© 2020 The Japan Society of Mechanical Engineers

Vol.7, No.3, 2020Bulletin of the JSME

Mechanical Engineering Journal

J-STAGE Advance Publication date : 18 March, 2020Paper No.19-00526

[DOI: 10.1299/mej.19-00526]

A simplified method for evaluating sloshing impact pressure on a flat roof based on Wagner’s theory

Shigeru TAKAYA* and Tatsuya FUJISAKI** *Japan Atomic Energy Agency

4002 Narita, O-arai, Ibaraki 311-1393, Japan E-mail: takaya.shigeru@jaea.go.jp

**NDD Corporation 1-1-6 Jonan, Mito, Ibaraki 310-0803, Japan

Abstract In sodium-cooled fast reactors, free liquid surfaces are found in several important components, including reactor vessels; sloshing due to earthquakes is one of the major concerns in design. Especially in cases where seismic isolation systems are installed to prevent or reduce damages to facilities due to earthquakes, periods of vibration are lengthened and become close to natural sloshing periods. As a result, sloshing is more likely to occur. In severe seismic conditions, sloshing waves are considered to even reach a roof slab of a reactor vessel. The structural integrity of roof slabs is required to be evaluated against sloshing impacts. However, there is no widely recognized evaluation method for sloshing impact pressure on flat roofs yet. Therefore, in this paper, a simplified evaluation method is proposed based on Wagner’s theory, which is a well-known classic theory for evaluating impact pressures on rigid wedges dropping on water surfaces. In the proposed method, we assume an equivalent wedge on a flat roof. The impact pressure on the equivalent wedge is evaluated by applying Wagner’s theory. Computational fluid dynamics analysis is conducted to confirm that a key assumption of Wagner’s theory is applicable to the evaluation of sloshing impact on a flat roof. In addition, the predictability of the proposed method is investigated by comparing literature data of sloshing experiments with the estimated values.

Keywords : Sloshing, Impact pressure, Flat roof, Wave crest velocity, Deadrise angle, CFD analysis

1. Introduction

In sodium-cooled fast reactors, we can find free liquid surfaces in several important components, including reactor

vessels; thus, sloshing effects need to be considered appropriately in the reactor design stage. Especially when seismic isolation systems are installed to prevent or reduce damages to facilities due to earthquakes, periods of vibration are lengthened and become close to natural sloshing periods. As a result, sloshing is more likely to occur. In severe seismic conditions, sloshing waves are considered to reach a roof slab of a reactor vessel. Therefore, structural integrity evaluation of the roof slab is required. Several experiments studying sloshing impact on flat roofs have been conducted (Asai et al., 1979; Kurihara et al., 1994; Toyoda and Tanaka, 2010), but there is no widely recognized evaluation method for sloshing impact pressure on a flat roof yet. For example, although Kurihara et al. proposed a method to evaluate impact pressure acting on a flat roof by considering temporal change in the momentum of the liquid (Kurihara et al., 1994), the spatial distribution of impact pressure observed in the experiment by Toyoda and Tanaka (Toyoda and Tanaka, 2010) was not taken into account in the method.

In contrast, many studies have evaluated the impact pressure on a rigid wedge dropping on a still water surface (Wagner, 1932; Bagnold, 1939; Takemoto, 1984; Otsubo and Kohno, 1985). The relative positional relationship between the side of a rigid wedge and a still water surface is very similar to that between a flat roof and a sloshing wave surface. Therefore, in this study, we focus on the similarity and propose a simplified method for evaluation of sloshing impact pressure on a flat roof based on Wagner’s theory, which is one of the most recognized theories for

Received: 25 October 2019; Revised: 25 November 2019; Accepted: 11 March 2020

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

evaluating impact pressures on rigid wedges dropping on water surfaces. Computer fluid dynamics (CFD) analysis is conducted to confirm that a key assumption of Wagner’s theory applies to the evaluation of a sloshing impact on a flat roof. Then, the predictability of the proposed method is investigated using literature data of sloshing experiments.

2. Classical theories for impact pressure on a rigid wedge dropping on a water surface

Wagner’s theory (Wagner, 1932) is one of the most recognized theories for evaluation of impact pressure on a rigid

wedge dropping on a water surface. The model is organized in two-dimensional planar field, as shown in Fig. 1. Wagner considered a pile-up of the water rising from the surface when a rigid wedge enters the water, and assumed a plate with a width of 2c. Then, the following formula was derived for the evaluation of pressure, p, on the surface of the rigid wedge by using the velocity potential of the plate in perpendicular uniform flow with a velocity of V,

(1)

where β is a deadrise angle, and ρ is the fluid density. In Wagner’s theory, the following relation was assumed between c and c’,

(2)

where c’ is half of the wetted width when pile-up of the water surface is not considered.

If V is constant, the maximum impact pressure, pmax, can be obtained as follows.

(3)

It is well known that Wagner’s theory tends to give conservative results compared to experiments when β is less

than about 15° (Chuang and Milne, 1971; Takemoto, 1984; Otsubo and Kohno, 1985). Especially when β is less than about 6°, values obtained by Wagner’s theory are much higher than the experiment results. Equation (3) shows that pmax diverges as β becomes close to 0. When β is close to 0, it is considered that the impact pressure is generated by compression of the air layer caught between a rigid body and the water surface. Bagnold’s theory is a classical theory applied to evaluations in such small β conditions (Bagnold, 1939).

Fig. 1 Geometrical relations of rigid wedge dropping on the water surface

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

3. Proposal of a simplified evaluation method for sloshing impact pressure on a flat roof A simplified method for evaluation of sloshing impact pressure on a flat roof is proposed based on Wagner’s

theory. Figure 2 shows a schematic model of the proposed method. The model is organized in two-dimensional planar field as well as Wagner’s theory. The sloshing surface is equated to the water surface in Wagner’s theory, and a virtual wedge perpendicular to the sloshing surface is considered. One side of the virtual wedge coincides with the flat roof. Then, the angle between the sloshing surface and the side of the virtual wedge, βs, is regarded as a deadrise angle, which is corresponding to β in Wagner’s theory. In sloshing waves, the vertical component of the flow velocity, Vs, is dominant at the edge region, where collisions between sloshing waves and a roof occur. Although the flow velocity in the direction perpendicular to the sloshing surface is smaller than Vs by cos(βs), for simplification, the vertical velocity, Vs, of the sloshing wave is used conservatively as a substitute for the dropping velocity of the wedge in Wagner’s theory. Based on these analogies with Wagner’s theory, the following equation is proposed for evaluating the sloshing impact pressure on a flat roof;

(4)

where R is the radius of a tank, and H is the height of the top space shown in Fig. 2. In the original Wagner’s theory, tan (β) is approximated by β, but the approximation is not used in the proposed method. Therefore, R/H is shown instead of 1/βs.

Fig. 2 Schematic model of the proposed simplified method for evaluation of impact pressure on a flat roof

4. Methods 4.1 CFD analysis procedures

The relationship described by Eq. (2) is a key assumption of Wagner’s theory. We need to confirm that this assumption is also applicable to the evaluation of a sloshing impact on a flat roof. Therefore, CFD analysis was conducted in this study.

Figure 3 shows a schematic of the analytical model. The dimensions are determined based on a medium-scale tank with a top space height of 37 cm, according to Kurihara et al.’s experiments (Kurihara et al., 1994), explained in the next subsection. Figures 4(a) and (b) show the side view and the top view of a mesh model of the tank shown in Fig. 3, respectively. A hexahedral mesh was employed. The height of the first mesh at the roof and wall was 1 mm. The total number of elements was about 1.3 million. In addition, a mesh model of the tank with a top space height of 77 cm was also prepared in order to simulate a sloshing waveform without collisions with the roof. The side view of the model is shown in Fig. 5. The upper part with a height of 40 cm was added to the model shown in Fig. 4. Mesh divisions in the radial direction and circumferential direction were the same as in the previous model. The total number of elements was approximately 1.7 million.

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3

2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

A commercial CFD code, STAR-CCM+, Ver. 12.04, was used. The main analytical options are summarized in Table 1. The time step was automatically controlled so that the Courant number was less than 0.1. The free surface is considered to be modeled appropriately by the VOF method if the number of the interface cell between liquid and gas phases is one. It was confirmed the simulation results in this study met the criterion.

The tank was shaken in the horizontal direction. The excitation period was 1.77 s, which agreed with the primary resonance period. The excitation amplitudes were 5.2 cm and 8.8 cm. The material properties are shown in Table 2.

Fig. 3 Schematic of the analytical model

(a) Side view

(b) Top view Fig. 4 Mesh model of the tank with the top space height of 37 cm

Fig. 5 Side view of the mesh model of the tank with the top space height of 77 cm

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

Table 1 Analytical options

Free surface VOF method Physic model (Water) Constant density

Physic model (Air) Ideal gas Turbulence model SST k-w model Solution method Implicit unsteady Wall boundary Non-slip

Wall model All-y+

Table 2 Material properties Fluid density (kg/m3) 997.561

Viscosity (Pa‧s) 8.8871e-4

4.2 Literature data for comparison

Kurihara, et al. conducted a series of sloshing experiments using three types of tanks, a small-scale rectangular

tank with the base length of 55 cm, a middle-scale cylinder tank with the radius of 111.5 cm, and a large-scale rectangular tank with the base length of 670 cm (Kurihara, et al., 1994). Impact pressures at the edges of flat roofs for various top space heights and excitation amplitudes were measured, and relationships between impact pressures and vertical wave velocity were reported. In this study, these experiment results were used to investigate the predictability of the proposed method.

5. Results and discussion 5.1 CFD analysis

Figure 6 shows comparison of time variations of impact pressure evaluated by the CFD analysis and measured by the experiment at the point close to the side wall on the excitation axis where the pressure gauge, RP1, was placed in the experiment. The excitation amplitude was 8.8 cm. The measurement data was read from other report including results of the same experiment (Masuko, et al., 1989). Although there was a difference between instantaneous peak values, general tendency agreed well and following three common features were recognized; Impact pressure increased rapidly after the collision, and then decreased with oscillating. After that, the negative pressure was caused. Sudden increase in impact pressure is a general feature of Wagner’s type impact pressure. The negative pressure indicates that air was hardly caught between the roof and the sloshing surface when the collision occurred. The reason of oscillation was not identified in the report, and further investigation is also needed for CFD analysis.

The assumption of the equivalent plate with a width equal to 2c, corresponding to the wetted width when taking pile-up of the water surface in consideration, in Fig. 1 is one of the key concepts of Wagner’s theory. As mentioned earlier, Wagner’s theory assumed that 2c is larger than 2c’, corresponding to the wetted width without the pile-up, by a factor of π/2 (Fig. 1). The method proposed in this study also employs this relationship. Therefore, the adequacy of the assumption of the relationship between c and c’ for our problems was checked by using the CFD analysis results.

Figure 7 shows an example of waveforms obtained by CFD analysis using the mesh models in Figs. 4 and 5. The dotted line corresponds to the waveform in the case when the height of the tank is 40 cm higher than in the experiment. In this case, the sloshing wave has not collided with the roof yet. In contrast, the solid line corresponds to the waveform in the case when the height of the tank is the same as it was in the experiment. The sloshing wave collides with the roof and the edge of the wave goes forward along the roof. A comparison of the solid and dotted lines shows that there is qualitative similarity in shape with the pile-up of the water surface in the case that ridged wedges drop on the water.

In this study, c and c’ considered in the sloshing impact pressure evaluation were defined as shown in Fig. 8, and time variation of c/c’ was evaluated. The result is shown in Fig. 9. In both cases of the excitation amplitudes of 5.2 cm and 8.8 cm, the values of c/c’ were almost constant at approximately π/2. These results indicate the possibility that the key assumption of Wagner’s theory is applicable to the evaluation of sloshing impact on a flat roof, although additional analysis for various different conditions is necessary.

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

Fig. 6 Time variation of impact pressure (Excitation amplitude : 8.8 cm)

Fig. 7 Example of waveforms obtained by mesh models in Figs. 4 and 5 (The excitation amplitude: 8.8 cm)

Fig. 8 Schematic for evaluation of c and c’

Fig. 9 Time variation of evaluated c/c’ from collision

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

5.2 Comparison with experiment results Figures 10(a)-(c) compare the measured impact pressures (Kurihara, et al., 1994) and the ones calculated using the

proposed method for the small-scale tank, medium-scale tank, and large-scale tank, respectively. It is found that, in general, the proposed method yielded conservative results. The conservativeness tends to be small if the top space height is relatively large, while it becomes large if the top space height of the tank is narrow. For example, for the small-scale tank, the calculated impact pressures in the cases of the top space height of 4 cm and 6 cm were within the range for a factor of about 2.5. In contrast, in the cases of the top space height of 2 cm, the calculated impact pressure is approximately ten times higher than the measured result at most. The results for a large-scale tank show the same tendency; however, the conservativeness was larger on the whole compared to the results for the small-scale and medium-scale tanks. The dependency of the conservativeness on the amplitude of impact pressure was not well recognized.

Then, the dependency of predictability on the wave crest velocity was examined. Figures 11(a)-(c) compare the wave crest velocity to the ratio of calculated to measured impact pressures for the small-scale tank, medium-scale tank, and large-scale tank, respectively. In the cases that the top space heights are relatively large and the predictability is relatively good, the dependency of the predictability on the wave crest velocity is not well recognized. For example, as for the medium-scale tank with the top space height of 37 cm, the ratios of calculated impact pressures by the proposed method to measured results were in the scatter band of a factor of 2.5 regardless of the wave crest velocity. In contrast, in the cases that the top space heights are relatively narrow and the conservativeness is relatively large, the dependency of the predictability on the wave crest velocity is slightly observed. For example, as for the medium-scale tank with the top space height of 7 cm and the large-scale tank with the top space height of 14 cm, the ratios of calculated impact pressures by the proposed method to measured results seem to increase with the wave crest velocity.

As mentioned earlier, it is well known that Wagner’s theory tends to give conservative results compared to experiments in small β conditions. Therefore, the dependence of predictability on the deadrise angle was examined. The deadrise angle in our problems was defined in Fig. 2. Figure 12 shows the relationship between the deadrise angle and the ratio of calculated impact pressures by the proposed method to the measured results. The plots are average values, and the error bars are standard deviations. The dependence of the ratio on the deadrise angle was observed regardless of the scale of the tanks. The ratios of the case with the largest deadrise angle, about 18°, scattered in the band of a factor of less than 2. Then, the ratios tend to increase as the deadrise angle decreases. When the deadrise angle was about 3.6°, the averaged ratio raise close to 10. In the case of that the deadrise angle decreased to about 2.4°, the averaged ratio exceeded 20. This tendency is similar to Wagner’s original theory. Bagnold’s theory is proposed to be used for small deadrise angle conditions in the cases of ridged wedge dropping on the water surface. It is considered that there is a lower limit for the application of the proposed method based on Wagner’s theory. If the overestimation up to 10 times is allowed in design, the lower limit is estimated to be around 5° by considering the variation.

It is noted that the difference in the flow velocity by cos(βs) was not taken into account when Eq. (4) was derived in Section 3. The maximum βs in the experiments was approximately 18.4°. The corresponding minimum cos(βs) is approximately 0.95, and impact pressure is estimated approximately 1.1 times higher at most. It is considered that the effect of neglecting the difference in the flow velocity on the predictability was not significant in the range discussed in this study.

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

(a) Small-scale tank

(b) Medium-scale tank

(c) Large-scale tank

Fig. 10 Comparison of the measured and calculated impact pressure

(a) Small-scale tank

(b) Medium-scale tank

(c) Large-scale tank

Fig.11 Relationship between wave crest velocity and ratio of calculated to measured impact pressure

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

Fig. 12 Relationship between deadrise angle and ratio of calculated to measured impact pressure

6. Conclusions

A simplified method for evaluation of sloshing impact pressure on a flat roof was proposed based on Wagner’s theory by focusing on the geometrical similarity between impact pressure on a wedge dropping on still water surface and sloshing impact pressure on a flat roof.

Consideration of pile-up of the water surface is a key concept of Wagner’s theory, and the ratio of width considering pile-up, c, to that without considering pile-up, c’, is assumed to be π/2. The applicability of this assumption to the proposed method was investigated by CDF analysis. Waveforms of sloshing wave colliding with the flat roof and those without colliding were simulated, and time variation of c/c’ was evaluated by using two waveforms mentioned. As a result, it was found that the time variation was small and the value was approximately π/2. This result indicates the possibility that the key assumption of Wagner’s theory is applicable to the evaluation of a sloshing impact on a flat roof, although it was a result of the limited conditions.

Finally, the proposed method was applied to the evaluation of a series of sloshing experiments, and the calculated impact pressures were compared with the experiment results. In general, the proposed method gave conservative results. It is found that the predictability of the proposed method strongly depends on the deadrise angle regardless of the scale sizes of tanks. The ratios of calculated impact pressures to the experiment results tend to increase as the deadrise angle decreases. The ratios of the case with the largest deadrise angle, about 18°, scattered in the band of a factor of less than 2 while the averaged ratio of the case with the smallest deadrise angle, about 2.4°, the averaged ratio exceeded 20. This tendency is the same as in the original Wagner’s theory. It is considered that there is a lower limit for the application of the proposed method based on Wagner’s theory. If the overestimation up to 10 times is allowed in design, the lower limit is estimated to be around 5° by considering the variation.

The model proposed in this study is organized in two-dimensional planar field as well as Wagner’s theory, and three-dimensional flow effects are not considered. Improvement in predictability is expected by taking account of these effects in the future.

References

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Bagnold, R. A., Interim report on wave-pressure research, Journal of the Institution of Civil Engineers, Vol.12 (1939), pp.202-226.

Chuang, S. L. and Milne, D. T., Drop test of cones to investigate the three-dimensional effects of slamming, NSRDC Report 3543 (1971).

Kurihara, C., Masuko, Y. and Sakurai, A., Sloshing impact pressure in roofed liquid tanks, Journal of Pressure Vessel

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2© 2020 The Japan Society of Mechanical Engineers

Takaya and Fujisaki, Mechanical Engineering Journal, Vol.7, No.3 (2020)

[DOI: 10.1299/mej.19-00526]

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CRIEPI Research Report, Report No. N10003 (2010) (in Japanese). Wagner, H., Über stoß‐ und gleitvorgänge an der oberfläche von flüssigkeiten, Journal of Applied Mathematics and

Mechanics, Vol.12, No.4 (1932), pp.193-215 (in German).

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