Proceedings of the XXVI Iberian Latin-American Congress on Computational Methods in Engineering CILAMCE 2005 Brazilian Assoc. for Comp. Mechanics & Latin American Assoc. of Comp. Methods in Engineering Guarapari, Esp´ırito Santo, Brazil, 19th –21st October 2005 Paper CIL05-230

NUMERICAL INERTIA AND DAMPING COEFFICIENTS DETERMINATION OF A TUBE-BUNDLE IN INCOMPRESSIBLE VISCOUS LAMINAR FLUID

Marcus V.G.de Morais Franck Baj Rene-Jean Gibert [email protected] or [email protected] [email protected] DRN/DM2S/SEMT/DYN - CEA Saclay - Bt.607 91191 Gif-sur-Yvette Cedex FRANCE Jean-Paul Magnaud [email protected] DEN/DM2S/SFME/LTMF - CEA Saclay - Bt.469 91191 Gif-sur-Yvette Cedex FRANCE

Abstract. In this paper, we compare the performances of ALE and T RANSPIRATION methods. The approach ALE is a powerful tool to treat coupled problems. We can mention for ALE, more precisely, the approach in finite elements of Donea and Hughes. However, the ALE performance for determining fluid-elastic forces to small vibrations amplitudes is still ignored. The T RANSPIRATION method is a simplified approach for calculation of fluid-elastic forces to relatively small vibrations amplitudes. Based on a first order development of velocity boundary conditions, this method allows the use of a fluid domain fixed in time during a dynamic computation, by avoiding hence the problems due to the mesh distortions. The methods have been validated with help of analytical solution of a vibrating cylinder immersed in stokes confined fluid medium. We made others analysis of dynamics characteristics in tube-bundle comparing with literature. This methods are implemented in C AST 3M a numerical platform of French Nuclear Agency - CEA-Saclay -. Keywords: Fluid structure interaction, tube bundle, ALE and T RANSPIRATION formulation, CFD, mass added and damping coefficient 1.

INTRODUCTION

Many industrial components are made of tube array which vibrate under flow fluid. These vibrations, if they are sufficiently intense, can generate troublesome phenomena on the level of the equipment. In a pressurized water nuclear reactor, the vibrations phenomena of a tube of a steam generator (GV) is a important problem. In fact, GV’s lifespan is directly dependent. Study are still necessary in order to optimize the operation of power plants and to fulfill the restrictive requirements of safety. Tube bundles subjected to flow constitute a fluid-elastic coupling problem which interests particularly the nuclear area. Since 60’s, several experimental studies were undertaken on reduced model (Gibert, 1988). In order to determine the coupling fluid-elastic behavior, there are mainly two experimental approaches used today:

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

⊲ direct method - imposed movement of a tube (Tanaka et al., 2000), and ⊲ indirect method - tube bundle excited by turbulence (Baj, 1998; Caillaud, 1999). We could be supposed to compare the results produced by these two approaches. On the other hand, we can compare these results if the experimental conditions are identical for both, because the fluid-elastic forces depend on the geometry of tube array (P/D and geometric disposition) and the flow characteristics (Reynolds Re, Stokes St, and reduced velocity V r) (Renou, 1998). As Caillaud (1999) shows, differents experimental flow conditions of the methods direct and indirect imply that the results obtained are not easily comparable. Methods Vr Re St direct 2 25 1540 4010 61 2005 indirect 0.5 3.5 8400 53100 16800 15200 Table 1: Experimental ranges of V r, Re and St by direct and indirect methods (Caillaud, 1999). The numerical study find than its place as a complementary means of study. The computational fluid dynamic(CFD) have progressed much these last twenty years, parallel to computers performance. The vibration analysis of cylindric obstacles traversed by monophasic flow, a typical problem of fluid-structure interaction (IFS), was already approached numerically by several auteurs (Huerta and Liu, 1988; Renou, 1998; Varella, 2001). The majority of the treated cases relate to high vibratory movements, for example, those of a fraction of diameter of tube (≥ 15% D). However our problem relates to small movements limited to the hundredth of diameter of a tube. The experiment shows that physics is not the same one, and numerical problems either (Gibert, 1988). In this paper, we compare the performances of ALE and T RANSPIRATION methods. The methods have been validated with help of analytical solution of a vibrating cylinder immersed in incompressible confined fluid medium. We made others analysis of dynamics characteristics in tube-bundle comparing with literature. This methods are implemented in C AST 3M a numerical platform of French Nuclear Agency - CEA Saclay -. 2.

PHYSICAL PROBLEM

Many industrial components comprise tube banks which vibrate under the effect of a flow of fluid. These vibrations, if they are sufficiently intense, can generate phenomena of troublesome fatigue and wear on the level of support’s devices of the tubes. Thus vibrations of the exchange tubes in steam generators (GV) of a pressurized water nuclear reactor must be analyzed carefully. External fluid forces may generate large vibrations amplitudes at tubular structures causing possible dramatic damages in terms of nuclear power plant. This vibrations results from four kinds of fluctuation: (i) random fluctuations generated by turbulence in fluid at large Reynolds numbers; (ii) fluctuations induced by structure-flow motion coupling due to fluid-elastic effects; (iii) resonance with flow periodicity due to vortex shedding; and (iv) possible acoustic excitation. Fluid-elastics forces coupling to tubes, case (ii), can affect the dynamics behavior of GVs, and be responsible of possible fluid-elastic instabilities. For industrial concerns, it

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

is necessary to be able to predict these fluid-elastic forces ant heir effects on tube bundle dynamic stability. After, a stage without flow (fluid-structure coupling) is necessary in order to improve the numerical modelling control of problem. 2.1 Fluid-structure tube bundle model In the first time, we are interesting in the study of vibrations of a flexible tube belonging to a square fixed tube bundle subjected to a fluid coupling without flow. This configuration is defined by known geometry and hydraulics parameters describing the system (Renou, 1998). Geometric parameters characterizing a regular square tube bundle ∂Ω T22

T23

T24

Γs,t T32

T33

p=0

p=0

Ωt

T34

P T42

D

T43

T44

D ey u=0

(a) Schematic of AMOVI model

ex

u=0

(b) Geometry and boundary conditions

Figure 1: Numerical model 2D of tube bundle AMOVI in a fluid at rest. are tube diameters D, tube gap P , and tube length L. From a mechanical point of view, the flexible tube motion can be modelized by: tube mass Ms , tube damping Cs , and tube stiffness Ks . Concerning the structural movement in the fluid at rest, first mode of vibration are affected by added mass Ma and fluid viscosity damping Ca . The equation of motion becomes: (Ms + Ma )¨s + (Cs + Ca )s˙ + Ks s = 0

(1)

where, ωf2s = Ks /(Ms +Ma ) and ξf s = (Cs +Ca )/[2ωf s (Ms +Ma )] are the pulsation and damping coefficients of the tube in fluid at rest. A procedure to determine the frequency ωf s and ratio damping ξf s of a spring-mass system vibrating in fluid environment without or with flow can be a least-square technique presented by Gharib et al. (2000) in FIV-2000. Another technique to identified de dynamic characteristics of a tube in fluid at rest is knows like Phase method. At each time step fluid forces acting on the tube are estimated. According to Equation (1), these forces are expressed as fonction of added mass and viscosity damping, Ff (t) = −Ma ¨s (t) − Ca s˙ (t). For an harmonic tube motion So sin ωt, fluid forces Ff are also periodic: Ff (t) = Fo sin(ωt + ϕ) , where

Fo =

p

(ω 2 So Ma )2 + (ωSo Ca )2 and tan ϕ =

(2) Ca ωMa

(3)

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

The in-phase and opposed phase coefficients are given by the expressions: ∴

Ma 1 Ca /2πSt Fo cos(ϕ) Ca Fo sin(ϕ) = = and =− 2 2 2 ρD ω So ρν ωf s ρD ωSo

(4)

where, 2πSt = ωf s D2 /ν is the Stokes number. This procedure (4) is very simple to implement: it’s indeed enough to obtain the values of maximum force Fo and the phase ϕ between displacement s(t) and force Ff (t). Indeed, the principal disadvantage of this method is that small errors in phase evaluation cause great variations in the determinations of damping Ca when St become elevate, as one will see later. 3.

ALE FORMULATION

First of all a system of rectangular cartesian axes is chooses in which the vectors position ξ, x and X will be expressed. EULERIEN

ALE

Ωξ

LAGRANGIEN

Ωτ

ξ

t=τ

x

ΩX

ψ

X

τ

matériel

α

Id

φ

t=0 Ωξ

ξ référence

Figure 2: Representation of Eulerian, Lagrangian and arbitraire reference domaines. Reference domaine Ωξ , Figure 2, is a field of R3 . An arbitrary movement, described by an application α, deforms this field independently of material particles mouvement. Supposed that αt is an homeomorphic geometrical transformation which at every instant t a point of domaine Ωτ is associated to a point of reference domaine Ωξ . ατ : Ωξ −→ Ωτ ξ 7−→ x = α(ξ, τ )

(5)

We consider that Ωτ is a field of R3 , Ωξ at the moment t = τ , be used as support of grid. The borders movement of this domaine is known and regular. The geometrical transformation ατ is constructed such that it is continuously differentiable bijective mapping (C 1 diffeomorphism class).

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

Now we consider that the ΩXτ domaine, at instant t = τ , contains all material points which were in ΩXξ domaine at instant t = 0. Same manner, we define a diffeomorphism φτ of material domaine ΩXτ on Ωξ domaine. φτ : ΩXτ −→ Ωτ

(6)

X 7−→ x = φ(X, τ )

The relations between the different domaines at t = τ are indicated on Figure 2. Let us pose: ∂φ(X, t) ∂α(ξ, t) . u= and w= (7) ∂t X ∂t ξ

where u is the velocity of material points and w is the velocity of reference points. Consequently: • if the reference volume Vξ is taken fixed by choosing α = Id, then ∀ τ ξ = x and w = 0, i.e. eulerian description;

Ωτ = Ωξ ,

• if the reference volume Vξ follows material volume VXτ in its movement, point by point, by choosing α = φ, then Ωξ = ΩXτ , ξ = X and w = u, i.e. lagrangian description. 3.1 Fluid Domaine The relations obtained in the preceding section make it possible to write the integral equations (on spatial domaine Ωt ) of the mass and momentum conservation equations on ALE description (Gounand, 1997)1 .   δ + (u − w) · ∇ ρ(x, t) = −ρ(x, t) ∇ · u (8) δt   δ ⇒ ρ(x, t) + (u − w) · ∇ u = ρ(x, t) b + ∇· σ (9) δt The fluid considered is an incompressible newtonian fluid with constant physical properties. Under this conditions, the conservation equations (Navier-Stokes) are written, on a spatial domaine Ωt :    ∇·u =0 (10) 1 δu   + ((u − w) · ∇) u = − ∇p + b + ν ∇2 u δt ρ

3.2 Domaine Solide

In a small displacements mode around the configuration of reference, the structural dynamics equations become, (11) Ms¨s + Cs s˙ + Ks s = F f,g R ⇒ R ⇒ with F f,g = Γt σ ·n ds = Γt [−p · I + µ τ ] · n ds vector of generalized fluid forces. 1

We adopted mixte derivative convention

δ δt

of Germain (1986).

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

3.3 Coupled Domaine The coupling between solid and fluid equations is operated by boundary conditions on the interface Γs,t : (a) the kinematic continuity speed, u = s˙ (fluid viscous); (b) the ⇒ ⇒ kinematic continuity of the efforts, σ ·n = σ s ·n. Using the equations (10) and (11) with boundary conditions above, the coupled problem is formulated as follows: δu + (u − w) · ∇u − ν∇2 u = −∇p/ρ + b δt ∇·u = 0

, in Ωt

u = uΓ u = s˙ Z Ms ¨s + Cs s˙ + Ks s =

Γf,t

(u, s, s˙ )| t=0 = 4.

T RANSPIRATION METHOD

, on ∂Ω − Γs,t

(12)

, on Γs,t h

i τ ·n · ϕi ds −p n+

u0 , s(0) , s(1)

⇒




We consider a kind of fluid-structure interaction problems characterized by small structural vibrations around the reference position. The formalism ALE has the disadvantages of appearing an excessive formulation for solving a problem of small vibration like the fluid-elastic coupling in tube bundles. Thus, before even the advent of the ALE formulation, aeronautics engineers developed a simplified technique, said method of T RANSPIRATION (Lighthill, 1958; Renou, 1998; Varella, 2001). Based on a first order development of velocity boundary conditions, this method allows the use of a fluid domain fixed in time, by avoiding hence the problems due to the mesh distortions and great adaptations of the fluid solvers. This technique allows simulate vibratory problems of fluid-structure interaction making use of the well-known, reliable and optimized euleriens methods with transpiration boundary conditions (Renou, 1998).

xm

um

real position of moving wall mesh boundary

up xp Figure 3: Development of velocity field at structure boundary Let us suppose a sensible linear velocity field near of moving walls. For notations, the index m, p, and o will correspond respectively to the fields (speed, pressure...) evaluated in limit of grid, at moving wall, and stationary position without disturbance of the wall.

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

By definition, xp = xm + s

(13)

The first order development of fluid-structure interface is given by, up = um + s · (∇u)m ,

in Γs,o

(14)

Let us suppose sufficiently regular at interface fluid-structure Γs,o , the gradient (∇u)m approximate to the gradient (∇u)o of steady velocity field uo in configuration non-deformed. Thus, the transpiration boundary condition (14) becomes: um = s˙ − s · (∇u)m ,

dans Γs,o

(15)

where, up = s˙ by adherence conditions. The forces determination on the structure remains a delicate point. Renou (1998) surmount this difficulty by another first order development of stress constraint fields at moving wall, like similar to expression (15): ⇒

⇒

⇒

σ (I + δX) = σ +s · (∇ σ)o ,

in Γs,o

(16)

4.1 T RANSPIRATION Coupled Domaine The solid and fluid coupling is similar to ALE formulation (12): ∂u + u · ∇u − ν∇2 u = ∇p/ρf ∂t ∇ · u = 0, u = 0,

dans Ωo sur ∂Ω − Γs,o

(17)

u = s − ∇o uo δx, sur Γs,o  ⇒ i R h⇒ · no · ϕi dso Ms ¨s + Cs s˙ + Ks s = Γs,o σ +s · ∇ σ o
(u, s, s˙ )| t=0 = u0 , s(0) , s(1) 5.

NUMERICAL IMPLEMENTATION

Now we fly over the numerical implementation of fluid-elastic coupling algorithm. Industrial applications often choose a kind of algorithms known as partitioned which integrate step-by-step each fluid and solid domaine independently. The present implementation is strongly inspired by the coupling algorithm improved serial stagered procedure (ISS) (Farhat and Lesoinne, 1997; Piperno and Farhat, 2001). For each time step n + 1, we should solve this coupled system:    n+ 21 n+ 21 n− 21 n− 12 n n+1  , w , x , x , p ) = NS v (v       n+ 12 n+1 n+1 n n ˙ n ¨n ˙ (18) , Fs (s , s ) = MR s , s , s , Ff      1 1   xn+ 2 = VM sn , s˙ n , xn− 2

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

where, NS is the projection algorithm for Navier-Stokes resolution, VM is a algorithm to determinate mesh velocity w, and MR is Newmark time-integration to solve spring-mass model (sn+1 , s˙ n+1 ). For more details, we can see Morais (2005). 6.

NUMERICAL RESULTS

6.1 Free vibration of a tube in fluid at rest We consider a tube of diameter D immersed in an infinite incompressible viscous fluid domaine. This tube is excited to an amplitude So and then the excitation is removed. We observed the tube vibrations decays in time. Simulations are made for a tube of diameter D = 13.30mm. the amplitude of releasing is limited to 2So /D = 0.001. The experimental tests give frequency ff s,exp. = 12.866Hz and ratio damping ξf s,exp. = 1.00%. The monophasic fluid domaine is considered water of density ρ = 1000kg/m3 , kinematic viscosity ν = 10−6 m2 /s and, by consequent, the number of Stokes St ⋍ 2274. The coarse mesh (raf = 0) is composed by two cylinder concentric of diameters D and De = 30D. Total number of elements is Nr × Nθ = 22 × 48, where Nr and Nθ is the numbers of nodes in radial and orthoradial directions. Grid refinements (raf = 1 and 2) were made dividing each elements into four, encase grid (fr. maillage embots). Frequency and damping ratio identification of the displacement signal is made by the Gharib’s least-square method. Numerical test carry out four periods of simulation of which the first is discarded of analysis. Time convergence of frequency ff s and damping ratio ξf s of each method is presented by Figure 4. The numerical results converge towards tests experimental. T RAN 1,50%

12,918

εfs (%)

ALE Lacher Nraf = 1

ffs (Hz)

ALE Lacher Nraf = 2

12,905

1,40%

ALE Lacher Nraf = 1 ALE Lacher Nraf = 2

Transp.Lacher Nraf = 2 12,892

Exp.AMOVI - Bas Vitesse

Transp.Lacher Nraf = 2 1,30%

Exp.AMOVI - Bas Vitesse Chen & Yeh(1976)

Chen & Yeh(1976)

1,20%

12,879 12,878

1,10%

12,866 12,866

12,853

1,00%

12,841

0,90%

12,828

0,80%

1,002%

0,905%

0

1000

2000

3000

4000

5000

(a) Frequency ff s (Hz)

6000

Nφ 7000

0

1000

2000

3000

4000

5000

6000

Nφ

7000

(b) Damping ratioξf s (%)

Figure 4: ALE and T RANSPIRATION time convergence of tubes + fluid system. SPIRATION method await the converged value by bottom, i.e., it overestimates the value of

mass added. And ALE method underestimates it, Figure 4a. Numerical simulations are very close to the analytical solution (Chen, 1987) and experimental tests. The numerical error doesn’t exceed ≃ 0.2%. From the point of view of damping ratio, Figure 4b, the time convergence observes a similar behavior of frequency. ALE method converge more quickly than T RANSPIRA TION method. This is necessary fonction of the exact description of moving wall by ALE formulation. According to Richardson extrapolation (Roy, 2005), the converged value φref is ob-

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

tained by the expression: φref = φNϕ /2 + [φNϕ /2 − φNϕ ]/[1 − 2αt ], where αt is the time convergence order, Nϕ and Nϕ /2 are respectively the numerical results with 2π/Nϕ and 4π/Nϕ times teps(ω∆t = 2π/Nϕ ). Knowing that the time convergence order of the present coupling algorithm is linear (αt ≃ 1), the expression of φref sounds: φref,t = 2φNϕ /2 + φNϕ

(19)

Figure 5 presents the evolution of relative error by report experimental test. We find a behavior in power function φref = φ6 − C N αt according to Richardson hypothesis. The convergence order αt of ALE and T RANSPIRATION method are all close to unit (αt ≃ 1), in conformity with the theoretical slope of our coupling algorithm. 100,0%

100,0%

εfs = 118,57Nφ-0,9715

εfs = 81,889Nφ-0,9755

R2 = 0,9999

R2 = 0,9999

10,0%

10,0%

1,0%

1,0%

0,1%

0,1%

ffs = 0,1278Nφ-0,9258

ffs = 0,2904Nφ-1,1248

2

R = 0,9992

R2 = 0,9957

0,01%

0,01% Erreur[F_fs; raf = 2; Num/Exp]

Erreur[F_fs; raf = 2; Num/Exp]

Erreur[Eps_fs; raf = 2; Num/Exp]

Erreur[Eps_fs; raf = 2; Num/Exp] 0,001% 100

1000

10000

0,001% 100

(a) T RANSPIRATION

1000

10000

(b) ALE

Figure 5: Relative error of frequency and damping ratio of coupled system. Table 2 synthesizes converged results by Richardson extrapolation comparing with experimental and analytical values. ALE raf = 1 raf = 2 ff s (Hz) ξf s (%)

12.878 0.905

12.866 1.002

T RANSPIRATION raf = 1 raf = 2 12.876 0.858

12.866 1.002

AMOVI 12.87 1.00

Chen&Yeh 12.84 0.99

Table 2: Summary of numerical, experimental and analytical results. 6.2 Fluid-structure coupling of AMOVI tube array model Fluid-structure coupling (vibration in fluid domaine without flow) of AMOVI tube array model is modelled in 2D by a square 9-tube bundle with tube-to-tube spacing P/D = 1.44 and diameter D = 12.15 mm, Figure 1. The flexible tube at central tube array has one vibration mode. All of other tubes are fixes. In order to avoided boundary effects, two lines of half-tubes at ex direction were added around of 9-tube configuration screw to the metal walls. Consequently, the other two lines of tubes at upstream and downstream flow direction (ey ) were also cut out forming a symmetrical unit. Numerical

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tests showed that the no-slip boundary conditions (u = 0) at walls sides and null pressure (p = 0) at upstream and downstream direction obtain same results by report a entire modelling of AMOVI model. The monophasic fluid is considered water with mass density ρ = 1000 kg/m3 and kinematic viscosity ν = 10−6 m2 /s. The flexible tube belonging to a fixed tube bundle in 9-tube configuration have the following vibratory characteristics in vacuum: circular frequency ωs = 2π · 14.30 = 89.85rad/s, ratio damping ξs = 0.25% and linear density Ms = 0.298kg/m. Consequently, stiffness and structure damping constants are respectively Ks = 2405.7kg/m/s2 and Cs = 0.130kg/m/s. According to experimental results with low Reynolds flow in AMOVI model (Baj, 2000), the vibratory characteristics under water are circular frequency ωf s = 2π · 11.90 = 74.76rad/s and ratio damping ξf s = 1.17%. Consequently, Stokes number is St = 1757. The flexible tube shake on ey direction with adimensional amplitude So /D = 0.001. Numerical tests with forced and free movement were carried out for each formulation in order to verifier the convergence of results. Numerical criteria In order to have the control of numerical modelling, the optimal grid and time-step characteristics for simulations will be treated in this item. First of all, it is necessary to define the characteristic length δ = δ ′ /R of boundary layer on the moving walls. For an oscillating cylinder p problem in laminar fluid, Figure 6a, the characteristic length can be defines as δ ≤ 1/ πSt/2. De δ′ ∂s = (ωSo )e[ωt] ey ∂t

Ωf

Γs

s = e[ωt] ey

D

(a) Characteristic length of fluid δ

De

∆xθ

Ωf

Γs D

∆xr

(b) Scheme of space discretization

Figure 6: Vibration of a cylinder rod in incompressible viscous laminar fluid. S PATIAL C RITERIA - The numerical description of the vibratory phenomena in boundary layer is essential for this kind of problem. An elementary spatial criteria is insurer a minimum of C elements in boundary layer, δD/2 ≥ C∆xr . By definition, the finite elements next fluid-structure walls Γs are square ∆xr ≈ ∆xθ = πD1 /Nδ√ , being Nδ the number of nodes around fluid-structure walls Γs . We obtain Nδ ≥ C · π 2πSt. Then, an estimation of nodes around the flexible tube is Nδ ≥ C · 330 orthoradial nodes. We use quadratic element Q2-Q1 . Others numerical tests show that a grid with a element in boundary layer limite (C = 1) (approximately 40840 elements and 126079 nodes) offer a good precision with a lower-cost computational, Figure 7. P HASE C RITERIA - The numerical time criteria is also necessary for vibratory phenomena. The numerical analysis of problem (Morais, 2005) shows that the criterion more restrictive is due to signal identification procedure. The error sensitivity analysis of signal

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

Mail Base

Mail Base

(a) raf. = 0 − Nδ = 168 nœuds orthoradiales

(b) raf. = 1 − Nδ = 336 nœuds orthoradiales

Figure 7: Encased fluid grids of AMOVI tube array model. identification procedure (4) gives the expressions of error propagation δMa and δCa : δCa 1 δMa δϕ ≥ tan(ϕ)δϕ and ≥ Ma Ca tan(ϕ)

(20)

We can estimate an optimal number of time-step per period Nϕ (∆ϕ = ω∆t = 2π/Nϕ ) by the analytical solutions of a cylinder rod vibrating in viscous fluid. But, a more correct estimate must take into account confinement effects of close tubes. By means an interpolation, Rogers et al. (1994) finds a relation between confinement ratio De /D and pitch-to-diameter ratio P/D, i.e., De /D = [1.07 + 0.56 P/D] · P/D. Then, the expressions of added mass Ma and damping coefficient Ca for a cylinder rod immersed in stokes confined fluid medium are given by: r Ma 1 + (D/De )3 π 1 + (D/De )2 π 1 Ca π √ (21) = + and = ρD2 4 1 − (D/De )2 St ω ρD2 πSt [1 − (D/De )2 ]2 Applicant the expressions (21) and (3b) in the equation of error propagation (20b), one obtains: √ Nϕ δCa πSt [1 + (D/De )2 ] · [1 − (D/De )2 ] · · (22) & 2π Ca 4 1 + (D/De )3 For present problem, the AMOVI tube array model (P/D = 1.44) have confinement ratio De /D = 2.702. One defines the number of time-steps per period Nϕ ≃ 2π/δϕ (∆ϕ . δϕ). Thus the optimal number of time-step is Nϕ · (δCa /Ca ) ≥ 109. For an error relative δCa /Ca ≃ 1%, the phase number Nϕ optimal is 10900 time-steps per period . Numerical results analysis The comparisons between ALE and of T RANSPIRATION methods are carried out under same geometrical and initial conditions. Using Pentium c 3GHz computers with 2Go of memory, simulations with forced movement and 4 Xeon

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

free vibration last three and four periods respectively. The signal identification procedures are the Phase method for forced movement and the Gharib’s least-square method for free vibration. Phase method analyse the last period, while Gharib’s method uses the last three periods to find the frequency and damping ratio.

(a) ALE

(b) T RANSPIRATION

Figure 8: Comparison of pressure fields at maximum velocity. The Figure 8 compares the pressure fields at maximum velocity obtained by ALE and T RANSPIRATION. The ALE and T RANSPIRATION pressure fields aren’t symmetric with the horizontal axis due to the confinement effects of close tubes. Figure 9 present the frequency ff s and damping ratio ξf s evolution in function of phase number Nϕ ∈ [300 − 4600] for T RANSPIRATION and ALE methods. Table 3 summarize numerical frequency and damping results, as well correspondents added mass Ma and damping coefficient Ca , identified by free vibration and forced movement procedures. This results are compared with Roger’s semi-analytical solution, low-Reynods experimental test of AMOVI model, and other numerical results - CREATIVE EdF-CEA co-operative work group (Bendjeddou, 2005)-. 1,80%

11,80

ffs (Hz)

ξfs (Hz)

ALE Lacher Nraf = 0

ALE Lacher Nraf = 0

1,70%

ALE Lacher Nraf = 1

11,75

ALE Lacher Nraf = 1

Transp.Lacher Nraf = 1

Transp.Lacher Nraf = 1 1,60%

Rogers et al (1994)

11,70

Exp. AMOVI - Bas Vitesse 11,65

Rogers et al (1994) Exp. AMOVI - Bas Vitesse

1,50%

11,628

1,40%

11,634

11,60

1,30% 11,55

1,254% 1,233%

1,20%

11,50 0

1000

2000

3000

4000

(a) Frequency ff s (Hz)

5000

Nφ φ

1,10% 6000

0

1000

2000

3000

4000

5000

Nφ φ

6000

(b) Damping ratio ξf s (%)

Figure 9: ALE and T RANSPIRATION time convergence of tube+fluide system. OBS: Each vertical scale subdivisions correspond to a relatif error δε ≃ 0.4% in frequency and δε ≃ 9% in damping. Table 3 synthesizes converged results by Richardson extrapolation comparing with

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

experimental and analytical values. Time convergence of coupling algorithms have a monotonous asymptotic behavior even with small Nϕ . 1,60%

ξfs (Hz)

Exp. AMOVI - Bas Vitesse ALE Nraf = 0

1,50%

ALE Nraf = 0 - Extr.Richardson Trs. Nraf = 1 Trs. Nraf = 1 - Extr.Richardson

1,40%

Trs. Nraf = 0 Trs. Nraf = 0 - Extr. Richardson 1,30%

1,254% 1,233%

1,20%

1,10% 0

1000

2000

3000

4000

5000

Nφ φ

6000

Figure 10: Richardson extrapolation (19) per report time convergence of damping ratio ξf s (%) of tube+fluide system. The ALE and T RANSPIRATION differences are only observable for damping ratio evolution that not exceeding 2.5% for T RANSPIRATION method. The frequency results are similar for both methods. A spatial discretization Nδ = 21 nodes offer alike results then to Nδ = 42 nodes. T RANSPIRATION raf = 0 raf = 1

ALE raf = 0 raf = 1

AMOVI - Bas Re -

Rogers

Creatif - EdF -

11.78 1.17

11.55 1.16

11.887 1.165

0.9561 0.05736 11.78 1.17

1.0769 0.05964 11.55 1.16

0.9028 0.05586 11.887 1.165

Free Vibration ff s (Hz) ξf s (%)

11.634 1.254

11.634 1.210

11.626 1.245

11.628 1.233

Forced Movement Ma Ca ff s (Hz) ξf s (%)

1.0886 0.06807 11.526 1.321

1.0359 0.06404 11.625 1.265

1.0886 0.06807 11.526 1.321

1.0359 0.06404 11.625 1.265

Table 3: Summary of converged numerical, experimental and semi-analytical results. 7.

CONCLUSIONS

The numerical performances of ALE and T RANSPIRATION methods are compared in this article. These methods have been validated with help of analytical solution and experiments data. And with help of Roger’s semi-analytical solution, experiment test and other numerical results, we made the analysis of dynamics characteristics of AMOVI square tube-bundle experimental model. In the present work, the two methods present of the similar converged results. The linearization on small vibrations, made by T RANSPIRATION method, is validated still that its convergence is slower by report ALE method.

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

Further developments will be carried out in order to improve the coupling process and the flow modelling in presence of moving boundaries. Acknowledgements The first author acknowledges the financial support by a Brazilian CAPES Fellowship, Process No. 1232/99-1. REFERENCES Baj, F., 1998. Amortissement et instabilite fluide-elastique d’un faisceau de tubes sous ecoulement diphasique. These de doctorat, Universite Paris VI. Baj, F., 2000. Adimensionnement des forces de couplage fluide-structure : etudes parametriques en monophasique (eau) et en diphasique (eau-air). influence des modeles cinematiques. Rapport d’etudes SEMT/DYN/RT/00-043/A, CEA Saclay - DMT. Bendjeddou, Z., 2005. Metodologie pour la simulation numerique des vibrations induites par ecoulement dans les faisceaux de tubes. These de doctorat, Universite de Lille. Caillaud, S., 1999. Excitation forcee et contrle actif pour la mesure des forces fluideelastiques. These de doctorat, Universite Pierre et Marie Curie - Paris VI. Chen, S. S., 1987. Flow-induced vibration of circular cylindrical structures. Hemisphere Publ. Corp., Washington. Farhat, C. & Lesoinne, M., 1997. Improuved staggered algorithms for the serial and parallel solution of three-dimensional non-linear transient aeroelastic problems. Report 97-11, Centre Aerospace Structures - University of Colorado, Boulder, Colorado. AIAA Journal, 36(9),pp.1774-1757,(1996). Germain, P., 1986. Mecanique, volume 1. Ellipses Marketing Edition. Gharib, M. R., Leonard, A., & Gharib, M., 2000. A fluid force deduction technique for vibrating structures in cross-flow. In Flow-Induced Vibration, Proceedings of the 7th International Conference on Flow-Induced Vibration - FIV2000 / Lucerne / Switzerland, pp. 85–89. Gibert, R.-J., 1988. Vibrations des Structures - Interactions avec les fluides - Sources d’excitation aleatoires, volume 69 of Collection de la Direction des Etudes et Recherches d’Eletricite de France - CEA-EDF-INRIA Ecole d’ete d’analyse numerique. Editions Eyrolles. Gounand, S., 1997. Simulation numerique d’ecoulements a surface libre. Technical report, CEA/Saclay - DRN/DMT/SEMT/TTMF. Huerta, A. & Liu, W. K., 1988. Viscous flow structure interaction. Journal of Pressure Vessel Technology, vol. 110, pp. 15–21. Lighthill, M., 1958. On displacement thickness. J. Fluid Mechanics, vol. 4, pp. 383–392.

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

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