FLUID-STRUCTURE INTERACTION
EFFECTS OF SLOSHING IN LIQUID-CONTAINING STRUCTURES
KTH Royal Institute of Technology
ABSTRACT
This report presents the work done within the framework of my master thesis in the program Infrastructure Engineering at KTH Royal Institute of Technology, Stockholm. This project has been proposed and sponsored by the French company Setec TPI, part of the Setec group, located in Paris.
The overall goal of this study is to investigate fluid-structure interaction1 and particularly sloshing in liquid-containing structures subjected to seismic or other dynamic action. After a brief introduction, the report is composed of three main chapters. The first one presents and explains fluid-structure interaction equations. Fluid-structure interaction problems obey a general flow equation and several boundary conditions, given some basic assumptions. The purpose of the two following chapters is to solve the corresponding system of equations. The first approach proposes an analytical solution: the problem is solved for 2D rectangular tanks. Different models are considered and compared in order to analyze and describe sloshing phenomenon. Liquid can be decomposed in two parts: the lower part that moves in unison with the structure is modeled as an impulsive added mass; the upper part that sloshes is modeled as a convective added mass. Each of these two added mass creates hydrodynamic pressures and simple formulas are given in order to compute them. The second approach proposes a numerical solution: the goal is to be able to solve the problem for any kind of geometry. The differential problem is resolved using a singularity method and Gauss functions. It is stated as a boundary integral equation and solved by means of the Boundary Element Method. The linear system obtained is then implemented on Matlab. Scripts and results are presented. Matlab programs are run to solve fluid-structure interaction problems in the case of rectangular tanks: the results concur with the analytical solution which justifies the numerical solution.
This report gives a good introduction to sloshing phenomenon and gathers several analytical solutions found in the literature. Besides, it provides a Matlab program able to model effects of sloshing in any liquid-containing structures.
Keywords: fluid-structure interaction, sloshing, tank, seismic action, hydrodynamic pressure, velocity potential, boundary element method, Housner, resonant sloshing
Author’s note: This version of the thesis, edited to be published on KTH Database, does not include the appendices and Matlab scripts.
TABLE OF CONTENT
1 Introduction ... 7 2 Fluid-structure interaction ... 8 2.1 Introduction ... 8 2.2 Fluid equations ... 8 2.2.1 Local equations ... 8 2.2.2 Velocity potential ... 9 2.3 Boundary conditions ... 102.3.1 Fluid-structure interface condition ... 11
2.3.2 Free surface condition ... 11
2.3.3 Fluid-ground interface condition ... 12
2.3.4 Inifite boundary condition ... 12
2.4 Conclusion ... 13
3 Analytical solution for 2D rectangular tanks ... 14
3.1 Introduction ... 14
3.2 Problem presentation ... 15
3.2.1 Model and assumptions ... 15
3.2.2 Sloshing equations ... 16
3.3 Study of sloshing: time-history analysis ... 17
3.3.1 Free oscillations ... 17
3.3.2 Forced displacement ... 19
3.4 Resolution of the sloshing problem: frequency domain analysis ... 20
3.4.1 Steady state response: Graham & Rodriguez’s method ... 20
3.4.1.1 Dynamic pressures acting on the tank walls ... 21
3.4.1.2 Hydrodynamic forces ... 22
3.4.2.1 Hydrodynamic pressures acting on the tank walls ... 24
3.4.2.1.1 Hydrodynamic impulsive pressure ... 26
3.4.2.1.2 Hydrodynamic convective pressure ... 28
3.4.2.1.3 Interpretation ... 31
3.4.2.2 Hydrodynamic pressure forces ... 31
3.4.2.2.1 Hydrodynamic impulsive pressure force ... 31
3.4.2.2.2 Hydrodynamic convective pressure force ... 32
3.4.2.3 Conclusion ... 34
3.4.3 Housner’s approached method ... 35
3.4.3.1 Impuslive part ... 36
3.4.3.2 Convective part... 37
3.4.3.3 Model ... 38
3.4.4 Conclusion ... 38
3.5 Study of resonant sloshing ... 40
3.5.1 Resonant amplification ... 40
3.5.2 Frequency domain analysis ... 45
3.5.3 Tuned sloshing dampers ... 48
3.5.4 Conclusion ... 48
3.6 Conclusion ... 50
4 Numerical solution for sloshing problems ... 51
4.3.2 Solution: Boundary Element Method ... 55
4.3.2.1 Green’s function for 2D problems ... 56
4.3.2.2 Green’s function for 3D problems ... 56
4.4 Examples and results ... 57
TABLE OF FIGURES
Figure 1: typical fluid-structure interaction problem ... 10
Figure 2: rectangular tank fixed to the ground ... 15
Figure 3: rectangular tank fixed to the ground ... 20
Figure 4: Graham & Rodriguez model – impulsive mass ... 23
Figure 5: Graham & Rodriguez's method - Impulsive mass ratio ... 23
Figure 6: Hunt & Priestley's method - impulsive pressure at the bottom of the tank ... 26
Figure 7: EN 1998-4 Figure A.6 - impulsive pressure at the bottom of the tank ... 27
Figure 8: Hunt & Priestley's method - Impulsive pressure ... 27
Figure 9: EN 1998-4 Figure A.7 - impulsive pressure ... 28
Figure 10: Hunt & Priestley's method - Convective pressure, sloshing mode 1 ... 29
Figure 11: Hunt & Priestley's method - Convective pressure, sloshing mode 2 ... 29
Figure 12: E EN 1998-4 Figure A.7 - convective pressure, sloshing mode 1 ... 30
Figure 13: EN 1998-4 Figure A.7 - convective pressure, sloshing mode 2 ... 30
Figure 14: Hunt & Priestley's method - Impulsive mass ratio for 1, 2, 3 and 10 sloshing modes ... 32
Figure 15: Hunt & Priestley's method – Convective mass ratio for the sloshing mode 1, 2 and 3 ... 33
Figure 16: Hunt & Priestley model – impulsive mass and convective mass (2 sloshing modes) ... 34
Figure 17: fundamental natural sloshing period for rectangular tanks ... 35
Figure 18: Housner's method - lamina fluid theory ... 36
Figure 19: Housner's method - Impulsive mass ratio, comparison with Graham & Rodriguez’s method ... 36
Figure 20: Housner's method - Convective mass ratio, comparison with Hunt & Priestley's method ... 37
Figure 21: Housner's model - Impulsive and convective mass ... 38
Figure 22: resonant sloshing ... 41
Figure 23: impulsive added mass for high frequencies as a function of L/h ... 42
Figure 24: resonant amplification coefficient as a function of L/h ... 43
Figure 26: natural sloshing frequency for different values of L ... 44
Figure 27: resonant amplification ... 45
Figure 28: time-history recording – ground acceleration ... 45
Figure 29: frequency domain description – power spectral density ... 46
Figure 30: rectangular tank fixed to the ground ... 46
Figure 31: amplitude spectrum of the earthquake (blue) and the transfer function (green) ... 47
Figure 32: inertia force caused by the seismic action ... 49
Figure 33: example of non-rectangular 2D tank ... 51
Figure 34: problem geometry ... 52
Figure 35: numerical and analytical added mass for 2D rectangular tanks ... 57
Figure 36: pressure distribution - vertical action ... 58
Figure 37: pressure distribution - horizontal action ... 58
Figure 38: pressure factor comparison of numerical and analytical solutions ... 59
Figure 39: horizontal added mass as a function of frequency ... 59
Figure 40: resonant sloshing for numerical and analytical solutions ... 60
Figure 41: slosh2D convergence test ... 60
Figure 42: pressure distribution for a 2D circular pipe ... 61
Figure 43: pressure factor for 2D rectangular tank - Eurocode and numerical solution ... 62
Figure 44: pressure acting on the walls for horizontal oscillation along x ... 63
Figure 45: pressure acting on the walls for horizontal oscillation along y ... 63
Figure 46: pressure acting on the walls for vertical oscillation ... 64
Figure 47: added mass as a function of frequency for oscillation along x ... 64
Figure 48: added mass as a function of frequency for oscillation along y ... 65
Figure 49: slosh3D convergence test ... 65
Figure 50: external problem geometry ... 66
1 INTRODUCTION
When subjected to external excitation like earthquake, liquid-containing structures are challenging to design due to sloshing effects. Indeed, fluid-structure interaction is the source of free surface fluctuation and hydrodynamic pressure loads that can cause unexpected instability or even failure of these structures. Thus, it is paramount to carry out a thorough investigation about sloshing phenomenon. The main purpose of this report is to characterize the dynamic response of liquid storage tanks.
In the first chapter of this study, equations that rule fluid-structure interaction are derived. The system to solve can be summarized to a differential problem with boundary conditions, often called Dirichlet problem. Different assumptions are discussed in order to obtain a simple model of the fluid behavior. Both analytical and numerical methods are then proposed in order to solve this problem.
The first approach proposes an analytical solution: the problem is solved for 2D rectangular tanks. Different models are considered and compared in order to analyze and describe sloshing phenomenon. This study is based on the results found in the literature dealing with this subject. It is interesting to focus on rectangular tanks because they represent the major part of practical cases. Besides, an analytical solution often provides a good understanding of the general aspects ruling the phenomenon. A detailed investigation of resonant sloshing is also conducted in this part.
The second approach proposes a numerical solution: the goal is to be able to solve the problem for any kind of geometry. The differential problem is resolved using a singularity method and Gauss functions. It is stated as a boundary integral equation and solved by means of the Boundary Element Method. Analytical and numerical solutions are then compared and performance is discussed.
The whole study assumes that the mass of liquid is lumped on the wall, based on rigid wall boundary condition in the calculation of hydrodynamic pressures. This assumption is widely used in practice, particularly for concrete structures which have relatively high stiffness.
2 FLUID-STRUCTURE INTERACTION 2.1 INTRODUCTION
The goal of this chapter is to present and explain fluid-structure interaction equations. Fluid-structure interaction is the interaction of a movable structure with an internal or external fluid.
In the first part, we will focus on the fluid. Different assumptions will be discussed in order to obtain a simple model of the fluid behavior. We will demonstrate that, under certain conditions, the fluid obeys Laplace’s equation. It will be done by considering the velocity potential function associated to the velocity field.
Then, in a second part, we will analyze several boundary conditions which are likely to occur in fluid-structure interaction problems. In particular, we will present free surface and fluid-structure interface boundary conditions. Finally, we will expose the system of equations to solve in order to model fluid-structure interaction.
2.2 FLUID EQUATIONS
In this part, we will derive the fundamental equations of the fluid. It will be done by writing local fluid equations. Then these equations will be simplified thanks to basic assumptions.
2.2.1 LOCAL EQUATIONS
We consider a fluid domain Ω described, for each point of Ω, by the following local functions:
Its velocity field ;
Its density ;
Its hydrodynamic pressure . Each function depends on space and time.
The fluid flow obeys two fundamental local equations of conservation:
Continuity equation, which expresses the conservation of mass [4]:
( )
Euler’s equation, which expresses the conservation of momentum [4]: (
( ) ( ) ) ( )
In this last equation, represents body forces acting on the fluid, such as the gravity force for example. is the stress tensor.
It is possible to assume that the flow is incompressible. Indeed, we study liquid with low flow velocity. This property implies that the density is constant and homogeneous. From the continuity equation, we obtain the well-known criterion of incompressibility:
The incompressible flow assumption is valid if [15]:
With is the Strouhal number and is the Mach number. In this study, we have and . Thus, the above condition becomes:
The frequency range for seismic action is between 0 and 50 rad.s-1 so the incompressible flow assumption is valid. The stress tensor can be expressed as a function of the pressure p, the viscosity and the strain gradient tensor
as follow [4]:
( ( ) ( ))
Euler’s equation can be simplified thanks to the perfect fluid model, which implies the following assumptions:
The fluid is assumed to be inviscid: ;
The flow is irrotational: ( ) . Euler’s equation becomes Navier-Stokes equation:
(
( )) ( )
2.2.2 VELOCITY POTENTIAL
The assumption of irrotational flow implies that there is a velocity potential function that satisfies: ( )
If we now use the property of incompressibility, expressed in the previous part by: ( )
It yields that:
In the case of an irrotational and incompressible flow, the fluid can be represented by its velocity potential function, which satisfies Laplace’s equation.
If we now consider a compressible flow, we can show that the velocity potential function verifies Helmholtz equation [6]:
This is the wave equation that expresses the pressure propagation in the fluid (with c the velocity of the sound in the fluid). When we use Laplace’s equation, we consider that this propagation is instantaneous. This assumption is correct when the velocity c is high before the dimension of the problem. It is the case for the study of water sloshing in tanks for example.
Body forces derive from a potential. For example, gravity field force gives ( ). Thus, Navier-Stokes’ equation becomes:
(
)
After integration of this equation in the fluid domain, we get Bernoulli’s equation:
( )
Fluid-structure interaction problems always consider small displacement. Thus, it is possible to neglect . It is possible to “include” the time-dependent function k in the velocity potential (we still have ( )because ( ) ) and we finally obtain linearized Lagrange’s equation where p is the total pressure (sum of the hydrodynamic and hydrostatic pressure):
This formula is paramount because we want to represent the fluid-structure interaction by the velocity potential function. So it will allow us to compute the value of the pressure with the knowledge of the velocity potential.
2.3 BOUNDARY CONDITIONS
Now that we have derived the equations that describe the fluid behavior (in particular Laplace’s equation), we need to express boundary conditions in order to solve fluid-structure interaction problems. A typical and general fluid-structure interaction problem is depicted below (Figure 1).
Figure 1: typical fluid-structure interaction problem
We can distinguish 4 borders:
Γs represents the interface between structure and fluid;
Γsf represents the free surface;
Γg represents the interface between ground and fluid;
Γi is an imaginary boundary to model an “infinite” domain of water. We have the following properties:
Fluid density:
Sound velocity in the fluid: c
Ground density:
Sound velocity in the ground: cg
This figure can describe, for example, a dam. In this case, it is of great important to be able to model fluid-structure interaction when a seismic action occurs in order to assess the response of the structure. The velocity of the ground due to the seismic action is .
2.3.1 FLUID-STRUCTURE INTERFACE CONDITION
The structure is animated by a movement . We want to express the continuity between the velocity field in the fluid and the loads in the structure. At the fluid-structure interface, if we consider a fluid and a solid particle, the normal velocity will be equal, so it holds that [3]:
This condition can be expressed in terms of velocity potential. With ( ) , it holds that:
is the velocity of the structure’s particles at the fluid-structure interface. In the case of a rigid structure, this velocity is equal to the ground velocity which is known. However, if we have a deformable structure, we have to add the velocity due to structure deformation and we get: . In this case, is unknown and we have a coupled system.
2.3.2 FREE SURFACE CONDITION
Free surface condition can be expressed by the means of gravity waves. For each point of the free surface, the pressure is given by the fluctuation of the vertical velocity field [3]:
In this formula, we can replace the pressure by its expression as a function of the velocity potential found in the previous part.
It is now possible to express the free surface condition with the velocity potential.
This condition allows us to study waves and surface fluctuation. In particular, we need to know the waves’ height if we want to avoid overflow. However, it is sometimes possible to take a simpler condition which does not involve time. In fact, we consider only high frequency response (for seismic action for example), so it holds that [6]:
2.3.3 FLUID-GROUND INTERFACE CONDITION
The ground is made of poroelastic material and can absorb pressure waves. The damping factor of the ground can be express as follow [8]:
And the fluid-ground interaction condition is given by [8]:
Of course, in the case of a rigid ground-water interface, and the condition is the same than the one for the fluid-structure interface.
2.3.4 INIFITE BOUNDARY CONDITION
At this interface, we have to model waves’ propagation from the fluid-structure interface toward the other side of the tank. We suppose that the tank is big enough so that there is no wave reflection. The most common condition is called the Sommerfeld radiation condition [8]:
2.4 CONCLUSION
Basically, we can summarize fluid-structure interaction problems to the mathematical resolution of Laplace’s equation with Neumann boundary conditons.
For a perfect fluid, it is convenient to model fluid-structure interaction with its velocity potential. Thus, we have the following system of equations:
{ { { {
This kind of system of differential equation can be really challenging to solve. That is what is done in detail in the following chapters of this paper.
3 ANALYTICAL SOLUTION FOR 2D RECTANGULAR TANKS 3.1 INTRODUCTION
In this chapter, we solve the fluid-structure interaction problem in the simple case of a 2D rectangular tank. The starting point is the Neumann problem derived in the previous part. Thanks to the simple geometry, it is possible to solve analytically the sloshing problem. Actually, an analytical solution is possible for rectangular and circular tanks only. These kinds of tanks represent most practical cases, that is why it is useful to focus on them. Moreover, an analytical solution will give us a good understanding of the solution we get at the end.
We are going to study fluid-structure interaction when some water, contained in a rectangular tank, is subjected to an external action (a seismic action for example). During an earthquake, the mass of water contained in the structure will move because of the displacement of the solid: that is what we call “sloshing”.
Water sloshing induces hydrodynamic (or fluctuating) pressures on the vertical tank walls because of the horizontal acceleration of the structure. The goal of this study is to assess the value of these hydrodynamic loads.
The simplest way to compute the horizontal pressure forces when the tank is subjected to a seismic action, represented by horizontal ground acceleration, is to “accelerate” the whole mass of water as if it were acting like a rigid body. Therefore, the hydrodynamic forces will be proportional to the total mass of water. However, the water does not behave like a rigid body: it is a conservative assumption that is way too detrimental to structure design. That is why we have to model the fluid flow and the interaction between the fluid and the solid that are both moving.
The literature presents several methods in order to express the value of the fluctuating pressure [1]. This part explains and details these different approaches and compares the assumptions and the results obtained. The whole study assumes that the mass of liquid is lumped on the wall, based on rigid wall boundary condition in the calculation of hydrodynamic pressures. This assumption is widely used in practice, particularly for concrete structures which have relatively high stiffness.
3.2 PROBLEM PRESENTATION
3.2.1 MODEL AND ASSUMPTIONS
The structure studied is modeled by a 2D rectangular tank fixed to the ground as depicted in the figure below (Figure 2). The tank is subjected to a horizontal acceleration ̈ (ground acceleration due to seismic action for example).
Figure 2: rectangular tank fixed to the ground
We define the dimensionless coefficient that describes the geometry of the system:
With:
L: inside length of the tank, in meter;
h: height of water in the tank, in meter.
The density of the water, expressed in t.m-3, is noted . Thus, the mass of water is: , in is the horizontal displacement of the structure, in meter. The acceleration of gravity is noted g.
We assume that the fluid has the following properties:
Incompressible fluid;
Inviscid flow;
Small displacement;
Non-turbulent regime;
Irrotational flow.
These are the usual assumptions for a perfect fluid. These assumptions are realistic in the case of water contained in a tank and they will allow us to solve analytically the problem.
We define everywhere in the fluid:
( ) ( ) ( ) : velocity of the fluid;
( ): pressure in the fluid.
L/2
h Xs
3.2.2 SLOSHING EQUATIONS
Given the assumptions, the velocity potential function verifies Laplace’s equation (see previous part): ( )
In this case, we have two types of “boundary” conditions: free surface and fluid-structure interface. With the gravity wave theory, the free surface condition is:
( )
( ) We have a minus sign because the z-axis is downward pointing.
The movement is horizontal, so given the tank’s geometry we have simple fluid-structure boundary conditions:
{ ( ) ̇( ) ( ) ̇( ) ( )
The first two conditions describe the interface with the edges of the tank (x = -L/2 and x = L/2). The last one is the condition at the bottom (z = h).
The problem to solve is described by the following equations which are respectively Laplace’s equation, the free surface condition and the boundary fluid-solid interface conditions:
3.3 STUDY OF SLOSHING: TIME-HISTORY ANALYSIS
3.3.1 FREE OSCILLATIONS
In this part, we are looking for the sloshing modes in the case of free oscillations. It means that: ̇( )
So the problem to solve becomes:
{ ( ) ( ) ( ) { ( ) ( ) ( )
This system is going to be solved with a modal decomposition of the velocity potential. So we are looking for solution of this type:
( ) ̇( ) ( ) We are looking for decoupled solutions, that is to say:
( ) ( ) ( ) The modal function verifies:
( ) So it yields: ( ) ( ) ( ) ( )
The functions cos, sin, sh and ch are the solutions of this equation. a and b have the same “frequency”. The interface conditions gives:
{ ( ) ( )
( )
The 2 first conditions give the expression of a:
And the last one gives b:
( ( )) We can choose A = B = 1 and we have:
( ) ( ( )) ( ( ))
The free surface condition becomes:
⃛ ( )
( )
( ) ̇ ( )
This is the equation of a harmonic oscillator without damping. It implies that ̇ is a sinusoidal function whose frequency satisfies: ( ) ( ) And we find: { ̇ ( ) ( ) ( )
n is sloshing mode of the fluid in the tank. The solution of the free oscillation problem is the sum of the contribution of every sloshing mode:
( ) ∑ ̇ ( ) ( )
And we know the fundamental sloshing frequency for a rectangular tank:
√ ( )
3.3.2 FORCED DISPLACEMENT
In this part, we consider that the tank is subjected to a horizontal displacement .
We suppose that the family of modal functions ( ) found previously forms a basis of functions of [ ]
[ ]. This is an orthogonal family for the usual inner product. We are looking for a velocity field of this type: ( ) ∑ ̇ ( ) ( )
̇ ( )
With:
( ) ( ( )) ( ( ))
So ( ) verifies the following system of equations:
{ ( ) ( ) ( ) { ( ) ̇( ) ( ) ̇( ) ( ) So is solution of the sloshing problem.
The projection of the free surface condition on the n sloshing modes (which are orthogonal) gives n equations:
[∫ ( ) ] ⃛ ( ) [ ∫ ( ) ( ) ] ̇ ( ) [ ∫ ( ) ] ⃛( )
After calculation, we obtain n equations of oscillators:
⃛ ( ) ̇ ( ) ⃛( ) ( ) With: { ( ) ( ) ( ) ( ) Each equation describes one sloshing mode.
3.4 RESOLUTION OF THE SLOSHING PROBLEM: FREQUENCY DOMAIN ANALYSIS
We still consider the 2D rectangular tank fixed to the ground (Figure 3):
Figure 3: rectangular tank fixed to the ground
This part proposes frequency-domain based solutions. Thus, the imposed displacement is a harmonic oscillation along ex:
( ) ( )
The frequency-domain based resolution will not give us access to the time response of the system but we will get the maximum deformation (and loads) induced by a given elastic ground acceleration response spectrum.
The problem is going to be solved using different method. In the first one, we neglect the transient state. We find the results proposed by Graham & Rodriguez [1]: the fluid is assimilated to an equivalent impulsive added mass that moves in unison with the tank. In a second part, when we take into account the transient state, an oscillating mass is added to the model: this is equivalent to the solution proposed by Hunt & Priestley [1]. The results are then compared with the one given in the Eurocode 8.4 [2]. And finally, we take a look at Housner’s method [1], which is an approached method that gives simple expressions for the added impulsive and oscillating masses. In the conclusion of this part, the results are compared and the differences are explained.
3.4.1 STEADY STATE RESPONSE: GRAHAM & RODRIGUEZ’S METHOD
In this method, we neglect the transient state. So the differential equations ( ) found in the previous part can be easily solved by finding a particular integral of the problem (by using a complex resolution for example):
̇ ( )
( ) ( ) And we have:
( ) ̇( )
̇ ( )
( ) ̇( ) And we get the expression of the velocity field:
( ) [∑ ( ( )) ( ( )) ( ) ] ̇( ) With: { ( ) ( ) ( ) ( )
3.4.1.1 DYNAMIC PRESSURES ACTING ON THE TANK WALLS Now we can express the value of the pressure, given by the usual formula:
( ) ( ) And we find: ( ) [∑ ( ( )) ( ( )) ( ) ] ̈( )
The expression of the pressure gives access to the pressure factor ( ) on the vertical tank wall (for x= L/2) defined by: ( ) ( ) ̈( ) So we it yields: ( ) [∑ ( ) ( ( )) ( ) ]
The seismic action has a huge importance for high frequencies. Besides, we will prove later that we only have to take into account the first sloshing modes (so low frequency modes) to have a good estimate of sloshing effects. Thus, we can consider that:
( )
And the pressure factor becomes: ( ) [ ∑ (( ) ( )) (( ) ) (( ) ) ] 3.4.1.2 HYDRODYNAMIC FORCES
What we want is the value of the force caused by this fluctuating pressure on the vertical tank walls. Thus, we integrate the expression of the pressure on the wall (from z = 0 to h) and we compute the value for x = L/2.
( ) ∫ ( )
We have to multiply by 2 because the pressure is acting on the 2 vertical tank walls. And we find the hydrodynamic pressure force on the vertical tank wall:
( ) [ ∑ (( ) ) ( ) ( ) ] ̈( ) With: ( ) (( ) )
The value of the horizontal pressure force applied on the vertical tank wall can be written: ( ) ( ) ̈( ) With: ( ) [ ∑ (( ) ) ( ) ( ) ]
This expression represents the virtual mass of water which exerts a force on the structure when subjected to a harmonic solicitation of frequency ω. The force caused by this mass and proportional to the acceleration ̈ ( ) is called the impulsive force.
It proves that it is not necessary to take the whole mass of water contained in the tank ( ), but we just need to consider a certain ratio of this mass. This is due to the interaction between the fluid and the structure. So finally, it is possible to model the phenomenon of sloshing in the tank by replacing the water by the added impulsive mass mi.
Figure 4: Graham & Rodriguez model – impulsive mass
The resonance aspects will be studied afterward.
We can express the ratio as a function of the dimensionless coefficient ⁄ for high oscillation frequencies (i.e. ( ) ):
∑ (
( ) ) ( )
The abacus that gives the percentage of impulsive mass to take into account for the calculation is presented below (Figure 5):
Figure 5: Graham & Rodriguez's method - Impulsive mass ratio
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 mi /m w α = L/h
Impulsive mass ratio
3.4.2 TRANSIENT AND STEADY STATE RESPONSE: HUNT & PRIESTLEY’S METHOD
In this part, we do not neglect the transient part of the solution. So we want the exact solutions of the differential equations ( ) which are given by Duhamel’s equation:
̇ ( ) ∫ ⃛( ) ( ( ))
This equation is valid because we can assume that the system is initially at rest. That gives initials conditions: { ̇ ( )
̈ ( ) We are still studying the case of harmonic oscillations:
( ) ( )
The solution is the sum of a particular and a general solution of the differential equation, which leads to: ̇ ( )
( )
( ( ) ( ))
So we have the expression of the velocity field:
( ) [∑ ( ( )) ( ( )) ( ) ] ̇ ( ) ∑ ( ( )) ( ( )) ( ) ( ) With: { ( ) ( ) ( ) ( )
3.4.2.1 HYDRODYNAMIC PRESSURES ACTING ON THE TANK WALLS Now we can express the value of the pressure, which is given by:
( )
( ) [∑ ( ( )) ( ( )) ( ) ] ̈ ( ) ∑ ( ( )) ( ( )) ( ) ( )
The hydrodynamic pressure consists of two terms; one proportional to the acceleration of the structure ̈ ( ), the other is proportional to ( ) for each element of the sum (i.e. for each sloshing mode).
Actually, if we compare this result to the one of Graham & Rodriguez’s method, we note that the term that is proportional to ̈ ( ) is the pressure factor ( ) on the vertical tank wall (for x= L/2) found previously. This is the impulsive pressure.
( ) ( ) ̈( )
With the impulsive pressure factor:
( ) [∑ ( ) ( ( ))
( )
]
The only difference with Graham & Rodriguez’s method is the presence of an oscillating term. For each mode, we have an oscillating hydrodynamic pressure with a frequency equal to the natural sloshing frequency of this mode. This is the convective pressure defined for each sloshing mode:
( ) ( ) ̈ ( )
With the convective pressure factor on the vertical tank wall (for x= L/2):
( ) ( ) (
( ))
( ) And the oscillating acceleration:
̈ ( ) ( ) Finally, the total hydrodynamic pressure can be written:
( ) ( ) ∑ ( )
It is possible to compare this expression with the abacus that we find in the Eurocode. It is however necessary to be careful with the notations. Indeed, the Eurocode considers a tank with a width of 2L and z = 0 corresponds to the bottom of the tank (while z = h corresponds to the free surface).
Let’s use these notations, that we write L’ and z’ (for the half width and the new vertical coordinate). In the Eurocode, the dimensionless parameter used is:
Besides, the results presented in the Eurocode consider only high frequencies, so: ( )
3.4.2.1.1 HYDRODYNAMIC IMPULSIVE PRESSURE In the Eurocode, the impulsive pressure is given by:
( ) ( ) ( ) Where ( ) is the ground acceleration.
Thus, we get the expression of the dimensionless impulsive pressure as a function of and :
( ) ∑ (
( ) ) (( ) ) (( ) )
Now, we can draw the graph that represents ( )as a function of (Figure 6). Here, z’ = 0 represents the bottom of the tank.
Figure 6: Hunt & Priestley's method - impulsive pressure at the bottom of the tank
0 0,2 0,4 0,6 0,8 1 0 1 2 3 4 5 q0 (0) β = h/L'
The abacus given by the Eurocode [2] is presented below (Figure 7):
Figure 7: EN 1998-4 Figure A.6 - impulsive pressure at the bottom of the tank
We can also draw the graph that gives the ratio (
) ( )
⁄ as a function of ⁄ for several value of ⁄ . This graph (Figure 8) represents the shape of the impulsive hydrodynamic pressure along the vertical wall of the tank.
Figure 8: Hunt & Priestley's method - Impulsive pressure
0 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 1 z'/h q0(z'/h)/q0(0)
Dimensionless impulsive pressure
The abacus given by the Eurocode [2] is presented below (Figure 9):
Figure 9: EN 1998-4 Figure A.7 - impulsive pressure
3.4.2.1.2 HYDRODYNAMIC CONVECTIVE PRESSURE In the Eurocode, the impulsive pressure is given by:
( ) ( ) ( )
Where ( ) is the acceleration response function of a simple oscillator having frequency of the n mode. Thus, we get the expression of the dimensionless impulsive pressure as a function of and :
( ) ( ( ) ) (( ) ) (( ) )
Figure 10: Hunt & Priestley's method - Convective pressure, sloshing mode 1
Figure 11: Hunt & Priestley's method - Convective pressure, sloshing mode 2
0 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 z'/h qc1(z'/h)
Dimensionless convective pressure: sloshing mode 1
h/L' = 0.3 h/L' = 0.5 h/L' = 1 h/L' = 2 h/L' = 5 0 0,2 0,4 0,6 0,8 1 0 0,02 0,04 0,06 0,08 z'/h qc2(z'/h)
Dimensionless convective pressure: sloshing mode 2
The abacuses given by the Eurocode [2] are presented below (Figure 12 and Figure 13):
Figure 12: E EN 1998-4 Figure A.7 - convective pressure, sloshing mode 1
3.4.2.1.3 INTERPRETATION
The different graphs depicted in the previous part give results in accordance with the Eurocode. The work done in this study gives analytical expressions for the impulsive and convective pressures. These formulas are much more accurate and easier to use than the abacus proposed by the Eurocode.
3.4.2.2 HYDRODYNAMIC PRESSURE FORCES
What we want is the value of the force caused by this fluctuating pressure on the vertical tank walls. Thus, we integrate the expression of the pressure on the vertical wall (from z = 0 to h) and we compute the value for x = L/2.
( ) ∫ ( )
We have to multiply by 2 because the pressure is acting on the 2 vertical canal-bridge walls. And we find: ( ) [ ∑ (( ) ) ( ) ( ) ] ̈( ) ∑ (( ) ) ( ) ( ) ( ) With: ( ) (( ) )
The transverse force consists of two terms; one proportional to the acceleration of the structure ̈( ), the other is proportional to ( ) for each element of the sum (i.e. for each sloshing mode).
3.4.2.2.1 HYDRODYNAMIC IMPULSIVE PRESSURE FORCE
Actually, if we compare this result to the one of Graham & Rodriguez’s method, we note that the term that is proportional to ̈ ( ) is the impulsive added mass found previously: it is the impulsive pressure force.
We can express the ratio as a function of the dimensionless coefficient ⁄ for high frequencies (i.e. ( ) ): ∑ ( ( ) ) ( )
The abacus that gives the percentage of impulsive mass to take into account for the calculation is presented below (Figure 14):
Figure 14: Hunt & Priestley's method - Impulsive mass ratio for 1, 2, 3 and 10 sloshing modes
The value of the added impulsive mass has been plotted for 1, 2, 3 and 10 sloshing modes in order to focus on the influence of the modes.
The first thing to note is that the more sloshing modes we take into account, the smaller the impulsive mass is. That is logical because if we do not consider any sloshing at all, the impulsive mass would be equal to the total mass of water contained in the tank.
The second observation is the rapid convergence of the value of the impulsive mass. The value obtained when we take into account 3 sloshing modes is almost the same than the one with 10 sloshing modes. Therefore, a good assessment of the added impulsive mass can be done by considering only a few sloshing modes, 3 for example. 3.4.2.2.2 HYDRODYNAMIC CONVECTIVE PRESSURE FORCE
The difference with Graham & Rodriguez’s method is the presence of an oscillating term in the expression of the hydrodynamic force. For each mode, we have an oscillating pressure force with a frequency equal to the natural sloshing frequency of this mode. It can be model as a spring with a stiffness defined by:
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 mi /m w α = L/h
Impulsive mass ratio
With the added convective mass of water (with an oscillating movement ( )
( )) given by: [ (( ) ) ( ) ( ) ] And the natural frequency of the sloshing mode n:
( )
(( ) )
It is the convective force:
∑
̈ ( )
We can express the ratio as a function of the dimensionless coefficient ⁄ for high frequencies (i.e. ( ) ):
(
( ) ) ( )
Depicted below is the convective mass ratio for the first 3 sloshing modes (Figure 15).
Figure 15: Hunt & Priestley's method – Convective mass ratio for the sloshing mode 1, 2 and 3
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 mc, n /m w α = L/h
Convective mass ratio
Once again, what stands out of this graph is the small contribution of the modes other than the fundamental mode. A good assessment of the convective force can be obtained by taking into account a few sloshing modes, like the first and second one for example.
3.4.2.3 CONCLUSION
For high frequencies, Hunt & Priestley’s method decomposes the horizontal pressure force into two distinct parts:
An impulsive force, the same than the one proposed by Graham & Rodriguez, which represents the ratio of water that moves in unison with the tank:
̈( )
An convective force, which is the sum of the oscillation of every sloshing modes:
∑
̈ ( )
The only difference with the previous approach is that the solution of the differential equation is composed of the general solution and a particular solution. That’s why we have the convective forces, which stand for the “transient state” of the response. And the hydrodynamic pressure force can be written as the contribution of these two forces:
Therefore, it is possible to model the system as depicted in the Figure 16. It is important to remember that this mechanical analogy is only possible when we consider that the added masses are independent of the oscillation frequency (i.e. for high frequencies compared to natural sloshing frequencies).
Figure 16: Hunt & Priestley model – impulsive mass and convective mass (2 sloshing modes)
The graphs that give the values of the impulsive and convective hydrodynamic pressures prove that the impulsive mass is mainly acting on the lower part of the tank while the added convective masses are mainly acting on the upper part of the tank. Indeed, the waves that cause the convective forces are developed on the surface of the fluid.
Besides, the graphs that give the values of the added impulsive and convective mass show that the shallower the tank is, the more important the convective forces are (it is the opposite for the impulsive mass). Once again, that can be explained by the fact that the waves that cause the convective forces are developed on the surface of the
fluid: their influence will be even more important than these waves represent a large part of the fluid. This is the case for shallow tanks.
In a practical case, it will be interesting to compare the impulsive and convective forces. In particular, is it possible to neglect one of them? When we look at the fundamental natural sloshing frequency of water for different tank geometry, we get the following graph (Figure 17):
Figure 17: fundamental natural sloshing period for rectangular tanks
The fundamental period of water sloshing is, for most of the cases, several seconds long. At such long periods, the generated hydrodynamic convective pressures are much smaller than the hydrodynamic impulsive pressures. The convective forces are generally negligible and can be ignored [9]. It is however necessary to be careful when L is small (less than 5m) even if the failure risks when an earthquake occurs for such a small tank are not very high. Besides, we know that the added convective mass ratio is important for shallow tanks, so when L is large in comparison with h.
3.4.3 HOUSNER’S APPROACHED METHOD
Housner’s method is a widely used approach to assess sloshing effects in a tank containing a liquid during an earthquake. The principle of this approach is based on the movement of the fluid. When the tank is subjected to a horizontal acceleration, a part of the water moves together with the tank walls: the pressure force caused by this
1 5 9 13 17 21 25 29 0 2 4 6 8 10 12 14 16 18 20 1 5 9 L (m) Fu n d ame n tal slo sh in g p e riod T ( s) h (m)
Fundamental natural sloshing periods as a function of the tank dimensions
movement is proportional to the acceleration of the tank. Besides, the upper part of the fluid starts to slosh and causes vibration forces on the tank walls.
Actually, this description of the fluid movement corresponds to the expression of the pressure forces found in the previous part with Hunt & Priestley’s method. Housner used the lamina fluid theory to calculate the impulsive and convective hydrodynamic pressure [1]. The liquid is assumed to be incompressible and undergo small displacements.
Figure 18: Housner's method - lamina fluid theory
Housner’s method is an approximate method that gives simple expressions for the impulsive and convective masses mi and mc.
3.4.3.1 IMPUSLIVE PART
The value of the added impulsive mass is given by [1]:
(√ ) √
We can express the ratio as a function of the dimensionless coefficient ⁄ . The abacus that gives the percentage of added impulsive mass to take into account for the calculation is presented below (Figure 19). The results of Housner’s method are compared with the results of Graham & Rodriguez’s method.
Figure 19: Housner's method - Impulsive mass ratio, comparison with Graham & Rodriguez’s method
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 mi /m w α = L/h
Impulsive mass ratio
Housner u(x,t)
L
The result of the impulsive mass given by Housner is very close to the one obtained previously, in particular for shallow tanks. For α < 2, it might be better to use Graham & Rodriguez’s method.
Housner method provides also a simple formula that gives the point of application of the resulting pressure force [1]:
3.4.3.2 CONVECTIVE PART
This method considers only the first sloshing mode for the value of the added convective mass. We have demonstrated previously that this is a valid assumption. This mass is given by [1]:
(√ )
We can express the ratio as a function of the dimensionless coefficient ⁄ . The abacus that gives the percentage of added convective mass to take into account for the calculation is presented below (Figure 20). The results of Housner’s method are compared with the results of Hunt & Priestley’s method (1st sloshing mode).
Figure 20: Housner's method - Convective mass ratio, comparison with Hunt & Priestley's method
Once again, the approximate approach of Housner gives really good results in comparison with Hunt & Priestley’s method.
The point of application of the resulting pressure force is given by the following formula [1]:
[ √ (√ ) √ (√ )] 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 mc /m w α = L/h
Convective mass ratio
Housner proposes an expression for the value of the natural frequency of the first sloshing mode [1]: √ (√ )
We can compare it to what we had with the analytical resolution of the sloshing problem: ( )
These expressions are similar, except for π which is replaced by √ in Housner’s method. Though, it is not a secret that √ .
3.4.3.3 MODEL
Housner’s method is a simple approach based on lamina fluid theory. It gives results in accordance with the ones found with the other methods. It takes into account only one sloshing mode for the added convective mass. Thus, the system can be modeled as follow (Figure 21):
Figure 21: Housner's model - Impulsive and convective mass
It will be interesting to specify in which case this approximate and simple method can be used.
3.4.4 CONCLUSION
In this part, we have presented 3 different methods that describe the sloshing phenomenon when a tank, fixed to the ground, is subjected to harmonic oscillations. Nevertheless, every signal can be decomposed into harmonic signals with the Fourier transform so it can be seen as a general solution.
Graham & Rodriguez and Hunt & Priestley’s methods are similar in the approach of the problem: assumptions and equations are the same. The difference lies in the resolution of the equation of sloshing.
Graham & Rodriguez neglect the “transient state” of the response of the fluid. Thus, with this model, it was like no waves were created due to the movement of the tank. The consequence is that the pressure force is proportional to the ground acceleration with a coefficient called the added impulsive mass. The impulsive mass represents the quantity of water that is moving in unison with the structure.
Hunt & Priestley take into account the “transient state”. The waves formed are oscillating in the tank without damping (so the transient state is not really transient…), which leads to a convective force acting on the tank walls. This convective force is the sum of n added convective mass (n being the number of sloshing modes) that are oscillating like a simple oscillator having the frequency of the sloshing mode considered. The steady state gives the same impulsive force than in Graham & Rodriguez’s method.
Housner proposes an approximate method that gives simples formulas to assess both of impulsive and convective forces. Actually, Housner takes into account only one sloshing mode for the computation of the added convective mass. The results are very close to the ones found with Hunt & Priestley’s method.
We observe in the abacus that gives the added impulsive and convective mass ratio that the shallower the tank is, the more important the oscillating part is. Indeed, the waves are formed on the surface and the oscillating force has an influence on the upper part of the fluid only. It will be interesting to compare the contribution of the impulsive and the oscillating force in order to conclude about the validity of Graham & Rodriguez’s method. The table below presents the 3 different approaches with their hypothesis, results, limitations and model. Method Graham & Rodriguez Hunt & Priestley Housner
Hypothesis →Perfect fluid described by velocity potential method →Solution of sloshing equation: steady state only
→Perfect fluid described by velocity potential method →Solution of sloshing equation: steady and transient state
→Perfect fluid described by the lamina fluid theory →Approximate method: fluid movement decomposed in two part
Results →Impulsive part of the solution only (no wave created on the free surface)
→Impulsive and convective part of the solution
→Impulsive and convective part of the solution, 1st sloshing mode only for the convective part
→Solution for high frequencies only Limits No convective part:
non-conservative assumption for shallow tanks (convective part can be important)
Contribution of many sloshing modes can be hard to
compute
Results independent of frequency: no analysis for low frequencies possible
Results less accurate for L/h < 2
Diagram
In most of the case, it is possible to neglect the convective forces. Indeed, for shallow tanks (with L large), the fundamental natural period of water sloshing is several seconds long. At such long periods, the generated hydrodynamic convective pressures are much smaller than the hydrodynamic impulsive pressures. The convective forces are generally negligible and can be ignored.
3.5 STUDY OF RESONANT SLOSHING
The results presented in the previous parts are only valid for high frequency. Indeed, formulas given for the added mass ratio consider that ( ) . It is paramount to show that, for seismic action, this assumption is correct. In this part, we analyze the response of a 2D rectangular tank subjected to a given seismic action. We compare the results of a frequency domain analysis with the “high frequency assumption”.
3.5.1 RESONANT AMPLIFICATION
Resonance phenomena can be studied with the introduction of a damping factor. Indeed, for an inviscid flow, we would have an infinite amplification of the added mass at the natural sloshing frequencies. In the case of a viscous flow, the sloshing equations become:
⃛ ( ) ̈ ̇ ( ) ⃛( )
Where is the liquid damping factor [5]. For water, we can consider that, for the first sloshing mode:
For the other sloshing modes (n ≥ 1), we take [12]:
When there is a damping factor, even very low, it is possible to neglect the transient state. Thus, the solution of this equation can be found easily (with a complex resolution for example) for a harmonic oscillation. We find:
̇
√( ( ) ) ( ) ̇
Therefore, the ratio impulsive mass over total mass of water becomes:
( ) ( ) ∑ (( )( ) )
√( ( ) ) ( )
With the natural sloshing frequency for the mode n given by:
√ ( ) (( ) )
Figure 22: resonant sloshing
The table below gives the first and second natural sloshing frequencies:
L (m) ω0 (rad.s-1) ω1 (rad.s-1)
1 5.6 9.7
5 2.5 4.3
20 1.0 2.2
The curves can be decomposed:
: oscillating frequency is very small and the water does not slosh. Thus, the whole water contained in the tank is moving in unison with the structure and the added mass ratio is equal to 100%;
: oscillating frequency is equal to the fundamental sloshing frequency of the tank. Resonance induces a huge fall of the added mass: this is the resonant amplification;
: oscillating frequency is equal to the second natural sloshing frequency. Resonance effect is much smaller than for the first sloshing mode;
: oscillating frequency becomes high and the added mass ratio reaches quickly an asymptotic value. It’s the impulsive added mass found in the previous part with the assumption of high frequencies. Two main conclusions can be extracted from these graphs for the different geometries. Firstly, we note that the shallower the tank is, the higher the fundamental sloshing frequency is. In the same time, amplification decreases when natural frequency increase.
The different methods presented in this document assume that it is possible to take the impulsive added mass for high oscillating frequencies. It is of great importance to justify this assumption. It can be done by studying the resonant amplification of the added impulsive mass. Indeed, this graph shows that resonant water in the tank can induce a huge impulsive mass for certain geometry. The force associated with this impulsive added mass would be huge as well. What are the resonance effects when our structure is subjected to a seismic action?
-1000 -800 -600 -400 -200 0 200 0 1 2 3 4 5 6 7 8 9 10 Added mass ra tio (% )
Oscillating frequency (rad/s)
Added impulsive mass as a function of frequency
The fundamental sloshing mode is predominant. The other ones are cushioned and do not have any significant effect on the impulsive mass value. Thus, it is possible to consider the first sloshing mode only for the calculation. The added mass ratio becomes:
( ) ( ) ( )
√( ( ) ) ( )
The fundamental natural sloshing frequency has the following expression:
√ ( )
The value of the ratio ( ) for can be expressed as a function of α: ( ) ( )
This is the value of the asymptotic added mass by taking into account only one sloshing mode (Figure 23). The resonant amplification has to be compared with this expression.
Figure 23: impulsive added mass for high frequencies as a function of L/h
The value of the ratio ( ) for can be expressed as a function of α: ( ) ( )
This is the resonant amplification value. It is necessary to be very careful with the magnitude found. Indeed, the model can be put in into question. In particular, sloshing equations established concerned a perfect inviscid fluid. In the resonant sloshing study, we have added an artificial damping coefficient in the final equation with a certain value that is purely theoretical. Somehow, using this formula, we get the following graph (Figure 24):
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 1 2 3 4 5 6 7 8 9 10 added mass r ati o 𝛼 = L/h
Figure 24: resonant amplification coefficient as a function of L/h
It is of course very unlikely to have a resonant amplification up to 80. It would mean that resonant sloshing could induce hydrodynamic forces 80 times larger than if we considered water as a rigid body. The useful information that can be extracted from this graph is that resonant amplification increase with α.
In order to now if it is reasonable to take the asymptotic value of the added mass ratio, it is important to have a frequency domain based description of seismic action. If the typical frequency window of earthquake corresponds to sloshing resonance, we will have to study this phenomenon very carefully.
So the first question to ask is: when do we have amplification due to resonance? What is the frequency window of resonant sloshing? In order to answer this question, we have to find the frequency range for which:
( ) ( ) After calculation we find the condition:
( ) ( ) With: ⁄ ( ) [ ⁄ √ ( ) ( ( ) ) ] ⁄
This coefficient k is the “resonant amplification peak width” (Figure 25). It characterizes the frequency window where there is resonant amplification.
Figure 25: frequency window coefficient of the resonant amplification
We note that ( ) is close to 1: it means that the amplification is significant in a thin frequency window. Resonance will be significant when the structure is subjected to a seismic action if oscillating frequencies of the earthquake are close to the tank fundamental sloshing frequency.
This fundamental sloshing frequency is given on the graph below for different geometries (Figure 26).
Figure 26: natural sloshing frequency for different values of L
To sum up, this is what is important to recall about resonant amplification:
Amplification increases en α increases;
Natural sloshing frequency decreases when α and L increase.
It means that for geometry with high water loads (i.e. α and L large), even if we have a big amplification, the natural sloshing frequency is very low. Thus, in order to know if earthquake can cause resonant sloshing, we have to analyze power spectral densities of earthquake and compare them to tanks natural sloshing frequencies. If the frequency window of the earthquake has a higher range than natural sloshing frequencies, it is possible to use the asymptotic value of the added mass.
0 0,5 1 1,5 2 0 1 2 3 4 5 6 7 8 9 10 k( 𝛼) 𝛼 = L/h
Resonant amplification peak width
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 w0 (ra d. s-1) 𝛼 = L/h
Natural sloshing frequency
The graph depicted below (Figure 27) sums up important features of resonant amplification.
Figure 27: resonant amplification
3.5.2 FREQUENCY DOMAIN ANALYSIS
In order to analyze resonance, we need to have a frequency domain based description of seismic action. If we have the accelerogram of an earthquake (time-history description [11]), we just need to compute the Discrete Fourier Transform (DFT) of the signal. It can be done with numerical software such as Matlab, using the Fast Fourier Transform (FFT) function [10]. The Matlab program is presented in Appendix 4.
Depicted below is an example with a time-history recording of the notorious Kobe’s earthquake of 1995 and its associated power spectral density (Figure 28 and Figure 29). The unit is g.
Figure 28: time-history recording – ground acceleration
frequency Added mass
Natural sloshing frequency
Asymptotic added mass Total mass
Resonant amplification
Figure 29: frequency domain description – power spectral density
Let’s now analyze the response of a tank filled with water subjected to this earthquake (Figure 30).
Figure 30: rectangular tank fixed to the ground
In the case of a tank fixed to the ground, it is very simple to get the harmonic response of the structure. The inertia force of water has the following expression:
( ) ̈ With ( ) the equivalent added mass of water:
( ) ( ) ( ) can be seen as the transfer function of our system.
The graphs depicted below compares amplitudes of the transfer function and the earthquake (Figure 31). We still have h = 5 m and L = 20, 5 and 1 m.
What stands out of these graphs is that the frequency of sloshing resonance is much lower than the frequency window of the earthquake when the amplification amplitude is significant (cases L = 5 and 20 m). When L = 1, the natural sloshing frequency corresponds to the frequency window of high energy of the earthquake. However, the resonant sloshing is not significant for this geometry.
Knowing the DFT, it is now possible to compute this inertia force [5]: ( ) [ ∑ ( ) ]
Depicted in the next page are the graphs that represent the inertia force for 2 cases:
1st graph: the equivalent mass considered is ( );
2nd graph: the equivalent mass considered is the asymptotic value of ( ). The 3 graphs represent respectively the case for L = 20, 5 and 1 m (Figure 32).
As expected, the 2 responses are almost identical. The maximum forces are equal which justifies the fact that it is possible to take the asymptotic value of ( ) in the calculations. This is true for seismic action, but for other external loads such as wind for example, it is not always possible to neglect resonant sloshing. In some other cases, it is even possible to use the resonant sloshing with “Tuned Sloshing Dampers”.
3.5.3 TUNED SLOSHING DAMPERS
A Tuned Sloshing Damper (TSD) is used to give supplemental dumping to a structure (a bridge or a high tower for example). It utilizes liquid waves to absorb energy from vibrating structures through waves and viscous action in a partially filled tank of liquid. The tank is designed so that the liquid surface wave has a frequency "tuned" to be near the fundamental frequency of the building.
The main practical applications of TDS concern damping of wind-induced vibrations of tower. TDS have been used for example in the design of the New European Court of Justice, Luxembourg. A study [13] has shown that, with an optimal design of the TDS, total maximum reduction in top-level acceleration is found to be in the range of 35-40%. Comparable results have been demonstrated in another study [14] which deals with the use of TDS at Nagasaki Airport Tower (height 42 m) and the Yokohama Marine Tower (height 101 m).
3.5.4 CONCLUSION
It is not relevant to take into account resonant sloshing when we study dynamic response of structure subjected to seismic actions. It is consequently possible to use the formula presented in this chapter.
3.6 CONCLUSION
In this part, we have thoroughly studied water sloshing in 2D rectangular tanks. We have used the equations of fluid-structure interaction derived in the first part. In this particular case, it is possible to solve the corresponding Dirichlet problem analytically. Actually, it is only possible to do so for rectangular and circular tanks.
Different models are studied and compared. In the end, we can propose a quite simple model that describes water sloshing in 2D rectangular tank. When the tank is subjected to a horizontal acceleration, the lower part of the water moves together with the tank walls: the pressure force induced by this movement is proportional to the acceleration of the tank. Besides, the upper part of the fluid starts to slosh and causes vibration forces on the tank walls. That is why we can decompose hydrodynamic pressure forces in two parts: an “impulsive” and a “convective” force that corresponds respectively to the water that moves in unison with the tank and the water that is sloshing. These two forces can be computed with the knowledge of their associated added mass. The models assume that the mass of liquid is lumped on the wall based on rigid wall boundary condition in the calculation of hydrodynamic pressures.
4 NUMERICAL SOLUTION FOR SLOSHING PROBLEMS 4.1 INTRODUCTION
Analytical solutions presented earlier are really useful in order to understand phenomenon that occur in sloshing problems. They provide good results and do not require any complicated theory: most of the time, the simple Housner formula is sufficient to estimate the impulsive added mass. However, it only allows solving problems in a very limited number of cases. Indeed, analytical solutions are only possible for simple geometry, namely rectangular and circular tanks. Even if these 2 cases encompass a great majority of practical cases, it is possible to encounter more specific geometry.
The purpose of a numerical method is to solve sloshing problems for every kind of structure. Let’s take the example of a tank as depicted below (Figure 33):
Figure 33: example of non-rectangular 2D tank
It can move vertically and horizontally along z and y and it can rotate around x. The goal of numerical methods is to obtain the added mass matrix. In other words, for each degree of freedom, we want to know the effective mass of water that we have to take into account to assess the loads.
Then, the hydrodynamic force torsor will be expressed as follow:
( ) [ ] ( ̈ ̈ ̈
)
Where [ ] is the so-called added mass matrix.
A singularity method will be used to compute this matrix. The radiation problem is solved by imposing elementary movement to the tank. Hydrodynamic pressures associated with each elementary movement are then integrated to build the added mass matrix.
z
4.2 GENERAL METHOD
We assume that the fluid is incompressible and we consider only the case of high frequency oscillations. In this case, the flow is characterized by its velocity potential which obeys the following equation:
{
̇( ) The notations are explained in the graph below (Figure 34):
is the domain of the fluid
is the free surface
is the fluid-solid interface
is the normal vector of the contour pointing toward the fluid
Figure 34: problem geometry
We define the surface that surrounds the fluid domain .
The system being linear, it is possible to solve elementary problems corresponding to each degree of freedom. For a 2D problem as presented in the introduction, we will resolve the 3 following problems, for i = 1, 2 and 3.
{ { ( ⃗⃗⃗⃗⃗⃗ )
The structure rotates around point O.
