A Composite Finite Element-Finite Difference Model Applied to Turbulence Modelling

Abstract

Turbulence has been modeled by a two equation k-ω turbulence model to investigate the wind induced circulation patterns in coastal waters. Predictions of the model have been
compared by the predictions of two equation k-ε turbulence model. Kinetic energy of turbulence is k, dissipation rate of turbulence is ε, and frequency of turbulence is ω. In the three dimensional modeling of turbulence by k-ε model and by k-ω model, a composite finite element-finite difference method has been used. The governing equations are solved by the Galerkin
Weighted Residual Method in the vertical plane and by finite difference approximations in the horizontal plane. The water
depths in coastal waters are divided into the same number of layers following the bottom topography. Therefore, the vertical
layer thickness is proportional to the local water depth. It has been seen that two equation k-ω turbulence model leads to better predictions compared to k-ε model in the prediction of wind induced circulation in coastal waters.

Full text

Y. Shi et al. (Eds.): ICCS 2007, Part I, LNCS 4487, pp. 1 – 8, 2007.
© Springer-Verlag Berlin Heidelberg 2007
A Composite Finite Element-Finite Difference Model
Applied to Turbulence Modelling
Lale Balas and Asu İnan
Department of Civil Engineering, Faculty of Engineering and Architecture
Gazi University, 06570 Ankara, Turkey
lalebal@gazi.edu.tr,
asuinan@gazi.edu.tr
Abstract. Turbulence has been modeled by a two equation k-ω turbulence
model to investigate the wind induced circulation patterns in coastal waters.
Predictions of the model have been compared by the predictions of two
equation k-ε turbulence model. Kinetic energy of turbulence is k, dissipation
rate of turbulence is ε, and frequency of turbulence is ω. In the three
dimensional modeling of turbulence by k-ε model and by k-ω model, a
composite finite element-finite difference method has been used. The governing
equations are solved by the Galerkin Weighted Residual Method in the vertical
plane and by finite difference approximations in the horizontal plane. The water
depths in coastal waters are divided into the same number of layers following
the bottom topography. Therefore, the vertical layer thickness is proportional to
the local water depth. It has been seen that two equation k-ω turbulence model
leads to better predictions compared to k-ε model in the prediction of wind
induced circulation in coastal waters.
Keywords: Finite difference, finite element, modeling, turbulence, coastal.
1 Introduction
There are different applications of turbulence models in the modeling studies of coastal
transport processes. Some of the models use a constant eddy viscosity for the whole flow
field, whose value is found from experimental or from trial and error calculations to match
the observations to the problem considered. In some of the models, variations in the
vertical eddy viscosity are described in algebraic forms. Algebraic or zero equation
turbulence models invariably utilize the Boussinesq assumption. In these models mixing
length distribution is rather problem dependent and therefore models lack universality.
Further problems arise, because the eddy viscosity and diffusivity vanish whenever the
mean velocity gradient is zero. To overcome these limitations, turbulence models were
developed which accounted for transport or history effects of turbulence quantities by
solving differential transport equations for them. In one-equation turbulence models, for
velocity scale, the most meaningful scale is k0.5, where k is the kinetic energy of the
turbulent motion per unit mass[1]. In one-equation models, it is difficult to determine the
length scale distribution. Therefore the trend has been to move on to two-equation models

2 L. Balas and A. İnan
which determine the length scale from a transport equation. One of the two equation
models is k-ε turbulence model in which the length scale is obtained from the transport
equation of dissipation rate of the kinetic energy ε [2],[3]. The other two equation model is
k-ω turbulence model that includes two equations for the turbulent kinetic energy k and
for the specific turbulent dissipation rate or the turbulent frequency ω [4].
2 Theory
The implicit baroclinic three dimensional numerical model (HYDROTAM-3), has been
improved by a two equation k-ω turbulence model. Developed model is capable of
computing water levels and water particle velocity distributions in three principal
directions by solving the Navier-Stokes equations. The governing hydrodynamic
equations in the three dimensional cartesian coordinate system with the z-axis vertically
upwards, are [5],[6],[7],[8]:
0 =
z
w
+
y
v
+
x
u
∂
∂
∂
∂
∂
∂ (1)
))
x
w
+
z
u
((
z
+))
x
v
+
y
u
((
y
+)
x
u
(
x
2+
x
p1
-fv =
z
u
w+
y
u
v+
x
u
u+
t
u
zhh
o∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
ννν
ρ
(2)
))
y
w
+
z
v
((
z
+))
y
u
+
x
v
((
x
+)
y
v
(
y
2+
y
p1
-fu- =
z
v
w+
y
v
v+
x
v
u+
t
v
zhh
o∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
ννν
ρ
(3)
)
z
w
(
z
+))
z
u
+
x
w
((
x
+)
z
v
y
w
(
y
g
z
p1
-=
z
w
w+
y
w
v+
x
w
u+
t
w
zhh
o∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
+
∂
∂
∂
∂
+−
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
ννν
ρ
)( (4)
where, x,y:horizontal coordinates, z:vertical coordinate, t:time, u,v,w:velocity
components in x,y,z directions at any grid locations in space, vz:eddy viscosity
coefficients in z direction, vh:horizontal eddy viscosity coefficient, f:corriolis coefficient,
ρ(x,y,z,t):water density, g:gravitational acceleration, p:pressure.
As the turbulence model, firstly, modified k-ω turbulence model is used. Model
includes two equations for the turbulent kinetic energy k and for the specific turbulent
dissipation rate or the turbulent frequency ω. Equations of k-ω turbulence model are
given by the followings.
k
y
k
y
+
x
k
x
P
z
k
zdt
dk
hhz
ϖβνσνσνσ
**** −
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
++
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
= (5)
2***
βϖ
ϖ
νσ
ϖ
νσ
ϖ
α
ϖ
νσ
ϖ
−
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
++
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
∂
∂
=yy
+
xx
P
kzzdt
dhhz (6)

A Composite Finite Element-Finite Difference Model 3
The stress production of the kinetic energy P, and eddy viscosity νz are defined by;
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
z
v
+
z
u
+
x
v
+
y
u
+
y
v
2+
x
u
2 = P
22
z
22
2
h
νν
;
ϖ
ν
k
z= (7)
At high Reynolds Numbers(RT), the constants are used as; α=5/9, β=3/40, β*=
9/100,σ=1/2 and σ*=1/2. Whereas at lower Reynolds numbers they are calculated as;
6/1
6/40/1
*
T
T
R
R
+
+
=
α
;1*)(
7.2/1
7.2/10/1
9
5−
+
+
=
αα
T
T
R
R;
ϖν
k
RT=;4
4
*
)8/(1
)8/(18/5
100
9
T
T
R
R
+
+
=
β
(8)
Secondly, as the turbulence model a two equation k-ε model has been applied.
Equations of k-ε turbulence model are given by the followings.
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
+−+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
y
k
yx
k
x
P
z
k
v
zz
k
w
y
k
v
x
k
u
t
khh
k
z
ννε
σ
(9)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
+−+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
yyxxk
C
k
PC
z
v
zz
w
y
v
x
u
thh
z
ε
ν
ε
ν
εεε
σ
εεεε
εε
ε
2
21 (10)
where, k :Kinetic energy, ε:Rate of dissipation of kinetic energy, P: Stress production
of the kinetic energy. The following universal k-ε turbulence model empirical
constants are used and the vertical eddy viscosity is calculated by:
ε
μ
2
k
Cvz=; Cμ=0.09, σε=1.3, C1ε=1.44, C2ε=1.92. (11)
Some other turbulence models have also been widely applied in three dimensional
numerical modeling of wind induced currents such as one equation turbulence model
and mixing length models. They are also used in the developed model HYROTAM-
3, however it is seen that two equation turbulence models give better predictions
compared to the others.
3 Solution Method
Solution method is a composite finite difference-finite element method. Equations are
solved numerically by approximating the horizontal gradient terms using a staggered
finite difference scheme (Fig.1a). In the vertical plane however, the Galerkin Method of
finite elements is utilized. Water depths are divided into the same number of layers
following the bottom topography (Fig.1b). At all nodal points, the ratio of the length
(thickness) of each element (layer) to the total depth is constant. The mesh size may be
varied in the horizontal plane. By following the finite element approach, all the variables
at any point over the depth are written in terms of the discrete values of these variables at
the vertical nodal points by using linear shape functions.

4 L. Balas and A. İnan
kk GNGNG 2211
~+= ;
k
l
zz
N−
=2
1;
k
l
zz
N1
2−
=;12 zzlk−= (12)
where G
~:shape function; G: any of the variables, k: element number; N1,N2: linear
interpolation functions; lk:length of the k’th element; z
1,z2:beginning and end
elevations of the element k; z: transformed variable that changes from z1 to z2 in an
element.
(a) (b)
Fig. 1. a) Horizontal staggered finite difference scheme, ○: longitudinal horizontal velocity, u;
□: lateral horizontal velocity, v; *: all other variables b) Finite element scheme throughout the
water depth
After the application of the Galerkin Method, any derivative terms with respect to
horizontal coordinates appearing in the equations are replaced by their central finite
difference approximations. The system of nonlinear equations is solved by the Crank
Nicholson Method which has second order accuracy in time. Some of the finite
difference approximations are given in the following equations.
()
(
)
()
()
(
)
()
⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
+
⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−+
+
+−
−+
−
−+
22
11
1
,,11
11
1
,1,1
,ii
iii
jijiii
ii
iii
jijiii
ji xx
xxx
llxx
xx
xxx
llxx
x
l (13)
(
)
(
)
()
(
)
(
)
()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+−
+
+−
+−
−
−+
22
11
1
,1,1
11
1
1,,1
,jj
jjj
jijijj
jj
jjj
jijijj
ji yy
yyy
llyy
yy
yyy
llyy
y
l (14)
()
(
)
()
()
(
)
()
⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
+
⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−+
+
+−
−+
−
−+
22
11
1
,,11
11
1
,1,1
,ii
iii
jijiii
ii
iii
jijiii
ji xx
xxx
CCxx
xx
xxx
CCxx
x
C (15)

A Composite Finite Element-Finite Difference Model 5
(
)
(
)
()
(
)
(
)
()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+ΔΔ+Δ
−Δ+Δ
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+−
+
+−
+−
−
−+
22
11
1
,1,1
11
1
1,,1
,jj
jjj
jijijj
jj
jjj
jijijj
ji yy
yyy
CCyy
yy
yyy
CCyy
y
C (16)
() ()()
22
22
(2
11
,
111
1,
,
2
2
+−
+−−
−Δ+ΔΔ+Δ
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+Δ
Δ+Δ
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
jjjj
ji
jj
j
jj
ji
ji yyyy
C
yy
y
yy
C
y
C
()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+Δ
+Δ
Δ+Δ
+
−++
+
22
111
1,
jj
j
jj
ji
yy
y
yy
C)
(17)
() ()(
)
2
xx
2
xx
C
2
xx
x
2
xx
C
(2
x
C
i1ii1i
j,i
1i1i
i
i1i
j,1i
j,i
2
2
ΔΔΔΔ
ΔΔ
Δ
ΔΔ
++
−
⎟
⎠
⎞
⎜
⎝
⎛+
+
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+−
+−−
−
()
)
22
111
,1
⎟
⎠
⎞
⎜
⎝
⎛Δ+Δ
+Δ
Δ+Δ
+
+−+
+
ii
i
ii
ji
xx
x
xx
C
(18)
()
() ()
1
1,1
1
1,
11
2
,1
,
2
12
2
2
2
4−
−+
−
−
−−
−
+Δ+Δ
⎟
⎠
⎞
⎜
⎝
⎛Δ+
Δ
+
Δ
⎟
⎠
⎞
⎜
⎝
⎛Δ+
Δ
+
Δ+ΔΔ
Δ
=
ii
i
i
ji
i
i
i
ji
iii
iji
ji xx
x
x
u
x
x
x
u
xxx
xu
u (19)
()
() ()
1
11,
1
1,
11
2
1,
2
1
,2
2
2
2
4−
−+
−
−
−−
−
+Δ+Δ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+
Δ
+
Δ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛Δ+
Δ
+
Δ+ΔΔ
Δ
=
jj
j
j
ji
j
j
j
ji
jjj
jji
ji yy
y
y
v
y
y
y
v
yyy
yv
v (20)
4 Model Applications
Simulated velocity profiles by using k-ε turbulence model, k-ω turbulence model have
been compared with the experimental results of wind driven turbulent flow of an
homogeneous fluid conducted by Tsanis and Leutheusser [9]. Laboratory basin had a
length of 2.4 m., a width of 0.72 m. and depth of H=0.05 meters. The Reynolds
Number,
μ
ρ
Hu
Rs
s= was 3000 (us is the surface velocity, H is the depth of the
flow, ρ is the density of water and μ is the dynamic viscosity). The velocity profiles
obtained by using k-ε turbulence model and k-ω turbulence model are compared with the
measurements in Fig.2a and vertical eddy viscosity distributions are given in Fig.2b.

6 L. Balas and A. İnan
(a) (b)
Fig. 2. a)Velocity profiles, b) Distribution of vertical eddy viscosity (solid line: k-ε turbulence
model, dashed line: k-ω turbulence model, *: experimental data)
The root mean square error of the nondimensional horizontal velocity predicted by the
k-ε turbulence model is 0.08, whereas it drops to 0.02 in the predictions by using k-ω
turbulence model. This basically due to a better estimation of vertical distribution of
vertical eddy viscosity by k-ω turbulence model.
Developed three dimensional numerical model (HYROTAM-3) has been
implemented to Bay of Fethiye located at the Mediterranean coast of Turkey. Water
depths in the Bay are plotted in Fig.3a. The grid system used has a square mesh size
of 100x100 m. Wind characteristics are obtained from the measurements of the
meteorological station in Fethiye for the period of 1980-2002. The wind analysis
shows that the critical wind direction for wind speeds more than 7 m/s, is WNW-
WSW direction. Some field measurements have been performed in the area. The
current pattern over the area is observed by tracking drogues, which are moved by
currents at the water depths of 1 m., 5 m and 10 m.. At Station I and at Station II
shown in Fig.3a, continuous velocity measurements throughout water depth, at
Station III water level measurements were taken for 27 days. In the application
measurement period has been simulated and model is forced by the recorded wind as
shown in Fig. 3b. No significant density stratification was recorded at the site.
Therefore water density is taken as a constant. A horizontal grid spacing of
Δx=Δy=100 m. is used. Horizontal eddy viscosities are calculated by the sub-grid
scale turbulence model and the vertical eddy viscosity is calculated by k-ε turbulence
model and also by k-ω turbulence model. The sea bottom is treated as a rigid
boundary. Model predictions are in good agreement with the measurements.
Simulated velocity profiles over the depth at the end of 4 days are compared with the
measurements taken at Station I and Station II and are shown in Fig.4. At Station I,
the root mean square error of the horizontal velocity is 0.19 cm/s in the predictions by k-
ε turbulence model and it is 0.11cm/s in the predictions by k-ω turbulence model. At
Station II, the root mean square error of the horizontal velocity is 0.16 cm/s in the
predictions by k-ε turbulence model and it is 0.09cm/s in the predictions by k-ω
turbulence model.

A Composite Finite Element-Finite Difference Model 7
1000 2000 3000 4000 5000 6000
X (m)
1000
2000
3000
4000
Y (m)
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
-0
E
Fig. 3. a)Water depths(m) of Fethiye Bay, +:Station I, •:Station II,∗ :Station III. b) Wind speeds
and directions during the measurement period.
012345
Horizo ntal velocit
y
(cm
/
s)
-5
-4
-3
-2
-1
0
Water depth (m)
-4-20246
Horizo ntal veloc it
y
(cm
/
s)
-8
-6
-4
-2
0
Water depth (m)
-9.65
(a) (b)
Fig. 4. Simulated velocity profiles over the depth at the end of 4 days; solid line: k-ε turbulence
model, dashed line: k-ω turbulence model, *: experimental data, a) at Station I, b) at Station II
5 Conclusions
From the two equation turbulence models, k-ε model and k-ω model have been used in
three dimensional modeling of coastal flows. The main source of coastal turbulence
production is the surface current shear stress generated by the wind action. In the
numerical solution a composite finite element-finite difference method has been
applied. Governing equations are solved by the Galerkin Weighted Residual Method
in the vertical plane and by finite difference approximations in the horizontal plane on
a staggered scheme. Generally, two-equation turbulence models give improved
estimations compared to other turbulence models. In the comparisons of model
predictions with both the experimental and field measurements, it is seen that the two
equation k-ω turbulence model predictions are better than the predictions of two equation
k-ε turbulence model. This is basically due to the better parameterizations of the non-
linear processes in the formulations leading a more reliable and numerically rather easy
to handle vertical eddy viscosity distribution in the k-ω turbulence model.

8 L. Balas and A. İnan
Acknowledgment. The author wishes to thank the anonymous referees for their
careful reading of the manuscript and their fruitful comments and suggestions.
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