3D Fully Nonlinear Sloshing Fluid in Tanks by a TIME INDEPENDENT FDM.PDF

7/21/2019 3D Fully Nonlinear Sloshing Fluid in Tanks by a TIME INDEPENDENT FDM.PDF

1/6

28 2006 11Proceedings of the 28th Ocean Engineering Conference in Taiwan

National Sun Yat-Sen University, November 2006

3D Fully Nonlinear Sloshing Fluid in Tanks by a

Time-independent Finite Different Method

Chih-Hua Wu1 Bang-Fuh Chen2

PH.D. student, Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung

Professor, Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung

ABSTRACT

Sloshing waves in moving tanks have been studied numerically, theoretically and

experimentally over the past several decades. Previous studies only examined sloshing waves in tanks

excited by limited exciting directions and with a fixed excitation frequency throughout the excitation.

In reality, the excited directions are actually multi-degree of freedoms (coupled surge / sway / heave /

pitch / roll /yaw) in earthquake induced tank sloshing. In the present study, a 3-D tank with various

fluid depths, excited amplitudes, excitation frequencies and multiple degrees of freedoms of

excitations were considered. The developed time-independent finite difference method was extended

to incorporate the incompressible and inviscid Navier-Stokes equations, fully nonlinear kinematic and

dynamic free surface conditions in the analysis of the seismic response of sloshing fluid in a 3-D

rectangular tank with square basin. The benchmark study confirms the acceptance and accuracy of the

proposed numerical scheme. The tank excited in horizontal ground motion (coupled surge-sway

motion) with various excited angle ( indicates the direction of horizontal ground motion) resulted

in five types of waves, which are diagonal wave, square-like wave, swirling-like, swirling

wave and irregular wave. The main focus of this paper was taking the heave motion coupling with

horizontal ground motion into account and discussing its effects. For the effect of coupling Surge,sway and heave motions, the results show an unstable influence when the excited frequency of heave

motion is twice as large as the fundamental natural frequency and these sloshing waves mentioned

above would change their wave type into swirling waves except for diagonal and irregular waves. But

we only choose three cases of sloshing waves presented in this paper. The developed numerical

scheme can easily simulate the coupled excitations of six-degree of freedom and also the real fluid

sloshing in the tanks.

3 Navier-Stokes equations

-433-


2/6

2

6

1. Introduction

Sloshing effects of free-surface motion in tanks

driven by external forces may have many serious

consequences in a range of engineering applications.

For examples, liquid oscillations in a large storage

tanks caused by earthquakes, tank trucks on highways,

the motion of liquid fuel in aircraft and spacecraft,

prismatic membrane tanks for transportation of

liquefied natural gas (LNG) , sloshing of liquid cargo

in ocean-going vessel, and the use of tuned liquiddampers designed to supress wind-induced structural

buildings. It is known that partially filled tanks are

prone to violent sloshing under certain motions,

especially when the near-resonant excitation occurs.

The large liquid movement would create highly

localized impact pressure on tank walls which in turn

causes structural damage and may even create large

rolling moments to affect the stability of the vehicle

that carries the container.

As we know, a liquid container or reservoir

undergoing an earthquake or a ships oil container is

subjected to the forcing waves which not only has

horizontal but also vertical oscillation. Faraday(1831)

studied the sloshing instability in vertical oscillating

container, known as faraday wave, and found it has a

parametric instability. Benjamin and Ursell(1954)

investigated the vertical motion theoretically and

concluded that the solutions solved from Mathieu

equation are always unstable for an external forcing

frequency equal to twice the sloshing frequency. The

heave motion coupling with surge or pitch motion was

discussed in resent years. Frandsen(2004) coupled

surge and heave motion together and found that

vertical excitation causes the instability associatedwith parametric resonance of the combined motion for

a certain set of frequencies and amplitudes of vertical

motion while the horizontal motion is related to

classical resonances. Chen and Nokes (2005) reported

the initial disturbance of the fluid surface of the tank

would result in huge difference of the surface

elevation when vertical excitation is coupled with

surge or pitch motion.

The 3D tank with different ratios of

depth/excited amplitude, multiple degrees of freedom

of excitation, excitation frequencies of near or far from

natural frequency and non-constant excitation

frequencies are considered in the present study. The

developed time-independent finite difference method

is extended to study the seismic response of sloshing

fluid in a 3D rectangular tank with square basin. The

invisid fluid and incompressible flow are considered

solving the Navier-Stokes equations and fully

nonlinear kinematic and dynamic free surface

conditions are taken into account as well in the present

numerical work. The main study of this paper is to

simulate a 3D tank undergoing different combination

of motions with varying vibrating directions,

especially for the case of coupled surge-sway-heave

motion. In horizontal ground forcing, five types of

sloshing waves have been discovered in tanks, which

are diagonal, square-like, swirling-like, and swirling

waves. As the tank under heave motion with disturbed

free surface and vertical excitation frequency is equal

or near to12 , the unstable influence of heave

motion is obviously found in previous studies. In the

present study, therefore, we discuss the effect of heave

motion coupled with horizontal ground motions under

unstable vertical excitation on the different sloshingwaves mentioned above. The effects of heave motion

on different sloshing waves are discussed in detail in

this paper.

2. Mathematical Formulation

A fully non-linear model of inviscid 3-D waves

in a numerical wave tank was developed. As shown in

Fig 1, a rigid tank with breadth L, width B and still

water depth d0 . As the coordinate system is chosen to

move with the tank motions (including surge, sway,

heave, yaw, roll and pitch motions, see Fig. 1), the

momentum equations can be derived and written as

CXx

pg

z

uw

y

uv

x

uu

t

u&&

=

+

+

+

1cos

)(2)2()( 22 vwxyz &&&&&&&&&& (1)

CY

y

pg

z

vw

y

vv

x

vu

t

v&&

=

+

+

+

1cos

)(2)2()( 22 wuyzx &&&&&&&&&& (2)

CZz

pg

z

ww

y

wv

x

wu

t

w&&

=

+

+

+

1cos

)(2)2()( 22 uvzxy &&&&&&&&&& (3)

-434-


3/6

-435-

)4()]()(

)([2)]()[2

)()(

)(

222

2

2

2

2

2

2

x

v

y

u

z

u

x

w

y

w

z

v

z

ww

y

wv

x

wu

zz

vw

y

vv

x

vu

y

z

uw

y

uv

x

uu

xz

p

y

p

x

p

+

+

++++

+

+

+

+

+

+

=

+

+

&&

&&&&&&&&&&

Surge

Rol

Swa

Pitch

Heavy Yaw

XY

Z

G

A

B

CD

Bd0

x

y

Z

),(1 zyb),(2 zyb

),,( tzxh

where u, vandware velocity components inx, y

andzdirections,c Cy and are the acceleration

components of tank in x, y and zdirections ; (

x&& && Cz&&

&

) and ( , ,& & && && && ) are the corresponding

angular velocities and accelerations with respect tox,y

and z axes, p is the pressure, is the fluid density

and g the acceleration of gravity. Taking partial

differentiation of Eqs.( 1), (2) and (3) with respect to x,

yandzrespectively, and summing the results, one canobtain the following pressure wave equation which is

used to solve for the pressure.

),(4 yxb

),(3 yxb

The continuity equation for incompressible flow

is

The kinematic free surface boundary condition is

and the dynamic free surface condition is p= 0

and the boundary condition at the solid walls must

satisfied momentum equations. Instead of using

boundary fitted coordinate (BFC) system, we used

simple mapping functions to remove the time-varied

boundary of the fluid domain. The irregular tank walls

and tank bottom can be mapped onto a cubic one by

the proper coordinate transformations as

where the instantaneous water surface,h(x,z,t), is

a single-valued function measured from tank bottom,

d(x,z) represents the vertical distance between stillwater surface and tank bottom, b1and b2are horizontal

distance from the x-axis to the west and east walls

respectively, and b3 and b4 are horizontal distance

from the z-axis to the north and south walls

respectively, (see Fig.2).The coordinates (x*, z*) can

be further transformed so that the layer near the free

surface, tank bottom and tank wall will be stretched by

the following exponential functions:

The constants 2 and 2 control the mesh sizesnear free surface and tank bottom. Similarly, the

constants 1,3 and1,3 map irregular finite difference

mesh sizes near tank wall to the regular ones in the

computational domain(X,Y,Z). Thus, the geometry of

the flow field and the meshes in the computational

domain (X-Y-Z system) become time-independent

throughout the computational analysis.

L

Fig. 1 Definition Sketch

0=

+

+

z

w

y

v

x

u

(5)

Fig.2 Definition Sketch of coordinate transformation

3. Finite Difference Method

The numerical method used in this study is based

on a finite difference method. There are many finite

difference and volume method for evaluating free

surface. The most famous schemes are SURF, MAC

and VOF methods and those methods need complicate

computer programming in treating the time varied free

surface boundary and updating computational meshes.

Alternately, in the present study, the time-varied free

surface boundary is transformed to a time-independent

free surface in the x*-y*-z* domain and no boundary

tracing is needed during the calculation. A single value

height function is assumed and is evaluated by solving

the kinematic free surface condition. In a

three-dimensional analysis, the fluid flow is solved in

a cubic mesh network of the transformed domain. The

staggered grid system is used in the analysis (see Fig

3). The Crank-Nicholson iteration scheme and

Gauss-Seidel Point successive over-relaxation

vz

hw

x

hu

t

h=

+

+

(6)

),(),(

),(

12

1*

zybzyb

zybxx

=

),,(

),(1*

tzxh

zxdyy

+= )7(

),(),(

),(

34

3*

yxbyxb

yxbzz

=

)1(

1

*

1

**1)(

+= xxk

exX )1(

2

*

2

**2)( +=

yykeyY

)8()()1(

3

*

3

**3 +=

zzkezZ


4/6

-436-

kjip ,,

kjiU ,,

kjiU ,,1+

kji ,1, +

Fig. 3 The staggered grid system

iteration procedures are used to calculate velocity and

pressure, respectively. The detailed iteration procedure

is similar that reported in Chen and Roger (2005) andis omitted in the manuscript.

V

kjiV ,,

kjiW ,,

1,, +kjiW

4. Result and Disscusion

In the present study, a rectangular tank

with ratio breadth / width = L / B = 1, still

water depth / breadth = d0 /L=0.25 are used in

most of the simulations. Although the surge,

sway, heave, pitch, roll, and yaw motions are

considered in this study, the main focus of this

paper is a tank under couple d su rge-sway -

heave motion. The ground acceleration of

surge, heave, and sway motion are given

as , ,

and , respectively, where X

tXX xxc = sin2

0&& tYY yyc = sin

20

&&

tXZ zzc = sin2

0&&

0,Y0 and Z0 are the maximum excited amplitudes

and x , y , and z are the corresponding

excited frequency with respect to surge, heave,

and sway motion.

4.1 Verification

In order to validate the accuracy of numerical

simulation, the results obtained from the present

numerical model are compared with those reported in

the literature for the benchmark tests. The present

results for the tank under only heave motion with

initial disturbed free surface agree greatly with

Frandens (2004) numerical results as shown in Fig.4.

The results for diagonal excitation produced by Kim

(2001) also show great agreement with the present

study as shown in Fig.5. Thus, the present numerical

model can be used by the benchmark tests mentioned

above.

4.2 Surge-sway-heave Motion

A tank may be simultaneously excited by

horizontal and vertical ground acceleration during

earthquakes. The heave motion is just a change in

gravitational acceleration, and its effect is expected to

be small if the free surface of the tank is initiallyundisturbed. However, if the free surface is initially

sloping or it is disturbed by other combination of

ground motions, the vertical excitation would enlarge

the free surface elevation during tank motion. In the

present study, five types of sloshing waves have been

found whena tank is under surge-sway motion with

various excited angles (see Fig. 6) and excitation

frequencies. To see the influence of heave motion on

different sloshing waves, the coupled

surge-sway-heave motion is taken into account with

various excitation frequencies in this paper. Only

selected cases are presented here as the excited

frequency of heave motion is twice as large as the first

natural frequency. The heave motion begins at

dimensionless time T=40 of each case described

below.

t 1

0 2 0 4 0 6 0 8 0 1

(x,

t)/ah

-1.0

-0.5

0. 0

0. 5

1. 0

1. 5

p res en t r es u l t

F ra n d s en (200 4)

00

Fig. 4 The wave history on tank's corner under

Heave motion, the ratio d0 (still water depth) / L and d0

/B=0.5. Tank's displacement a0/L =0.272, 18.0 =y

0 1 0 2 0 3 0 4 0

/d0

-1 .0

-0 .5

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

P re s e n t re s u lt

K im (2 0 0 1 )

Fig. 5 The wave history on tank's corner under Surge-

Sway (diagonal) motion, the ratio d0/ L and d0 / B =

0.25, tank's displacement a0 / L=0.0093,

199.0 == zx

The effect of heave motion on diagonal waves is

shown in Fig 7. The obvious influence of heave


5/6

motion starts after T=300 depicted in Fig 7-(a). Fig

7-(b) presents the peaks distribution and its found

that the original diagonal waves seems not to becomeswirling wave in the end after the intense heave

motion effect. This is because the swirling wave is

insignificantly found in diagonal forcing in the present

study. Fig. 8 shows the results of square-like waves

affected by heave motion. Fig. 8-(a) depicts the wave

history of point A and we can find that the apparent

influence of heave motion starts approximately at

T=320. Then, the sloshing displacement becomes

larger and larger and the original square-like wave turn

into swirling waves after T=370. Fig 8-(b) and (c)

show the peaks distribution of surge-sway-heave

motion and surge-sway motion, respectively, and can

be found that square-like waves change to

counterclockwise swirling waves due to heave motion

in the end.

-437-

Fig. 6 The definition of excited angle

The effect of heave motion on swirling-like

waves is shown in Fig. 9. Fig. 9-(a) presents the wave

history of point A and shows the result that the

influence of heave motion comparing with only

surge-sway motion is clear. Through Fig 9-(b) and (c),

showing the peaks distribution of only surge-sway

motion and coupled surge-sway-heave motion, the

phenomenon is found that the original swirling-like

waves finally turn into swirling waves because of the

effect of heave motion. We also can discover that the

maximum peaks of free surface appear at the corner B

and D; this means the rotation direction of swirling

waves is clockwise in the end. Thus, we keep an eye

on the swirling direction of this case in detail. Fig. 10

illustrates the rotation directions in different time

regions and we can find that the switching direction of

swirling waves repeats over and over again. This

phenomenon cant be discovered under the condition

of a tank is merely excited in horizontal ground

motion and excitation frequency is equal to19.0 .

Another phenomenon found is that the period of

swirling waves (clockwise or counterclockwise) is

shorter than the pervious cases without the influence

of heave motion. Thus, the short period of

quickly-switching swirling waves would cause more

instability of the moving vehicles or structures.

5. Conclusion

The developed time-independent finite difference

method is extended to solve the incompressible and

inviscid Navier-Stokes equations, fully nonlinear

kinematic and dynamic free surface conditions in a

rectangular tank with a square basin. The main study

of this research simulates a 3-D tank undergoing

different combination of motions with varying

vibrating directions. The coupling effect of surge,

sway and heave motions is also made in the present

study and the results shows an unstable influence

when the excited frequency of heave motion is twice

as large as the fundamental natural frequency. The

heave motion effect on diagonal waves shows that it

causes large wave elevation in the end. The

square-like waves finally trun into swirling waves due

to being affected by heave motion. The effect of

heave motion on swirling-like waves shows unstable

switch of swirling directions that would cause more

instability of the moving vehicles or structures. The

developed numerical scheme can easily simulate the

coupled excitations of six-degree of freedom and also

the real fluid sloshing in the tanks.

Fig. 7 Effect of heave motion on diagonal waves . (a)

The wave history of point A; (b) The distribution of

peaks.

Z

(a)

(b)

15.0 == zx 005.0,45, 0== yo12=y


6/6

-438-

Fig. 8 Effect of heave moton on square-like waves;

(a) The wave history of point A; (b) The distributionof peaks of surge-sway-heave motion; (c) The

distribution of peaks of surge-sway motion.

Fig. 9 Effect of heave motion on swirling-like waves

(a) The wave history of point A; (b) The distribution

of peaks of surge-sway-heave motion; (c) The

distribution of peaks of surge-sway motion.

Fig. 10 The time regions of clockwise or

counterclockwise swirling waves under coupled

surge-sway-heave motion.

Reference1. Benjamin T. B. and Ursell F. (1954) The

stability of the plane free surface of a liquid in a

vertical periodic motion, Proc. R. Soc. Lond. Ser.

A,Vol. 225, pp. 505-515.

2. Chen, B. F. and Nokes, R. (2005) Time-

independent finite difference analysis of fully

non-linear and viscous fluid sloshing in a

rectangular tank, J. Comput. Phys., Vol. 209,

pp. 47-81.

3. Faraday, M. (1831) On the forms and states of

fluids on vibrating elastic surfaces, Philos.

Trans. Roy. Soc. London,Vol. 52, pp. 319-340.

4. Frandsen, J. B. (2004) Sloshing motions in

excited tanks, J. Computational Physics., Vol.

196, pp. 53-87

5. Kim, Y. (2001) Numerical simulation of

sloshing flows with impact loads, Appl. Ocean

Res., Vol. 23, pp. 53-62

(a) (c)

(b)

11 2,9.0 === yzx 005.0,5, 0== yo

(c)

11 2,8.0 === yzx 005.0,5, 0== yo

(a)

(b)

3D Fully Nonlinear Sloshing Fluid in Tanks by a TIME INDEPENDENT FDM.PDF

Documents

Transcript of 3D Fully Nonlinear Sloshing Fluid in Tanks by a TIME INDEPENDENT FDM.PDF