This study investigates, both theoretically and numerically, the hydrodynamic interaction between two parallel circular cylinders undergoing small-amplitude forced harmonic oscillations in an initially quiescent viscous two-dimensional fluid. The theoretical framework builds on a Helmholtz decomposition of the fluid velocity field combined with a bipolar coordinate system. This decomposition leads to a coupled system comprising a Laplace equation and a Helmholtz equation with a variable Lam & eacute; coefficient. While similar in spirit to our previous work on the same configuration, the present approach fundamentally differs in the resolution of the Helmholtz equation. In the earlier study, the Helmholtz equation was replaced by an ad hoc version with a constant Lam & eacute; coefficient, introducing a residual in the linearized Navier-Stokes equations and providing only an approximate description of the fluid forces. In contrast, the present formulation solves both the Laplace and Helmholtz equations in their full form, thereby preserving the complete structure of the viscous flow problem. This refined model now makes it possible to analyze both axial and transverse oscillations of the cylinders, whereas the previous study was limited to axial motion (i.e., motion aligned with the centers of the cylinders). The fluid forces remain expressible as linear combinations of the cylinder velocities and accelerations, with the coefficients of these combinations corresponding to viscous self-and cross-added mass and damping terms. The variations of these coefficients with the dimensionless separation distance, the Stokes number, and the oscillation direction are explicitly quantified through parametric studies. The accuracy of the improved theory is assessed by direct comparisons with numerical simulations performed using the Arbitrary Lagrangian-Eulerian method implemented in our open-source TrioCFD software, as well as with reference results from the literature. The new theoretical approach yields more accurate estimates of the fluid-added coefficients over a wide range of Stokes numbers, although some deviations between theory and numerics persist for very small Stokes numbers. To foster reproducibility and further investigations, a Python implementation of the present theory is made available for computing the fluid-added coefficients.

This paper investigates turbulence-induced vibrations in a confined annular flow configuration representative of pressurized water reactor internals. The experimental setup consists of a rigid inner cylinder mounted on two flexible rectangular plates inside a cylindrical vessel. Building on our previous work, analytical expressions for the fluid-elastic forces in quiescent fluid are derived, accounting for added mass, damping, partial immersion, viscous, and confinement effects. Theoretical predictions of the natural frequencies for various filling heights are obtained and systematically validated against experimental measurements. A new nondimensional formulation of the governing equations further yields an analytical expression for the root mean square displacement under turbulent forcing. The root mean square prediction is based on an effective homogeneous turbulence model characterized by a prescribed pressure power spectral density and correlation lengths drawn from the literature. Although dedicated measurements indicate that the turbulence in the present configuration is not strictly homogeneous, introducing a calibrated cut-off reduced frequency in the pressure spectrum yields a consistent and quantitatively accurate description of the root mean square response as a function of reduced velocity.

Pressure vessels submitted to turbulent flows are prone to fluid-structure interactions and vibrations. The design of a nuclear power plant comes along with the prediction of the large scale vibration pattern generated by turbulent flows exerted upon large areas of the core barrel containing the fuel assemblies.The present paper focuses on turbulent forcing in annular gaps with impinging inlets, in view of assessing the relevance of traditional models of reactor vessel studies and of improving future calculations. An analytical reference case is designed to test the pressure field homogeneity hypothesis of the literature models. Pressure fluctuations associated to the turbulent flow are measured in an experimental mock-up and calculated in CFD simulations, at a gap Reynolds number of 105. The global flow pattern in the annular gap is first provided. The Power Spectrum Density of the pressure field and its convection and coherence parameters are obtained both experimentally and numerically. A fair agreement is found between the measurements and the simulations, and the flow pattern appears inhomogenous in large proportions, contrary to the traditional representation. Furthermore, the first mode of vibration of the inner cylinder is measured under turbulent forcing, and compared to the predictions of the simplified model and of CFD calculations: a fair agreement is observed. Finally, the literature model is revisited in the light of these findings, and some potential improvements are discussed.

This study, motivated by applications in nuclear engineering, examines the fluid-induced vibrations of a flexible inner cylinder concentrically positioned within a rigid outer cylinder, separated by a quiescent Newtonian viscous fluid. Building on our previous work, which focused on forced motions, we extend the theoretical formulation to account for vibrations induced by fluid forces. A new expression for the linear fluid force is derived, introducing a fluid transfer function that depends on key dimensionless numbers such as the aspect ratio, radius ratio, and Stokes number. The vibration frequency and viscous decay rate are predicted by coupling this force with an Euler–Bernoulli beam model and using a modal decomposition based on a complex angular frequency. A computational approach is also developed, integrating an explicit partitioned coupling scheme into the open-source triocfd software. This method enables efficient numerical simulations of the fluid–structure interaction, ensuring internal coupling without excessive data exchange. Comparisons between theoretical predictions and numerical results show excellent agreement for pinned-pinned and clamped-clamped configurations across specific Stokes number and density ratio values. Additionally, the validity of the proposed theory and numerical simulations is further confirmed by comparisons with experimental data from the literature, showing strong consistency for a clamped-free configuration. A parametric study including variations of the Stokes number is eventually also performed, demonstrating the robustness and applicability of the theoretical and numerical approaches.

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.

The in-flow fluidelastic instability of tube bundles prompted renewed interest since the recent unanticipated failure of the replacement steam generators at the San Onofre nuclear power station. A literature review on the topic discloses contrasting views, depending on the tube bundle and flow configuration addressed. In a recent paper, the authors reported experiments using square bundles, subjected to single and two-phase flows. No streamwise instability was observed, for the tested bundle configurations and the flow velocity ranges explored. In the present paper, experimental results obtained at CEA-Saclay for a rotated triangular tube bundle are presented, providing new in-flow fluidelastic data for both single-phase and two-phase transverse flows. The bundle consists of 50 tubes, with reduced pitch P/D = 1.44 and tube diameter D = 30 mm. It was subjected to single-phase (air or water) and two-phase air-water (with homogeneous void fraction in the range 40% to 98%) transverse flows. In the upper region of the bundle, several different flexibility configurations were tested, with up to 15 flexible tubes, mounted using anisotropic supports, which allow for in-flow vibrations. Results presented in the paper include in-flow fluidelastic stability data obtained for both single-phase and two-phase transverse flows. A detailed complex modal identification of the bundle under natural turbulence excitation was performed, at several flow velocities, highlighting the modeshapes which are prone to in-flow instability. Moreover, local void fraction and identified flow regimes are also presented. These results are discussed and compared to those obtained by previous authors, for similar tested configurations.

In this paper, we investigate the fluid-elastic instability of a square tube bundle subject to two-phase cross-flow. A dimensional analysis is carried out, leading to a new criterion of instability. This criterion establishes a direct link with the instability thresholds in single-phase flows. In parallel to the dimensional analysis, experimental work is carried out to i) determine the instability thresholds in single-phase flows (new relation between the Scruton, Stokes and Reynolds number), ii) to test the validity of the two-phase flow instability criterion, derived from the dimensional analysis. The experiments are carried out on a square tube bundle (pitch ratio of 1.5) consisting on 5 rows of 3 tubes (plus two end-rows of half tubes). The central tube is mounted on two flexible blades allowing a vibration in the transverse direction only, whereas all the other tubes are rigid. The instability threshold in single-phase (water) flow is obtained from a method of direct measurement of the fluid-elastic forces, in which the motion of the central tube is imposed. The instability threshold in two-phase (air–water) flow is obtained from a method of indirect measurement, with an active system stability control, of the fluid-elastic forces, in which the central tube vibrates freely. Three sets of blades with different stiffnesses are tested to investigate the stability of the central tube. The criterion of instability is in very good agreement with air–water experiments, predictive for all homogeneous void fractions, and so for all flow regimes (identified in our experiments with an optical probe). This new criterion is of theoretical interest for the understanding of complex two-phase flow excitations, as well as of practical significance for the predictive analysis of industrial components.

This article addresses the small-amplitude forced beam vibrations of two coaxial finite-length cylinders separated by a viscous Newtonian fluid. A new theoretical approach based on an Helmholtz expansion of the fluid velocity vector is carried out, leading to a full analytical expression of the fluid forces and subsequently of the modal added mass and damping coefficients. Our theory shows that the fluid forces are linear combinations of the Fourier harmonics of the vibration modes. The coefficients of the linear combinations are shown to depend on the aspect ratio of the cylinders, on the separation distance, and on the Stokes number. As a consequence, the linear fluid forces do not have, in general, the same shape as the forced vibration mode, so that the fluid makes it possible to couple vibration modes with different wave numbers. Compared to the previous works, the present theory includes the viscous effects of the fluid, accounts for the finite length of the cylinders, does not rely on the assumption of a narrow annulus, and covers in a unique formulation all types of classical boundary conditions for an Euler-Bernoulli beam. The theoretical predictions for the modal added mass and damping coefficients (self and cross) are corroborated numerically, considering rigid, pinned-pinned, and clamped-free vibrations.

This work deals with the hydrodynamic interaction of two parallel circular cylinders, with identical radii, immersed in a viscous fluid initially at rest. One cylinder is stationary while the other one is imposed a harmonic motion with a moderate amplitude of vibration. The direction of motion is parallel to the line joining the centers of the two cylinders. The two dimensional fluid–structure problem is numerically solved by the Arbitrary Lagrangian–Eulerian method implemented in the open-source CFD code TrioCFD. First, we show that the fluid forces on the two cylinders are aligned with the direction of the imposed motion. Second, we show that the moderate oscillations of the moving cylinder create nonlinear effects in the fluid that strongly affect the characteristics (Fourier harmonics) of the hydrodynamic force acting on the stationary cylinder. The fluid force on the moving cylinder is shown to be poorly affected by the nonlinear effects, which makes it possible to extend the linear concept of self-added mass and damping coefficients. First, we show that the self-added coefficients decrease as Sk−1/2, with Sk the Stokes number (dimensionless number constructed from the imposed vibration frequency). Second, we show that the self-added mass (resp. damping) decreases (resp. increases) as −KC3 (resp. +KC3), with KC the Keulegan–Carpenter number (ratio between the imposed amplitude vibration and the separation distance between the cylinders). These variations are included in new power laws derived from nonlinear regressions of the numerical results. These new power laws for the self-added coefficients combine the effect of both Sk and KC, covering the viscous (Sk≥500) and weakly nonlinear (KC≤0.3) regimes.

In this paper, we consider the small-amplitude forced beam vibrations of two coaxial finite-length cylinders separated by an inviscid Newtonian fluid. The three-dimensional fluid problem is solved by introducing a potential function from which the pressure field in the gap is derived. It yields a full analytical expression of the self and cross-fluid forces, which are shown to depend on the Fourier components of the forced vibrations, the aspect ratio, and the radius ratio of the cylinders. Unlike previous theories, the present formulation does not rely on the slender-body approximation nor on the assumption of a narrow gap. Also, the theory is valid whatever the profile of the forced beam vibrations. The theoretical predictions are successfully compared to our numerical simulations, considering clamped-sliding, free-pinned vibration modes and various geometrical configurations (narrow, medium, and wide fluid gaps).

This paper considers the fluid–structure interaction problem of two coaxial cylinders separated by a thin layer of fluid. The flexible inner cylinder is imposed a small amplitude harmonic displacement corresponding to a dry vibration mode of an Euler–Bernoulli beam, while the external cylinder is rigid. A new theoretical formulation based on the assumption of a narrow fluid annulus is derived to estimate the modal added-mass matrix of the vibrating cylinder. This formulation accounts for the finite length of the flexible cylinder, clearly highlights the effect of the aspect ratio of the vibrating cylinder on the structure of the added-mass matrix, and covers all types of classical boundary conditions in the same theory and can easily be implemented in any numerical computing environment. The diagonal coefficients of the added-mass matrix are shown to increase with the confinement, with the aspect ratio of the flexible cylinder, and are sensitive to the wave-number of the vibration mode. Also importantly, we show that the dry vibration modes generate off-diagonal coefficients that vanish for an infinitely long cylinder. Our theoretical observations are corroborated by an extensive set of CFD numerical simulations, covering all types of classical boundary conditions, different confinement configurations, and different aspect ratios of the vibrating cylinder. The results obtained are presented in graphical form, which can be directly applied in engineering applications.

In this paper, we assess the capabilities of the Arbitrary Lagrangian-Eulerian (ALE) method imple-mented in the open-source code TrioCFD to tackle down two fluid-structure interaction problems involv-ing moving boundaries. To test the code, we first consider the bi-dimensional case of two coaxial cylinders moving in a viscous fluid. We show that the two fluid forces acting on the cylinders are in phase opposition, with amplitude and phase that only depend on the Stokes number, the dimensionless separation distance and the Keulegan-Carpenter number. Throughout a detailed parametric study, we show that the self (resp. cross) added mass and damping coefficients decrease (resp. increase) with the Stokes number and the sepa-ration distance. Our numerical results are in perfect agreement with the theoretical predictions of the litera-ture, thereby validating the robustness of the ALE method implemented in TrioCFD. Then, we challenge the code by considering the case of a vibrating cylinder located in the central position of a square tube bundle. In parallel to the numerical investigations, we also present a new experimental setup for the measurement of the added coefficient, using the direct method introduced by Tanaka. The numerical predictions for the self-added coefficients are shown to be in very good agreement with a theoretical estimation used as a refer-ence by engineers. A good agreement with the experimental results is also obtained for moderate and large Stokes numbers, whereas an important deviation due to parasitic frequencies in the experimental setup ap-pears for low Stokes number. Still, this study clearly confirms that the ALE method implemented in TrioCFD is particularly efficient in solving fluid-structure interaction problems. As an open-source code, and given its ease of use and its flexibility, we believe that TrioCFD is thus perfectly adapted to engineers who need simple numerical tools to tackle down complex industrial problems.

This article addresses the interaction of two coaxial cylinders separated by a thin fluid layer. The cylinders are flexible, have a finite length, and are subject to a vibration mode of an Euler-Bernoulli beam. Assuming a narrow channel, an inviscid and linear theoretical approach is carried out, leading to a new simple and tractable analytical expression of the fluid forces. We show that the dimensionless form of this matrix reduces to a single coefficient whose properties (sign and variations) strongly depend on the boundary conditions, the wave number of the vibration modes, and the aspect ratio of the cylinders. All these properties are made explicit in our formulation, which applies to all classical types of boundary conditions. A numerical approach based on an arbitrary Lagrange-Eulerian method is also presented and successfully compared to the theoretical predictions.

In this paper, we assess the capabilities of the Arbitrary Lagrangian-Eulerian method implemented in the open-source code TrioCFD to tackle down two fluid-structure interaction problems involving moving boundaries.

This paper deals with the small oscillations of two circular cylinders immersed in a viscous stagnant fluid. A new theoretical approach based on an Helmholtz expansion and a bipolar coordinate system is presented to estimate the fluid forces acting on the two bodies. We show that these forces are linear combinations of the cylinder accelerations and velocities, through viscous fluid added coefficients. To assess the validity of this theory, we consider the case of two equal size cylinders, one of them being stationary while the other one is forced sinusoidally. The self-added mass and damping coefficients are shown to decrease with both the Stokes number and the separation distance. The cross-added mass and damping coefficients tend to increase with the Stokes number and the separation distance. Compared to the inviscid results, the effect of viscosity is to add a correction term which scales as Sk(-1/2) . When the separation distance is sufficiently large, the two cylinders behave as if they were independent and the Stokes predictions for an isolated cylinder are recovered. Compared to previous works, the present theory offers a simple and flexible alternative for an easy determination of the fluid forces and related added coefficients. To our knowledge, this is also the first time that a numerical approach based on a penalization method is presented in the context of fluid-structure interactions for relatively small Stokes numbers, and successfully compared to theoretical predictions. y(C) 2019 Elsevier Ltd. All rights reserved.

The importance of fluid-elastic forces in tube bundle vibrations can hardly be over-emphasized, in view of their damaging potential. In the last decades, advanced models for representing fluid-elastic coupling have therefore been developed by the community of the domain. Those models are nowadays embedded in the methodologies that are used on a regular basis by both steam generators providers and operators, in order to prevent the risk of a tube failure with adequate safety margins. From an R&D point of view however, the need still remains for more advanced models of fluid-elastic coupling, in order to fully decipher the physics underlying the observed phenomena. As a consequence, new experimental flow-coupling coefficients are also required to specifically feed and validate those more sophisticated models. Recent experiments performed at CEA-Saclay suggest that the fluid stiffness and damping coefficients depend on further dimensionless parameters beyond the reduced velocity. In this work, the problem of data reduction is first revisited, in the light of dimensional analysis. For single-phase flows, it is underlined that the flow-coupling coefficients depend at least on two dimensionless parameters, namely the Reynolds number Re and the Stokes number Sk. Therefore, reducing the experimental data in terms of the compound dimensionless quantity Vr = Re/Sk necessarily leads to impoverish results, hence the data dispersion. In a second step, experimental data are presented using the dimensionless numbers Re and Sk. We report experiments, for a 3 × 5 square tube bundle subjected to water transverse flow. The bundle is rigid, except for the central tube which is mounted on a flexible suspension allowing for translation motions in the lift direction. The evolutions of the flow-coupling coefficients with the flow velocity are determined using two different experimental procedures: (1) In the direct method, an harmonic motion of increasing frequency is imposed to the tube. (2) In the indirect method, the coefficients are obtained from the modal response of the tube (frequency, damping). The coefficient identification was performed well beyond the system instability boundary, by using active control, allowing an exploration of a significant range of flow velocity. For a given Sk, the results show that: (a) at low Re, the flow-coupling coefficients are close to zero; (b) at intermediate Re, the flow stabilizes the tube; (c) at high Re, the flow destabilizes the tube, leading to a damping-controlled instability at a critical Re. Reducing the data in terms of Re and Sk clarifies the various experimental “branches”, which are mixed when using Vr. The two identification techniques lead to reasonably compatible fluid-elastic coefficients.

A potential theory is presented for the problem of two moving circular cylinders, with possibly different radii, large motions, immersed in an perfect stagnant fluid. We show that the fluid force is the superposition of an added mass term, related to the time variations of the potential, and a quadratic term related to its spatial variations. We provide new simple and exact analytical expressions for the fluid added mass coefficients, in which the effect of the confinement is made explicit. The self-added mass (resp. cross-added mass) is shown to decrease (resp. increase) with the separation distance and increase (resp. decreases) with the radius ratio. We then consider the case in which one cylinder translates along the line joining the centers with a constant speed. We show that the two cylinders are repelled from each other, with a force that diverges to infinity at impact. We extend our approach to the case in which one cylinder is imposed a sinusoidal vibration. We show that the force on the stationary cylinder and the vibration displacement have opposite (resp. identical) axial (resp. transverse) directions. For large vibration amplitudes, this force is strongly altered by the nonlinear effects induced by the spatial variations of the potential. The force on the vibrating cylinder is in phase with the imposed displacement and is mainly driven by the added mass term. The results of this paper are of particular interest for engineers who need to understand the essential features associated with the vibration of a solid body in a still fluid.

We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal, and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling, or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test bed for exploring the effects of geometrically induced forces on assemblies of topological charges.

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.