 FM results at detector locations

4.2 Joint flooding events4.3 Impact of buildings on flooding simulations4.4 Sensitivity to the forcing inputs and mesh resolution4.5 Performance of floodity modelling

5 ConclusionsAcknowledgementsReferences

advances in water resources - spiral: home...domingo et al., 2010). recently various eﬀorts to...

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