turbulent squeeze-film forces

Journal of Fluids and Structures 33 (2012) 1–18

Contents lists available at SciVerse ScienceDirect

Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs

A theoretical model and experiments on the nonlinear dynamics of parallel plates subjected to laminar/turbulent squeeze-film forces Philippe Piteau a, Jose´ Antunes b,n ´ tudes de Dynamique, Commissariat a l’Energie Atomique et aux Energies Alternatives, F-91191 Gif-Sur-Yvette Cedex, France Laboratoire d’E Applied Dynamics Laboratory, Instituto Tecnologico e Nuclear, Instituto Superior Te´cnico, Universidade Te´cnica de Lisboa, Estrada Nacional 10, 2686 Sacavem Codex, Portugal

a

b

a r t i c l e i n f o

abstract

Article history: Received 20 February 2010 Accepted 30 April 2012 Available online 13 June 2012

Squeeze film dynamical effects are relevant in many industrial contexts, bearings and seals being the most conspicuous applications, but also in other industrial contexts, for instance when dealing with the seismic excitation of spent fuel racks. The significant nonlinearity of the squeeze-film forces which arise prevents the use of linearized flow models, and a fully nonlinear formulation must be used for adequate computational predictions. Because it can easily accommodate both laminar and turbulence flow effects, a simplified bulk-flow model based on gap-averaged Navier–Stokes equations, incorporating all relevant inertial and dissipative terms was previously developed by the authors, assuming a constant skin-friction coefficient. In this paper we develop an improved theoretical formulation, where the dependence of the friction coefficient on the local flow velocity is explicitly accounted for, such that it can be applied to laminar, turbulent and mixed flows. Numerical solutions for both the basic and improved nonlinear one-dimensional time-domain formulations are presented in the paper. Furthermore, we present and discuss the results of an extensive series of experiments performed at CEA/Saclay, which were performed on a test rig consisting on a long gravity-driven instrumented plate of rectangular shape colliding with a planar surface. Theoretical results stemming from both theoretical flow models are confronted with the experimental measurements, in order to assert the strengths and drawbacks of the simpler original model, as well as the improvements brought by the new but more involved flow formulation. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Squeeze-film Flow–structure interaction Nonlinear dynamics Impacts

1. Introduction Many systems of practical interest are subjected to intense fluid/structure interaction forces when a thin layer of fluid is interposed between two vibrating structures. A typical example is provided by immersed structural components, which may impact if an external excitation is imposed. Nonlinear effects can then become dominant and should not be neglected. Significant work in this field has been performed at CEA/Saclay during the last two decades, in connection with nuclear facilities—see Esmonde et al. (1990a, 1990b) and Antunes and Piteau (2001, 2010). The extensive literature on the fluid/structure dynamics of industrial squeeze-film problems is mostly concerned with ~ linearized analysis, when the vibratory motions are such that, at each location r, the fluctuating part hðr,tÞ of the local fluid

n

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

0889-9746/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.04.012

2

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

~ gap hðr,tÞ ¼ hðrÞ þ hðr,tÞ is small compared to the mean gap value hðrÞ—see, for instance, works by Fritz (1970), Hobson (1982), Mulcahy (1980, 1988) or Moreira et al. (2000a), as well as the review books by Paı¨doussis (1998) and Kaneko et al. (2008). One of the most thorough analyse along these lines was achieved by Inada and Hayama (1990a, 1990b), who evaluated the fluidelastic force under steady flow, including added mass, damping and stiffness flow terms for a onedimensional tapered leakage channel. More recently, Porcher and De Langre (1997) evaluated the dynamical effects of changing the loss coefficients at the channel inlet and outlet boundary conditions. The linearized approach is obviously unable to provide answers at larger vibratory amplitudes, when the fluctuating ~ gap hðr,tÞ is of the same order of magnitude as hðrÞ. Finding a suitable analytical formulation for the fully nonlinear problem is the main subject of the present paper. Here we will address the case of squeeze-film dynamics under no permanent flow, which in a sense restricts our previous efforts in this field. However, our previous dynamical solution will be now extended in order to encapsulate, in the analytical formulation of the nonlinear fluid coupling force, the flow velocity dependence of the dissipative terms. Overall, the analytical formulation developed here may be connected with our approach in work focused on immersed rotor dynamics—see Axisa and Antunes (1992), Grunenwald (1994), Antunes et al. (1996, 1999), and Moreira et al. (2000a, 2000b). Indeed, the successful ‘‘bulk-flow’’ approach started by Fritz (1970) and Hirs (1973), and later used by many authors – see Childs (1993) – appears also well suited to deal with the small-tomoderate fluctuating gaps, such as found in the configuration of the present problem. We start by recalling our nonlinear flow model obtained by assuming constant (velocity independent) flow friction coefficients, see Antunes and Piteau (2001). Apart from the specific dissipative terms connected with our quadratic-invelocity pressure drop formulation, we obtained fluid forces which are similar to those previously presented by Esmonde et al. (1990a). Then we refine our formulation in order to incorporate the dependence on flow velocity of the loss coefficients, which are conveniently expressed in terms of a single formulation, suitable for laminar or turbulent conditions, as well as mixed flows. We thus produce a unique analytical solution for the coupling fluid force applied to the plates, irrespectively of the nature of the flow inside the gap. It should be mentioned that Guo et al. (2010) and Hellum et al. (2010) recently addressed the problem of dealing with laminar and turbulent flows, in the context of fluid-conveying pipe dynamics. However, the approaches developed by these authors are directed toward modeling flows that may be either laminar or turbulent, whereas the configuration of the flow/structure problem dealt here is such that, depending on time and location, the flow may be partly laminar and partly turbulent within the system. The second part of this paper relates to experimental results performed at CEA/Saclay, on the motion of a gravity-driven plate colliding with a rigid plane surface. This example is motivated by impact problems which may arise between immersed nuclear components, such as fuel racks. The colliding plate has dimensions such that the width W is much larger than the length L, therefore the flow may be assumed approximately one-dimensional, along the shorter dimension L. Extensive experiments have been performed, in particular by changing the fluid temperature, hence its viscosity. These experimental results are reported and compared with both the basic and refined models of the fluid force, with general satisfying agreement. Furthermore, the strengths and drawbacks of the older model are discussed, and the improvements brought by the new but more involved flow formulation are highlighted.

2. Theoretical flow model 2.1. Flow equations To derive the bulk-flow equations for the flow within a small-to-moderate fluctuating gap, the control volume shown in Fig. 1 applies, where the gap h(r,t) between the vibrating plates 1 and 2 is filled with an incompressible fluid of volumic mass r.

Fig. 1. Control volume used for deriving the bulk-flow equations.

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

3

~ As mentioned before, at each location r  {x,z} we define hðr,tÞ ¼ hðrÞ þ hðr,tÞ, where hðrÞ is the average value of the local gap between the non-vibrating plates under steady flow and load conditions. Because the transverse gap h(r,t) is assumed small with respect to the longitudinal plate dimensions L (along x) and W (along z), in the bulk-flow modeling approach one neglects the changes in pressure and velocity across the fluid gap, so that we may conveniently replace the pressure field P(x,y,z,t) and the components Vx(x,y,z,t) and Vz(x,y,z,t) of the velocity field by their gap-averaged values Z hðx,z,tÞ Pðx,y,z,tÞ dy, ð1Þ pðr,tÞ  pðx,z,tÞ ¼ 0

uðr,tÞ  uðx,z,tÞ ¼

Z

hðx,z,tÞ

V x ðx,y,z,tÞ dy,

vðr,tÞ  vðx,z,tÞ ¼

0

Z

hðx,z,tÞ

V z ðx,y,z,tÞ dy:

ð2Þ

0

Then, considering a small control volume CV across the fluid gap thickness, with whole boundary CS, the continuity equation is written as ZZZ ZZ @ r dV þ rVUn dS ¼ 0, ð3Þ @t CV CS where V is the velocity of the fluid crossing CS and n is the local unit vector normal to the boundary. The momentum equation is written as ZZZ ZZ X @ Vr dV þ VrVUn dS ¼ Fn , ð4Þ @t CV CS n where Fn are the external and volumic forces acting on the fluid in CV. Referring to the elementary control volume shown in Fig. 1, one obtains from (3) the continuity equation—see, for instance, Childs (1993), Antunes et al. (1996) or Antunes and Piteau (2010) @h @ @ þ ðhuÞ þ ðhvÞ ¼ 0, @t @x @z

ð5Þ

and, from the momentum equation (4), the following scalar equations:     @ @ @ @p , r ðhuÞ þ ðhu2 Þ þ ðhuvÞ ¼  tx1 þ tx2 þ h @t @x @z @x    @ @ @ @p ðhvÞ þ ðhuvÞ þ ðhv2 Þ ¼  tz1 þ tz2 þ h , @t @x @z @z

ð6Þ



r

ð7Þ

where, as discussed in Section 2.2, the tx,z 1,2 are the shear stresses at the fluid/wall interfaces of plates 1 and 2, respectively, along the orthogonal directions of the unit vectors i and k. For the geometry of interest here, shown in Fig. 2, we are interested in the flow between two plates such that L5 W, therefore we may postulate v 5 u. Then, Eqs. (5)–(7) reduce to the following 1D formulation: @h @ þ ðhuÞ ¼ 0, @t @x     @ @ @p : r ðhuÞ þ ðhu2 Þ ¼  tx1 þ tx2 þh @t @x @x

ð8Þ

ð9Þ

Now, because the plate surfaces will in this paper be assumed parallel, we have h(x,t)¼Y(t), 8x, where for simplicity it will be assumed that the lower plate is fixed and the fluid gap Y(t) is given by the displacement of the upper plate. Finally,

Fig. 2. Geometry of the flow–structure system.

4

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Eqs. (8) and (9) become @Y @u þY ¼ 0, @t @x   @ @ @p r ðYuÞ þY ðu2 Þ þ tx1 þ tx2 þY ¼ 0, @t @x @x

ð10Þ

ð11Þ

and the dynamic force exerted by the fluid per unit width of the plate – e.g., the total force is W Ff(t) – is given by Z L=2 Z L=2 pðx,tÞ dx ¼ 2 pðx,tÞ dx, ð12Þ F f ðtÞ ¼ 0

L=2

where the last form is justified by the problem symmetry, which enables working on the positive half-space region only. One should note that a uniform velocity profile is implicit when deriving the bulk-flow formulation. In particular, the quadratic convection acceleration term derived in the momentum equations (9) and (11) is, in a strict sense, correct only for turbulent flow. Actually, if a parabolic velocity profile is assumed under laminar conditions, the convective acceleration term in (9) and (11) is affected by the coefficient 6/5 (instead of one), see Porcher and De Langre (1997). However, for the sake of simplicity, we decided to adopt the straight convection term used here, because the magnitude difference between these two results is comparatively small with respect to the significant differences between the laminar and turbulent flow friction terms, which will be addressed next in greater detail. 2.2. Dissipative phenomena Following the bulk-flow approach, the tangential stresses tx1 and tx2 will be formulated using a loss-of-head model: 1 2

tx1,2 ¼ ru9u9f 1,2 ,

ð13Þ

where the Fanning friction coefficients at the walls f1 and f2 are established based on empirical correlations—see Idel’Cik (1960) and Blevins (1984). Typically, f changes with the flow Reynolds number as (Hirs, 1973) f ¼ n ðReÞm ,

ð14Þ

where parameters m and n depend on the flow and the Reynolds number is, for 1-D channels, given by Reðx,tÞ ¼

DH 9uðx,tÞ9

n

¼

ð4S=PÞ9uðx,tÞ9

n

¼

ð4Whðx,tÞ=2WÞ9uðx,tÞ9

n

¼

2hðx,tÞ9uðx,tÞ9

n

,

ð15Þ

with DH being the equivalent hydraulic diameter for non-circular cross-sections, S being the flow-conveying cross-area and P the corresponding fluid/wall contact perimeter. For turbulent flows between plane smooth surfaces, Hirs suggest the following values: 103 oRe o3  104 ) m ¼ 0:25,

0:055 ffiffiffiffiffiffi , n ¼ 0:055 ) f ¼ p 4 Re

ð16Þ

otherwise, the Blasius friction correlation can be used with almost similar results 0:0665 ffiffiffiffiffiffi : f¼ p 4 Re

ð17Þ

On the other hand, for laminar flows, the friction coefficient decreases very fast for increasing flow velocities. Between two parallel plates, it can be shown that the Fanning friction coefficient at each fluid/wall interface is given by, see Blevins (1984) Re o 4  103 ) f ¼

12 : Re

ð18Þ

Beyond these distributed stresses, other singular dissipative effects arise in the boundaries x ¼ 7L/2 of the moving plate. Very complex phenomena arise here and, in the absence of reliable and extensive loss data obtained from unsteady flow configurations, we will adopt – as in many other investigations – a classic quasi-static Bernoulli formulation with a loss term at the channel/reservoir interfaces. Hence, at the boundary x ¼ L=2     2          L 1 L 1 L L L ð19Þ p  ,t þ r u  ,t ¼ P ext  ru  ,t u  ,t K in,out Re  ,t , 2 2 2 2 2 2 2 and, at the boundary x ¼ L=2     2         L  L 1 L 1 L L p ,t þ r u ,t ,t u ,t K in,out Re ,t , ¼ P ext þ ru 2 2 2 2 2 2 2

ð20Þ

where Pext is a static reference pressure in an external reservoir, far from the singularity, and the loss coefficients Kin,out depend on the local geometry and Reynolds number. Idel’Cik (1960) and Blevins (1984) give extensive values for the

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

5

singular loss coefficient Kin,out (Re), as a function of the flow direction (entering or leaving the channel), of the Reynolds number and of the channel/reservoir interface geometry. For turbulent flows and an abrupt gap change, typical values are Kin E 0.5 (when Y_ ðtÞ 40) and Kout E1 (when Y_ ðtÞ o 0). In short, this means that about half the kinetic energy of the inflow and the full kinetic energy of the outflow is lost through irreversible phenomena. In general, smooth (rounded) corners of the plates at x¼ 7L/2 will lead to lower values of Kin. On the other hand, low velocity flows will lead to increased values of the loss coefficient. 2.3. Analytical approximation for the friction losses We start by discussing the distributed losses. The dichotomy of the formulation for the friction coefficients – Eqs. (16) or (17) and (18) – is one of the difficulties with using the bulk-flow formulation in the context of the present analytical model: turbulent regimes lead to tangential stresses which depend almost quadratically on the average flow velocity, through a friction coefficient fT which is almost independent of the flow velocity. On the other hand, because the laminar tangential stresses increase proportionally to the flow velocity, then for consistence the quadratic formulation (13) implies a laminar friction coefficient decreasing in f L  Oð1=uÞ. In general terms, from Eqs. (14)–(18), we may write for turbulent flows  m 2h9u9 T 1 2mT 1 nT r mT m þ1 tðTÞ ¼ r u9u9n ¼ h u9u9 T , Re4Re0 , ð21Þ T 1,2 2 n nmT and for laminar flows 1 2

tðLÞ ru9u9nL 1,2 ¼



m 2h9u9 L

n

¼

2mL 1 nL r

nmL

mL

h

u9u9

mL þ 1

,

ReoRe0 ,

ð22Þ

where Re0 is the Reynolds number separating the laminar and turbulent flow ranges. ðLÞ For any realistic values of the constants nL, mL, nT and mT, it is clear that tðTÞ 1,2 =t1,2 becomes negligible in the range ðTÞ Re o Re0 , while tðLÞ = t becomes negligible in the range Re 4Re . Then, a very simple and natural choice for a continuous 0 1,2 1,2 function emulating the behaviour of (21) and (22) in the full range of laminar and turbulent flows is  m  m 2h9u9 L 1 2h9u9 T 1 t1,2 ¼ ru9u9nL þ ru9u9nT , 2 2 n n ¼

2mL 1 nL r mL

mL

h

mL þ 1

u9u9

þ

2mT 1 nT r

mT

h

mT þ 1

u9u9

n nmT mL þ 1 m þ1 mL mT þC T rh u9u9 T 8Re, ¼ C L rh u9u9

, ð23Þ

with CL ¼

2mL 1 nL mL

n

,

CT ¼

2mT 1 nT

nmT

:

ð24Þ

As an illustration, Fig. 3 displays the result of approximating the Hirs friction model using formulation (23), which highlights the adequacy of the approach adopted.

Fig. 3. Comparison between the Hirs friction model (16) and (18) with the continuous analytical approximation (23).

6

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

We now turn to the singular loss coefficients at the channel boundaries, which change in a more drastic manner according to the direction of the flow (either entering or being ejected from the channel), on the boundary geometry (abrupt or smooth change of fluid gap), on the nature of the flow (laminar or turbulent), as well as on the Reynolds number. Idel’Cik (1960) and Blevins (1984) provide guidelines for a number of standard geometries, as a function of the Reynolds number, following given assumptions on the flow velocity profiles. For any actual nonlinear time-domain computation, it is easy to compute relevant value of Kin[Re(L/2,t)] and Kout[Re(L/2,t)], by interpolating through the values shown in their tables. Unfortunately, our experimental geometry does not conform exactly to any of the cases detailed in these references. Therefore, we decided to use in our numerical simulations a constant value for the singular loss coefficient, which both fits our experimental data and is generally consistent with these references—see Section 4.3. 3. Formulation of the dynamical flow force 3.1. Simplified solution with constant friction coefficients Here, the external static pressure on reservoir at the left side of the plate is equal to the static pressure on the right side, Pext. We also postulate that the skin-friction coefficients are independent of the flow velocity and that the plate surfaces are similar, so that f 1 ¼ f 2  f ð8ReÞ. We thus obtain from Eqs. (10) and (11) Y_ @u ¼ , Y @x " # @p @u u9u9 Y_ ¼ r u_ þ u þ 2u þ f , @x @x Y Y

ð25Þ

ð26Þ

with Y(t) 40, 8t. It appears immediately that, when Y-0, the flow-generated force will tend to infinity. Accounting for the problem symmetry, the direct integration of (25) provides the velocity field uðx,tÞ ¼ x

Y_ , Y

ð27Þ

and, from (26) and (27), we obtain " ! ! ! ! Y_ Y_ Y_ Y_ @ Y_ @p @ 1 x ¼r x x þ 2 x þ Y Y Y Y @x Y @x @t Y or

x

Y_ Y

!  #  Y_    x f ,  Y

2 !   3 € _ 2 Y_ Y_  Y Y @p 2 ¼ r4x 2x þx f 5, Y Y @x Y3

ð28Þ

ð29Þ

where the last term was modified accounting that Y(t) is always positive, and also that we will perform integrations only in the positive half-space 0 r x rL=2. Integrating (29) we obtain the pressure field 2 3 ! € _ 2 1 Y_ 9Y_ 9 Y Y 1 pðx,tÞ ¼ r4 x2 x2 ð30Þ þ x3 3 f 5 þCðtÞ, Y 2 Y 3 Y and the integration constant C(t) is computed by applying the boundary condition (20) to 2 3 !2     L L Y_ L L2 Y€ L2 Y_ L3 Y_ 9Y_ 9 5 4 u ; p  ,t ¼  ,t ¼ r þ f þ CðtÞ, 2 2Y 2 8Y 4 Y 24 Y 3

ð31Þ

whence L2 Y€ L2 CðtÞ ¼ P ext r þr 8Y 8

Y_ Y

!2 r

L3 Y_ 9Y_ 9 L2 Y_ 9Y_ 9 f r K in,out ½ReðL=2,tÞ, 24 Y 3 8 Y2

so that the pressure field reads "  # "   # "  # !2 Y_ 9Y_ 9 1 L 2 2 Y€ 1 L 2 2 Y_ 1 L 3 3 Y_ 9Y_ 9 1 þr x x  r x f  rL2 2 K in,out ½ReðL=2,tÞ: pðx,tÞ ¼ Pext  r 3 2 2 2 2 3 2 8 Y Y Y Y

ð32Þ

ð33Þ

Note that, as discussed before, in the previous integration the friction coefficient f was assumed constant in space, which is not actually true. However, experimental evidence show that for turbulent flows the exponent m in (16) and (17) is quite small, so that the change of the friction coefficient with flow velocity becomes second order. Hence, for turbulent flows, the preceding simplification is well justified. However, if the flow velocity is low enough, laminar flows will arise

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

7

and the convenient simplification adopted here will lead to some error. Notice that such is the case for the geometry of Fig. 2, as symmetry imposes that near the middle of the plate u(0, t) ¼0, therefore a (smaller or larger) region of the channel will always display laminar flow, whatever the values of Y(t) and Y_ ðtÞ which may well lead to turbulent flows in the outer regions of the channel. From (12) and (33), we obtain the following explicit form for the total dynamic force of the fluid on the plates, per unit width: F f ðtÞ ¼ P ext L

rL3 Y€ 12 Y

þ

rL3 Y_

2

24 Y 2



rL4 Y_ 9Y_ 9 32 Y 3

f

rL3 Y_ 9Y_ 9 8

Y2

K in,out ½ReðL=2,tÞ,

ð34Þ

where one can recognize, beyond the trivial constant force term stemming from Pext, four vibration-induced nonlinear dynamical terms which are related to: (a) the local fluid inertia, (b) the convective inertia, (c) the distributed wall interface stresses, and (d) the singular losses at the boundaries. Notice that the magnitude of all these force terms increase dramatically as the fluid gap closes, because of the various powers of Y(t) in the denominators of (34)—hence the squeezefilm effect. Most terms in solution (34) are analogous to those published by Esmonde et al. (1990a). However, our quadratic dissipative term related to flow/wall stresses is different from theirs, due to the distinct assumptions involved. Both coefficients Kin and Kout can be used when performing a numerical simulation, the first one during aspiration ðY_ ðtÞ 4 0Þ and the other during ejection ðY_ ðtÞ o0Þ. 3.2. General solution for mixed laminar/turbulent flows Due to the geometry of system, (23) may be written

t1,2 ¼ C L rY mL u9u9mL þ 1 þC T rY mT u9u9mT þ 1 , 8uðx,tÞ:

ð35Þ

Then, replacing the friction stress formulation (35) into (11), the momentum equation to be integrated becomes " # @p @u Y_ m þ1 m þ1 ¼ r u_ þ u þ 2u þ 2C L Y mL 1 u9u9 L þ 2C T Y mT 1 u9u9 T , ð36Þ @x @x Y with CL and CV are given by (24) and the velocity field is given by (27). We thus obtain 2 3 ! ! ! ! !  !  _ _ _ _ @ _ _  Y_ mL þ 1 _  Y_ mT þ 1 Y Y Y Y Y Y Y @p @ m 1 m 1 L T 5, x ¼ r4 x x x þ 2C T Y x þ 2 x þ 2C L Y x  x  Y Y Y Y @x Y Y  Y Y  Y @x @t or

2 3 ! _ 9Y_ 9mL þ 1 _ 9Y_ 9mT þ 1 € _ 2 Y Y @p Y Y m þ 2 m þ 2 5, ¼ r4x 2x þ 2C L x L þ 2C T x T @x Y Y Y3 Y3

ð37Þ

ð38Þ

where, again, the last two terms were modified accounting that Y(t) is always positive, and also that we will perform integrations only in the positive range 0 r x rL=2. Integrating (38) we obtain the pressure field 2 # ! _ _ mL þ 1 _ _ mT þ 1 _ 2 1 2 Y€ 2 2 2 Y mL þ 3 Y 9Y 9 mT þ 3 Y 9Y 9 4 pðx,tÞ ¼ r x ð39Þ CLx CT x x þ þ þ CðtÞ, 2 Y mL þ 3 mT þ3 Y Y3 Y3 and the integration constant C(t) is computed by applying the boundary condition (20) to 8   L L Y_ > > , > ,t ¼  u > > Y 2 2 < 2 3 !   m þ1 m þ1 2 € 2 _ 2 Y_ 9Y_ 9 L Y_ 9Y_ 9 T 5 > LmL þ 3 C L LmT þ 3 C T > p L ,t ¼ r4L Y  L Y > þ CðtÞ, þ m þ2 þ m þ2 > > 8Y 4 Y : 2 Y3 Y3 2 L ðmL þ 3Þ 2 T ðmT þ3Þ

ð40Þ

whence CðtÞ ¼ P ext r

L2 Y€ L2 þr 8Y 8

Y_ Y

!2 r

LmL þ 3 C L mL þ 2

2

ðmL þ 3Þ

m þ1 Y_ 9Y_ 9 L

Y

3

,r

LmT þ 3 C T mT þ 2

2

ðmT þ 3Þ

m þ1 Y_ 9Y_ 9 T

Y

3

r

L2 Y_ 9Y_ 9 K in,out ½ReðL=2,tÞ, 8 Y2 ð41Þ

and the pressure field becomes !2 "  # "   # 1 L 2 2 Y€ 1 L 2 2 Y_ þr x x pðx,tÞ ¼ Pext  r Y Y 2 2 2 2

8

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

r

2C L mL þ 3

"  # "  # m þ1 m þ1 L mL þ 3 mL þ 3 Y_ 9Y_ 9 L 2C T L mT þ 3 mT þ 3 Y_ 9Y_ 9 T x  r x 3 2 2 mT þ 3 Y Y3

Y_ 9Y_ 9 1  rL2 2 K in,out ½ReðL=2,tÞ: 8 Y

ð42Þ

Finally, from (12) and (42), we obtain an explicit form for the total dynamic force of the fluid on the plates, per unit width F f ðtÞ ¼ P ext L

rL3 Y€ 12 Y

þ

rL3 Y_ 24 Y

2

 2

m þ1 m þ1 Y_ 9Y_ 9 2mL 2 C L rLmL þ 4 Y_ 9Y_ 9 L 2mT 2 C T rLmT þ 4 Y_ 9Y_ 9 T 1   rL3 2 K in,out ½ReðL=2,tÞ, 3 3 8 mL þ 4 m þ 4 T Y Y Y

ð43Þ where we recall the various parameters from the friction formulation (using Hirs correlation for the turbulent region) CL ¼

2mL 1 nL

nmL

,

CT ¼

2mT 1 nT

nmT

,

nL ¼ 12,

mL ¼ 1,

nT ¼ 0:055,

mT ¼ 0:25:

ð44Þ

In (44), the singular loss coefficient K in,out ½ReðL=2,tÞ can be computed at each time-step, according to the system geometry, the fact that the flow is being ejected or sucked from the channel, as well as of the actual Reynolds number   8   _       < Y_ ðtÞ 40-Rein L ,t ¼ LY_ 2Y 2L YY  2h 2L ,t u 2L ,t  L9Y_ 9 L n 2 ,t ¼ ¼ ¼ ) ð45Þ Re  : Y_ ðtÞ o0-Reout L ,t ¼  LY_ : 2 n n n 2 n Then, K In,Out ½ReðL=2,tÞ is computed according to the experimental data discussed in Section 2.3. 3.3. Fluid/structure coupled model For a simple structure with a single degree of freedom, we have M s Y€ ðtÞ þ C s Y_ ðtÞ þK s ½YðtÞH ¼ W F f ðYðtÞ, Y_ ðtÞ, Y€ ðtÞÞ þ F e ðtÞ,

ð46Þ

with the fluid force given by either (34) or (43) and Fe(t) is any given externally applied force. On the other hand, H is the reference value of the fluid gap, for the non-excited system. In state-space form, Eq. (46) is equivalent to a system of two first-order equations X_ 1 ¼ X 2 , 1 fC s X 2 K s ½X 1 H þ W F f ðX 1 ,X 2 , X_ 2 Þ þ F e ðtÞg, X_ 2 ¼ Ms

ð47Þ

where X1 ¼ Y and X 2 ¼ Y_ . One can notice that the inertial term X_ 2 appears in both sides of the second equation (47), which must be modified. Accounting that both (34) and (43) may be written in the form X_ 2 rL3 X_ 2 þGf ðX 1 ,X 2 Þ ¼ Mf þ Gf ðX 1 ,X 2 Þ, F f ðX 1 ,X 2 , X_ 2 Þ ¼  12 X 1 X1

ð48Þ

then the second equation (47) reads

 X1 C s X 2 K s ½X 1 H þ W Gf ðX 1 ,X 2 Þ þF e ðtÞ , X_ 2 ¼ M s X 1 þWM f

ð49Þ

For instance, for the simpler flow force (34), Eqs. (47) and (48) become X_ 1 ¼ X 2 , X_ 2 ¼

(

X1 3

L M s X 1 þW r12

"

C s X 2 K s ½X 1 H þW Pext Lþ

rL3 X 22

rL4 X 2 9X 2 9

24 X 1

32

, 2

X 31

f

rL3 X 2 9X 2 9 8

X 21

# )

K in,out ReðL=2,tÞ þ F e ðtÞ ,

ð50Þ

Then, the first order differential system (50) – or the alternative formulation obtained by inserting the general fluid force (43) into (47) – is time-step integrated using any adequate algorithm. We used an explicit Runge–Kutta method, with variable time-step controlled by an estimate of the local integration error. 4. Experimental validation 4.1. Test rig The experimental set-up, sketched and pictured in Fig. 4, consists on an immersed rectangular plate with dimensions L  W¼55 mm  214 mm, which is dropped from a height Y0 ¼15 mm. The plate is free to move along the vertical direction, when subjected to gravity, so that Ks ¼0, the total mass of the mobile fixture being Ms ¼36.6 kg. However, there is some damping Cs ¼200 N s/m in a plate-guiding fixture. The fluid is water, with volumic mass r ¼1000 kg/m3 and

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

9

Fig. 4. Experimental rig and detail of the rigid instrumented plate.

kinematic viscosity n ¼10  6 m2/s at normal temperature. However, several tests were also performed at other temperatures, in order to change the viscosity (and hence the Reynolds number), as discussed in the following. The fluid force was measured using two force piezo-transducers KISTLER 9117AB (with charge amplifiers KISTLER 5011), mounted symmetrically between the plate and the sustaining fixture, at locations x ¼ 795 mm. The plate motion was sensed using a non-contact displacement transducer KAMAN KD2300-6C, mounted along the vertical direction, providing a linear response in the range Y ¼0–6 mm. Furthermore, two acceleration transducers ENDEVCO 224C (with charge amplifiers B&K 2635) where used, one of them being processed with an analogue integrator, in order to obtain the velocity response of the plate.

10

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Fig. 5. Sample measured forces by the two plate transducers, demonstrating the parallelism of the plate/base assembling.

Fig. 6. Comparison of the experimental plate motion (displacement, velocity and acceleration) and dynamical fluid force with the theoretical solution from the general formulation with Hirs empirical correlation f(Re) encapsulated, and using the singular loss coefficient Kout ¼ 0.85 (drop test at temperature 20 1C).

4.2. Preliminary tests As a preliminary verification of the plate/base parallelism, we show in Fig. 5 a sample response of the left and right force transducers, stemming from a typical drop test. A correction was performed on the measured forces in order to plot only the effects stemming from the flow/structure coupling. This was achieved by subtracting from the measured forces the inertia of the rectangular plate that is located between the two force transducers and the fluid, see Fig. 4. It can be seen that the traces are

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

11

almost indistinguishable, confirming a near-perfect symmetrical distribution of the dynamical fluid pressure, hence a satisfying parallel geometry of the experimental fluid gap. Notice however that, after the large force spike related to the fluid force Ff(t), some other complex phenomenon arises. This secondary feature in the experimental responses is not related to fluid forces, but to a solid contact between the plate ant the base arising at the very end of the drop motion, which leads to a plate rebound. Such behaviour, which was consistently observed, is a direct and unavoidable consequence of any residual lack of planarity of the plate and base surfaces, which ultimately becomes dominant as Y(t)-0. Our theoretical model obviously does not apply to such phenomenon, which we did not try to emulate. 4.3. Experimental results and theoretical predictions The width-to-length ratio W=L ¼ 3:9 of the tested plate is quite significant. Nevertheless, it may not be sufficient to rule-out an experimental ‘‘error’’ connected with the 2D flow component under the plate squeezing, which is not addressed in our theoretical formulations. Even so, the 1D flow models addressed here will be confronted with these experimental results, on the reasonable ground that such 2D effect should be of second order magnitude. In Fig. 6 we compare the experimental results of a plate drop at temperature 20 1C due to gravity acceleration with the theoretical predictions stemming from our general model for mixed laminar/turbulent flow, which was developed in Section 3.2, including an external excitation Fe ¼ Msg, with the gravity acceleration g ¼9.81 m/s2. Here, the theoretical results were obtained using parameters (44), pertaining to Hirs empirical correlation for distributed skin-friction stresses. On the other hand, as mentioned in Section 2.3, our experimental geometry does not conform exactly to any of the geometries for which the inlet/outlet singular loss coefficients are tabulated in reference works—see Idel’Cik (1960) and Blevins (1984). Therefore some incertitude exists concerning the value(s) of Kout which apply to our experiments—note that in these tests the fluid is always ejected (at least before rebounding occurs), so only the outlet coefficient is relevant. After some numerical testing, we tentatively opted for a straightforward engineering approach, which proves effective. Actually, we use in our numerical simulations a constant value for Kout which both fits well our experimental data and is generally consistent with the referenced values. For all tests, the value Kout ¼0.85 fulfilled both conditions. Notice that the experimental displacement signal in Fig. 6 is saturated when Y(t) 46 mm, because the beginning of the motion lays beyond the transducer range. Also notice the complex motion which arises at the end of the motion, because

Fig. 7. Space-time plots of the Reynolds number and of the corresponding friction coefficient, computed from the general formulation with Hirs empirical correlation f(Re), in particular at two different locations along the channel (all parameters as in Fig. 6).

12

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Fig. 8. Comparison of the experimental dynamical fluid force with theoretical solutions from the simplified model with constant friction coefficient, for values f ¼0.01 (upper plot) and f ¼0.1 (lower plot), and using the singular loss coefficient Kout ¼ 0.85 (drop test at temperature 20 1C).

of spurious solid contact, as discussed before. In general, the motion of the dropping plate and the dynamical fluid force are well predicted by the theoretical model, except for the acceleration at near-contact, and obviously also beyond nearcontact, when comparing the theoretical predictions and the experimental results is meaningless. For future comparison, it will prove interesting to show illustrative space-time plots of the Reynolds number (15) and of the corresponding variable friction coefficient, as computed from Hirs formulation (16). These are plotted in Fig. 7, first using full space–time representations, and then also detailed at two specific locations along the channel (x ¼ 0:25L=2 and x ¼ 0:75L=2). The horizontal planes in the upper plots highlight the borders between the space–time laminar and turbulent flow regimes. These plots pertain to the computation results shown in Fig. 6. Notice that, as expected from the velocity field (27), the Reynolds number increases as approaching the plate boundary. Also note that, during most of the time, the friction coefficient approaches fE0.01, although a two-decade increase is experienced at the beginning and near the end of the motion. We now turn to the theoretical predictions from the simplified model (34), with constant friction coefficient. The fluid forces predicted are shown by the plots in Fig. 8, respectively for constant friction values of f¼0.01 and 0.1 (and using, as before, Kout ¼0.85), which from the results of Fig. 7 might be reasonable guesses. Actually, the results obtained when using f¼ 0.01 are quite usable, however those stemming from f¼0.1 clearly underestimate the fluid force. One can thus conclude that the simplified model (34) may well produce good predictions, provided an adequate value of the ‘‘constant’’ friction coefficient is supplied. However, by incorporating the velocity dependence f(Re), our new improved analytical solution eliminates incertitude and guesswork concerning this parameter. At the cost of a modest increase in the model complexity, we believe that the gains from the refined solution far surpass this slight drawback. Fig. 9 shows the time-histories of the Reynolds numbers, corresponding to the computations of Fig. 8. Notice that these Reynolds numbers are quite insensitive to the actual value of the friction coefficient. Actually, both results are also quite similar to the plot connected with the variable friction coefficient, shown in Fig. 7. This suggests that, for this fluid/ structure coupled system, the flow pressure field is more sensitive to friction effects than the velocity field.

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

13

Fig. 9. Time-histories of the Reynolds number, computed from the simplified model with constant friction coefficients f ¼ 0.01 (upper plot) and f¼ 0.1 (lower plot), at two different locations along the channel (all parameters as in Fig. 8).

Fig. 10. Decomposition of the dynamical fluid force into inertia and dissipative terms, from the improved theoretical model with velocity-dependent friction coefficient (all parameters as in Fig. 6). FK: force term from singular outlet losses; Ff: force term from distributed flow friction; FC: force term from convective flow acceleration; FL: force term from local flow acceleration; FTot: total flow force.

4.4. Time-domain decomposition of the fluid forces In Figs. 10 and 11 we have decomposed the total fluid force, to highlight the various inertia (local and convective) and dissipative (distributed and singular losses) terms, as per equation (43). Shown are the force term FK due to singular outlet

14

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Fig. 11. Decomposition of the dynamical fluid force into inertia and dissipative terms, from the simplified model with constant friction coefficients, respectively for f ¼0.01 (upper plot) and f¼ 0.1 (lower plot) (all parameters as in Fig. 8). FK: force term from singular outlet losses; Ff: force term from distributed flow friction; FC: force term from convective flow acceleration; FL: force term from local flow acceleration; FTot: total flow force.

Table 1 Physical data for the tests at several temperatures. Fluid temperature [1C] 2 20 59.5 72

Volumic mass [kg/m2] 1000 998 983 977

Kinematic viscosity [m2/s] 6

1.6  10 1.0  10  6 4.7  10  7 4.0  10  7

Maximum Reynolds (x ¼ 7 L=2) E 7000 E12000 E25000 E 30000

losses, the force term Ff due to distributed flow friction, the force term FC from convective flow acceleration, the force term FL from local flow acceleration, as well as the total flow force FTot. Fig. 10 pertains to the computation results shown in Fig. 6, with velocity-dependent friction coefficient, from the general model (43). The plots in Fig. 11 pertain to the computations shown in Fig. 8, with constant friction coefficient, from the simplified model (34). In Fig. 10 the dissipative terms display similar magnitudes, although at different phases of the motion. Both have the same order of magnitude of the local inertia term, which is for this system the only term with opposite sign. The convective inertia term is the least significant of all. In Fig. 11, the terms of the upper plot (f¼0.01) follow similar trends, although with some quantitative differences. Not unexpectedly, the lower plot (f¼0.1) is dominated by the skin-friction force term.

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

15

Fig. 12. Effect of the fluid temperature: experimental plate motions (displacement, velocity and acceleration) and dynamical fluid forces.

Fig. 13. Effect of the fluid temperature: theoretical plate motions and dynamical fluid forces, computed from the general formulation with Hirs correlation f(Re) and the singular loss coefficient Kout ¼ 0.85.

16

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Fig. 14. Effect of the fluid temperature: decomposition of the dynamical fluid force into inertia and dissipative terms, from the improved theoretical model with velocity-dependent friction coefficient, respectively for T¼2 1C (upper plot) and T¼72 1C (lower plot) (all parameters as in Fig. 13). FK: force term from singular outlet losses; Ff: force term from distributed flow friction; FC: force term from convective flow acceleration; FL: force term from local flow acceleration; FTot: total flow force.

4.5. Influence of the fluid temperature Our last results concern the influence of the fluid viscosity on the plate dynamics. Experiments were performed at four different temperatures, as shown in Table 1. Notice that the corresponding changes in viscosity enable a significant range of Reynolds numbers. Fig. 12 presents the experimental results obtained, the plots of the four tests being shown superposed. The nearinsensitivity of the system dynamics to the fluid temperature is striking. Furthermore, such behaviour is also displayed by the theoretical results shown in Fig. 13, were the plots shown were computed from the improved model with velocitydependence of the friction coefficient, Eq. (43). This interesting behaviour is better understood by looking at the force decompositions for two extreme temperatures, as shown in Fig. 14. Notice that, in spite of the total forces being near-identical, there are significant differences in the various force terms – in particular the loss terms – which somehow self-compensate, leading to the same net result. Notice that the friction force term Ff decreases as temperature increases, while the force FK due to the outlet losses and the inertia forces FL and FC increase with the temperature. A possible explanation for this behavior is that, because the friction coefficient decreases as temperature increases, then the flow velocity also increases, leading to an increase of the outlet singular loss force term FK. Also, the magnitude of the inertia force terms FL and FC will also increase, because of the increased deceleration stemming from the higher approaching velocity of the plate. In other words, if less energy is

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

17

dissipated through the skin-friction term (because of a higher Reynolds number), then such ‘‘excess’’ energy leads to enhancing of other terms, which tend to compensate the difference.

5. Conclusions We have developed in this paper a nonlinear analytical solution for the squeeze-film dynamical forces between parallel plates. Our approach is based on a 1D classical bulk-flow formulation and accommodates any kind of flow conditions along the channel gap, either laminar, turbulent or mixed flows. The extensive time-domain computations presented show that a simpler formulation, using a constant skin-friction coefficient, may be able to produce satisfying predictions, provided a ‘‘good’’ average value of the friction coefficient is used. Nevertheless, because the improved formulation takes into account the space–time variation of f ½Reðx,tÞ, it effectively eliminates all guesswork concerning the friction stresses along the fluid channel. This considerable advantage comes at the modest cost of a slight increase in the complexity of the improved flow solution. Computations performed on an immersed dropping plate show that all inertia and dissipative terms in the fluid force solution can be of significant magnitude. However the maximum amplitude of each term usually occurs at a different stage of the motion. For the specific system addressed, the distributed and singular friction losses lead to fluid force terms of similar magnitude. On the other hand, the least significant force term is related to the convective inertia. Analysis of the Reynolds number along the fluid channel indicates that, as expected, the flow is laminar near the centre of the moving plate but most often quite turbulent near the plate boundary. On the other hand, for the system studied in this paper, large increases in the velocity-dependent friction coefficient are experienced in the beginning as well as at the end of the plate drop motion. All these features stress the practical usability of the new improved flow force solution. In the second part of this paper we presented and discussed the results of an extensive series of experiments performed at CEA/Saclay. These consisted on a long gravity-driven instrumented plate of rectangular shape colliding with a planar surface. Among the experimental results obtained, we present a series of tests performed at various fluid temperatures, leading to a significant range of the flow Reynolds number. Interestingly, we consistently observed in experiments and computations that – although the various fluid force terms display some sensitivity to the actual Reynolds number along the squeeze-film – a kind of self-compensation actually arises between the various force terms, leading to a net fluid force which is nearly the same, irrespectively of the fluid viscosity. In all cases, our theoretical predictions were confronted with the experimental measurements with satisfying agreement. Nevertheless, one should keep in mind that some incertitude exists on the value of the singular loss coefficient which should be applied to our system. Here, we pragmatically overcame this problem by using a constant value for Kout which both fits well our experimental results, while being generally consistent with the published data. Even if such approach produced quite acceptable results, further effort is needed to refine the manner in which this force term should be dealt. In a broader sense, modeling of the flow dissipative terms will certainly be improved when reliable loss coefficients for unsteady flows become available. As a final remark we recall that, for many systems of interest, 2D flow effects cannot be safely neglected. Acknowlegements We gladly acknowledge the valuable contribution of THIERRY VALIN, from CEA/DEN/DM2S/SEMT (Saclay, France), concerning the experimental part of this work. Also, the careful reading and constructive criticisms from the anonymous reviewers, which contributed to a more thorough and clarified revised manuscript, are gratefully acknowledged. References Antunes, J., Axisa, F., Grunenwald, T., 1996. Dynamics of rotors immersed in eccentric annular flows: Part 1—theory. Journal of Fluids and Structures 10, 893–918. Antunes, J., Mendes, J., Moreira, M., Grunenwald, T., 1999. A theoretical model for nonlinear planar motions of rotors under fluid confinement. Journal of Fluids and Structures 13, 3–21. Antunes, J., Piteau, P., 2001. A nonlinear model for squeeze-film dynamics under axial flow. In: Proceedings of the ASME Pressure Vessel and Piping Conference—Symposium on Flow-Induced Vibration, Atlanta, July 2001, USA. Antunes, J., Piteau, P., 2010. A nonlinear analytical model for the squeeze-film dynamics of parallel plates subjected to axial flow. International Journal of Mechanical Sciences 52, 1491–1504. Axisa, F., Antunes, J., 1992. Flexural vibrations of rotors immersed in dense fluids: Part 1—theory. Journal of Fluids and Structures 6, 3–21. Blevins, R., 1984. Fluid Dynamics. Van Nostrand Reinhold Company, New York, USA. Childs, D., 1993. Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis. John Wiley & Sons, New York, USA. Esmonde, H., Fitzpatrick, J., Rice, H., Axisa, F., 1990a. Analysis of nonlinear squeeze-film dynamics: i—physical theory and modeling. In: Proceedings of the ASME Pressure Vessel and Piping Conference, Nashville, June 1990, USA. Esmonde, H., Axisa, F., Fitzpatrick, J., Rice, H., 1990b. Analysis of nonlinear squeeze-film dynamics: ii—experimental measurement and model verification. In: Proceedings of the ASME Pressure Vessel and Piping Conference, June 1990, Nashville, USA. Fritz, R., 1970. The effects of an annular fluid on the vibrations of a long rotor: Part 1—theory. ASME Journal of Basic Engineering 92, 923–929. Grunenwald, T., 1994. Comportement Vibratoire d’Arbres de Machines Tournantes dans un Espace Annulaire de Fluide de Confinement Mode´re´. Doctoral Thesis. Paris University, September 1994. Guo, C.Q., Zhang, C.H., Paı¨doussis, M.P., 2010. Modification of equation of motion of fluid-conveying pipe for laminar and turbulent flow profiles. Journal of Fluids and Structures 26, 793–803.

18

P. Piteau, J. Antunes / Journal of Fluids and Structures 33 (2012) 1–18

Hellum, A.M., Mukherjee, R., Hull, A.J., 2010. Dynamics of pipes conveying fluid with non-uniform turbulent and laminar velocity profiles. Journal of Fluids and Structures 26, 804–813. Hirs, G., 1973. A bulk-flow theory for turbulence in lubricant films. ASME Journal of Lubrication Technology 95, 137–146. Hobson, D.E., 1982. Fluid-elastic instabilities caused by flow in an annulus. In: Proceedings of the Third International Conference on Vibration in Nuclear Plant, Keswick, UK, pp. 440–463 (BNES, London). Idel’Cik, I., 1960. Memento des Pertes de Charge. Editions Eyrolles, Paris, France. CRC Press, Boca Raton, USA (Also published in 1994 as Handbook of Hydraulic Resistance). Inada, F., Hayama, S., 1990a. A study of leakage-flow-induced vibrations: Part 1—fluid dynamic forces and moments acting on the walls of a narrow tapered passage. Journal of Fluids and Structures 4, 395–412. Inada, F., Hayama, S., 1990b. A study of leakage-flow-induced vibrations: Part 2—stability analysis and experiments for two-degree-of-freedom systems combining translational and rotational motions. Journal of Fluids and Structures 4, 413–428. Kaneko, S., Nakamura, T., Inada, F., Kato, M., 2008. Flow Induced Vibrations: Classifications and Lessons from Practical Experiences. Elsevier, Amsterdam, The Netherlands. Moreira, M., Antunes, J., Pina, H., 2000a. An improved linear model for rotors subjected to dissipative annular flows. In: Proceedings of the Seventh International Conference on Flow-Induced Vibration (FIV2000), June 2000, Lucerne, Switzerland. Moreira, M., Antunes, J., Pina, H., 2000b. A theoretical model for nonlinear orbital motions of rotors under fluid confinement. Journal of Fluids and Structures 14, 635–668. Mulcahy, T.M., 1980. Fluid forces on rods vibrating in a finite length annular region. ASME Journal of Applied Mechanics 47, 234–240. Mulcahy, T.M., 1988. One-dimensional leakage-flow vibration instabilities. Journal of Fluids and Structures 2, 383–403. Paı¨doussis, M.P., 1998. Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1. Academic Press, San Diego, USA. Porcher, G., De Langre, E., 1997. A friction-based model for fluidelastic forces induced by axial flow. In: Proceedings of the Fourth International Symposium on Fluid–Structure Interaction, November 1997, Dallas, USA.