Comparison of 2D triangular C-grid shallow water models
Hamidreza Shirkhani a , Abdolmajid Mohammadian
a , Ousmane Seidou
a , Hazim Qiblawey
b
a Department of Civil Engineering, University of Ottawa, CBY D216, 161 Louis Pasteur, Ottawa, ON, K1N 6N5, Canada b Department of Chemical Engineering, Qatar University, P.O. Box 2713, Doha, Qatar
a r t i c l e i n f o
Article history:
Received 6 May 2016
Revised 14 October 2017
Accepted 21 November 2017
Available online 24 November 2017
Keywords:
Shallow water equations
Dispersion relation analysis
Triangular C-grid
Source terms
a b s t r a c t
An ideal two-dimensional (2D) shallow water model should be able to simulate correctly various types
of waves including pure gravity and inertia-gravity waves. In this paper, two different triangular C-grid
methods are considered, and their dispersion of pure gravity waves, frequencies of inertia-gravity waves
and geostrophic balance solutions are investigated. The proposed C-grid methods employ different spa-
tial discretization schemes for coupling shallow water equations together with the various reconstruc-
tion techniques for tangential velocity estimation. The proposed reconstruction technique for the second
method, which is analogous to a hexagonal C-grid scheme, is shown to be energy conservative and sat-
isfies the geostrophic balance exactly while it supports the unphysical geostrophic modes for hexagonal
C-grid. Because of the importance of the application of 2D shallow water models on fully unstructured
grids, particular attention is also given to various types of isosceles triangles that may appear in such
grids. For the gravity waves, the results of the phase speed ratio of the computed phase speeds over the
analytical one are shown and compared. The non-dimensional frequencies of various modes for inertia-
gravity waves are also investigated and compared in terms of being monotonic and isotropic respect to
the continuous solution. The analyses demonstrate some advantages of the first method in phase speed
behaviour for gravity waves and monotonicity of inertia-gravity dispersion. The results of the dispersion
analysis are verified through a number of numerical tests. The first method, which is shown to have
a better performance, examined through more numerical tests in presence of various source terms and
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 137
Fig. 1. The normal velocity components along the axes of the non-orthogonal coordinate system.
n
t
h
s
i
fi
g
S
C
f
m
t
o
s
s
a
m
d
w
i
c
t
a
c
t
e
c
g
W
e
h
o
l
g
i
i
c
2
s
e
o
w
p
a
t
r
e
f
S
w
s
t
b
t
a
d
s
3
c
r
t
p
v
i
v
x
n
e
t
S
w
m
d
T
c(w
b
i
f
a
m
umerical modes are sensitive to the details of the discretiza-
ion of the Coriolis terms. The performance of the C-grid on the
exagonal grids for different reconstruction techniques was also
tudied [24,40] . Thuburn et al. [39] proposed a tangential veloc-
ty reconstruction such that the steady geostrophic mode is satis-
ed. In addition, the issues with reconstruction techniques in C-
rid were studied [16,45] . Using the dispersion relation analysis,
hirkhani et al. [33] investigated the behaviour of the triangular
-grid scheme for the shallow water equations. They investigated
ast gravity waves and reported good performance of the numerical
odel for both linear and non-linear cases.
One of the important modelling aspect for an ideal shallow wa-
er model is to account correctly for different source terms. Most
f the finite-volume schemes for shallow water equations were
hown to lead to numerical oscillations in the presence of the
ource terms, and several special treatments were proposed to bal-
nce the source and flux term [12,15,32,35] .
In this paper, we investigate two different 2D shallow water
odels based on the triangular C-grid method. Different spatial
iscretizations for continuity and momentum equations together
ith different velocity reconstruction techniques are implemented
n proposed models. Using the dispersion relation analysis, the dis-
rete frequencies of the models compared with continuous solu-
ions. Moreover, we investigate the effect of fully unstructured tri-
ngular grid cells. To this end, we consider an isosceles triangular
ell with various vertex angles that can represent the deviation of
he triangular grid cell from the equilateral shape. We study the
ffect of the grid on the isotropy and monotonicity of the frequen-
ies solutions. The discrete frequencies are also investigated for the
eostrophic solution as well as gravity and gravity-inertia waves.
e verify the theoretical analyses through a number of numerical
xamples. As per dispersion analyses, the first method is shown to
ave a better performance and is further examined in the presence
f various source terms in shallow water equations.
The current paper is organized as follows. In Section 2 , the shal-
ow water equations are introduced. Section 3 describes 2D trian-
ular C-grid models. The dispersion relation analysis is presented
n Section 4 , and the proposed models are examined through var-
ous numerical examples in Section 5 . We complete the paper by
oncluding remarks in Section 6 .
. Shallow water equations
In this section, the linear 2D shallow water equations are pre-
ented. The two-dimensional linear form of the shallow water
quations in Cartesian coordinates and, in the presence of the Cori-
lis term can be written in the following form:
∂η
∂t + H∇ . u = 0 (1)
∂u
∂t + f k × u + g∇η = S (2)
here η stands for the surface elevation, u and v are the com-
onents of the depth-averaged velocity vector u = ( u, v ) in the x -
nd y -directions, f is the Coriolis parameter, k is a unit vector in
he vertical direction, g is the gravitational acceleration, H is the
eference depth of the water, and ∇ is the two-dimensional gradi-
nt operator. S is the vector of the source terms and in a general
orm can be written as follows:
= τw
− τb (3)
here τw
= ( τwx , τwy ) and τb = ( τbx , τby ) are the surface wind
tress and the bottom friction vectors, respectively. The source
erms are not considered for the theoretical analysis and will
e used in Section 5 , where we examine the proposed method
hrough a number of numerical examples. The Coriolis term plays
n important role in the derivation of inertia-gravity waves. For the
ispersion relation analysis, we consider the f -plane where a con-
tant Coriolis parameter is assumed f = f 0 .
. Triangular C-grid methods
In this part, two triangular C-grid methods, used for spatial dis-
retization of shallow water Eqs. (1) , (2) , together with their cor-
esponding reconstruction techniques are introduced. We refer to
he first and second methods as method A and B throughout the
aper, respectively. In the C-grid approach, the water surface ele-
ation is located at the cell’s circumcenter, while the normal veloc-
ties are considered at the cell mid-edges. Fig. 1 shows the normal
elocity components U 1 , U 2 and U 3 along the defined x 1 , x 2 and
3 directions, respectively. In order to obtain the equation for the
ormal component of velocity at each edge, the dot product of the
dge normal vector n = ( n x , n y ) will be taken with the momen-
um Eq. (2) ,
∂U
∂t + f u n y − f v n x + g
∂η
∂n
= S n
n = n . ( τw
− τb )
(4)
here ∂ ∂n
is the edge normal gradient, and U = n . ( u, v ) is the nor-
al velocity defined at the cell edge. The edge normal vector n is
efined at the middle of the three different edge types of each cell.
he edge normal gradient of the water surface elevation at each
ell mid-edge is defined as follows:
∂η
∂n
)j
=
η f − ηb
D j
(5)
here D j is the distance between the circumcenters of two neigh-
ouring cells and j = 1 , 2 , 3 is the edge type index. f provides the
ndex of the cell in the direction of n j while b provides the index
or the cell in the opposite direction; see Fig. 2 .
The u and v components of Cartesian velocity vector u = ( u, v )re required at the cell faces for the Coriolis terms in the mo-
entum Eq. (4) . Therefore, a reconstruction method is required to
138 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 2. Estimation of normal gradient of water surface elevation at face j .
Fig. 4. The finite-difference discretization along x 2 direction ( δx 2 ): − shows the nor-
mal velocities participating in finite-difference discretization along x 2 direction at
cell center shown by �.
3
o
a
g
t
w
r
s
t
g
fi
t
t
m
b
r
a
a
w
w
t
obtain the tangential velocities at the cell faces. In the following
section, discretization methods for continuity (1) and momentum
(4) equations along with the corresponding reconstruction tech-
nique for each presented C-grid method will be described.
3.1. Method a
In the first method, the continuity Eq. (1) is discretized in the
following form:
∂η
∂t +
H
A i
3 ∑
j=1
U j d j = 0 (6)
where A i is the cell area, d j is the length of the face j , and U j is
the normal velocity at the j th side. For the momentum Eq. (4) , the
edge normal gradient of the water surface elevation is estimated
using (5) . For the estimation of the Coriolis term reconstruction of
tangential velocities is required.
Following the work of Fringer et al. [14] and Perot [26] , the re-
construction technique for method A will result in the following
velocity vector at the j th edge,
u j =
1
2
(u 1 j + u 2 j
)(7)
where u 1 j and u 2 j are the velocity vectors at the cell centers of the
two sides of face j and are obtained as follows,
u i =
1
A i
3 ∑
j=1
n j U j d j d ∗i j (8)
where d ∗1j
and d ∗2j
are the distances between the face j and the cir-
cumcenters of cells 1 and 2 which share face j , respectively. Fig. 3 a
shows the stencil of the first method for the velocity reconstruc-
tion at one of the cell faces, and it shows the normal velocity com-
ponents that take part in velocity reconstruction of this face.
Fig. 3. Stencil for reconstruction techniques: − shows the normal velocities participating i
B.
.2. Method b
The second method proposed in this paper for discretization
f the continuity (1) and momentum (4) equations is, indeed, an
nalogous of a method in Ni ckovic et al. [24] for the hexagonal C-
rid scheme. In this method, the continuity Eq. (1) for equilateral
riangular cells can be written as,
∂η
∂t +
2
3
H
3 ∑
j=1
δx j U j = 0 (9)
here δxj is the finite-difference discretization applied to the cor-
esponding normal velocities in the j th direction. Fig. 4 , for in-
tance, shows the stencil for δx 2 , the finite-difference discretiza-
ion along x 2 axes. As for the momentum Eq. (4) , the edge normal
radient of the water surface elevation is estimated similar to the
rst method using (5) . The reconstruction technique employed in
his method is based on the normal velocity averaging. Indeed, the
angential velocity at the j th edge is obtained by averaging the nor-
al velocities along the two other directions j 1 and j 2 . It should
e noted that j 1 and j 2 directions can be specified based on the
ight-handed coordinates. For instance, if j = x 1 , we have j 1 = x 2 nd j 2 = x 3 . This reconstruction technique results in the following
pproximation of the Coriolis term in the momentum equation,
( f v n x − f u n y ) j = − f √
3
(U
j 2 j 1
− U
j 1 j 2
)(10)
here superscript ¯ j shows the averaging along the j direction
hile the subscript j shows the j th edge of the cell. For instance,
he estimation of Coriolis term at edge j = 1 can be written as
n the velocity reconstruction at face shown by for (a) Method A and, (b) Method
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 139
t
A
i
s
4
4
t
B
c
v
t
p
r
t
z
a
l
ω
T
t
t
ω
i
t
t
d
4
m
l
o
t
v
m
e
t
(
4
s
2
e
u
t
U
a
t
t
o
v
U
U
U
w
t
p
t
η
η
w
t
E
s⎛⎝w
m
A
μ
0
μ
w
a
μ
s
d
n
b
( f v n x − f u n y ) 1 = − f √
3 ( U
x 3 2
− U
x 2 3
) . The stencil of this reconstruc-
ion technique is illustrated in Fig. 3 b for one of the cell edges.
s it can be seen in this figure, there are two normal velocities
n each j 1 and j 2 direction which are averaged using the ¯ j super-
cript.
. Dispersion relation analysis
.1. Continuous case
For the purpose of analysing the dispersion relation of the con-
inuous case, the linear shallow water Eqs. (1) , (2) are considered.
y considering the behaviour of one Fourier mode, the solution
an be studied. Thus, the solutions in the form of u = ˜ u e i ( kx + ly + ωt ) ,
=
v e i ( kx + ly + ωt ) , and η = ˜ ηe i ( kx + ly + ωt ) are sought, where k and l are
he wave numbers in the x − and y − directions, respectively. Re-
lacing the wave form of u, v , and η in the governing Eqs. (1) , (2)
esults in a square matrix for the amplitudes ˜ u , v , and ˜ η. For a non-
rivial solution, the determinant of the matrix should be equal to
ero, which leads to a relationship between the wave numbers k
nd l and the frequency ω. The relationship is called dispersion re-
ation and is obtained as follows:
(ω
2 − f 2 − gH
(k 2 + l 2
))= 0 (11)
he first solution of ω = 0 corresponds to the geostrophic mode
hat matches to the slow Rossby mode on a β-plane, and the other
wo solutions are as follows:
AN = ±√
f 2 + gH
(k 2 + l 2
)(12)
n which ω AN stands for the analytical solution and corresponds to
he inertia-gravity mode. As Eq. (12) shows, ω AN is purely real, and
herefore, all modes are naturally stable and neither amplify nor
ecay.
.2. Semi-discrete triangular C-grid methods
In this section, we analyse two proposed triangular C-grid
ethods for shallow water equations. As for Method A, a triangu-
ar grid made up of isosceles triangles with arbitrary vertex angle
f 2 α and side length of h is considered; see Fig. 5 . We use the ob-
ained dispersion relation to investigate the effect of cell grid de-
iation from an equilateral one on the performance of the C-grid
ethod. For Method B, we only consider grids made up of equilat-
ral triangles. We then compare the results of the dispersion rela-
ion analysis of Methods A and B for the case of equilateral grids
i.e. 2 α = 60 o ).
.2.1. Method a
For dispersion analysis of the semi-discrete method A, we con-
ider (acute) isosceles triangular cell with arbitrary vertex angle of
α, as is shown in Fig. 5 a.
Considering the discrete form of the continuity and momentum
quations, one can employ Fourier analysis similar to the contin-
ous case in order to find the dispersion relation corresponding
o the discrete form. The discrete solutions corresponding to U =˜ e i ( k x j + l y j + ωt ) and η = ˜ ηe i ( k x j + l y j + ωt ) are sought, where ˜ U and ˜ η are
D =
1
2
⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0
μ1 cos ( α) cos
(k h x + 2 l h y
4
)
−μ2 cos ( α) cos
(k h x − 2 l h y
4
)
mplitudes, and ( x j , y j ) coordinates are expressed in terms of dis-
ance to a reference circumcenter.
The normal velocity U j may be located on three possible edge
ypes; i.e., either on the horizontal one or the other two biased
nes. Using (5) , (7) and (8) , discrete forms of three possible normal
elocities are obtained as follows,
t 1 +
(f v n 1 n x 1 − f u
n 1 n y 1
)+
g
h D 1
(η f 1 − ηb1
)= 0
t 2 +
(f v n 2 n x 2 − f u
n 2 n y 2
)+
g
h D 2
(η f 2 − ηb2
)= 0
t 3 +
(f v n 3 n x 3 − f u
n 3 n y 3
)+
g
h D 3
(η f 3 − ηb3
)= 0
(13)
ith D 1 =
cos ( 2 α) cos α and D 2 = D 3 = tan (α) .
Fig. 5 b shows a stencil for the momentum equation discretiza-
ion in Method A. Similarly, the water surface elevation η can be
laced at two different locations, i.e. either on the upper or lower
ip triangles. Using Eq. (6) , one obtains:
t b1
+
H
| T | h ( U 1 2 sin α + U 2 + U 3 ) = 0
t f 1
+
H
| T | h
(−U 1 2 sin α − U
′ 2 − U
′ 3
)= 0
(14)
here | T | = h 2 sin α cos α is the cell area and superscript t shows
he time derivative.
Substituting the periodic solution of ˜ U and ˜ η into the discrete
qs. (13) and (14) , one can obtain a square matrix form for the
ystem of amplitudes as:
iω I 2 H
T A
g
h
B iω I 3 − f D
⎞
⎠
( −→
˜ η−→
˜ U
)
=
(0
0
)(15)
here −→
˜ U = ( U 1 , ˜ U 2 , ˜ U 3 ) , −→
˜ η = ( ηb , ˜ η f ) , and I n is the n × n identity
atrix, and we also have:
= h
(
a 2 sin α b c
−1
a 2 sin α −1
b −1
c
)
,
B =
⎛
⎜ ⎜ ⎜ ⎝
1
D 1
0 0
0
1
D 2
0
0 0
1
D 3
⎞
⎟ ⎟ ⎟ ⎠
⎛
⎜ ⎜ ⎜ ⎝
−1
a a
−1
b b
−1
c c
⎞
⎟ ⎟ ⎟ ⎠
,
2 cos ( α) cos
(k h x + 2 l h y
4
)−μ3 cos ( α) cos
(k h x − 2 l h y
4
)
−μ3 sin ( 2 α) cos
(k h x
4
)
2 sin ( 2 α) cos
(k h x
4
)0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
,
ith:
= e −il h y D 1
2 cos α b = e i 2
(k h x
2 −l h y D 1 )
c = e −i 2
(k h x
2 + l h y D 1 )
1 =
cos ( 2 α)
2 cos 2 ( α) ; μ2 = μ3 =
1
4 cos 2 ( α)
It should be noted that matrices A , B and D are related to the
tencil of velocity and water surface elevation as they represent the
istance of periodic solutions of � U and
� η from an arbitrary coordi-
ates center in the stencil. The dispersion relation is then obtained
y setting the determinant of the 5 × 5 matrix system (15) to zero.
140 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 5. (a) Isosceles triangular cell grid with vertex angle 2 α and side length of h , (b) Stencils used for momentum equation discretization in Method A, and (c) Method B.
(dashed vectors show the reconstructed velocities at cell edges).
4
g
m
f
U
U
U
t
E
t
E
t
o
s
The dispersion relation will be a polynomial of degree 5 with the
following form:
c 5 ω
5 + c 4 ω
4 + c 3 ω
3 + c 2 ω
2 + c 1 ω + c 0 = 0 (16)
where the coefficients c 0 , c 1 , . . . , c 5 are functions of kh x and lh y .
For Method A, coefficients c 4 , c 2 and c 0 are zero in (16) that
implies ω = 0 is one of the roots which corresponds to the
geostrophic solutions and is identical to the continuous case (11) .
It demonstrates the capability of this method in reproducing the
stationary character of the geostrophic motion. The other roots are
obtained as follows:
ω 1 = 0 , ω 2 , 3 = ±O ( 1 ) , ω 4 , 5 = ±O
(1
h
).
for which in the limit as mesh spacing h → 0 we can obtain:
ω 2 , 3 = ω AN + O
(h
2 ), ω 4 , 5 = ± 2
√
gH
tan ( α) √
cos ( 2 α) h
+ O ( h ) (17)
As one can observe, ω 2, 3 correspond to the continuous solu-
tion, while ω 4, 5 can be considered as spurious modes. The explicit
form of the dispersion relation ω 2, 3 for the equilateral triangular
grids, i.e. 2 α = π/ 3 , is as follows:
ω 2 , 3 =
1
6 h
√
R A 1 − 6 h
√
R A 2
R A 1 = 432 gH + 6 h
2 (1 + pq + p 2
)R A 2 = h
4 (1 + 2 pq + 2 p 2 + 2 p 3 q + p 2 q 2 + p 4
)+48 gH h
2 (1 − 5 pq − 5 p 2
)+ 576 g 2 H
2 (1 + 4 pq + 4 p 2
) (18)
p = cos ( k h x ) ; q = cos ( l h y ) c
.2.2. Method b
For this method, we just consider the equilateral triangular
rids i.e. 2 α = π/ 3 . Using (5) and (10) the discrete forms of mo-
entum equation for three possible normal velocities at the cell
aces are obtained as follows,
t 1 −
f √
3
(U
3 2 − U
2 3
)+
g
h D 1
(η f 1 − ηb1
)= 0
t 2 −
f √
3
(U
1 3 − U
3 1
)+
g
h D 2
(η f 2 − ηb2
)= 0
t 3 −
f √
3
(U
2 1 − U
1 2
)+
g
h D 3
(η f 3 − ηb3
)= 0
(19)
Fig. 5 c shows a stencil for the momentum equation discretiza-
ion in Method B. Similarly, using discrete form of the continuity
q. (9) for Method B, the following equations are obtained for the
wo possible places of the water surface elevation,
∂ η f 1
∂t +
2
3
H
3 ∑
j=1
δx j U j = 0
∂ ηb1
∂t +
2
3
H
3 ∑
j=1
δx j U j = 0
(20)
Substituting the periodic solution of ˜ U and ˜ η into the discrete
qs. (19) and (20) , the square matrix form is obtained. Similar to
he procedure for the Method A, the dispersion relation is then
btained by setting the determinant of the matrix to zero. For the
ake of brevity, we do not present the tedious mathematical pro-
ess for obtaining the dispersion relation. The resulting relation,
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 141
Fig. 6. Selected directions for the triangular grid cell.
w
c
f
i
a
t
[
c
a
ω
f
ω
c
o
A
t
(
a
ω
hich is a polynomial of degree 5, can be presented as follows:
5 ω
5 + c 4 ω
4 + c 3 ω
3 + c 2 ω
2 + c 1 ω + c 0 = 0 (21)
or Method B, coefficients c 4 , c 2 and c 0 are zero that implies ω = 0
s one of the roots which corresponds to the geostrophic solutions
Fig. 7. Phase speed ratio for Method A with (a) 2 α = π/ 3 , (b) 2 α
nd is identical to the continuous case (11) . It should be mentioned
hat the original reconstruction method proposed in Ni ckovic et al.
24] for the hexagonal C-grid does fail to reproduce the stationary
haracter of the geostrophic solution. The other roots are obtained
s follows:
1 = 0 , ω 2 , 3 = ±O ( 1 ) , ω 4 , 5 = ±O ( 1 ) .
or which in the limit as mesh spacing h → 0 we can obtain:
2 , 3 = ω AN + O
(h
2 ), ω 4 , 5 = ±
√
gH
(k 2 + l 2
)+ O
(h
2 )
(22)
Roots ω 2, 3 correspond to the continuous solution while ω 4, 5
an be considered as spurious modes. As it can be seen, the spuri-
us mode for this method has different form compared to Method
. Indeed, this spurious mode corresponds to the continuous solu-
ion of pure gravity waves, i.e. if the Coriolis parameter is f = 0 in
12) . The explicit form of the dispersion relation ω 2, 3 is obtained
s follows,
2 , 3 =
1
6 h
√
R B 1 + 6 h
√
R B 2
R B 1 = 96 gH
(2 − qp − p 2
)+ 6 h
2 (1 − 3 pq + q 2 + 4 q p 3
)R B 2 = 128 gH
(p 2 + q 2 + 2 q p 3 − p 4 − pq − q 2 p 2
)
= π/ 4 , (c) 2 α = 5 π/ 12 , and (d) Method B with 2 α = π/ 3 .
142 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 7. Continued
a
P
w
a
i
c
c
t
t
w
s
O
k
a
s
t
P
c
a
+ h
2 (16 q 2 p 6 + 8 q 3 p 3 + 8 q p 3 − 6 qp − 24 q 2 p 4
− 6 q 3 p + 2 q 2 + q 4 + 9 q 2 p 2 + 1
)(23)
4.3. Analysis of semi-discrete methods
In this section, we consider the discrete wave frequencies of
both methods as an important modelling aspect. Indeed, we com-
pare the behaviour of two methods with continuous solutions. In
order to investigate the frequency behaviour, three different wave
types are considered, namely, gravity waves, inertia-gravity waves
and geostrophic balance. For method B, we consider equilateral
grids while we study the isosceles triangular grids with different
vertex angles for method A.
4.3.1. Gravity waves
In order to analyse the gravity waves, we set f = 0 in the
continuous form (12) as well as the obtained dispersion relations
(18) and (23) for methods A and B. We investigate the phase speed
ratio which, indeed, is the relative gravity wave frequencies de-
fined as a ratio of the discrete frequencies and the continuous one
s a function of kh and lh ,
h r =
ω
G dis
ω
G AN
(24)
here Ph r is the phase speed ratio and ω
G dis
, ω
G AN
are the discrete
nd, analytical (continuous) frequencies corresponding to the grav-
ty wave. It can be seen that for Method A, the modes of type O (1)
oincide with the continuous solution and those of types O (1/ h )
an be considered as the spurious modes. However, for Method B,
here are two double roots and all the roots coincide with the con-
inuous solution. Therefore, there is no spurious mode associated
ith the gravity waves. We investigate the frequencies along the
elected axes which are presented in Fig. 6 . The directions OX and
Y represent the waves travelling in x − and y − directions where
h x = 0 and l h y = 0 , respectively. We select the diagonal axes OT 1
nd OT 2 for the triangular grid with k h x = l h y , k h x = −l h y . The re-
ults of phase speed ratio are illustrated in Fig. 7 as a surface func-
ion versus lh and kh as well as plots along the selected axis.
For an ideal discretization scheme, the phase speed ratio is
h r = 1 , which would be the case in the absence of numeri-
al dispersion. Since ( −π, π) corresponds to the shortest resolv-
ble wave, the surface functions are shown over a range of (0, π ).
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 143
Fig. 8. Dispersion relation results of non-dimensional frequency ω/ f for (a) continuous case, (b) Method A, and (c) Method B with equilateral triangular grids for the high-
resolution case ( λ/h = 2 ).
C
i
M
t
s
T
m
t
s
d
e
omparing Fig. 7 a and d, it can be seen that the phase speed ratio
s much more closer to 1 for both long and intermediate waves in
ethod A. It shows that the long and intermediate waves, which
ransfer most of the energy in the domain, travel at more closest
peed as continuous waves in Method A compared to Method B.
he phase speed ratios suddenly decrease for shorter waves which
ean that in both methods, the shorter waves are traveling slower
han expected. However, it can be seen that the resulting phase
peeds of Method A hold higher values, particularly in OX and OY
irections, compared to Method B.
In order to investigate the effect of grid cells deviation from the
quilateral form in Method A, one can consider the phase speed
144 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 9. Dispersion relation results of non-dimensional frequency ω/ f for method A with (a) 2 α = π/ 4 , (b) 2 α = 5 π/ 12 for the high-resolution case ( λ/h = 2 ).
t
t
w
a
h
t
t
t
t
M
a
c
e
g
2
i
i
t
2
q
l
h
i
a
l
f
c
ratios presented in Fig. 7 b and c. It can be seen that for the tri-
angular cell with vertex angle of 2 α = π/ 4 , the phase speed be-
haviour of Method A is improved compared to the equilateral case.
However, the isotropy of the frequencies is much lower. As for the
triangle with vertex angle of 2 α = 5 π/ 12 , one can see that the
phase speed ratio drops significantly compared to the equilateral
case.
4.3.2. Inertia-gravity waves
In this section, we consider the performance of the proposed
models in the presence of the Coriolis term. We compare the be-
haviour of their non-dimensional frequencies ( ω/ f ) with those cor-
respond to the continuous case. To this end, by defining the Rossby
radius as λ =
√
gH / f , we can rewrite the continuous dispersion re-
lation (12) as follows: (w
f
)2
= 1 +
(λ
h
)2 [( kh )
2 + ( lh ) 2 ]
(25)
We consider two cases, namely, a low-resolution case with λ/h =3 / 8 and a high-resolution one with λ/h = 2 . The non-dimensional
frequency ( ω/ f ) for Methods A and B can also be obtained from
Eqs. (18) and (23) . While the values of kh x and lh y vary over (0, π ),
the non-dimensional frequencies ( ω/ f ) are illustrated in Figs. 8–
11 as surface functions, contour lines and plots along the selected
axes. Fig. 8 shows the results of the high-resolution case ( λ/h = 2 )
for the continuous solutions as well as those of Methods A and B
for equilateral triangles. As it can be seen, the exact frequencies
increase monotonically with increasing wave numbers in all direc-
ions and reach to the value of more than 8 along the OT direc-
ions. One can see that the true frequency of the inertia–gravity
aves is also fully isotropic. In terms of isotropy, both Methods A
nd B hold a high degree of isotropy. However, Method A has a
igher isotropy. The non-dimensional frequencies increase mono-
onically in all directions for both methods. They have lower values
han the continuous solution, however, Method A results are closer
o the analytical solutions compared to Method B.
In order to investigate the effect of cell grid deviation from
he equilateral shape, the results of non-dimensional frequency for
ethod A with isosceles triangles with vertex angles of 2 α = π/ 4
nd 2 α = 5 π/ 12 are shown in Fig. 9 . As it can be seen, for both
ases, the level of isotropy is relatively high but lower than the
quilateral case. One can see that in terms of isotropy, the trian-
ular cell with 2 α = π/ 4 has a better behaviour compared to the
α = 5 π/ 12 case. Moreover, for the case of 2 α = π/ 4 , frequencies
ncrease monotonically in all directions while for 2 α = 5 π/ 12 , the
ncreasing behaviour is not monotonically in OT directions. In addi-
ion, the values of ω/ f are closer to the continuous solution for the
α = π/ 4 case.
As for the low-resolution λ/ 8 = 3 / 8 , the inertia–gravity fre-
uencies for the continuous case, and Methods A and B with equi-
ateral grids are shown in Fig. 10 . The exact solution, similar to the
igh-resolution case, is entirely isotropic and increasing monoton-
cally in all directions. The highest value of the frequency is also
bout 1.8 and reached along the OT directions. A high isotropy
evel is indicated for Method A while there is a high anisotropy
or Method B. Although the fact that Method B has closer frequen-
ies to the continuous case, its behaviour is not monotonic which
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 145
Fig. 10. Dispersion relation results of non-dimensional frequency ω/ f for (a) continuous case, (b) Method A, and (c) Method B with equilateral triangular grids for the
low-resolution case ( λ/h = 3 / 8 ).
m
i
M
n
f
n
s
t
w
g
s
i
h
w
w
b
continuous case.
ay result in a false numerical simulation. It should be noted that
n the OT directions, the increasing behaviour of the frequency for
ethod A is not monotonic for a very slight range of large wave
umbers close to π .
Similar to the high-resolution case, we may investigate the ef-
ect of grid shape for Method A. To this end, the corresponding
on-dimensional frequencies are shown in Fig. 11 . As it can be
een, for both cases, there is a high level of anisotropy. Moreover,
he frequencies do not increase in all directions with increasing the
ave numbers. It indicates that the performance of the triangular
rids with deviation from the equilateral case may lead to false
imulation of the physical process in the low-resolution case.
The results of dispersion relation analysis show that frequencies
n all investigated methods lose their monotonically increasing be-
aviour, except for Method A with equilateral grids. Numerically,
e found that for λ/ h < 3/8 the wave frequencies of inertia-gravity
aves for Method A starts decreasing with increasing wave num-
ers and it is not anymore monotonic in all directions as for the
146 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 11. Dispersion relation results of non-dimensional frequency ω/ f for method A with (a) 2 α = π/ 4 , (b) 2 α = 5 π/ 12 for the low-resolution case ( λ/h = 3 / 8 ).
s
a
w
2
s
w
a
λ
t
h
t
f
w
o
f
a
a
r
t
b
q
c
m
5. Numerical tests
In this section, we verify the theoretical analyses and evalu-
ate the ability of the selected method, Method A with equilat-
eral grids. Indeed, Method A employs a finite-volume discretization
technique for the continuity equation and finite differencing for
the momentum equation. Two first numerical tests verify the theo-
retical analyses of semi-discrete methods. As Method A with equi-
lateral grids is shown to have a better performance based on the
theoretical analyses, in the rest of the numerical examples, we test
the performance of this method. We examine the selected model
with various source terms as well as nonlinear terms. It should
be noted that in all numerical tests, the second-order Leap-Frog
time-stepping technique is used. As for the boundary conditions,
where the boundaries are periodic, the velocity on the cell edges
are reconstructed using the periodic boundary cells. For the no-
flow solid boundaries, the velocity reconstruction is done consid-
ering a zero velocity for the neighbouring cell.
5.1. Geostrophic balance
This test was proposed by Batteen and Han [4] and carried out
for various methods. In this test, we verify the theoretical analy-
sis of the dispersion relation of the inertial-gravity waves. We con-
ider a rectangular large domain with no-flow boundary conditions
nd a source point of 0.1 m at the middle of the domain and the
ater is initially at rest. We set the triangular grid resolution to
00 m and keep CF L = 0 . 1 for all cases. The Coriolis parameter is
et to f 0 = 10 −4 s −1 , for which with
√
gH = 40 , 7 . 5 and 2 . 5 m s −1
e have different resolutions of λ/h = 2 , 3 / 8 and 1/8, respectively.
The results of the free surface elevation η for various methods
re presented in Fig. 12 . As one can see, for all the methods, for
/h = 1 / 8 the negative elevations appear in the domain around
he source point which can be related to the non-monotonic be-
aviour of the non-dimensional frequency ω/ f , as was discussed in
he previous section. For λ/h = 3 / 8 , however, there is no negative
ree surface elevation around the source point for the Method A
ith equilateral grids ( 2 α = π/ 3 ). It is in agreement with the the-
retical analysis presented in Fig. 10 , where the non-dimensional
requency is monotonic along all the directions. For Method A tri-
ngles with 2 α = 5 π/ 12 and 2 α = π/ 4 , one can see the appear-
nce of very small negative values which are mostly along the di-
ections for which the behaviour of ω/ f is non-monotonic. Note
hat the negative elevations are clearer for Method B, which can
e related to its non-monotonic behaviour of non-dimensional fre-
uency in all directions. For the high-resolution case λ = 2 , all the
ases show good performance which is expected since the ω/ f is
onotonic in all directions for all the methods.
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 147
Fig. 12. Water surface elevation around the source point computed using various methods with various grid resolution: λ/h = 1 / 8 (left column), λ/h = 3 / 8 (middle), and
λ/h = 2 (right column).
148 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 13. Water surface elevation of the seiche wave for the domain midpoint during
6th period with wave number of (a) π /8, (b) π /4, and (c) π /2.
w
g
b
w
m
t
2
t
p
t
2
g
N
b
5.2. Seiche wave (standing wave)
In the second numerical test, a seiche is simulated as an ex-
ample of a gravity wave. In fact, seiche is a standing wave formed
by the superposition of two waves of equal wavelength and propa-
gating in opposite directions. Such a situation happens in confined
water bodies such as a lake by reflection from lateral boundaries.
The water surface elevation of a seiche in a closed, long and non-
rotating rectangular basin of length L and uniform depth H has a
simple trigonometric exact solution:
η( x, t ) = Acos ( kx ) cos ( ωt ) (26)
where A is the wave amplitude, x is the along-basin coordinate, t
is time, k = 2 π/λ is the wave number, λ is the wavelength, ω =2 π/T is the wave angular frequency, and T is the wave period. The
angular frequency and wave number can be calculated based on
the wave propagation speed c =
√
gH ,
ω = kc (27)
In this numerical test, the fourth oscillation mode of the seiche
is considered, where k = 4 π/L and λ = 2 L/ 4 . Indeed, in the fourth
mode of the seiche, the wavelength is equal to half of the basin
length. Different values for the closed basin with no-flow boundary
conditions, and of length L results in a seiche with a specific wave
number and length. The length of the triangles selected is h =2 . 5 km and the initial water surface η( x, 0 ) = 0 . 01 cos ( kx ) cos ( ωt )
is prescribed, with H = 10 m and g = 9 . 81 m
s 2 .
For various wave numbers, the results of the numerical test are
presented in Fig. 13 for both numerical models using equilateral
triangular grids. We keep the CFL number very small in order to
attenuate the effect of temporal discretization. The results illus-
trate oscillation of the midpoint of the domain for one period with
various wave numbers. Computed results using proposed numeri-
cal models are compared with exact solution in Fig. 13 . As it can
be seen, for smaller wave number e.g. π /8 the results computed
by both methods are in good agreement with the exact solution.
However, by increasing the wave number to π /4 and π /2 signif-
icant higher phase speed error can clearly be seen for Method
B. This confirms the results of the dispersion relation analysis in
Fig. 7 where higher phase speed error was predicted for Method B.
5.3. The Stommel problem
This numerical example is a wind-driven large-scale circulation
test originally proposed by Stommel [36] and widely used in the
literature [2,20] . In this test case, as was mentioned previously, we
further examine the performance of the selected Method A. Wind
is considered in this example, as one of the main factors in gener-
ating the surface waves. In addition to the surface wind, we con-
sider linearized bottom friction as well as the Coriolis force in a
square basin with a flat bottom. Since we employ the β−plane
approximation f = f 0 + βy , the solution is expected to be a con-
centrated crowding of streamlines toward the boundary due to
changes in the Coriolis force through the basin.
We consider a square 40 0 0 km × 40 0 0 km domain and set a no-
flow boundary condition all around the basin. The mean depth of
the water is assumed to be H = 400 m and the reduced gravity
acceleration is g ′ = 0 . 01 m s −2 . For the β−plane approximation of
f = f 0 + βy , we set f 0 = 10 −4 s −1 and β = 10 −11 m
−1 s −1 , which
leads to the radius of deformation R d =
√
g ′ H f 0
= 20 km .
A zonal wind is considered, which results in a surface stress of
τw
= ( τwx , τwy ) with: {
τwx = τ0 sin
(πy
L
)τwy = 0
(28)
here − L 2 < y < L/ 2 and τ0 = 2 × 10 −7 m s −2 , spinning up a sin-
le anticyclonic gyre. The bottom friction is given by τb = ξu . The
ottom drag coefficient of ξ = 10 −6 s −1 , leads to a Stommel layer
idth of δ =
ξβ
= 100 km . Two cases, of low and high-resolution
esh grids, are considered in this numerical experiment. We set
he computational triangular cells at size h = 20 km and h =00 km for the low-and high-resolution grids respectively, and we
ake CF L = 0 . 1 .
The main objective of this test is to examine and compare two
roposed triangular C-grid shallow water models with the simula-
ion of the evolution of the velocity divergence field. For an ideal
D shallow water model an accurate estimation of velocity diver-
ence is essential for a good resolution of the vertical velocity field.
ote that in this test, we consider the effect of the wind in com-
ination with the Coriolis force and bed shear stress as important
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 149
Fig. 14. Contour plots of the divergence field in a 40 0 0 km × 40 0 0 km domain computed by unstructured C-grid scheme after 100 days using (a) low-resolution and (b)
high-resolution mesh grids. Contour interval is 1 × 10 −9 s −1 .
Fig. 15. As of Fig. 14 but after 500 days . Contour interval is 1 × 10 −9 s −1 .
f
g
[
g
l
f
o
t
s
b
l
s
c
i
m
[
p
o
r
fi
e
n
o
t
I
w
t
p
p
t
w
5
a
c
l
w
e
w
t
actors affecting the surface oceanic waves. The structured rectan-
ular C-grid scheme has already been examined with this test by
2,20] and the results reported for both low- and high-resolution
rids.
The obtained divergence of the velocity field using a triangu-
ar C-grid scheme after 100 days of simulation is shown in Fig. 14
or both low and high-resolution grids. In the high-resolution case,
ne can see a good agreement between the obtained results of the
riangular C-grid scheme and those reported for a structured C-grid
cheme in [2,20] .
More precisely, one can see the appearance of the eddy at the
ottom-right corner of the basin. Moreover, a zoom of the bottom-
eft corner of the basin ( Fig. 14 b) is presented in Fig. 16 a, which
hows that the results are free of numerical oscillations, even very
lose to the basin wall. With respect to the results obtained us-
ng low-resolution grids ( Fig. 14 a), one can see a great improve-
ent compared to the results for the structured C-grid reported in
2] which confirms the dispersion relation analysis results, as ex-
lained in Section 4 of this paper.
The obtained divergence of the velocity field after 500 days
f simulation is also shown in Fig. 15 for both low- and high-
esolution grids. It can be seen that the evolution of the divergence
eld is resolved well by the triangular C-grid scheme. Again, the
f
ddy appears and is developed accurately at the bottom right cor-
er of the basin. For the high-resolution grid, one can see a zoom
f the bottom-left corner of the basin in Fig. 16 b, which shows
hat the results are free of numerical noises close to the boundary.
n the low-resolution case ( Fig. 15 a), the results are in agreement
ith the high-resolution one, and one can see the appearance of
he eddy at the bottom-left corner of the basin. One can also com-
are the results with those reported for low-resolution grids com-
uted by a CD-grid in [2,20] . Altogether, the results of the Stommel
est show that the proposed triangular C-grid method performs
ell and is accurate even when using low-resolution mesh grids.
.4. Equatorial Rossby waves
In this numerical example, we further test Method A with ex-
mining its ability in resolving the slow Rossby waves. Indeed, we
onsider the exact solution of the linear β-plane equatorial shal-
ow water equations, which is called symmetric equatorial Rossby
aves of Index 1. As these waves are long and slow, they have an
ssential role in transferring energy in the ocean.
When we assume the exact solution of an equatorial Rossby
ave as the initial condition, a numerical method should be able
o preserve the solution. However, many of the available schemes
ail to do so.
150 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 16. Zoom of the bottom left corner of the basin for divergence of velocity field after (a) 100 days and (b) 500 days . Contour interval is 0 . 4 × 10 −9 s −1 .
U
w
w
f
i
o
g
t
l
s
s
o
a
o
o
s
o
d
o
I
c
v
f
f
w
f
t
i
p
a
t
l
5
u
[
I
m
p
e
Shallow water Eqs. (1) and (2) on an equatorial β-plane can
be stated in the dimensionless form using the dimensionless vari-
ables x ′ = x/ L ∗, y ′ = y/ L ∗, t ′ = t/ T ∗, u
′ = u / U
∗ and η′ = η/H. Using
the Lamb parameter E =
4 �2 a 2
gH , the characteristic length ( L ∗), time
( T ∗), and velocity ( U
∗) can be written as:
L ∗ =
a
E 1 4
T ∗ =
E 1 4
2�
∗ =
√
gH
(29)
where � is the angular frequency of the Earth’s rotation and a is
the mean radius of the Earth.
First, we write the analytical solution of linear shallow wa-
ter Eqs. (1) , (2) for symmetric equatorial Rossby waves of Index 1.
We assume a Rossby wave with a wavelength of W L = 11 , 0 0 0 km
which propagates in a 2 W L × W L domain with periodic boundary
conditions. We set the mean depth H = 300 m and gravity g =0 . 03 m s −2 , and we use (29) to estimate the characteristic length,
time, and velocity. The non-dimensional domain 2 L × L is obtained
by L = 16 . The dispersion relation for this wave is as follows:
ω
2 − k 2 − k
ω
= 3 (30)
where k = 2 π/L is the wave number. The analytical solution of an
equatorial Rossby wave with an amplitude of A w
is given by:
r + = −A w
2 y 2 − 1
k − ω R
sin ( kx − ω R t ) e − y 2
2
r − = −A w
−1
k + ω R
sin ( kx − ω R t ) e − y 2
2
η =
r − − r +
2
u = − r − ∓ r +
2
(31)
where r ± are the Riemann invariants and ω R is the smallest root
in the dispersion relation (30) . The water surface elevation for a
Rossby wave with A w
= 1 obtained from analytical solution (31) is
shown in Fig. 17 (right) at dimensionless time T = 250 , which is
equivalent to 5 wave periods.
Now, in order to examine the triangular C-grid scheme, the an-
alytical solution (31) is used as an initial condition for simulat-
ing a large-scale equatorial Rossby wave. To this end, a mesh grid
ith cell area approximately equal to 1500 km
2 is employed, and
e set CF L = 0 . 1 . The contour plot of the water surface elevation
or Rossby waves obtained from the proposed scheme at T = 250
s shown in Fig. 17 (left). Compared to the analytical results, the
btained results are promising. Fig. 17 shows that the proposed C-
rid triangular method preserves the symmetric form of the solu-
ion well. It also presents accurate results compared with the ana-
ytical one, without any significant damping.
A 1-D slice of the computed water surface elevation is also
hown and compared to the analytical one in Fig. 18 . As one can
ee, neither significant damping nor any substantial phase error is
bserved in the solution, and they are in good agreement.
By decreasing the mesh grid resolution, we also examine the
ccuracy of the method in terms of predicting the maximum value
f the water surface elevation. We compare the maximum value
f the water surface for Rossby waves computed using various cell
izes, and results showed that the damping is negligible for a range
f grid resolutions. Indeed, the relative true errors of damping for
ifferent cases is less than 1%.
Using this numerical test with an analytical solution, the spatial
rder of accuracy for the triangular C-grid scheme is also obtained.
n order to numerically determine the rate of accuracy for spatial
onvergence, the CFL number is set to a small value in order to
anish the error due to the temporal discretization. We use dif-
erent mesh grids with various resolutions and obtain the results
or water surface elevation for a Rossby wave at t = 250 . Equipped
ith the corresponding analytical solution, we obtain the L 2 error
or each case. The evolution of the L 2 error versus the length of
he computational triangle cells ( x ) is plotted in a l og − l og scale
n Fig. 19 . It shows that the spatial order of accuracy for the pro-
osed scheme is 1.95, which is close to the formal order of spatial
ccuracy of 2 for the C-grid. The results show the capability of the
riangular C-grid method in the simulation of large-scale slow and
ong oceanic waves.
.5. Non-linear wind induced circulation in a circular basin of
neven topography
This numerical test was originally proposed by Kranenburg
19] and then has been broadly used in the literature [8,29,34] .
n this numerical example, we test the capability of Method A in
odelling the non-linear shallow water equations with topogra-
hy. In this test case, simulation of the topographic gyres is consid-
red due to a uniform wind field. In addition to the surface wind
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 151
Fig. 17. Water surface elevation for Rossby waves in a dimensionless 32 × 8 domain at dimensionless time T = 250 : Using proposed scheme with CFL = 0.1 (left) and
analytical solution (right).
Fig. 18. Comparison of numerical and analytical solution of water surface elevation
for Rossby wave after 5 periods ( T = 250 ).
s
g
c
l
c
w
t
I
e
g
[
m
d
w
i
t
C
f
h
w
a
b
p
e
C
w
c
c
b
g
i
o
a
l
s
t
p
tress term in the momentum equation, the non-uniform topo-
raphical changes of the bottom as well as the bottom friction are
onsidered. We modify this test by taking into account the non-
inear velocity advection term in the momentum equation. In this
ase, the non-linear shallow water equations for C-grid read:
∂η
∂t +
h t
A i
3 ∑
j=1
U j d j = 0 (32)
∂ ( U h t )
∂t + n · ∇ · ( h t uu ) − f v h t n x + f u h t n y + g h t
∂ ( η)
∂n
=
1
ρ( S n )
(33)
here h t = h s + η is the total depth, h s is the still water depth in
he basin, and S n stands for the source terms as described in (4) .
n order to take into account the bottom topographical changes, we
stimate the gradient term in (40) as follows [34] :
h t ∂η
∂n
= gη∂ h s
∂n
+
1
2
g ∂ (η2 + 2 ηh s
)∂n
(34)
Following the work of Fringer et al. [14] , Casulli and Zanolli
11] and Perot [26] , Casulli [9] , the nonlinear advection term of the
omentum equation for equilateral grids at the cell face j can be
iscretized as follows:
( n · ∇ · ( h t uu ) ) j =
1
2
( C 1 + C 2 ) (35)
here C is defined as the component of advection of momentum
n the direction of the normal vector n j within cell i and is ob-
ained by integrating the advection term over the cell area to yield:
i =
( h t ) i A i
3 ∑
m =1
n j u m
U m
d m
(36)
In this numerical test, the axially-symmetric expression is used
or the still water depth as follows:
s =
1
1 . 3
(
0 . 5 +
√
0 . 5 − 0 . 5 r
R
)
(37)
here r is the radial distance from the center of the circular basin
nd R is the radius of the basin. The bed friction terms are given
y τbx = ρC f u √
u 2 + v 2 and τby = ρC f v √
u 2 + v 2 where C f is an em-
irical coefficient based on bed roughness. We use the following
quation based on the Chezy friction law:
f =
g
C 2 ch
here C ch is the Chezy friction coefficient. We assume a closed cir-
ular basin with R = 192 m . The triangular grids with approximate
ell area of 29 m
2 is used and CFL is set to CF L = 0 . 2 . The uneven
athymetry contour lines and the unstructured triangular mesh
rid is shown in Fig. 20 . No-flow boundary with free-slip condition
s applied at the basin wall. Mean depth is set to H = 0 . 769 m and
ther parameters are taken as τwx = 0 . 02 N/ m
2 , C ch = 33 . 11 m
1 2 /s
nd f = 0 .
The water surface elevation results obtained from the triangu-
ar C-grid scheme at steady state condition at t = 50 0 0 s are pre-
ented in Fig. 21 . One can see that the proposed scheme resolved
he flow pattern well and they are in agreement with those re-
orted in [34] . The steady state velocity field can also be seen in
152 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
Fig. 19. Evolution of L 2 error for Rossby wave after 5 periods ( T = 250) in Log-log scale.
Fig. 20. Unstructured triangular mesh grids of the circular basin along with the still
water depth contour lines.
Fig. 21. Water surface elevation at t = 50 0 0 s .
M
r
o
a
o
e
s
h
l
i
h
a
i
c
s
t
fi
T
Fig. 22 . The results obtained using the triangular C-grid scheme is
in agreement with those reported by [34] . As can be seen, the re-
sults are free of spurious modes and demonstrate the ability of the
proposed scheme in the simulation of wind-driven surface waves
with an uneven bathymetry.
6. Discussion
As for the geostrophic solutions, both methods are shown to
be capable of producing the exact stationary geostrophic motion.
Indeed, the dispersion relation admits the exact zero numerical
frequencies for both methods. This is also shown to be valid for
the isosceles triangles with different vertex angles for Method A.
It should be noted that the proposed reconstruction method for
Method B was shown to fail to reproduce the geostrophic solution
for the original hexagonal C-grid method. In regards with the grav-
ity waves, the phase speed ratios of different discrete cases respect
to the continuous solution are studied. The results show that for
ethod B, gravity waves travel slower than expected for a wider
ange of wave numbers. In other words, gravity wave frequencies
f Method A is closer to the continuous solutions which result in
more accurate simulation of gravity waves, particularly, in terms
f phase speed. The effect of grid cells deviation from the equilat-
ral triangle is also analysed. It is shown that the results of phase
peed are improved for the vertex angle of 2 α = π/ 4 . However, a
igh level of non-isotropy of the solutions will cause a false simu-
ation of gravity wave dispersion.
In order to investigate the behaviour of the methods for the
nertia-gravity waves, two different cases of low ( λ/h = 3 / 8 ) and
igh ( λ/h = 2 ) resolutions were considered for dispersion relation
nalysis. The analyses showed that for Method A, there is a mode
n O (1) that for infinitesimal grid size h → 0 coincides with the
ontinuous case for all vertex angles. In addition, we found the
purious mode in O (1/ h ) for all cases of Method A. For Method B,
here are two pairs of numerical frequencies in O (1) that for in-
nitesimal grid size h → 0 coincide with the continuous solutions.
he discrete frequencies compared to the continuous ones in terms
H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154 153
Fig. 22. Velocity field component of the solution at t = 50 0 0 s .
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f being isotropic, monotonically increasing with increasing wave
umbers and having values closer to the analytical solutions.
In the high-resolution case ( λ/h = 2 ), there is a high level of
sotropy for both Methods A and B. However, by deviation of
riangular cells from the equilateral form, the isotropy becomes
ower in Method A. The frequencies for all cases are monotonic
nd those of Method A for equilateral and isosceles triangles with
α = π/ 4 have closer values to the continuous solutions. For the
ow-resolution case ( λ/h = 3 / 8 ), we found that all the cases except
or the Method A with equilateral grids losing their isotropy dra-
atically. In addition, only Method A with equilateral grids keeps
he monotonic behaviour in all directions. However, results showed
hat for the λ/ h < 3/8, for all cases, the non-dimensional frequency
ill not have a monotonic behaviour as the continuous case along
ll directions.
The results of the Seiche wave for both methods compared to
he analytical ones for various wave numbers. As suggested by
he theoretical analysis, the higher phase error is observed for
ethod B with increasing the wave number. The numerical test
f geostrophic balance is also considered for all cases for vari-
us resolutions. As is predicted by the dispersion relation analy-
is, the negative water elevation may appear around the source
erm for all methods with the resolution of λ/h = 1 / 8 . It can be
ue to the non-monotonic behaviour of the discrete frequencies.
or λ/h = 3 / 8 , results computed using Method B show the checker-
oard which can be related to the non-monotonic and low level of
sotropy observed in the gravity-inertia frequencies. The checker-
oard pattern is also visible for Method A with non-equilateral
rids, particularly, in the directions for which the behaviour of the
requencies is not monotonic. However, at this resolution, the re-
ults of Method A with equilateral triangular grids show no nega-
ive water level around the source point.
Then, Method A with equilateral grids as the selected model
as further examined with a long slow equatorial Rossby wave.
he results obtained after 5 periods were compared to the analyt-
cal solution, and they were in good agreement. The symmetrical
hape of the wave was preserved by the model while the maxi-
um water surface elevation was well predicted. The maximum
alues of the water surface elevation were compared to the ana-
ytical ones for various grid resolutions. The results confirmed that
ven by using a relatively coarse mesh, there is no considerable
amping associated with the solution. Moreover, the spatial order
f accuracy for the triangular C-grid scheme was sought by inves-
igating the evolution of the L 2 error of water surface elevation. The
btained rate was very close to the formal spatial order of accuracy
or the C-grid.
In order to more examine the scheme, we applied the proposed
odel to the Stommel problem for two low- and high-resolution
rids. The results demonstrated the ability of the model in predict-
ng the divergence of the velocity field. In this test, we examined
he proposed triangular C-grid for low and high-resolution cases.
he results of the high-resolution case were satisfactory. With re-
ard to the low-resolution scenario, a great improvement was ob-
ained in comparison to the structured C-grid schemes which con-
rmed the theoretical dispersion analysis. The model was also ex-
mined with uneven bottom topography and wind-induced circu-
ation in a circular basin. The water surface elevation and velocity
eld were computed by Method A using an unstructured grid. The
esults confirmed the ability of the model to correctly account for
he surface wind stress as well as a variable bottom topography.
. Conclusion
Two different triangular 2D shallow water models based on C-
rid discretization are investigated. Each of the proposed models
as its own spatial discretization as well as velocity reconstruction
echniques. Velocity reconstruction is required in order to estimate
he tangential velocity at cell faces. The reconstruction method
sed for Method A is based on the work of [26] . The proposed
econstruction technique for Method B is, indeed, analogous to the
exagonal C-grid method. In addition, in order to investigate the
ffect of the fully unstructured triangular grids, various isosceles
riangles with different vertex angles are considered.
Dispersion relation analysis is implemented in this paper in or-
er to investigate the performance of the proposed 2D triangular
-grid models in shallow water equations. The behaviour of wave
requencies corresponding to the proposed schemes is compared
o those of continuous solutions. We investigate the performance
f the proposed methods for geostrophic solutions as well as grav-
ty and inertia-gravity waves.
Both methods showed a good performance in reproducing the
xact stationary geostrophic motion as well as gravity waves. Re-
ardless of the triangular grid angle, both Methods A and B were
hown to have a good performance in a high-resolution case. How-
ver, for the medium-resolution scenario the results of Method A
ith equilateral grids were better.
Through a number of numerical tests, we verified the theoreti-
al analysis of the dispersion relation. We also examined Method A
with equilateral grids), which was shown to have a better perfor-
ance based on the analyses, with a number of numerical tests.
n the numerical tests, we examined the proposed scheme with
arious source terms such as the Coriolis force, wind stress, bot-
om friction and bottom topographical changes. The behaviour of
ethod A was also investigated for simulation of non-linear shal-
ow water equations. In sum, the results confirmed the theoreti-
al analyses and capability of the model in solving shallow water
quations with various source terms.
cknowledgement
The authors acknowledge the support by NPRP grant # 4-935-
-354 .
154 H. Shirkhani et al. / Computers and Fluids 161 (2018) 136–154
[
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