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## Frequency and damping coefficient

FIGURE 2.28 shows the vibration frequency and damping coefficient evolution with different water heights.

In FIGURE 2.28, vibration frequency and damping coefficient of both experiments and numerical simulations are plotted. Range of experimental results from repeated tests has been plotted as well as numerical results with implicit method.

As water height increases, vibration frequency decreases and damping coefficient increases due to increase of fluid force acting on the assembly which is proportional to water height. Comparing the case when water height at the top of the assembly with a thin foil cover (WH=500mm) and that when at full water height with upper water tank filled (WH=700mm) , edge effect of the recirculation from the top shows little influence on damping coefficient and vibrating frequency. There is a trend of decreasing frequency and increasing damping coefficient from WH=167mm to WH=500mm, both in numerical results and experiments. This is due to the increasing added mass with the increase of water height.

However, when WH=333mm, the damping coefficient of experiments doesn’t increase comparing with that when WH=167mm, which is different from both the expected behaviour and the numerical simulation. Therefore, it is supposed that some errors may involved in this experiment when WH=333mm.

During experiments, the outer container is supposed to be fixed, however, in reality, it may be affected by the vibration of the hexagonal assembly. Coupling of the movement of outer container and the movement of the assembly will affect each other mutually. Asimplified model with coupled 2D cylindrical model has been elaborated in AppendixB. FIUGRE B.2 shows the spectra for the vibration of coupled cylindrical assembly and cylindrical outer container.

### Free-vibration experiments on PISE-2C

There were four groups of free-vibration experiments with different physical conditions and varying modes of vibration:

Experiments in air with sole active assembly.

Experiments in water with sole active assembly.

Experiments in water with crown grouped assemblies activated (partial flowering).

Experiments in water with whole mock-up active (total flowering).

As introduced in Chapter 3, big initial displacement as 3 mm will introduce unwanted affect from initial shock of release. Therefore, an initial displacement around 1 mm will be imposed at the beginning of each test both for experiments with sole assembly and multi assemblies. This initial displacement is at 1/3 of the inter-assembly channel width, therefore, non-linearity effects are expected to happen especially during the first several periods. The crab will pull the chosen assembly or assemblies along the outward radial direction and then release it or them in order to start the vibration.

For experiments in water, the container will be filled with water at a height above the assembly around 200 mm to assure that the whole mockup is immersed in water. To facilitate the experiments, calibration factor of strain-gauge, structural stiffness and damping of each assembly have been calibrated (see FIGURE 3.4, 3.5). These calibrations were performed on the base of PISE-1A, difference of calibration factor for the assemblies installed on PISE-2C base may exist.

#### Methodologies of analysis on global behaviour of PISE-2C

To carry out the analysis on global behaviour of PISE-2C during the free-vibration movements, several global indicators have to be introduced.

**Displacement**

Displacements have been measured for each assembly by strain gauge. Since there are different delay time for the crab to release all the assemblies, the signals of displacements have been cut at the beginning in order to only take the duration when all the assemblies are in movement. Same as signal processing of PISE-1A, displacement signal of PISE-2C experiments have also been assessed with filtering to remove the unwanted high-frequency noise.

**Velocity of assembly**

Velocity of each assembly was deducted from the time evolution of displacement with 2ndorder time scheme. x_ n(i) = 3xn+1(i) 4xn(i) + xn1(i) 2t [m/s]; (3.51). where x_ n and xn are the velocity and displacement of Assembly No.i at tn respectively.

**Indicators of symmetry**

The symmetry of PISE-2C installation is an important indicator of the complexity of PISE- 2C (the level of order in oscillations of the assemblies) which justifies if the deterministic approach is acceptable in analysis.

To get an idea of the level of order in the experiments, several indicators of symmetry have been examined. In addition, this is also necessary for establishing the pertinence of the reticulate model. We can divide the whole plan into 6 sections (see FIGURE 3.27).

. Section 1 :

– Assemblies : 4, 12, 13, 14 .

– Complete channels : 3-4, 14-15, 15-16, 16-17, 15-38, 16-41 .

– Half channels : 3-14, 4-17 .

. Section 2 :

– Assemblies : 3, 10, 11, 12 .

– Complete channels : 2-3, 11-12, 12-13, 13-14, 12-33, 13-36 .

– Half channels : 2-11, 3-14 .

. Section 3 :

– Assemblies : 2, 8, 9, 10 .

– Complete channels : 1-2, 8-9, 9-10, 10-11, 9-28, 10-31 .

– Half channels : 1-8, 2-11 .

**Volume of liquid contained in the mockup**

Volume of liquid in the mockup can be calculated as the multiplication results of horizontal cross-section surface and height of the mockup. Horizontal cross-section surface can be calculated by the sum of horizontal surface area of all the internal channels. V (t) = H Scross(t) [m3]: (3.54). V (t) is the volume of liquid contained in the mockup, H is the height of the assembly, Scross(t) is the area of the horizontal cross-section surface.

**Average outflow velocity**

A virtual average outflow velocity v(t) is introduced to characterise the average movement of the whole mockup. It will be the result of total liquid volume contained in the mock-up divided by the area of all the outflow surface. The outflow surface includes the top and bottom cross-section surface with the vertical gap surface Sside(t) between the assemblies in outer crown. v(t) = V (t)=(2 Scross(t) + Sside(t)) [m/s]: (3.55).

As seen in free-vibration experiments of PISE-1A by PIV [26] and also the 3D NAVIERSTOKES simulation of PISE-1A, there are three flow regions existed: 2 recirculation regions and 1 2D region. It means that the outflow and inflow at the edge don’t affect the flow in the 2D region. This observation can be extrapolated to PISE-2C. Since the volume variations in recirculation regions at upper and lower edges are small with regard to that in 2D region, the velocities coming from the upper and lower surface of PISE-2C are negligible comparing to the velocities coming from the vertical external faces. Therefore, the average flow velocity is mainly in representation of the horizontal velocity coming from the vertical external sides.

**Surface confined by assemblies’ center of external crown**

To describe the global behaviour of the PISE-2C, another global quantity is introduced here as the surface area defined by the center of the assemblies on the external crown ([m2]). As shown section 3.2.2.4.1, it is the area defined by blue dash line.

**Total flowering**

To analyse the dynamic behaviour of PISE-2C as an ensemble facility, an experiment with all the assemblies in the mockup activated (total flowering) was performed with five repeated tests. All the assemblies were pulled to a position where the initial displacement is 1 mm outward radially by the crab. They were released at approximately the same time to start the vibrations.

**Table of contents :**

Abstract

Résumé

Acknowledgements

**1 Introduction **

1.1 Background

1.2 Objectives

1.3 State of the Art

1.3.1 Existing methods

1.3.2 Previous works

1.4 Scheme and Approaches

**2 Experimental Approaches on PISE-1A and Corresponding Numerical Interpretation**

2.1 Introduction to PISE-1A

2.2 Free-vibration experiments on PISE-1A

2.2.1 Free-vibration experiments in air

2.2.2 Free-vibration experiments in water

2.2.3 Free-vibration experiments in water-glycerol mixtures

2.3 Numerical simulations based on PISE-1A experiments

2.3.1 Introduction to the numerical methodologies

2.3.1.1 Governing equations

2.3.1.2 Boundary conditions

2.3.1.3 Algorithms

2.3.1.4 Mesh

2.3.1.5 Stability

2.3.1.6 Comments

2.3.2 Introduction to data analysis methodologies for frequency and damping coefficient

2.3.2.1 Regression method

2.3.2.2 ERA method

2.3.2.3 FFT method

2.3.3 Methodologies of signal processing

2.3.3.1 Butterworth filter

2.3.3.2 Hanning filtering

2.4 Methodologies of analysis on velocity and energy of PISE-1A

2.4.1 Velocity

2.4.2 Energy

2.5 Free-vibration experiments in air

2.5.1 Frequency and damping coefficient

2.5.2 Velocity and energy

2.5.3 Homogeneous problem

2.6 Experiments with different water heights

2.6.1 Flow behaviour

2.6.2 Frequency and damping coefficient

2.6.3 Velocity and energy

2.7 Experiments with water-glycerol mixture

2.7.1 Frequency and damping coefficient

2.7.2 Velocity and energy

2.8 Conclusions

**3 Experimental and Analytical Approaches on PISE-2C **

3.1 Introduction to PISE-2C

3.2 Reticulate model

3.2.1 Position of problem

3.2.1.1 General hypothesis

3.2.1.2 Dimensions

3.2.1.3 Fluid

3.2.1.4 Solid

3.2.2 Coupling

3.2.2.1 Kinematic coupling

3.2.2.2 Dynamic coupling

3.2.2.3 Phenomenological analysis

3.2.2.4 Mesh

3.2.2.5 Countings

3.2.3 Resolution

3.2.3.1 Notations

3.2.3.2 Trivial simplifications

3.2.3.3 Kinematic constraints

3.2.3.4 Symmetries

3.2.3.5 Counting

3.2.3.6 States

3.2.3.7 Equations

3.2.3.8 Integrations

3.2.3.9 Characteristics of flow

3.2.4 Energetic balance

3.2.4.1 Initial formulation

3.2.4.2 Geometrical elements

3.2.4.3 Evaluation of balance

3.2.4.4 Equations

3.2.4.5 Energy of assemblies

3.2.4.6 Kinetic energy of fluid

3.2.4.7 Dissipation of the structure

3.2.4.8 Dissipation of fluid

3.2.4.9 Flux at outlets

3.2.4.10 Phenomenological analysis

3.2.5 Conclusions

3.3 Free-vibration experiments on PISE-2C

3.4 Methodologies of analysis on global behaviour of PISE-2C

3.4.1 Displacement

3.4.2 Velocity of assembly

3.4.3 Indicators of symmetry

3.4.4 Energy of assembly

3.4.5 Volume of liquid contained in the mockup

3.4.6 Average outflow velocity

3.4.7 Surface confined by assemblies’ center of external crown

3.5 Total flowering

3.5.1 Displacements

3.5.2 Velocity

3.5.3 Energies of assembly

3.5.4 Volume contained in whole mockup

3.5.5 Average outflow velocity

3.5.6 Surface confined by centres on external crown

3.5.7 Indicators of symmetry

3.6 Partial flowering : Internal crown

3.6.1 Displacements

3.6.2 Velocity

3.6.3 Energies of assembly

3.6.4 Volume contained in whole mockup

3.6.5 Average outflow velocity

3.6.6 Surface confined by centres on external crown

3.6.7 Indicators of symmetry

3.7 Partial flowering : External crown

3.7.1 Displacements

3.7.2 Velocity

3.7.3 Energies of assembly

3.7.4 Volume contained in whole mockup

3.7.5 Average outflow velocity

3.7.6 Surface confined by centres on external crown

3.7.7 Indicators of symmetry

3.8 Conclusions

**4 Conclusions **

**A Analytical Analysis Based on Added Mass and Damping **

A.1 Position of the problem

A.1.1 Geometry

A.1.2 Solid

A.1.3 Fluid

A.1.4 Equations and boundary conditions

A.2 General properties

A.2.1 Dynamic approaches

A.2.1.1 Imposed movement

A.2.1.2 Free movement

A.2.1.3 Fluid force

A.2.2 Energy approach

A.2.2.1 Imposed movement

A.2.2.2 Free movement

A.3 Linearisation

A.3.1 Fluid flow

A.3.1.1 Scaling

A.3.1.2 Equations

A.3.1.3 Boundary conditions

A.3.2 Interaction force

A.3.3 Kinetic energy

A.3.4 Viscous dissipation

A.4 Resolution

A.4.1 Perfect fluid

A.4.2 Real fluid

A.4.2.1 Imposed movement

A.4.2.2 Free movement

A.5 Conclusion

**B Oscillations of two cylinders coupled by fluid **

B.1 Introduction

B.2 Position of problem

B.2.1 Geometry

B.2.2 Fluid

B.2.3 Solid

B.2.4 Boundary conditions

B.2.4.1 Non-penetration condition

B.2.4.2 No-slip condition

B.3 Scaling

B.3.1 Geometry

B.3.2 Fluid

B.3.2.1 Equations

B.3.2.2 Stress tensor

B.3.2.3 Pressure force

B.3.3 Solid

B.3.4 Boundary conditions

B.3.4.1 Non-penetration condition

B.3.4.2 No-slip condition

B.4 Phenomenological analysis

B.4.1 Hypothesis

B.4.2 Dynamics of the assemblies

B.4.3 Boundary conditions

B.5 Resolution

B.5.1 Integration

B.6 External fixed cylinder

B.7 Spectra

B.8 Conclusion

**C System of DOF at 2 **

C.1 Position of the problem

C.2 Initial equilibrium

C.3 Transient equations

C.3.1 Time domain

C.3.2 Frequency domain

C.4 Energies

C.4.1 Kinetic energy

C.4.2 Potential energy

C.4.3 Total energy

C.4.4 Energy balance

C.5 Resolution

C.6 Conclusion

**D 3D Effects: Recirculation Flow **

D.1 Geometry and kinematics

D.1.1 Geometry

D.1.1.1 Boundary condition

D.2 Fluid and flow

D.2.1 Equations

D.2.2 Boundary conditions

D.2.3 Scaling

D.2.3.1 Independent variables

D.2.4 Velocity and pressure

D.2.5 Non-dimensionalised formulation

D.2.5.1 Mass conservation

D.2.5.2 Momentum conservation

D.2.5.3 Boundary conditions

D.2.5.4 Geometrical developments

D.3 Perfect fluid

D.3.1 Statement of problem

D.3.1.1 Equations

D.3.1.2 Slipping condition

D.3.1.3 Scaling of pressure

D.3.2 External scaling, first approximation

D.3.2.1 Small amplitude, 1

D.3.3 Internal scaling

D.3.3.1 First approximation

D.4 Conclusions

**E Analytical and Numerical Analysis on Two-dimensional Fluid Channel Model with Oscillating Wall and Continuous Injection **

E.1 Phenomenological analysis

E.1.1 Two-dimensional geometry and basic conditions

E.1.2 Dimensioned equations

E.1.3 Decomposition

E.1.4 Scaling

E.1.5 Non-dimensioned equations

E.1.6 Thin-layer approximation

E.1.7 Reference solution

E.1.7.1 Mass conservation

E.1.7.2 Momentum conservation

E.1.7.3 Boundary conditions

E.1.8 Perturbation

E.1.8.1 Mass conservation

E.1.8.2 Momentum conservation

E.1.8.3 Boundary conditions

E.1.8.4 First approximation

E.1.9 Kinetic energy theorem

E.1.9.1 Stationary solution

E.1.9.2 Perturbation

E.1.10 Conclusions

E.2 Numerical analysis

E.2.1 Conditions of the Simulations

E.2.1.1 Simulation conditions

E.2.1.2 Governing equations solved by Cast3M

E.2.1.3 Non-dimensional parameters and scalings

E.2.2 Data analysis

E.2.2.1 Velocity and pressure profiles

E.2.2.2 Time evolutions

E.2.2.3 Average pressure on the oscillating plate

E.2.2.4 Average inlet pressure

E.2.2.5 Dissipation and pressure work

E.2.3 Conclusions

E.3 Conclusions

**Bibliography**