Time-domain numerical simulations of a loosely supported tube subjected to frequency-dependent fluid–elastic forces

Journal of Fluids and Structures 81 (2018) 383–398

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Time-domain numerical simulations of a loosely supported tube subjected to frequency-dependent fluid–elastic forces Philippe Piteau a , Laurent Borsoi a , Xavier Delaune a , Jose Antunes b, * a

Den-Service d’Études Mécaniques et Thermiques (SEMT), CEA, Université Paris-Saclay, F-91191, Gif-sur-Yvette, France Centro de Ciências e Tecnologias Nucleares (C2TN), Instituto Superior Técnico (IST), Universidade de Lisboa, 2695-066 Bobadela LRS, Portugal b

article

info

Article history: Received 23 October 2017 Received in revised form 18 April 2018 Accepted 4 May 2018

Keywords: FIV Tube bundle Transverse flow Turbulence Fluid–elastic instability Vibro-impact dynamics

a b s t r a c t Flow-induced vibrations of heat-exchanger tubes are extensively studied in the nuclear industry for safety reasons. Adequate designs, such as anti-vibration bars in PWR steam generators, prevent excessive vibrations provided the tubes are well supported. Nevertheless, degraded situations where the tube/support gaps would widen, must also be considered. In such a case, the tubes become loosely supported and may exhibit vibro-impacting responses due to both turbulence and fluid–elastic coupling forces induced by the crossflow. This paper deals with the predictive analysis of such a nonlinear situation, given the necessity of taking into account both the strong impact nonlinearity due to the gap and the linearized fluid–elastic forces. In time-domain numerical simulations, computation of flow-coupling forces defined in the frequency-domain is a delicate problem. We recently developed an approach based on a hybrid time–frequency method. In the present paper a more straightforward and effective technique, based on the convolution of a flow impulse response pre-computed from the frequency-domain coefficients, is developed. Illustrative results are presented and discussed, in connection with the previous hybrid method and with experiments. All results agree in a satisfactory manner, validating both computational methods, however the convolutional technique is faster than the hybrid method by two orders of magnitude. Finally, to highlight the subtle self-regulating frequency effect on the stabilization of such system, additional demonstrative computations are presented. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Flow-induced vibrations of heat-exchanger tubes are a major concern in the nuclear industry for safety reasons and repair costs, and are thus extensively studied. Adequate designs, such as anti-vibration bars in PWR steam generators, generally prevent any problem. Nevertheless some degraded situations, although not very likely, must also be considered, as for example ill positioned or worn supports, which would widen the tube/support gaps. In such a case the tube becomes loosely supported and may exhibit vibro-impacting responses due to both turbulence and fluid–elastic coupling forces induced by the cross-flow. Predictive analysis is an essential step in component design, as well as for diagnosing anomalous behaviour. Significant efforts have been directed into the development of time-domain computational methods and tools to deal with the nonlinear dynamics of the flow-excited tube bundles — see, for instance, Axisa and Antunes (1988), Eisinger et al. (1995),

*

Corresponding author. E-mail address: [email protected] (J. Antunes).

https://doi.org/10.1016/j.jfluidstructs.2018.05.003 0889-9746/© 2018 Elsevier Ltd. All rights reserved.

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Nomenclature cd (fr ), cd (Vr ) Dimensionless fluid–elastic damping coefficient ck (fr ), ck (Vr ) Dimensionless fluid–elastic stiffness coefficient Cd (fr ), Cd (Vr ) fluid–elastic damping coefficient Ck (fr ), Ck (Vr ) fluid–elastic stiffness coefficient CD Dimensionless drag coefficient CL Dimensionless lift coefficient Flow damping matrix Cf (fr ) Cs Structural damping matrix D Tube diameter Dref Reference tube diameter EFCplg (t) Energy from the fluid–elastic coupling EFImpact (t) Energy connected with the impacts at the clearance support EFTurb (t) Energy from the turbulence excitation EVibr (t) Vibration energy EζTube (t) Energy dissipated through modal damping f Frequency fn Frequency of mode n fr = fD/V Reduced frequency fR Rice frequency Fc (t) Contact force x FFE (t), FFE (t) Lift fluid–elastic force y

FFE (t)

Drag fluid–elastic force

FnFE (t)

Modal force due to the fluid–elastic coupling for mode n

FnTurb (t)

Modal force due to the turbulence excitation for mode n fFE (x, x˙ ) Motion-dependent fluid–elastic forces fNL (x, x˙ ) Motion-dependent contact forces at the clearance supports fTurb (t) Turbulence force vector ˙ (t) Dimensionless time-derivative of the flow transient function g(t) = Φ G(s) Laplace transform of g(t) hFE (t) fluid–elastic impulse response HFE (s) fluid–elastic transfer function ˆ FE (s) H Experimentally identified fluid–elastic transfer function Heaviside step function H(t) ∫t ∫∞ I(t i ) Normalized integral I(t i ) = 0 i g(t)dt / 0 g(t)dt kc Contact stiffness of the clearance support Kf (fr ) Flow stiffness matrix Ks Structural stiffness matrix L Tube length Lref Reference tube length Ms Structural mass matrix Mf Fluid added mass matrix P Square bundle pitch ⟨ ⟩ PCplg Time-averaged power from the fluid–elastic coupling

⟨P ⟩ ⟨ Turb ⟩

Time-averaged power from the turbulence excitation

Pζ Tube Time-averaged power dissipated through modal damping ˙ , q¨ (t) Modal displacement, velocity and acceleration vectors q(t), q(t) s Laplace variable s = sD/V Reduced Laplace variable eq ˜ Sff (fr ) Turbulence excitation envelope spectrum eq ˜ Sref (fr )

Turbulence reference envelope spectrum t Time t = tV /D Dimensionless time ti Time-window of flow ‘‘memory’’ effects t i = ti V /D Dimensionless time-window of flow ‘‘memory’’ effects V Pitch (inter-tube) flow velocity

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385

V∞ Upstream flow velocity Vr = V /fD = 1/fr Reduced velocity x Coordinate along lift direction x = x/D Reduced lift coordinate X (t), X˙ (t), X¨ (t) Lift displacement, velocity and acceleration X (t) = X (t)/D Reduced lift displacement x(t), x˙ (t), x¨ (t) Displacement, velocity and acceleration vectors y Coordinate along drag direction Y (t), Y˙ (t), Y¨ (t) Drag displacement, velocity and acceleration Y (t) = Y (t)/D Reduced drag displacement z Coordinate along the tube axis zc Coordinate of the clearance support αi , δi Dimensionless coefficients in Φ (t) and g(t) δ (t) Dirac delta function δc Gap size of the clearance support ω Circular frequency ω = ωD/V = 2π fr Reduced circular frequency ωn Circular frequency of mode n ΦX (f ) Spectrum of tube response X (t) Dimensionless flow transient function Φ (t) φn Modeshape of mode n Φ = [φ1 φ2 · · · φN ] Modal matrix ρf Fluid mass density τ Time delay τ = τ V /D Dimensionless time delay ζn Reduced damping of mode n Mohany et al. (2012), Hassan and Mohany (2012), Sawadogo and Mureithi (2013) and Piteau et al. (2015). Modelling fluid–elastic forces in the time domain is complicated by the requirement of taking into account both the strong impact nonlinearity due to the gap and the linearized fluid–elastic forces, typically defined in the frequency domain. The paper deals with such predictive analysis, following our previous work in which the aforementioned difficulties were tackled using a time–frequency hybrid computational method (Piteau et al., 2015). Analysis in the frequency domain is often used in elasto-dynamics and linear structural dynamics. The question of extending such methods in the nonlinear case has naturally arisen in the past, given the practical interest for treating nonlinear structures, an essential part of which is linear but defined in the frequency domain through a dynamic stiffness (impedance) or flexibility (receptance). Amongst others, some classical examples are soil–structure interaction (SSI) in seismic engineering, structure isolation with viscoelastic devices, or fluid–structure interaction (FSI) — see Kawamoto (1983), Hillmer and Schmid (1988), Wolf (1988), Darbe and Wolf (1988) and Politopoulos et al. (2014). The impedance or receptance functions may be derived from theory, calculation or experiment, but in any case represent a sub-structure through a transfer function that produces an output (‘‘reaction’’) from an input (‘‘action’’), under assumed linear conditions. Semi-infinite spatial domains are particularly well reduced and modelled by such sub-structuring in the frequency domain. However, on the other hand, dynamic analysis of structures displaying material or geometric nonlinear behaviour usually require time-history computations, when equivalent linearization is impossible or inadequate. Our previous efforts focused on a time–frequency hybrid approach, based on iterative recursion going back and forth from the frequency-domain to the time-domain, until convergence of the nonlinear numerical solution (Piteau et al., 2015). As easily understood, the stronger the nonlinearity is, the harder it is to achieve convergence. Therefore, even if the results obtained by Piteau et al. (2015) are entirely adequate, we address in the present paper a more straightforward and effective technique for computing the time-domain fluid–elastic force. The approach developed here, which is possibly the most intuitive and physically plausible, is based on the convolution of a flow impulse response pre-computed from the frequency-domain fluid–elastic coefficients. In the first part, the paper briefly recalls the experimental campaign carried out at CEA-Saclay for this purpose, using a rigid square bundle surrounding a flexible cantilever tube which vibrates in the lift direction under water cross-flow. In the second part, the paper presents the convolution method used to generate the time-domain fluid–elastic forces originally defined in the frequency domain in order to calculate the dynamic response of the nonlinear system. Focus is on the computational efficiency of the present approach, when compared with the time–frequency hybrid technique, as well as on the similarity of the results obtained through both methods. Also, we address the issue of non-causality of the timedomain fluid–elastic forces. As discussed by Piteau et al. (2015), this important aspect is related to the need for frequencyextrapolation of band-limited experimental data. Because an entirely satisfying manner of overcoming this problem is not yet available, a suitable workaround was devised by fitting the band-limited experimental fluid–elastic data to a well-known theoretical model.

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Fig. 1. Experimental rig, showing the test section, tube bundle and the instrumented clearance supports.

In the last part, the paper compares experimental and computational results, for a large range of flow velocities and three values of the support gap. As in our previous work, special attention is devoted to energy balances of the computed responses. A satisfying overall agreement is obtained between the results stemming from the two computational approaches, as well as between these and the nonlinear experiments, thus validating both proposed methods for generating the fluid–elastic time-domain forces. However, because it does not entail convergence requirements, the convolutional technique proved faster by two orders of magnitude. 2. Experiments The experimental setup is shown in Fig. 1. It consists of a rigid square bundle (3 × 5) surrounding a flexible tube vibrating along the lift direction (D = 30 mm, L = 0.3 m, reduced pitch P /D = 1.5). The experiments were carried out with water flow, first without the loose support in order to identify the linearized fluid–elastic coupling coefficients as well as the turbulence excitation. This was achieved for a significant range of reduced velocity, using an active control method — see Caillaud et al. (2000, 2003). The fluid–elastic coefficients thus identified were presented and discussed by Piteau et al. (2012). Then experiments were performed after installing an instrumented loose support approximately located at mid-span, leading to nonlinear vibro-impact motions. For each test configuration, both the tube response and the impact forces were measured, as described by Piteau et al. (2015). The tested flexible tube was, for the nonlinear tests, instrumented with a displacement optical probe pointed to the tube free extremity, and two force transducers at both stops of the clearance support. 3. Computational method With the usual assumption that the added-mass fluid term is independent of the flow conditions and can be encapsulated with the structural mass, the dynamic equations of the clearance-supported tube are written as: Ms + Mf x¨ (t) + Cs x˙ (t) + Ks x(t) = fTurb (t) − fFE (x(τ ≤ t), x˙ (τ ≤ t)) − fNL (x(t), x˙ (t))

(

)

(1)

where the right hand-side terms stand respectively for the turbulence force field, the flow-coupling fluid–elastic forces (see hereafter) and the nonlinear contact forces at the clearance-supports. The writing in (1), a bit unusual, stresses the fluid–elastic forces are ‘‘hereditary’’ in the sense they depend on the past time-history response (τ ≤ t) of the tube, as they are frequency-dependent through the fluid–elastic coupling coefficients which vary according to the reduced frequency fr = fD/V . For a harmonic motion of given frequency f , we would have: fFE (t) = Kf (fr ) x(t) + Cf (fr ) x˙ (t)

(2)

In the general case, the fluid–elastic forces can be written in terms of convolution products: fFE (t) = Kf ∗ x (t ) + Cf ∗ x˙ (t )

(

)

(

)

(3)

where, for each component in (3):

(

)

∫

t

Kf (t − τ )x(τ )dτ

Kf ∗ x (t) = −∞

;

(

) Cf ∗ x˙ (t) =

∫

t

Cf (t − τ )x˙ (τ )dτ

(4)

−∞

As discussed in the following, the ‘‘memory’’ effects connected with the flow-coupling coefficients have a restricted timewindow ti , therefore the convolutions (4) are in practice computed using t − ti as lower integration boundary, instead of −∞.

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387

4. Modelling the fluid–elastic force and the issue of causality It might seem trivial to speak about causality, but the artificial non-causality of fluid–elastic forces is often masked, in particular because the fluid–elastic coefficients of the coupling matrices Kf (fr ) and Cf (fr ) are usually experimentally identified within limited frequency ranges. Actually, when converting from the frequency to the time-domain, the converted quantities must be known on the whole FFT frequency range, which implies the need for some form of interpolation and/or extrapolation, given the limited frequency range of the experiments which provide the fluid–elastic coefficients. In general, careless procedures will artificially introduce non-causality of the corresponding time-domain force terms. Such is the case, for instance, when the fluid–elastic coefficients are simply assumed nil within the frequency range(s) where they are unknown. A physical system is always causal because the effect cannot anticipate the cause. In order to obtain a physically plausible frequency interpolation/extrapolation of the coupling coefficients and achieve causality, the approach adopted in this work is to fit our frequency-domain experimental fluid–elastic data to a suitable theoretical model, and then use such fitted model in the nonlinear time-domain computations to produce the fluid–elastic forces. Contextual information on fluid–elastic models is provided by Price (1995) and Weaver (2008), in their thorough reviews. For the bundle configuration addressed in this paper, the theoretical model proposed by Granger and Païdoussis (1996), inspired by unsteady aerodynamic theory and later revisited by Meskell (2009) using a vortex wake model, proved to be a suitable framework for the extrapolation of our experimental data. In their ‘‘quasi-unsteady’’ formulation, causal by construction, for the case of a single flexible tube orbiting within a rigid square bundle subjected to transverse flow, without the added mass term already included in (1), the fluid–elastic lift (x) and drag (y) forces are written as:

{

x FFE (t) y FFE (t)

} =

1 2

ρf V LD 2

[(

{ })

X Kf ∗ Y

(

{ }) ] ˙ X (t) + Cf ∗ ˙ (t) Y

(5)

˙

˙

with the dimensionless time t = t V /D and parameters X (t) = X (t)/D, Y (t) = Y (t)/D, X (t) = X˙ (t)/V and Y (t) = Y˙ (t)/V , where X (t) and Y (t) stand respectively for the tube displacements along the lift and drag directions, while X˙ (t) and Y˙ (t) are the corresponding tube velocities. Then, as shown by Granger and Païdoussis (1996), for the case of a single flexible tube vibrating solely along the lift direction, Eq. (5) simplifies substantially and only the stiffness-related term is convolved. The fluid–elastic force reads: FFE (t) =

1 2

ρf V 2 LD

[

) ∂ CL ( ˙ g ∗ X (t) − CD X (t) ∂x

] (6)

with the dimensionless lift coordinate x = x/D, while ∂ CL /∂ x and CD are dimensionless experimental coefficients for the square bundle tested. On the other hand, g(t) is the derivative (in the sense of distributions) of a transient function:

( Φ (t) =

1−

N ∑

) αi e

−δi t

(7)

H(t)

i=1

where H(t) is the Heaviside step function, while the dimensionless coefficients αi and δi , which are discussed by Granger and Païdoussis (1996), may be fitted from experimental data. Then: g(t) =

dΦ (t) dt

=

N ∑

αi δi e−δi t H(t) + Φ (0)δ (t)

(8)

i=1

where δ (t) is the Dirac delta-function. Notice that the relaxation term g(t) in (6) acts as a memory function which, as discussed by Granger and Païdoussis (1996), physically encapsulates the reorganization of the flow, following the diffusion– convection of the vorticity layer created on the surface of the moving body. As should be expected, this physically implies Φ (t → ∞) = 1. In the limit case of assuming an instantaneous flow recovering, then Φ (t) = H(t) and g(t) = δ (t) in (6), hence a quasi-steady solution is obtained, meaning that the flow would reestablish immediately following the tube motion perturbations. Under such extreme simplification no fluid–elastic instability can arise, but the intermediate simplified option of assuming full flow recovery after a delay τ f leads to Φ (t) = H(t − τ f ) and g(t) = δ (t − τ f ), which revert the improved fluid–elastic model to the basic one by Price and Païdoussis (1986). Once developed, (6) leads to a standard convolution of the tube response with the modelled ‘‘fluid–elastic impulse response’’ hFE (t): FFE (t) = (hFE ∗ X ) (t)

(9)

where: hFE (t) =

1 2

ρf V 2 L

[

∂ CL g(t) − CD δ˙ (t) ∂x

] (10)

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P. Piteau et al. / Journal of Fluids and Structures 81 (2018) 383–398

Fig. 2. Modal frequency and damping, fitted by the model of Granger and Païdoussis (1996).

which enables calculating the corresponding ‘‘fluid–elastic transfer function’’ HFE (ω) by Laplace transforming (8) to (10) with respect to the reduced variable s = s D/V : HFE (s) =

1 2

ρf V 2 L

[

∂ CL G(s) − CD s ∂x

] (11)

with:

[

[

]

G(s) = L g(t) = L

dΦ (t) dt

[(

]

+ Φ (0)δ (t) = sL

1−

N ∑

) αi e

−δi t

] H(t)

=1−s

i=1

N ∑ αi δi + s

(12)

i=1

Hence, from (11) and (12) taken along the frequency axis s = iω, with ω = ω D/V = 2π fr , we obtain the following theoretical solution in terms of the reduced frequency:

[

∂ CL HFE (fr ) = ρf V L 2 ∂x 1

2

(

N ∑ 4π 2 fr2 αi 1− 2 δ + 4π 2 fr2 i=1 i

)

( − i2π fr

N ∂ CL ∑ α i δi + CD 2 ∂x δ + 4π 2 fr2 i=1 i

)] (13)

ˆ FE (ω) as usually, in terms of the identified On the other hand, one may write an experimental transfer function H dimensionless fluid–elastic coefficients cd , ck : ˆ FE (fr ) = − 1 ρf V 2 L [ck (fr ) + i2π fr cd (fr )] H 2

(14)

ˆ FE (ω), Eqs. (13) and (14), one obtains: and, by identifying the real and imaginary parts of HFE (ω) with those of H ∂ CL ck (fr ) = − ∂x

(

N ∑ 4π 2 fr2 αi 1− δ 2 + 4π 2 fr2 i=1 i

) ;

cd (fr ) =

N ∂ CL ∑ αi δi + CD 2 ∂x δ + 4π 2 fr2 i=1 i

(15)

The experimental coefficients ck (fr ) and cd (fr ) have been fitted to the theoretical formulae (15), within the identification frequency range, by assuming the coefficients ∂ CL /∂ x = −73/r 2 = −8.11 and CD = 2.3/r 2 = 0.256, with r = V /V∞ = 3, following the values obtained by Price and Païdoussis (1986) for this array geometry, referred to the pitch (inter-tube) flow velocity V used here, instead of the upstream velocity V∞ of their paper. Actually, fitting was performed by optimally adjusting the unknown coefficients αi and δi using a two-term expansion in the theoretical model, in order to minimize the model errors on the predicted modal frequency and damping, with respect to the measured values of f1 and ζ1 , for each flow velocity, as shown in Fig. 2. The measured fluid–elastic coefficients, their fitting and extrapolation by Eqs. (15), which have been used in this study, are plotted as a function of the reduced velocity Vr = 1/fr in Fig. 3, as well as the corresponding plots in terms of ck (Vr )Vr2 and cd (Vr )Vr . To understand the relevance of such representations, one should bear in mind that the dimensionless coupling

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389

Fig. 3. Experimental flow-coupling coefficients ck (Vr ) and cd (Vr ) compared to the corresponding identified model of Granger and Païdoussis (1996).

coefficients ck (Vr ) and cd (Vr ) are related to the identified (dimensional) force coefficients per unit length, Ck (Vr ) and Cd (Vr ), as: Ck (Vr ) Cd (Vr ) ck (Vr ) = (16) ; cd (Vr ) = (1/2) ρf Vr2 (1/2) ρf DVr It is clear that, as Vr → 0, even small errors in the identified coefficients Ck (Vr ) and Cd (Vr ) lead to large errors in the dimensionless coupling coefficients ck (Vr ) and cd (Vr ). However, because the dimensional coefficients are actually used for computing the fluid–elastic forces, for clarity it makes sense to plot the dimensionless coefficients in terms of ck (Vr )Vr2 and cd (Vr )Vr . These apparent large errors are artificial, their origin is well understood and with no consequences whatsoever when computing the fluid–elastic forces. Also notice the trend changes in the fluid–elastic coefficients beyond Vr = 5, a range for which we have no experimental data available and therefore the nonlinear time-domain computations follow what is predicted by the chosen theoretical model fitted to our experimental data. One should note that, as discussed for instance by Granger (1991), for a fully flexible tube bundle, any simulation approach based on a single tube can only lead to approximate results. Strictly speaking, one cannot simulate the stiffness-coupling mechanism with a single tube, and the dynamics of a suitable cell of flow-coupled tubes must be computed. Therefore, in a more general context (which is not the case of the present paper of demonstrative nature), Eq. (5) could be implemented for several coupled tubes, having previously identified all the flow-coupling coefficients of matrices Kf and Cf . In conclusion, in this work convolutional computations take care of the frequency-dependence of the identified fluid– elastic coupling coefficients, while use of the Granger–Païdoussis theoretical model is aimed at overcoming causality issues. Within the described framework, any bundle/motion configurations can be addressed, provided that computations are performed using the adequate flow-coupling coefficients, identified from the relevant tube bundle and motion direction. 5. Computational details The purpose of the computation is to simulate the nonlinear experiments with the clearance stops, modelling the turbulence excitation and using the fluid–elastic coefficients previously identified from the linear experiments. The turbulence excitation and nonlinear vibro-impact support forces have been modelled as extensively discussed by Piteau et al. (2012) and Antunes et al. (2015). Computations were performed using the following reference parameters: - Four cantilever beam modes (free at the clearance support, as shown in Fig. 4), including the water added mass, the first at f1 ≈ 18 Hz with a measured damping ζ1 ≈ 1%, the viscous damping of higher modes was estimated – see Chen et al. (1976) and Rogers et al. (1984) – at ζ2 = 0.16%, ζ3 = 0.09% and ζ4 = 0.06%; - Stop stiffness kc = 106 N/m (1st frequency of the tube with closed gap at about 200 Hz) and no stop damping; - Three increasing gap sizes δc = ±0.5 mm, δc = ±1.0 mm and δc = ±1.5 mm; - Zero initial conditions of tube displacement and velocity; - Pitch flow velocity V = 1 ∼ 4 m/s. - Computations were performed, as described in the work by Piteau et al. (2015), for 26 s tube response time-histories, using an integration time-step 5 · 10−5 s, from which relevant average quantities were computed.

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Fig. 4. Modal parameters and modeshapes of the modes used in the dynamical computations.

Fig. 5. Turbulence spectrum, as well as the reference envelope spectrum proposed by Axisa et al. (1990).

Even if all parameters were fixed without uncertainty, the tube response is stochastic, given the random excitation and the chaotic physics induced by the impacts, see Borsoi et al. (2012, 2014). To cope with such a process, 10 time-realizations have been produced for each computed configuration. This was found sufficient to make tendencies emerge. As shown in Fig. 5, the turbulence excitation generated by the water cross-flow was found in the experiments to be very close to the logarithmic bi-slope reference envelope spectrum proposed by Axisa et al. (1990) for design:

{ 4 · 10−4 fr−0.5 eq ˜ Sref (fr ) = 3 · 10−6 fr−3.5

(0.01 ≤ fr ≤ 0.2) (0.2 ≤ fr ≤ 3)

(17)

which is referred to Lref = 1 m and Dref = 0.02 m, see Axisa et al. (1990) for details. This reference spectrum has been taken for our computations, or more precisely the equivalent spectrum corrected for the actual length and diameter of the present tube bundle, given by: Lref D eq eq ˜ ˜ Sff (fr ) = Sref (fr ) Dref L

(18)

Time-histories of the modal turbulence forces were generated using the efficient technique developed by Antunes et al. (2015), which properly accounts for the space correlation of the turbulence random field. Concerning the fluid–elastic forces, when (9) to (10) are replaced into (1), the integro-differential equation obtained for the flow-coupled system can be conveniently projected on the modal basis of the unconstrained tube, while physical responses are computed by modal ˙ superposition, x(t) = Φ q(t) and x˙ (t) = Φ q(t), as usual. As illustrated in Fig. 6, for the relevant experimental parameters, a plot of the normalized integral I(t i ) shows a convenient stabilization of the results for t i ≥ 20, from which an inferred integration time-window of ti ≈ 1 s proved more than adequate, as illustrated in Fig. 7. 6. Results Sample experimental results and computations are shown in Figs. 8 and 9, respectively, for pitch velocity V = 2.1 m/s and support gap δc = ±0.5 mm. Further results are shown in Figs. 10 and 11, for the higher velocity V = 3.4 m/s and larger clearance δc = ±1.0 mm, leading to a more energetic vibro-impact response of the tube. Both cases display a credible

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Fig. 6. Convergence of the function: I(t i ) =

∫ ti 0

∫∞

g(t)dt /

0

391

g(t)dt.

Fig. 7. Illustrative plot of a typical convolution with the memory function.

qualitative comparison between the experimental results and the numerical simulations using the convolution technique to model frequency-dependent fluid–elastic forces. Fig. 12 presents a quantitative comparison of the average Rice response frequency:

fR =

1 σX˙ 2π σX

with

√ ⎧ ∫ T ⎪ [ ⟨ ⟩]2 1 ⎪ ⎪ X˙ (t) − X˙ dt σ = ⎪ ⎨ X˙ T 0 √ ∫ T ⎪ ⎪ 1 ⎪ ⎪ ⎩ σX = [X (t) − ⟨X ⟩]2 dt T

0

;

;

⟨ ⟩ 1 X˙ = T

⟨X ⟩ =

1 T

T

∫

X˙ (t)dt 0

(19) T

∫

X (t)dt 0

which is extracted from the experimental and computed results, as a function of the flow velocity, for three values of the support clearance. The corresponding results for the average impact force are displayed in Fig. 13. Concerning the numerical simulations, the results shown were computed using the hybrid time–frequency technique (Piteau et al., 2015) as well as the present convolutional approach to model fluid–elastic forces. One can note that both computational approaches lead to identical results, which are reasonably compatible with the experimental data. Quite important is the fact that the convolution approach is faster by two orders of magnitude than the hybrid time– frequency computation method. This significant gain in computational efficiency is easily understood from Fig. 7, which shows that the ‘‘memory’’ time-scale of the fluid–elastic forces is short, when compared to the tube motion time-history. This

392

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Fig. 8. Experimental nonlinear responses for V = 2.1 m/s and δc = ±0.5 mm.

Fig. 9. Computed nonlinear responses for V = 2.1 m/s and δc = ±0.5 mm, using the convolution technique.

enables the use of a comparatively short convolution time-window, leading to an efficient computation of the fluid–elastic coupling force. Fig. 14 illustrates the self-regulating effect of vibro-impacts on the linearly unstable tube. As the flow velocity increases, the average frequency response fR (V ) of the vibro-impacting tube also increases, so that the actual average reduced velocity V r = V /fR (V )D tends to an asymptotic value. This process limits the magnitude of the actual destabilizing fluid–elastic forces, creating an effective self-regulation mechanism for the vibro-impacting tube. However, for the gap of 1.5 mm and the highest flow velocity, experimental reduced velocity keeps on growing slightly.

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393

Fig. 10. Experimental nonlinear responses for V = 3.4 m/s and δc = ±1.0 mm.

Fig. 11. Computed nonlinear responses for V = 3.4 m/s and δc = ±1.0 mm, using the convolution technique.

Notice that we have chosen to compute the reduced velocity in Fig. 14 with respect to the nonlinear motion-dependent Rice frequency fR (V ), as defined in Eq. (19) and shown in Fig. 12, because such definition of Vr proved to be physically relevant for vibro-impact responses. This aspect will be further developed in the next section, where related computations – using only the first mode of the more complex system, devoid of turbulence excitation – will further explain the self-regulating mechanism of the fluid–elastic forces leading to the asymptotic results of Fig. 14. In particular, such analysis will highlight: (1) the importance of the Rice frequency, as an essential quantifier of the vibro-impacting system responses, and (2) the energy balance requirements underlying the resulting limit cycles. Fig. 15 shows a power balance of the tube dynamics, for the responses computed using the convolution technique. The relevant time-averaged terms of the power balance, obtained from the computed modal excitations and responses, stem

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Fig. 12. Experimental Rice frequency compared to the computed result using both the hybrid method (Piteau et al., 2015) and the convolution technique.

from the turbulence excitation: N ∫ 1∑

⟨PTurb ⟩ =

T

n=1

T

FnTurb (t)q˙ n (t)dt

(20)

0

the fluid–elastic coupling forces:

⟨

⟩

PCplg =

N ∫ 1∑

T

n=1

T

FnFE (t)q˙ n (t)dt

(21)

0

and the modal damping dissipation:

⟨

⟩

Pζ Tube =

N ∫ 1∑

T

n=1

T

2mn ωn ζn [q˙ n (t)]2 dt

(22)

0

Other terms pertaining to the system vibrational and the impact energy are neglected in Fig. 15, as they are much smaller than (20)–(22). Interestingly, for small support gaps, when the turbulence excitation dominates the injected energy, the fluid–elastic forces are dissipative. However for large support gaps, when flow-coupling forces dominate, the energy is supplied by the fluid–elastic forces while the turbulence excitation becomes dissipative. These effects are further explored and discussed in two recent papers by Borsoi et al. (2017a, b). 7. Discussion As pointed before, the results shown in Fig. 14 clearly indicate that the frequency response of the vibro-impacting system increases with the flow velocity in an asymptotic manner. Actually, because of the frequency-dependence of the fluid–elastic coefficients, for each flow velocity the system dynamical response self-adapts in order to balance the energy dissipated by the structure and the one injected by the fluid–elastic coupling. As pointed before, the physics leading to such behaviour can be better illustrated by stripping the problem of all non-essential features and simulating a simple SDOF system – actually the first mode of the more complex system previously computed, which clearly dominates the dynamical responses – subjected only to the fluid–elastic forces, without turbulence excitation. Fig. 16 illustrates the response limit-cycles obtained for the pitch velocity V = 3.4 m/s and support gap δc = ±1.5 mm, for three values of the structural damping coefficient Cs = 0, Cs0 and 2Cs0 , with Cs0 = 0.35 Ns/m. For these three conditions, stable limit cycles are obtained, the first two leading to near-similar impact forces. The third computation leads here to a single unilateral impact per cycle, with higher impact forces — however, for other initial conditions, a symmetrical solution with two impacts per cycle was also obtained.

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395

Fig. 13. Experimental mean impact force compared to the computed results using both the hybrid method (Piteau et al., 2015) and the convolution technique.

Fig. 14. Experimental average reduced velocity compared to the computed results using both the hybrid method (Piteau et al., 2015) and the convolution technique.

The various terms of the corresponding energy plots (beyond the nil turbulence term EFTurb (t) ≡ 0), displayed in Fig. 17, were computed as follows. For the energy connected with the fluid–elastic coupling force: t

∫

F1FE (τ )q˙ 1 (τ )dτ

EFCplg (t) = 0

(23)

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Fig. 15. Energy balance of the tube dynamics obtained using the convolution technique (PTurb = Turbulence excitation, PCplg = Flow coupling, Pζ Tub = Tube damping).

Fig. 16. Response limit cycles of a SDOF vibro-impacting tube, without turbulence excitation and subjected to a fluid–elastic force, for three values of the structural damping coefficient Cs .

the energy dissipated through modal dissipation: EζTube (t) = 2m1 ω1 ζ1

t

∫

[q˙ 1 (τ )]2 dτ

(24)

0

the energy connected with the impacts at the clearance support: EFImpact (t) = φ1 (zc )

t

∫

Fc (τ )q˙ 1 (τ )dτ 0

(25)

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397

Fig. 17. Energy balance of the limit cycles of a SDOF vibro-impacting tube, without turbulence excitation and subjected to a fluid–elastic force, for three values of the structural damping coefficient Cs .

Fig. 18. Change of the frequency-dependent fluid–elastic coupling coefficient Cf (fD/V ) and Rice estimation fR of the vibration response, for three values of the structural damping coefficient Cs .

where φ1 (zc ) is the tube modeshape at the clearance support location, and finally the mechanical energy of the oscillator: 1

1

m1 ω12 [q1 (t)]2 (26) 2 2 These plots show a perfect balance between the injected and dissipated energy per cycle of the motion, for the three computations, while the Rice response frequencies are respectively 29.6 Hz, 28.6 Hz and 27.6 Hz. Now, if one plots the change of the fluid–elastic coefficient Cf (f D/V ) as a function of frequency (for D = 30 mm and V = 3.4 m/s), it becomes clear that in each case the system self-regulates to achieve an average Rice response frequency fR such that Cf (fR D/V ) = Cs . This general conclusion substantiates the trends displayed by the results of the preceding section, highlighting the stabilizing effect of vibro-impact motion regimes after the linear instability of fluid–elastic ally coupled systems (see Fig. 18). EVibr (t) =

m1 [q˙ 1 (t)]2 +

8. Conclusion In this paper we proposed a convolutional technique for dealing with frequency-dependent fluid–elastic forces when performing nonlinear time-domain simulations. Results from this approach are fully compatible with those from the hybrid time–frequency approach previously developed, while achieving a very significant improvement in computational efficiency. The delicate issue of preserving causality of the fluid–elastic forces when performing time-domain simulations, which may be rooted to the limited frequency range of the experimentally identified flow-coupling coefficients, has been addressed. The proposed approach for achieving causality was based on frequency extrapolation by fitting the experimental coefficients to the causal theoretical fluid–elastic model by Granger and Païdoussis (1996). Such model proved quite representative of the experimental data in the frequency range explored by our tests. Overall, both modelling approaches and the experimental results are in reasonable agreement. The numerical results obtained show several interesting features, in particular by disclosing a physical mechanism which limits the magnitude of the actual destabilizing fluid–elastic forces, hence a self-regulation of the vibro-impacting tube. Also, a somewhat puzzling

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behaviour is displayed by the nonlinear system, as either the turbulence or the fluid/elastic forces can supply or dissipate energy, depending on the magnitude of the support gap. Finally, supplementary computations performed on a SDOF vibroimpacting system subjected only to fluid–elastic forces clarify the self-regulating frequency effect leading to the stabilization of the nonlinear tube responses. Acknowledgements The authors acknowledge financial support for this work, which was performed in the framework of a joint research program co-funded by AREVA NP, EDF and CEA (France). A valuable contribution by T. Valin (CEA-Saclay) to the experimental work is gladly acknowledged. References Antunes, J., Piteau, P., Delaune, X., Borsoi, L., 2015. A new method for the generation of representative time-domain turbulence excitation. J. Fluids Struct. 58, 1–19. Axisa, F., Antunes, J. Villard, B., 1988. 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