BARC/2015/E/001B

AR

C/2015/E

/001

LIQUID SLOSHING IN GRAVITY DRIVEN WATER POOL OFADVANCED HEAVY WATER REACTOR:

POOL LIQUID UNDER DESIGN SEISMIC LOAD AND SLOSH CONTROL STUDIES

byM. Eswaran and G.R. Reddy

Structural and Seismic Engineering Section,Reactor Safety Division

BARC/2015/E/001

GOVERNMENT OF INDIAATOMIC ENERGY COMMISSION

BHABHA ATOMIC RESEARCH CENTREMUMBAI, INDIA

2015

BA

RC

/201

5/E

/001

LIQUID SLOSHING IN GRAVITY DRIVEN WATER POOL OFADVANCED HEAVY WATER REACTOR:

POOL LIQUID UNDER DESIGN SEISMIC LOAD AND SLOSH CONTROL STUDIES

byM. Eswaran and G.R. Reddy

Structural and Seismic Engineering Section,Reactor Safety Division

BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT(as per IS : 9400 - 1980)

01 Security classification : Unclassified

02 Distribution : External

03 Report status : New

04 Series : BARC External

05 Report type : Technical Report

06 Report No. : BARC/2015/E/001

07 Part No. or Volume No. :

08 Contract No. :

10 Title and subtitle : Liquid sloshing in gravity driven water pool of Advanced Heavy Water Reactor - Pool liquid under design seismic load and slosh control studies

11 Collation : 84 p., 67 figs., 13 tabs.

13 Project No. :

20 Personal author(s) : M. Eswaran; G.R. Reddy

21 Affiliation of author(s) : Structural and Seismic Engineering Section, Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai

22 Corporate author(s) : Bhabha Atomic Research Centre, Mumbai - 400 085

23 Originating unit : Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai

24 Sponsor(s) Name : Department of Atomic Energy

Type : Government

Contd...

BARC/2015/E/001

BARC/2015/E/001

30 Date of submission : January 2015

31 Publication/Issue date : February 2015

40 Publisher/Distributor : Head, Scientific Information Resource Division, Bhabha Atomic Research Centre, Mumbai

42 Form of distribution : Hard copy

50 Language of text : English

51 Language of summary : English

52 No. of references : 39 refs.

53 Gives data on :

60

70 Keywords/Descriptors : HWLWR TYPE REACTORS; SEISMIC EFFECTS; EARTHQUAKES; FLUID MECHANICS; FOURIER TRANSFORMATION

71 INIS Subject Category : S21

99 Supplementary elements :

Abstract : Sloshing phenomenon is well understood in regular cylindrical and rectangular liquidtanks subjected to earthquake. However, seismic behaviour of water in complex geometry suchas a sectored annular tank, e.g., Gravity Driven Water Pool (GDWP) which is located in AdvancedHeavy Water Reactor (AHWR) need to be investigated in detail in the view of safety significance.Initially, for validation of Computational Fluid Dynamics (CFD) procedure, square and foursectored square tanks are taken. Slosh height and liquid pressure are calculated over time throughtheoretical and experimental procedures. Results from theoretical and experimental approachesare compared with CFD results and found to be in agreement. The present work has two mainobjectives. The first one is to investigate the sloshing behaviour in an un-baffled and baffledthree dimensional single sector of GDWP of AHWR under sinusoidal excitation. Other one is tostudy the sloshing in GDWP water using simulated seismic load along the three orthogonaldirections. This simulated seismic load is generated from design basis floor response spectrumdata (FRS) of AHWR building. For this, the annular tank is modelled along with water andnumerical simulation is carried out. The sinusoidal and earthquake excitations are applied asacceleration force along with gravity. For the earthquake case, acceleration-time history isgenerated compatible to the design FRS of AHWR building. The free surface is captured byVolume of Fluid (VOF) technique and the fluid domain is solved by finite volume method whilethe structural domain is solved by finite element approach. Un-baffled and baffled tankconfigurations are compared to show the reduction in wave height under excitation. Theinteraction between the fluid and pool wall deformation is simulated using a partitioned fluid–structure coupling. In the earthquake case, a user subroutine function is developed to convertFRS in to time history of acceleration in three directions. Wavelet analysis is performed to findthe frequency variation of sloshing with respect to time. Results such as sloshing heights andhydrodynamic pressure considering with and without structure interaction effects have beenpresented

ii

ABSTRACT

Sloshing phenomenon is well understood in regular cylindrical and rectangular liquid

tanks subjected to earthquake. However, seismic behaviour of water in complex geometry

such as a sectored annular tank, e.g., Gravity Driven Water Pool (GDWP) which is located

in Advanced Heavy Water Reactor (AHWR) need to be investigated in detail in the view

of safety significance. Initially, for validation of Computational Fluid Dynamics (CFD)

procedure, square and four sectored square tanks are taken. Slosh height and liquid

pressure are calculated over time through theoretical and experimental procedures. Results

from theoretical and experimental approaches are compared with CFD results and found

to be in agreement.

The present work has two main objectives. The first one is to investigate the sloshing

behaviour in an un-baffled and baffled three dimensional single sector of GDWP of

AHWR under sinusoidal excitation. Other one is to study the sloshing in GDWP water

using simulated seismic load along the three orthogonal directions. This simulated seismic

load is generated from design basis floor response spectrum data (FRS) of AHWR

building. For this, the annular tank is modelled along with water and numerical simulation

is carried out. The sinusoidal and earthquake excitations are applied as acceleration force

along with gravity. For the earthquake case, acceleration-time history is generated

compatible to the design FRS of AHWR building. The free surface is captured by Volume

of Fluid (VOF) technique and the fluid domain is solved by finite volume method while

the structural domain is solved by finite element approach. Un-baffled and baffled tank

configurations are compared to show the reduction in wave height under excitation. The

interaction between the fluid and pool wall deformation is simulated using a partitioned

fluid–structure coupling. In the earthquake case, a user subroutine function is developed to

convert FRS in to time history of acceleration in three directions. Wavelet analysis is

performed to find the frequency variation of sloshing with respect to time. Results such as

sloshing heights and hydrodynamic pressure considering with and without structure

interaction effects have been presented.

iii

CONTENTS

Contents

ABSTRACT ........................................................................................................................................ ii 

CONTENTS ....................................................................................................................................... iii 

LIST OF FIGURE .............................................................................................................................. v 

1.  INTRODUCTION ...................................................................................................................... 1 

1.1 Background ................................................................................................................................ 1 

1.2 Problem Definition ...................................................................................................................... 2 

1.3 Objectives .................................................................................................................................. 3 

1.4 Report Organisation .................................................................................................................... 4 

2  MATHEMATICAL FORMULATIONS FOR FLUID, STRUCTURE AND INTERACTION ..... 5 

2.1 Fluid Formulation ....................................................................................................................... 5 

2.2 Structural Formulation ................................................................................................................. 6 

2.3 Coupling between Fluid Flows and Structural Media ..................................................................... 7 

3  NUMERICAL METHODOLOGY ............................................................................................. 9 

3.1 Introduction ................................................................................................................................ 9 

3.2 Grid Generation and Boundary Conditions .................................................................................... 9 

3.3 Volume of Fluid (VOF) Method ................................................................................................. 11 

4  SLOSHING ANALYSIS USING SEISMIC DESIGN CODES .................................................. 13 

4.1 Introduction .............................................................................................................................. 13 

4.2 Equivalent Rectangular Method (ERM): ..................................................................................... 14 

4.3 Spring-Mass Damper Model ...................................................................................................... 15 

4.4 Result and Discussions .............................................................................................................. 18 

5  ESTIMATION OF SLOSH FREQUENCIES USING CFD ....................................................... 24 

5.1 Introduction .............................................................................................................................. 24 

5.2 Modal Analysis of the GDWP .................................................................................................... 25 

5.3 Effect of Excitation Frequency ................................................................................................... 26 

6  EXPERIMENTAL AND ANALYTICAL VALIDATION OF CFD SIMULATIONS .................. 29 

iv

6.1 Case Study 1: Analytical Validation ............................................................................................ 29 

6.2 Case Study 2: Experimental Validation ....................................................................................... 30 

7  CFD SIMULATION OF LIQUID SLOSHING CONTROL IN GDWP UNDER SINUSOIDAL

EXCITATION .................................................................................................................................. 34 

7.1 Effect of Amplitude of GDWP ................................................................................................... 34 

7.2 Structural Analysis .................................................................................................................... 37 

7.3 Effect of Baffles in GDWP ........................................................................................................ 37 

7.4 Liquid Elevation in Higher Modes of GDWP .............................................................................. 41 

8  CFD SIMULATION OF SLOSHING IN GDWP UNDER SEISMIC EXCITATION ................. 44 

8.1 Introduction .............................................................................................................................. 44 

8.2 Development of Random Waves ................................................................................................. 45 

8.3 Wavelet Analysis ....................................................................................................................... 45 

8.4 GDWP under Sinusoidal Excitation in Multi-Direction (Case 1): .................................................. 48 

8.5 GDWP under Random Excitation in Multi-Direction (Case 2) ...................................................... 49 

9  CONCLUSIONS ...................................................................................................................... 57 

9.1 General Conclusions from above Study ....................................................................................... 57 

9.2 Some more Observations between ERM and CFD ....................................................................... 59 

APPENDIX- A.................................................................................................................................. 69 

NON-LINEARITY EFFECT DUE TO WAVE BREAKING .......................................................... 69 

A-1. Tank with Wave Breaking Device (Passive baffle) ..................................................................... 71 

A-2 Discussion and Conclusions ...................................................................................................... 72 

v

LIST OF FIGURE

Fig. 1-1 Advanced Heavy Water Reactor Building (Sinha and Kakodkar, 2006) ............................. 2 

Fig. 1-2 GDWP located top of the primary containment .................................................................. 3 

Fig. 2-1 Feedback loop of fluid structure interaction (Eswaran et al., 2009) .................................... 7 

Fig. 3-1 One compartment of GDWP. ............................................................................................. 9 

Fig. 3-2 Numerical grid of un-baffled GDWP. ................................................................................ 10 

Fig. 3-3 Block structured mesh on fluid and structure portions. ..................................................... 10 

Fig. 4-1 Fully curved outer wall GDWP. ......................................................................................... 13 

Fig. 4-2 Partially curved outer wall GDWP. ................................................................................... 13 

Fig. 4-3 Sectional view of GDWP (updated dimensions) ............................................................... 14 

Fig. 4-4 Plan of GDWP .................................................................................................................. 14 

Fig. 4-5 Equivalent rectangular model ............................................................................................ 15 

Fig. 4-6 Spring –mass model ........................................................................................................... 15 

Fig. 4-7 Impulsive pressure on wall ................................................................................................ 23 

Fig. 4-8 Convective pressure on wall .............................................................................................. 23 

Fig. 5-1 Free surface elevation of liquid under free vibration. ........................................................ 25 

Fig. 5-2 Spectrum analysis of signal. .............................................................................................. 25 

Fig. 5-3 Dominant mode shapes of structure. ................................................................................ 26 

Fig. 5-4 Time history of free surface elevation at 0.1 m excitation amplitude and for different

excitation frequencies. ..................................................................................................................... 28 

Fig. 6-1 The sketch of the 2-D rigid rectangular tank. .................................................................... 30 

Fig. 6-2 Comparisons of free surface elevation. ............................................................................. 30 

Fig. 6-3 Details of the experimental setup. ..................................................................................... 30 

Fig. 6-4 Experimental setup. ........................................................................................................... 31 

Fig. 6-5 Isometric view of 4-sectored square tank .......................................................................... 31 

Fig. 6-6 Comparison of experimental and CFD pressure data at 200 mm from tank bottom

(position 2). ..................................................................................................................................... 31 

Fig. 6-7 Actuator with shake table................................................................................................... 32 

Fig. 6-8 Water oscillation during the excitation .............................................................................. 32 

Fig. 6-9 Water spill out snapshots during base excitation in an 4-sectored rectangular tank .......... 32 

vi

Fig. 7-1 Power spectral density for amplitude 0.1 m and 1ω =0.312 Hz. ...................................... 35 

Fig. 7-2 Time history of non-dimensional free surface elevation at 1ω =0.312 Hz for different

excitation amplitudes ....................................................................................................................... 35 

Fig. 7-3 Phase-plane diagram for amplitude 0.1 m and 1ω =0.312 Hz. ......................................... 35 

Fig. 7-4 Pressure point locations. .................................................................................................... 36 

Fig. 7-5 Time history of pressure at various locations for 0.1 m amplitude sec and 1ω =0.312 Hz.

......................................................................................................................................................... 36 

Fig. 7-6 Wall horizontal displacement contour at time 1.417 sec ................................................... 36 

Fig. 7-7 Outer and inner walls x-displacement versus time ............................................................ 36 

Fig. 7-8 VanMises stress contour at time 12.4 sec. ......................................................................... 37 

Fig. 7-9 Sectional view of velocity magnitude at time 12.4 sec. ..................................................... 37 

Fig. 7-10 Baffles shape and its position in the water pool. ............................................................. 39 

Fig. 7-11 Comparison of liquid elevations with no baffle, annular baffle and cap-plate baffle cases

at 1ω =0.312 Hz and amplitude 0.1 m. ............................................................................................ 40 

Fig. 7-12 Effect of baffles at right corner of the water pool case at 15.05 sec and 1ω =0.312 Hz. . 40 

Fig. 7-13 Comparisons of free surface profile at different time instant for un-baffled and baffled

water pool for 0.1 m amplitude sec and 1ω =0.312 Hz. (Figs. (a)-(d), (e)-(h), (i)-(l) show un-

baffled, annular baffled, cap-plate baffled water pools respectively).............................................. 42 

Fig. 7-14 Comparison of free surface elevation of liquid at 0.1 m amplitude and 5 1ω excitation

frequency. ........................................................................................................................................ 43 

Fig. 7-15 Comparison of free surface elevation of liquid at 0.1 m amplitude and 10 1ω excitation

frequency. ........................................................................................................................................ 43 

Fig. 8-1 Three sectors in GDWP ..................................................................................................... 46 

Fig. 8-2 FRS data for AHWR building at 137 m ............................................................................. 47 

Fig. 8-3 Acceleration- time history for GDWP of AHWR building at 137 m ................................. 48 

Fig. 8-4 Multi directional sinusoidal excitation (Case 1) (a) –(c) Non-dimensional slosh height for

sector 1 to 3 (d) Phase-plane diagram (e)Wavelet diagram for slosh height signal at right corner of

sector 1 ............................................................................................................................................ 49 

Fig. 8-5 Non-dimensional slosh height for sector 1 to 3 of the GDWP subjected to design

excitation as given in Fig. 8.3 (Case 2) ........................................................................................... 50 

Fig. 8-6 FFT from slosh height at sector 1 of GDWP subjected to design excitation as given in Fig.

vii

8.3 (Case 2) ..................................................................................................................................... 51 

Fig. 8-7 FFT from slosh height for sector 2 of GDWP subjected to design excitation as given in

Fig. 8.3 (Case 2) .............................................................................................................................. 51 

Fig. 8-8 FFT from slosh height for sector 3 of GDWP subjected to design excitation as given in

Fig. 8.3 (Case 2) .............................................................................................................................. 51 

Fig. 8-9 Phase-Plane diagram of sector 1 slosh height .................................................................... 51 

Fig. 8-10 Time –frequency curve for sector 1 of GDWP slosh height subjected to design excitation

as given in Fig. 8.3 (Case 2) ............................................................................................................ 52 

Fig. 8-11 Time –frequency curve for sector 2 of GDWP slosh height subjected to design excitation

as given in Fig. 8.3 (Case 2) ............................................................................................................ 52 

Fig. 8-12 Time –frequency curve for sector 3 of GDWP slosh height subjected to design excitation

as given in Fig. 8.3 (Case 2) ............................................................................................................ 52 

Fig. 8-13 Base shear at convective mode for sector 1 through 3 .................................................... 53 

Fig. 8-14 Total force (convective + impulsive) at sector wall 1 ..................................................... 54 

Fig. 8-15 Total moment (convective + impulsive) at sector wall 1 ................................................. 55 

Fig. 8-16 Tank wall displacement versus time ................................................................................ 55 

Fig. 8-17 Wall pressure (convective + impulsive) at sector 1 through 3 ......................................... 56 

Fig. 9-1 Comparison between CFD and ERM for fully curved outer wall GDWP. ........................ 60 

Fig. 9-2 Comparison between CFD and ERM for partially curved outer wall GDWP. .................. 60 

Fig. A-1 Slosh height elevation computed from linear and nonlinear equations ............................ 69 

Fig. A-2 Snap shot of maximum slosh height at left end ................................................................ 69 

Fig. A -3 Snap shot of maximum slosh height at right end ............................................................. 69 

Fig. A -4 Tank with wave breaking device (Case 3) ........................................................................ 70 

Fig. A -5 Fine mesh arrangement (Case 3) ...................................................................................... 70 

Fig. A -6 Snap shot of Non-linear violent flow at 0.99 with baffle arrangement (case 4) ........ 71 

viii

LIST OF TABLE

Table 3-1: Material properties ......................................................................................................... 10 

Table 4-1: Slosh height estimation through seismic codes for sector 1 of fully curved outer wall

GDWP ............................................................................................................................................. 17 

Table 4-2: Slosh height estimation through seismic codes for sector 1 of partially curved outer

wall GDWP ..................................................................................................................................... 18 

Table 4-3: X and Y direction values of sector 1 of fully curved outer wall GDWP. ...................... 19 

Table 4-4: X and Y direction values of sector 1 of partially curved outer wall GDWP. ................. 21 

Table 5-1: Fully curved outer wall GDWP natural frequency in Hz for different sectors ............... 25 

Table 5-2: Partially curved outer wall GDWP natural frequency in Hz for different sectors .......... 26 

Table 5-3: Numerical cases taken for investigation. ....................................................................... 27 

Table 8-1: Sectors frequencies in hertz computed by CFD simulations .......................................... 47 

Table 8-2: Numerical case studies ................................................................................................... 47 

Table 8-3: Total force and moment in sectors 1, 2 and 3 ................................................................. 54 

Table 9-1: X and Y direction values of sector 1 of fully curved wall GDWP. ................................ 61 

Table 9-2: X and Y direction values of sector 1 of partially curved wall GDWP. ........................... 62 

v

LIST OF ABBREVIATION

ACI American Concrete Institute

AHWR - Advanced Heavy Water Reactor

BIS - Bureau of Indian Standards

CFD - Computational Fluid Dynamics

DFT - Discrete Fourier Transformation

ERM - Equivalent Rectangular Method

FFT - Fast Fourier Transformation

FNPP - Floating Nuclear Power Plant

FRS - Floor Response Spectrum

FSI - Fluid-Structure Interaction

GBS - Gravity-Based Structure

GDWP - Gravity Driven Water Pool

LOCA - Loss of Coolant Accident

NPP - Nuclear Power Plant

ONPP - Offshore Structures of Nuclear Power Plant

STFT - Short Term Fourier Transform

TLD - Tuned Liquid Damper

UDF - User-Defined Function

VOF - Volume of fluid Method

1

1. INTRODUCTION

1.1 Background

For the last few decades, liquid sloshing is an important problem in several areas including

nuclear, aerospace, and seismic engineering. Considering the safety aspects of the liquid

storage structures, liquid sloshing is one of the significant problems needs to be addressed.

Particularly, the load generated on the structure due to liquid sloshing is very important to

ensure structural integrity and liquid spillage due to oscillations. Such an oscillatory

motion of the liquid in its container is termed as sloshing. Under the seismic load, severe

accidents might be possible due to this kind of oscillatory motions. This sloshing can

cause possible leakage (Malhotra, 1997), pollution to the surrounding area, and elephant-

foot buckling due to bucking of the tank (Ibrahim, 2005) wall of NPPs.

To generate the nuclear power safely and continuously, technology development for

enhanced safety is vital for future nuclear power plants. As discussed by Lee at al. (2013),

at the beginning of the 1950s, the USA and USSR began to develop Floating Nuclear

Power Plants (FNPPs). Now days, various countries are focusing their studies towards

floating, Gravity-Based Structure (GBS), and submerged Offshore Structures of Nuclear

Power Plants (ONPP) (Gerwick, 2007, Lee et al., 2011). To enhance the safety of nuclear

power, the conventional nuclear power plant (NPP) can be moved from land to ocean.

However, most of the present working nuclear reactors are land-based. And some of the

future land-based reactors are also being designed with improved safety features by

various countries.

The typical Indian advanced Heavy water reactor (AHWR) as shown in Fig. 1.1 is

land-based with enhanced safety features which have an annular-sectored water pool

called as Gravity Driven Water Pool (GDWP),on its dome region of the primary

containment of reactor building (Sinha and Kakodkar, 2006). This present work is focused

on the liquid sloshing in a GDWP of AHWR. The water in the pool serves as a heat sink

2

of the residual heat removal system and several other passive systems. As reported by

Sinha and Kakodkar (2006), this GDWP is divided into eight compartments as shown in

Fig. 1.2, which are interconnected to each other. Each compartment of water pool contains

an isolation condenser for core decay heat removal during shutdown. Water in the pool is

used to condense the steam flowing through the isolation condenser during reactor

shutdown and also functions as a suppression pool to cool the steam and air mixture

during Loss of Coolant Accident (LOCA). This water pool provides cooling to the fuel in

passive mode during first fifteen minutes of LOCA by high pressure injection from

advanced accumulators and later for three days.

Fig.1.1 AHWR building (Sinha and Kakodkar, 2006)

1.2 Problem Definition

To ensure the safety of the reactor against the seismic load, the GDWP located on the

primary containment needs to be investigated for liquid sloshing under simulated seismic

excitations. And also needs to be studied the control of sloshing using passive bafflesin

GDWP of AHWR. From this study, one can be estimated theslosh height, dynamic

pressure on the wall, water spilled out conditions (i.e., possibility of water spillage),

differe

rectan

slosh

proced

determ

regula

geome

Two c

and ot

the pr

and Eq

1.3 Ob

The ob

(i)

(ii)

ent type o

ngular and

frequency,

dure for the

mine. Many

ar geometri

etries.

configuratio

ther one is p

resent case,

quivalent R

bjectives

bjectives of

To estim

7024, AC

) To eval

conditio

of bafflesar

cylindrical

, slosh ind

e baffled GD

y numerical

ies. Althou

on of GDWP

partially cur

fully and p

Rectangular

f this presen

mate liquid n

CI 350.3 (2

luate slosh

ns, the wate

(a) Isometr

Fig.1.2 GD

rrangement

tanks have

duced press

DWP with

l and experi

ugh only fe

P’s outer w

rved. Furthe

partially cu

Method (ER

nt work are,

natural slos

2001) usingE

heights an

er will be sp

ric view

DWP locate

3

and its e

e a detailed

sure and s

considerab

imental stu

ew studies

wall are cons

er details ar

urved wall G

RM) (ACI 3

hing freque

ERM in GD

nd conditio

pilled out du

ed top of the

effect onliq

d analytical

slosh heigh

le interactio

udies have b

have stud

sidered in th

re discussed

GDWP geo

350.3 (2001

enciesusing

DWP.

ns for wat

uring sloshi

e primary co

quid oscill

l solution p

ht. Howeve

on effects i

been report

died the ba

his work. O

d in the follo

metries ana

1)).

CFD and d

ter spilled

ing).

(b) Pl

ontainment

ation.The

procedure t

er, the ana

s very diffi

ed on slosh

affled asym

One is fully c

owing chapt

alysed using

design code

(i.e., under

lan

simple

to find

alytical

icult to

hing in

mmetric

curved

ters. In

g CFD

s TID-

r what

4

(iii) To calculate water pool wall displacement and stress induced in the wall due to

the liquid load.

(iv) To estimate the slosh height using sinusoidal excitation for first mode and higher

modes along with the combined three dimensional excitations.

(v) To determine the slosh height under seismic load for given design seismic

motion.

(vi) To study the effect of vertical excitation and also combined (Horizontal and

vertical)excitation on the fluid behaviour.

(vii) To compute the liquid sloshing design parameters usingERM and comparing with

CFDresults.

(viii) To study effect of annular and cap-plate baffles to prevent excessive sloshing.

1.4 Report Organisation

In chapter 2, theoretical formulation has been established for fluid and structural domains

to explore the fluid–structure interaction phenomena for a moving liquid water pool.

Chapter 3 explains the numerical approach and the frequency modes of liquid and

structural domains. The estimated slosh frequencies and slosh height using international

seismic codes TID-7024, ACI 350.3 (2001) and Euro 8 are reported in chapter 4.

Estimated slosh frequencies for all the sectors of GDWP using CFD is shown in Chapter

5. Chapter 6 discusses the validation of present CFD work using experimental and

analytical methods for which a sectored rectangular tank has been taken for analysis.

Chapter 7 explains the effect of amplitude in slosh height and effect of higher modes

under sinusoidal excitation while Chapter 8 converses the seismic load on GDWP sectors.

Chapter 9 discusses the slosh control techniques for the fluid in GDWP tanks. Chapter 10

depicts the conclusions from the above completed work.

5

2 MATHEMATICAL FORMULATIONS FOR FLUID, STRUCTURE

AND INTERACTION

In this report,numerical approach is followed to solve the sloshing problem in GDWP. So

that mathematical formulation for the fluid and structural domains and coupling methods

are discussed in this chapter. The formulations for fluid domain is based on the three-

dimensional time-dependent conservation equations of mass and momentum to determine

the sloshing characteristics. For the structure domain, the equation of motion is utilized to

simulate the displacement of the concrete wall for balancing at the fluid and concrete wall

boundaries.

2.1 Fluid Formulation

The forces acting on the fluid in order to conserve momentum must balance the rate of

change of momentum of fluid per unit volume. For the laminar transient, incompressible

flow with constant fluid properties over the computational domain, the mass continuity

and Navier–Stokes (NS) equations are given as follows.

0=⋅∇ ur (2.1)

FuPuutu rrrrr

+∇+−∇=⎟⎠⎞

⎜⎝⎛ ∇⋅+

∂∂ 2μρ (2.2)

where Fr

is the external force vector and μ , ρ are the dynamic viscosity and density

of the fluid respectively. The external force is the sum of the gravitational and applied

forces. The ur and P denote the velocity vector and pressure of the oscillating fluid.

The aforementioned governing equations are discretized by finite volume approach to

replace the partial differential equations with the resulting algebraic equations for the

entire domain. Using the staggered-grid arrangement, grids of velocities are segregated

from grids of scalars and laid directly on the surfaces of the control volumes for estimating

those convective fluxes across cell surfaces. The well-known Semi-Implicit Method for

6

Pressure-Linked Equations Consistent (SIMPLEC) numerical algorithm is employed for

the velocity–pressure coupling. In SIMPLEC, an equation for pressure-correction is

derived from the continuity equation which governs mass conservation. It is an inherently

iterative method. The under-relaxation technique is also implemented to circumvent

divergence during iterations. The velocities and local pressure can be determined until

convergent criteria are satisfied.

2.2 Structural Formulation

In the structure model, the linear elasticity approach is utilized. After the finite element

analysis of the solid wall under the slosh loading condition, the solid wall displacement

caused by the fluid–structure interactions is assumed to be small and linear. Hence to

simulate the motion of solid portions, the governing equations are written as below.

xszxyxxx

s gzyxt

sρ

ττσρ +

∂∂

+∂

∂+

∂∂

=∂

∂2

2

(2.3)

yszyxyyy

s gzxyt

sρ

ττσρ +

∂

∂+

∂

∂+

∂

∂=

∂

∂2

2

(2.4)

zsyzxzzz

s gyxzt

s ρττσρ +∂

∂+

∂∂

+∂

∂=

∂∂

2

2

(2.5)

Here, sρ and S are the structure density and structure displacement respectively. The g

is the gravity due to acceleration. The linear stress– strain relation can be expressed as

( ) 0σεεσ +−= oDrr . Here,

0ε and 0σ are initial strains and stresses respectively. Here, the

symbol D is the elasticity matrix containing the material properties. A finite element

method is used to solve the solid model with the principal of virtual work. Structural

domains are meshed using similar first order quadrilateral elements.

7

2.3 Coupling between Fluid Flows and Structural Media

The coupling of the fluid and structural response can be attained numerically in different

ways, however in all cases, of course, the conditions of displacement compatibility and

traction equilibrium along the structure–fluid interfaces ( si ) must satisfy the following

conditions.

(i) The fluid and solid wall move concurrently (displacement compatibility).

ssf iondd = (2.6)

(ii) The fluid force (pressure and shear stress) applied on the solid wall is identical to

the wall force exerted to the liquid side (Traction equilibrium).

ssf ionff = (2.7)

where, fd and sd are the displacements, ff and sf are the tractions of the fluid and

solid, respectively, and si is the interface of the fluid and solid domains. These conditions

must be imposed efficiently in the numerical solution.

Fig.2-1Feedback loop of fluid structure interaction (Eswaran et al., 2009)

Input (External Disturbances) like Force.

Output (Dynamic Variables) like Displacement, Stresses.

Structural Response

Fluid Response

Interface Pressure Interface Acceleration

Output (Dynamic Variables) like Velocities and Pressures.

8

In this problem, the solutions are based on partitioned method where separate solutions for

the different domains are obtained. One solution is for fluid and other is for structure from

the independent solvers. At the fluid-structure interface, information for the solution is

shared between the fluid solver and structure solver. The information is exchanged at

interface in a coupled manner. Two way coupling is adopted for calculations (Benra et al.,

2011). Here, portioned strong coupling is used. Interaction (or coupling) between the fluid

and solid response can be viewed as a feedback loop illustrated in Fig. 2.1.

9

3 NUMERICAL METHODOLOGY

3.1 Introduction

This chapter discusses the numerical formulation for liquid sloshing in GDWP. Here, one

sector of GDWP is taken for analysis from eight-sectored water pool (i.e., GDWP) as

shown in Fig. 3.1. Three domains are modelled viz., water pool wall, liquid and air

domains. The dimensions of the water pool are shown in Fig. 3.1. The side wall thickness

is 500 mm and bottom is 1000mm. Height of the water and air is 8 m and 1 m respectively.

Fig.3-1One compartment of GDWP.

3.2 Grid Generation and Boundary Conditions

A grid is an artificial geometric construction that assists governing equations to be solved.

Block structured grid (BSG) has been used to generate the grid, i.e., the flow domain is

split up into a number of topographically simpler domains and each domain is meshed

separately and joined with neighbours. The BSG arrangement for un-baffled and baffled

water pool has been shown respectively in Figs. 3.2 and 3.3. The fluid-structure

interaction is considered by appropriately coupling the nodes that lie in the common

Tank wall

Air

Water

16000

R 27800

R 7000

10000

10

element faces of the two (i.e., fluid and structure) domains. Fig. 3.3 clearly shows that the

fluid and structure domains share the common element face at fluid-structure interface

which may lead to better data transformation between domains. The fluid domain is

divided into 18,000 sub domains and structure wall into 8000.

Fig.3-2Numerical grid of un-baffled GDWP.Fig. 3-3Block structured mesh on fluid and

structure portions.

The NS equations(Eqs. 2.1 and 2.2) are solved in each sub volumes of fluid domains

(liquid and air). Cell centered average value is taken into consideration. In calculations,

the material properties used for Fluid-structure Interaction (FSI) simulations are

summarized in Table 3.1. Top boundary of the air is fixed pressure condition (at

atmosphere condition). Implicit pressure and implicit shear wall condition is applied on

the fluid-structure interfaces. Iteration loops are continued until the corrections are small

enough to satisfy convergence criteria (10-6< R< 10-8).

Table 3-1: Material properties

1 Water Kinematic

Viscosity 1 E-6 m2/sec Density 1000 kg/m3

2 Air Dynamic

Viscosity 1.846E-05 Kg/m

sec Density 1.1614 kg/m3

3 Concrete

wall

Poisson's ratio

0.2 Density 2500 kg/m3

Young Modulus

33E+09 N/m2

11

The choice of method used for the solution of the assembled system of equations can

have a major impact on the overall solution time and solution quality. Here, the algebraic

multigrid is used for solving the system of linear equations. The basic idea of a multigrid

solution is to use a hierarchy of grids, from fine to coarse, with each grid being

particularly effective for smoothing the errors at the characteristic wavelength of the mesh

spacing on that grid. Other iterative solvers are non-optimal in the sense that as the mesh

resolution increases, the convergence rate degrades. (ESI CFD Inc, 2011). The simulation

was carried out on an Intel Xeon, 2.8GHz six core processor workstation and the

simulation ran for approximately 32 CPU hours for 20 seconds sloshing simulation. The

implicit scheme is used for temporal integration and the higher order upwind schemes are

used for the spatial discretization. The free surface elevation (ζ ) has been captured every

0.05 second.

3.3 Volume of Fluid (VOF)Method

For modeling free surface flows, marker and cell (Chen et al., 1997), VOF, level set

method, sigma-transformation (Frandsen, 2004, Chen and Nokes, 2005, Eswaran and

Saha, 2009, 2010, 2011) and meshless method based on smoothed particle hydrodynamics

(Vorobyev et al., 2011) are known methods. Nevertheless, this work adopts VOF method.

The VOF method is developed by Hirt and Nichols (1981) and refined thereafter by

various authors. Since the method is designed for two or more immiscible fluids, a portion

of air is filled above the liquid level for all cases. The air portion is also modeled and

discretized using the 3-D fluid element. In this method, the term, first fluid and second

fluid indicate the air and water domains respectively. It is based on tracking a scalar field

variable f which stands for the distribution of the second fluid in the computational grid. f

specifies the fraction of the volume of each computational cell in the grid occupied by the

second fluid. All cells containing only fluid 2 will take the valuef = 1 and cells completely

filled with fluid 1 is represented by f= 0. Cells containing an interface between air and

water take on a value of fbetween 0 and 1. For a given flow field with the velocity vector

ur and the volume fraction distribution f(and hence the distribution of fluid two) is

12

determined by the passive transport equation:

( ) 0=⋅∇+∂∂ fu

tf r

(3.1)

This equation must be solved jointly with the primary equations of conservation of

mass and momentum, to achieve computational coupling between the velocity field

solution and the liquid distribution. From the f distribution the interface between the two

fluid phases has to be reconstructed at every time step. As depicted in the manual (ESI

CFD Inc, 2011, Glatzel et al., 2008), the VOF method in CFD-ACE+ offers some

additional features like an algorithm to remove the so called flotsam and jetsam caused

due to numerical errors. It is characterized by the generation of tiny isolated droplets of

liquids or gas in the regions of the other medium, especially in regions of high swirl.

13

4 SLOSHING ANALYSIS USING SEISMIC DESIGN CODES

Earthquake prone countries across the world rely on “codes of practise” to mandate that

all structuressatisfy at least a minimum level of safety requirements against future

earthquakes. Indian seismic code IS 1893:1984 had very limited provisions on seismic

design of elevated tanks. In 2002, revised part 1 of IS 1893 has been brought out by the

bureau of Indian Standards (BIS). The ERM approach and use of design codes, slosh

frequencies and slosh height are found and slosh design terms are discussed in detail in

this chapter.

4.1 Introduction

As mentioned in introduction chapter, two GDWP models have taken for analysis. One

is fully curved outer wall and the other one is partially curved wall. Figs. 4.1 and 4.2 are

showing fully curved outer wall and partially curved wall GDWP. The sketch and

dimensions of the partially curved outer wall GDWP are depicted in Fig. 4.3. The ERM

analysis is performed to design against the seismic load.

Fig. 4-1 Fully curved outer wall GDWP. Fig. 4-2Partially curved outer wall GDWP.

Fully curvedouter wall

Inner wallPartially curvedouter wall

14

Fluid in each sector has different dynamic characteristics, however some of the sectors

are symmetric. As shown in sector arrangement Fig. 4.4, the each sector has different

slosh frequencies. The slosh frequencies for sectors, 1 and 5 are same. Also, similarly

sectors 2, 4, 6 and 8 are same and remaining sectors 3 and 7 are same slosh frequencies.

These frequencies are found using ERM approach and listed in Chapter 5 along with

frequencies from CFD approach for comparison.

Fig. 4-3 Sectional view of GDWP (updated

dimensions) Fig. 4-4Plan of GDWP

4.2 Equivalent Rectangular Method (ERM): Since the design code for GDWP (8- sectored annular water pool with outer spherical

wall) geometry is not available directly in seismic design code, as instructed [31], the one

sector s taken alone and ERM analysis is made. The rectangular domain is constructed as

shown in Fig. 4.5. While converting from GDWP fluid domain to equivalent rectangular

domain the following assumptions are taken.

(i) Liquid volume should be equal in GDWP sector and rectangular model.

(ii) Ratio between liquid height (h) and length of the tank (L) is taken almost equal to

the GDWP sector dimensions.

Hydrodynamic forces exerted by liquid on tank wall shall be considered for analysis in

addition to hydrostatic forces. These hydrodynamic forces are evaluated with the help of

spring mass model of rectangular tanks as shown in Fig. 4.6.

Excitation direction

1

23

4

5

67

8

15

Fig. 4-5Equivalent rectangular model Fig. 4-6Spring –mass model

4.3 Spring-Mass Damper Model

For the purposes of incorporating the dynamic effects of sloshing in the pools, it is

convenient to replace the liquid conceptually by an equivalent linear mechanical system.

The equations of motion of oscillating masses and rigid masses are included more easily in

the analysis than are the equations of fluid dynamics. Fig. 4.6 illustrates generalized

spring-masses model for the rectangular tank and the symbols used in the analysis. The

width of the tank are 2L and height of the liquid is denoted as h. The center of the mass of

the liquid is represented as C.G, while, the locations (Hn) of the masses are references to

the C.G. The tank is excited by a small time-varying linear acceleration . Rigidly

attached mass is denoted as m0, while the convective (slosh) masses are showed as m1

through mn .The deflection of the mass is represented as xn which is relative to the tank

walls as a result of the tank motion.

The mathematical equations can be derived from static and dynamic properties of spring-

mass model. These equations and derivations can be found in ACI, Housner (1963) and

Dodge (2000) for simple geometries like rectangular and cylindrical tanks. According to

static properties, the sum of all the masses must be equal to the liquid mass and center of

mass of the model must be same elevation as the liquid. These can be derived as follows,

m0+ m1+ m2+…..+ mn = mliq (4.1)

Z

H = 8m

W=11 mL= 11.5 m

x

yRectangular model

GDWP/ sector

(11.5 x11 x 8) = 1012 m3

1000 m3

16

. m0H0+ m1H1+ m2H2+ ….+ mnHn = 0 at C.G (4.2)

Equation of motion can be derived by inserting the acceleration terms and applying

static properties into the force equation. The equations of motion for each of the spring-

masses is expressed as

0 (4.3)

From the above equations forces acting on the rectangular tank can be estimated. The

slosh height is estimated using the well derived equations as listed here (Eqs. 4.1 to 4.8).

The convective and impulsive masses are,

.)12(

/)12tanh(833 hn

LhnLmm liqn −−

=π

π (4.4)

.)12(

/)12tanh(81

330 ⎥⎦

⎤⎢⎣

⎡−−

−=hn

LhnLmm liq ππ

(4.5)

Height of the convective masses and natural frequency,

.)12(

/)12tanh(8)12(

sinh

12/)12tanh(

)12(2 33 hnLhnL

Lhn

Lhnn

LhH n −−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

−−=

ππ

ππ

π (4.6)

⎟⎠⎞

⎜⎝⎛=

LH

Lg

58.1tanh58.1ω (4.7)

The first mode slosh height,

1

)58.1coth(527.0

2

max,1

−==

Lg

LhLd

h

n

θω

where, n =1, (4.8)

Based on the above formula, the slosh height and other parameters have been calculated

and tabulated for sector 1 of fully curved outer wall GDWP and partially curved outer wall

GDWP in Tables 4.1 and 4.2 respectively.However the detailed computations for both

models are shown in Tables 4.3 and 4.4.

17

Table 4-1: Slosh height estimation through seismic codes for sector 1 of fully curved outer

wall GDWP

Length direction Width direction unit

Length of pool 2L 11.5 11 m

Width of pool W 11 11.5 m

Height of pool 10 10 m

Height of water 8 8 m

Density of water 1000 1000 kg/m3

Mass of water 1.01E+06 1.01E+06 kg

Volume of water 1012 1012 m3

Convective acceleration 0.13 0.145 g

Impulsive acceleration 1.11 1.88 g

Convective acceleration 1.275 1.422 m/sec2

Displacement (A) 0.485 0.515 m

Convective

frequency

ω 1.622 1.662 rad/sec

f 0.258 0.265 Hz

Slosh height

TID-7024 0.778 0.856 m

Slosh height

ACI 350.3 (2001) 0.748 0.798 m

18

Table 4-2: Slosh height estimation through seismic codes for sector 1 of partially curved

outer wall GDWP

Length direction Width direction unit

Length of pool 2L 13.5 12.5 m

Width of pool W 12.5 13.5 m

Height of pool 8 8 m

Height of water 6 6 m

Density of water 1000 1000 kg/m3

Mass of water 1.01E+06 1.01E+06 kg

Volume of water 1012.5 1012.5 m3

Convective acceleration 0.13 0.145 g

Impulsive acceleration 1.11 1.88 g

Convective acceleration 1.275 1.422 m/sec2

Displacement (A) 0.627 0.632 m

Convective

frequency

ω 1.427 1.501 rad/sec

f 0.227 0.239 Hz

Slosh height

TID-7024 0.893 0.953 m

Slosh height

ACI 350.3 (2001) 0.878 0.906 m

4.4 Result and Discussions

The fully curved wall GDWP slosh height and other calculations are shown in Table 4.2,

while slosh calculations are shown for partially curved wall GDWP in Table 4.3.The

approved model of GDWP’s outer wall is fully curved (Fig. 4.5). During internal review

meeting, a partially curved wall GDWP model (Fig. 4.6) is proposed (at AHWR Review

Meeting 201, BARC). To address the both cases, two cases are studied and tabulated in

Tables 4.2 and 4.3.

19

Table 4-3: X and Y direction values of sector 1 of fully curved outer wall GDWP.

Sl. No

Term

Longitudinal

(x)direction

Lateral (y) direction

Unit Comments

1 Impulsive mass (mi) 6.88E+05 7.06E+05 kg Sum of Impulse and convective mass is slightly higher than total mass of fluid. 2 Convective mass (mc) 3.75E+05 3.60E+05 kg

3

Height of the impulsive mass above the bottom of the tank wall (hi) (without considering base pressure)

3.00 3.00 m See the Fig. 4.7

4

Height of the convective mass above the bottom of the tank wall (hc)(without considering base pressure)

5.09 5.15 m See the Fig. 4.6

5

Height of the impulsive mass above the bottom of the tank wall (hi*)(with considering base pressure)

4.88 4.73 m See the Fig. 4.7

6

Height of the convective mass above the bottom of the tank wall (hc*)(with considering base pressure)

5.91 5.87 m See the Fig. 4.6

7 Wall deflection (d) due to load

0.0192 0.0180 m Considered as fixed at three edges and free at top

8 Impulsive frequency (fi)

3.600 3.715 hz ACI 350.3 (2001)

9 Convective frequency (fc)

0.243 0.255 hz ACI 350.3 (2001)

10

Seismic co-efficient (Ah)

FRS at 137 m height of AHWR building. Impulsive (Ah)i 1.11g 1.88g

Convective (Ah)c 0.13g 0.145g

20

11

Total shear force (V) at bottom of the wall

13765.85 23402.02 KN Lateral base shear 23.2 % of total seismic weight in x direction while same in 24 % in y direction

Impulsive Vi 13757.55 23396.42 KN

Convective Vc 477.86 512.11 KN

12

Total bending moment at bottom of the wall (M)

50.72 85.79 MN SRSS rule as followed in all international code except Eurocode 8 (1988).

Impulsive Mi 50.66 131.71 MN Convective Mc 2.43 3.52 MN

13 Over turning moment at bottom of base slab. (M*)

78.59 131.76 MN-m Housner(1963)

14 Impulsive time period (Ti)

0.28 0.27 sec ACI 350.3 (2001) and NZS 3106 (1986)

15 Convective time period (Tc)

4.11 3.92 sec ACI 350.3 (2001) Housner (1963)

16 Slosh height (hs) 0.748 0.798 m Free board is 1 m (Importance factor 1 )

17

Impulsive pressure on wall (y=0) (Piw)

63.89 106.15 KN/m2 ACI 350.3 Housner(1963) See the Fig. 4.7

Impulsive pressure on top of base (y=0) (Pib)

30.74 51.83 KN/m2

18

Convective pressure on wall (y=0) (Pcw)

1.34 1.29 KN/m2 ACI 350.3 Housner(1963) See the Fig. 4.7

Convective pressure on wall (y=h) (Pcwt) 6.11 6.52 KN/m2

Convective pressure on top of base (y=0) (Pcb)

1.34 1.29 KN/m2

19

Pressure due to wall inertia (Pww)

13.61 23.05 KN/m2 ACI 350.3 ; Housner(1963) Pressure due to

vertical excitation (Pv) 41.20 41.20 KN/m2

20 Maximum hydrodynamic pressure (P)

87.78 135.62 KN/m2 Hydro static pressure is 78.8KN/m2.

21

Table 4-4: X and Y direction values of sector 1 of partially curved outer wall GDWP.

Sl. No

Term

Longitudinal

(x)direction

Lateral (y) direction

Unit Comments

1 Impulsive mass (mi) 4.99E+05 5.32E+05 kg

Sum of Impulse and convective mass is slightly higher than total mass of fluid.

2 Convective mass (mc) 5.33E+05 5.06E+05 kg

3

Height of the impulsive mass above the bottom of the tank wall (hi) (without considering base pressure)

2.25 2.25 m See the Fig. 4.7

4

Height of the convective mass above the bottom of the tank wall (hc)(without considering base pressure)

3.41 3.47 m See the Fig. 4.8

5

Height of the impulsive mass above the bottom of the tank wall (hi*)(with considering base pressure)

5.34 4.96 m See the Fig. 4.7

6

Height of the convective mass above the bottom of the tank wall (hc*)(with

considering base pressure)

5.67 5.31 m See the Fig. 4.8

7 Wall deflection (d) due to

load 0.0068 0.0060 m Considered as fixed at

three edges and free at top

8 Impulsive frequency (fi) 6.063 6.433 hz ACI 350.3 (2001)

9 Convective frequency (fc) 0.224 0.239 hz ACI 350.3 (2001)

10

Seismic co-efficient (Ah) FRS at 137 m height of AHWR building.

Impulsive (Ah)i 1.11g 2.16g

Convective (Ah)c 0.13g 0.145g

11

Total shear force (V) at bottom of the wall

11224.95 22294.37 KN

Lateral base shear 23.2 % of total seismic weight in x direction while same in 24 % in y direction

Impulsive Vi 11204.35 22282.76 KN

Convective Vc 679.80 719.36 KN

22

12

Total bending moment at bottom of the wall (M)

32.51 63.96 MN SRSS rule as followed in all international code except Eurocode 8(1988).

Impulsive Mi 32.42 116.76 MN

Convective Mc 2.32 4.54 MN

13 Over turning moment at

bottom of base slab. (M*) 60.57 116.85 MN-m Housner (1963)

14 Impulsive time period (Ti) 0.16 0.16 sec ACI 350.3 (2001) and NZS 3106 (1986)

15 Convective time period

(Tc) 4.46 4.18 sec

ACI 350.3 (2001) Housner (1963)

16 Slosh height (hs) 0.878 0.906 m Free board is 1 m (Importance factor 2 )

17

Impulsive pressure on wall (y=0) (Piw)

54.33 104.29 KN/m2

ACI 350.3 Housner (1963) See the Fig. 4.7

Impulsive pressure on top of base (y=0) (Pib)

20.73 41.95 KN/m2

18

Convective pressure on wall (y=0) (Pcw)

3.32 3.10 KN/m2

ACI 350.3 Housner (1963) See the Fig. 4.7

Convective pressure on wall (y=h) (Pcwt) 7.17 7.41 KN/m2

Convective pressure on top of base (y=0) (Pcb)

3.32 3.10 KN/m2

19

Pressure due to wall inertia (Pww)

13.61 26.49 KN/m2 ACI 350.3 ; Housner (1963)

Pressure due to vertical excitation (Pv)

30.90 30.90 KN/m2

20 Maximum hydrodynamic

pressure (P) 74.71 134.42 KN/m2 Hydro static pressure

(58.86KN/m2).

23

Fig.4-7 Impulsive pressure on wall Fig. 4-8 Convective pressure on wall

Formulas used here are taken from IS 1893 (Part 1):2002, ACI 350.3, Housner (1963) and

NZS 3106 (1986). The deflection of the wall due to this pressure, it can be considered to

be fixed at three edges and free at top. Deflection of wall can be obtained by performing

analysis of wall or by classical analysis using theory of plates. However, here, simple

approach is used as suggested in international codes. Sum of Impulse and convective mass

is slightly higher than total mass of fluid. But however up to 3% is allowed.

24

5 ESTIMATION OF SLOSH FREQUENCIES USING CFD

The estimation of modal frequency through numerical investigation is discussed in the

chapter and followed by the test cases are made for the complete numerical investigation.

5.1 Introduction

The total fluid dynamic pressure for a flexible tank partially filled with liquid undergoing

a seismic motion consists of three components. The first pressure component is called the

impulsive pressure which varies synchronously with input tank base motion and the tank

wall is assumed to be rigid, moving together with the tank base. The second component is

caused by the fluid sloshing motion. This pressure is generally referred to as the

convective pressure or non-impulsive pressure, while the third component is induced by

the relative motion of the flexible tank wall with respect to the tank base (Chang et al.,

1989). This fluid-structure interaction effect results in the dynamic characteristics of the

tank-liquid system to be notably different from that of a rigid tank has led to the inclusion

of a third hydrodynamic component to quantify the dynamic response of flexible tanks

namely the ‘flexible-impulsive’ component. Methods for determining the contribution of

the flexible-impulsive component to the total response such as base shear, overturning

moments, wall stresses of tanks under seismic excitations have been proposed by various

researchers (Haroun and Housner, 1981, Tedesco et al., 1989).

In the present study, the problems are restricted to the water tank under surge motion.

During analysis, the complete flow regime is assumed to be laminar. Some localized

turbulence effects may be caused at the sheared interface during sloshing which should not

affect the fluid-wall interactions and global fluid behaviour. This type of gravity force

dominant flow problems are mainly considered as inviscid. It had been proved that the

free surface oscillations of low viscosity fluids in partly-filled tank persist over long

durations (Kandasamy et al., 2010).

25

5.2 Modal Analysis of the GDWP

The sinusoidal excitation is applied on the fluid domain in terms of acceleration force via

user subroutine functions. As discussed earlier, GDWP is divided into eight

compartments. Among the eight, one compartment has taken for analysis. The sketch and

dimensions of the water pool are depicted in Fig. 3.1. The modal analysis is required to

find the inherent dynamic properties of the any domain in terms of its natural frequencies.

In the beginning, the first mode natural frequency of water pool has been calculated by

free vibration i.e., free surface elevation (ζ ) of liquid shown in Fig. 5.1.

Fig. 5-1Free surface elevation of liquid

under free vibration. Fig. 5-2 Spectrum analysis of signal.

Table 5-1: Fully curved outer wall GDWP natural frequency in Hz for different sectors

Mode

number Sectors 1 and 5 Sectors 2,4, 6 and 8 Sectors 3 and 7

Analytical CFD Analytical CFD Analytical CFD

1 0.257 0.312 0.212 0.292 0.264 0.309

2 0.368 0.393 0.313 0.412 0.377 0.421

3 0.451 - 0.384 - 0.461 -

4 0.521 - 0.443 - 0.533 -

0 2 4 6 8 10 12 14-1500

-1000

-500

0

500

1000

1500

Free

sur

face

ele

vatio

n (m

m)

Time (Sec)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

50

100

150

200 ω1=0.312 Hz

Frequency (Hz)

Pow

er

26

Table 5-2: Partially curved outer wall GDWP natural frequency in Hz for different sectors

Mode

number Sectors 1 and 5 Sectors 2,4, 6 and 8 Sectors 3 and 7

Analytical CFD Analytical CFD Analytical CFD

1 0.226 0.242 0.201 0.231 0.228 0.247

2 0.338 0.342 0.310 0.332 0.407 0.390

3 0.416 - 0.383 - 0.499 -

4 0.48 - 0.442 - 0.576 -

The first mode natural frequency is observed as 0.312 Hz by spectrum analysis using

Fast Fourier Transformation (FFT) (Fig. 5.2). The FFT is a faster version of the Discrete

Fourier Transform (DFT). The FFT utilizes some clever algorithms in much less time than

DFT. The FFT is extremely important in the area of frequency (spectrum) analysis since it

takes a discrete signal in the time domain and converts that signal into its discrete

frequency domain representation. Tables 5.1 and 5.2 show the comparison of slosh modes

computed through analytical and CFD approaches for fully and partially curved outer wall

GDWP respectively. The dominant modes of the structure i.e., the first four mode shapes

are illustrated in Fig. 5.3. The material properties are listed in Table 3.1.The case studies

for numerical simulation were summarized in Table 5.3.

Mode 1 (11.21 Hz) Mode 2 (17.61 Hz) Mode 3 (21.16 Hz) Mode 4 (27.89 Hz) Fig. 5-3 Dominant mode shapes of structure.

5.3 Effect of Excitation Frequency

Fig. 5.4 shows the numerical results for the liquid heights at the left, right and center point

in a three-dimensional GDWP subject to harmonic motions under 0.1 m amplitude for

different frequencies.

27

Table 5-3: Numerical cases taken for investigation.

Sl. No Excitation

Sectors consider

ed in GDWP

tank

Baffle Excitation Amplitude

(m)

Excitation

direction

Excitation Frequency

(Hz)*

Condition

1

Sinusoidal excitation

1 (Fully curved outer wall

GDWP)

No baffle

0.01

x

1ω

Fixed wall

2 0.02 1ω

3 0.06 1ω

4

0.1

1ω 5 0.5 1ω

6 0.8 1ω

7 1.2 1ω

8 0.1 1ω Flexible wall

9 Annul

ar 0.1

1ω Flexible wall

10 5 1ω

11 10 1ω

12 Cap-Plate

0.1 1ω Flexible

wall 13 5 1ω

14 10 1ω

15 Sinusoidal excitation

3 (partially curved outer wall

GDWP)

No baffle

0.03 xyz 1ω Flexible wall

16 Earthquake excitation

3 (partially curved outer wall

GDWP)

No baffle

Max. floor response acceleration

xyz Design FRS data

Flexible wall

*First mode natural frequency of liquid is 1ω =0.312 Hz.

When the excitation frequencies are close to the natural frequency as shown in Figs. 5.4

(a) and (c), the beat phenomena are noticeable (Eswaran et al., 2009). It can be observed

from Fig. 5.4 (d) that when the excitation frequency is far-off from the natural frequency,

i.e., 1.92 rad/sec, the liquid heights are very small and frequency is equal to excitation

28

frequency. When the frequency is almost near to the first mode natural frequency, i.e. Fig.

5.4 (b), the amplitude grows monotonically with time. There is a slight difference in liquid

elevation between right and left corner of the water pool.

Fig.5-4 Time history of free surface elevation at 0.1 m excitation amplitude and

for different excitation frequencies.

0 2 4 6 8 10 12 14 16 18 20 22 24-2

-1

0

1

2

Right Left Center

Time (Sec)

-2

-1

0

1

2

Right Left Center

Free

sur

face

Ele

vatio

n (m

)

-2

-1

0

1

2

Right-2

-1

0

1

2

ωx/ω1 = 0.5

ωx/ω1 = 0.8

ωx/ω1 = 0.99

ωx/ω1 = 1.2

(d)

(c)

(b)

(a)

Right Left Center

29

6 EXPERIMENTAL AND ANALYTICAL VALIDATION OF CFD

SIMULATIONS

In this chapter, slosh height is computed through experimental and analytical methods

and compared with CFD procedure for validation.

6.1 Case Study 1: Analytical Validation

In this section, a 2D model partially filled tank has been taken and the liquid elevation has

been captured under sinusoidal excitation by numerical as well as analytical relation. The

2-D rigid tank which is 570 mm long and 300 mm high is excited with )sin( tA ω as shown

in Fig. 6.1. The water depth is 150 mm and excited amplitude is 5 mm. The lowest natural

frequency 1ω for this case is 6.0578 rad/sec. The natural frequency is calculated from

Equation 6.2. Liquid free surface elevation has been calculated from the following third

order analytical relations (Faltinsen et al., 2000) and compared with the present numerical

simulation results for frequency ratio 0.583.

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−−

++

+= )2cos(

)cos()4(

3

2

1)2cos(*

)4(2

33

8

1

8

1

)cos()cos(),(

222

22

22422

2

22

22

224

4

224

4

224

2

xk

tkgt

kgkgkg

gaxktAtx n

nnnn

nnnnn

nnn

nn

n

nn

n

nn

nnn

ωωωω

ωωωω

ωωωω

ωω

ωω

ωωζ (6.1)

where, the linear sloshing frequencies

)tanh( snnn hkgk=ω and )2tanh(22 snnn hkkg=ω . (6.2)

The initial conditions are )cos(),(0

xkanx xt=

=ζ and 0),(

0=

=tzxφ , where A is the

amplitude of the initial wave profile, bnk n /π= is the wave profile for n = 0,1,2... , hs is

still water level and x is the horizontal distance from the left wall. This analytical result is

30

compared with the present numerical approach. From Fig. 6.2, it is observed that the free

surface elevation of analytical and present numerical coincides with each other.

Fig.6-1 The sketch of the 2-D rigid rectangular

tank. Fig.6-2 Comparisons of free surface elevation.

Fig.6-3 Details of the experimental setup.

6.2 Case Study 2: Experimental Validation

This case study shows the comparison of experimental and numerical results. For this

purpose, a model square tank with sectored arrangement was built and experiments were

conducted. The experiments were performed on a shake table (1.2 m x 1.0 m) coupled

with a servo-controlled hydraulic actuator of 250 KN capacity. The test setup is specially

designed for the sloshing experiments as shown in Fig. 6.3. The perspective view of the

setup is shown in Fig. 6.4 with the actuator coupling arrangement.

570

Probe

20

All dimensions are in mm

Water

Air

300xy

Time (Sec)

Free

surfa

ceel

evat

ion

(m)

0 2 4 6 8 10-0.015

-0.01

-0.005

0

0.005

0.01

0.015

AnalyticalPresent Numerical

Platform

Liquid A sin (ω t)

DAS

10020

0

500250

1000

A

arrang

all dir

install

Water

(a) S

Fig.6

10

20

30

40

50

60

70

80

Pres

sure

(Pa)

square tan

gement as s

rections. Pr

led at 100 m

r fill level in

Fig.6-4

Square tank

6-6 Compari

0 1 20

00

00

00

00

00

00

00

00

nk has 1 m

shown in Fi

ressure var

mm (positio

n the tank is

4 Experimen

at excitatio

1ω .

ison of expe

3 4 5Time (Sec

m length an

ig. 6.5 for th

riations are

on 1) and 2

s maintained

ntal setup.

on frequency

erimental an

(

6 7 8

Numerical Experimenta

c)

31

nd 0.5 m h

he experim

e sensed by

200 mm (po

d as 250 mm

y 0.57 (

nd CFD pre

(position 2)

9 10

al

Pres

sure

(Pa)

height alon

ments in orde

y two flush

osition 2) f

m.

Fig.6-5 Is

(b) Sectored

fr

essure data a

).

0 1 20

100

200

300

400

500

600

700

800

Pres

sure

(Pa)

ng with rem

er to allow

h type pre

from the bo

sometric vie

square ta

d square tan

requency 0.

at 200 mm f

2 3 4 5

Time (

movable 4

liquid moti

ssure trans

ottom of the

ew of 4-sect

ank

nk with exci

99 1ω .

from tank b

6 7 8

Sec)

Numerical Experimen

Baffle

Wate

-sector

ions in

sducers

e tank.

tored

tation

ottom

9 10

ntal

er

32

Fig.6-7Actuator with shake table Fig.6-8 Water oscillation during the

excitation

Fig.6-9Water spill out snapshots during base excitation in an 4-sectored rectangular tank

33

For this case the liquid natural frequency has been calculated as 0.71 Hz through

analytical relation and sine sweep experiments. The comparison of experimental and

numerical time history of pressure at position 2 for square tank and 4- sectored square tank

cases are shown in Fig. 6.6 (a) and (b) respectively. For these cases, the excitation

frequency ratio is taken as 0.57 and 0.99 of the first mode square tank. The comparisons of

numerical with experiment results are shown and found the CFD results are good in

agreement with the experiment. Figs. 6.7 through 6.9 show the snap shots of the

experimental setup to display the tank and actuator arrangement and water spill outs.

In this chapter, experimental and analytical studies are also performed to validate present

numerical results. For experimental validation, a simple square and four-sectored squared

tanks was taken. The pressure variations were captured at different locations under the

surge motions of the tank and found the CFD results are good in agreement with the

experiment.

34

7 CFD SIMULATION OF LIQUID SLOSHING CONTROL IN

GDWP UNDER SINUSOIDAL EXCITATION

In this chapter, the detailed sloshing studies are carried out for studying the effect of

amplitude on liquid sloshing and structural analysis to compute the wall displacement and

induced stress. Un-baffled and baffled tank configurations are compared to show the

reduction in wave height under sinusoidal excitation. For this, annular baffle and cap-plate

baffles are taken for analysis. The slosh height is compared between un-baffled and

baffled configurations under design acceleration.

7.1 Effect of Amplitude of GDWP

Fig. 7.2 depicts the effect of excitation amplitude (A) under its first mode natural

frequency. For this purpose, the non-dimensional free surface elevation is captured at the

right corner of the water pool for 20 seconds. If the excitation amplitude is increased, the

fluid response becomes large. Fig. 7.2 is plotted with the assumption of with and without

fluid-structure interaction conditions. In the case of without FSI, the boundaries are

considered as rigid wall. It is also found in the Fig. 7.3 that FSI consideration has slight

more free surface elevation than the without FSI. It is caused due to the interaction of the

fluid domain with structure produces the relative pressure component. However, there is a

large gap between the first mode frequency of the structure and fluid portions. Deviations

are not high as the excitation frequency is low and faraway from the structural first mode

frequency. It is also observed that the amplification of the fluid motion is relatively larger

at lower amplitude while at the higher amplitude; the amplification of free surface

elevation is less than the lower amplitude case. Fig. 7.1 shows the power spectral density

of liquid elevation wave at 0.1 m excitation amplitude. Closer to natural frequency, a

single dominant frequency is absorbed. The phase-plane diagram is plotted in Fig. 7.3,

shows that non-linearity exists in the flow.

35

Fig.7-1Power spectral density for amplitude 0.1 m and 1ω =0.312 Hz.

Fig.7-2Time history of non-dimensional free surface elevation at 1ω =0.312 Hz for different excitation amplitudes

Fig.7-3Phase-plane diagram for amplitude 0.1 m and 1ω =0.312 Hz.

Pressure waves are captured in different locations of the water pool and the locations

(i.e., A through E) are depicted in Fig. 7.4. Positions A through C are 1 m below from the

liquid free surface and positions D and E are in 5 m and 8 m from the free surface

respectively. The time histories of pressure at the different places of the water pool for un-

baffled water pool are plotted in Fig. 7.5 for 0.1 m amplitude and 1ω =0.312 Hz. It can be

seen from the Fig. 7.5 that when the water pool is excited, the impulse pressures occur

because of the relatively large amplitude of the external excitation. If liquid oscillation is

not controlled efficiently, sloshing of liquids in storage water pools may lead to large

dynamic stress to cause structural failure. On the other hand, if the baffle exists in the

water pool, the dynamic pressure will be minimal. The horizontal displacement histories

0 2 4 6 8 10 12 14 16 18 20-40

-20

0

20

40

Amplitute = 0.01 m

Time (Sec)

-40

-20

0

20

40

Amplitute = 0.02 m

ξ / A -40

-20

0

20

40

(d)

(c)

(b)

(a)

Amplitute = 0.06 m

-40

-20

0

20

40

Amplitute = 0.1 m (FSI) Amplitute = 0.1 m (without FSI)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.000

0.005

0.010

0.015

0.020

0.025

Frequency (Hz)

Pow

er

ζ/A

(∂ζ/

∂t)/(

A*ω

)

-15 -10 -5 0 5 10 15 20-1500

-1000

-500

0

500

1000

1500

36

of the container inner and outer walls are drawn in Figs. 7.6 and 7.7. The Fig. 7.1 shows

that the steady state values are reached from around 12 sec for 0.1 m excitation amplitude.

The displacement is captured at 12.04 sec. The displacement frequency is almost equal to

wave frequency. And it can be seen that the horizontal displacement is symmetric in both

side walls as shown in Fig. 7.6.

Fig.7-4Pressure point locations. Fig.7-5Time history of pressure at various locations for 0.1 m amplitude sec and 1ω=0.312 Hz.

Fig.7-6Wall horizontal displacement contour at time 1.417 sec

Fig.7-7Outer and inner walls x-displacement versus time

0 2 4 6 8 10 12 14 160

20

40

60

80

100

120 Position A Position B Position C Position D Position E

Pres

sure

(Kpa

)

Time (Sec)

0 5 10 15 20

-0.0010

-0.0005

0.0000

0.0005

0.0010 Displacement of outer wall Displacement of inner wall

Wal

l dis

plac

emen

t in

x di

rect

ion

(m)

Time (Sec)

37

7.2 Structural Analysis

Fig. 7.8 shows the VonMises stress on water pool wall at 12.4 sec. The maximum stress

created on the side wall is observed as 7 X 106 N/m2. Sloshing occurs primarily at the

liquid surface and oscillates, exerting forces on the tank structure walls. The liquid free

surface profile has a positive gradient when it moves towards right side. As soon as the

free surface elevation reaches its peak at the right wall, fluid vertical velocity will become

zero. Then, due to its own gravity and external applied forces move the liquid free surface

to down. Now, the direction of the fluid velocity switches from right to left and the

magnitudes of these velocities continue to increase until they reach their maximum. This

cycle will continue until the free surface stops its oscillations which may happen due to

the removal of external excitation to the system. Fig. 7.9 shows the velocity magnitude

from right to left at the time of 12.4 sec. During the surge motion of the water pool, a

single directional standing wave is moving upward and downward direction inside the

water pool.

Fig. 7-8VanMises stress contour at time 12.4 sec.

Fig. 7-9 Sectional view of velocity

magnitude at time 12.4 sec.

7.3 Effect of Baffles in GDWP

Tanks of asymmetric shapes and tanks with baffles, give rise to complications in fluid-

structure interaction, which is not amenable to analytical solution. The studies of liquid

sloshing in a tank with baffles are still very necessary (Eswaran and Saha, 2011; Xue et

al., 2012). Baffle is a passive device which reduces sloshing effects by dissipating kinetic

38

energy due to the production of vortices into the fluid. The linear sloshing in a circular

cylindrical tank with rigid baffles is being investigated by many authors in the context of

spacecraft and ocean applications. The shapes and positions need to be designed with the

use of either numerical model or experimental approaches. Nonetheless, the damping

mechanisms of baffle are still not fully understood. The effects of baffle on the free and

forced vibration of liquid containers were studied by Gedikli and Erguven (1999), Biswal

et al., (2006). To the author’s knowledge, there is a very limited set of analytically

oriented approaches to the sloshing problem in baffled tanks.

Here, two types of baffles are taken for analysis. First one is an annular baffle as

depicted in Fig. 7.10 (a) and (b). Few authors worked on this annular baffle for their own

geometries mainly two dimensional. This article is focused on annular baffle for a three

dimensional annular cylindrical water pool. Biswal et al. (2006) found that the baffle has

significant effect on the non-linear slosh amplitude of liquid when placed close to the free

surface of liquid. The effect is almost negligible when the baffle is moved very close to

the bottom of the tank. Past investigations also convey that the performance of the annular

baffle is better when it is near to the liquid free surface (Eswaran et al., 2009). Second one

is cap-plate baffle or shroud as shown in Fig. 7.10 (c) and (d) which is fixed at center of

the water pool. (More details about baffle for thermal stratification can be found in

Vijayan, 2010). Under reactor shutdown conditions, natural convection process starts due

to the strong heat source at the IC wall. Long-time effect of this natural convection

process leads to warm fluid layers floating on the top of gradually colder layers. This

results in a thermally stratified pool having steep temperature gradient along the vertical

plane. Over a period of time, the substantial part of this pool gets thermally stratified

except for the region close to the heat source where there is horizontal temperature

gradient as well (Gupta et al., 2009). Cap-plate model has been proposed to satisfy the

thermal stratification inside the water pool, since this water pool is mainly designed to

perform as a suppression pool to cool the steam and air mixture during LOCA in the

reactor vessel.

39

(a) Annular baffle. (b) Annular baffle in the water pool.

(c) Cap-plate baffle. (d) Cap-plate baffle (or shroud) in the water

pool.

Fig.7-10Baffles shape and its position in the water pool.

Liquid sloshing is violent near free surface and the liquid motion at the bottom of the

tank is almost zero. Here, one could assume that the mounting of the cap-plate does not

disturb the liquid sloshing as it is placed at bottom of the tank. The present work also

estimates and compares the annular and cap-plate baffle performance against the liquid

sloshing under the regular excitation. Fig. 7.11 illustrates that the comparison of liquid

elevations for un-baffle, annular baffle and cap-plate baffle cases. As expected, both the

baffle cases reduce the liquid oscillations as well. It is found that cap-plate baffle is more

effective in reducing the sloshing oscillations and sloshing pressure.

Volume = 7.3 m3

Thickness = 200 mmHeight = 250 mm

Volume = 36 m3

Thickness = 200 mmHeight = 5 m

40

Fig.7-11Comparison of liquid elevations with no baffle, annular baffle and cap-plate baffle cases at 1ω =0.312 Hz and amplitude 0.1 m.

Fig.7-12Effect of baffles at right corner of the water pool case at 15.05 sec and 1ω=0.312 Hz.

To elucidate the performance of baffles, at near right corner the liquid height is

captured and shown in Fig. 7.12. The liquid height is captured near the right corner of

water pool at 14.85 sec under the excitation frequency ( 1ω ) of 0.312 Hz. The liquid height

deviation for no baffle case is found at excitation amplitude between 0.01 m and 0.1 m is

0 5 10 15 20-2

-1

0

1

2

No baffle Annular baffle Cap-plate baffle

ζ (m

)

Time (Sec)

0.01 0.02 0.06 0.17.5

8.0

8.5

9.0

9.5

10.0

Line equivalent to design acceleration

Line of mean water level

Line of the top of the tank

Max

imum

Liq

uid

Hei

ght (

m)

Excitation Amplitude (m)

No baffle with FSI No baffle without FSI Annular Baffle with FSI Cap-plate Baffle with FSI

41

around 1.23 m. At the same time, this value for annular baffle and cap - plate baffle is

around 0.563 m and 0.182 m respectively. Moreover, it is found from the numerical

investigation that the liquid from the GDWP will spill out around 0.06 m excitation

amplitude ( ≈0.023 g acceleration) under liquid first mode frequency. The response

spectrum for the structure will give us design acceleration corresponding to first mode

frequency. Here, design acceleration is 0.16g at 0.312 Hz and corresponds to 0.028 m

equivalent harmonic amplitude. This line is shown as vertical in Fig. 7.12 to mark free

surface elevations for all cases. Baffle reduces the liquid slosh height 0.7 m to 0.3 m at

design acceleration as shown in Fig. 7.12. The Fig. 7.13 is drawn for qualitative

comparison between no baffle, annular and cap-plate baffle case. Here, snap shots of

liquid water pool (under regular excitation of 0.1 m amplitude) for different time step has

been shown.

7.4 Liquid Elevation in Higher Modes of GDWP

To study the effect of higher modes, both the annular and cap-plate baffled water pools

studied previously are analysed in this section. When the tanks are subjected to motions at

higher than first mode, the fluid in the tank will tend to undergo sloshing motions under

near to the same tank frequency. At the beginning of the disturbance, the fluid dynamic

pressure is dominated by the impulsive pressure. After few seconds, sloshing pressure or

convective pressure becomes the dominant component pressure. The small oscillations on

the pressure curve are the impulsive pressures. Figs. 7.14 and 7.15 show the free surface

elevation at 5 and 10 times of 1ω excitation frequency respectively for annular and cap-

plate baffle cases. Due to the strong impact forces at beginning, i.e., 0 - 2 seconds, liquid

rise is more, it reaches the steady state around 10 second on wards.

42

(a) Time at 0 sec (e) Time at 0 sec (i) Time at 0 sec

(b) Time at 14.6 sec (f) Time at 14.6 sec (j) Time at 14.6 sec

(c) Time at 15.6 sec (g) Time at 15.6 sec (k) Time at 15.6 sec

(d) Time at 16.6 sec (h) Time at 16.6 sec (l) Time at 16.6 sec

Fig.7-13Comparisons of free surface profile at different time instant for un-baffled and baffled water pool for 0.1 m amplitude sec and 1ω =0.312 Hz. (Figs. (a)-(d), (e)-(h), (i)-(l) show un-

baffled, annular baffled, cap-plate baffled water pools respectively).

43

Fig.7-14Comparison of free surface elevation of liquid at 0.1 m amplitude and 5

1ω excitation frequency.

Fig.7-15Comparison of free surface elevation of liquid at 0.1 m amplitude and

10 1ω excitation frequency.

0 2 4 6 8 10 12 14 16

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

ζ (m

)

Time (Sec)

Annular Baffle Cap-plate baffle

0 2 4 6 8 10 12 14 16

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

ζ (m

)

Time (Sec)

Annular Baffle Cap-plate Baffle

44

8 CFD SIMULATION OF SLOSHING INGDWP UNDER SEISMIC

EXCITATION

In this chapter, a random wave is created using time histories compatible to design floor

response spectrum (FRS). For that, the floor response spectrum (FRS) at 137m height of

reactor building is used to generate the acceleration time history. A User-Defined Function

(UDF) subroutine has been developed to apply the random acceleration as a volume force.

The slosh height and forces on tank wall have been calculated for different excitation

directions. The combined effects in longitudinal and lateral directions are studied.

8.1 Introduction The sloshing studies are usually performed to ensure the safety of plants and to avoid

consequences of any seismic induced accidents.The mechanical mass–spring model

(Chapter 4) based on linear theory is generally employed by design standards to predict

the free surface displacement as well as other seismic design parameters of the liquid

storage tanks. However, nonlinear effects are always present and they occasionally

dominate the sloshing response. These nonlinear slosh effects arise mostly as a

consequence of large wave amplitudes (Chapter 7). Large amplitude waves may appear

when the great earthquakes are accompanied withpretty long period (3 to 10 sec)

components of seismic wave which coincide with the primary natural period of the

contained liquid (Goudarzia and Sabbagh-Yazdi, 2012). In this chapter, the floor response

spectrum (FRS) at 137m height of AHWR building is used to generate the acceleration

time history. This simulated earthquake accelearation data is used to study the sloshing

beheavior in GDWP.

The need to include nonlinearity in the hydrodynamics of the tank–liquid system arises

whenever high amplitude sloshing waves form on the liquid surface, leading to a nonlinear

influence of sloshing wave on the dynamic response of a tank. To estimate the non-linear

sloshing, the fluid momentum equations are solved as discussed in chapters 2 and 3. To

45

estimate the frequencies and slosh height in each sector of GDWP under seismic

excitation through CFD, three sectors are modelled among eight-sectored water pool as

depicted in Fig. 8.1. In each sector, three domains are modelled viz., water pool wall,

liquid and air domains. The sketch of the water pool are depicted as in Fig. 8.1.

8.2 Development of Random Waves

To study the response of GDWP under seismic load a time history is generated from FRS

along three orthogonal directions separately. FRS for AHWR building at 137 m is shown

in Fig. 8.2. The 5% broaden spectrum is generated from FRS. Acceleration time history is

generated using SIMQKE code [33] in three directions separately. These graphs are shown

in Fig. 8.3. A user subroutine function is developed to call the random acceleration data

and applied on the all fluid in terms of gravity force. Implicit pressure and implicit shear

stress conditions have been applied on the fluid solid interfaces. Air at top is at fixed

pressure condition (at atmosphere condition). The free surface elevation has been captured

every 0.005 sec.

8.3 Wavelet Analysis

In the past, the wavelet transform has been used to detect the frequencies at different

regions. In the field of fluid mechanics the wavelet analysis has been used to detect the

multi stable flow regions. In this problem, wavelet tool is used to find the sloshing

frequencies information. The Fourier transforms provide the spectral coefficients which

are independent of time i.e. they can give the amplitude-frequency information and donot

have any information about frequency with respect to time. Thus, it is useful only for a

stationary signal where the amplitude-frequency does not change with time. But, in real

life cases the signals are time dependent and also non-stationary. In such cases a scan

analysis using the Short Term Fourier Transform (STFT) is used but it has its limitations

like it can give information only about the amplitude and frequency, but not anything

about the time and frequency relation. The limitations of STFT are overcome by the

46

wavelet transform which gives a better idea about the time-frequency information about

the signals. The wavelet transform is a linear convolution of a given one dimensional

signal which is to be analysed and the mother wavelet (t). Mathematically a wavelet

transform is as shown below:

( ) dts

bttps

bsW ⎟⎠⎞

⎜⎝⎛ −

∫= *1),( ψ (8.1)

where, W(s, b) is the wavelet coefficient, the asterisk sign denotes the complex conjugate,

‘b’ is the translation parameter and ‘s’ is the scale parameter.There is a number of mother

wavelet which is used in practise but only some of the mother wavelets such as Mexican

hat wavelet, Gabor wavelet and Morlet wavelet are used in the field of fluid dynamics.

Wavelet has been used which is given by,

2/)/(4

12

)( γωωπψ tti oo eet −−= (8.2)

Where 2 2⁄ and is the number of wave in the wavelets. In practise the

value of varies from 5 to 12 and generally it is taken as 6. A frequency resolution of 12

is chosen when frequency of resolution of a signal is more important than time resolution.

(a) Plan (b) Isometric view

Fig. 8-1 Three sectors in GDWP

Sector 1

Sector 2

Sector3

Excitation direction

47

Table 8-1:Sectors frequencies in hertz computed by CFD simulations

Mode

number Sectors 1 and 5 Sectors 2,4, 6 and 8 Sectors 3 and 7

1 0.262 0.231 0.247

2 0.332 0.332 0.394

3 0.4738 0.460 0.447

4 0.532 0.542 0.548

Table 8-2: Numerical case studies

Case Excitation

Sectors

considered in

GDWP tank

Excitation

direction

Excitation

Amplitude

(m)

Excitation

Frequency

(Hz)

Condition

1 Sinusoidal

Excitation 3 xyz 0.03 First mode

Flexible

wall

2 Random

excitation 3 xyz 137m FRS data

Flexible

wall

(a) Longitudinal (x) (b) Vertical (y) (c) Lateral (z)

Fig.8-2FRS data for AHWR building at 137 m

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.000.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

12.00

13.00

14.00

15.00

Floor Response Spectra at GDWP SLAB LEVELEL. 137.00 m (Node 19) in X-Direction

ζ=1% ζ=2% ζ=4%

S a /g

Frequency (Hz)

0.00 10.00 20.00 30.00 40.00 50.000.000.501.001.502.002.503.003.504.004.505.005.506.006.507.007.508.008.50

Floor Response Spectra at GDWP SLAB LEVELEL. 137.00 m (Node 19) in Y-Direction

ζ=1% ζ=2% ζ=4%

Sa /

g

Frequency (Hz)

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.000.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

Floor Response Spectra at GDWP SLAB LEVELEL. 137.00 m (Node 19) in Z - Direction

ζ=1% ζ=2% ζ=4%

S a /g

Frequency (Hz)

48

(a) Longitudinal (x) (b) Vertical (y) (c) Lateral (z)

Fig. 8-3 Acceleration- time history for GDWP of AHWR building at 137 m

8.4 GDWP under Sinusoidal Excitation in Multi-Direction (Case 1):

Table 8.1 shows the sectors frequencies in hertz computed by CFD simulations. The first

mode frequency is taken for sinusoidal excitation study. Table 8.2 shows the details of

case studies considered to understand the sloshing behaviour in GDWP under harmonic

and random excitations. First, GDWP is excited with harmonic loading having frequency

equal first mode natural frequency in three directions (x, y, z). Non-dimensional slosh

height is obtained and shown in Figs. 8.4 (a) through (c). The phase-plane diagram for

regular excitation is presented in Fig 8.4 (d). As it is the horizontal sinusoidal excitation

case, single frequency is observed with respect to time as depicted in Fig 8.4 (e) which is

obtained from wavelet analysis.

‐0.8

‐0.6

‐0.4

‐0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25

Acceleration (g)

Time (Sec)

‐0.6

‐0.4

‐0.2

0

0.2

0.4

0.6

0 5 10 15 20 25

Acceleration (g)

Time (sec)

‐0.8

‐0.6

‐0.4

‐0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25

Acceleration (g)

Time (Sec)

49

Fig. 8-4 Multi directional sinusoidal excitation (Case 1) (a) –(c) Non-dimensional slosh

height for sector 1 to 3 (d) Phase-plane diagram (e)Wavelet diagram for slosh height signal at right corner of sector 1

8.5GDWP under Random Excitation in Multi-Direction (Case 2)

Slosh height is computed in three directional excitation, viz., two horizontal and one

vertical directions. Non-dimensional slosh height for sector 1- 3 is captured and shown in

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26-12

-8

-4

0

4

8

12

(a)

Sector 1 Right sector 1 center

Non

-dim

ensi

onal

slo

sh h

eigh

t

Time(Sec)-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26

-12

-8

-4

0

4

8

12

(d)(c)

(b) Sector 2 Right sector 2 center

Non

-dim

ensi

onal

slo

sh h

eigh

t

Time(Sec)

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26-12

-8

-4

0

4

8

12

Sector 3 Right sector 3 center

Non

-dim

ensi

onal

slo

sh h

eigh

t

Time(Sec)-15 -10 -5 0 5 10 15

-20

-15

-10

-5

0

5

10

15

20

dζ/d

t / (A

ω2 )

ζ/A

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Sec)

Freq

uenc

y (H

z)

(e)

50

Fig. 8.5. It is found from Fig. 8.5,slosh height at sector 1 is more than other two sectors

(i.e., sectors 2 & 3). The FFT is computed from the steady state signal. FFT of slosh

height varying with time for sector 1 through 3 is shown in Figs. 8.6 to 8.8. Dominant

frequency is near to first mode frequency (i.e., 0.312 hz). During random excitation, the

frequency for the each sector is varies with respect to the length of the free surface. Fig 8.9

shows the phase plane diagram for sector1. The phase plane diagram is visualize the

presences of nonlinearity in the curve.

Fig. 8-5 Non-dimensional slosh height for sector 1 to 3 of the GDWP subjected to

design excitation as given in Fig. 8.3 (Case 2)

0 5 10 15 20 25 30 35 40-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5 Inner Right Inner Center

Time (Sec)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Inner Right Inner Center

Non-

dim

ensi

onal

Slo

sh h

eigh

t

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Sector 2

Sector 3

Sector 1

Inner Right Inner Center

51

Fig. 8-6 FFT from slosh height at sector 1 of GDWPsubjected to design excitation as

given in Fig. 8.3 (Case 2)

Fig. 8-7 FFT from slosh height for sector 2 of GDWP subjected to design excitation as

given in Fig. 8.3 (Case 2)

Fig. 8-8 FFT from slosh height for sector 3

of GDWP subjected to design excitation as

given in Fig. 8.3 (Case 2)

Fig. 8-9 Phase-Plane diagram of sector 1

slosh height

The slosh height signals are analysed to get the frequency information along time using

wavelet analysis. Figs. 8.10 through 8.12 show the wavelet analysis in sector1, sector 2

and sector 3 respectively during random excitation. Due to this random excitation, the

sector 1 through 3 have the multi-frequencies data. These frequencies are mainly first few

sloshing frequencies of GDWP in three directions. Slosh height is found more at the tank

wall corners of direction excitation.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.000

0.002

0.004

0.006

0.008

Frequency (Hz)

Ampl

itude

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.01

0.02

0.03

0.04

Frequency (Hz)

Ampl

itude

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00

0.01

0.02

0.03

0.04

Frequency (Hz)

Ampl

itude

-3 -2 -1 0 1 2 3-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(dζ

/ d t)

/ (A

/ ω

1)

ζ / A

52

Fig. 8-10 Time –frequency curve for sector 1 of GDWP slosh height subjected to design excitation as given in Fig. 8.3 (Case 2)

Fig.8-11 Time –frequency curve for sector 2 of GDWP slosh heightsubjected to design

excitation as given in Fig. 8.3 (Case 2)

Fig.8-12 Time –frequency curve for sector 3 of GDWP slosh heightsubjected to design excitation as given in Fig. 8.3 (Case 2)

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Sec)

Freq

uenc

y (H

z)

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Sec)

Freq

uenc

y (H

z)

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Sec)

Freq

uenc

y (H

z)

53

Fig. 8-13 Base shear at convective mode for sector 1 through 3

Base shear is the estimate of the maximum expected lateral force that will occur due to

seismic ground motion at the base of the structure. An accurate prediction of the base

shear and moment is essential to ensure the safety of the tanks against the shell bucking

and uplift. Base shear at sector 1, 2 and 3 are shown in Figs. 8. 13 (a) through (c). The sum

of convective and impulsive forces (pressure and shear force) on tank wall is shown in

Fig. 8.14. The average total force in longitudinal and lateral directions are 1.02 MN and

2.25 MN respectively. Pressure moment and viscous moment also calculated at the bottom

of the tank and depicted in Fig. 8.15. Maximum force and moment values for sector 1

through 3 are also shown in Table 8.3. Due to the liquid load during its oscillation, the

(a)

(b)

(c)

54

wall deflection is also varying with respect to time. Wall deflection due to liquid sloshing

for different time has been depicted in Fig. 8.11. Finally, Fig. 8.17 shows the liquid

pressure near wall at free surface and base. The dynamic pressure is computed as 2.7

KN/m2 and 7 KN/m2 respectively.

Fig. 8-14 Total force (convective + impulsive) at sector wall 1

Table 8-3: Total force and moment in sectors 1, 2 and 3

Condition Sector 1 Sector 2 Sector 3

Total force at GDWP in MN

(convective + impulsive)

Maximum 13.2 13.4 15.7

Minimum 7 6.69 4.2

Average 9.91 9.9 9.8

Total momentin bottom of

GDWPMN-m

(convective + impulsive)

Maximum 205.2 207.8 241

Minimum 65 64 54

Average 146 146 146

(a)

(b)

55

Fig. 8-15 Total moment (convective + impulsive) at sector wall 1

Fig.8-16Tank wall displacement versus time

0 5 10 15 20 250

1x107

2x107

3x107

4x107

5x107

6x107

7x107

8x107

My

Mom

ent a

t tan

k bo

ttom

due

to s

ecto

r1 fo

rce

Time(Sec)

‐4

‐2

0

2

4

6

8

10

0 0.0005 0.001 0.0015 0.002

Vertical len

gth in m

Inner wall displacement in m

0 Sec 0.05 Sec 0.25 sec

56

Fig. 8-17 Wall pressure (convective + impulsive) at sector 1 through 3

The convective frequency, slosh height, dynamic wall pressure near free surface, and

impulsive and convective (total) pressure on wall near base are compared between CFD

and ERM approaches for fully curved outer wall GDWP and partially curved outer wall

GDWP separately in chapter 10.

0 5 10 15 20 25 30 3530

40

50

60

70

80

Sector 1 Sector 2 Sector 3

Wal

l pre

ssur

e ne

ar b

ase

KN/m

2

Time (Sec)

0 5 10 15 20 25 30 350

2

4

6

8

(b)

(a)

Maximum wave amplitute is 17 KN/m2

Maximum wave amplitute is 2.7 KN/m2

Sector 1 Sector 2 Sector 3

Wal

l pre

ssur

e ne

ar fr

ee s

urfa

ce K

N/m

2

57

9 CONCLUSIONS

This work focused on the numerical investigation of liquid sloshing in GDWP of AHWR. The

sloshing behaviour in an un-baffled and baffled GDWP of AHWR are studied under sinusoidal

excitation. Effect of baffles in the annular water pool is studied for different excitation and

amplitude. If liquid oscillation is not controlled efficiently, sloshing of liquids in storage water

pools may lead to water spilled out from water pool or large dynamic stress to cause structural

failure. Hence, the study of sloshing and measures to suppress it are well justified with two types

of baffles for this kind of annular water pools. To estimate the slosh height and other design

parameters sloshing in GDWP is studied using simulated seismic load along the three orthogonal

directions.

From the above numerical investigation, the following observations are made.

Maximum slosh height under design seismic load is found around 0.62 m for fully curved

outer wall GDWP (existing model) and 0.72 m partially curved outer wall GDWP. It is

also observed that GDWP has sufficient free board.

X and Y directions of convective frequencies for GDWP are 0.265 hz and 0.315 hz for

fully curved wall and 0.231hz and 0.262hzfor partially curved wall of GDWP. It is also

noted that the convective frequencies are varies with respect to the direction of excitation.

Seismic design codes (through ERM approach) for tanks and CFD results are compared.

It is found that ERM has around 25% difference in slosh height from CFD results.

Cap-plate baffle is more effective than annular in reducing the sloshing oscillations.

9.1 General Conclusions from above Study

1) The liquid free surface elevation has been captured for different excitations and different

amplitudes. Since the liquid first mode frequency is very less than the structure first mode

58

frequency, the fluid-structure interaction effect is found to be negligible. As a result, the fluid

pressure on the free surface is dominated by the sloshing pressure or convective pressure and

the relative or fluid-structure interaction pressures are negligibly small for the case of first

mode excitation.

2) When the water pools are subjected to higher modes (i.e., higher than first mode excitation),

the fluid in the water pool will tend to undergo sloshing motions. In the higher modes the

liquid elevation is lower than the first mode frequency. At the beginning of the disturbance,

the fluid dynamic pressure is dominated by the impulsive pressure. After few seconds,

sloshing pressure or the convective pressure becomes the dominant component of pressure.

3) Experimental and analytical studies are also performed to validate present numerical results.

For experimental validation, a simple square and four-sectored squared tanks was taken. The

pressure variations were captured at different locations under the surge motions of the tank

and found the CFD results are good in agreement with the experiment.

4) It is found from the numerical investigation that the liquid will spill out around 0.06 m

excitation amplitude ( ≈0.023g acceleration) from the fully curved wall GDWP under first

mode sloshing frequency. However, design acceleration is 0.16g at 0.312Hz and corresponds

to 0.028 m equivalent harmonic amplitude. At this amplitude, liquid slosh height is found 0.7

m.

5) Further, the tank with different baffles was studied. As expected, annular baffle and cap-plate

baffle cases were reducing the liquid oscillations as well. However, cap-plate baffle was more

effective in reducing the sloshing oscillations for this kind of water pool geometry. This baffle

was reduced the liquid slosh height from 0.7 m to 0.3 m under design acceleration.

6) During random excitation, frequency for the each sector is varies with respect to the length of

the free surface. At the end of the seismic force, free vibration shows the natural frequency of

the tank. So that, maximum slosh height was found at the end of the seismic force. It is also

observed that more slosh height found at tank wall corners in the direction of excitation.

59

9.2Some more Observations between ERMand CFD

GDWP is an annular 8-sectored tank as depicted in Fig. 1.2. The outer wall of GDWP is

spherical in shape while the inner one is in cylindrical shape. Since seismic codes are not

available for these kind of sectored geometries, here CFD and ERM approaches are used to

estimate the sloshing behaviour in GDWP. For the effective comparison, two GDWP models

have been chosen as presented in Figs. 4.1 and 4.2. They are as follows,

(i) Fully curved outer wall GDWP (existing model)

(ii) Partially curved outer wall GDWP (proposed model).

The partially curved wall model is proposed for thermal hydraulic issues (in AHWR review

meeting 201). Slosh height and dynamic pressure is obtained through CFD for both GDWP

models. CFD results are compared with corresponding ERM as shown in Tables 9.1 and 9.2,

while making an equivalent rectangle the following assumptions are considered.

(i) Liquid volume should be equal in GDWP sector and rectangular model.

(ii) Ratio between liquid height (h) and length of the tank (L) is taken almost equal in

both cases.

60

Fig.9-1Comparison between CFD and ERM for fully curved outer wall GDWP.

Fig.9-2Comparison between CFD and ERM for partially curved outer wall GDWP.

0.258 0.265

0.7780.856

0.312 0.315

0.6150.692

00.10.20.30.40.50.60.70.80.9

x y x y

Convective frequency in Hz Slosh height in m

Fully curved outer wall GDWP

ERM CFD

0.227 0.239

0.893 0.953

0.262 0.27

0.721 0.782

0

0.2

0.4

0.6

0.8

1

1.2

x y x y

Convective frequency in Hz Slosh height in m

Partially curved outer wall GDWP

ERM CFD

61

Tabl

e 9-

1:X

and

Y d

irec

tion

val

ues

of s

ecto

r 1

of f

ully

cur

ved

wal

l GD

WP.

D

irec

tion

Equ

ival

ent

rect

angl

e m

etho

d

(ER

M)

CF

D

Dif

fere

nce

in %

& C

omm

ents

|(CF

D-E

RM

)|/C

FD

Geo

met

ry

(Equ

ival

ent r

ecta

ngle

vs

full

y cu

rved

wal

l)

Slo

pe w

all w

ill s

uppr

ess

the

slos

hing

.

Liq

uid

volu

me

in m

3

1012

10

12

Con

vect

ive

freq

uenc

y in

Hz

x 0.

258

0.31

2 17

.3

y 0.

265

0.31

5 15

.8

Slo

sh h

eigh

t in

m

x 0.

778

0.61

5 26

.5

y 0.

856

0.69

2 19

.1

Dyn

amic

wal

l pre

ssur

e

near

fre

e su

rfac

e in

KN

/m2

(Con

vect

ive)

x 6.

11

4.2

45.4

y 6.

52

4.25

53

.4

Tot

al p

ress

ure

on w

all i

n

KN

/m2

(Im

puls

ive

+ C

onve

ctiv

e)

x 63

.89+

6.11

=70

.0

58.2

20

.2

y 10

6.15

+6.

52

=16

6.67

54

.8

204

8 m

62

Tabl

e 9-

2: X

and

Y d

irec

tion

val

ues

of s

ecto

r 1

of p

arti

ally

cur

ved

wal

l GD

WP

.

D

irec

tion

Equ

ival

ent

rect

angl

e m

etho

d

(ER

M)

CF

D

Dif

fere

nce

in %

& C

omm

ents

|(CF

D-E

RM

)|/C

FD

Geo

met

ry

(Equ

ival

ent r

ecta

ngle

vs

part

ially

cur

ved

wal

l)

Slo

pe r

egio

n is

loca

ted

near

the

free

sur

face

. It

wil

l sup

pres

s th

e

slos

hing

.

Liq

uid

volu

me

in m

3

1012

10

12

Con

vect

ive

freq

uenc

y in

Hz

x 0.

227

0.26

2 13

.3

y 0.

239

0.27

11

.4

Slo

sh h

eigh

t in

m

x 0.

893

0.72

1 23

.8

y 0.

953

0.78

2 18

.36

Dyn

amic

wal

l pre

ssur

e

near

fre

e su

rfac

e in

KN

/m2

(Con

vect

ive)

x 3.

32

2.7

5.6

y 3.

1 2.

5 51

.9

Tot

al p

ress

ure

on w

all i

n

KN

/m2

(Im

puls

ive

+ C

onve

ctiv

e)

x 54

.33+

7.17

=61

.5

74

16.8

y 10

4.29

+7.

41

=11

1.7

69

61.8

6m

4 m

6m

63

From the Fig. 9.1 and 9.2 and Tables 9.1 and 9.2 show the difference between the

ERM and CFD slosh analysis results for fully curved outer wall GDWP and partially

curved outer wall GDWP respectively. For comparison between ERM and CFD, the

convective frequency, slosh height, pressure are taken. While comparing the fully and

partially curved wall GDWP cases, the partially curved wall has very close to the CFD

results. Since length of the free surface is become very less in the case of fully curved

outer wall, the rectangular assumption leads slightly more error in these slosh results.

However, the partially curved wall has close relation between ERM and CFD except the

pressure in y direction.

Acknowledgement

Authors express their sincere gratitude and graceful acknowledgement to our colleagues

Shri. Ravi Kiran, Shri. PN Dubey, Dr. Parulekar and Mr. Piyanshu Goyal, (Scientific

Officers, Reactor Safety Division) and we sincerely thankDr. RK Singh (AD, RDDG &

Head, RSD) forhis time, careful work and valuable comments and suggestions on the

report.

64

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(www.iaea.org/INPRO/CPs/AWCR/3rd_Meeting/Indian-3-1.pdf ).

[38] Vorobyev, A., Kriventsev, V., Maschek, W. (2011), “Simulation of central sloshing

experiments with smoothed particle hydrodynamics (SPH) method”, Nuclear Engineering

and Design, 241, 3086– 3096.

[39] Xue, M.A., Zheng, J., Lin, P. (2012), “Numerical simulation of sloshing phenomena in cubic

tank with multiple baffles”, Journal of Applied Mathematics, Article ID 245702.

69

APPENDIX- A

NON-LINEARITY EFFECT DUE TO WAVE BREAKING

The nonlinear nature of sloshing is the greatest hindrance in solving such a problem analytically

and even computationally. Sloshing flows are sturdily nonlinear in most cases. Wave breaking,

impact, and splashes can be observed in violent sloshing flows. In general, the nonlinearity of

sloshing flow becomes dominant at any of the following situations.

(i) Tank under angular excitations (ex. Ship cargo)

(ii) Tank with wave breaking device/mechanism. (TLD, passive baffle)

(iii) Tank with lower filling conditions (Shallow depth)

(iv) External unusual loads (Thermal loads, Suppression of high velocity steam)

The slosh-induced impact pressure is affected by wave breaking phenomena, therefore the

observation and understanding of physical phenomena on these flows are important to develop

the proper models. This phenomena dissipate part of the slosh energy induced by earthquakes. It

is used as vibration control mechanism in Tuned Liquid Damper (TLD). The fundamental

frequency of the fluid in the TLD should be close to the natural frequency of the structure if the

TLD is to dissipate energy efficiently. The study on dissipation induced by a free surface flow is

arduous, especially in the presence of a wave breaking flow. The local wave breaking is not

likely to disturb the global motion of sloshing flow (i.e., the local splashes and wave breaking

may be ignored when global fluid motion is concerned).

To study the wave braking, a dominant nonlinearity condition needs to be made. As

discussed in above paragraph, all the situations can be omitted except second one for seismic

studies (i.e. Tank with wave breaking device/mechanism). Here, three case studies are shown to

demonstrate the nonlinearity effects as indicated below. Simple rectangular tank is taken for this

study. Tank dimensions are 10.8 m length and water fill height is 8.4 m. Water is used as tank

fluid. For, simulation, 0.1 m is used as an excitation amplitude. Since the non-linearity need to be

captured from simulation, around 40,000 elements have been created in the computational

domain. VOF technique is used to track the interface between liquids.

70

Fig. A-1 Slosh height elevation computed from linear and nonlinear equations

(a) case1 (b) case2 (c) case3

Fig. A-2 Snap shot of maximum slosh height at left end (a- linear violent flow; b- non-linear normal flow at 0.5 ; c- Non-linear violent

flow at 0.99 )

(a) case1 (b) case2 (c) case3

Fig.A-3 Snap shot of maximum slosh height at right end (a- linear violent flow; b- non-linear normal flow at 0.5 ; c- Non-linear violent

flow at 0.99 )

‐3

‐2

‐1

0

1

2

3

0 5 10 15 20 25 30

slos

h he

igh

in m

Time in Sec

Linear violent flow (case 1) Non-linear violent flow (case 3)

Non-linear normal flow (case 2)

To see t

(i) Cas

freq

(ii) Cas

freq

(iii) Cas

freq

(iv) Cas

Fig. A-

flows. T

the snap

A-1. Ta

Tank w

Fig. A5

baffle s

Fig.A-

the slosh be

se 1 – Linea

quency)

se 2 – Non-

quency)

se 3- Non-li

quency)

se 4 -Tank w

-1 shows th

There is no

p shots of li

ank with Wa

with passive

5 respective

ize is 1 m le

-4 Tank with

ehaviour du

ar equations

-linear equa

inear equati

with wave b

hat slosh h

much non-

inear and no

ave Breakin

baffle com

ely. To capt

ength and 0

h wave brea

e to linear a

s and high e

ations and lo

ions and hig

breaking de

height captu

linearity is

on-linear eq

ng Device (P

mputational d

ture the no

0.08 m thick

aking devic

71

and nonline

excitation fr

ow excitatio

gh excitatio

evice (Passiv

ured from l

found in th

quations slos

Passive baff

domain and

n-linearity

k. Baffles ar

e (Case 3)

ar equation

requency (A

on frequency

n frequency

ve baffle)

linear and

his simple st

sh profile.

fle)

d grid arran

a very fine

e fixed at 1

Fig.A-5 F

s are solved

Around 100

y (Around 5

y (Around 1

non-linear

tructures. Fi

ngement are

e mesh is cr

.6 m from li

Fine mesh a

d with four c

% of natura

50 % of natu

100 % of na

equation fr

igs. A-2 and

shown in F

reated near

iquid free su

arrangemen

cases.

al

ural

atural

rom violent

d A-3 show

Fig. A4 and

r walls. The

urface.

nt (Case 3)

t

w

d

e

72

The passive baffles are used to dissipate the sloshing motion energy by breaking a main sloshing

flow into several weaker sub-streams to induce the non-linearity. Fig.A-6 shows the snapshots of

non-linear free surface profiles. This wave breaking causes due to the presence of the passive

baffles.

Fig. A -6Snap shot of Non-linear violent flow at 0.99 with baffle arrangement (case 4)

A-2 Discussion andConclusions

Sloshing flows are sturdily non-linear in most cases. Wave breaking, impact, and splashes can be

observed in violent sloshing flows. However, non-linearity will occur due to any one or

combined, situations such as tank under angular excitations, tank with wave breaking

device/mechanism, tank with lower filling conditions, external unusual loads, etc. The local

wave breaking is not likely to disturb the global motion of sloshing flow. So that, the local

splashes and wave breaking may be ignored when global fluid motion is concerned. During the

non-linear violent slosh flow at 0.99 , some local wave breaking is observed, but however, it

is insignificant, compare to global motions of liquid. It is also observed that the slosh height from

linear and non-linear formulations is around 2% difference near trough and crest of the wave.

Tank with baffles case, a high non-linear slosh flow is found near corner of the tank. Baffle

dissipates sloshing energy by breaking a main sloshing flow into several weaker sub-streams.