Analytic, Simulation and Experimental Analysis of Fluid-Pipe Systems

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Fluid-Pipe Systems

SIDDHARTH VENKATARAMAN

Course code: SD211X

Master in Engineering Mechanics Date of Approval: December 6, 2018

Academic Supervisor and Examiner: Mats Åbom

Industry Supervisors: Anders Daneryd and Johannes Kocher Swedish title: Analytisk, Simulerings och Experimentell analys av fluid-rör system

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Abstract

Inviscid fluid inside thin pipe system is first analytically solved for eigenfrequencies and eigenmodes using Modal Interaction Model method with fluid-structure interaction condition at boundary. Shear-diaphragm boundary condition is used for comparing and validating Analytic re-sults with Simulation using COMSOL Multiphysics. Effect of viscosity is also compared using Newtonian fluid model. Experiment is per-formed using simple pipe geometry and fluid to measure transfer ac-celerance which is post-processed to extract cirumferential modes up to order 4; this is used to compare and validate Experiemental results with Simulation. Good correlation is obtained between Analytic, Ex-periment and Simulation results with n=0 breathing modes requiring modification of governing equations to incorporate compressibility ef-fects due to changing pipe cross-section area.

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iv

Sammanfattning

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Acknowledgement

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3 COMSOL Simulation 46 3.1 COMSOL modules . . . 46 3.2 Parameters . . . 46 3.3 Geometric model . . . 47 3.4 Meshing . . . 48 3.5 Studies performed . . . 48 3.6 Multiphysics . . . 49 3.7 Simulation of Pipe . . . 49 3.7.1 COMSOL module . . . 49 3.7.2 Material Properties . . . 49 3.7.3 Meshing . . . 49

3.7.4 Choice of Pipe representation in 3D . . . 51

3.8 Simulation of ideal fluid . . . 53

3.8.1 COMSOL module . . . 53

3.8.2 Material Properties . . . 53

3.8.3 Meshing . . . 54

3.9 Simulation of Viscous fluid . . . 54

3.9.1 COMSOL module . . . 54

3.9.2 Material Properties . . . 54

3.9.3 Computational optimization . . . 55

3.9.4 Meshing . . . 56

3.10 Mode order classification . . . 56

3.11 Energy ratio and Impedance . . . 56

4 Analytic vs. Simulation 60 4.1 In-vacuo pipe modes . . . 60

4.2 Rigid walled fluid modes . . . 63

4.3 Coupled fluid-pipe modes . . . 64

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viii CONTENTS

4.3.2 Impedance Comparison for n=0 . . . 67

4.3.3 Convergence test . . . 69

4.3.4 Conclusion . . . 69

4.4 Effect of Viscosity . . . 72

4.4.1 Radial forcing on pipe . . . 78

4.4.2 Conclusion . . . 80

5 Vibration Experiments on Fluid-Pipe Systems 83 5.1 Aim . . . 83

5.2 Experimental setup . . . 84

5.2.1 Instrument specifications . . . 84

5.2.2 Pipe suspension . . . 84

5.2.3 Excitation and Response measurement positions . 87 5.2.4 Wiring and Signal type . . . 91

5.3 Excitation methods . . . 91

5.3.1 Impact Hammer: . . . 91

5.3.2 Shaker: . . . 91

5.3.3 Excitation method for Steel . . . 93

5.3.4 Excitation method for PVC . . . 93

5.4 Signal acquisition and preprocessing . . . 95

5.5 Test cases . . . 96

5.5.1 Pipe sub-systems . . . 96

5.5.2 Fluid sub-systems . . . 97

5.5.3 Combinations tested . . . 99

5.6 Remarks . . . 100

6 Experiment vs. Simulation Analysis 103 6.1 Processing of Experimental Data for Analysis . . . 103

6.1.1 Variables used . . . 103

6.1.2 Circumferential Mode decomposition . . . 103

6.1.3 Axial mode number estimation . . . 114

6.2 COMSOL simulation of experiment . . . 115

6.2.1 Test Cases simulated . . . 115

6.2.2 Complexities in Simulating Experiment . . . 116

6.2.3 Modal sensitivity to Fluid properties . . . 117

6.3 Results . . . 120

6.3.1 Complementary improvement in COMSOL mod-eling and Experiment data analysis . . . 120

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6.3.3 Fluid Density Determination . . . 122

6.3.4 Steel pipe . . . 127

6.3.5 PVC with flange . . . 135

7 Misc. experimental data analysis 147 7.1 Influence of Air bubbles . . . 147

7.2 Acosense Equipment . . . 149

7.2.1 Acosense Accelerometer . . . 150

7.2.2 Acosense Emitter . . . 151

8 Analytical reformulation for n=0 modes 153 8.1 Error in original Analytic formulation . . . 153

8.2 Modification of pipe Governing equation . . . 154

8.2.1 Derivation of Coupled pipe governing equation . 154 8.2.2 Previous implementation of fluid mode influence 156 8.2.3 Modified implementation of fluid mode influence for n=0 . . . 157

8.3 Modified Results . . . 158

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x CONTENTS

List of parameters

Parameter Description p acoustic pressure c speed of sound B bulk modulus ρ density

R radius of pipe at mid-surface h thickness of pipe

ν Poisson’s ratio ωr ring frequency

β pipe thickness parameter (r,θ, z) cylindrical coordinates

s non-dimensional axial coordinate = z/R

(u,v,w) axial, tangential and circumferential displacement of pipe

E Young’s Modulus

Jn() nth order Bessel function of the first kind

J′

n() first derivative for nthorder Bessel function of the first kind

k acoustic wavenumber

(n, m, p) circumferential, axial and radial mode order cp specific heat at constant pressure

γ ratio of specific heats µ dynamic viscosity µB bulk viscosity

κ coefficient of thermal conductivity

lvor / lent acoustic boundary layer thickness due to vorticity/entropy modes

pN contribution of Nthuncoupled fluid mode

ψN mode shape of Nthuncoupled fluid mode

wP contribution of Pthuncoupled pipe mode

φP mode shape of Pthuncoupled pipe mode

ΛN / ΛP norm of Nth/ Pthuncoupled fluid/pipe mode shape

CN P coupling coefficient between Nth uncoupled fluid mode and Pthuncoupled pipe

Φ velocity potential representation of fluid mode

α attenuation of pressure wave of the form Nepers/length H51’ to H55’ complex accelerance function from accelerometers 51 to 55 n0 amplitude of n = 0 mode

n1pand n1n amplitude of n = 1 mode on 2 perpendicular projections

n1 amplitude of n = 1 mode

n2 amplitude of n = 2 mode

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Document Description

The report is a result of Master Thesis work performed for Acosense AB in co-operation with KTH and Synerleap - Powered by ABB. The aim of the thesis as well as content of this report is to understand and compare simple fluid-pipe systems through analytic, simulation and experimental results.

The main aspects of this report are centered around:

• Analytically implementing Fluid-structure interaction between fluid and pipe

• Simulating fluid-pipe system vibrations using COMSOL FEM software

• Experimentation and Data Analysis on simple fluid-pipe systems • Comparing results from above mentioned methods

• Influence of Viscosity in fluid-pipe vibrations

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Literature Study

To obtain relevant knowledge and understanding of acoustic interac-tion in fluid-pipe systems, extensive material from books and journal articles is utilized.

1.1 Basics

For establishing the basics in sound propagation in structures and flu-ids in-vacuo (i.e., uncoupled), Chapter (1) in Fahy’s book [1] is rele-vant. Any acoustics textbook has similar information, but Fahy’s book is used to maintain continuity as this book will further be used for the preliminary study of Fluid-Structure interaction. The concepts are highlighted below:

1.1.1 Acoustic Wave equation in fluids

For an unbounded Ideal fluid medium with no acoustic sources, the wave equation is homogeneous and is written as:

1 c ∂2_p ∂t2 − ∇ 2_{p = 0} _(1.1) Where: • p = acoustic pressure

• c = speed of sound in fluid medium =pB/ρin terms of the Bulk Modulus ’B’ and density ’ρ’ of the fluid

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2 CHAPTER 1. LITERATURE STUDY

• ∇2_{is the Laplacian of the pressure field and has different}

formu-lation for Cartesian and Cylindrical coordinates.

1.1.2 Structural vibration in a pipe

Structural vibrations in a pipe is more complicated that structural vi-brations in a plate. In the case of a thin plate: longitudinal and flexural (bending) motion is uncoupled from each other and so they can exist independently of each other with each being described by a separate PDE (Partial differential equation). Whereas, in the case of a pipe the curvature introduces mid-plane membrane effects which effectively couple the 3 orthogonal motions (Radial, Axial and Tangential) lead-ing to pipe vibrations belead-ing described by a matrix of 3 PDEs coupled to each other.

Interaction with a fluid medium at the surface is affected only by the radial component, (assuming effect of fluid viscosity negligible: the other 2 motions do not influence the fluid field) and for the case of including influence of viscosity in Section (3.9), simulation of fluid-pipe system includes the appropriate fluid model which takes into ac-count non-radial motion of the pipe . In this report, the pipe is consid-ered thin (h/R<0.1) and so other complicating effects like rotary inertia and shear deformation which is proporti can be ignored.

The force balance equation of pipe system can be written as shown in equation (1.2): (LDM + β2LM OD)   u v w  =   Faxial Ftangential FRadial   (1.2) Where:

• u, v and w are the axial, tangential and radial displacement fields, being functions of z and θ

• β is the pipe thickness parameter (see equation (1.7))

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Figure 1.1: Diagram of pipe section with labels for coordinates and displacement directions

For simplicity, the pipe matrix is represented as [Lpipe] (equation

(1.4)) and the resulting equation is shown in (1.3). Lpipe   u v w  =   Faxial Ftangential FRadial   (1.3) Where: Lpipe = (LDM + β2LM OD) (1.4)

Extensive work on vibrations in shells is available from literature by AW Leissa [2] which offers numerous ways to implement coupling in the system matrix. The ’base’ matrix is the one developed by Donnell-Mushtari and ’mod’ matrix is used to incorporate effects of bending and membrane stress. Flügge’s shell theory is selected for the ’mod’ matrix on the basis of overall performance, a symmetric system matrix and popular use in many published papers related to pipe vibrations.

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4 CHAPTER 1. LITERATURE STUDY LDM =    (_∂s∂22 + 1−ν₂ ∂ 2 ∂θ − 1 ω2 r ∂2 ∂t2) 1+ν₂ ∂ 2 ∂s∂θ ν ∂ ∂s 1+ν 2 ∂2 ∂s∂θ (1−ν2 ∂2 ∂s2 + ∂2 ∂θ2 − 1 ω2 r ∂2 ∂t2) ∂ ∂θ ν ∂ ∂s ∂ ∂θ 1 + β 2_∇4₊ 1 ω2 r ∂2 ∂t2    (1.5) The Flügge’s shell theory ’mod’ matrix is shown in equation (1.6):

LM OD = LF lugge =    1−ν 2 ∂2 ∂θ2 0 −∂ 3 ∂s3 + 1−ν₂ ∂ 3 ∂s∂θ2 0 3(1−ν)₂ _∂s∂22 −3−ν₂ ∂ 3 ∂s2_∂θ −∂3 ∂s3 +1−ν₂ ∂ 3 ∂s∂θ2 0 1 + 2∂ 2 ∂θ2    (1.6) Where the terms mentioned are:

• z, r and θ are the cylindrical co-ordinates with z along the pipe axis. Refer to figure (1.1).

• R and h are the radius and thickness of the pipe • s = z / R , to non-dimesionalize the axial component • ν the Poisson’s ratio of pipe material

• ωris the ring frequency of the pipe (see equation (1.8))

1.1.3 Pipe parameter -

β

The pipe’s thickness to radius ratio is conveniently represented by the pipe parameter β. This is given in equation (1.7):

β =√h

12R (1.7)

The parameter β’s influence can be seen in the L33 term of ’base’

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1.1.4 Ring frequency -

ω

ring

An important frequency when considering acoustics in pipe systems is the ring frequency. It is the cut-on frequency of the radial axisym-metric mode (n=0) also called the breathing mode. Physically, it is the frequency at which the wavelength of longitudinal wave in the pipe matches its circumference. It is convenient to normalize the angular frequency with the ring frequency. The equation for ring frequency is given below 1.8: ωring = 1 R s E ρ(1 − ν2₎ (1.8)

All the parameters in the equation (1.8) are with regards to the pipe-material.

Below ring frequency, influence of the membrane strain dominates and above ring frequency the bending strain dominates. Thus it can also be said that above ring frequency, the pipe will behave like a plate. This leads to different dispersion curves on either side of the ring fre-quency. This can be seen in the changing behavior of dispersion curves near ω = ωring in figure (1.5).

1.1.5 Normalized Frequency -

Ω

The ring frequency ωring plays an important role in the pipe

dynam-ics and so it is convenient to normalize the frequency ω with the ring frequency and represent this as shown below:

Ω = ω

ωring (1.9)

1.1.6 Mode order m, n and p

Throughout this report the mode order will be defined using the letters m, n and p. This is described below:

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2. n - Circumferential order of fluid or pipe mode. Defined as the number of full sine waves of fluid-pressure/pipe-displacement around pipe circumference

3. p - Radial order of fluid mode. Defined as number of pressure nodes in fluid mode between r=0 and r=R

1.1.7 Fluid-pipe coupling

A brief of fluid-pipe coupling will be mentioned here. The pressure field of a fluid in a circular cylindrical cavity will have shapes defined by the Bessel Function of the first kind. The boundary condition be-tween the fluid and pipe is defined by the linearized fluid momentum equation (neglecting effects of advection and viscosity) relating pres-sure in the fluid to radial displacement of the pipe shown in equation (1.10). This equation is the backbone of all subsequent fluid-structure interaction methods.

∂p

∂r = −ρ0 ∂w

∂t (1.10)

Fluid external to the pipe is assumed to be of low density and thus its influence is neglected. The equations presented will thus represent a fluid filled pipe in a vacuum and the results are accurate as long as the external fluid density is low and pipe bending stiffness is high.

The radial force on the pipe from the fluid pressure field can thus be incorporated in L33term, i.e. 3rdcolumn which represents pipe radial

motion with 3rd _{row which represents external radial force (equation}

1.3).

For a particular circumferential mode order ’n’, the modification is shown in equation (1.11) and (1.12) taken from [3]:

L33= 1 + β2∇4 + 1 ω2 r ∂2 ∂t2 − F L (1.11)

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Where the terms in equation (1.12) are defined as: • kr

s is the radial wavenumber for branch s

• Jn(x)is the Bessel function of First kind of order ’n’ at ’x’

• J′

n(x)is the derivative of Bessel function at ’x’

The coupled system can thus be represented through system equa-tion (1.3) by modifying L33. The method presented here is the direct

application of physics to the interface. Other methods are presented in section (1.4) and the most useful method for analytic solution is chosen with justification.

1.1.8 Characteristic Equation and Dispersion relation

Many of the results that will be obtained from the system equations (1.3) will involve solving the equation for determinant ([Lpipe]) = 0.

This is called the characteristic equation of the system.

For the in-vacuo case, the characteristic equation has order of ks

equal to 8 and order of Ω equal to 6.

A particular wavenumber will have 6 different frequency solutions satisfying the system equation. These 6 solutions represent 3 pairs of waves with different combinations of Radial, Tangential and Axial contribution propagating in positive and negative axial direction.

For the fluid-filled pipe case, the presence of the FL term in equa-tion (1.11) makes the characteristic equaequa-tion non-linear because the Bessel function is of oscillating nature and can provide an infinite num-ber of solutions. Thus a single frequency can have an infinite numnum-ber of wavenumber solutions, corresponding to fluid modes with increas-ing radial nodes.

Relation between axial wavenumber and frequency gives the dis-persion relation and this is obtained from the system characteristic equation. It is shown using a kzvs. Ω plot (example in figure 1.5).

In this pipe system, complex valued wavenumbers arise and a clever way of representing the dispersion relation (as can be seen in papers like [3], [4]) is to plot the Real part of kz above the x-axis and

Imagi-nary part of kz below the x-axis. This provides information about the

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1.1.9 Energy calculation

Energy in pipe and fluid can be compared by calculating their poten-tial (elastic for pipe and pressure for fluid) and kinetic energy. Since a harmonic system without losses is being considered, the two ener-gies will be equal at resonant frequency and thus for the end purpose of determining ratio of pipe and fluid energy of different modes, just the rms (root means square) kinetic energy is calculated as it is more straightforward. These equations for pipe and fluid are shown in (1.13) and (1.14) respectively. KEpipe= h 2 Z S ρpipe(vrms(z, θ))2dS (1.13)

Where h is the pipe thickness, ρpipe is pipe material density, vrms is

root-mean-square velocity at pipe mid-surface and S is mid-surface of the pipe. KEf luid= 1 2 Z V ρf luid(vrms(z, r, θ))2dV (1.14)

Where ρf luid is fluid density, vrms is root-mean-square velocity of

fluid and V is volume enclosed by fluid.

1.2 Fluid filled pipe characteristics

This section is a description of relevant fluid-filled pipe systems ob-tained from Chapters 4 and 7 in [1], and more detailed information from the papers [3] and [5].

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1.2.1 Influence of pipe thickness and density ratios

From the ’Fluid loading’ term in equation (1.12), it is clear that de-creasing the pipe thickness or inde-creasing the fluid density will increase magnitude of fluid-loading term. The influence of ’FL’ magnitude on the coupled fluid-pipe system is not very direct and can be better un-derstood through calculation of energy ratios in the fluid and pipe as done in [3]. The following conclusions are drawn:

1. If there is a pipe and a fluid wave propagating at a particular frequency, the energy is either mostly in the fluid or mostly in the pipe depending on the type of excitation.

2. Increasing pipe thickness or reducing fluid density will make the energy predominantly exist in one of the systems, resulting in rigid-walled fluid waves and radially stiff pipe waves.

3. Decreasing pipe thickness or increasing fluid density will lead to pressure-release type fluid waves and increasingly slow flexural

pipewaves.

1.2.2 Influence of pipe material properties

Stiffness of pipe material determines the phase velocity of different waves. Remarks about wave behavior for pipe made of less stiff mate-rial than steel is shown below:

1. Choosing a material with longitudinal or torsional wave speed lesser that the speed of sound in the fluid (e.g. rubber pipe with water) completely removes occurrence of coincidence between fluid and pipe waves.

2. Energy is mostly concentrated in the pipe at low frequencies due to higher cut-on frequencies of acoustic modes.

3. High impedance from fluid waves which are still not cut-on at Ω = 1lead to n = 0 pipe breathing mode to be suppressed. 4. Fluid modes cut-on and propagate as pressure-release waves

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1.2.3 Mobility towards radial excitation

The mobility of fluid-pipe systems towards radial excitation is ana-lyzed in [5]. Some relevant statements are listed below:

1. Propagation of acoustically slow pipe waves generates a decay-ing pressure field close to the shell wall. This fluid acts as an added mass to the system.

2. Presence of rigid-walled fluid waves increases the stiffness expe-rienced by the pipe.

1.3 Natural in-vacuo modes of vibration

In this section, theory from subsection (1.1) will be used to determine natural frequencies of pipe and fluid in-vacuo (uncoupled).

1.3.1 Infinite pipe

An infinite pipe unlike an infinite plate, is bounded in the circumfer-ential dimension. Interference of waves with equal but opposite cir-cumferential wavenumbers allow for integer wavelengths along the circumference (n=0,1,2..) where n is the circumferential order of the mode.

For each circumferential mode, there are an infinite number of nat-ural modes associated with the unbounded axial dimension. Of partic-ular interest are the natural frequencies of the circumferential modes with infinite axial wavelength kz = 0. These frequencies can be viewed

as cut-on frequencies of circumferential modes and can be calculated from the system matrix 1.3 by applying the following solution ansatz and parameter values:

The solution ansatz of u, v and w are taken as:   u v w  =   Acos(kss)cos(nθ)cos(ωt) Bsin(kss)sin(nθ)cos(ωt) Csin(kss)cos(nθ)cos(ωt)   (1.15)

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The specific reason for the sine and cosine terms for circumferential variation in equation (1.15) is to ensure a vertically symmetric mode throughout. An example with circumferential order =1 is shown in figure (1.2).

Figure 1.2: Vertically symmetric modes using appropriate trigonomet-ric functions forθ- example shown for n=1 circumferential order

The axial variation is represented with sine and cosine terms to al-low for a smooth transition into finite pipe case that will folal-low in sub-section 1.3.2.

Characteristic Equation

By applying the solution ansatz given in equation (1.15) to the pipe-system equation (1.3) with the following parameter values:

• Free-vibration by setting RHS of (1.3) = 0

• Infinite axial wavelength by setting kz = ks/R = 0 to represent

cut-on of mode

This leads to an eigenvalue problem with 3 solutions for a given mode number n. Three solutions arise due to the coupling between the axial, tangential and radial motions in a pipe. Eigenvectors for each eigenvalue will reveal the primary contributor for each solution and thus eigenvalues with significant radial contribution (i.e., A/C and B/C less than 1) are readily identified.

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Figure 1.3: Normalized natural frequencies for circumferential modes of an infinite pipe in-vacuo, h/R = 0.053

Figure (1.3) shows different behavior of the radial, tangential and axial modes. Not that a mode is assigned to a category based on the primary contributing motion; this means other 2 motions also exist in that mode.

Important features from figure (1.3):

1. 0 Hz for n=0 Axial (and) Tangential and n=1 Radial mode is be-cause of Rigid Body motion of the infinite pipe and will not arise in the finite pipe case.

2. Radial modes begin with a natural frequency Ωradial,n=0= 1 which

corresponds to the ring frequency and subsequent modes be-gin at a much lower frequency Ωradial,n=1 << Ωradial,n=0.

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1.3.2 Finite pipe

The method used to calculate natural frequencies of infinite pipe in subsection (1.3.1) can be easily extended to the case of a finite pipe by considering Close Shell / Shear Diaphragm boundary conditions.

Close Shell / Shear Diaphragm:

This boundary condition is the physical equivalent of closing the end of a pipe with a thin circular sheet such that radial and circumfer-ential displacement is restricted but axial displacement is still possible. This is by virtue of a large stiffness of the thin sheet in the radial and tangential direction but negligible stiffness in the axial direction. From the solution ansatz given in equation (1.15), Radial and Tangential mo-tion is dependent on axial posimo-tion through the sine funcmo-tion and Axial motion through the cosine function. Thus when Radial and Tangen-tial motion is zero at an axial position, Axial motion is maximum and vice-versa. This can thus represent the Close Shell/ Shear diaphragm condition.

This boundary condition is analytically tractable because the solu-tion ansatz in equasolu-tion (1.15) for the infinite pipe case can be directly applied to this case. Thus a given pipe of length L and axial mode number m (’m’ half sinusoidal wavelengths along the length) implies an axial wavenumber kz = mπ/L.

Characteristic EquationSimilar to infinite pipe case, by using the

solution ansatz in equation (1.15) in equation (1.3) and applying the following parameter values:

• Free-vibration by setting RHS of (1.3) = 0

• Axial mode number ’m’ by setting kz = ks/R= mπ/L

Setting determinant of the following matrix = 0 yields the char-acteristic equation with 3 eigenvalues for each combination of (m,n) (similar to the infinite pipe case). Ratio of the axial, tangential and radial components can be extracted from the eigenvectors.

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Figure 1.4: Natural frequencies and radial contributions of finite pipe for different n with m=1 , h/R = 0.053

Remarks on figure (1.4):

1. In figure (1.4), each colored line has 3 markings representing the 3 eigenvalues for that particular (m=1,n) combination. The x-axis and y-axis represent its frequency and fraction of radial motion. 2. Higher values of m produce a very similar plot and no new

in-formation. Thus just a single m value is taken for plotting figure (1.4).

3. It is clear that increasing ’n’ also increases the maximum nor-malized natural frequency obtained. This is expected because higher circumferential order involves larger gradients and thus more stiffness.

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5. All other modes apart from n=0, have the radial mode (i.e., mode with largest radial contribution) occurring at the smallest root of Ω. An exception is n=1, which has good coupling between the radial and tangential mode thus giving 2 modes with almost equal radial contribution (as well as tangential, not shown here). 6. As the circumferential order increases, radial contribution to the higher roots steadily decreases. This implies that at higher modes (n ≥ 3, only the smallest root of Ω is relevant for fluid-Structure coupling.

Dispersion Curve

The dispersion curve for in-vacuo finite pipe is calculated using the system matrix and shown in figure (1.5).

Figure 1.5: Dispersion curve for in-vacuo finite pipe, L = 1 m, h/R = 0.053

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frequencies (Ω < 1) is seen where slope of the curves do not rise as rapidly as expected from literature. Possible reasons can be differences in numerical methods used and also differences in pipe parameters.

Other Boundary Conditions

The shear diaphragm is not common in physical situations and thus serves as a tool for understanding fluid-pipe interactions. Imple-menting other boundary conditions like clamped, free, etc. involves using a solution ansatz with an exponential representation for the axial variation as shown in equation (1.16). It is to be noted that Boundary condition at the end of the pipe should be independent of θ and t.

  u v w  =   Aeλs_{cos(nθ)cos(ωt)} Beλs_{sin(nθ)cos(ωt)} Ceλs_{cos(nθ)cos(ωt)}   (1.16)

The characteristic equation is now solved for the λ instead of ω and this yields an 8th order differential equation. The solutions to λ have the form given in equation (1.17) for n>=1.

λ = ±λ1, ±iλ2, ±(λ3± iλ4) (1.17)

Taking this λ, the displacement fields (u,v,w) can be expressed as a combination of the 8 different λ’s and involve 8 Real-valued unknowns (A1, A2 .. A8). Boundary conditions at the ends are then applied to this

system of equations and a coefficient matrix is obtained from which the 8 unknowns (A1, A2 .. A8) can be determined. By setting

determi-nant of this coefficient matrix = 0, natural frequencies can be solved for which holds true for the particular boundary condition. Thus the mode shape and the natural frequency can be obtained. These cal-culations are not performed as they are not relevant for this project. Further information can be found in [2] Chapter 2.4.

1.3.3 Fluid volume

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natural frequencies of fluid modes in a pipe, wave equation (equa-tion (1.1) in cylindrical coordinates is necessary. It is given in equa(equa-tion (1.18): ∂2_p ∂t2 + 1 r ∂p ∂r + 1 r2 ∂2_p ∂θ2 + ∂2_p ∂z2 + k 2_{p = 0} _(1.18) Where

• (r,θ,z) are the cylindrical coordinates • p is the pressure field

• k is the acoustic wavenumber = ω/c

Fluid boundary conditions will be specified using the linearized fluid-momentum equation given in equation (1.10). This equation is derived from the momentum equation of fluids after neglecting effects of advection and viscosity and retaining only linear terms.

Velocity field at the surface of the pipe due to pipe vibration is taken to be periodic in axial and circumferential direction as men-tioned in the ansatz equation (1.15). Thus the pressure field inside the fluid will also be of the same periodicity because this allows for equation (1.10) to be satisfied everywhere on the surface.

Given a radial velocity distribution on the plate of the form in equa-tion (1.19):

w(z, θ, t) = wz(z)wθ(θ)eωt (1.19)

The pressure field can be taken to be of the form in equation (1.20): p(r, z, θ, t) = pr(r)pz(z)pθ(θ)eiωt (1.20)

Where the periodicity of z and θ terms in equations (1.19) and (1.20) are the same.

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Here n and kz define the circumferential and axial wavenumber.

Applying this periodicity in equation (1.21) to the pressure field (equation (1.20)) and substituting this in the cylindrical wave equation (1.18) yields equation (1.22): ∂2_p r(r) ∂r2 + 1 r ∂pr(r) ∂r + (k 2 − k2z− n r 2 )pr(r) = 0 (1.22)

Solution to pr(r)in equation (1.22) is the Bessel function of the first

(Jn(r)) and second kind (Yn(r)). Bessel function of the second kind

tends to ∞ at r = 0 and since this is unrealistic, it is not considered for fluid modes inside pipes. Solution for pr(r) is as shown in equation

(1.23).

pr(r) = AJn[(k2− k2z)

1

2r] (1.23)

Where

• Jnis the nthorder Bessel function of the first kind

• (k2_{− k}2 z)

1

2 can be seen as the radial wavenumber k_r

• A is amplitude of pr(r)

Surface boundary condition

The final step is to determine pressure gradient at the surface of the pipe. A simple MATLAB code is used that can approximate values of krr that yield the required value of Jn(x)or its derivative Jn′(x). It is

important to note that, given the periodic nature of Bessel functions more than one solution can be expected corresponding to different ’p’ values (p is the radial order, which is the number of concentric circular nodal lines).

For the purpose of representing fluid pressure as a combination of in-vacuo modes, the rigid walled assumption is used: this implies

∂pr(r)

∂r = 0at the wall. The plot in figure (1.6) shows for required slope

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Figure 1.6: Plot of different order Bessel functions: p-value mentioned for n=1 with marked points

Natural Frequencies

Given surface boundary condition, circumferential order ’n’ and radial order ’p’, the value x where Jn(x) satisfies the conditions (e.g.

for rigid boundary, J′

n(x) = 0) is found using the MATLAB code. It is

then a straightforward mathematical task to convert this value to the natural frequency as shown in equation (1.24):

ω(n, p, R) = c

x/R (1.24)

Where R is radius of the pipe and c is speed of sound in the fluid.

Mode Shape

The radial component of the fluid mode shape for a given n, p and pipe radius R can be generated once the value x where Jn(x)satisfies

the conditions is found. This is shown in equation (1.25). pr(r) = Jn(x

r

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The circumferential and axial variation depends on variation in pipe displacement defined in equation (1.19) .

Dispersion curve

Continuing to use the rigid-wall modes, frequencies are calculated and resulting dispersion plot is shown in figure (1.7). Nature of the curves are similar to that from literature ([1] Section 5.14).

Figure 1.7: Dispersion curve for rigid-wall fluid modes. ωring for

nor-malization is taken from pipe data (= 22kHz)

1.4 Fluid-structure coupling principles

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1.4.1 Interaction analysis by Green’s function

This method is mentioned in [1] Section 7.5. The idea is to represent displacement as a combination of its in-vacuo pipe modes. Beginning with the governing equation of pipe displacement and incorporating effects of the fluid, appropriate integration is performed making use of the orthogonality property of in-vacuo mode shapes to obtain the governing equation for each pipe mode which now includes the effects of direct and cross fluid loading.

The direct fluid loading represents the influence of fluid modes with same circumferential order ’n’ as the pipe mode. Cross fluid load-ing represents the influence of other pipe modes of order ’n’ that indi-rectly influence wp through the coupling fluid medium.

1.4.2 L33 modification

The method used in the paper [3] involves modifying the pipe system matrix [L] by adding a Fluid Loading term FL to L33mentioned before

in equation (1.11). This is the term corresponding to radial direction (row = 3) influenced by radial forces (column = 3), precisely how fluids influence the pipe.

The method converts a relatively easy system matrix into a non-linear system because the FL term involves the Bessel functions. Thus complex root-finding algorithms are required to obtain solutions.

1.4.3 Modal Interaction Model

This is the method mentioned in [1] Section 7.6 and is similar to Green’s representation mentioned in subsection (1.4.1). In this method, both the pipe and fluid displacement is represented as a combination of their respective in-vacuo modes (in-vacuo here refers to set of orthog-onal modes when fluid and pipe sub-systems are uncoupled).

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modes have no velocity and therefore no displacement at the pipe sur-face whereas pressure-release fluid modes have no pressure at the pipe surface. On the other hand, a coupled fluid-mode will have both pipe-induced displacement and pipe-pipe-induced pressure on the surface. A so-lution using this method will thus produce results which are accurate in a least-squares sense. Theoretically, infinite set of in-vacuo modes are required but only a limited set will be used. The limit is chosen when calculated eigenfrequency does not change by more than 0.01% with addition of more in-vacuo modes.

Rigid-walled fluid modes are used as the set of orthogonal fluid modes because the pipe-fluid combination considered is a steel pipe with low density fluid, which resembles a rigid pipe more than a flex-ible pipe. Thus convergence is expected to be quicker if rigid-walled fluid modes are used.

Following the usual procedure of integration and applying orthog-onality rules, a coupled system matrix is obtained with a governing equation for each pipe and fluid mode. Governing equation for each mode has terms to describe interaction with the other fluid and pipe modes.

1.4.4 Chosen Method for FSI

The Modal interaction method (MIM) mentioned in subsection (1.4.3) is chosen to conduct analytical fluid-pipe interaction analysis. The rea-sons for this choice are:

1. Representation of displacement and pressure as combination of in-vacuo modes allows for analyzing coupled modes in terms of un-coupled modes. The modes can then be categorized as ’pipe’, ’fluid’ or ’coupled’.

2. In-vacuo modes needed for solving using MIM can be easily cal-culated as mentioned in the Section (1.3).

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1.5 Viscosity and its influence

Incorporating effect of Viscosity in the Acoustic wave in fluids involve replacing the Ideal fluid model with a Newtonian fluid model done by including terms which are previously considered negligible when lin-earizing the equations governing fluid particles. The important steps and assumptions involved is mentioned in this section. Chapter 10 from [6] is used as reference for the content that follows.

Viscosity is a commonly occurring term in fluid mechanics and is related to forces arising from shear in the liquid. It plays an important role in incorporating realistic boundary conditions at surfaces and also in including dissipation effects in fluid volume. With respect to acous-tic waves, the attenuation effect of viscosity can be brought up in wave propagation through the fluid medium as well as near surfaces.

The mathematics used for derivations in the following sub-sections require prior knowledge of tensors, material derivative, concept of perturbations and linearization. The general principle is to apply mass, momentum and energy conservation principles to a fluid particle. Af-terwards, using thermodynamic relations and assuming small harmonic variations (perturbations) in the fluid property, the higher order terms are neglected resulting in linear acoustic equations.

1.5.1 Fluid Dynamics equations

To reach the Acoustic wave equation (1.1) requires using mass, mo-mentum and energy conservation equations of a fluid particle assum-ing adiabatic process and neglectassum-ing viscosity. In this section, the same conservation equations are used including viscosity and advection. In the final equations, effect of advection is removed by taking thermal conductivity to be 0.

Mass equation:

The mass equation is unaffected by viscosity terms to begin with and is shown in equation (1.26).

∂ρ ∂t +

∂ρvi

∂xi

= 0 (1.26)

Where v is the velocity vector and ρ is fluid density.

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Viscosity incorporates the no-slip boundary condition at the sur-face (vsurf ace = 0) . The basic wave equation incorporates only the

no-penetration boundary condition (v.n = 0, where n is the surface normal vector) but due to the no-slip condition, force on the surface is no longer restricted to be along its normal. Thus the stress tensor σij is

required for representing the momentum equation and this is shown in equation (1.27). ρDvi Dt = ∂σij ∂xj (1.27) Where D/Dt is the material derivative that incorporated convec-tive effects.

Energy Equation:

The basic wave equation assumes adiabatic process and so Ds/Dt = 0. Including effects of viscosity (as well as thermal effects) lead to the equation of energy balance shown in equation (1.28).

ρDu Dt = σij

∂vi

∂xj − ∇.q

(1.28) Where u is the internal energy and q is the heat flux vector.

Constitutive Relation:

To establish the constitutive relations, the following assumptions are made:

• Assuming shear stress is proportional to the rate of shear (i.e., Newtonian Fluid) implies stress tensor can only be modeled in a particular way and this relation is shown in equations (1.29) and (1.30).

• An assumption is made that for low frequency and for small spa-tial variation the fluid is assumed to be in thermodynamic equi-librium. This gives rise to the relation between thermodynamic quantities shown in equation (1.31).

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σij = −pδij + µφij (1.29) φij = ∂vi ∂xj +∂vj ∂xi − 2 3∇.vδij (1.30)

µis the fluid’s dynamic viscosity (units [Pa.s]) and φij is also called

the rate-of-shear tensor.

ds = 1 Tdu +

p

Td(1/ρ) (1.31)

q = κ_∇T (1.32)

Where T is the temperature and κ is coefficient of thermal conduc-tivity.

Navier Stokes equation:

Applying constitutive relations to momentum equation (1.27) leads to the famous N.S. equation (1.33).

ρDvi Dt = − ∂p ∂xi +∂µφij ∂xj (1.33)

Kirchhoff Fourier equation:

The equation that arises when applying constitutive relations to Energy balance equation (1.28) is the Kirchhoff Fourier equation shown in (1.34).

ρTDs Dt =

µ

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1.5.2 Linear Acoustic Equations

Perturbations are introduced as follows in the variables : p′_{, ρ}′_{, T}′_{, s}′_{, v}′.

These perturbations are related to each other through the equilibrium state and parameters like cp, c and β which are the Specific heat at

constant pressure, speed of sound and coefficient of thermal expansion respectively. These relations are not explicitly mentioned here as they are not immediately relevant but the interested reader can find it in Chapter 10 from [6].

Applying perturbations to equations (1.26), (1.33) and (1.34) and using thermodynamic relations between the perturbations lead a set of 3 equations. These equations when linearized by ignoring 2nd_order

terms gives equations (1.35), (1.36) and (1.37). ∂ρ′ ∂t + ρ0∇.v = 0 (1.35) ρ0 ∂vi ∂t = − ∂p ∂xi + µ∂φij ∂xj (1.36) ρ0T0 ∂s ∂t = κ∇ 2_T′ (1.37)

It is to be noted that the super script "’" is removed from perturba-tion quantities v, p and s. The subscript "0" denotes the equilibrium state of the quantity.

1.5.3 Modal Wave fields

Applying a harmonic time dependence on perturbed variables in equa-tions (1.35), (1.36) and (1.37) and afterwards deriving dispersion rela-tion by setting Real part of the equarela-tion = 0 lead to 3 types of waves with their distinct dispersion relation and relationship between per-turbed variables.

The 3 waves are called Vorticity, Entropy and Acoustic modes. Equation of Acoustic mode is shown in (1.38).

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Here, δcl is given in equation (1.39). δcl = µ 2ρ0 (4 3 + γ − 1 P r ) (1.39)

Here γ is the ratio of specific heats (γ = cp/cv) and Pr is the Prandtl

number ( P r = cpµ/κ).

Some features about these waves are listed below:

• These waves are linearly independent of each other and can exist simultaneously in the same space. Therefore, superposition of these waves is allowed.

• The Acoustic and Entropy modes have v k k while for vorticity mode v ⊥ k

• At the surface, all 3 waves must exist together in order to satisfy the no-slip and isothermal boundary condition.

• The imaginary part of wavenumber for Vorticity and Entropy modes are very large and thus these modes die out rapidly leaving only the Acoustic mode to propagate. Their presence can only be felt close to boundaries.

Acoustic Boundary Layer:

Close to a surface, all 3 modes exist to satisfy the no-slip and isother-mal boundary condition given by v = 0 and T’=0 respectively. The isothermal condition at the surface is valid provided heat capacity and heat conductivity of the solid is much larger than the fluid which is true for most cases. The length scale normal to the surface up to which the Vorticity and Entropy modes are significant is called the boundary-layer thickness and is shown in equation (1.40) and (1.41) respectively.

lvor = r 2µ ωρ (1.40) lent = s 2κ ωρcp = √lvor P r (1.41)

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1.5.4 Wave attenuation due to volume

For an acoustic wave propagating in free space in direction x1 free of

any boundaries, attenuation does occur if viscous effects are included because of dissipation of energy due to momentum transfer (by virtue of viscosity). This attenuation is represented by αcl which can be

un-derstood through equation (1.42) and its magnitude is given in equa-tion (1.43).

|ˆp| = |ˆp|x=0e−αclx (1.42)

Where |ˆp| is amplitude of pressure perturbation and units of αcl is

per unit length.

αcl =

ω2_δ cl

c3 (1.43)

1.5.5 Wave attenuation due to surface

Plane wave through rigid walled pipe:

The attenuation of greater magnitude and concern is that which occurs at surface boundary.

For pipes which are the focus of this project, wavenumber "k" of plane waves through circular cross-sections has been derived in ([6]) and shown in equation (1.44) and αwallsin equation (1.45) after

apply-ing assumptions listed below:

1. Radius of pipe should be much larger than acoustic boundary layer thickness (R >> lvor, lent)

2. Pipe cross-section is constant

3. Attenuation due to pipe surface is much larger than attenuation due to fluid volume (αwalls>> αcl). This is calculated and shown

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Figure 1.8: Plane wave propagating in rigid walled pipe with water -αdue to viscous effects in fluid volume vs. at pipe surface

k = ω c + (1 + i)αwalls (1.44) αwalls = 1 √ 2R r ωµ ρc2(1 + γ + 1 √ P r) (1.45)

Higher Order waves through rigid walled pipe:

Literature from [7] provides equations for determining αwalls (here

termed as αc) for higher order fluid modes propagating through rigid

walled circular ducts after they cut-on. These are shown below:

For axisymmetric modes i.e. n = 0, equation (1.46) and for asym-metric modes equation (1.47) applies.

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30 CHAPTER 1. LITERATURE STUDY α0 = 1 R s (πµf ρc2 ) (1.48)

The variables used in equations (1.46), (1.47) and (1.48) are defined below:

• fc: cut on frequency for higher order mode

• n: circumferential order of mode • R: Radius of circular duct

• µ, ρ, c: Dynamic viscosity, density and speed of sound of fluid • kRnp : (acoustic wavenumber)x(radius) at cut-on of the higher

order mode with circumferential order ’n’ and radial order ’p’ Plots of α calculated using these formulas is shown in figure (1.9). It is to be noted that equation (1.46) reduces to equation (1.45) for n=0.

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The case of higher order modes below cut-on is not considered be-cause these waves even without viscous effects considered are already heavily attenuated by their complex valued wavenumber. Further de-tails about their wave attenuation due to viscosity can be found in pa-per [8].

Fluid waves through Flexible circular pipe:

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Chapter 2 Analytical solution for coupled

fluid-pipe system

2.1 Modal interaction model - MIM

2.1.1 Method

The steps in MIM are as follows:

1. Determine in-vacuo natural frequencies and mode shapes for fi-nite pipe and fluid volume with specified pipe geometry and ma-terial properties for fluid and pipe.

2. Calculate norm of pipe and fluid mode-shapes as well as the cou-pling coefficient

3. Generate system matrix for the pipe and fluid modes.

4. Solve for the coupled natural frequencies (eigenvalues) and mode-shapes (eigenvectors) by setting determinant of system matrix = 0

2.1.2 Procedure

In the following section, steps to apply MIM will be elaborated with required formulas and parameters. A more comprehensive derivation can be found in Section 7.6 in [1].

In-Vacuo Pipe and Fluid natural frequencies and mode shapes

Given the following parameters:

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• Pipe Geometry - Radius (R), Length (L), thickness (h)

• Pipe Material Properties - Young’s Modulus (E), Poisson’s ratio (ν) and density (ρp)

• Fluid Medium Properties - Speed of sound (c), Density (ρ0)

Natural frequencies and mode shapes can be calculated for any given (m,n) for pipe and (m,n,p) for fluid as explained in Section (1.3). Mode shapes for sample modes are shown in figure (2.1) for fluid pres-sure field and figure (2.2) for pipe displacement field.

Representation of fluid pressure ˆp and pipe displacement ˆw The method involves representing fluid pressure ˆp and plate dis-placement ˆw of a coupled mode as a combination of their respective in-vacuo modes, as shown in equations (2.1) and (2.2). The in-vacuo mode shapes are orthogonal to each other and this comes useful when deriving governing equation for each coupled mode.

ˆ p(V ) =X N pNψN(V ) (2.1) ˆ w(S) = X P wPφP(S) (2.2)

Where in equations (2.1) and (2.2):

• V is a point in the enclosed fluid volume • S is a point on surface of the pipe

• pN is contribution of fluid mode N

• wP is contribution of pipe mode P

• ψN is mode shape of fluid mode N

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34 CHAPTER 2. ANALYTICAL SOLUTION FOR COUPLED FLUID-PIPE SYSTEM

(a) n=0,p=1

(b) n=1,p=1

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(a) n=0

(b) n=1

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Norm and Coupling Coefficient

Given the 2-D pipe mode shape φP and 3-D fluid mode shape ψN,

the norm Λ and coupling coefficient CN P can be calculated as shown

in equations (2.3), (2.4) and (2.5). ΛP = Z S φP(S)2dS (2.3) ΛN = Z V ψN(V )2dV (2.4) CN P = Z S ψN(S)φP(S)dS (2.5)

Where S is surface of the pipe and V is volume of fluid enclosed by the pipe

System equations

For the fluid mode, obtaining the coupled governing equation in-volves the following steps:

1. Begin with inhomogeneous wave equation, i.e., equation (1.1) with an unknown mass source term in the RHS.

2. Implement influence of pipe modes as a mass source term through pipe displacement.

3. Represent fluid pressure ˆp and pipe displacement ˆwas a combi-nation of in-vacuo modes as shown in equations (2.1) and (2.2). 4. For fluid mode ’N’, multiply equation with ψNand integrate over

V. Orthogonality of mode shapes lead to a simplified governing equation shown in equation (2.6).

ω_N2pN + ¨pN = − c2_ρ 0 ΛN X P CN Pw¨P (2.6)

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1. Begin with force balance equation for pipes as shown in equation (1.3) including Radial pressure forces on the RHS.

2. Implement influence of fluid modes as a radial pressure.

3. Represent fluid pressure ˆp and pipe displacement ˆwas a combi-nation of in-vacuo modes as shown in equations (2.1) and (2.2). 4. For pipe mode ’P’, multiply equation with φP and integrate over

S. Orthogonality of mode shapes lead to a simplified governing equation shown in equation (2.7).

ω2PwP + ¨wP = 1 mΛP X N pNCN P (2.7)

where m is the mass per unit area of the pipe.

System Matrix assembly

N in-vacuo fluid modes and P in-vacuo pipe modes lead to (N+P) governing equations each with (N+P) terms that can be represented with a single system matrix in the following way:

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• (i,j) is the row and column index (1-indexed) • Nnois the number of fluid modes

• Ni refers to ith_{fluid mode and Pj refers to j}th_{pipe mode}

Modified System Matrix Assembly

Due to orthogonality of the fluid and pipe modes, it is more conve-nient to group fluid and pipe modes of the same (m,n) order together. This is done by individually assembling each sub-matrix like as show in equation (2.8) but only for uncoupled fluid and pipe modes with same (m,n) values. These sub-matrices are then either solved indi-vidually and later combined, or assembled together along the main diagonal and solved as a whole. This leads to easier eigenvector calcu-lations and in the case of individually solving sub-matrices the ability to increase number of in-vacuo fluid modes.

Choice of variable to represent fluid field

To represent the fluid field, either pressure or velocity potential can be used. Velocity potential is related to pressure as shown in equation (2.9).

p = −ρf luid

∂Φ

∂t (2.9)

NOTE: Pipe mode shape is represented using φ and velocity

po-tential as Φ.

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2.2 Convergence test

The number of uncoupled fluid modes used to represent the coupled fluid mode is limited by computational and practical reasons. Let pupper represent the number of uncoupled fluid mode shapes (p = 0,

1, .. (pupper-1)) used to represent the coupled fluid mode.

A relative convergence test is performed by comparing results ob-tained from cases with different pupper. The results are shown in figure

(2.4).

Figure 2.4: p Relative Convergence test. p = pupper

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Figure 2.5: Plot of modal contributions from uncoupled fluid modes for coupled fluid mode

Keeping in mind that uncoupled fluid modes are all of the rigid-walled type, the plot in figure (2.5) can be understood as follows. A ’fast’ coupled fluid mode closely resembles an uncoupled rigid-walled fluid mode and thus reaches convergence quickly. A ’slow’ coupled fluid mode is in-between a rigid-wall and a pressure-release fluid mode and thus needs many uncoupled modes to be well represented, thus slowly converging with increasing pupper.

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2.3 Results

2.3.1 Coupled Natural frequencies

For natural frequencies, RHS of the matrix in equation (2.8) is set to 0. This implies solving ω for determinant of system matrix = 0 yield natural frequencies of the coupled system.

Comparing the uncoupled and coupled frequencies in low (Ω < 3) and high frequency (Ω > 3) regions is shown in figure (2.6).

NOTE:For this analysis, the maximum (m,n) is limited to (5,4) and

p takes values up to 5. This limitation is applied only for generating figures (2.6) so that number of coupled modes that arise are limited and this allows for making remarks about modes without loss of gen-erality.

The figures in (2.6) give a qualitative understanding of frequency regions where coupling effect can be expected.

1. At very low frequencies (Ω < 0.2), coupled modes exist as pipe modes with negligible radial motion (due to stiffness of pipe to undergo radial motion of long wavelength) and rigid walled fluid modes (also due to stiffness of pipe). The fluid acts as an added mass to the pipe vibration, thereby reducing the coupled natural frequency.

2. Around ring frequency (Ω = 1), the pipe becomes very compli-ant to radial motion and thus couples with fluid modes. This leads to mode shapes with natural frequencies different from the uncoupled modes.

3. At very high frequencies (Ω > 3), fluid modes with larger ’p’ values cut-on while pipe modes of same (m,n) values are of com-paratively low-frequency. This frequency mismatch lead to cou-pled modes with predominantly rigid walled fluid modes and negligible pipe motion. Thus there is minor difference in natural frequency of coupled and uncoupled modes.

2.3.2 Uncoupled mode contributions

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(a) Ω < 3

(b) Ω > 3

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be used to understand the extent of fluid-pipe coupling, generate the coupled mode shapes and also generate other useful expressions like ratio of Fluid to Pipe Kinetic energy.

The plot for 2 coupled modes’ pressure and displacement fields are shown in figures in (2.7).

Pressure field in the coupled mode shape shown in figure (??) is seen to be a combination of (n=0,p=0) and (n=0,p=1) uncoupled fluid modes and their contribution as seen from the eigenvector is in the ratio 1:0.2.

The eigenvector also reveals the coupled fluid mode in figure (??) to be primarily made of (n=1,p=0) and (n=1,p=1) uncoupled fluid modes with contributions in the ratio 1:0.01. Thus this mode is predominantly that of p=1 as can be seen from the mode shape.

Studying the eigenvector reveals many expected features of cou-pled modes mentioned below:

• For a given (m,n), only in-vacuo pipe and fluid modes with the same (m,n) will effectively couple and the remaining modes can be neglected.

• The contribution to a coupled mode at a particular frequency pri-marily comes from uncoupled modes with similar natural fre-quencies.

• Furthermore when including uncoupled fluid modes with fquencies very high compared to uncoupled pipe modes, the re-sulting coupled mode is predominantly the uncoupled fluid mode itself with negligible pipe motion.

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(a) Ω = 0.4

(b) Ω = 0.84

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Chapter 3 COMSOL Simulation

The COMSOL Finite Element Modeling software is used for perform-ing numerical simulations of fluid-pipe vibrations. Aspects of Model-ing, in-built modules and material properties involved in the simula-tion is presented in this secsimula-tion.

3.1 COMSOL modules

COMSOL offers many modules to analyze a fluid-pipe system.

For the pipe, Solid Mechanics module is used for 2D, 2D-axisymmetric and 3D. The Shell module is used for 3D model when representing the pipe as a planar surface. Further details are mentioned in Section (3.7). For the fluid, if convection, viscous and thermal effects are ne-glected the Pressure Acoustics module is sufficient for capturing fluid behavior. To incorporate viscosity the Thermoviscous Acoustics module is used. Further details are mentioned in Section (3.8) and (3.9).

3.2 Parameters

The first step in the goal towards simulation is to assign variables and their quantities which will be used for successfully setting up, running and manipulating a simulation. The variables can be used for the fol-lowing aspects:

• Model geometry: To define parameters like Radius, pipe thick-ness, radial force position, etc.

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• Mesh control: To define parameters than ensure a mesh of suffi-cient fineness. eg. points per wavelength, number of mesh layers in pipe walls, etc.

• Material properties: To define the material itself (solid or fluid), eg. steel’s Young’s Modulus, fluid’s density, etc.

• Study control: Study settings like parametric sweep involve ma-nipulating these variables to understand their effects on simula-tion.

3.3 Geometric model

The required geometry is created using variables listed in ’Parame-ters’. The types of components used depend on whether it is 2D or 3D model. The 3 types of geometric models used for representing pipe-fluid systems are shown in figure (3.1).

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48 CHAPTER 3. COMSOL SIMULATION

3.4 Meshing

Meshing of a model is critical for obtaining reliable results. In acous-tics, rule of thumb is to have at least 6 nodes per wavelength and this applies to fluid as well as solid domain.

Variables like points per wavelength, λmin, maximum axial order,

maximum circumferential order and number of mesh layers are used for ensuring mesh has sufficient fineness to capture the wave.

For viscosity modeling, extra effort goes into defining the acous-tic boundary layer separately and creating a Boundary layer mesh to capture variations close to the surface. Figures of these meshes will be shown in Section (3.9).

3.5 Studies performed

Eigenfrequency:

Eigenfrequency study is the most used study in this project and is performed on the model to extract eigenfrequencies and mode shapes. Only the geometry and boundary conditions are incorporated and not forces on the model. This study is efficient at identifying modes in the system and for understanding their properties. The number of eigenfrequencies to be solved for is case-dependent and varies from 20 for preliminary checks to 200 for extended analysis.

Frequency Domain:

The frequency domain study is performed when a spectrum across different frequencies is desired for a given radial force input. The com-putation is done for each frequency separately and time consumed de-pends on the number of frequency steps.

Parametric Sweep:

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3.6 Multiphysics

A model which studies the interaction between elements from dif-fernt modules (like pipe element in Solid Mechanics and fluid element in Pressure Acoustics) requires a Multiphysics implementation at the interface. This feature couples the modules using equations like the Fluid-structure interaction equation (1.10) for the case of fluid and pipe elements at their interface.

Similar multiphysics interfaces are available between all combina-tions of modules listed in Section (3.1).

3.7 Simulation of Pipe

Pipe elements follow equations that govern behavior of Linear elastic materials.

3.7.1 COMSOL module

In 3D component set-up, COMSOL offers 2 possible ways to simulate the pipe: As a volume domain element using Solid Mechanics module or as a shell surface element using Shell module. They are shown in figure (3.2):

In 2D and 2D axisymmetric, only Solid Mechanics module is used for simulating the pipe.

3.7.2 Material Properties

The parameters required to describe pipe material are: • Density - ρ

• Young’s Modulus - E • Poisson’s ratio - ν

3.7.3 Meshing

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(a) Volume element

(b) Shell element

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define this and an example is shown in figure (3.3). More examples are shown in figure (4.16).

Figure 3.3: Meshing of pipe with layers

For the case of using ’Shell’ to represent pipe, sufficient nodes should be present on the surface such that at least 6 nodes per wavelength are present for maximum circumferential and axial order considered.

3.7.4 Choice of Pipe representation in 3D

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Figure 3.4: Comparison of Eigenvalues - Volume vs. Shell To choose a method to represent the pipe, a convergence study is performed. The relative error vs. computational time is shown for volume and shell element in figure (3.5) . Convergence is obtained by refining the mesh and the relative error is calculated using eigenvalues of the most-refined mesh.

Figure (3.4) shows that both the representations give almost iden-tical results. From figure (3.5), it is clear that the shell element is more reliable and also less computationally expensive. Thus for 3D pipe

representation ’shell’ elements will be utilized.

NOTE: The shell representation is only available in COMSOL 3D

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Figure 3.5: Relative error in 3D invacuo-pipe Eigenvalues vs. compu-tational time

3.8 Simulation of ideal fluid

Most of the simulations of fluids in this report are performed assuming it is an ideal fluid. Theory behind this is from the basic wave equation (1.1) with appropriate boundary conditions and sources applied on the RHS by COMSOL.

3.8.1 COMSOL module

If fluid domain can be considered stationary (Mach number Ma < 0.03) and also viscous and thermal effects can be neglected then the Pressure

Acousticsmodule can reliably simulate the fluid.

3.8.2 Material Properties

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• Speed of sound - c

3.8.3 Meshing

The rule of thumb is to ensure there are at least 6 nodes per wavelength for the maximum frequency considered. If ppw is the points per wave-lenght and fmax the maximum frequency considered, the minimum

element size - dminis given by equation (3.1).

dmin =

c fmax∗ ppw

(3.1)

3.9 Simulation of Viscous fluid

Incorporating viscous effects complicate the wave equation because of adding terms previously neglected. The literature in Section (1.5) mentions the equations that are used in COMSOL simulations. It must be noted that for Ideal fluids: Pressure (1 unknown) alone is to be solved for each node. In the case of Viscous simulation: Pressure (1 unknown), velocity (3 unknowns) and Temperature (1 unknown) are required.

3.9.1 COMSOL module

The Thermoviscous Acoustics module is used along with Pressure

Acous-tics module with a Multiphysics interface between them. Reason for

including Pressure Acoustics module is given in Section (3.9.3).

3.9.2 Material Properties

To include thermal and viscous effects in the fluid domain, additional fluid properties apart from those listed for the Pressure Acoustics mod-ule are required to describe the fluid domain. These are listed below:

• Dynamic viscosity - µ • Bulk viscosity - µb

(67)

• Specific heat at constant pressure - cp

• Ratio of specific heats - γ

Note: For ensuring accurate representation of viscous effects, γ is

required only if the equilibrium density (ρ) of the fluid is not defined as a function of equilibrium pressure (p0). Similarly for representing

thermal effects, γ is required if (ρ) is not defined as a function of equi-librium temperature T0.

3.9.3 Computational optimization

As mentioned before, the Thermoviscous Acoustics module solves for 5 unknowns leading to considerable increase in computational time. This can be optimized by understanding that viscous effects are only dominant near the surface. Thus by appropriately splitting the fluid domain into 2 parts i.e, one narrow region near the surface with

Ther-moviscous Acousticsand remaining region with Pressure Acoustics the

computation power can be better used. This splitting is shown in fig-ure (3.6) with the narrow region exaggerated for visual convenience.

Analytic, Simulation and Experimental Analysis of Fluid-Pipe Systems

Fluid-Pipe Systems

SIDDHARTH VENKATARAMAN

Abstract

Sammanfattning

Acknowledgement

Contents

List of parameters

Document Description

Literature Study

1.1

Basics

1.1.1

Acoustic Wave equation in fluids

1.1.2

Structural vibration in a pipe

1.1.3

Pipe parameter -

β

1.1.4

Ring frequency -

ω

1.1.5

Normalized Frequency -

Ω

1.1.6

Mode order m, n and p

1.1.7

Fluid-pipe coupling

1.1.8

Characteristic Equation and Dispersion relation

1.1.9

Energy calculation

1.2

Fluid filled pipe characteristics

1.2.1

Influence of pipe thickness and density ratios

1.2.2

Influence of pipe material properties

1.2.3

Mobility towards radial excitation

1.3

Natural in-vacuo modes of vibration

1.3.1

Infinite pipe

1.3.2

Finite pipe

1.3.3

Fluid volume

1.4

Fluid-structure coupling principles

1.4.1

Interaction analysis by Green’s function

1.4.2

L33 modification

1.4.3

Modal Interaction Model

1.4.4

Chosen Method for FSI

1.5

Viscosity and its influence

1.5.1

Fluid Dynamics equations

1.5.2

Linear Acoustic Equations

1.5.3

Modal Wave fields

1.5.4

Wave attenuation due to volume

1.5.5

Wave attenuation due to surface

Chapter 2

Analytical solution for coupled

fluid-pipe system

2.1

Modal interaction model - MIM

2.1.1

Method

2.1.2

Procedure