# Turbulence-induced vibration in annular flow of a rigid cylinder mounted on a cantilever beam

Romain Lagrange, Loucas Plado Costante, Maud Kocher

► To cite this version:

Romain Lagrange, Loucas Plado Costante, Maud Kocher. Turbulence-induced vibration in annular flow of a rigid cylinder mounted on a cantilever beam. Journal of Fluids and Structures, 2024, 131, pp.104213. 10.1016/j.jfluidstructs.2024.104213. cea-04767560

HAL Id: cea-04767560

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Submitted on 5 Nov 2024

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# Turbulence-induced vibration in annular flow of a rigid cylinder mounted on a cantilever beam.

Romain Lagrange<sup>a</sup>, Loucas Plado Costante<sup>a</sup>, Maud Kocher<sup>b,c</sup>

<sup>a</sup>Université Paris-Saclay, CEA, Service d'Etudes Mécaniques et Thermiques, F-91191, Gif-sur-Yvette, France

<sup>b</sup>EDF Lab Paris-Saclay, 7 bd Gaspard Monge, 91120 Palaiseau, France

<sup>c</sup>IMSIA UMR CNRS-EDF-CEA-ENSTA 9219, France

## Abstract

This study investigates the fluid-structure interaction of two coaxial cylinders separated by a Newtonian fluid under turbulent axial flow. The theoretical framework treats the inner cylinder as a rigid body mounted on a flexible blade modeled as a Rayleigh beam. The goals of this study are to determine the free vibration modes and frequencies, identify the fluid-elastic instability threshold, and establish an analytical expression for the mean-square displacement of the structure. The approach integrates various fluid forces and torques, such as Archimedean thrust, fluid-elastic forces for a quiescent fluid, fluid-elastic forces due to flow, and the effects of fluid turbulence. The new approach reveals that vibration modes, frequencies, instability thresholds, and mean-square displacement each depend on a different set of dimensionless parameters: 8, 11, and 12, respectively. These parameters include the cylinder aspect ratio and fluid gap radius ratio. By incorporating models from the literature for viscous friction coefficients, turbulent pressure power spectral density, and coherence function, the study demonstrates stability conditions and the scaling of mean-square displacement with Reynolds number squared. The study, presented in a fully dimensionless formulation, aims to assist engineers in constructing small-scale experiments representative of pressure vessel vibrations. To facilitate this, a Python code for system stability determination and mean-square displacement calculation is provided.

**Keywords:** Turbulence-induced vibration; Annular flow; Fluid-elastic instability; Power spectral density; Coherence function; Root mean-square displacement

## Nomenclature

$\tilde{P}$ Turbulent pressure

$\tilde{p}$ Dimensionless turbulent pressure

**U** Displacement vector of the neutral axis of the flexible blade

**Q** Column vector containing the dimensionless modal amplitudes $q_j$

$\chi$ Coherence function

$\delta$ Dirac function

i Unit imaginary number

$\kappa$ Dimensionless quadratic moment of area of the flexible blade

$\Lambda_{\eta}, \Lambda_{\phi}$ Correlation lengths

$C_1$ Inner cylinder

$C_2$ Outer cylinder

$G$ Gravitational acceleration

$\nu$ Kinematic viscosity of the fluid

$\omega_0^{(j)}$ Dimensionless angular frequency in an inviscid still fluid of the j-th free vibration mode of the blade

$\omega_s^{(j)}$ Dimensionless structural angular frequency of the j-th free vibration mode of the blade

$\bar{\alpha}$ Complex conjugate of $\alpha$

Email address: romain.lagrange@cea.fr (Romain Lagrange)
$\partial C_i$ Boundary of the cylinder $C_i$

$\phi$ Orthoradial coordinate angle

$\Re, \Im$ Real and imaginary part operators

$\rho$ Volume mass density of the flexible blade

$\rho_F$ Volume mass density of the fluid

$\Sigma^2$ Mean-square value of the structure displacement

$\sigma^2$ Dimensionless mean-square value of the structure displacement

$\tilde{\Gamma}$ Random turbulence excitation fluid torque acting on the inner cylinder

$\tilde{F}$ Random turbulence excitation fluid force acting on the inner cylinder

$F_0$ Fluid-elastic force for a fluid at rest

$F_a$ Archimedean thrust

$F_U$ Fluid-elastic force due to fluid flow

$F$ Resulting fluid force acting on the inner cylinder

$R$ Internal resultant force in the flexible blade

$\Gamma_0$ Fluid-elastic torque for a fluid at rest

$\Gamma_U$ Fluid-elastic torque due to fluid flow

$\theta$ Angular deviation of the neutral axis of the flexible blade

$\theta_L$ Angular deviation at the tip of the flexible blade

$\varepsilon$ Radius ratio of the annular fluid gap

$\xi, \zeta$ Dimensionless Rayleigh damping coefficients of the flexible blade, proportional to mass and stiffness

$\Xi, Z$ Rayleigh damping coefficients of the flexible blade, proportional to mass and stiffness

$\{e_x, e_y, e_z\}$ Direct orthonormal basis

$C_f$ Darcy friction factor

$C_T, C_N$ Tangential and normal friction coefficients

$C_U, A_U, K_U, D_U$ Added coefficients due to fluid flow

$c_u, a_u, k_u, d_u$ Dimensionless added coefficients due to fluid flow

$E$ Young's modulus of the flexible blade

$F_d$ Viscous drag force acting on the inner cylinder

$f_d$ Dimensionless viscous drag force

$f_r$ Reduced frequency

$G$ Center of mass of the inner cylinder

$g$ Dimensionless gravitational acceleration

$H$ Length of the inner cylinder

$h$ Cylinder-to-blade aspect ratio

$I$ Quadratic moment of area of the flexible blade, with regard to the axis $e_z$

$J_{cyl}$ Moment of inertia of the inner cylinder about the axis ($G, e_z$)

$j_{cyl}$ Dimensionless moment of inertia of the inner cylinder

$L$ Length of the flexible blade

$l$ Aspect ratio of the inner cylinder

$M$ Internal bending moment in the flexible blade

$M_0, C_0, J_0, D_0$ Added coefficients for a fluid at rest

$m_0, c_0, j_0, d_0$ Dimensionless added coefficients for a fluid at rest
$M_{cyl}$ Mass of the inner cylinder

$m_{cyl}$ Dimensionless cylinder mass

$m_F$ Dimensionless fluid mass

$N$ Internal normal force in the flexible blade

$O$ Origin point of coordinate $X = 0$

$Q$ Volume flow rate of the axial flow

$q_j$ Dimensionless modal amplitude of the j-th vibration mode $w_j$

$R_H$ Hydraulic radius

$R_i$ Radius of the cylinder $C_i$

$Re$ Reynolds number of the axial flow

$S$ Section area of the flexible blade

$Sk$ Stokes number

$T$ Dimensional time

$t$ Dimensionless time

$U$ Axial displacement of the neutral axis of the flexible blade (component of $U$ along $e_x$)

$u$ Dimensionless mean flow velocity

$U_c$ Convective speed

$U_m$ Mean flow velocity of the axial flow

$V$ Internal shear force in the flexible blade

$W$ Transverse displacement of the neutral axis of the flexible blade (component of $U$ along $e_y$)

$w$ Dimensionless transverse displacement

$w_0^{(j)}$ j-th free vibration mode of the blade in an inviscid still fluid

$W_G$ Transverse displacement of the center of mass of the inner cylinder

$X$ Axial coordinate

$x$ Dimensionless axial coordinate

$\Gamma$ Moment about point $G$ of the resulting fluid force acting on the inner cylinder

$\mathbf{n}$ Unit normal vector of the neutral axis of the flexible blade

$\mathbf{n}_i$ Outward unit normal vector of $\partial C_i$

$\mathbf{t}$ Unit tangent vector of the neutral axis of the flexible blade

$\mathcal{P}_j, w^\dagger, \theta^\dagger, w_G^\dagger$ Basis and test functions

$\tilde{\gamma}$ Component along $e_z$ of the dimensionless random turbulence excitation fluid torque

$\tilde{f}$ Component along $e_y$ of the dimensionless random turbulence excitation fluid force

$S_{\tilde{P}}$ Turbulent pressure power spectral density

$s_{\tilde{p}}$ Dimensionless turbulent pressure power spectral density

$\Omega$ Angular frequency of vibration

$\omega$ Dimensionless angular frequency of vibration

$\Omega_0$ Characteristic angular frequency
# 1. Introduction

This work is part of a comprehensive effort led by Electricité de France (EDF) in partnership with the French Alternative Energies and Atomic Energy Commission (CEA) to develop a numerical tool capable of simulating vibrations in the primary circuits of nuclear Pressurized Water Reactors (PWRs) during regular operation. To achieve this, theoretical models must be developed, tested against simple analytical experiments that isolate different physical effects, and eventually incorporated into the future calculation code. Here, our focus is on the vessel and the internal core support cylinder of a PWR, a simplified sketch of which is depicted in Figure 1. Inside a PWR, the coolant enters the pressure vessel through two, three, or four cold legs, with inlets evenly spaced around the circumference. The coolant flows down the annular passage between the reactor vessel and the core support cylinder, reverses its direction in the lower plenum, exchanges heat with the fuel assembly and exits the vessel through two, three of four hot legs. The cold leg jets impacting the top of the core barrel and the flow within the annular space generate pressure fluctuations and unsteady fluid forces, leading to random vibrations of the core support cylinder. To prevent fatigue failure due to these flow-induced vibrations, it is crucial to develop theoretical models that estimate the root mean-square (RMS) amplitude of the core support cylinder’s response. These models are grounded in prototypical fluid-structure interaction problems, using simplified geometries and turbulence spectrum models to simulate complex real interactions, such as jet impacts. One widely studied simplified model is that of a flexible cylinder subjected to axial flow, as extensively investigated in works such as Paidoussis (1966); Païdoussis (1966); Chen and Wambsganss (1972); Païdoussis (1973); Païdoussis and Ostoja-Starzewski (1981); Païdoussis (2024). Previous studies indicate that at low flow velocities, the cylinder experiences small, random vibrations, while at higher velocities, it may encounter fluid-elastic instabilities (divergence or flutter). These instabilities arise from interactions with the mean flow field, characterized by the mean axial flow velocity. Vibrations at low velocities are termed sub-critical vibrations. Although often overlooked at a fundamental level, sub-critical vibrations are crucial for practical engineering considerations, as fluid-elastic instabilities typically occur beyond normal operating conditions Païdoussis (1980). While fluid-elastic instabilities have received significant attention, sub-critical vibration, though less explored at the fundamental level, holds paramount importance in engineering systems Païdoussis (1980). It is well-established Païdoussis (1980); Chen (1974); Païdoussis (1974a,b, 1981, 1983) that any mean flow entails a perturbation flow field with pressure fluctuations originating from turbulence and far-field disturbances. This perturbation field, acting on the cylinder’s surface, constitutes the excitation for sub-critical vibration. Although other mechanisms have been proposed in the past Païdoussis (1974a), sub-critical vibration is commonly described as buffeting, similar to the buffeting experienced by cylinders in cross-flow. Sub-critical vibration typically involves small amplitudes, generally ranging from $10^{-2}$ to rarely exceeding $10^{-1}$ of the cylinder radius. While vibrations of such minuscule amplitudes might be disregarded under normal circumstances, their significance arises in practical applications, especially when dealing with confined cylinders. Turbulence-induced vibrations, though not causing immediate structural failure, can lead to long-term damage, such as fatigue and fretting wear. The treatment of turbulence-induced vibration often centers on viewing it as a result of random forces inherent to turbulent flow. The phenomenological characterization of turbulence excitation by axial flows has been addressed by many authors in the past, since the pioneering work of Corcos (1964); Clinch (1969); Willmarth and Wooldridge (1962); Bull (1967); Bakewell (1968); Chase (1987); Bull (1996), among others, and also in more recent times, see Au-Yang (2001); Ciappi et al. (2012). These works represent turbulence excitation as a force $\mathbf{F}(M, T)$ acting at location point $M$ of the structure, as a random function of time $T$. Convenient formulations for describing such the turbulence excitation and for computing the structure response can be developed, starting with the classical random linear vibration theory. Assuming that $\mathbf{F}(M, T)$ is stationary Gaussian and ergodic, it may be completely characterized by its cross-correlation spectrum $[\mathbf{F}|\mathbf{F}]$, formally defined as

$$[\mathbf{F}|\mathbf{F}](M, M', \Omega) = \int_{-\infty}^{\infty} \left[ \lim_{\Delta \to \infty} \frac{1}{\Delta} \int_{0}^{\Delta} \mathbf{F}(M, T) \cdot \mathbf{F}(M', T - T') \, dT \right] \exp(-i\Omega T') \, dT' \quad (1)$$

The mean-square value of the structure displacement can then be obtained from

$$\Sigma^2(M) = \frac{1}{2\pi} \int_{0}^{\infty} [\mathbf{U}|\mathbf{U}](M, \Omega) \, d\Omega. \quad (2)$$

This response is characterized by the auto-correlation spectrum

$$[\mathbf{U}|\mathbf{U}](M, \Omega) = \int_{\partial C} \int_{\partial C} H(M, M_1, \Omega) \bar{H}(M, M_2, \Omega) [\mathbf{F}|\mathbf{F}](M_1, M_2, \Omega) \, dS_1 dS_2, \quad (3)$$
where $H(M, M_i, \Omega)$ (with complex conjugate denoted as $\bar{H}$) represents a transfer function that correlates the structure displacement of point $M$ with the turbulent fluid force at point $M_i$. While this statistical approach does not provide access to the detailed time history of the response, it enables the inference of the spectral distribution of quantities linearly related to displacement, particularly stress and strain within the structure. The present study aims to expand this general approach to the case of a simplified model of a PWR, featuring coaxial cylinders; see Fig. 2. The outer cylinder represents the rigid reactor vessel, while the inner cylinder simulates the core support, which is modeled as a rigid cylinder mounted on a flexible blade and exposed to annular axial flow. This configuration allows for the capture of some of the vibrational characteristics of a PWR's core support by varying the properties of the blade. The fluid turbulence excitation is modeled using an auto-correlation spectrum of pressure applied to the inner flexible cylinder. This turbulence spectrum is intended to encompass various sources of turbulence present in the actual configuration, including those originating from cold leg jets. The geometrical and material properties characterizing the simplified model are provided in § 2.

<figure><img src="image_2.png" /><figcaption>Figure 1: Sketch of the vessel and the internal core support cylinder of a PWR; from Tong et al. (2021).</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Schematic diagram of coaxial cylinders (axes along $e_x$) in turbulent axial fluid flow. The inner cylinder $C_1$ (length $H$, radius $R_1$) is rigid, has a center of mass $G$ (acceleration of gravity is $G$) and is mounted on a cantilever flexible blade (length $L$). The mean flow velocity is $U_m$. The conventional (resp. reverse) direction flow corresponds to $U_m > 0$ (resp. $U_m < 0$), i.e. flow in the $e_x$ (resp. $-e_x$) direction. The system vibrates in the $e_y$ direction.</figcaption></figure>

The paper is organized as follows. In § 2, we introduce the problem, the governing equations, their dimensionless formulations, perform a modal analysis, establish an analytical expression for the mean-square structure displacement, and determine the fluid-elastic stability conditions. In § 3, we present models from the literature needed to complement our theory and establish theoretical predictions, considering three sets of dimensionless parameters. Finally, some conclusions are drawn in § 4.

## 2. Theoretical analysis

### 2.1. Presentation of the problem

Let $C_i$, $i \in \{1, 2\}$, be two coaxial rigid circular cylinders, with axes along $e_x$ (axial coordinate $X$), radius $R_i$ and boundary $\partial C_i$ (outward unit normal vector $n_i = \cos(\phi)e_x + \sin(\phi)e_y$, $\{e_x, e_y, e_z\}$ being a direct orthonormal basis and $\phi$ being the orthoradial angle), see Fig. 2. The two cylinders are subject to an axial flow whose volume flow rate is $Q(T)$, with $T$ the dimensional time. The inner cylinder $C_1$ has a length $H$, a mass $M_{cyl}$ and is subject to the acceleration of gravity $G$. Its center of mass is denoted $G$ and its moment of inertia about the axis ($G$, $e_z$) is $J_{cyl}$. The cylinder $C_1$ is mounted on a flexible blade, assimilated to a beam clamped at the origin point $O$ ($X=0$). It has a length $L$, a quadratic moment of area $I$ with regard to the axis $e_z$, a section area $S$, a Young's modulus $E$ and a volume mass density $\rho$. The blade and the cylinder $C_1$ are subject to a turbulent axial flow of a Newtonian fluid of volume mass density $\rho_F$ and kinematic viscosity $\nu$. The effect of gravity on the fluid is not taken into consideration. The mechanical loading of the fluid on the blade is disregarded, whereas its action on $C_1$ is modeled by a resulting fluid force $F$, whose moment about point $G$ is $\Gamma$. The kinematics of the neutral axis of the blade is modeled by a displacement vector $U(X, T) = U(X)e_x + W(X, T)e_y$, $X \in [0, L]$. In the instantaneous configuration, this neutral axis has tangent and normal vectors $t = \cos(\theta)e_x + \sin(\theta)e_y$ and $n = -\sin(\theta)e_x + \cos(\theta)e_y$, respectively. The angular deviation $\theta(X, T)$ at the tip of the blade ($X = L$) is denoted $\theta_L(T)$. The internal resultant force (resp. moment) is $\mathbf{R} = N\mathbf{t} + V\mathbf{n}$ (resp. $\mathbf{M} = M\mathbf{e}_z$), with $N$ the normal force, $V$ the shear force and $M$ the bending moment. Cross-sections of the blades are assumed to remain plane and orthogonal to the neutral axis (Euler-Bernoulli assumption, i.e. $M = EI\partial\theta/\partial X$). As the cylinder $C_1$ is rigid, the $e_y$ component of the displacement of its neutral axis is of the form $W(X, T) = W(L, T) + \sin(\theta_L(T))(X - L)$, $X \in ]L, L+H]$. Therefore, the transverse displacement of the center of mass $G$ ($X = L + H/2$) is $W_G(T) = W(L, T) + \sin(\theta_L(T))H/2$.

In what follows, $i$ refers to the unit imaginary number, $\mathbb{R}$ (resp. $\mathbb{C}$) to the real (resp. imaginary) part operator,
$\bar{\alpha}$ to the complex conjugate of $\alpha$. The cross-correlation spectrum $[S_1|S_2]$ between two random stationary ergodic fields $S_i$ is defined as

$$[S_1|S_2](X, \phi, X', \phi', \Omega) = \int_{-\infty}^{\infty} \left[ \lim_{\Delta \to \infty} \frac{1}{\Delta} \int_{0}^{\Delta} S_1(X, \phi, T) S_2(X', \phi', T-T') dT \right] \exp(-i\Omega T) dT'. \quad (4)$$

The hydraulic radius is defined as $R_H = (R_2 - R_1)/2$ and the mean flow velocity is

$$U_m = \lim_{\Delta \to \infty} \frac{1}{\Delta} \int_{0}^{\Delta} \frac{Q(T)}{\pi (R_2^2 - R_1^2)} dT. \quad (5)$$

The conventional (resp. reverse) direction flow corresponds to $U_m > 0$ (resp. $U_m < 0$), i.e. flow in the $\mathbf{e}_x$ (resp. $-\mathbf{e}_x$) direction.

## 2.2. Dimensional equations

The conservation of linear and angular momentums, applied to a weightless infinitesimal element of the blade, yields the equilibrium equations

$$0 = \frac{\partial N}{\partial X} \cos(\theta) - N \frac{\partial \theta}{\partial X} \sin(\theta) - \frac{\partial V}{\partial X} \sin(\theta) - V \frac{\partial \theta}{\partial X} \cos(\theta), \quad (6a)$$

$$\rho S \frac{\partial^2 W}{\partial T^2} = \frac{\partial N}{\partial X} \sin(\theta) + N \frac{\partial \theta}{\partial X} \cos(\theta) + \frac{\partial V}{\partial X} \cos(\theta) - V \frac{\partial \theta}{\partial X} \sin(\theta), \quad (6b)$$

$$\rho I \frac{\partial^2 \theta}{\partial T^2} = \frac{\partial M}{\partial X} + V. \quad (6c)$$

Similarly, the equilibrium equations for the rigid cylinder $C_1$ are

$$0 = -N(L) \cos(\theta_L) + V(L) \sin(\theta_L) - M_{cyl}G + \mathbf{F} \cdot \mathbf{e}_x, \quad (7a)$$

$$M_{cyl} \frac{d^2 W_G}{dT^2} = -N(L) \sin(\theta_L) - V(L) \cos(\theta_L) + \mathbf{F} \cdot \mathbf{e}_y, \quad (7b)$$

$$J_{cyl} \frac{d^2 \theta_L}{dT^2} = -M(L) + \frac{H}{2} V(L) + \mathbf{\Gamma} \cdot \mathbf{e}_z. \quad (7c)$$

To first order in $\theta$, Eq. (6a) is automatically satisfied, whereas Eq. (7a) yields $N(L) = \mathbf{F} \cdot \mathbf{e}_x - M_{cyl}G$, with $G$ the gravitational acceleration. Plugging Eq. (6c) into Eq. (6b), with $\theta = \partial W / \partial X$, yields

$$\rho S \frac{\partial^2 W}{\partial T^2} + \frac{\partial}{\partial T} \left( 2\Xi\rho S W + 2Z EI \frac{\partial^4 W}{\partial X^4} \right) + EI \frac{\partial^4 W}{\partial X^4} + M_{cyl}G \frac{\partial^2 W}{\partial X^2} - \rho I \frac{\partial^4 W}{\partial T^2 \partial X^2} = \frac{\partial^2 W}{\partial X^2} \mathbf{F} \cdot \mathbf{e}_x. \quad (8)$$

In Eq. (8), the second term on the left was introduced a posteriori to model the structural damping of the blade. Terms $\Xi$ and $Z$ refer to Rayleigh damping coefficients proportional to mass and stiffness; see Rayleigh (1896).

The boundary conditions (7) at $X = L$ reduce to

$$M_{cyl} \left( \frac{\partial^2 W}{\partial T^2} + \frac{H}{2} \frac{\partial^3 W}{\partial T^2 \partial X} \right) = - \left( \mathbf{F} \cdot \mathbf{e}_x - M_{cyl} G \right) \frac{\partial W}{\partial X} - \left( \rho I \frac{\partial^3 W}{\partial T^2 \partial X} - EI \frac{\partial^3 W}{\partial X^3} \right) + \mathbf{F} \cdot \mathbf{e}_y, \quad (9a)$$

$$J_{cyl} \frac{\partial^3 W}{\partial T^2 \partial X} = -EI \frac{\partial^2 W}{\partial X^2} + \frac{H}{2} \left( \rho I \frac{\partial^3 W}{\partial T^2 \partial X} - EI \frac{\partial^3 W}{\partial X^3} \right) + \mathbf{\Gamma} \cdot \mathbf{e}_z. \quad (9b)$$

In this study, it is assumed that the resultant fluid force $\mathbf{F}$ and torque $\mathbf{\Gamma}$ acting on $C_1$ expand as

$$\mathbf{F} = \mathbf{F}_a + \mathbf{F}_0 + \mathbf{F}_U + \tilde{\mathbf{F}}, \quad (10a)$$

$$\mathbf{\Gamma} = \mathbf{\Gamma}_0 + \mathbf{\Gamma}_U + \tilde{\mathbf{\Gamma}}, \quad (10b)$$

with $\mathbf{F}_a = \rho_F \pi R_1^2 H G \mathbf{e}_x$ the Archimedean thrust, $\mathbf{F}_0$ (resp. $\mathbf{\Gamma}_0$) the fluid-elastic force (resp. torque) for a fluid at rest, $\mathbf{F}_U$ (resp. $\mathbf{\Gamma}_U$) the fluid-elastic force (resp. torque) due to fluid flow and $\tilde{\mathbf{F}}$ (resp. $\tilde{\mathbf{\Gamma}}$) the random turbulence excitation fluid force (resp. torque).
According to Lagrange and Puscas (2023), the fluid force $\mathbf{F}_0$ and torque $\mathbf{\Gamma}_0$ are

$$ \mathbf{F}_0 = \left( -M_0 \frac{d^2 W_G}{dT^2} - C_0 \frac{d W_G}{dT} \right) \mathbf{e}_y, \qquad (11a) $$

$$ \mathbf{\Gamma}_0 = \left( -J_0 \frac{d^2 \theta_L}{dT^2} - D_0 \frac{d \theta_L}{dT} \right) \mathbf{e}_z. \qquad (11b) $$

The terms $M_0$, $C_0$, $J_0$ and $D_0$ refer to the added coefficients for a fluid at rest. These coefficients are given in Appendix B.

According to Taylor (1952); Hawthorne (1961); Païdoussis (1973); Païdoussis and Ostoja-Starzewski (1981); Païdoussis (2024), the fluid force $\mathbf{F}_U$ and torque $\mathbf{\Gamma}_U$ are

$$ \mathbf{F}_U = \left( -C_U \frac{dW_G}{dT} - A_U \frac{d\theta_L}{dT} - K_U \theta_L \right) \mathbf{e}_y + F_d \mathbf{e}_z, \qquad (12a) $$

$$ \mathbf{\Gamma}_U = -D_U \frac{d\theta_L}{dT} \mathbf{e}_z. \qquad (12b) $$

The terms $C_U$, $A_U$, $K_U$ and $D_U$ refer to the added coefficients due to fluid flow. These coefficients, along with the viscous drag force $F_d$, are given in Appendix C.

In this study, it is assumed that the fluid force $\tilde{\mathbf{F}}$ and torque $\tilde{\mathbf{\Gamma}}$ derive from a turbulent pressure $\tilde{P}$ field, i.e.

$$ \tilde{\mathbf{F}}(T) = \int_{L}^{L+H} \int_{0}^{2\pi} -\tilde{P}(X, \phi, T) \cos(\phi) R_1 \, d\phi \, dX \, \mathbf{e}_y, \qquad (13a) $$

$$ \tilde{\mathbf{\Gamma}}(T) = \int_{L}^{L+H} \int_{0}^{2\pi} -\left(X - L - \frac{H}{2}\right) \tilde{P}(X, \phi, T) \cos(\phi) R_1 \, d\phi \, dX \, \mathbf{e}_z, \qquad (13b) $$

whereas the term $\tilde{\mathbf{F}} \cdot \mathbf{e}_x$ is neglected. The turbulent pressure is characterized by its cross-correlation spectrum $[\tilde{P}|\tilde{P}](X, \phi, X', \phi', \Omega)$.

## 2.3. Dimensionless equations

Let $L$ and $\Omega_0 = \sqrt{EI/(\rho SL^4)}$ be a characteristic length and angular frequency. We define $t = \Omega_0 T$, $x = X/L$ and $w = W/L$ as the dimensionless time, axial coordinate and displacement, respectively. The dimensionless cross-correlation spectrum $\langle s_1|s_2 \rangle$ between two dimensionless random stationary ergodic fields $s_i = S_i/S_i^{(0)}$ is $\langle s_1|s_2 \rangle = \Omega_0 [S_1/S_1^{(0)}|S_2/S_2^{(0)}]$, i.e.

$$ \langle s_1|s_2 \rangle (x, \phi, x', \phi', \omega) = \int_{-\infty}^{\infty} \left[ \lim_{\delta \to \infty} \frac{1}{\delta} \int_0^{\delta} s_1(x, \phi, t) s_2(x', \phi', t-\tau) dt \right] \exp(-i\omega\tau)d\tau, \qquad (14) $$

with $\omega = \Omega/\Omega_0$ the dimensionless angular frequency. According to Eq. (2), the dimensionless mean-square structure displacement, $\sigma^2 = \Sigma^2/(R_2-R_1)^2$, is

$$ \sigma^2(x) = \left(\frac{l}{h\varepsilon-1}\right)^2 \frac{1}{2\pi} \int_0^\infty \langle w|w \rangle(x, \omega) d\omega. \qquad (15) $$

In dimensionless form, Eq. (8) are

$$ \frac{\partial^2}{\partial t^2} \left[ w - \kappa^2 \frac{\partial^2 w}{\partial x^2} \right] + 2 \frac{\partial}{\partial t} \left[ \xi w + \zeta \frac{\partial^4 w}{\partial x^4} \right] + \frac{\partial^4 w}{\partial x^4} - \left[ (m_F - m_{cyl}) g + f_d \right] \frac{\partial^2 w}{\partial x^2} = 0, \qquad (16) $$

whereas Eq. (9) reduces to

$$ \frac{\partial^2}{\partial t^2} \left[ (m_{cyl} + m_0) \left( w + \frac{h}{2} \frac{\partial w}{\partial x} \right)_1 + \kappa^2 \frac{\partial w}{\partial x}_1 \right] + \frac{\partial}{\partial t} \left[ (c_0 + c_u) \left( w + \frac{h}{2} \frac{\partial w}{\partial x} \right)_1 + a_u \frac{\partial w}{\partial x}_1 \right] \\ + \left[ (m_F - m_{cyl}) g + k_u + f_d \right] \left. \frac{\partial w}{\partial x} \right|_1 - \left. \frac{\partial^3 w}{\partial x^3} \right|_1 = \tilde{f}, \qquad (17a) $$

$$ \frac{\partial^2}{\partial t^2} \left[ \left( j_{cyl} + j_0 - \frac{h}{2} \kappa^2 \right) \frac{\partial w}{\partial x} \right]_1 + (d_0 + d_u) \left. \frac{\partial}{\partial t} \frac{\partial w}{\partial x} \right|_1 + \left. \frac{\partial^2}{\partial x^2} \left( w + \frac{h}{2} \frac{\partial w}{\partial x} \right) \right|_1 = \tilde{\gamma}. \qquad (17b) $$
Equations (16)-(17) involve 8 independent dimensionless parameters :

$$g = \frac{G}{L\Omega_0^2}, \quad m_{cyl} = \frac{M_{cyl}}{L\rho S}, \quad m_F = \frac{\rho_F \pi R_1^2 H}{L\rho S}, \quad h = \frac{H}{L}, \quad \kappa = \sqrt{\frac{I}{L^2 S}}, \quad j_{cyl} = \frac{J_{cyl}}{L^3 \rho S}, \quad \xi = \frac{\Xi}{\Omega_0}, \quad \zeta = Z\Omega_0, \quad (18)$$

referred to as the dimensionless gravitational acceleration, the dimensionless cylinder mass, the dimensionless fluid mass, the cylinder-to-blade aspect ratio, the dimensionless quadratic moment of area of the blade, the dimensionless inertia of the cylinder, the reduced mass and stiffness Rayleigh damping coefficients, respectively. The dimensionless added coefficients for a fluid at rest, appearing in Eqs. (16)-(17), are defined as

$$m_0 = \frac{M_0}{L\rho S}, \quad c_0 = \frac{C_0}{L\rho S\Omega_0}, \quad j_0 = \frac{J_0}{L^3\rho S}, \quad d_0 = \frac{D_0}{L^3\rho S\Omega_0}. \quad (19)$$

The calculation detailed in Appendix B demonstrates that the added coefficients may be expressed as

$$m_0 = m_F \Re\{F^{(1)}\}, \quad c_0 = -m_F \omega \Im\{F^{(1)}\}, \quad j_0 = h^2 m_F \Re\{F^{(2)}\}, \quad d_0 = -h^2 m_F \omega \Im\{F^{(2)}\}. \quad (20)$$

In Eq. (20), $F^{(i)}$ is a complex-valued function derived from integrating the linear fluid force $f_0$ over the length of the inner cylinder; see Eq. (B.1). The calculation of $F^{(i)}$ builds on our previous work, Lagrange and Puscas (2023), which provides an analytical expression for $f_0$. In Lagrange and Puscas (2023), we examined small-amplitude forced beam vibrations of two coaxial, finite-length cylinders separated by a viscous Newtonian fluid. We derived a comprehensive analytical expression for $f_0$ using a Helmholtz expansion of the fluid velocity vector. We demonstrated that $F^{(i)}$ is a function of the three dimensionless numbers, $l$, $\varepsilon$ and $\omega Sk$, defined as

$$l = \frac{H}{R_1}, \quad \varepsilon = \frac{R_2}{R_1}, \quad Sk = \frac{(R_2 - R_1)^2 \Omega_0}{\nu}, \quad (21)$$

and referred to as the aspect ratio of the inner cylinder, the radius ratio of the annular fluid gap and the Stokes number, respectively. Figures 6, 7, and 8 in Lagrange and Puscas (2023) illustrated the variations of $m_0$ and $c_0$ with $l$, considering rigid, pinned-pinned or clamped-free vibration modes of the inner cylinder. Figure 3 in the present paper extends this analysis by showing, via the real and imaginary parts of $F^{(i)}$, the variations of $m_0$, $c_0$, as well as $j_0$ and $d_0$, considering small rigid displacements of the inner cylinder, see Eq. (B.2). First, we observe that $\Re\{F^{(i)}\}$ and $-\Im\{F^{(i)}\}$ strongly depend on $l$. Both $\Re\{F^{(i)}\}$ and $-\Im\{F^{(i)}\}$ are positive monotonic increasing functions of the aspect ratio, with a horizontal asymptote corresponding to the slender-body limit, $l \to \infty$. Second, the comparison of the curves traced out for $\omega Sk = 250$ and $Sk \to \infty$ (inviscid fluid) shows that the added coefficients $m_0$ and $j_0$ are weakly influenced by the viscosity, contrary to the damping coefficients $c_0$ and $d_0$ (which obviously vanish as $Sk \to \infty$).
<figure><img src="image_4.png" /><figcaption>Figure 3: Real and imaginary parts of $F^{(i)}$, see Eq. (20), as functions of the aspect ratio, $l$. The solid lines correspond to the model of Lagrange and Puscas (2023), see also Appendix B.1, traced out for $\omega Sk = 250$. The dashed lines correspond to the inviscid model $Sk \to \infty$ of Lagrange et al. (2023), see also Appendix B.2. The radius ratio is $\epsilon = 1.14$.</figcaption></figure>

The dimensionless coefficients due to fluid flow, appearing in Eqs. (16)-(17), are defined as

$$c_u = \frac{C_U}{L \rho S \Omega_0}, \quad f_d = \frac{F_d}{L^2 \rho S \Omega_0^2}, \quad k_u = \frac{K_U}{L^2 \rho S \Omega_0^2}, \quad a_u = \frac{A_U}{L^2 \rho S \Omega_0}, \quad d_u = \frac{D_U}{L^3 \rho S \Omega_0}. \qquad (22)$$

In Appendix C, we show that these coefficients write

$$c_u = m_F \frac{\epsilon - 1}{4\pi^2} |u| C_N, \quad f_d = m_F \frac{h}{l} \frac{1}{\pi} \left(\frac{\epsilon - 1}{4\pi}\right)^2 |u| u C_T, \quad k_u = m_F \frac{h}{l} \frac{1}{\pi} \left(\frac{\epsilon - 1}{4\pi}\right)^2 |u| u (C_N - C_T), \\ a_u = 2m_0^{(Sk \to \infty)} \frac{h \epsilon - 1}{l 4\pi} u, \quad d_u = m_F h^2 \frac{\epsilon - 1}{48\pi^2} |u| C_N. \qquad (23)$$

In Eq. (23), $m_0^{(Sk \to \infty)}$ (and by extension $j_0^{(Sk \to \infty)}$) corresponds to the limit of $m_0$ (by extension $j_0$), given by Eq. (20), as $Sk \to \infty$, (inviscid regime, see Lagrange et al. (2023) or also Appendix B.2), whereas $C_N$ (resp. $C_T$) is the normal (resp. tangent) friction coefficient. The dimensionless mean flow velocity is defined as

$$u = \frac{2\pi U_m}{\Omega_0 R_H}, \qquad (24)$$
and is related to the Reynolds number as $Re = 2(R_2 - R_1)U_m/\nu = uSk/(2\pi)$.

Finally, according to Eq. (13), the dimensionless turbulent fluid force $\tilde{f} = \tilde{\mathbf{F}} \cdot \mathbf{e}_y/(L^2 \rho S \Omega_0^2)$ and torque $\tilde{\gamma} = \tilde{\mathbf{\Gamma}} \cdot \mathbf{e}_z/(L^3 \rho S \Omega_0^2)$ write

$$ \tilde{f}(t) = \frac{h^2}{l} \int_0^1 \int_0^{2\pi} -\tilde{p}(\eta, \phi, t) \cos(\phi) \, d\phi d\eta, \qquad (25a) $$

$$ \tilde{\gamma}(t) = \frac{h^3}{l} \int_0^1 \int_0^{2\pi} -\left(\eta - \frac{1}{2}\right) \tilde{p}(\eta, \phi, t) \cos(\phi) \, d\phi d\eta, \qquad (25b) $$

with $\eta = (X - L)/H$ and $\tilde{p} = \tilde{P}/(\rho S \Omega_0^2)$ the dimensionless turbulent pressure. The dimensionless turbulent pressure is characterized by its dimensionless cross-correlation spectrum $\langle \tilde{p} | \tilde{p} \rangle (\eta, \phi, \eta', \phi', \omega)$.

## 2.4. Modal analysis

The set of equations (16)-(17), along with clamped conditions at the origin, i.e. $B_0[w] = (w(0,t), \partial w/\partial x(0,t)) = (0,0)$, defines a linear boundary value problem with nonzero dimensionless fluid forcing terms $\tilde{f}$ and $\tilde{\gamma}$. For convenience, we rewrite this boundary value problem, which is fully detailed in Appendix A, as

$$ \mathcal{L}[w] = 0, \quad B_0[w] = (0,0), \quad B_1[w] = \tilde{f}, \quad B_2[w] = \tilde{\gamma}, \qquad (26) $$

with $\mathcal{L}$ (resp. $B_1$ and $B_2$) the linear operator associated to the left hand side of Eq. (16) (resp. Eq. (17a) and Eq. (17b)). The undamped free vibration modes of the flexible blade in an inviscid still fluid correspond to the non-trivial solutions of the homogeneous problem $\tilde{f} = \tilde{\gamma} = 0$, with $\xi = \zeta = 0$ (undamped structure), $m_0 = m_0^{(Sk \to \infty)}$, $j_0 = j_0^{(Sk \to \infty)}$, $c_0 = 0$, $d_0 = 0$ (inviscid fluid) and $f_d = c_u = a_u = k_u = d_u = 0$ (still fluid). These solutions are obtained by introducing $w(x,t) = \Re\{w_0(x)e^{i\omega_0 t}\}$ into $\mathcal{L}[w] = 0$, leading to a linear homogeneous ordinary differential equation for $w_0$. Accounting for the clamped conditions at the origin, i.e. $B_0[w] = (0,0)$, the general solution expresses as a linear combination of two basis functions. Applying the boundary conditions $B_i[w] = 0, i \in \{1,2\}$, then yields a linear homogeneous system of algebraic equations for the two constants of integration. A solvability condition (vanishing determinant) yields an infinite countable number of real solutions $\omega_0^{(j)}$ corresponding to each free vibration mode $w_0^{(j)}$; see Appendix A for more details. The modal equations (26), detailed in Appendix A and given by Eqs. (A.1)-(A.2), involve 8 dimensionless parameters. Consequently, both $\omega_0^{(j)}$ and $w_0^{(j)}$ depend on the parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, l, \varepsilon\}$, which are defined by Eqs. (18) and (21). The first four vibration modes are shown in Fig. 4 and their angular frequencies<fnref n="1" /> are given in Table 1.

The dimensionless parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, l, \varepsilon\}$ used to trace out this figure are given in the set $\Pi_1$ of Table 2. The set $\Pi_1$ (resp. $\Pi_2$ and $\Pi_3$) would correspond to a configuration with a rectangular blade of thickness $b_1 = 14.5$ mm (resp. $b_2 = 12$ mm and $b_3 = 10$ mm). Apart from that, all sets would share the same other dimensional parameters. They would correspond to the case of a cylinder $C_1$ (resp. $C_2$) of radius $R_1 = 1.32 \cdot 10^{-1}$ m (resp. $R_2 = 1.50 \cdot 10^{-1}$ m), length $H = 5.15 \cdot 10^{-1}$ m, mass $M_{cyl} = 30.0$ kg (acceleration of gravity $G = 9.81$ m/s²), moment of inertia $J_{cyl} = 9.24 \cdot 10^{-1}$ kg m² (cylinder treated as a thin vessel), attached to a flexible blade of length $L = 5.00 \cdot 10^{-2}$ m, rectangular section $a \times b$, with $a = 1.00 \cdot 10^{-1}$ m (quadratic moment of area $I = ab^3/12$), Young modulus $E = 2.10 \cdot 10^{11}$ Pa and volume mass density $\rho = 7.85 \cdot 10^3$ kg/m³. The Rayleigh damping coefficient $\zeta$ is set so that the reduced structural dampings of modes 1 and 2 are less than or equal to 1%, see Eq. (D.9). The fluid would be Newtonian of volume mass density $\rho_F = 1.00 \cdot 10^3$ kg/m³ and kinematic viscosity $\nu = 1.00 \cdot 10^{-6}$ m²/s. The characteristic angular frequency for the set $\Pi_1$ (resp. $\Pi_2$ and $\Pi_3$) would be $\Omega_0 = 8.66 \cdot 10^3$ Hz. (resp. $\Omega_0 = 7.17 \cdot 10^3$ Hz. and $\Omega_0 = 5.97 \cdot 10^3$ Hz). For the set $\Pi_1$, the dimensional structural (resp. inviscid still fluid) frequencies $F_s^{(j)} = \Omega_0 \omega_s^{(j)} / (2\pi)$ (resp. $F_0^{(j)} = \Omega_0 \omega_0^{(j)} / (2\pi)$) would be $F_s^{(1)} = 28.5$ Hz, $F_s^{(2)} = 1230$ Hz, $F_s^{(3)} = 29.7 \cdot 10^3$ Hz and $F_s^{(4)} = 74.0 \cdot 10^3$ Hz (resp. $F_0^{(1)} = 14.5$ Hz, $F_0^{(2)} = 819$ Hz, $F_0^{(3)} = 29.6 \cdot 10^3$ Hz and $F_0^{(4)} = 73.8 \cdot 10^3$ Hz).

<footnote n="1">1. The corresponding dimensionless structural angular frequencies $\omega_s^{(j)}$, which depend on the 7 dimensionless parameters $\{g, m_{cyl}, h, \kappa, j_{cyl}, l, \varepsilon\}$, are given in Table A.3.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 4: Normalized first four undamped free vibration modes $w_0^{(j)}/||w_0^{(j)}||_\infty$ of the blade ($x \in [0, 1]$) - cylinder ($x \in [1, 1+h]$) system in an inviscid still fluid. Black solid line: $j = 1$, red dashed line: $j = 2$, blue dotted line: $j = 3$, green dash-dotted line: $j = 4$. For $x \in [1, 1+h]$ all curves are straight lines due to rigid body motion of the cylinder. These straight lines are only drawn up to $x = 3$ instead of $x = 1+h$, to avoid squashing the curves between $x = 0$ and $x = 1$. Therefore, the infinite norm used to normalize the modes is $||w||_\infty = \sup_{x \in [0,3]} (|w|)$. The dimensionless parameters {$g, m_{cyl}, m_F, h, \kappa, j_{cyl}, l, \varepsilon$} used to trace out this figure are given in the set $\Pi_1$ of Table 2.</figcaption></figure>

<table><caption>Table 1: Dimensionless angular frequencies $\omega_0^{(j)}$ of the first four free vibration modes. The dimensionless parameters {$g, m_{cyl}, m_F, h, \kappa, j_{cyl}, l, \varepsilon$} used to calculate $\omega_0^{(j)}$ are given in Table 2.</caption><thead><tr><th>Set</th><th>ω<sub>0</sub><sup>(1)</sup></th><th>ω<sub>0</sub><sup>(2)</sup></th><th>ω<sub>0</sub><sup>(3)</sup></th><th>ω<sub>0</sub><sup>(4)</sup></th></tr></thead><tbody><tr><td>Π<sub>1</sub></td><td>1.06 · 10<sup>-2</sup></td><td>5.95 · 10<sup>-1</sup></td><td>21.5</td><td>53.6</td></tr><tr><td>Π<sub>2</sub></td><td>9.61 · 10<sup>-3</sup></td><td>5.41 · 10<sup>-1</sup></td><td>21.8</td><td>55.8</td></tr><tr><td>Π<sub>3</sub></td><td>8.77 · 10<sup>-3</sup></td><td>4.95 · 10<sup>-1</sup></td><td>22.0</td><td>57.4</td></tr></tbody></table>

<table><caption>Table 2: Sets of dimensionless parameters. All the sets are completed with $h = 10.3$, $l = 3.90$ and $\varepsilon = 1.14$.</caption><thead><tr><th>Set</th><th>g</th><th>m<sub>cyl</sub></th><th>m<sub>F</sub></th><th>κ</th><th>j<sub>cyl</sub></th><th>Sk</th><th>ξ</th><th>ζ</th><th>u or Re</th></tr></thead><tbody><tr><td>Π<sub>1</sub></td><td>2.61 · 10<sup>-6</sup></td><td>52.7</td><td>49.5</td><td>8.37 · 10<sup>-2</sup></td><td>650</td><td>2.81 · 10<sup>6</sup></td><td>0</td><td>0.469</td><td>variable</td></tr><tr><td>Π<sub>2</sub></td><td>3.81 · 10<sup>-6</sup></td><td>63.7</td><td>60.0</td><td>6.93 · 10<sup>-2</sup></td><td>785</td><td>2.32 · 10<sup>6</sup></td><td>0</td><td>0.623</td><td>variable</td></tr><tr><td>Π<sub>3</sub></td><td>5.50 · 10<sup>-6</sup></td><td>76.4</td><td>71.8</td><td>5.77 · 10<sup>-2</sup></td><td>942</td><td>1.93 · 10<sup>6</sup></td><td>0</td><td>0.819</td><td>variable</td></tr></tbody></table>

## 2.5. Power spectral density of vibration displacement

Having established the equilibrium equations of the blade and the inner rigid cylinder, we now proceed with deriving an expression of the power spectral density of the vibration displacement. For a given external fluid force $\tilde{f}$ and a torque $\tilde{\gamma}$, find a solution to Eq. (26) is equivalent to finding a function $w$ such that $B_0[w] = (0, 0)$ and such that

$$ \int_0^1 w^\dagger \mathcal{L}[w] + \delta(x-1) [w_G^\dagger (B_1[w] - \tilde{f}) + \theta^\dagger (B_2[w] - \tilde{\gamma})] dx = 0, \quad (27) $$

for any arbitrary functions $w^\dagger$, $\theta^\dagger$ and $w_G^\dagger$; $\delta$ being the Dirac function. Whereas obtaining such a solution is out of reach, one can still find an approximate function $w$ that would satisfy Eq. (27) for a specific set of test
functions $w^\dagger$, $\theta^\dagger$ and $w_G^\dagger$. In the present study, an approximate solution is sought in the form

$$w(x,t) = \sum_{j=1}^{K} q_j(t) w_j(x) = \sum_{j=1}^{K} q_j(t) \frac{1}{N_j} \left| \begin{array}{ll} \mathcal{P}_j(x) & \text{if } x \in [0, 1] \\ \mathcal{P}_j(1) + \frac{d\mathcal{P}_j}{dx} \bigg|_1 & \text{if } x \in ]1, 1+h] \end{array} \right. , \quad (28)$$

with $\{\mathcal{P}_j\}$ a set of K basis functions, satisfying $B_0[\mathcal{P}_j] = (0,0)$ for all $j \in [1, K]$. In Eq. (28), $N_j \neq 0$ is a scaling factor defined as

$$\mathcal{N}_j = \mathcal{P}_j(1) + h \frac{d\mathcal{P}_j}{dx} \bigg|_1, \quad (29)$$

introduced to normalize the displacement of the j-th mode $w_j$ at the tip of the cylinder, i.e. $\forall j \in [1, K]$, $w_j(1+h) = 1$.

Plugging the expansion (28) into Eq. (27) eventually yields

$$\forall i \in [1, K], \quad \sum_{j=1}^{K} m_{i,j} \frac{d^2 q_j}{dt^2} + c_{i,j} \frac{dq_j}{dt} + k_{i,j} q_j = \alpha_i \tilde{f} + \beta_i \tilde{\gamma}, \quad (30)$$

with

$$m_{i,j} = m_{i,j}^{(s)} + (m_{cyl} + m_0) m_{i,j}^{(m)} + (j_{cyl} + j_0) m_{i,j}^{(J)}, \quad (31a)$$

$$c_{i,j} = c_{i,j}^{(s)} + (c_0 + c_u) c_{i,j}^{(c)} + (d_0 + d_u) c_{i,j}^{(d)} + a_u c_{i,j}^{(a)}, \quad (31b)$$

$$k_{i,j} = k_{i,j}^{(s)} + [(m_F - m_{cyl})g + f_d] k_{i,j}^{(f)} + k_u k_{i,j}^{(k)}. \quad (31c)$$

In Eq. (31), the mass $m_{i,j}^{(X)}$, damping $c_{i,j}^{(X)}$ and stiffness $k_{i,j}^{(X)}$ coefficients depends only on the dimensionless numbers $\kappa$ and $h$, characterizing the geometry of the problem, and on the Rayleigh coefficients $\xi$ and $\zeta$, characterizing the structural damping of the flexible blade. These coefficients along with the factors $\alpha_i$ and $\beta_i$ are reported in Appendix D.

Applying $\langle \cdot | \tilde{f} \rangle$ and $\langle \cdot | \tilde{\gamma} \rangle$ to Eq. (30) yields a set of $2K$ equations, i.e.

$$\forall i \in [1, K], \quad \sum_{j=1}^{K} (-\omega^2 m_{i,j} - i\omega c_{i,j} + k_{i,j}) \langle q_j | \tilde{f} \rangle = \alpha_i \langle \tilde{f} | \tilde{f} \rangle + \beta_i \langle \tilde{\gamma} | \tilde{f} \rangle, \quad (32a)$$

$$\forall i \in [1, K], \quad \sum_{j=1}^{K} (-\omega^2 m_{i,j} - i\omega c_{i,j} + k_{i,j}) \langle q_j | \tilde{\gamma} \rangle = \alpha_i \langle \tilde{f} | \tilde{\gamma} \rangle + \beta_i \langle \tilde{\gamma} | \tilde{\gamma} \rangle. \quad (32b)$$

In matrix form, Eq. (32) may be written as

$$\overline{[\mathbf{B}] \langle \mathbf{Q} | \tilde{f} \rangle} = \alpha \langle \tilde{f} | \tilde{f} \rangle + \beta \langle \tilde{\gamma} | \tilde{f} \rangle, \quad (33a)$$

$$\overline{[\mathbf{B}] \langle \mathbf{Q} | \tilde{\gamma} \rangle} = \alpha \langle \tilde{f} | \tilde{\gamma} \rangle + \beta \langle \tilde{\gamma} | \tilde{\gamma} \rangle, \quad (33b)$$

with $\mathbf{Q}$ (resp. $\alpha$ and $\beta$) the real column vector containing the modal amplitudes $q_j$ (resp. $\alpha_j$ and $\beta_j$) and $\langle \mathbf{Q} | \tilde{f} \rangle$ (resp. $\langle \mathbf{Q} | \tilde{\gamma} \rangle$) the complex column vector containing $\langle q_j | \tilde{f} \rangle$ (resp. $\langle q_j | \tilde{\gamma} \rangle$), $j \in [1, K]$. The $(i, j)$ coefficient of the $K \times K$ complex matrix $[\mathbf{B}]$ is $[\mathbf{B}]_{i,j} = -\omega^2 m_{i,j} + i\omega c_{i,j} + k_{i,j}$. Assuming an invertible matrix $[\mathbf{B}]$, Eq. (33) may be written as

$$\langle \mathbf{Q} | \tilde{f} \rangle = \overline{[\mathbf{B}]^{-1} \alpha} \langle \tilde{f} | \tilde{f} \rangle + \overline{[\mathbf{B}]^{-1} \beta} \langle \tilde{\gamma} | \tilde{f} \rangle, \quad (34a)$$

$$\langle \mathbf{Q} | \tilde{\gamma} \rangle = \overline{[\mathbf{B}]^{-1} \alpha} \langle \tilde{f} | \tilde{\gamma} \rangle + \overline{[\mathbf{B}]^{-1} \beta} \langle \tilde{\gamma} | \tilde{\gamma} \rangle. \quad (34b)$$

Applying $\langle q_k | \cdot \rangle$ to Eq. (30) yields a set of $K^2$ equations, i.e.

$$\forall i, k \in [1, K], \quad \sum_{j=1}^{K} (-\omega^2 m_{i,j} + i\omega c_{i,j} + k_{i,j}) \langle q_k | q_j \rangle = \alpha_i \langle q_k | \tilde{f} \rangle + \beta_i \langle q_k | \tilde{\gamma} \rangle. \quad (35)$$

In matrix form and under the assumption of an invertible matrix $[\mathbf{B}]$, Eq. (35) may be written as

$$\forall k \in [1, K], \quad \langle q_k | \mathbf{Q} \rangle = [\mathbf{B}]^{-1} \alpha \langle q_k | \tilde{f} \rangle + [\mathbf{B}]^{-1} \beta \langle q_k | \tilde{\gamma} \rangle, \quad (36)$$
with $\langle q_k | \mathbf{Q} \rangle$ the column vector of length $K$ whose $j$-th coefficient is $\langle q_k | q_j \rangle$. This coefficient may be expressed as

$$ \begin{aligned} \langle q_k | q_j \rangle &= [\overline{[\mathbf{B}]}^{-1} \alpha \langle q_k | \tilde{f} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta \langle q_k | \tilde{\gamma} \rangle]_j]_k \\ &= [\overline{[\mathbf{B}]}^{-1} \alpha]_j \langle q_k | \tilde{f} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta]_j \langle q_k | \tilde{\gamma} \rangle. \end{aligned} \quad (37) $$

The substitution of Eq. (34) into the last equation of (37) yields

$$ \begin{aligned} \langle q_k | q_j \rangle &= [\overline{[\mathbf{B}]}^{-1} \alpha]_j [\overline{[\mathbf{B}]}^{-1} \alpha \langle \tilde{f} | \tilde{f} \rangle + \overline{[\mathbf{B}]}^{-1} \beta \langle \tilde{\gamma} | \tilde{f} \rangle]_k + [\overline{[\mathbf{B}]}^{-1} \beta]_j [\overline{[\mathbf{B}]}^{-1} \alpha \langle \tilde{f} | \tilde{\gamma} \rangle + \overline{[\mathbf{B}]}^{-1} \beta \langle \tilde{\gamma} | \tilde{\gamma} \rangle]_k \\ &= [\overline{[\mathbf{B}]}^{-1} \alpha]_k [\overline{[\mathbf{B}]}^{-1} \alpha]_j \langle \tilde{f} | \tilde{f} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta]_k [\overline{[\mathbf{B}]}^{-1} \alpha]_j \langle \tilde{\gamma} | \tilde{f} \rangle \\ &\quad + [\overline{[\mathbf{B}]}^{-1} \alpha]_k [\overline{[\mathbf{B}]}^{-1} \beta]_j \langle \tilde{f} | \tilde{\gamma} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta]_k [\overline{[\mathbf{B}]}^{-1} \beta]_j \langle \tilde{\gamma} | \tilde{\gamma} \rangle. \end{aligned} \quad (38) $$

In matrix form, the last equation of (38) may be written as

$$ \langle \mathbf{Q} | \mathbf{Q} \rangle = \overline{[\mathbf{B}]}^{-1} \alpha ( [\overline{[\mathbf{B}]}^{-1} \alpha \langle \tilde{f} | \tilde{f} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta \langle \tilde{f} | \tilde{\gamma} \rangle]^{\top} + [\overline{[\mathbf{B}]}^{-1} \beta ( [\overline{[\mathbf{B}]}^{-1} \alpha \langle \tilde{\gamma} | \tilde{f} \rangle + [\overline{[\mathbf{B}]}^{-1} \beta \langle \tilde{\gamma} | \tilde{\gamma} \rangle]^{\top} ) ]^{\top} ) \quad (39) $$

with $\top$ the transpose operator and $\langle \mathbf{Q} | \mathbf{Q} \rangle$ the $K \times K$ matrix whose $(i, j)$ coefficient is $\langle q_i | q_j \rangle$. Note that $\alpha$ and $\beta$ are real column vectors so that $[\overline{[\mathbf{B}]}^{-1} \alpha] = [\overline{[\mathbf{B}]}^{-1} \alpha]$ and $[\overline{[\mathbf{B}]}^{-1} \beta] = [\overline{[\mathbf{B}]}^{-1} \beta]$. Subsequently, introducing column vectors $[\mathbf{H}_1] = [\overline{[\mathbf{B}]}^{-1} \alpha]$ and $[\mathbf{H}_2] = [\overline{[\mathbf{B}]}^{-1} \beta]$, Eq. (39) eventually may be written as

$$ \langle \mathbf{Q} | \mathbf{Q} \rangle = [\mathbf{H}_1] [\overline{[\mathbf{H}_1]}^{\top} \langle \tilde{f} | \tilde{f} \rangle + [\mathbf{H}_1] [\overline{[\mathbf{H}_2]}^{\top} \langle \tilde{f} | \tilde{\gamma} \rangle + [\mathbf{H}_2] [\overline{[\mathbf{H}_1]}^{\top} \langle \tilde{\gamma} | \tilde{f} \rangle + [\mathbf{H}_2] [\overline{[\mathbf{H}_2]}^{\top} \langle \tilde{\gamma} | \tilde{\gamma} \rangle]^{\top} \quad (40) $$

## 2.6. Power spectral density of the random fluid force and torque

In this section, we relate the dimensionless power spectral densities $\langle \tilde{f} | \tilde{f} \rangle$, $\langle \tilde{f} | \tilde{\gamma} \rangle = \langle \tilde{\gamma} | \tilde{f} \rangle$ and $\langle \tilde{\gamma} | \tilde{\gamma} \rangle$ to the dimensionless power spectral density of the dimensionless random fluid pressure $\tilde{p}$.
Introducing Eq. (25) in Eq. (14) yields

$$ \langle \tilde{f} | \tilde{f} \rangle (\omega) = \frac{h^4}{l^2} \int_{-\infty}^{\infty} \left[ \lim_{\delta \to \infty} \frac{1}{\delta} \int_0^{\delta} \left( \int_0^{1} \int_0^{2\pi} \tilde{p}(\eta, \phi, t) \cos(\phi) \, d\phi \, d\eta \int_0^{1} \int_0^{2\pi} \tilde{p}(\eta', \phi', t) \cos(\phi') \, d\phi' \, d\eta' \right) \right] dt \exp(-i\omega\tau)d\tau, \quad (41) $$

which after permutation of integral signs may be written as

$$ \langle \tilde{f} | \tilde{f} \rangle (\omega) = \frac{h^4}{l^2} \int_0^1 \int_0^{2\pi} \int_0^1 \int_0^{2\pi} \langle \tilde{p} | \tilde{p} \rangle (\eta, \phi, \eta', \phi', \omega) \cos(\phi) \cos(\phi') \, d\phi \, d\eta \, d\phi' \, d\eta'. \quad (42) $$

Assuming homogeneous turbulence, it is considered that the cross-correlation spectrum $\langle \tilde{p} | \tilde{p} \rangle (\eta, \phi, \eta', \phi', \omega)$ depends on $(\eta - \eta', \phi - \phi')$ and is of the form

$$ \langle \tilde{p} | \tilde{p} \rangle (\eta, \phi, \eta', \phi', \omega) = s_{\tilde{p}}(\omega) \chi(\eta - \eta', \phi - \phi', \omega), \quad (43) $$

with $s_{\tilde{p}}$ the turbulent pressure power spectral density (PSD, dimensional form $S_{\tilde{p}} = (\rho S \Omega_0^2)^2 s_{\tilde{p}} / \Omega_0$) and $\chi$ a coherence function. Introducing Eq. (43) into Eq. (41), the power spectral density $\langle \tilde{f} | \tilde{f} \rangle$ is

$$ \langle \tilde{f} | \tilde{f} \rangle = \frac{h^4}{l^2} s_{\tilde{p}} I_{\tilde{f}}, \quad (44) $$

with

$$ I_{\tilde{f}}(\omega) = \int_0^1 \int_0^{2\pi} \int_0^1 \int_0^{2\pi} \chi(\eta - \eta', \phi - \phi', \omega) \cos(\phi) \cos(\phi') \, d\phi \, d\eta \, d\phi' \, d\eta', \quad (45) $$

the dimensionless joint acceptance function of $\tilde{f}$.

Similarly, the power spectral densities $\langle \tilde{f} | \tilde{\gamma} \rangle = \langle \tilde{\gamma} | \tilde{f} \rangle$ and $\langle \tilde{\gamma} | \tilde{\gamma} \rangle$ are

$$ \langle \tilde{f} | \tilde{\gamma} \rangle = \frac{h^5}{l^2} s_{\tilde{p}} I_{\tilde{f} \tilde{\gamma}}, \quad (46a) $$

$$ \langle \tilde{\gamma} | \tilde{\gamma} \rangle = \frac{h^6}{l^2} s_{\tilde{p}} I_{\tilde{\gamma}}, \quad (46b) $$
with

$$I_{\tilde{f}\tilde{\gamma}}(\omega) = \int_0^1 \int_0^{2\pi} \int_0^1 \int_0^{2\pi} \chi(\eta - \eta', \phi - \phi', \omega) \left(\eta' - \frac{1}{2}\right) \cos(\phi) \cos(\phi') \, d\phi \, d\eta \, d\phi' \, d\eta', \quad (47a)$$

$$I_{\tilde{\gamma}}(\omega) = \int_0^1 \int_0^{2\pi} \int_0^1 \int_0^{2\pi} \chi(\eta - \eta', \phi - \phi', \omega) \left(\eta - \frac{1}{2}\right) \left(\eta' - \frac{1}{2}\right) \cos(\phi) \cos(\phi') \, d\phi \, d\eta \, d\phi' \, d\eta'. \quad (47b)$$

## 2.7. mean-square of displacement vibration

From the bilinearity of $\langle \cdot | \cdot \rangle$, the power spectral density of the displacement is $\langle w | w \rangle = \sum_{i,j=1}^K \langle q_i | q_j \rangle w_i w_j$, which, given that $\langle q_i | q_j \rangle = \langle q_j | q_i \rangle$, may be written as

$$\langle w | w \rangle (x, \omega) = \Re \left\{ \sum_{i=1}^{K} \sum_{j=1}^{K} \langle q_i | q_j \rangle (\omega) w_i(x) w_j(x) \right\}, \quad (48a)$$

$$= \Re \{ \mathbf{w}^{\top}(x) \langle \mathbf{Q} | \mathbf{Q} \rangle (\omega) \mathbf{w}(x) \}, \quad (48b)$$

with $\mathbf{w}$ the column vector containing the displacements $w_j$.

Plugging Eq. (48) into Eq. (15), the dimensionless mean-square structure displacement may be written as

$$\forall x \in [0, 1 + h], \quad \sigma^2(x) = \left(\frac{l}{h} \frac{1}{\varepsilon - 1}\right)^2 \frac{1}{2\pi} \Re \left\{ \mathbf{w}^{\top}(x) \left[ \int_{0}^{\infty} [\mathbf{H}] [\overline{\mathbf{H}}]^{\top}(\omega) s_{\tilde{p}}(\omega) \, d\omega \right] \mathbf{w}(x) \right\}, \quad (49)$$

with

$$[\mathbf{H}] [\overline{\mathbf{H}}]^{\top} = \frac{h^4}{l^2} [\mathbf{H}_1] [\overline{\mathbf{H}_1}]^{\top} I_{\tilde{f}} + \frac{h^5}{l^2} [\mathbf{H}_1] [\overline{\mathbf{H}_2}]^{\top} I_{\tilde{f}\tilde{\gamma}} + \frac{h^5}{l^2} [\mathbf{H}_2] [\overline{\mathbf{H}_1}]^{\top} I_{\tilde{\gamma}\tilde{f}} + \frac{h^6}{l^2} [\mathbf{H}_2] [\overline{\mathbf{H}_2}]^{\top} I_{\tilde{\gamma}}. \quad (50)$$

According to Eqs. (28)-(29), $\mathbf{w}$ ($1+h$) is a column vector with all elements equal to 1. From Eq. (49), it follows that the dimensionless mean-square displacement at the cylinder tip is

$$\sigma^2 (1 + h) = \left(\frac{l}{h} \frac{1}{\varepsilon - 1}\right)^2 \frac{1}{2\pi} \sum_{i,j=1}^{K} \Re \left\{ \int_{0}^{\infty} \left( [\mathbf{H}] [\overline{\mathbf{H}}]^{\top} \right)_{i,j} (\omega) s_{\tilde{p}}(\omega) \, d\omega \right\}. \quad (51)$$

In conclusion, Eq. (49) provides a general expression for the local mean-square structure displacement $\sigma$. This equation involves the column vector $\mathbf{w}$, whose elements $w_j$ are defined by a selection of $K$ basis functions $\{P_j\}$; see Eq. (28). Equation (49) also includes an integral with the integrand $[\mathbf{H}] [\overline{\mathbf{H}}]^{\top} s_{\tilde{p}}$. To compute this integral, the term $[\mathbf{H}] [\overline{\mathbf{H}}]^{\top}$ must be known. This term is provided in Eq. (50), which comprises subterms $[\mathbf{H}_i] [\overline{\mathbf{H}_i}]^{\top}$ and joint acceptance functions $I_{\tilde{f}}$, $I_{\tilde{f}\tilde{\gamma}}$, and $I_{\tilde{\gamma}}$. Since $[\mathbf{H}_1] = [\overline{\mathbf{B}}]^{-1}\alpha$ and $[\mathbf{H}_2] = [\overline{\mathbf{B}}]^{-1}\beta$, $[\mathbf{H}_i]$ is directly related to the matrix $[\mathbf{B}]_{i,j} = -\omega^2 m_{i,j} + i\omega c_{i,j} + k_{i,j}$. The mass, damping, and stiffness coefficients $m_{i,j}$, $c_{i,j}$, and $k_{i,j}$ depend on the fluid added coefficients due to fluid flow $c_u$, $a_u$, $k_u$, $d_u$, which themselves depend on the friction coefficients $C_N$ and $C_T$. Thus, a model for $C_N$ and $C_T$ must be selected to calculate the integral for the root mean-square $\sigma$. The joint acceptance functions are expressed as integrals whose integrands depend on a coherence function $\chi$; see Eqs. (45) and (47). A model for $\chi$ is therefore required to compute the joint acceptance functions. Finally, to calculate the integral in Eq. (49), a model for the turbulent pressure power spectral density $s_{\tilde{p}}$ is necessary. Ultimately, incorporating all these models into Eq. (49) enables the numerical evaluation of the integral for the root mean-square. Particular choices for the different models are presented in subsections 3.1, 3.2, 3.3 and 3.4.

## 2.8. Fluid-elastic instability

Equation (49) is the general analytical expression for the dimensionless mean-square structure displacement. This expression is valid as long as the homogeneous solution of the linear system of ODEs (30) remains stable, that is to say vanishes as $t \to \infty$. The stability of the homogeneous solution can be determined from the eigenvalues $\lambda^{(j)}$ of the Jacobian matrix of the linear system. This real matrix is of size $(2K) \times (2K)$ and may be written as

$$[\mathbf{A}] = \begin{bmatrix} [0] & [1] \\ -[\mathbf{M}]^{-1} [\mathbf{K}] & -[\mathbf{M}]^{-1} [\mathbf{C}] \end{bmatrix}. \quad (52)$$

In Eq. (52), $[\mathbf{M}]$ (resp. $[\mathbf{C}]$ and $[\mathbf{K}]$) is the mass (resp. damping and stiffness) matrix, of size $K \times K$, with real elements $m_{i,j}$ (resp. $c_{i,j}$ and $k_{i,j}$), see Eq. (31) and Appendix D. Hence, a flutter (resp. divergence) instability
occurs when $\Re\{\lambda^{(j)}\} = 0$ and $\Im\{\lambda^{(j)}\} \neq 0$ (resp. $\Im\{\lambda^{(j)}\} = 0$). The instability threshold is determined numerically. For a given set of dimensionless numbers $\Pi$, a vector of Reynolds numbers, $[Re_1, Re_2, \dots, Re_n]$, is defined. For each Reynolds number $Re_i$, the code calculates the matrices $[M]$, $[K]$, and $[C]$ and a Jacobian matrix $[A(Re_i)]$ is constructed from Eq. (52). The eigenvalue Python function is used to obtain the complex conjugate eigenvalues $\{\lambda_i^{(1)}, \dots, \lambda_i^{(2K)}\}$ of this matrix. The smallest index $i$ for which the real part of $\lambda_i^{(j)}$ changes sign is then determined, leading to the identification of the critical Reynolds number $Re_{crit}^{(j)}$ and the critical dimensionless mean flow velocity $u_{crit}^{(j)} = 2\pi Re_{crit}^{(j)}/Sk$.

Whereas the vibration modes $w_0^{(j)}$ and angular frequencies $\omega_0^{(j)}$ depend on the 8 dimensionless parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, l, \varepsilon\}$, $\sigma^2(x)$ depends on a set $\Pi$ of 12 independent dimensionless numbers

$$ \Pi = \{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, \xi, \zeta, l, \varepsilon, Sk, u\}, \qquad (53) $$

defined by Eqs. (18), (21) and (24). The instability threshold depends on the first 11 dimensionless numbers listed in the set $\Pi$. Eventually, equations (49) and (52) can be used to calculate $\sigma^2(x)$ and determine the instability threshold, provided that i) a choice of basis and test functions $P_j$, $w^\dagger$, $\theta^\dagger$, $w_G^\dagger$ is made, ii) a model for the tangential and normal friction coefficients $C_T$ and $C_N$ is defined, iii) a model for the turbulent pressure PSD $s_p$ is chosen, and iv) a model for the coherence function $\chi$ is adopted.

### 3. Theoretical predictions

#### 3.1. Choice of basis and test functions

As stated in § 2.5, the basis functions $P_j$ appearing in Eq. (28) must satisfy the boundary condition $B_0[P_j] = (0,0)$ for all $j \in [1, K]$. A convenient choice for $P_j$ is a monomial in $x$, of degree higher or equal to 2, i.e.

$$ \forall j \in [1, K], \quad P_j(x) = x^{j+1}. \qquad (54) $$

Plugging Eq. (54) into Eq. (28), the displacement $w$ is searched as $w(x) = \sum_{j=1}^K q_j(t) w_j(x)$ with

$$ w_j(x) = \frac{1}{1 + (j + 1)h} \begin{vmatrix} x^{j+1} & \text{if } x \in [0, 1] \\ 1 + (j + 1)(x - 1) & \text{if } x \in ]1, 1 + h] \end{vmatrix}. \qquad (55) $$

In the Galerkin method, the test functions have the same form as the basis functions. Therefore, following this method, the test functions used in the present study are

$$ w_j^\dagger = w_j, \quad \theta_j^\dagger = \frac{dw_j}{dx}, \quad w_{G,j}^\dagger = w_j + \frac{h}{2} \frac{dw_j}{dx}. \qquad (56) $$

#### 3.2. Model for the tangential and normal friction coefficients

The tangential friction factor $C_T$ is related to the Darcy<fnref n="2" /> friction factor $C_f$ through the relation $C_T = \pi C_f / 4$. For circular ducts, $C_f$ in turbulent flow ($Re > 4000$) is generally a function of both $Re = 2(R_2 - R_1)U_m/\nu$ and the equivalent surface roughness relative to the hydraulic diameter. All correlations for $C_f$ in turbulent flow are based on experimental data. The phenomenological Colebrook-White equation Colebrook and White (1937, 1939) gives a transcendental equation for $C_f$, fitting the data of experimental studies of turbulent flow in smooth and rough pipes. This equation can be solved numerically due to its implicit nature, or approximations can be used to evaluate $C_f$. For smooth surface, Blevins (2003) proposes the approximation

$$ C_f = \left[ 1.14 - \frac{2}{\ln(10)} \ln\left(\frac{21.25}{Re^{0.9}}\right) \right]^{-2}, \qquad (57) $$

which approaches the solution of the Colebrook-White equation within 0.5%, see Fig. 5. Regarding the normal friction factor $C_N$, it is difficult to come up with a recommendation, but a reasonable suggested range is $0.5 < C_N/C_T < 2$; see Païdoussis (2016) for a full discussion. In this study it is considered that $C_N = C_T$. Eventually, the model for the tangent and normal friction coefficients used in this study is

$$ C_T = C_N = \frac{\pi}{4} \left[ 1.14 - \frac{2}{\ln(10)} \ln\left(\frac{21.25}{Re^{0.9}}\right) \right]^{-2}. \qquad (58) $$

According to Eq. (58), $C_N$ and $C_T$ are expressed in terms of $Re$ only. Since $Re$ is related to $u$ and $Sk$ through the relation $Re = uSk/(2\pi)$, the Reynolds number is not a new independent dimensionless number. It follows

<footnote n="2">2. Also called the Darcy-Weisbach or Moody friction factor.</footnote>
that the model chosen for $C_N$ and $C_T$ does not affect the list Π of independent dimensionless numbers; see Eq. (53).

<figure><img src="image_6.png" /><figcaption>Figure 5: Evolution of the tangential friction factor, $C_T$, as a function of the Reynolds number, $Re$, for turbulent flow on a smooth surface. The red solid line corresponds to the exact solution of the Colebrook-White equation. The black dashed line corresponds to the approximation (58).</figcaption></figure>

### 3.3. Model for the turbulent pressure power spectral density

Based on data from two separate experiments, the following empirical equations for $S_{\tilde{P}}$ are recommended by Au-Yang and Jordan (1980); Au-Yang et al. (1995); Weaver et al. (2000) for flows without cavitation<fnref n="3" />

$$ S_{\tilde{P}}(\Omega) = \rho_F^2 U_m^3 R_H \begin{cases} 0.310 \exp(-6.0 \frac{\Omega R_H}{2\pi U_m}) & \text{if } 0 \le \frac{\Omega R_H}{2\pi U_m} < 0.5 \\ 0.054 \exp(-2.52 \frac{\Omega R_H}{2\pi U_m}) & \text{if } 0.5 \le \frac{\Omega R_H}{2\pi U_m} \le 2.5 \end{cases}. \quad (59) $$

In dimensionless form, Eq. (59) may be written as

$$ s_{\tilde{p}}(\omega) = u^3(\varepsilon - 1)^4 m_F^2 \frac{1}{h^2} \frac{1}{2^7 \pi^5} \Re\left(\frac{\omega}{u}\right), \quad (60) $$

with

$$ \Re\left(\frac{\omega}{u}\right) = \begin{cases} 0.310 \exp(-6.0 \frac{\omega}{u}) & \text{if } 0 \le \frac{\omega}{u} < 0.5 \\ 0.054 \exp(-2.52 \frac{\omega}{u}) & \text{if } 0.5 \le \frac{\omega}{u} \le 2.5 \end{cases}, \quad (61) $$

a function of the reduced frequency $f_r = \omega/u = \Omega R_H / (2\pi U)$, whose graph is shown in Fig. 6. According to Eqs. (60)-(61), $s_{\tilde{p}}$ is expressed in terms of $m_F$, $h$, $\varepsilon$, $u$ and the integration variable $\omega$. It follows that the model chosen for $s_{\tilde{p}}$ does not affect the list Π of independent dimensionless numbers; see Eq. (53).

<footnote n="3">3. In Au-Yang and Jordan (1980); Au-Yang et al. (1995); Weaver et al. (2000) the power spectral density $S_{\tilde{P}}$ is expressed in terms of a characteristic length $\delta = R_2 - R_1$, while it is expressed in terms of $R_h = (R_2 - R_1)/2$ in the present study. It follows that the numerical values appearing in Eq. (59) are twice those given in Au-Yang and Jordan (1980); Au-Yang et al. (1995); Weaver et al. (2000), while the variation range of $f_r$ is reduced by a factor of 2.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 6: Evolution of the dimensionless pressure PSD function $\aleph$, given by Eq. (61), as a function of the reduced frequency, $f_r = \omega/u$.</figcaption></figure>

### 3.4. Model for the coherence function

The coherence function $\chi$, as elucidated by Powell (2005), accentuates two distinct physical phenomena. The first pertains to the convection of turbulent eddies by the mean flow. The second phenomenon arises from the coherence of pressure, dependent on the proximity of the two points into consideration. In parallel turbulent flows over surfaces of structures, it is generally assumed that the coherence function $\chi$ can be written in the form

$$ \chi(\eta - \eta', \phi - \phi', \Omega) = \exp\left(i\frac{\Omega}{U_c}H(\eta - \eta')\right) \exp\left(-\frac{H}{\Lambda_{\eta}}|\eta - \eta'|\right) \exp\left(-\frac{R_1}{\Lambda_{\phi}}|\phi - \phi'|\right), \quad (62) $$

with $U_c$ the convective speed and $\Lambda_\eta$, $\Lambda_\phi$ two correlation lengths. In the model of Corcos (1963) it is also considered that

$$ U_c = \alpha_c U_m, \quad \Lambda_\eta = \alpha_\eta \left(\frac{\omega}{u}\right)^{-1} R_H, \quad \Lambda_\phi = \alpha_\phi \left(\frac{\omega}{u}\right)^{-1} R_H, \quad (63) $$

so that Eq. (62) may be written as

$$ \chi(\eta - \eta', \phi - \phi', \omega) = \exp\left(i\frac{\eta - \eta'}{a_1(\omega)}\right) \exp\left(-\frac{|\eta - \eta'|}{a_2(\omega)}\right) \exp\left(-\frac{|\phi - \phi'|}{a_3(\omega)}\right), \quad (64) $$

with $a_1$, $a_2$ and $a_3$ functions of $\omega$, defined as

$$ a_1(\omega) = \alpha_c \frac{1}{2\pi} \left(\frac{\omega}{u}\right)^{-1} \frac{\varepsilon - 1}{2l}, \quad a_2(\omega) = \alpha_\eta \left(\frac{\omega}{u}\right)^{-1} \frac{\varepsilon - 1}{2l}, \quad a_3(\omega) = \alpha_\phi \left(\frac{\omega}{u}\right)^{-1} \frac{\varepsilon - 1}{2}. \quad (65) $$

A large body of work has been produced for quantifying the various parameters in Eq. (64), for internal pipe flows Corcos (1964); Clinch (1969), external flows over plates Willmarth and Wooldridge (1962); Bull (1967) and cylinders Bakewell (1968). Typically, the three parameters $\alpha_c$, $\alpha_\eta$ and $\alpha_\phi$ suggested by Corcos (1963); Clinch (1969); Willmarth and Wooldridge (1962); Bull (1967); Bakewell (1968); Chen and Wambsganss (1972) and used in the current study are

$$ \alpha_c = 0.6, \quad \alpha_\eta = 1.6\alpha_c, \quad \alpha_\phi = 0.29\alpha_c. \quad (66) $$

According to Eq. (64), $\chi$ is expressed in terms of $l$, $\varepsilon$, $u$ and the integration variable $\omega$. It follows that the model chosen for $\chi$ does not affect the list $\Pi$ of independent dimensionless numbers, see Eq. (53).

Eventually, plugging Eq. (64) into Eq. (49) yields a simplified expression for the dimensionless mean-square structure displacement, which reduces to

$$ \sigma^2(x) = u^3(\varepsilon - 1)^2 m_F^2 \frac{1}{2^7 \pi^5} \frac{1}{2\pi} \Re \left\{ \mathbf{w}^\top(x) \left[ \int_0^\infty I_\eta(\omega) I_\phi\left(\frac{\omega}{u}\right) \aleph\left(\frac{\omega}{u}\right) d\omega \right] \mathbf{w}(x) \right\}, \quad (67) $$
with

$$I_{\eta} = \sum_{i,j=1}^{K} \left( \overline{[\mathbf{H}_1][\mathbf{H}_1]^T} I_{\tilde{f},\eta} + h[\mathbf{H}_1][\overline{\mathbf{H}_2}]^T I_{\tilde{f},\eta} + h[\mathbf{H}_2][\overline{\mathbf{H}_1}]^T I_{\tilde{\gamma},\eta} + h^2[\mathbf{H}_2][\overline{\mathbf{H}_2}]^T I_{\tilde{\gamma},\eta} \right)_{i,j} \quad (68)$$

Analytical expressions for $I_{\tilde{f},\eta}$, $I_{\tilde{f},\tilde{\gamma},\eta} = \overline{I_{\tilde{\gamma},\tilde{f},\eta}}$, $I_{\tilde{\gamma},\eta}$ and $I_{\phi}$ are given in Appendix E.

Finally, note that Kocher et al. (2022, 2023), who studied the case $x = 1 + h$ only, obtained the same equation as (67), but with a different function $I_{\eta}$. Indeed, in Kocher et al. (2022, 2023), this function writes

$$I_{\eta} = \frac{\frac{2}{3} \left(\frac{1}{a_2} + \frac{i}{a_1}\right)^{-1} - \left(\frac{1}{a_2} + \frac{i}{a_1}\right)^{-2} - 2 \left[ \left(\frac{1}{a_2} + \frac{i}{a_1}\right)^{-3} + \left(\frac{1}{a_2} + \frac{i}{a_1}\right)^{-4} \right] \exp\left(-\left(\frac{1}{a_2} + \frac{i}{a_1}\right)\right) + 2\left(\frac{1}{a_2} + \frac{i}{a_1}\right)^{-4}}{\left|-m^{(\text{ref})}\omega^2 + i\eta \frac{\sqrt{m^{(\text{ref})}}}{h}\omega + \frac{1}{h^2}\right|^2}, \quad (69)$$

with

$$m^{(\text{ref})} = m_{cyl} \left( \frac{1}{3} + \frac{1}{2l^2} \right) + m_F \frac{\varepsilon^2 + 1}{3 \varepsilon^2 - 1}. \quad (70)$$

The differences between the two formulations (68) and (69) come from the fact that Kocher et al. (2022, 2023) considered the case of a balancing cylinder, i.e. rigid cylinder in a pinned-free configuration, with a dimensionless displacement $w(x,t) = q(t)P(x)$. This kinematics actually corresponds to the restriction of Eq. (28) to a single test function for the cylinder, $P(x) = (x-1)/h$, $x \in ]1, 1+h]$, while the blade is assimilated into an angular spring. Also, in Kocher et al. (2022, 2023), the Archimedean thrust, the fluid-elastic force due to fluid flow and the fluid torques are neglected, i.e. $F_a = F_u = \Gamma = 0$. An inviscid model for the fluid-elastic force $F_0$ is used, i.e. $m_0 = m_F (\varepsilon^2 + 1) (\varepsilon^2 - 1)^{-1}$, $c_0 = 0$, and a global reduced damping $\eta$ is introduced to encompass all possible causes of damping (of structural origin, due to fluid viscosity or due to fluid flow)$^4$. Thus formulated, the mean-square displacement of the cylinder tip predicted by Kocher et al. (2022, 2023) is a function of 7 independent dimensionless numbers, $\{m_{cyl}, m_F, h, \eta, l, \varepsilon, u\}$, compared to the 12 listed in the equation (53) of the present theory. A comparative study between the predictions of the two models is carried out in § 3.5.

### 3.5. Predictions

Having chosen $K = 5$ basis and test functions in § 3.1, a model for the friction coefficients in § 3.2, a model for the turbulent pressure PSD in § 3.3 and a model for the coherence function in § 3.4, we now proceed to determine the fluid-elastic instability thresholds and evaluate the root mean-square structure displacement, $\sigma$, see Eq. (49).

In Fig. 7, for the three sets $\Pi_i$, we show that the first vibration mode is stable for positive Reynolds numbers (flow in the conventional direction), whereas it exhibits a flutter instability for negative Reynolds number (flow in the reverse direction). The instability threshold for the first mode is $Re_{crit}^{(1)} \approx 0$, i.e. $u_{crit}^{(1)} \approx 0$. In fact, $Re_{crit}^{(1)}$ is very slightly negative because fluid flow is required to compensate for the structural damping of the first mode (whose reduced damping is less than 1%). On his side, the second vibration mode is stable for negative Reynolds numbers, whereas it turns unstable for positive Reynolds numbers. For the set $\Pi_1$, the instability threshold of mode 2 is $Re_{crit}^{(2)} = 1.74 \cdot 10^5$, i.e. $u_{crit}^{(2)} = 0.39$. For the set $\Pi_2$ (resp. $\Pi_3$), the instability threshold of mode 2 is $Re_{crit}^{(2)} = 1.18 \cdot 10^5$, i.e. $u_{crit}^{(2)} = 0.32$ (resp. $Re_{crit}^{(2)} = 7.70 \cdot 10^4$, i.e. $u_{crit}^{(2)} = 0.25$). In terms of dimensional quantities, the critical mean flow velocities are $U_{m,crit} = 4.84 \text{ m/s}$ (set $\Pi_1$), $U_{m,crit} = 3.29 \text{ m/s}$ (set $\Pi_2$) and $U_{m,crit} = 2.22 \text{ m/s}$ (set $\Pi_3$). The corresponding critical mean volume flow rates are $Q_{m,crit} = 7.71 \cdot 10^{-2} \text{ m}^3\text{/s}$, $Q_{m,crit} = 5.24 \cdot 10^{-2} \text{ m}^3\text{/s}$ and $Q_{m,crit} = 3.55 \cdot 10^{-2} \text{ m}^3\text{/s}$. In Fig. 7, we also show that the fluid flow weakly affects (resp. does not significantly affect) the angular frequency $\Im\{\lambda^{(j)}\}$ of the first (resp. second) vibration mode. In fact, $\Im\{\lambda^{(j)}\}$ remains within a few percents (less than 2%) of $\omega_0^{(j)}$ (angular frequency in inviscid still fluid) for $Re \in [-2.23 \cdot 10^5, 2.23 \cdot 10^5]$, i.e. $u \in [-0.5, 0.5]$ (set $\Pi_1$) or $Re \in [-1.85 \cdot 10^5, 1.85 \cdot 10^5]$, i.e. $u \in [-0.5, 0.5]$ (set $\Pi_2$) or $Re \in [-1.54 \cdot 10^5, 1.54 \cdot 10^5]$, i.e. $u \in [-0.5, 0.5]$ (set $\Pi_3$). For $Re = 0$, the ratio $\Im\{\lambda^{(1)}\}/\omega_0^{(1)}$ reaches its maximum and deviates from 1 due to the influence of $\Im\{\lambda^{(1)}\}Sk$ on the added coefficients $m_0, c_0, j_0$, and $d_0$ for a fluid at rest. Conversely, due to the high frequency of mode 2, $\Im\{\lambda^{(2)}\}Sk$ is sufficiently large so that the added coefficients approach their inviscid limit, as depicted in Fig. 3. Therefore, $\Im\{\lambda^{(2)}\}/\omega_0^{(2)} \approx 1$ for $Re = 0$.

Having determined the instability thresholds, we now proceed to evaluate the PSD $\langle Q|Q \rangle$ of the modal amplitudes. In Fig. 8, we illustrate the evolution of $\langle Q|Q \rangle_{1,1}$ and $\langle Q|Q \rangle_{2,2}$ as functions of the dimensionless

<footnote n="4">4. A value $\eta = 5\%$ is recommended by Kocher et al. (2022, 2023).</footnote>
angular frequency $\omega$, considering the first set of dimensionless parameters provided in Table 2. The spectrum $\langle \mathbf{Q}|\mathbf{Q} \rangle_{i,i}$ displays a peak at an angular frequency $\omega$ close to $\omega_0^{(i)}$, representing the angular frequency of the i-th undamped free mode vibrating in an inviscid still fluid. Therefore, the primary angular frequency of system vibration remains largely unaffected by fluid-elastic forces and torques ($F_U$, $\Gamma_U$; see Eq. (10)) due to fluid flow, or even by the turbulence excitations $\tilde{F}$ and $\tilde{\Gamma}$ represented through the pressure PSD $\mathcal{N}$, see Eq. (61). These latter factors mainly influence the frequency distribution within the spectrum and the amplitude of $\langle \mathbf{Q}|\mathbf{Q} \rangle_{i,i}$. Notably, as $\langle \mathbf{Q}|\mathbf{Q} \rangle_{1,1} \ge \langle \mathbf{Q}|\mathbf{Q} \rangle_{2,2}$, the system primarily responds to the flow effects of the fluid on its first mode of vibration. This observation is corroborated by Fig. 9, which shows that the dimensionless RMS structure displacement, $\sigma(x)$, exhibits the same shape as the first test function, $w_1(x) = x^2$; see Fig. 4.

Finally, in Fig. 10, we demonstrate that the dimensionless mean-square displacement at the cylinder tip, denoted as $\sigma(1+h)$ and governed by Eqs. (67)-(68), exhibits a scaling behavior $\sigma(1+h) \sim Re^2$ across all three sets of dimensionless parameters $\Pi_i$ given in Table 2. Our prediction, which incorporates analytical formulations accounting for all sources of structural and fluid-added damping, validates the recommended value of $\eta = 5\%$ proposed by Kocher et al. (2022, 2023) to encompass all damping influences, see Eq. (69). In fact, Fig. 10 shows that the predictions for $\sigma(1+h)$ obtained from both the present theory and the theory proposed by Kocher et al. (2022, 2023) utilizing $\eta = 5\%$, exhibit good agreement, as long as the Reynolds number remains significantly lower than the critical threshold $Re_{crit}^{(2)}$.
<figure><img src="image_8.png" /><figcaption>Figure 7: Stability diagrams of the blade-cylinder system for the three sets of dimensionless parameters {$g, m_{cyl}, m_F, h, \kappa, j_{cyl}, \xi, \zeta, l, \varepsilon, Sk$}, as given in Table 2. The top (resp. middle, bottom) two subfigures correspond to set $\Pi_1$ (resp. $\Pi_2$, $\Pi_3$). For a given set, the left (resp. right) figure shows the evolution of $\operatorname{sgn}(\mathbb{R}\{\lambda^{(j)}\})$ (resp. $\mathfrak{S}\{\lambda^{(j)}\}/\omega_0^{(j)}$) as function of the Reynolds number, for $j = 1$ (mode 1, black lines) and $j = 2$ (mode 2, red lines). The j-th mode is stable if $\operatorname{sgn}(\mathbb{R}\{\lambda^{(j)}\}) = -1$ (solid lines) and unstable if $\operatorname{sgn}(\mathfrak{S}\{\lambda^{(j)}\}) = 1$ (dashed lines). The conventional (resp. reverse) direction flow corresponds to $Re > 0$ (resp. $Re < 0$).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Evolution of the dimensionless PSD of the modal amplitude, $\langle |Q|^2 \rangle_{j,j}$, given by Eq. (40), as a function of the dimensionless angular frequency $\omega \in ]0, 2.5u]$. Black solid line: $j=1$, red dashed line: $j=2$. The dotted vertical line corresponds to $\omega = 0.5u$, see Eq. (61). The dimensionless parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, \xi, \zeta, l, \varepsilon, Sk\}$ used to trace out this figure are given in the set $\Pi_1$ of Table 2, with $u = 0.32$ (i.e. $Re = 1.37 \cdot 10^5$).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9: Evolution of the root mean-square structure displacement, $\sigma$, given by Eq. (49), as a function of the dimensionless axial coordinate $x \in [0, 1+h]$. The dimensionless parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, \xi, \zeta, l, \varepsilon, Sk\}$ used to trace out this figure are given in the set $\Pi_1$ of Table 2, with $u = 0.32$ (i.e. $Re = 1.37 \cdot 10^5$).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 10: Evolution of the root mean-square structure displacement at the cylinder tip, $\sigma(1+h)$, as a function of the Reynolds number, $Re$. The dimensionless parameters $\{g, m_{cyl}, m_F, h, \kappa, j_{cyl}, \xi, \zeta, l, \varepsilon, Sk\}$ used to trace out this figure are given in the sets $\Pi_1$ (black line), $\Pi_2$ (red line) and $\Pi_3$ (blue line) of Table 2. Dashed lines correspond to the predictions of Kocher et al. (2022, 2023), given by Eq. (67) and Eq. (69), with $\eta = 5\%$. The vertical dotted lines correspond to the mode 2 fluid-elastic instability threshold $Re_{crit}^{(2)}$, presented in Fig. 7.</figcaption></figure>

4. Conclusion

In this study, we investigate the fluid-structure interaction problem involving two coaxial cylinders separated by a Newtonian fluid under turbulent axial flow conditions. The inner cylinder is mounted on a flexible blade, allowing transverse vibrations. We develop a theoretical approach based on the dynamical equilibrium equations for the blade (treated as a Rayleigh beam with structural damping coefficients) and the inner cylinder (treated as a rigid body). Specifically, we aim to: i) determine the undamped free vibration modes and corresponding frequencies of the system in an inviscid still fluid, ii) determine the fluid-elastic instability threshold of the system, and iii) establish a full analytical expression for the mean-square displacement of the structure. This novel approach incorporates various components of fluid forces and torques, including an Archimedean thrust component, a fluid-elastic component for a fluid at rest (through the inclusion of viscous added coefficients that account for the three-dimensionality of the problem), a fluid-elastic component due to fluid flow, a fluid component due to turbulence, characterized by a pressure power spectral density and coherence function. Notably, this approach significantly expands upon our previous theory Kocher et al. (2022, 2023), which was limited to a single balancing mode for the cylinder kinematics and did not consider fluid torques. Additionally, the previous model did not account for Archimedean thrust and represented fluid-elastic forces through inviscid added coefficients without considering three-dimensional effects or fluid flow velocity. Accounting for the aforementioned effects, the dynamical equilibrium equations for the blade and the inner cylinder are derived and expressed in fully dimensionless forms. The new theoretical approach reveals that the vibration modes and frequencies, as well as the instability threshold and mean-square structure displacement, depend on 8, 11, and 12 dimensionless parameters, respectively. These parameters include the aspect ratio of the cylinder, the radius ratio of the annular fluid gap, and the blade-to-cylinder aspect ratio. By incorporating models from the literature for the viscous friction coefficients, turbulent pressure power spectral density, and coherence function into our new theory, we demonstrate that: i) the first vibration mode is stable (resp. unstable) for conventional (resp. reverse) flow directions, ii) the second vibration mode becomes unstable at a positive critical Reynolds number, and iii) the root mean-square structure displacement scales as the square of the Reynolds number. The first objective of this study, presented in a fully dimensionless formulation, is to assist engineers in constructing
small-scale experiments representative of vibrations in pressure vessels. To facilitate this, the Python code<fnref n="5" /> for determining the stability of the system and calculating the mean-square displacement from the present theory is made available free of charge; please refer to Plado Costante and Lagrange (2024). The theoretical predictions presented in this paper, along with those from Kocher et al. (2022, 2023), have led EDF and CEA to design a small-scale rig based on the coaxial cylinders configuration. This rig is primarily intended to enhance our understanding of turbulence-induced vibrations and fluid-structure interactions within a pressure vessel under controlled experimental conditions. In the short term, it will allow us to test the assumption of homogeneous turbulence and evaluate the accuracy of the PSD and coherence function models from Au-Yang and Jordan (1980); Au-Yang et al. (1995); Weaver et al. (2000) and Powell (2005). Based on these future tests, updated models may be proposed and integrated into the present theoretical analysis to refine the predictions in § 3. However, note that the theoretical analysis in § 2 uses arbitrary PSDs for the turbulent fluid force $\vec{f}$ and torque $\tilde{\gamma}$. Thus, the expression (48) for the PSD $\langle w, w \rangle$ of the displacement, which constitutes the main result of this study, will remain valid even for non-homogeneous turbulence. Additionally, since this expression is derived by recasting the equilibrium equations of our simplified problem into the general form of a mass-damper-spring equation (Eq. (30)), it is not limited to our specific simplified model of a PWR but also applies to more complex models. At some point, one could even envision a more complex representative system of a PWR core vessel, where the mass, damping, and stiffness matrices are determined numerically, for example, using the finite element method, and then incorporated into our Eq. (48) to obtain new predictions for the RMS value of the structural displacement. In the longer term, this rig will support the development of a numerical tool capable of simulating vibrations in the primary circuits of nuclear PWRs, as outlined in the introduction of this paper.

<footnote n="5">5. The user is prompted to enter their values for $L$, $a \times b$ or $I$, $R_1$, $H$, the thickness of cylinder $C_1$ or $J_{cyl}$, $R_2$, $E$, $\rho$, $\Xi$, $Z$, $M_{cyl}$, $\rho_F$, $\nu$, $U_m$, $G$ and $X$ in the file named *MainCode.py*. Compiling this file indicates whether the system is stable by checking the sign of the real part of the eigenvalues of the Jacobian matrix $[\mathbf{A}]$ (see Eq. (52)), and it also provides the value of $\Sigma(X)$ (dimensional root mean-square value of the structure displacement), from the dimensionless equation (49). The user can implement their own models for the turbulent pressure PSD and for the coherence function, these defaulting to the $\kappa$ and $\chi$ functions defined in Eqs. (61) and (64), respectively. The integral (49) is calculated with integration limits $10^{-5}$ and $2.5u$, corresponding to the limits of the spectrum; see Eq. (61). A square integration method is used to calculate the integral. A parameter *Resol*, whose default value is 1000, is used to define the number of evaluations of the integrand. Also, the infinite sum of Eq. (B.6) is truncated to some parameter *NFi*, whose default value is 50.</footnote>
## Appendix A. Modal analysis

In this appendix, we detail the calculation of the undamped free vibration modes $w_0^{(j)}$ in inviscid still fluid, and the associated frequencies $\omega_0^{(j)}$. As stated in §2.4, these modes correspond to the non-trivial solutions of Eqs. (16)-(17), along with clamped conditions at the origin, considering $\tilde{f} = \tilde{\gamma} = 0$ (no turbulent forcing), $\xi = \zeta = 0$ (undamped structure), $m_0 = m_0^{(Sk \to \infty)}$, $j_0 = j_0^{(Sk \to \infty)}$, $c_0 = 0$, $d_0 = 0$ (inviscid fluid) and $f_d = c_u = a_u = k_u = d_u = 0$ (still fluid). Under these conditions, Eq. (16) may be written as

$$ \frac{\partial^2}{\partial t^2} \left( w - \kappa^2 \frac{\partial^2 w}{\partial x^2} \right) + \frac{\partial^4 w}{\partial x^4} - (m_F - m_{cyl}) g \frac{\partial^2 w}{\partial x^2} = 0, \quad (A.1) $$

and Eq. (17) may be written as

$$ \frac{\partial^2}{\partial t^2} \left[ \left( m_{cyl} + m_0^{(Sk \to \infty)} \right) \left( w + \frac{h}{2} \frac{\partial w}{\partial x} \right) \Big|_1 + \kappa^2 \frac{\partial w}{\partial x} \Big|_1 \right] + (m_F - m_{cyl}) g \frac{\partial w}{\partial x} \Big|_1 - \frac{\partial^3 w}{\partial x^3} \Big|_1 = 0, \quad (A.2a) $$

$$ \frac{\partial^2}{\partial t^2} \left( j_{cyl} + j_0^{(Sk \to \infty)} - \frac{h}{2} \kappa^2 \right) \frac{\partial w}{\partial x} \Big|_1 + \frac{\partial^2}{\partial x^2} \left( w + \frac{h}{2} \frac{\partial w}{\partial x} \right) \Big|_1 = 0. \quad (A.2b) $$

Seeking solutions $w(x,t) = \Re\{w_0(x)e^{i\omega_0 t}\}$, Eq. (A.1) may be written as

$$ \frac{d^4 w_0}{dx^4} = \omega^2 w_0 + \left[ (m_F - m_{cyl}) g - \omega^2 \kappa^2 \right] \frac{d^2 w_0}{dx^2}. \quad (A.3) $$

In matrix form, the solutions of Eq. (A.3) with clamped conditions at the origin, i.e. $B_0[w_0] = (w_0(0), dw_0/dx(0)) = (0, 0)$, are

$$ \begin{bmatrix} w_0 \\ dw_0/dx \\ d^2 w_0/dx^2 \\ d^3 w_0/dx^3 \end{bmatrix} = \exp([\mathbf{A}] x) \begin{bmatrix} 0 \\ 0 \\ C_1 \\ C_2 \end{bmatrix}, \quad (A.4) $$

with $C_1$ and $C_2$ two constants and

$$ [\mathbf{A}] = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \omega^2 & 0 & (m_F - m_{cyl}) g - \omega^2 \kappa^2 & 0 \end{pmatrix}. \quad (A.5) $$

Introducing $w(x,t) = \Re\{w_0(x)e^{i\omega_0 t}\}$, with $w_0$ given by (A.4), into Eq. (A.2) yields a linear homogenous system of two equations for unknown $C_1$ and $C_2$. In matrix form, this system may be written as

$$ \begin{bmatrix} -\omega^2 (m_{cyl} + m_0^{(Sk \to \infty)}) & -\omega^2 \left[ \left(m_{cyl} + m_0^{(Sk \to \infty)}\right) \frac{h}{2} + \kappa^2 \right] + (m_F - m_{cyl}) g & 0 & -1 \\ 0 & -\omega^2 \left(j_{cyl} + j_0^{(Sk \to \infty)} - \frac{h}{2} \kappa^2\right) & 1 & \frac{h}{2} \end{bmatrix} \exp([\mathbf{A}]) \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix} \times \begin{bmatrix} C_1 \\ C_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \quad (A.6) $$

A solvability condition (vanishing determinant) for this linear system yields an infinite countable number of real solutions $\omega_0^{(j)}$ corresponding to each free vibration mode $w_0^{(j)}$ (non trivial solution of Eq. (A.6) with $\omega = \omega_0^{(j)}$, determined up to a scaling factor). The dimensionless structural angular frequencies $\omega_s^{(j)}$ and modes $w_s^{(j)}$ are determined similarly, setting $m_0^{(Sk \to \infty)} = j_0^{(Sk \to \infty)} = m_F = 0$ into Eq. (A.6). Values for $\omega_s^{(j)}$ are given in Table A.3.

<table><caption>Table A.3: Dimensionless angular structural frequencies $\omega_s^{(j)}$ of the first four free vibration modes. The dimensionless parameters $\{g, m_{cyl}, h, \kappa, j_{cyl}, l, \varepsilon\}$ used to calculate $\omega_s^{(j)}$ are given in Table 2.</caption><thead><tr><th>Set</th><th>ω<sub>s</sub><sup>(1)</sup></th><th>ω<sub>s</sub><sup>(2)</sup></th><th>ω<sub>s</sub><sup>(3)</sup></th><th>ω<sub>s</sub><sup>(4)</sup></th></tr></thead><tbody><tr><td>Π<sub>1</sub></td><td>2.07 · 10<sup>-2</sup></td><td>8.94 · 10<sup>-1</sup></td><td>21.6</td><td>53.7</td></tr><tr><td>Π<sub>2</sub></td><td>1.88 · 10<sup>-2</sup></td><td>8.15 · 10<sup>-1</sup></td><td>21.8</td><td>55.9</td></tr><tr><td>Π<sub>3</sub></td><td>1.72 · 10<sup>-2</sup></td><td>7.45 · 10<sup>-1</sup></td><td>22.0</td><td>57.4</td></tr></tbody></table>
## Appendix B. Added coefficients for a fluid at rest

In this appendix the main results of Lagrange and Puscas (2023) are reproduced, leading to an expression of the added coefficients appearing in Eq. (11). According to Lagrange and Puscas (2023) the linear fluid force $\mathbf{f}_0$ associated to a displacement $\Re\{\underline{W}(X) \exp(i\Omega T) \mathbf{e}_y\}$ is

$$ \mathbf{f}_0(\underline{W}, X) = \Re\{\pi\rho_F R_1^2 \Omega^2 g_0(\underline{W}, X) \exp(i\Omega T) \mathbf{e}_y\}, \quad (B.1) $$

with $g_0$ a function that is linear in $\underline{W}$, given in Eq. (B.6). Assuming small displacements of the rigid cylinder, i.e.

$$ \underline{W}(X) = \underline{W}_G + \underline{\theta}_L H \left( \frac{X - L}{H} - \frac{1}{2} \right), \quad (B.2) $$

the resulting fluid force $\mathbf{F}_0 = \int_L^{L+H} \mathbf{f}_0(\underline{W}, X) dX$ is

$$ \begin{aligned} \mathbf{F}_0 &= \Re \left\{ \pi \rho_F R_1^2 H \Omega^2 \underline{W}_G \int_0^1 g_0(1, \eta) \, d\eta \exp(i\Omega T) \mathbf{e}_y \right\} \\ &= \Re \left\{ (M_0 \Omega^2 \underline{W}_G - i C_0 \Omega \underline{W}_G) \exp(i\Omega T) \mathbf{e}_y \right\}. \end{aligned} \quad (B.3) $$

Considering $\mathbf{G}\mathbf{X}$ along $\mathbf{e}_x$, the resulting fluid torque $\mathbf{\Gamma}_0 = \int_L^{L+H} \mathbf{G}\mathbf{X} \times \mathbf{f}_0(\underline{W}, X) dX$ is

$$ \begin{aligned} \mathbf{\Gamma}_0 &= \Re \left\{ \pi \rho_F R_1^2 \Omega^2 H^3 \underline{\theta}_L \int_0^1 \left(\eta - \frac{1}{2}\right) g_0 \left(\eta - \frac{1}{2}, \eta\right) d\eta \exp(i\Omega T) \mathbf{e}_z \right\} \\ &= \Re \left\{ (J_0 \Omega^2 \underline{\theta}_L - i D_0 \Omega \underline{\theta}_L) \exp(i\Omega T) \mathbf{e}_z \right\}. \end{aligned} \quad (B.4) $$

It follows that the dimensionless added coefficients are given by Eq. (20) with

$$ F^{(1)} = \int_0^1 g_0(1, \eta) \, d\eta, \quad (B.5a) $$

$$ F^{(2)} = \int_0^1 \left( \eta - \frac{1}{2} \right) g_0 \left( \eta - \frac{1}{2}, \eta \right) d\eta. \quad (B.5b) $$

The function $g_0$ is defined as

$$ g_0(w, \eta) = (\varepsilon - 1) \sum_{n=1}^{\infty} h_n b_n(w) \sin(n\pi\eta), \quad (B.6) $$

with $b_n(w) = 2 \int_0^1 w(\eta) \sin(n\pi\eta) d\eta$ the n-th Fourier coefficient of $w$.

### Appendix B.1. Viscous case

In the case of a viscous fluid, Lagrange and Puscas (2023) has shown that $h_n$ writes

$$ h_n = i \left[ \Phi_n(r_1) + \frac{\gamma_n^2 \Phi_n(r_1) - \alpha_n^2 \Psi_n(r_1)}{\gamma_n^2 - \alpha_n^2} \right], \quad (B.7) $$

with

$$ \gamma_n = i \frac{n\pi}{l} (\varepsilon - 1), \quad \alpha_n^2 = \gamma_n^2 - iSk\omega, \quad r_1 = \frac{1}{\varepsilon - 1}. \quad (B.8) $$

In Eq. (B.7), $\Phi_n$ and $\Psi_n$ are linear combinations of Hankel functions of the first and second kind ($H^{(1)}$ and $H^{(2)}$), given by

$$ \begin{bmatrix} \Phi_n(r) \\ \Psi_n(r) \\ A_n(r) \end{bmatrix} = i \begin{bmatrix} \mathbf{B}(\gamma_n r, \alpha_n r) \end{bmatrix} \begin{bmatrix} [\mathbf{M}(r_1, \gamma_n, \alpha_n)] \\ [\mathbf{M}(r_2, \gamma_n, \alpha_n)] \end{bmatrix}^{-1} \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad (B.9) $$

with $r_2 = \varepsilon / (\varepsilon - 1)$ and

$$ [\mathbf{B}(\gamma r, \alpha r)] = \begin{bmatrix} H_1^{(1)}(\gamma r) & H_1^{(2)}(\gamma r) & 0 & 0 & 0 & 0 \\ 0 & 0 & H_1^{(1)}(\alpha r) & H_1^{(2)}(\alpha r) & 0 & 0 \\ 0 & 0 & 0 & 0 & H_0^{(1)}(\alpha r) & H_0^{(2)}(\alpha r) \end{bmatrix}, \quad (B.10a) $$

$$ [\mathbf{M}(r, \gamma, \alpha)] = \begin{bmatrix} \gamma H_1^{(1)\prime}(\gamma r) & \gamma H_1^{(2)\prime}(\gamma r) & H_1^{(1)}(\alpha r)/r & H_1^{(2)}(\alpha r)/r & -\gamma^2 H_0^{(1)}(\alpha r) & -\gamma^2 H_0^{(2)}(\alpha r) \\ H_1^{(1)}(\gamma r)/r & H_1^{(2)}(\gamma r)/r & \alpha H_1^{(1)\prime}(\alpha r) & \alpha H_1^{(2)\prime}(\alpha r) & -\gamma^2 H_0^{(1)}(\alpha r) & -\gamma^2 H_0^{(2)}(\alpha r) \\ H_1^{(1)}(\gamma r) & H_1^{(2)}(\gamma r) & 0 & 0 & \alpha H_0^{(1)\prime}(\alpha r) & \alpha H_0^{(2)\prime}(\alpha r) \end{bmatrix}. \quad (B.10b) $$
In Eq. (B.10b), the prime subscript notation on the Hankel functions stands for the derivative with the r variable.

Appendix B.2. Inviscid case, $Sk \to \infty$

The particular case of an inviscid fluid is recovered from the previous viscous case by taking the limit $Sk \to \infty$. In such a case, Lagrange and Puscas (2023); Lagrange et al. (2023) have shown that $h_n \to h_n^{(Sk \to \infty)}$, where

$$h_n^{(Sk \to \infty)} = - \begin{bmatrix} H_1^{(1)}(\gamma_n r_1) & H_1^{(2)}(\gamma_n r_1) \end{bmatrix} \begin{bmatrix} \gamma H_1^{(1)'}(\gamma r_1) & \gamma H_1^{(2)'}(\gamma r_1) \\ \gamma H_1^{(1)'}(\gamma r_2) & \gamma H_1^{(2)'}(\gamma r_2) \end{bmatrix}^{-1} \begin{bmatrix} 1 \\ 0 \end{bmatrix}. \quad (B.11)$$
## Appendix C. Added coefficients due to fluid flow

In this appendix we give an expression of the added coefficients due to fluid flow, appearing in Eq. (12). According to Taylor (1952); Hawthorne (1961); Païdoussis (1973); Païdoussis and Ostoja-Starzewski (1981); Païdoussis (2024), the linear fluid force $f_U$ due to axial fluid flow and associated to a displacement $\Re\{\underline{W}(X) \exp(i\Omega T) \mathbf{e}_y\}$ is<fnref n="6" />

$$f_U = \Re \left\{ \left( -2 \frac{M_0^{(Sk \to \infty)}}{H} U_m i \Omega \frac{dW}{dX} - \frac{M_0^{(Sk \to \infty)}}{H} U_m^2 \frac{d^2 W}{dX^2} - F_N + F_T \frac{dW}{dX} \right) \exp(i\Omega T) \mathbf{e}_y \right\} + F_T \mathbf{e}_x, \quad (C.1)$$

with

$$F_N = \rho_F R_1 |U_m| C_N \left( i \Omega \underline{W} + U_m \frac{dW}{dX} \right), \quad (C.2a)$$

$$F_T = \rho_F R_1 |U_m| U_m C_T. \quad (C.2b)$$

In Eq. (C.1), the first two terms refer to the inviscid added damping and added stiffness due to fluid flow. The inviscid added mass $M_0^{(Sk \to \infty)}$ corresponds to the limit of $M_0$, given by Lagrange and Puscas (2023), as $Sk \to \infty$ (inviscid regime, see also Lagrange et al. (2023)). The third term $F_N$ is the normal viscous force, with $C_N$ the normal friction coefficient. The term $F_T$ is the tangent viscous force, with $C_T$ the tangent friction coefficient.

Assuming small displacements of the rigid cylinder, i.e.

$$\underline{W}(X) = \underline{W}_G + \theta_L H \left( \frac{X - L}{H} - \frac{1}{2} \right), \quad (C.3)$$

the resulting fluid force $\mathbf{F}_U = \int_L^{L+H} \mathbf{f}_U(\underline{W}, X) dX$ is

$$\begin{aligned} \mathbf{F}_U &= \Re \left\{ \left( -H \rho_F R_1 |U_m| C_N i \Omega \underline{W}_G - 2 M_0^{(Sk \to \infty)} U_m i \Omega \theta_L - H \rho_F R_1 |U_m| U_m (C_N - C_T) \theta_L \right) \exp(i\Omega T) \mathbf{e}_y \right\} \\ &= \Re \left\{ \left( -C_U i \Omega \underline{W}_G - A_U i \Omega \theta_L - K_U \theta_L \right) \exp(i\Omega T) \mathbf{e}_y \right\} + F_d \mathbf{e}_x. \end{aligned} \quad (C.4)$$

Considering $\mathbf{G}\mathbf{X}$ along $\mathbf{e}_x$, the resulting fluid torque $\mathbf{\Gamma}_U = \int_L^{L+H} \mathbf{G}\mathbf{X} \times \mathbf{f}_U(\underline{W}, X) dX$ is

$$\begin{aligned} \mathbf{\Gamma}_U &= \Re \left\{ -\frac{1}{12} \rho_F R_1 |U_m| C_N H^3 i \Omega \theta_L \exp(i\Omega T) \right\} \\ &= \Re \left\{ -D_U i \Omega \theta_L \exp(i\Omega T) \right\}. \end{aligned} \quad (C.5)$$

Equation (C.4) (resp. Eq. (C.5)) yields analytical expressions of coefficients $C_U$, $A_U$, $K_U$ and $F_d$ (resp. $D_U$). By plugging these expressions into Eq. (22) finally produces expressions for the dimensionless coefficients $c_u$, $a_u$, $k_u$, $f_d$ and $d_u$, see Eq. (23).

<footnote n="6">6. The expression for the linear fluid force proposed by Taylor (1952); Hawthorne (1961); Païdoussis (1973); Païdoussis and Ostoja-Starzewski (1981); Païdoussis (2024) contains a term $\rho_F DC_D i \Omega \underline{W}/2$, called normal viscous force in a quiescent surrounding fluid. This term is not reported in Eq. (C.1) as it is actually taken into account in our model, via the added damping coefficient $C_0$; see Appendix B.</footnote>
## Appendix D. Mass, damping and stiffness coefficients

In this appendix, the expressions of the mass $m_{i,j}$, damping $c_{i,j}$, stiffness $k_{i,j}$ coefficients, along with the factors $\alpha_i$ and $\beta_i$, appearing in Eq. (30) are provided. We also provide an expression relating the dimensionless Rayleigh damping coefficient $\zeta$ to the reduced structural modal damping $\xi_T$.

The mass coefficient is $m_{i,j} = m_{i,j}^{(s)} + (m_{cyl} + m_0) m_{i,j}^{(m)} + (j_{cyl} + j_0) m_{i,j}^{(J)}$ with

$$m_{i,j}^{(s)} = \int_{0}^{1} \left( w_j - \kappa^2 \frac{d^2 w_j}{dx^2} \right) w_i^{\dagger} + \delta(x-1) \kappa^2 \frac{dw_j}{dx} w_{G,i}^{\dagger} - \frac{h}{2} \kappa^2 \delta(x-1) \frac{dw_j}{dx} \theta_i^{\dagger} dx, \quad (D.1a)$$

$$m_{i,j}^{(m)} = \int_{0}^{1} \left( \delta(x-1) w_j + \frac{1}{2} \delta(x-1) h \frac{dw_j}{dx} \right) w_{G,i}^{\dagger} dx, \quad (D.1b)$$

$$m_{i,j}^{(J)} = \int_{0}^{1} \delta(x-1) \frac{dw_j}{dx} \theta_i^{\dagger} dx. \quad (D.1c)$$

The damping coefficient is $c_{i,j} = c_{i,j}^{(s)} + (c_0 + c_u) c_{i,j}^{(c)} + (d_0 + d_u) c_{i,j}^{(d)} + a_u c_{i,j}^{(a)}$ with

$$c_{i,j}^{(s)} = \int_{0}^{1} \left( 2\xi w_j + 2\zeta \frac{d^4 w_j}{dx^4} \right) w_i^{\dagger} dx, \quad (D.2a)$$

$$c_{i,j}^{(c)} = \int_{0}^{1} \left( \delta(x-1) w_j + \frac{h}{2} \delta(x-1) \frac{dw_j}{dx} \right) w_{G,i}^{\dagger} dx, \quad (D.2b)$$

$$c_{i,j}^{(d)} = \int_{0}^{1} \delta(x-1) \frac{dw_j}{dx} \theta_i^{\dagger} dx, \quad (D.2c)$$

$$c_{i,j}^{(a)} = \int_{0}^{1} \delta(x-1) \frac{dw_j}{dx} w_{G,i}^{\dagger} dx. \quad (D.2d)$$

The stiffness coefficient is $k_{i,j} = k_{i,j}^{(s)} + [(m_F - m_{cyl})g + f_d] k_{i,j}^{(f)} + k_u k_{i,j}^{(k)}$ with

$$k_{i,j}^{(s)} = \int_{0}^{1} \frac{d^4 w_j}{dx^4} w_i^{\dagger} - \delta(x-1) \frac{d^3 w_j}{dx^3} w_{G,i}^{\dagger} + \left( \delta(x-1) \frac{d^2 w_j}{dx^2} + \frac{h}{2} \delta(x-1) \frac{d^3 w_j}{dx^3} \right) \theta_i^{\dagger} dx, \quad (D.3a)$$

$$k_{i,j}^{(f)} = \int_{0}^{1} -\frac{d^2 w_j}{dx^2} w_i^{\dagger} + \delta(x-1) \frac{dw_j}{dx} w_{G,i}^{\dagger} dx, \quad (D.3b)$$

$$k_{i,j}^{(k)} = \int_{0}^{1} \delta(x-1) \frac{dw_j}{dx} w_{G,i}^{\dagger} dx. \quad (D.3c)$$

The factors $\alpha_i$ and $\beta_i$ are

$$\alpha_i = \int_{0}^{1} \delta(x-1) w_{G,i}^{\dagger} dx, \quad (D.4a)$$

$$\beta_i = \int_{0}^{1} \delta(x-1) \theta_i^{\dagger}. \quad (D.4b)$$

Finally, to determine the Rayleigh damping coefficients $\xi$ and $\zeta$, we first return to the equation of the structural vibration modes, non-trivial solutions of

$$\frac{d^2 \mathbf{q}}{dt^2} + [\mathbf{m}]^{-1} [\mathbf{c}^{(s)}(\xi, \zeta)] \frac{d\mathbf{q}}{dt} + [\mathbf{m}]^{-1} [\mathbf{k}] \mathbf{q} = \mathbf{0}, \quad (D.5)$$

with

$$[\mathbf{m}] = [\mathbf{m}^{(s)}] + m_{cyl} [\mathbf{m}^{(m)}] + j_{cyl} [\mathbf{m}^{(J)}], \quad (D.6a)$$

$$[\mathbf{k}] = [\mathbf{k}^{(s)}] - m_{cyl} g [\mathbf{k}^{(f)}]. \quad (D.6b)$$

Then, by introducing $[\mathbf{p}]$ as a matrix containing the eigenvectors of $[\mathbf{m}]^{-1} [\mathbf{k}]$ and defining $\mathbf{r} = [\mathbf{p}]^{-1} \mathbf{q}$, Eq. (D.5) is rewritten in the form

$$\frac{d^2 \mathbf{r}}{dt^2} + [\mathbf{p}]^{-1} [\mathbf{m}]^{-1} [\mathbf{c}^{(s)}(\xi, \zeta)] [\mathbf{p}] \frac{d\mathbf{r}}{dt} + [\omega_s^2] \mathbf{r} = \mathbf{0}. \quad (D.7)$$

Obviously, it is not possible to find two Rayleigh damping coefficients that would satisfy the $K \times K$ conditions $[\mathbf{f}] (\xi, \zeta, \xi_T) = [\mathbf{p}]^{-1} [\mathbf{m}]^{-1} [\mathbf{c}^{(s)} (\xi, \zeta)] [\mathbf{p}] - 2 [\xi_T \omega_s] = [\mathbf{0}]$ for any arbitrary value of the reduced damping $\xi_T$. To
get around this problem, a least squares method could be used to determine fit values for $\xi$ and $\zeta$ that would minimize a norm of $[f]$. In the present study, a simpler approach is chosen, in which $\xi$ is set to zero and $\zeta$ is determined by requiring that $[f]_{2,2} = 0$, i.e.

$$ ([\mathbf{p}]^{-1} [\mathbf{m}]^{-1} [\mathbf{c}^{(s)} (0, \zeta)] [\mathbf{p}])_{2,2} = 2\xi_T \omega_s^{(2)}. \qquad (D.8) $$

Plugging Eq. (D.2a) into Eq. (D.8), the Rayleigh damping coefficient $\zeta$ is

$$ \zeta = \frac{\xi_T \omega_s^{(2)}}{\left( [\mathbf{p}]^{-1} [\mathbf{m}]^{-1} \left[ \int_0^1 \frac{d^4 w_i}{dx^4} w_i^\dagger dx \right] [\mathbf{p}] \right)_{2,2}}. \qquad (D.9) $$

The values of $\zeta$ given in Table 2 are calculated for $\xi_T = 1\%$.
Appendix E. Joint acceptance functions

In this appendix we give analytical expressions of the dimensionless joint acceptance functions $I_{\tilde{f}}$, $I_{\tilde{f}\tilde{\gamma}}$ and $I_{\tilde{\gamma}}$ appearing in Eq. (45) and Eq. (47), for a coherence function $\chi$ given by Eq. (64). Separating integrals on $(\eta, \eta')$ and $(\phi, \phi')$ in Eq. (45) and Eq. (47), the joint acceptance functions are

$$I_{\tilde{f}} = I_{\tilde{f},\eta} I_{\phi}, \quad I_{\tilde{f}\tilde{\gamma}} = I_{\tilde{f}\tilde{\gamma},\eta} I_{\phi}, \quad I_{\tilde{\gamma}} = I_{\tilde{\gamma},\eta} I_{\phi}, \qquad (E.1)$$

with

$$I_{\tilde{f},\eta} = \int_{0}^{1} \int_{0}^{1} \exp\left(i\frac{\eta-\eta'}{a_1}\right) \exp\left(-\frac{|\eta-\eta'|}{a_2}\right) d\eta d\eta', \qquad (E.2a)$$

$$I_{\tilde{f}\tilde{\gamma},\eta} = \int_{0}^{1} \int_{0}^{1} \exp\left(i\frac{\eta-\eta'}{a_1}\right) \exp\left(-\frac{|\eta-\eta'|}{a_2}\right) \left(\eta' - \frac{1}{2}\right) d\eta d\eta', \qquad (E.2b)$$

$$I_{\tilde{\gamma},\eta} = \int_{0}^{1} \int_{0}^{1} \exp\left(i\frac{\eta-\eta'}{a_1}\right) \exp\left(-\frac{|\eta-\eta'|}{a_2}\right) \left(\eta - \frac{1}{2}\right) \left(\eta' - \frac{1}{2}\right) d\eta d\eta', \qquad (E.2c)$$

$$I_{\phi} = \int_{0}^{2\pi} \int_{0}^{2\pi} \exp\left(-\frac{|\phi - \phi'|}{a_3}\right) \cos(\phi) \cos(\phi') \, d\phi d\phi'. \qquad (E.2d)$$

Analytical calculations show that the double integrals in Eq. (E.2) are

$$I_{\tilde{f},\eta} = \frac{2a_1^2 a_2}{(a_1^2 + a_2^2)^2} \left[ \left( -i a_1 a_2^2 + \frac{1}{2} a_1^2 a_2 - \frac{1}{2} a_2^3 \right) \exp\left( -\frac{i a_2 - a_1}{a_1 a_2} \right) + \left( i a_1 a_2 + \frac{1}{2} a_1^2 - \frac{1}{2} a_2^2 \right) a_2 \exp\left( \frac{i a_2 - a_1}{a_1 a_2} \right) - a_1^2 a_2 + a_2^3 + a_1^2 + a_2^2 \right], \qquad (E.3a)$$

$$I_{\tilde{f}\tilde{\gamma},\eta} = \frac{-3a_1^2 a_2^2}{(a_1^2 + a_2^2)^3} \left\{ \left[ \left( -\frac{a_2}{3} - \frac{1}{6} \right) a_1^4 + i \left( a_2 + \frac{1}{3} \right) a_2 a_1^3 + a_1^2 a_2^3 - \frac{i}{3} (a_2 - 1) a_2^3 a_1 + \frac{1}{6} a_2^4 \right] \exp\left( -\frac{i a_2 - a_1}{a_1 a_2} \right) + \left[ \left( \frac{a_2}{3} + \frac{1}{6} \right) a_1^4 + i \left( a_2 + \frac{1}{3} \right) a_2 a_1^3 - a_1^2 a_2^3 - \frac{i}{3} (a_2 - 1) a_2^3 a_1 - \frac{1}{6} a_2^4 \right] \exp\left( \frac{i a_2 - a_1}{a_1 a_2} \right) - 2i a_1 \left[ (a_2 - \frac{1}{3}) a_1^2 - \frac{1}{3} a_2^2 (a_2 + 1) \right] a_2 \right\}, \qquad (E.3b)$$

$$I_{\tilde{\gamma},\eta} = \frac{4a_1^2 a_2}{(a_1^2 + a_2^2)^4} \left\{ \left[ \left( -\frac{i}{4} a_1 - \frac{1}{4} a_1^2 + \frac{1}{16} \right) a_2^6 - \left( i a_1^2 - \frac{i}{8} - \frac{3}{4} a_1 \right) a_1 a_2^5 + \frac{1}{2} \left( i a_1 + 3 a_1^2 + \frac{1}{8} \right) a_1^2 a_2^4 + a_1^3 \left( i a_1^2 + \frac{i}{4} + \frac{a_1}{2} \right) a_2^3 + \left( \frac{3}{4} i a_1^5 - \frac{1}{4} a_1^6 - \frac{1}{16} a_1^4 \right) a_2^2 + \frac{1}{8} \left( i - 2 a_1 \right) a_1^5 a_2 - \frac{1}{16} a_1^6 \right] a_2 \exp\left( -\frac{i a_2 - a_1}{a_1 a_2} \right) - a_2 \left[ \left( \frac{1}{4} a_1^2 - \frac{i}{4} a_1 - \frac{1}{16} \right) a_2^6 - a_1 \left( i a_1^2 - \frac{i}{8} + \frac{3}{4} a_1 \right) a_2^5 + \frac{1}{2} \left( i a_1 - 3 a_1^2 - \frac{1}{8} \right) a_1^2 a_2^4 + a_1^3 \left( -\frac{a_1}{2} + i a_1^2 + \frac{i}{4} \right) a_2^3 + \left( \frac{1}{16} a_1^4 + \frac{1}{4} a_1^6 + \frac{3}{4} i a_1^5 \right) a_2^2 + \frac{1}{8} (2 a_1 + i) a_1^5 a_2 + \frac{1}{16} a_1^6 \right] \exp\left( \frac{i a_2 - a_1}{a_1 a_2} \right) + \left( \frac{1}{8} + \frac{1}{2} a_1^2 \right) a_2^7 + \frac{1}{24} a_2^6 + \left( -3 a_1^4 + \frac{1}{8} a_1^2 \right) a_2^5 + \frac{1}{8} a_1^2 a_2^4 + \left( \frac{1}{2} a_1^6 - \frac{1}{8} a_1^4 \right) a_2^3 + \frac{1}{8} a_1^4 a_2^2 - \frac{1}{8} a_2 a_1^6 + \frac{1}{24} a_1^6 \right\}, \qquad (E.3c)$$

$$I_{\phi} = \frac{2a_3}{(a_3^2 + 1)^2} \left( a_3^2 \pi - a_3 + \pi + a_3 \exp\left( -\frac{2\pi}{a_3} \right) \right). \qquad (E.3d)$$
## Acknowledgements

This work was performed within the framework of a joint research program co-funded by EDF and CEA (France). A valuable contribution by P. Moussou and A. Joly (EDF) to the support of this work is gladly acknowledged. The authors also wish to thank D. Panunzio (CEA) for his insightful comments on the development of a structural beam model for the flexible blades.

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