Content uploaded by Asu Inan

Author content

All content in this area was uploaded by Asu Inan on Nov 17, 2014

Content may be subject to copyright.

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.

References

1. Li, Z., Davies, A.G.: Turbulence Closure Modelling of Sediment Transport Beneath Large

Waves. Continental Shelf Research (2001) 243-262

2. Bonnet-Verdier,C., Angot P., Fraunie, P., Coantic, M.: Three Dimensional Modelling of

Coastal Circulations with Different k-ε Closures. Journal of Marine Systems (2006) 321-

339

3. Baumert, H., Peters, H.: Turbulence Closure, Steady State, and Collapse into Waves.

Journal of Physical Oceanography 34 (2004) 505-512

4. Neary, V.S., Sotiropoulos, F., Odgaard, A.J.: Three Dimensional Numerical Model of Lateral

Intake Inflows. Journal of Hyraulic Engineering 125 (1999) 126-140

5. Balas,L., Özhan, E.: An Implicit Three Dimensional Numerical Model to Simulate

Transport Processes in Coastal Water Bodies, International Journal for Numerical Methods

in Fluids 34 (2000) 307-339

6. Balas, L., Özhan, E.: Three Dimensional Modelling of Stratified Coastal Waters, Estuarine,

Coastal and Shelf Science 56 (2002) 75-87

7. Balas, L.: Simulation of Pollutant Transport in Marmaris Bay. China Ocean Engineering,

Nanjing Hydraulics Research Institute (NHRI) 15 (2001) 565-578

8. Balas, L., Özhan, E: A Baroclinic Three Dimensional Numerical Model Applied to Coastal

Lagoons. Lecture Notes in Computer Science 2658 (2003) 205-212

9. Tsanis,K.I., Leutheusser, H.J.:The Structure of Turbulent Shear-Induced Countercurrent

Flow, Journal of Fluid Mechanics 189 (1998) 531-552