20th International Conference on Structural Mechanics in Reactor Technology (SMiRT 20) Espoo, Finland, August 9-14, 2009 SMiRT 20-Division 3, Paper 1898
Fluid-Structure Interaction Calculations Using a Linear Perturbation Method
Antti Timperi
Structural Performance, VTT Technical Research Centre of Finland, Espoo, Finland, e-mail: [email protected]
Keywords: Fluid-structure interaction, CFD, added mass effect, pressure load, condensation pool.
1
ABSTRACT
Two-way coupled Fluid-Structure Interaction (FSI) calculations become numerically unstable in certain cases. If the structural displacements are relatively small, it may be sufficient to use only one-way mapping of fluid pressure to the structure. However, often the fluid still affects frequency of the structural motion considerably through the added mass effect.
This work presents FSI calculations with a linear perturbation method where only one-way coupling is used and the added mass effect is accounted for. Commercial Computational Fluid Dynamics (CFD), structural analysis and coupling codes are used. Modeling of condensation pools of boiling water reactors during injection of gas into the pool water is considered as a specific application.
Basis of the method and its validity for modeling condensation pools are first examined mathematically with an order of magnitude analysis. The method is then validated against numerical data by using a simplified model of a condensation pool. Finally, validation is carried out against scaled-down condensation pool experiments where air was blown into the pool. Results obtained with the method are reasonably close to the experiments whereas in one-way coupled calculations absence of the added mass effect is clearly seen.
2
INTRODUCTION
It is well known that an explicit Fluid-Structure Interaction (FSI) coupling scheme is numerically unstable in some applications. In these cases, stable calculations may be achieved with an implicit scheme where boundary data is exchanged between the fluid and structure several times during iteration inside a time step (Vierendeels et al., 2005; Sigrist and Abouri, 2006). With some commercial codes, however, only the explicit scheme is available. If the structural displacements are relatively small, it may be sufficient to use only one-way mapping of fluid pressure to the structure. However, often the presence of the fluid still affects the frequency of the structural motion considerably through the added mass effect.
During a hypothetical Loss-Of-Coolant Accident (LOCA) in a Boiling Water Reactor (BWR), a large amount of steam and non-condensable gas would be injected into the pool water which causes pressure loads on the pool walls. FSI has significance in certain situations of the blowdown event (Gienke, 1980; Björndahl and Andersson, 1998). The pressure loads have a large range of frequency components and they exert a continuous excitation on the walls during the blowdown (McCauley et al., 1981; Puustinen et al., 2009a). Using only one-way FSI coupling is not necessarily conservative in this kind of situation because the water lowers eigenfrequencies of the walls. Two-way coupled FSI calculations of a scaled-down condensation pool test facility have been numerically unstable with explicit coupling (Kauppinen et al., 2006).
3
PPOOLEX EXPERIMENTS
The PPOOLEX facility shown in Fig. 1 is a scaled-down model of a BWR containment and consists of closed drywell and wetwell compartments (Laine and Puustinen, 2008; Puustinen et al., 2009a,b). The compartments are connected with a blowdown pipe whose outlet is submerged in the pool water. Diameters of the pool and blowdown pipe are about 2.4 m and 0.2 m, respectively.
In the SLR-05-02 experiment, air was blown into the drywell from air tanks. This causes the drywell to pressurize and air is injected through the blowdown pipe into the pool water. In the experiment, water level from the bottom centre was about 2.14 m and the submergence depth of the pipe was about 1.05 m. An experiment with air blowdown was chosen for this work as the condensation of steam in the pool water can not be currently modelled accurately. The test facility and experiment are described in detail in another article of this publication (Puustinen et al., 2009b).
Figure 1. Scaled-down PPOOLEX test facility. (Laine and Puustinen, 2008)
4
DESCRIPTION OF THE METHOD
Schematic of the linear perturbation method is presented in Fig. 2. The pressure load is transferred from the CFD model to the structural model, but no displacement feedback is sent back. The mass of the fluid is accounted for in the structural motion through a separate acoustic fluid which has two-way coupling with the structure. The method eliminates the numerical instability as the coupling between the CFD and structural models is only one-way and a stable monolithic approach is used for the acoustic-structural system (see e.g. Cook et al., 2002). In the method, two different flow fields are solved and superposed: flow field that would occur if the walls were rigid and flow field due to wall motion.
4.1 Order of magnitude analysis
Basis of the linear perturbation method is analyzed in the following mathematically with guide lines taken from Sonin (1980). For the water in the pool, we have the conservation equations of mass and momentum for compressible fluid:
0 ) ( = ! " + # # V $ $
t (1)
g T V V V !
! %=$# +#" +
& ' ( ) * # " + + + p
t ( ) (2)
where T is the viscous stress tensor T=µ("V+("V)T)+#("!V)I
. Here constant viscosity and negligible bulk viscosity are assumed which makes the viscous terms linear. The following linear pressure-density relation is used for the pool water:
2 c p = ! !
" (3)
The variables of the flow solution of the coupled FSI problem are separated into components due to the stagnant hydrostatic initial state (0), flow that would occur if the walls were rigid (1) and flow resulting from the flexure of the walls (2):
) , ( ) , ( 0 ) ,
(x t V1 x t V2 x t
V = + +
) , ( ) , ( ) ( ) ,
( t p0 p1 t p2 t
p x = x + x + x (4)
) , ( ) , ( ) ,
(x t !0 !1 xt !2 xt
! = + +
By definition, V0 is zero and p0 is function of position only. In addition, 0 is assumed constant. Note that in (4) no assumption of linearity has been made, but the variables are merely separated into the above-mentioned components. For the rigid wall flow, since by definition the components 2 are zero, we have
0 )
( 0 1 1 1 1
1 = ! " + " ! + + # # $ $ $ $ V V
t (5)
g T V V V 1 1 1 1 1 1 1
0 ) ( )
(! ! %=$# +#" +!
& ' (
)
* + "#
+ +
+ p
t (6)
where p0 has been eliminated by observing that p0(x) = 0g·x + constant. Substituting (4) into (1) and (2) and subtracting from the resulting equations the rigid wall equations (5) and (6), respectively, we obtain equations for the flow due to wall flexure:
0 )
( 0 1 2 2 2 1 1 2 2 1 2 2
2 = ! " + ! " + ! " + " ! + " ! + + + # # $ $ $ $ $ $ $
$ V V V V V
t (7)
g T V V V V V V V V V V 2 2 2 1 1 1 2 2 2 1 2 2 1 2 2 1 0 ) ( ) ( ) ( ) ( ) ( ! ! ! ! ! + " # + $# = % & ' ( ) * # " + + + + % & ' ( ) * # " + # " + # " + + + + + p t
t (8)
Next, we nondimensionalize (7) and (8) in order to examine the relative magnitudes of their terms. The variables are nondimensionalized as
0 * j j j V V V = 0 * j j j p p p = 0 * j j j ! ! ! = ! ! ! 0
0*= g g g*=
0 *
j t
t
t = !*=L! (9)
L = characteristic length of the pool
w = maximum wall displacement
0 2
t = periodic time of pool wall oscillation
0 1
V = flow velocity in the pool
c = speed of sound in water
! = density of water
µ = viscosity of water
g = acceleration of gravity
Scale of the rigid wall flow pressure is estimated from the Bernoulli equation:
2 0 1 0
1 V
p =! (10)
and time scale of the rigid wall flow is
0 1 0 1 V L
t = (11)
For the flow velocity due to wall flexure it is obvious to choose
0 2 0 2 t w
V = (12)
Pressure scale for the wall flexure flow is estimated as the pressure resulting from an acceleration magnitude 2
0 2 /t
w acting on a water column of length L:
2 0 2 0 2 t wL
p =! (13)
Note that in estimating 0 2
p , the assumption of incompressible water has been used which is, however, a good approximation in the examined case as shown later. Changes in water density may be estimated from the pressure-density relation (3):
2 0 0 c pj j =
! (14)
where j = 1,2. We first nondimensionalize (7) and (8) according to (9) and make the obvious assumptions 0
1 !
!>> and 0 2 !
!>> already at this stage. Then by substituting the estimates of the scales, we obtain the following dimensionless equations:
0 * * * * * * *) * * * * * ( * * * * * 2 2 4 1 1 2 3 2 1 1 2 3 1 2 0 2 1 = !" # # + !" # + !" + ! " # # + ! " + $ $ # % % % % % % V V V V V
t (15)
* * * * * * * *) * ( * * * * *) * ( *) *) * ( * *) * (( * * * 2 6 2 5 2 1 1 1 2 3 2 2 4 1 2 2 1 2 2 0 g T V V V V V V V V V V ! ! ! " + # $ " + %$ = & ' ( ) * + #$ + , , " + & ' ( ) * + #$ " + #$ + #$ " + , , p t
t (16)
2
0 2
1 !!
" # $$ % &
=
' ct
L
L V t20 10 2=
!
2 0 1
3 !!
" # $$ % &
=
' c V
L w
=
!4 2
0 2 5
L t !
µ =
"
2 6
c gL
=
! (17)
Estimated values of the reference constants and the resulting values of the dimensionless parameters for the PPOOLEX facility are listed in Tables 1 and 2.
Table 1. Values of the reference constants for the PPOOLEX facility.
L [m] w [m] 0
2
t [s] V10 [m/s] c [m/s] ! [kg/m 3
] µ [Pas] g [m/s2]
1 0.001 0.1 1 1000 1000 0.001 10
Table 2. Values of the dimensionless parameters (17) for the PPOOLEX facility.
1
! !2 !3 !4 !5 !6
10-4 10-1 10-6 10-3 10-7 10-5
Values of the dimensionless parameters are small compared to unity which indicates that we can to a good approximation simplify (7) and (8) to
0 2
2 =
! "
+
# #
V
$ $
t (18)
2
2 p
t =!"
# #V
$ (19)
These are basic equations for compressible, inviscid fluid with convection terms neglected. Eqs. (3), (18) and (19) can be combined to obtain the wave equation for p2:
0 2 2 2 2
2 2
=
! " # #
p c t
p
(20)
which represents an acoustic fluid. Further simplification could be obtained by dropping also the time derivative from (7), i.e. by assuming the wall flexure flow incompressible. It is found that then p2 could be described with the simple Laplace equation
0 2
2 =
! p (21)
The above approximations decouple the solutions of the flow with rigid walls and flow due to wall flexure as the terms containing the rigid wall solution components 1 are neglected from (7) and (8). It should be then allowed to solve the initial flow problem with rigid walls, transfer the transient wall pressure on the coupled acoustic-structural problem and solve the acoustic-structural problem separately from the initial flow problem. After application of the transient wall pressure from flow solution 1, the motion of the walls drives the flow solution 2 through boundary condition n!#p="$n!u&&. Note, however, that the above analysis is not a complete proof for the validity of the method but describes its main features.
At least the first four dimensionless parameters 1 - 4 have clear physical meanings. For example, 1 is the ratio of system length to wave length and 4 is the ratio of wall displacement to system length. The small value of 1 shows that for the present case p2 could be described by the Laplace equation, but we nonetheless solve it in Abaqus by using the wave equation. That is, we solve also the wave propagation phenomena in the wall flexure problem, but the effect of these phenomena should be small.
It is worth noting that the parameter 2, which multiplies the cross terms (V1"!)V2 and (V2"!)V1, has the highest value in Table 2 and that its value was estimated a little nonconservatively. Velocity 0
near moving walls, even a small velocity wall motion may induce significant pressure variations. In such a case then the method would obviously become invalid. However, it seems that a local high-velocity region away from the moving walls does not invalidate the method as shown in Sec. 5.
4.2 Boundary conditions
It is assumed here that the effect of the wall flexure on the pressure applied on the free surfaces of the fluid, i.e. on the gas pressure on the pool surface and inside the blowdown pipe, is negligible. Therefore, pressure boundary condition p2 = 0 is used on the free surfaces of the wall flexure flow problem (20). If the gas is assumed incompressible, however, more appropriate boundary condition inside the blowdown pipe is that of a wall n"!p=0. It is also assumed that the effect of wall flexure on locations of the free surfaces at different instants of time is negligible. Therefore, the boundary condition should be applied at the current locations of the free surfaces of the rigid wall flow. This introduces additional approximation to the method used here, where the locations of the free surfaces of the acoustic fluid remain at their initial location during the calculation. Error due to this depends on the size of the bubbles forming in the pool and should not be very large due to relatively small bubbles as indicated by calculations in Sec. 5.1. Magnitude of this error could also be examined by modeling bubbles of various sizes at the pipe exit in the acoustic domain and by examining effect of the bubbles on the eigenmodes of the coupled acoustic-structural system.
5
VALIDATION CALCULATIONS
5.1 Comparison with two-way coupling
The method is first compared with a two-way coupled FSI calculation by using a simplified axisymmetric model of the PPOOLEX facility. The model has a rigid side wall and a bottom plate that experiences only vertical rigid-body motion. This kind of system resembles the real facility in that the most dominant mode is the vertical oscillation of the whole pool. Mass of the bottom plate was set to 700 kg which is considerably lighter than structures of the real facility; a light structure was chosen to have a more pronounced added mass effect. The spring stiffness and damping were set to 123 MN/m and 58 kNs/m, respectively. Eigenfrequencies of this system with and without the pool water are about 18 Hz and 67 Hz, respectively.
In the CFD calculation, the Volume Of Fluid (VOF) model was used for tracking the free surface and the k- model was used for modelling turbulence. Air was assumed incompressible and a logarithmic equation of state suitable for compressible liquid was used for water. A short time step of 40 s , determined by stability of the two-way coupled calculation, was used in all calculations. Air velocity at the blowdown pipe inlet was ramped smoothly to a constant value of 30 m/s which is of the same order as in the experiments.
An explicit coupling scheme was used in the two-way FSI calculation. The flow field is solved by Star-CD and the fluid pressure is interpolated on the structure by MpCCI. The structural deformations and stresses are in turn solved by Abaqus and the wall displacement is interpolated on the CFD model. The surface nodes of the CFD model are moved accordingly and the internal mesh is smoothed to preserve the mesh quality. In the present simulation, the data exchange was performed once in each time step.
t = 0 s 0.07 s 0.1 s 0.2 s 0.3 s
Figure 3. Volume fraction of water in a calculation with axisymmetric model.
-7.0E+04 -5.0E+04 -3.0E+04 -1.0E+04 1.0E+04 3.0E+04 5.0E+04 7.0E+04
0.0 0.1 0.2 0.3
Time [s]
P
re
s
s
u
re
[
P
a
]
p20_ 1way ps p20p s+p3 89 p20p s_fsi
-1.8E-03 -1.2E-03 -6.0E-04 0.0E+00 6.0E-04 1.2E-03 1.8E-03 2.4E-03
0.0 0.1 0.2 0.3
Time [s]
D
is
p
la
c
e
m
e
n
t
[m
]
LPM Two-way One-way
Figure 4. Wall pressure below pipe and wall displacement in calculations with axisymmetric model.
5.2 Comparison with PPOOLEX experiment
Numerical models used for the calculations are presented in Fig. 2. The CFD mesh had about 135 000 hexahedral cells. For the structure, a fairly detailed finite-element model consisting mainly of about 15 000 4-noded shell elements was used. Flexibility of disc springs and base structures under the four vertical support columns were modeled with linear springs. The VOF and k- models were used in the CFD calculation. Air was treated as ideal gas and a logarithmic equation of state was used for water. Mass flow of air at the drywell inlet was set to a constant value of 805 g/s.
obtained with one-way coupling may be caused by the higher eigenfrequency of the empty pool. In addition, damping of the pool motion is faster when the mass of water is neglected.
t = 1.53 s 1.63 s 1.74 s
1.84 s 1.94 s 2.04 s
Figure 5. Formation of air bubble at the vent outlet in the PPOOLEX experiment and in the calculation.
1.20E+05 1.25E+05 1.30E+05 1.35E+05 1.40E+05
1.5 1.7 1.9 2.1 2.3 2.5
Time [s]
P
re
s
s
u
re
[
P
a
]
LPM One-way Experiment
-4.0E-04 -2.0E-04 0.0E+00 2.0E-04 4.0E-04
1.5 1.7 1.9 2.1 2.3 2.5
Time [s]
D
is
p
la
c
e
m
e
n
t
[m
]
LPM One-way Experiment
Figure 6. Wall pressure below pipe and wall displacement in the PPOOLEX experiment and in the calculations.
6
CONCLUSIONS
FSI calculations have been presented by using a linear perturbation method which circumvents the numerical instability present with explicit two-way coupling. The commercial Star-CD CFD and Abaqus structural analysis codes were used. The MpCCI middleware was used for coupling the CFD and structural analysis tools. Basis and limitations of the method were examined mathematically and validation calculations for modeling blowdown of air into a scaled-down BWR condensation pool were performed.
The method assumes the flow field due to wall motion as a linearized perturbation on the rigid wall flow. The most important restriction is that the structural displacements have to be sufficiently small. The method may also become inaccurate in some cases if the flow velocity is very large near moving walls. In the method, the complex three-dimensional added mass effects of fluid are accounted for through a separate (acoustic) fluid.
A reasonable agreement in wall pressure and displacement was found between the experiment and LPM calculation. Calculation with one-way coupling showed qualitatively incorrect results for the wall pressure. In addition, the structural displacements were smaller compared to the experiment and to those obtained with LPM.
The method should be in general suitable also for modeling real condensation pools because of the expectedly small structural displacements. In other words, it is expected that the CFD geometry need not to be updated during simulation and that FSI occurs practically only through the added mass effects which lower the pool eigenfrequencies. Because of the added mass effects, using only one-way FSI coupling is not necessarily conservative in condensation pool simulations. Furthermore, the wall motion induces considerable pressure fluctuations in the pool water which are not present with one-way coupling. These pressure fluctuations have significance for example in determining the so-called pressure source at the blowdown pipe outlet from the experimental data.
Acknowledgements. This work is part of the SAFIR2010 programme (The Finnish Research Programme on
Nuclear Power Plant Safety 2007 - 2010).Experimental data from the members of the PPOOLEX project at Lappeenranta University of Technology is also gratefully acknowledged.
Symbols
c speed of sound m/s
g acceleration of gravity m/s2
g acceleration of gravity vector m/s2
I identity tensor -
L length m
n unit normal vector -
p pressure Pa
t time s
T viscous stress tensor Pa
u displacement vector m
V velocity m/s
V velocity vector m/s
w wall displacement m
x position vector m
molecular viscosity Pas
bulk viscosity Pas
density kg/m3
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