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

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Free-vibration experiments on PISE-1A

Two series of free-vibration experiments have been performed to study the behaviour of flow and structure during free vibrations under different physical conditions. Also, to calibrate the structural parameters including stiffness and damping ratio, a series of freevibration tests in air have been conducted .
The assembly will be pulled to a given initial position by a rope at the top of the assembly and then released to start the oscillation by cutting the rope suddenly. To determine the magnitude of initial displacement, several tests have been carried with initial displacement at around 1 mm and 3 mm respectively.

Free-vibration experiments in water

To characterise the structural behaviour during free vibrations with increasing added mass by enhancing water heights, also in purpose of examining the edge effects, a series of freevibration tests with different water heights was conducted on PISE-1A. Several different water levels have been chosen as:
full water height with upper water tank filled.
full water height with empty water tank.
2/3 of the assembly’s height from the bottom of the assembly.
1/3 of the assembly’s height from the bottom of the assembly.
If we take the reference plane at the bottom of the assembly, the four tests can be indexed as WH = 700 mm, 500 mm, 333 mm and 167 mm.

Free-vibration experiments in air

As introduced above, four repeated tests of free-vibration experiments in air have been conducted to calibrate the structural stiffness and damping ratio with a duration of 5 s and initial displacement at around 1 mm.

Frequency and damping coefficient

To get a more accurate prediction of the physical structural parameters, for free vibration tests in air, all three data analysis methodologies have been applied to calculate the vibration frequency and damping coefficient for all four repeated tests. The range of the data for the same group of tests are shown in FIGURE 2.18.
For frequency, all three methods have similar performances. While for damping coefficient, the results of the three different methodologies show big discrepancies. Comparingto ERA method and FFT method, regression method has a much larger range of damping coefficient. This may due to the high dependancy of regression method on choices of the peak points. ERA method slightly outperforms FFT method when comparing the range of damping coefficient. The spectrum of experimental data after fast Fourier transform is not as smooth as shown in FIGURE 2.11 but with several lower peaks distributed (see FIGURE 2.19) . This may be the factor that compromises the performance of FFT methods when calculating the damping coefficient. Therefore, when choosing the reference value for calculating the structural parameters, results from ERA method were considered.

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.

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 

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