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Measurements of fluid fluctuations around an oscillating nuclear fuel assembly
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Measurements of fluid fluctuations around an oscillating nuclear fuel assembly Guillaume Ricciardi n, Eric Boccaccio CEA CADARACHE DEN/DTN/STCP/LHC, 13108 Saint-Paul-Lez-Durance Cedex, France
a r t i c l e i n f o
abstract
Article history: Received 7 February 2013 Accepted 30 March 2014
In this paper, dynamic measurements of fluid velocity in the by-passes of a test-section representing a nuclear fuel assembly are presented. The test-section was designed to identify stiffness, damping and mass coefficients of a fuel assembly under axial flow, and previous studies have shown that the by-passes have an influence on the identified coefficients. The results presented in this paper show that the motion of the fuel assembly induces fluctuations in the axial fluid velocity in the by-passes. These fluctuations depend on the excitation frequency and position. A delay has been observed between the fuel assembly displacement and the fluid velocity fluctuations. The delay decreases when the axial velocity increases which means that it is a convection driven phenomenon. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Fuel assembly Dynamic flow measurements Fluid–structure interaction
1. Introduction Earthquakes can irreversibly damage nuclear power plants especially in the core, where the nuclear fuel assemblies containing enriched uranium have to be particularly resistant. Before building a nuclear power plant, it is necessary to make sure that the core will resist the worst possible earthquake conditions liable to occur at the reactor site. Therefore, when Pressurized Water Reactors (PWR) are subjected to seismic loading, the fuel assembly spacer grids strike each other, and safety measures are necessary to ensure the dropping of control rods and also the cooling of the reactor core. A way to ensure these two criteria is to prevent the spacer grids from buckling. Engineers need special tools for designing and maintaining reactor cores. The reactor core made of fuel assemblies is subjected to an axial water flow to cool the reactor. The flow strongly modifies the dynamical behavior of the fuel assemblies (Collard et al., 2004), therefore the identification of the fluid forces is important in order to provide a relevant model of the fuel assemblies behavior. The first approximation of the fluid forces is to consider them as added mass and damping (Rigaudeau, 1997; Viallet et al., 2003). A more complex expression of these fluid forces is given by Païdoussis (2003) in which the velocity and the relative direction of the flow with respect to the fuel assembly are accounted for. Ricciardi et al. (2009a,b) proposed a porous media approach based on the Païdoussis theory. All these models involve parameters that need to be identified by experiments or numerical simulations. However, reliable numerical predictions are difficult to obtain for such a complex fluid–structure problem, thus experimental results are needed. The LHC (Laboratory of Core and Circuit Hydrodynamics) has facilities to perform dynamic tests on a fuel assembly under axial flow. In a previous study (Ricciardi and Boccaccio, 2012), tests dedicated to the identification of the fluid forces acting on a full scale fuel assembly were performed. These tests highlighted an added stiffness effect under axial flow, and that the
n
Corresponding author. E-mail addresses: [email protected] (G. Ricciardi), [email protected] (E. Boccaccio).
http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016 0889-9746/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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added mass effect seemed to depend on the axial velocity. Identified coefficients of stiffness, damping and mass showed a strong dependence on the lateral by-passes. These by-passes are necessary to allow the displacement of the fuel assemblies, but they are not representative of the distance between two fuel assemblies in a core. Therefore it is needed to understand the effects of the by-passes on the identified coefficient, and thus on the fluid forces. In this paper, measurements of fluid velocity in the by-passes during static and dynamic tests are presented and discussed.
2. Experimental apparatus HERMES T is a single phase hydraulic loop that can handle full scale PWR 1300 MW fuel assemblies. The pump can supply 1200 m3/h in axial flow and 400 m3/h in cross-flow, at 35 bar and 170 1C. Therefore, the flow rate is similar to the PWR condition, the lower temperature (PWR operates at 315 1C) allows to provide accurate measurement devices to the test-section. In the present study, only axial flow is considered at 50 1C. The fuel assembly used is made up of 25 guide tubes and 264 fuel rods, each having a height of 4.5 m and linear density of about 200 kg/m. Fuel rods contain uranium pellets and have a diameter of 9.5 mm. The pitch of the fuel bundle is 12.5 mm. The fuel assembly is clamped to the test-section at the top and the bottom. The test-section is about 40 mm larger than the fuel assembly in the excitation direction and 10 mm larger in the orthogonal direction. Grids of the fuel assembly are around 200 mm wide. The displacement of the fifth grid is imposed with a hydraulic jack (Fig. 1). A plexiglass window allows making optical fluid measurements. The displacements of the grids number 2–9 noted dg2, dg3, dg4, dg5, dg6, dg7, dg8 and dg9 are measured with LVDT sensors. The movable portion of each sensor is a stainless steel rod with a diameter of 2.5 mm and is placed across by-pass 2. Static and dynamic tests are carried out. For static tests, the axial fluid velocity is measured at three altitudes in the fuel assembly and in the by-passes (Fig. 2), namely, between grids 2 and 3, between grids 4 and 5 and between grids 8 and 9. For each altitude, measurements are made every 2 mm along a line in each by-pass, and 6 lines in the fuel assembly, homogeneously distributed. Static tests are performed with the fuel assembly at rest, and with a 10 mm imposed displacement of the fifth grid toward by-pass 1. For dynamic tests, a swept sine ranging from 0 to 3 Hz with an amplitude of 6 mm is imposed. Tests are performed under axial flow for three axial velocities (1.5 m/s, 3 m/s and 5 m/s). The axial component of fluid velocity is measured at three altitudes in the by-pass 2 (Fig. 1), between grids 2 and 3, between grids 4 and 5 and between grids 8 and 9, at two depth from the window. Also, measurements are performed at five altitudes at the fourth grid level (Fig. 3). In spite of the disturbance of the LVDT sensors, measurements are made in the by-pass 2 because the fastener device that link the hydraulic jack to the grid is more perturbing the flow in the by-pass 1 than the LVDT sensors in the by-pass 2. The LDV device (TSI TRx60 probe) allows to measure the fluid velocity at only one point, so the same grid excitation is repeated for each measurement location. LDV device measures the velocity of one particle when it meets the measurement volume which is about 1 mm3, thus it is a random process and data are not regularly sampled. The sample
Fig. 1. Experimental apparatus.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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fuel rods grid LDV line measurement
2 mm window
Fig. 2. Indicative diagram of measurement locations for static test in the fuel assembly.
G5
G4
H5 H4 LDV H3 measurement H2 locations H1
G3 Fig. 3. Measurement locations of the fourth grid for dynamic tests.
Fig. 4. LDV measurements in the fuel assembly and the by-passes between the grids 8 and 9 at 1.5 m/s axial velocity.
rate is about 2.5 kHz. To perform spectral analysis, data are resampled by an interpolation method for 2 kHz. For static tests, the mean velocity is calculated from 1024 samples. For confidential reason, all the velocity values are given dimensionless and normalized by the same quantity. 3. Static tests For static tests, the fuel assembly is still and the displacement of the fifth grid is imposed. LDV measurements are made for two values of displacement: 0 mm, the fuel assembly is in the middle of the test-section, and 10 mm toward by-pass 1. The fifth grid is located approximately at the middle of the fuel assembly, thus when a displacement of 10 mm is imposed, the shape of the fuel assembly looks like the one of a clamped–clamped Euler–Bernoulli beam with a located force (Fig. 5). Fig. 4 shows an example of fluid velocity profile measured in the cross-section of the test-section. One can observe that the velocity is homogeneous in the fuel assembly, and that the fluid velocity is higher in the by-passes. This is a consequence of the difference of pressure loss coefficient between the fuel assembly and the by-passes. The hydraulic diameter in the fuel assembly is significantly smaller than in the by-passes, so it is easier for the fluid to flow in the by-passes than in the fuel assembly. The velocity in the by-passes seems to increase linearly with the axial velocity (Fig. 6). Fig. 7 shows the velocity profile in the by-passes with the fuel assembly deformed (displacement of 10 mm) and not deformed (displacement of 0 mm). One can observe that the deformation induces a difference of velocity between the by-passes. Without deformation the test-section is symmetric, the by-passes have the same size, and the velocities are the same. When a deformation is imposed to the fuel assembly, the by-passes have no longer the same size and the velocities Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 5. Shape of the fuel assembly for an imposed displacement of 0 mm (not def) and 10 mm (def) on the fifth grid under an axial velocity of 1.5 m/s.
Fig. 6. Mean of fluid velocity along the measurement line in the by-pass 1 for static tests with the fuel assembly not deformed.
are changing. The size of the by-pass 2 increases and so does the velocity, whereas in the by-pass 1 the velocity decreases since it is smaller. This difference of velocity will induce fluid forces. The fluid velocity in the fuel assembly does not seem to be significantly modified by the fuel assembly deformation. One can observe oscillations of the fluid velocity along the line measurement, these are due to a specific design at the bound of the grids. Mixing vanes are located between rods on every alternate location, and these locations are out of phase between the two sides of the grid which explain the different locations of minima and maxima velocity observed between the by-passes without deformation. The difference of fluid velocity between the by-passes increases with the axial velocity and the altitude (Fig. 8). The fluid velocity between grids 4 and 5, and between grids 8 and 9 are quite similar, whereas the difference of velocity is much higher between grids 8 and 9 than between grids 4 and 5. Moreover, between grids 2 and 3 no significant difference of velocity is observed. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 7. Fluid velocity profile in the fuel assembly and in the by-passes between the grids 8 and 9 at 5 m/s axial velocity with the fuel assembly deformed and not deformed.
Fig. 8. Difference between the mean of fluid velocity along the measurement line in the by-pass 1 and by-pass 2 for static tests with the fuel assembly deformed.
Fig. 9. PSD of the fifth grid displacement imposed by the hydraulic jack for dynamic tests.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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4. Dynamic behavior of the structure For dynamic tests, the displacement of the fifth grid is imposed. The frequency range 0–3 Hz is swept (Fig. 9) for a peak amplitude of 6 mm. The aim is to mainly excite the first natural mode. This can be verified by a Proper Orthogonal Decomposition (POD) analysis. POD is a mathematical tool to identify the modes that contain the most energy. It applies to numerical and experimental results. Let us consider the spatial field at a time t WðtÞ ¼ ðwðx1 ; tÞ; …; wðxNn ; tÞÞ where Nn is the number of unknowns or discrete ~ sensors. We subtract from W its average value to obtain its fluctuations W: Ns ~ ðt i Þ ¼ Wðt i Þ 1 ∑ Wðt k Þ; W Ns k ¼ 1
ð1Þ
where N s is the number of time steps. ~ That is to find an orthogonal basis, POD aims to get the most characteristic fields Φ in terms of energy of the dataset W. maximizing the time average of the inner product (Liang et al., 2002): ~ ΦÞj〉; Maximize〈jðW;
with J Φ J ¼ 1;
ð2Þ
where 〈:〉 is the time average, (.,.) is the vector product, j:j is the absolute value and J : J is a L2 norm. We can then write the field as 1
~ WðtÞ ¼ ∑ qi ðtÞΦi ; i¼1
where Φk are proper orthogonal modes (POM), and qk(t) is the temporal evolution of the ith POM. Consider the result matrix X, such that h i ~ 1 ÞT ; …; Wðt ~ N ÞT : X ¼ Wðt s
ð3Þ
ð4Þ
Kerschen et al. (2005) shows that the problem (2) is equivalent to factorizing the matrix X using the singular value decomposition (SVD): X ¼ USVT ;
ð5Þ
where U is an orthogonal matrix (N n N n ) columns of which are the POM Φi, S is a pseudo-diagonal matrix (N n Ns ), elements of which are the singular values si and V is an orthogonal matrix (Ns N s ). The amount of energy captured by each mode Φi is given by ei ¼
s2i : Nn ∑k ¼ 1 s2k
ð6Þ
The qi(t) are given by the rows of the matrix R ¼ SVT : qi ðt j Þ ¼ Rij :
ð7Þ
Fig. 10. First three normalized POM shape obtained by the POD analysis for various axial velocities.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Table 1 Percentage of energy captured by the POM. Num
1.5 m/s
3 m/s
5 m/s
POM 1 POM 2 POM 3
92.7 5.9 1.2
91.5 7.0 1.4
87.6 9.9 2.0
Fig. 11. PSD of the fluid velocity in the by-pass 2 between grids 4 and 5 for various axial velocities.
For more information on the POD method, Graham and Kevrekidis (1996), Païdoussis et al. (2005), Bellizzi and Sampaio (2006), and Mastroddi et al. (2012) could be consulted. In the present study, Wðt i Þ ¼ ðdg2 ðt i Þ; dg3 ðt i Þ; dg4 ðt i Þ; dg5 ðt i Þ; dg6 ðt i Þ; dg7 ðt i Þ; dg8 ðt i Þ; dg9 ðt i ÞÞ. Fig. 10 shows the first three POM obtained on the dynamic test for the three axial velocities. POM are normalized so they are dimensionless, the dimension of the data is contained in the temporal evolution qi. One can observe that the first POM seems to correspond to the real part of the first hydroelastic mode. One can also observe that this POM is modified as the axial velocity increases. Table 1 shows the percentage of energy captured by the POM, it can be observed that a great part of the energy is captured by the first POM , although this percentage seems to decrease as the axial velocity increases. This could be due to the increase of damping that tends to broaden the resonance peaks and thus to overlap them. 5. Velocity fluctuations The flow in the by-passes is highly turbulent, considering the width of the by-pass as characteristic length, the Reynolds number is 290 000, 180 000 and 100 000 respectively for axial velocities of 5, 3 and 1.5 m/s. Fig. 11 shows the Power Spectral Density of the fluid velocity in the by-pass 2 between grids 4 and 5 for the three axial velocities. The low frequency part between 0 and 3 Hz shows the effect of the fuel assembly displacement on the fluid velocity fluctuation, the high frequency part is due to the turbulent flow. A classical broad band turbulent behavior increasing with the axial velocity can be observed. One can also observe that the cutoff frequency over which the turbulent energy is attenuated increases with the axial velocity which is a classical result (Lesieur, 1993). Some spectra at low axial velocity show peaks beyond 3 Hz (Fig. 12), which was unexpected. This can be explained by the observation of the imposed displacement in log scale. It shows that the displacement has some spectral component beyond 3 Hz. Although this component is very small, the coherence shows that the peaks are related to the structure displacement. However this shows that fluid velocity fluctuations highly depend on the structure velocity at frequencies beyond 3 Hz for low axial velocities. 6. By-pass flow Fig. 13 shows the mean value of the fluid velocity in the by-pass. As in the static case, the velocity increases linearly with the axial velocity, but in the dynamic case the mean velocity between grids 8 and 9 is slightly smaller than between grids 4 and 5. Moreover, the velocity is more important at the grid 4 level. This is explained by the restriction of the cross section at grids level. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 12. PSD of the fluid velocity in the by-pass 2 at altitude H5 of grid 4 (top), PSD of the fifth grid displacement imposed by the hydraulic jack (centre), and coherence between the fluid velocity and the displacement imposed (bottom).
Fig. 13. Mean fluid velocity in the by-pass 2 for the dynamic tests versus the axial velocity (left) and the position along the fuel assembly (right).
Fig. 14 shows the PSD of the fluid velocity in the by-pass. The fluctuations increase with the axial velocity at every level. The relative values show that fluctuations seem to be proportional to the mean value. Fluctuations induced by the fuel assembly displacement between grids 2 and 3 are barely distinguishable from the turbulent fluctuations. Fluctuations between grids 4 and 5 show a flat profile, they do not seem to depend on the excitation frequency and thus on the structure velocity. One could extrapolate and conclude that the velocity fluctuations only depend on the structure displacement, i.e. on the instantaneous size of the by-passes, however, this statement is contradicted by the measurements made between grids 8 and 9 where the pattern seems to depend on the axial velocity, and fluctuations can increase with the excitation frequency. Fluid velocity fluctuations seem to be homogeneous at the fourth grid level except at H1 level which corresponds to the inlet (Fig. 15). One can observe a slight increase downstream. At each level, fluctuations increase with the excitation frequency, and the relative velocity is more important at low axial velocity (Fig. 16). Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 14. PSD of fluid velocity in the by-pass 2 for various axial velocities, between the grids 2 and 3 (top), between the grids 4 and 5 (centre), between the grids 8 and 9 (bottom), absolute (left) and relative values (right).
Fig. 17 compares fluid velocity fluctuations at a fixed axial velocity. One can observe that fluctuations between grids 2 and 3 and at the fourth level grid are small compared to the other locations between grids 4 and 5 and between grids 8 and 9. Fluctuations between grids 4 and 5 and between grids 8 and 9 are very close at 3 m/s axial velocity, but in the 5 m/s axial velocity case, fluctuations are more important between grids 8 and 9. The coherence γ between two signals x(t) and y(t) can be calculated from the formula (Ewins, 1995): γ2 ¼
CSDðx; yÞ2 ; PSDðxÞPSDðyÞ
ð8Þ
where CSD stands for Cross Spectral Density. Fig. 18 shows the coherences between the hydraulic jack displacement and the fluid velocity. Between grids 2 and 3 the coherence reaches 0.9 and increases with the axial velocity. Between grids 4 and 5 the coherence is very close to 1 for the three axial velocities tested. Between grids 8 and 9, the coherence is also very close to 1 for 3 and 5 m/s axial velocities, but it only reaches 0.9 for 1.5 m/s around 1.5 Hz. At the fourth grid location the coherence is weaker and barely reaches 0.95. Fig. 19 shows the PSD of the fluid velocity between grids 4 and 5 and between grids 8 and 9 for two distances from the window. One can observe that qualitative results are not modified by the depth but values are quite different. This is a Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 15. PSD of fluid velocity in the by-pass 2 for various altitude of grid 4, at 5 m/s axial velocity.
Fig. 16. PSD of fluid velocity in the by-pass 2 for various axial velocities, at altitude H2 of grid 4 (top), at altitude H3 of grid 4 (bottom), absolute (left) and relative values (right).
consequence of the space variation of the mean velocity in the by-passes (Fig. 7) since it has been observed that the fluctuations depend on the axial velocity. 7. Flow delay The phase between the hydraulic jack displacement and the fluid velocity (Fig. 20) increases with the excitation frequency. One could make the approximation that the phase φ linearly increases with the excitation frequency f: φ ¼ cp f ;
ð9Þ
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 17. PSD of fluid velocity in the by-pass 2 for various position, at 3 m/s axial velocity (top), and 5 m/s axial velocity (bottom), absolute (left) and relative values (right).
Fig. 18. Coherence between the imposed displacement and the fluid velocity in the by-pass 2 for various axial velocities, between grids 2 and 3 (top left), between grids 4 and 5 (top right), between grids 8 and 9 (bottom left), and at altitude H3 of grid 4 (bottom right).
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Fig. 19. PSD of fluid velocity in the by-pass 2 at 5 m/s axial velocity, between grids 4 and 5, between grids 8 and 9, for two depth 78 mm and 84 mm, absolute (left) and relative values (right).
Fig. 20. Phase between the imposed displacement and the fluid velocity in the by-pass 2 for various axial velocities, between grids 2 and 3 (top left), between grids 4 and 5 (top right), between grids 8 and 9 (bottom left), and at altitude H3 of grid 4 (bottom right).
where cp is a proportional coefficient. A linear relation between φ and f would mean that there is a constant delay Δt between the fluid velocity and the structure displacement: Δt ¼
cp : 360
ð10Þ
Δt can be approximated by the least square method (Fig. 21). The correlations coefficients (Fig. 22) are very close to 1, thus the linear approximation seems to be appropriate. The delay is greater between grids 2 and 3 than between grids 4 and 5, and even greater between grids 8 and 9. This may be related to the distance from the fifth grid where the amplitude of the structure reaches its maximum. Nevertheless, the greatest delay is observed at the fourth grid location. Thus it strongly depends on the geometry. The delay decreases as the axial velocity increases, considering the width of the test-section as the characteristic length the theoretical convection time based on the axial velocity is close to experimental values. Thus one can Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 21. Time delay between the imposed displacement and the fluid velocity in the by-pass 2.
Fig. 22. Correlation for the linear interpolation of phase.
conclude that this delay is a convection driven effect and that it is related to cross-flow in the fuel assembly between the by-passes. 8. Theoretical analysis The by-passes dimensions allows us to consider the leakage flow theories. Extensive work has been done in this field (Païdoussis, 2003; Kang et al., 2012; Piteau and Antunes, 2012). Based on the work of Hobson (1982) and Spurr and Hobson (1984) Païdoussis proposed equations governing the motion of a cylinder in a narrow annulus which has some similarity with the present study. The cylinder is subjected to axial flow Uðx; tÞ ¼ U þ uðx; tÞ and a cross-flow vðx; tÞ is accounted for. The displacement is noted hðx; tÞ, the pressure is noted Pðx; tÞ ¼ P ðxÞ þ pðx; tÞ, the annular clearance is noted H and the mean radius is noted a. The coordinate x refers to the flow direction. The perturbation components are expressed in terms of modal decomposition: uðx; tÞ ¼ ∑un ðxÞeiot ;
ð11Þ
vðx; tÞ ¼ ∑vn ðxÞeiot ;
ð12Þ
pðx; tÞ ¼ ∑pn ðxÞeiot ;
ð13Þ
hðx; tÞ ¼ ∑φn ðxÞeiot :
ð14Þ
n
n
n
n
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Linearization and substitution of (11)–(14) into continuity and momentum equations leads to the following set of equations: ! dun 1 U d iω þ þ vn ¼ φn ; ð15Þ dx a H dx H
ρ iωþ 2U
ρ iωþ U
! ! 2 d U ρU dp ρU ρU d 1 dP þ 2C f þ un þ vn þ n ¼ iω þ φn ; dx a dx H H H dx H dx
! d U 1 þC f vn pn ¼ 0; dx a H
ð16Þ
ð17Þ
where Cf is the friction coefficient. One can use (15) and (17) to eliminate vn(x) and pn(x) from (16): " ! ! !# 2 2 2 d d d U d 1 a2 2 þ C f iω 1 a2 2 þ U 2 a2 2 un dx H dx dx dx " ¼ a2 iω
U H
2
! # 2 2 Cf d U d d Cf 1 dP 21 d þ þ ω þ φn : 2 H dx H dx2 dx H H dx a2 ρH dx dx d
2
ð18Þ
Linearization and neglecting friction terms gives a simple relationship between fluid velocity fluctuations and the displacement of the structure: 2ω 1þi U d=dx φn : un ¼ a H dx2 1 þ i ω U d=dx 2U
d
2
Assuming that the structure velocity is negligible compared to the mean fluid velocity, one can write 0 !2 1 2 U d 3 ω @1 þ Aeiω=ðU d=dxÞ φn : un ¼ a2 2 U d=dx H dx2
ð19Þ
ð20Þ
In expression (20) it clearly appears that fluid fluctuations are proportional to the axial velocity and that they increase with the angular frequency. It also appears that the phase is proportional to the angular frequency and decreases with the increase of the axial velocity. This is in agreement with the assumption made in the previous section, of a delay between the fuel assembly displacement and the fluid fluctuations proportional to the inverse of the axial velocity. One may conclude that the cross-flow has a significant influence on the fuel assembly behavior.
9. Discussion It has been shown that the displacement of the fuel assembly induces fluctuations of the fluid velocity in the by-passes that are responsible for the added stiffness effect observed in previous study. Nevertheless, fluid forces responsible for what we called an added stiffness effect may be more complicated than that. In static tests, one can observe that the difference of fluid velocity in the by-passes are two times greater between grids 8 and 9 than between grids 4 and 5 for the three axial velocities tested. In the dynamic case the difference is not that obvious, for 5 m/s axial velocity, fluctuations are more important between grids 8 and 9 than between grids 4 and 5, but at 3 m/s axial velocity, fluctuations are similar. Moreover, the mean value of the fluid velocity is more important between grids 8 and 9 than between grids 4 and 5, whereas it is the opposite in the dynamic case. If one could expect that fluctuations between grids 2 and 3 would be small since the flow is not established at this level, it is more surprising to observe that fluctuations at the fourth grid level are negligible compared to those between grids 4 and 5. This may be a consequence of a turbulent flow pattern induced by the abrupt restriction of cross section for the fluid. Considering the conservation of mass and the velocity of the structure that can reach 0.07 m/s one could expect that the fluid velocity fluctuations would increase with the structure velocity and thus with the excitation frequency as the simple theory developed in Section 8 testifies. This behavior is clearly observed at the fourth grid level, whereas between grids 4 and 5 at 3 m/s axial velocity fluctuations do not seem to depend on the frequency. The theory developed is very simple and a lot of assumptions and approximations have been made, thus it is not surprising to observe that it does not explain all the phenomenon observed. Nevertheless, this illustrates that friction and nonlinear effects have to be accounted for in order to develop an accurate model. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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The delay values obtain range from 0.05 s to 0.25 s. At very low frequencies the delay does not have significant influence, but it becomes significant when it is no longer negligible compared to the oscillation period. The fluid forces induced by the fluid velocity fluctuations will have different effects depending on the structure oscillation frequency. It has been shown that the results strongly depend on the position (Fig. 19). All these observations lead us to state that the flow in the fuel assembly and in the by-passes is very complex and that an effort has to be made to improve the understand of its dynamics and the consequences on the fluid forces. 10. Conclusion Dynamic flow measurements of the fluid velocity in a by-pass during dynamic tests of a fuel assembly were performed. Results showed that the motion of the fuel assembly induces fluid velocity fluctuations. These fluctuations depend on the axial velocity and the position and showed a very different behavior at the grid level and at the rod bundle level. Frequency distributions of fluctuation showed two patterns, either fluctuations are constant and only depend on the fuel assembly position, or fluctuations increase with the frequency and thus depend on the fuel assembly velocity. A delay between the fuel assembly displacement and the fluid velocity has been observed. The delay decreases with the increase of the axial velocity, thus the phenomenon involved should be related to the fluid convection. A simple model based on a cylinder subjected to an annular axial flow has been developed. It is in agreement with the delay observations, and it showed that the cross-flow was significant, and had to be accounted for. These observations will be useful to establish a model accounting for the by-pass effect on the fuel assembly behavior. Nevertheless, extra measurements should be undertaken with various fuel assembly excitations, by changing the amplitude and excitation level. Also, the complexity of the flow and the dependency to the position would justify a more global measurement device as particle image velocimetry. Acknowledgments The authors are thankful to T. Gay-Peiller, D. Allard, J-C. Petronin, F. Bazin and N. Eyrard who ably assisted in the measurements. The authors are grateful for the financial support of the Electicité de France (EDF) and AREVA. References Bellizzi, S., Sampaio, R., 2006. POMs analysis obtained from Karhunen–Loève expansion for randomly vibrating systems. Journal of Sound and Vibration 297, 774–793. Collard, B., Pisapia, S., Broc, D., Bellizzi, S., 2004. Flow induced damping of a PWR fuel assembly. In: Proceedings of the 8th International Conference on Flow-Induced Vibration – FIV 2004. Paris, France, 6–9 July 2004, pp. 307–312. Ewins, D.J., 1995. Modal Testing Theory and Practice. Research Studies Press Ltd, Taunton, Somerset, England. Graham, M.D., Kevrekidis, I.G., 1996. Alternative approaches to the Karhunen–Love decomposition for model reduction and data analysis. Computers and Chemical Engineering 20, 495–506. Hobson, D.E., 1982. Fluid-elastic instabilities caused by flow in an annulus. In: Proceedings of 3rd International Conference on Vibration in Nuclear Plant. Keswick, BNES, London, pp. 440–463. Kang, H.S., Mureithi, N.W., Pettigrew, M.J., 2012. Analytical solution for a vibrating simply-supported cylinder subjected to 2-D concentric annular flow, considering friction. Journal of Fluids and Structures 35, 1–20. Kerschen, G., Golinval, J.C., Vakakis, A., Bergman, L., 2005. The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dynamics, Special issue on Reduced Order Models: Methods and Applications 41, 141–170. Lesieur, M., 1993. Turbulence in Fluids, Third revised and enlarged ed. Kluwer Academic Publishers, Dordrecht, The Netherlands. Liang, Y.C., Lee, H.P., Lim, S.P., Lin, W.Z., Lee, K.H., Wu, C.G., 2002. Proper orthogonal decomposition and its applications, Part I: theory. Journal of Sound and Vibration 252, 527–544. Mastroddi, F., Dessi, D., Eugeni, M., 2012. POD analysis for free response of linear and nonlinear marginally stable aeroelastic dynamical systems. Journal of Fluids and Structures 33, 85–108. Païdoussis, M.P., Sarkar, A., Semler, C., 2005. A horizontal fluid-conveying cantilever: spatial coherent structures, beam modes and jumps in stability diagram. Journal of Sound and Vibration 280, 141–157. Païdoussis, M.P., 2003. Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 2. Elsevier Academic Press, London. Piteau, P., Antunes, J., 2012. A theoretical model and experiments on the non linear dynamics of parallel plates subjected to laminar/turbulent squeeze-film forces. Journal of Fluids and Structures 33, 1–18. Ricciardi, G., Bellizzi, S., Collard, B., Cochelin, B., 2009. Modelling pressurized water reactor cores in terms of porous media. Journal of Fluids and Structures 25, 112–133. Ricciardi, G., Bellizzi, S., Collard, B., Cochelin, B., 2009. Row of fuel assemblies analysis under seismic loading: modelling and experimental validation. Nuclear Engineering and Design 239, 2692–2704. Ricciardi, G., Boccaccio, E., 2012. Stiffening of a fuel assembly under axial flow. In: Proceedings of the 10th International Conference on Flow-Induced Vibration (& Flow-Induced Noise)- FIV2012. Dublin, Ireland, 2–6 July 2012, pp. 391–397. Rigaudeau, J., 1997. Grid modelling and strength criterion in the lateral response of PWR fuel assemblies under accident conditions. In: 5th International Conference on Nuclear Engineering Nice, France, ICONE5-2568. Spurr, A., Hobson, D.E., 1984. Forces on the vibrating centrebody of an annular diffuser. In: Païdoussis, M.P., Au-Yang, M.K. (Eds.), Proceedings Symposium on Flow-Induced Vibration, vol. 4. , ASME, New York, pp. 41–52. Viallet, E., Bolsee, G., Ladouceur, B., Goubin, T., Rigaudeau, J., 2003. Validation of PWR core seismic models with shaking table tests on interacting scale 1 fuel assemblies. In: Transactions of the 17th International Conference on Structural Mechanics in Reactor Technology (SMIRT 17). Prague, Czech Republic.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Measurements of fluid fluctuations around an oscillating nuclear fuel assembly Guillaume Ricciardi n, Eric Boccaccio CEA CADARACHE DEN/DTN/STCP/LHC, 13108 Saint-Paul-Lez-Durance Cedex, France
a r t i c l e i n f o
abstract
Article history: Received 7 February 2013 Accepted 30 March 2014
In this paper, dynamic measurements of fluid velocity in the by-passes of a test-section representing a nuclear fuel assembly are presented. The test-section was designed to identify stiffness, damping and mass coefficients of a fuel assembly under axial flow, and previous studies have shown that the by-passes have an influence on the identified coefficients. The results presented in this paper show that the motion of the fuel assembly induces fluctuations in the axial fluid velocity in the by-passes. These fluctuations depend on the excitation frequency and position. A delay has been observed between the fuel assembly displacement and the fluid velocity fluctuations. The delay decreases when the axial velocity increases which means that it is a convection driven phenomenon. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Fuel assembly Dynamic flow measurements Fluid–structure interaction
1. Introduction Earthquakes can irreversibly damage nuclear power plants especially in the core, where the nuclear fuel assemblies containing enriched uranium have to be particularly resistant. Before building a nuclear power plant, it is necessary to make sure that the core will resist the worst possible earthquake conditions liable to occur at the reactor site. Therefore, when Pressurized Water Reactors (PWR) are subjected to seismic loading, the fuel assembly spacer grids strike each other, and safety measures are necessary to ensure the dropping of control rods and also the cooling of the reactor core. A way to ensure these two criteria is to prevent the spacer grids from buckling. Engineers need special tools for designing and maintaining reactor cores. The reactor core made of fuel assemblies is subjected to an axial water flow to cool the reactor. The flow strongly modifies the dynamical behavior of the fuel assemblies (Collard et al., 2004), therefore the identification of the fluid forces is important in order to provide a relevant model of the fuel assemblies behavior. The first approximation of the fluid forces is to consider them as added mass and damping (Rigaudeau, 1997; Viallet et al., 2003). A more complex expression of these fluid forces is given by Païdoussis (2003) in which the velocity and the relative direction of the flow with respect to the fuel assembly are accounted for. Ricciardi et al. (2009a,b) proposed a porous media approach based on the Païdoussis theory. All these models involve parameters that need to be identified by experiments or numerical simulations. However, reliable numerical predictions are difficult to obtain for such a complex fluid–structure problem, thus experimental results are needed. The LHC (Laboratory of Core and Circuit Hydrodynamics) has facilities to perform dynamic tests on a fuel assembly under axial flow. In a previous study (Ricciardi and Boccaccio, 2012), tests dedicated to the identification of the fluid forces acting on a full scale fuel assembly were performed. These tests highlighted an added stiffness effect under axial flow, and that the
n
Corresponding author. E-mail addresses: [email protected] (G. Ricciardi), [email protected] (E. Boccaccio).
http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016 0889-9746/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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added mass effect seemed to depend on the axial velocity. Identified coefficients of stiffness, damping and mass showed a strong dependence on the lateral by-passes. These by-passes are necessary to allow the displacement of the fuel assemblies, but they are not representative of the distance between two fuel assemblies in a core. Therefore it is needed to understand the effects of the by-passes on the identified coefficient, and thus on the fluid forces. In this paper, measurements of fluid velocity in the by-passes during static and dynamic tests are presented and discussed.
2. Experimental apparatus HERMES T is a single phase hydraulic loop that can handle full scale PWR 1300 MW fuel assemblies. The pump can supply 1200 m3/h in axial flow and 400 m3/h in cross-flow, at 35 bar and 170 1C. Therefore, the flow rate is similar to the PWR condition, the lower temperature (PWR operates at 315 1C) allows to provide accurate measurement devices to the test-section. In the present study, only axial flow is considered at 50 1C. The fuel assembly used is made up of 25 guide tubes and 264 fuel rods, each having a height of 4.5 m and linear density of about 200 kg/m. Fuel rods contain uranium pellets and have a diameter of 9.5 mm. The pitch of the fuel bundle is 12.5 mm. The fuel assembly is clamped to the test-section at the top and the bottom. The test-section is about 40 mm larger than the fuel assembly in the excitation direction and 10 mm larger in the orthogonal direction. Grids of the fuel assembly are around 200 mm wide. The displacement of the fifth grid is imposed with a hydraulic jack (Fig. 1). A plexiglass window allows making optical fluid measurements. The displacements of the grids number 2–9 noted dg2, dg3, dg4, dg5, dg6, dg7, dg8 and dg9 are measured with LVDT sensors. The movable portion of each sensor is a stainless steel rod with a diameter of 2.5 mm and is placed across by-pass 2. Static and dynamic tests are carried out. For static tests, the axial fluid velocity is measured at three altitudes in the fuel assembly and in the by-passes (Fig. 2), namely, between grids 2 and 3, between grids 4 and 5 and between grids 8 and 9. For each altitude, measurements are made every 2 mm along a line in each by-pass, and 6 lines in the fuel assembly, homogeneously distributed. Static tests are performed with the fuel assembly at rest, and with a 10 mm imposed displacement of the fifth grid toward by-pass 1. For dynamic tests, a swept sine ranging from 0 to 3 Hz with an amplitude of 6 mm is imposed. Tests are performed under axial flow for three axial velocities (1.5 m/s, 3 m/s and 5 m/s). The axial component of fluid velocity is measured at three altitudes in the by-pass 2 (Fig. 1), between grids 2 and 3, between grids 4 and 5 and between grids 8 and 9, at two depth from the window. Also, measurements are performed at five altitudes at the fourth grid level (Fig. 3). In spite of the disturbance of the LVDT sensors, measurements are made in the by-pass 2 because the fastener device that link the hydraulic jack to the grid is more perturbing the flow in the by-pass 1 than the LVDT sensors in the by-pass 2. The LDV device (TSI TRx60 probe) allows to measure the fluid velocity at only one point, so the same grid excitation is repeated for each measurement location. LDV device measures the velocity of one particle when it meets the measurement volume which is about 1 mm3, thus it is a random process and data are not regularly sampled. The sample
Fig. 1. Experimental apparatus.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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fuel rods grid LDV line measurement
2 mm window
Fig. 2. Indicative diagram of measurement locations for static test in the fuel assembly.
G5
G4
H5 H4 LDV H3 measurement H2 locations H1
G3 Fig. 3. Measurement locations of the fourth grid for dynamic tests.
Fig. 4. LDV measurements in the fuel assembly and the by-passes between the grids 8 and 9 at 1.5 m/s axial velocity.
rate is about 2.5 kHz. To perform spectral analysis, data are resampled by an interpolation method for 2 kHz. For static tests, the mean velocity is calculated from 1024 samples. For confidential reason, all the velocity values are given dimensionless and normalized by the same quantity. 3. Static tests For static tests, the fuel assembly is still and the displacement of the fifth grid is imposed. LDV measurements are made for two values of displacement: 0 mm, the fuel assembly is in the middle of the test-section, and 10 mm toward by-pass 1. The fifth grid is located approximately at the middle of the fuel assembly, thus when a displacement of 10 mm is imposed, the shape of the fuel assembly looks like the one of a clamped–clamped Euler–Bernoulli beam with a located force (Fig. 5). Fig. 4 shows an example of fluid velocity profile measured in the cross-section of the test-section. One can observe that the velocity is homogeneous in the fuel assembly, and that the fluid velocity is higher in the by-passes. This is a consequence of the difference of pressure loss coefficient between the fuel assembly and the by-passes. The hydraulic diameter in the fuel assembly is significantly smaller than in the by-passes, so it is easier for the fluid to flow in the by-passes than in the fuel assembly. The velocity in the by-passes seems to increase linearly with the axial velocity (Fig. 6). Fig. 7 shows the velocity profile in the by-passes with the fuel assembly deformed (displacement of 10 mm) and not deformed (displacement of 0 mm). One can observe that the deformation induces a difference of velocity between the by-passes. Without deformation the test-section is symmetric, the by-passes have the same size, and the velocities are the same. When a deformation is imposed to the fuel assembly, the by-passes have no longer the same size and the velocities Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 5. Shape of the fuel assembly for an imposed displacement of 0 mm (not def) and 10 mm (def) on the fifth grid under an axial velocity of 1.5 m/s.
Fig. 6. Mean of fluid velocity along the measurement line in the by-pass 1 for static tests with the fuel assembly not deformed.
are changing. The size of the by-pass 2 increases and so does the velocity, whereas in the by-pass 1 the velocity decreases since it is smaller. This difference of velocity will induce fluid forces. The fluid velocity in the fuel assembly does not seem to be significantly modified by the fuel assembly deformation. One can observe oscillations of the fluid velocity along the line measurement, these are due to a specific design at the bound of the grids. Mixing vanes are located between rods on every alternate location, and these locations are out of phase between the two sides of the grid which explain the different locations of minima and maxima velocity observed between the by-passes without deformation. The difference of fluid velocity between the by-passes increases with the axial velocity and the altitude (Fig. 8). The fluid velocity between grids 4 and 5, and between grids 8 and 9 are quite similar, whereas the difference of velocity is much higher between grids 8 and 9 than between grids 4 and 5. Moreover, between grids 2 and 3 no significant difference of velocity is observed. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 7. Fluid velocity profile in the fuel assembly and in the by-passes between the grids 8 and 9 at 5 m/s axial velocity with the fuel assembly deformed and not deformed.
Fig. 8. Difference between the mean of fluid velocity along the measurement line in the by-pass 1 and by-pass 2 for static tests with the fuel assembly deformed.
Fig. 9. PSD of the fifth grid displacement imposed by the hydraulic jack for dynamic tests.
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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4. Dynamic behavior of the structure For dynamic tests, the displacement of the fifth grid is imposed. The frequency range 0–3 Hz is swept (Fig. 9) for a peak amplitude of 6 mm. The aim is to mainly excite the first natural mode. This can be verified by a Proper Orthogonal Decomposition (POD) analysis. POD is a mathematical tool to identify the modes that contain the most energy. It applies to numerical and experimental results. Let us consider the spatial field at a time t WðtÞ ¼ ðwðx1 ; tÞ; …; wðxNn ; tÞÞ where Nn is the number of unknowns or discrete ~ sensors. We subtract from W its average value to obtain its fluctuations W: Ns ~ ðt i Þ ¼ Wðt i Þ 1 ∑ Wðt k Þ; W Ns k ¼ 1
ð1Þ
where N s is the number of time steps. ~ That is to find an orthogonal basis, POD aims to get the most characteristic fields Φ in terms of energy of the dataset W. maximizing the time average of the inner product (Liang et al., 2002): ~ ΦÞj〉; Maximize〈jðW;
with J Φ J ¼ 1;
ð2Þ
where 〈:〉 is the time average, (.,.) is the vector product, j:j is the absolute value and J : J is a L2 norm. We can then write the field as 1
~ WðtÞ ¼ ∑ qi ðtÞΦi ; i¼1
where Φk are proper orthogonal modes (POM), and qk(t) is the temporal evolution of the ith POM. Consider the result matrix X, such that h i ~ 1 ÞT ; …; Wðt ~ N ÞT : X ¼ Wðt s
ð3Þ
ð4Þ
Kerschen et al. (2005) shows that the problem (2) is equivalent to factorizing the matrix X using the singular value decomposition (SVD): X ¼ USVT ;
ð5Þ
where U is an orthogonal matrix (N n N n ) columns of which are the POM Φi, S is a pseudo-diagonal matrix (N n Ns ), elements of which are the singular values si and V is an orthogonal matrix (Ns N s ). The amount of energy captured by each mode Φi is given by ei ¼
s2i : Nn ∑k ¼ 1 s2k
ð6Þ
The qi(t) are given by the rows of the matrix R ¼ SVT : qi ðt j Þ ¼ Rij :
ð7Þ
Fig. 10. First three normalized POM shape obtained by the POD analysis for various axial velocities.
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Table 1 Percentage of energy captured by the POM. Num
1.5 m/s
3 m/s
5 m/s
POM 1 POM 2 POM 3
92.7 5.9 1.2
91.5 7.0 1.4
87.6 9.9 2.0
Fig. 11. PSD of the fluid velocity in the by-pass 2 between grids 4 and 5 for various axial velocities.
For more information on the POD method, Graham and Kevrekidis (1996), Païdoussis et al. (2005), Bellizzi and Sampaio (2006), and Mastroddi et al. (2012) could be consulted. In the present study, Wðt i Þ ¼ ðdg2 ðt i Þ; dg3 ðt i Þ; dg4 ðt i Þ; dg5 ðt i Þ; dg6 ðt i Þ; dg7 ðt i Þ; dg8 ðt i Þ; dg9 ðt i ÞÞ. Fig. 10 shows the first three POM obtained on the dynamic test for the three axial velocities. POM are normalized so they are dimensionless, the dimension of the data is contained in the temporal evolution qi. One can observe that the first POM seems to correspond to the real part of the first hydroelastic mode. One can also observe that this POM is modified as the axial velocity increases. Table 1 shows the percentage of energy captured by the POM, it can be observed that a great part of the energy is captured by the first POM , although this percentage seems to decrease as the axial velocity increases. This could be due to the increase of damping that tends to broaden the resonance peaks and thus to overlap them. 5. Velocity fluctuations The flow in the by-passes is highly turbulent, considering the width of the by-pass as characteristic length, the Reynolds number is 290 000, 180 000 and 100 000 respectively for axial velocities of 5, 3 and 1.5 m/s. Fig. 11 shows the Power Spectral Density of the fluid velocity in the by-pass 2 between grids 4 and 5 for the three axial velocities. The low frequency part between 0 and 3 Hz shows the effect of the fuel assembly displacement on the fluid velocity fluctuation, the high frequency part is due to the turbulent flow. A classical broad band turbulent behavior increasing with the axial velocity can be observed. One can also observe that the cutoff frequency over which the turbulent energy is attenuated increases with the axial velocity which is a classical result (Lesieur, 1993). Some spectra at low axial velocity show peaks beyond 3 Hz (Fig. 12), which was unexpected. This can be explained by the observation of the imposed displacement in log scale. It shows that the displacement has some spectral component beyond 3 Hz. Although this component is very small, the coherence shows that the peaks are related to the structure displacement. However this shows that fluid velocity fluctuations highly depend on the structure velocity at frequencies beyond 3 Hz for low axial velocities. 6. By-pass flow Fig. 13 shows the mean value of the fluid velocity in the by-pass. As in the static case, the velocity increases linearly with the axial velocity, but in the dynamic case the mean velocity between grids 8 and 9 is slightly smaller than between grids 4 and 5. Moreover, the velocity is more important at the grid 4 level. This is explained by the restriction of the cross section at grids level. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 12. PSD of the fluid velocity in the by-pass 2 at altitude H5 of grid 4 (top), PSD of the fifth grid displacement imposed by the hydraulic jack (centre), and coherence between the fluid velocity and the displacement imposed (bottom).
Fig. 13. Mean fluid velocity in the by-pass 2 for the dynamic tests versus the axial velocity (left) and the position along the fuel assembly (right).
Fig. 14 shows the PSD of the fluid velocity in the by-pass. The fluctuations increase with the axial velocity at every level. The relative values show that fluctuations seem to be proportional to the mean value. Fluctuations induced by the fuel assembly displacement between grids 2 and 3 are barely distinguishable from the turbulent fluctuations. Fluctuations between grids 4 and 5 show a flat profile, they do not seem to depend on the excitation frequency and thus on the structure velocity. One could extrapolate and conclude that the velocity fluctuations only depend on the structure displacement, i.e. on the instantaneous size of the by-passes, however, this statement is contradicted by the measurements made between grids 8 and 9 where the pattern seems to depend on the axial velocity, and fluctuations can increase with the excitation frequency. Fluid velocity fluctuations seem to be homogeneous at the fourth grid level except at H1 level which corresponds to the inlet (Fig. 15). One can observe a slight increase downstream. At each level, fluctuations increase with the excitation frequency, and the relative velocity is more important at low axial velocity (Fig. 16). Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 14. PSD of fluid velocity in the by-pass 2 for various axial velocities, between the grids 2 and 3 (top), between the grids 4 and 5 (centre), between the grids 8 and 9 (bottom), absolute (left) and relative values (right).
Fig. 17 compares fluid velocity fluctuations at a fixed axial velocity. One can observe that fluctuations between grids 2 and 3 and at the fourth level grid are small compared to the other locations between grids 4 and 5 and between grids 8 and 9. Fluctuations between grids 4 and 5 and between grids 8 and 9 are very close at 3 m/s axial velocity, but in the 5 m/s axial velocity case, fluctuations are more important between grids 8 and 9. The coherence γ between two signals x(t) and y(t) can be calculated from the formula (Ewins, 1995): γ2 ¼
CSDðx; yÞ2 ; PSDðxÞPSDðyÞ
ð8Þ
where CSD stands for Cross Spectral Density. Fig. 18 shows the coherences between the hydraulic jack displacement and the fluid velocity. Between grids 2 and 3 the coherence reaches 0.9 and increases with the axial velocity. Between grids 4 and 5 the coherence is very close to 1 for the three axial velocities tested. Between grids 8 and 9, the coherence is also very close to 1 for 3 and 5 m/s axial velocities, but it only reaches 0.9 for 1.5 m/s around 1.5 Hz. At the fourth grid location the coherence is weaker and barely reaches 0.95. Fig. 19 shows the PSD of the fluid velocity between grids 4 and 5 and between grids 8 and 9 for two distances from the window. One can observe that qualitative results are not modified by the depth but values are quite different. This is a Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 15. PSD of fluid velocity in the by-pass 2 for various altitude of grid 4, at 5 m/s axial velocity.
Fig. 16. PSD of fluid velocity in the by-pass 2 for various axial velocities, at altitude H2 of grid 4 (top), at altitude H3 of grid 4 (bottom), absolute (left) and relative values (right).
consequence of the space variation of the mean velocity in the by-passes (Fig. 7) since it has been observed that the fluctuations depend on the axial velocity. 7. Flow delay The phase between the hydraulic jack displacement and the fluid velocity (Fig. 20) increases with the excitation frequency. One could make the approximation that the phase φ linearly increases with the excitation frequency f: φ ¼ cp f ;
ð9Þ
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 17. PSD of fluid velocity in the by-pass 2 for various position, at 3 m/s axial velocity (top), and 5 m/s axial velocity (bottom), absolute (left) and relative values (right).
Fig. 18. Coherence between the imposed displacement and the fluid velocity in the by-pass 2 for various axial velocities, between grids 2 and 3 (top left), between grids 4 and 5 (top right), between grids 8 and 9 (bottom left), and at altitude H3 of grid 4 (bottom right).
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Fig. 19. PSD of fluid velocity in the by-pass 2 at 5 m/s axial velocity, between grids 4 and 5, between grids 8 and 9, for two depth 78 mm and 84 mm, absolute (left) and relative values (right).
Fig. 20. Phase between the imposed displacement and the fluid velocity in the by-pass 2 for various axial velocities, between grids 2 and 3 (top left), between grids 4 and 5 (top right), between grids 8 and 9 (bottom left), and at altitude H3 of grid 4 (bottom right).
where cp is a proportional coefficient. A linear relation between φ and f would mean that there is a constant delay Δt between the fluid velocity and the structure displacement: Δt ¼
cp : 360
ð10Þ
Δt can be approximated by the least square method (Fig. 21). The correlations coefficients (Fig. 22) are very close to 1, thus the linear approximation seems to be appropriate. The delay is greater between grids 2 and 3 than between grids 4 and 5, and even greater between grids 8 and 9. This may be related to the distance from the fifth grid where the amplitude of the structure reaches its maximum. Nevertheless, the greatest delay is observed at the fourth grid location. Thus it strongly depends on the geometry. The delay decreases as the axial velocity increases, considering the width of the test-section as the characteristic length the theoretical convection time based on the axial velocity is close to experimental values. Thus one can Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Fig. 21. Time delay between the imposed displacement and the fluid velocity in the by-pass 2.
Fig. 22. Correlation for the linear interpolation of phase.
conclude that this delay is a convection driven effect and that it is related to cross-flow in the fuel assembly between the by-passes. 8. Theoretical analysis The by-passes dimensions allows us to consider the leakage flow theories. Extensive work has been done in this field (Païdoussis, 2003; Kang et al., 2012; Piteau and Antunes, 2012). Based on the work of Hobson (1982) and Spurr and Hobson (1984) Païdoussis proposed equations governing the motion of a cylinder in a narrow annulus which has some similarity with the present study. The cylinder is subjected to axial flow Uðx; tÞ ¼ U þ uðx; tÞ and a cross-flow vðx; tÞ is accounted for. The displacement is noted hðx; tÞ, the pressure is noted Pðx; tÞ ¼ P ðxÞ þ pðx; tÞ, the annular clearance is noted H and the mean radius is noted a. The coordinate x refers to the flow direction. The perturbation components are expressed in terms of modal decomposition: uðx; tÞ ¼ ∑un ðxÞeiot ;
ð11Þ
vðx; tÞ ¼ ∑vn ðxÞeiot ;
ð12Þ
pðx; tÞ ¼ ∑pn ðxÞeiot ;
ð13Þ
hðx; tÞ ¼ ∑φn ðxÞeiot :
ð14Þ
n
n
n
n
Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
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Linearization and substitution of (11)–(14) into continuity and momentum equations leads to the following set of equations: ! dun 1 U d iω þ þ vn ¼ φn ; ð15Þ dx a H dx H
ρ iωþ 2U
ρ iωþ U
! ! 2 d U ρU dp ρU ρU d 1 dP þ 2C f þ un þ vn þ n ¼ iω þ φn ; dx a dx H H H dx H dx
! d U 1 þC f vn pn ¼ 0; dx a H
ð16Þ
ð17Þ
where Cf is the friction coefficient. One can use (15) and (17) to eliminate vn(x) and pn(x) from (16): " ! ! !# 2 2 2 d d d U d 1 a2 2 þ C f iω 1 a2 2 þ U 2 a2 2 un dx H dx dx dx " ¼ a2 iω
U H
2
! # 2 2 Cf d U d d Cf 1 dP 21 d þ þ ω þ φn : 2 H dx H dx2 dx H H dx a2 ρH dx dx d
2
ð18Þ
Linearization and neglecting friction terms gives a simple relationship between fluid velocity fluctuations and the displacement of the structure: 2ω 1þi U d=dx φn : un ¼ a H dx2 1 þ i ω U d=dx 2U
d
2
Assuming that the structure velocity is negligible compared to the mean fluid velocity, one can write 0 !2 1 2 U d 3 ω @1 þ Aeiω=ðU d=dxÞ φn : un ¼ a2 2 U d=dx H dx2
ð19Þ
ð20Þ
In expression (20) it clearly appears that fluid fluctuations are proportional to the axial velocity and that they increase with the angular frequency. It also appears that the phase is proportional to the angular frequency and decreases with the increase of the axial velocity. This is in agreement with the assumption made in the previous section, of a delay between the fuel assembly displacement and the fluid fluctuations proportional to the inverse of the axial velocity. One may conclude that the cross-flow has a significant influence on the fuel assembly behavior.
9. Discussion It has been shown that the displacement of the fuel assembly induces fluctuations of the fluid velocity in the by-passes that are responsible for the added stiffness effect observed in previous study. Nevertheless, fluid forces responsible for what we called an added stiffness effect may be more complicated than that. In static tests, one can observe that the difference of fluid velocity in the by-passes are two times greater between grids 8 and 9 than between grids 4 and 5 for the three axial velocities tested. In the dynamic case the difference is not that obvious, for 5 m/s axial velocity, fluctuations are more important between grids 8 and 9 than between grids 4 and 5, but at 3 m/s axial velocity, fluctuations are similar. Moreover, the mean value of the fluid velocity is more important between grids 8 and 9 than between grids 4 and 5, whereas it is the opposite in the dynamic case. If one could expect that fluctuations between grids 2 and 3 would be small since the flow is not established at this level, it is more surprising to observe that fluctuations at the fourth grid level are negligible compared to those between grids 4 and 5. This may be a consequence of a turbulent flow pattern induced by the abrupt restriction of cross section for the fluid. Considering the conservation of mass and the velocity of the structure that can reach 0.07 m/s one could expect that the fluid velocity fluctuations would increase with the structure velocity and thus with the excitation frequency as the simple theory developed in Section 8 testifies. This behavior is clearly observed at the fourth grid level, whereas between grids 4 and 5 at 3 m/s axial velocity fluctuations do not seem to depend on the frequency. The theory developed is very simple and a lot of assumptions and approximations have been made, thus it is not surprising to observe that it does not explain all the phenomenon observed. Nevertheless, this illustrates that friction and nonlinear effects have to be accounted for in order to develop an accurate model. Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i
G. Ricciardi, E. Boccaccio / Journal of Fluids and Structures ] (]]]]) ]]]–]]]
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The delay values obtain range from 0.05 s to 0.25 s. At very low frequencies the delay does not have significant influence, but it becomes significant when it is no longer negligible compared to the oscillation period. The fluid forces induced by the fluid velocity fluctuations will have different effects depending on the structure oscillation frequency. It has been shown that the results strongly depend on the position (Fig. 19). All these observations lead us to state that the flow in the fuel assembly and in the by-passes is very complex and that an effort has to be made to improve the understand of its dynamics and the consequences on the fluid forces. 10. Conclusion Dynamic flow measurements of the fluid velocity in a by-pass during dynamic tests of a fuel assembly were performed. Results showed that the motion of the fuel assembly induces fluid velocity fluctuations. These fluctuations depend on the axial velocity and the position and showed a very different behavior at the grid level and at the rod bundle level. Frequency distributions of fluctuation showed two patterns, either fluctuations are constant and only depend on the fuel assembly position, or fluctuations increase with the frequency and thus depend on the fuel assembly velocity. A delay between the fuel assembly displacement and the fluid velocity has been observed. The delay decreases with the increase of the axial velocity, thus the phenomenon involved should be related to the fluid convection. A simple model based on a cylinder subjected to an annular axial flow has been developed. It is in agreement with the delay observations, and it showed that the cross-flow was significant, and had to be accounted for. These observations will be useful to establish a model accounting for the by-pass effect on the fuel assembly behavior. 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Please cite this article as: Ricciardi, G., Boccaccio, E., Measurements of fluid fluctuations around an oscillating nuclear fuel assembly. Journal of Fluids and Structures (2014), http://dx.doi.org/10.1016/j.jfluidstructs.2014.03.016i