- Email: [email protected]
Dynamic pressure change in a rotating, laterally oscillating cylindrical container
Available online at www.sciencedirect.com
Journal of Ocean Engineering and Science 3 (2018) 91–95 www.elsevier.com/locate/joes
Original Article
Dynamic pressure change in a rotating, laterally oscillating cylindrical container Yusuke Saito, Tatsuo Sawada∗ Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan Received 20 January 2018; accepted 25 April 2018 Available online 5 May 2018
Abstract We examined wave phenomena pertinent to water in a rotating, laterally oscillating cylindrical container. In particular, we measured the time-dependent dynamic water pressure and pressure change by fast Fourier transform analysis. The swirling of water in the container had three frequency components; the frequency responses of each frequency component are reported herein. When swirling occurs in a rotating cylindrical container, it was found that the wave rotating in the same direction as the rotation of the cylindrical container and the wave rotating in the opposite direction to the cylindrical container exist at the same time. The swirling direction was determined by the relationship of these magnitude. © 2018 Shanghai Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Sloshing; Dynamic pressure; Cylindrical container; Rotation; Oscillation; Swirling.
1. Background Sloshing, which is severe liquid agitation in a container, is a consequence of externally applied oscillation to the liquid. This is problematic in petroleum tanks and liquefied natural gas tankers; for example, the aftereffect of an earthquake or rolling tankers. Sloshing may lead to swirling [1], wherein a free surface rotates around the central axis of an axisymmetric container. Ibrahim [2] reported the theoretical, experimental, and numerical research pertinent to sloshing. The stability and control of a rocket (or spacecraft) depends on swirling and other fluid dynamic behavior. Swirling is extremely dangerous in heavily fuel-laden rockets and missiles because it diverts trajectory. Yam et al. [3] investigated the stability of a spinning axisymmetric rocket exhibiting dissipative internal fluid motion. Bauer and Eidel [4] and Zhang et al. [5] examined free-surface oscillations in a slowly spinning cylindrical container partially filled with a viscous fluid. Ohaba et al. [6] investigated the frequency response of a liquid surface in a rotating, laterally oscillating cylindrical con∗
Corresponding author. E-mail address: [email protected] (T. Sawada).
tainer using a capacitance wave-height meter. These studies indicate that cylindrical container rotation stabilizes swirling; nevertheless, detailed experiments have not yet been conducted. In this study, we measure the time-dependent dynamic pressure of water in a rotating, laterally oscillating cylindrical container. We verified that the time-dependent dynamic pressure is proportional to free surface displacement from our previous measurements [7]. Moreover, we investigated the frequency components of swirling through fast Fourier transform (FFT) analysis of the time-dependent dynamic pressure change. 2. Experimental configuration Figs. 1 and 2 show a schematic of the experimental apparatus and a cylindrical container, respectively. The cylindrical container is composed of Plexiglas (99 and 200 mm inner diameter and height, respectively). The rotating cylindrical container is connected to a motor fixed to the oscillating table. The table sinusoidally oscillates in the horizontal direction. Therefore, the cylindrical container laterally oscillates while rotating. We embedded the pressure sensor (12 mm above the
https://doi.org/10.1016/j.joes.2018.04.004 2468-0133/© 2018 Shanghai Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
92
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
3. Theoretical approach
Fig. 1. Experiment apparatus.
Researchers have only begun to theoretically investigate sloshing-pertinent phenomena in a rotating cylindrical container. As a first step toward understanding these phenomena, we analyzed sloshing in the absence of rotation. Fig. 2 shows our analytical model. Herein, we report z relative to the static level of the free surface. θ = 0 indicates the direction of the forced oscillation. R is the radius of the cylindrical container, and ω is the angular frequency of the oscillating table. Assuming irrotational flow and an incompressible fluid, the unsteady irrotational Bernoulli equation for z = η(r, θ , t ) is given by ∂φ 1 p + |∇φ|2 + + gz = aω2 r cos θ sin ωt (1) ∂t 2 ρ where φ, p, ρ, and g are velocity potential, water pressure, water density, and gravitational acceleration, respectively. For infinitesimally small waves, we assumed negligible |∇φ|2 . Using Eq. (1), the kinematic and dynamic free surface conditions are given by the following equations: ∂φ ∂η = (2) ∂t ∂z z=0 2 ∂ φ ∂φ +g = aω3 r cos θ cos ωt (3) ∂t 2 ∂z z=0 Boundary conditions on the bottom and side walls are given by ∂φ =0 (4) ∂z z=−h ∂φ =0 (5) ∂r r=R We solved the continuity equation ∇ 2 φ = 0 using boundary conditions (2)–(5). Eq. (6) describes the velocity potential ∞ ∞ Am cosh [kmn (z + h )]Jm (km r ) cos(mθ φ= M=1
N=1
2 ωmn ω3 2 − ω2 ωmn ra cos θ cosh [kmn (z + h )] (6) × cos ωt gkmn sinh (kmn h ) where m and n are natural numbers (m = 1, 2, 3 · ··; n = 1, 2, 3 · ··) that represent vibrational modes. Jm is the Bessel function of the first kind of order m. Amn , δ mn , and mn are arbitrary constants. Herein, kmn is a constant that satisfies the following equation: d Jm (kmn r ) =0 (7) dr r=R In addition, ωmn is the characteristic angular frequency given by ωmn = gkmn tanh (kmn h ) (8)
+δmn ) cos (ωmnt + mn ) +
Fig. 2. Cylindrical container.
bottom wall) in the inner wall of the cylindrical container (Fig. 2). We measured the fluid dynamic pressure over the course of 50 oscillations of forcing frequency f. We increased f in increments of 0.01 Hz and measured the fluid dynamic pressure as previously described. The fluid dynamic pressure fluctuates when sloshing occurs. In one period of lateral oscillation of the cylindrical container, the difference between the maximum and minimum pressure is P, measured 50 times for each f. We varied the forcing frequency f from 1.0 to 5.0 Hz. The amplitude of the lateral oscillation a = 1.0 mm, the rotating frequency of the cylindrical container = 1.0 Hz, and the water depth h = 50 mm.
Eqs. (7) and (8) yield the resonant frequency, 2.97 Hz; R = 49.5 mm and h = 50 mm.
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
93
Fig. 3. Stable and unstable swirling. Table 1 Frequency components as a function of relative swirling direction. Swirling direction (relative to cylinder rotation)
Frequency components (excluding strong swirling)
Opposite
f + (major) f − (minor) f −
Same
Fig. 4. Experimental frequency response of pressure fluctuation.
4. Results and discussion A clear understanding of our experimental results first requires an elaboration of swirling (Fig. 3). When the forcing frequency is close to the resonant frequency, the crest of the free surface rotates around the center axis of the cylindrical container. Unstable swirling is observed when the direction of the rotation of the crest is not fixed and the clockwise (CW) and counterclockwise (CCW) swirling irregularly repeat. Stable swirling is observed when the direction of the rotation of the crest is fixed. When the cylindrical container rotates at a certain speed, the result is solely stable swirling. All swirling subsequently described herein is “stable.” Fig. 4 shows the frequency responses of pressure change with respect to the forcing frequency f. Herein, is the rotating frequency of the cylindrical container. Near the forcing frequency f = 2.0 ∼ 2.7 Hz, if the rotational direction of the cylindrical container is CW, the swirling direction is always CCW. Near the forcing frequency f = 3.0 Hz, the free surface laterally oscillates, yet its os-
Additional frequency components (at strong swirling) 2f f +
cillating direction is vertical to that of the forced oscillation. Near the forcing frequency f = 3.1 ∼ 4.1 Hz, if the rotational direction of the cylindrical container is CW, the swirling direction is always CW. When the rotation direction of the cylindrical container is reversed, each swirling direction with respect to forcing frequency f is also reversed. In both cases, the relative swirling direction changes near the resonance frequency were predicted by linear theory ( f = 2.97 Hz). Fig. 5 shows the results of FFT analysis of the pressure change at each forcing frequency f. Fig. 6 shows the frequency response of three main frequency components detected by FFT analysis. Table 1 summarizes our frequency component results as a function of relative swirling direction (i.e., in comparison to container rotation). Because the pressure sensor is fixed to the cylindrical container, it rotates at together with the cylindrical container. The rotation speed of the swirling is the forcing frequency f. Therefore, the pressure sensor detects swirling in the direction opposite to the rotation of the cylindrical container as the frequency component f + and detects swirling in the same direction to the rotation of the cylindrical container as the frequency component f − . Thus, swirling in a rotating container includes waves that rotate at f in the same direction and the opposite direction relative to the rotation of the cylindrical container. The apparent direction of the swirling is the rotation direction of the larger wave. With respect to the frequency component 2f, we could not identify the wave type. Because we detected the wave
94
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
Fig. 5. Power spectra of P.
5. Concluding remarks
Fig. 6. Experimental results for the frequency response of the frequency components.
at frequency 2f irrespective of the rotation direction and rotation speed of the cylindrical container, it is possible that the swirling includes a wave that has no node in the circumferential direction, i.e., nodes only in the radial direction. However, when we detected the frequency component 2f, the swirling appeared in the direction opposite to the rotation of the cylindrical container, precluding visual confirmation.
We examined the frequency component of the dynamic pressure change of water in an upright, rotating, laterally oscillating cylindrical container. Our goal was to understand the relation between sloshing and swirling of water. The wave that appears when applying rotational motion to the lateral oscillation of the container contains three frequency components. Two of these components are waves that rotate in the same direction and opposite direction relative to the rotation of the container. The apparent swirling direction depends on the relative magnitude of both frequency components. We experimentally elucidated the approximate physical mechanism of the free-surface behavior in an axially symmetric rotating container oscillating in the lateral direction through measurements of time-dependent dynamic pressure. We have yet to clarify the wave type that corresponds to the third frequency component. A unifying theoretical framework remains undeveloped. We are continuing our efforts in these directions.
Acknowledgments We would like to thank Mr. T. Seki of Tokushu Keisoku Co. Ltd. for his assistance in the experiments. We also wish to acknowledge the contribution of Mr. R. Maejima, who was a student of Keio University.
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
References [1] R.E. Hutton, An investigation of resonant, nonlinear, non-planar free surface oscillations of a fluid, NASA Tech Note d-1870, (1963). [2] R.A. Ibrahim, Liquid Sloshing Dynamics, Part I, Cambridge University Press, Cambridge, 2005. [3] Y. Yam, D.L. Mingori, D. Halsmer, J. Guid. Control Dyn. 20 (1997) 306–312.
95
[4] H.F. Bauer, W. Eidel, Forschung Im Ingenieurwesen 67 (2002) 93–99. [5] E. Zhang, J. Tang, M. Tao, J. Appl. Mech. 63 (1996) 101–105. [6] M. Ohaba, T. Sawada, S. Sudo, T. Tanahashi, in: Proceedings of JSME Fluids Engineering Conference, 1995, pp. 203–204. (in Japanese). [7] T. Ishiyama, S. Kaneko, S. Takemoto, T. Sawada, Mater. Sci. Forum 792 (2014) 33–38.
Journal of Ocean Engineering and Science 3 (2018) 91–95 www.elsevier.com/locate/joes
Original Article
Dynamic pressure change in a rotating, laterally oscillating cylindrical container Yusuke Saito, Tatsuo Sawada∗ Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan Received 20 January 2018; accepted 25 April 2018 Available online 5 May 2018
Abstract We examined wave phenomena pertinent to water in a rotating, laterally oscillating cylindrical container. In particular, we measured the time-dependent dynamic water pressure and pressure change by fast Fourier transform analysis. The swirling of water in the container had three frequency components; the frequency responses of each frequency component are reported herein. When swirling occurs in a rotating cylindrical container, it was found that the wave rotating in the same direction as the rotation of the cylindrical container and the wave rotating in the opposite direction to the cylindrical container exist at the same time. The swirling direction was determined by the relationship of these magnitude. © 2018 Shanghai Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: Sloshing; Dynamic pressure; Cylindrical container; Rotation; Oscillation; Swirling.
1. Background Sloshing, which is severe liquid agitation in a container, is a consequence of externally applied oscillation to the liquid. This is problematic in petroleum tanks and liquefied natural gas tankers; for example, the aftereffect of an earthquake or rolling tankers. Sloshing may lead to swirling [1], wherein a free surface rotates around the central axis of an axisymmetric container. Ibrahim [2] reported the theoretical, experimental, and numerical research pertinent to sloshing. The stability and control of a rocket (or spacecraft) depends on swirling and other fluid dynamic behavior. Swirling is extremely dangerous in heavily fuel-laden rockets and missiles because it diverts trajectory. Yam et al. [3] investigated the stability of a spinning axisymmetric rocket exhibiting dissipative internal fluid motion. Bauer and Eidel [4] and Zhang et al. [5] examined free-surface oscillations in a slowly spinning cylindrical container partially filled with a viscous fluid. Ohaba et al. [6] investigated the frequency response of a liquid surface in a rotating, laterally oscillating cylindrical con∗
Corresponding author. E-mail address: [email protected] (T. Sawada).
tainer using a capacitance wave-height meter. These studies indicate that cylindrical container rotation stabilizes swirling; nevertheless, detailed experiments have not yet been conducted. In this study, we measure the time-dependent dynamic pressure of water in a rotating, laterally oscillating cylindrical container. We verified that the time-dependent dynamic pressure is proportional to free surface displacement from our previous measurements [7]. Moreover, we investigated the frequency components of swirling through fast Fourier transform (FFT) analysis of the time-dependent dynamic pressure change. 2. Experimental configuration Figs. 1 and 2 show a schematic of the experimental apparatus and a cylindrical container, respectively. The cylindrical container is composed of Plexiglas (99 and 200 mm inner diameter and height, respectively). The rotating cylindrical container is connected to a motor fixed to the oscillating table. The table sinusoidally oscillates in the horizontal direction. Therefore, the cylindrical container laterally oscillates while rotating. We embedded the pressure sensor (12 mm above the
https://doi.org/10.1016/j.joes.2018.04.004 2468-0133/© 2018 Shanghai Jiaotong University. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
92
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
3. Theoretical approach
Fig. 1. Experiment apparatus.
Researchers have only begun to theoretically investigate sloshing-pertinent phenomena in a rotating cylindrical container. As a first step toward understanding these phenomena, we analyzed sloshing in the absence of rotation. Fig. 2 shows our analytical model. Herein, we report z relative to the static level of the free surface. θ = 0 indicates the direction of the forced oscillation. R is the radius of the cylindrical container, and ω is the angular frequency of the oscillating table. Assuming irrotational flow and an incompressible fluid, the unsteady irrotational Bernoulli equation for z = η(r, θ , t ) is given by ∂φ 1 p + |∇φ|2 + + gz = aω2 r cos θ sin ωt (1) ∂t 2 ρ where φ, p, ρ, and g are velocity potential, water pressure, water density, and gravitational acceleration, respectively. For infinitesimally small waves, we assumed negligible |∇φ|2 . Using Eq. (1), the kinematic and dynamic free surface conditions are given by the following equations: ∂φ ∂η = (2) ∂t ∂z z=0 2 ∂ φ ∂φ +g = aω3 r cos θ cos ωt (3) ∂t 2 ∂z z=0 Boundary conditions on the bottom and side walls are given by ∂φ =0 (4) ∂z z=−h ∂φ =0 (5) ∂r r=R We solved the continuity equation ∇ 2 φ = 0 using boundary conditions (2)–(5). Eq. (6) describes the velocity potential ∞ ∞ Am cosh [kmn (z + h )]Jm (km r ) cos(mθ φ= M=1
N=1
2 ωmn ω3 2 − ω2 ωmn ra cos θ cosh [kmn (z + h )] (6) × cos ωt gkmn sinh (kmn h ) where m and n are natural numbers (m = 1, 2, 3 · ··; n = 1, 2, 3 · ··) that represent vibrational modes. Jm is the Bessel function of the first kind of order m. Amn , δ mn , and mn are arbitrary constants. Herein, kmn is a constant that satisfies the following equation: d Jm (kmn r ) =0 (7) dr r=R In addition, ωmn is the characteristic angular frequency given by ωmn = gkmn tanh (kmn h ) (8)
+δmn ) cos (ωmnt + mn ) +
Fig. 2. Cylindrical container.
bottom wall) in the inner wall of the cylindrical container (Fig. 2). We measured the fluid dynamic pressure over the course of 50 oscillations of forcing frequency f. We increased f in increments of 0.01 Hz and measured the fluid dynamic pressure as previously described. The fluid dynamic pressure fluctuates when sloshing occurs. In one period of lateral oscillation of the cylindrical container, the difference between the maximum and minimum pressure is P, measured 50 times for each f. We varied the forcing frequency f from 1.0 to 5.0 Hz. The amplitude of the lateral oscillation a = 1.0 mm, the rotating frequency of the cylindrical container = 1.0 Hz, and the water depth h = 50 mm.
Eqs. (7) and (8) yield the resonant frequency, 2.97 Hz; R = 49.5 mm and h = 50 mm.
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
93
Fig. 3. Stable and unstable swirling. Table 1 Frequency components as a function of relative swirling direction. Swirling direction (relative to cylinder rotation)
Frequency components (excluding strong swirling)
Opposite
f + (major) f − (minor) f −
Same
Fig. 4. Experimental frequency response of pressure fluctuation.
4. Results and discussion A clear understanding of our experimental results first requires an elaboration of swirling (Fig. 3). When the forcing frequency is close to the resonant frequency, the crest of the free surface rotates around the center axis of the cylindrical container. Unstable swirling is observed when the direction of the rotation of the crest is not fixed and the clockwise (CW) and counterclockwise (CCW) swirling irregularly repeat. Stable swirling is observed when the direction of the rotation of the crest is fixed. When the cylindrical container rotates at a certain speed, the result is solely stable swirling. All swirling subsequently described herein is “stable.” Fig. 4 shows the frequency responses of pressure change with respect to the forcing frequency f. Herein, is the rotating frequency of the cylindrical container. Near the forcing frequency f = 2.0 ∼ 2.7 Hz, if the rotational direction of the cylindrical container is CW, the swirling direction is always CCW. Near the forcing frequency f = 3.0 Hz, the free surface laterally oscillates, yet its os-
Additional frequency components (at strong swirling) 2f f +
cillating direction is vertical to that of the forced oscillation. Near the forcing frequency f = 3.1 ∼ 4.1 Hz, if the rotational direction of the cylindrical container is CW, the swirling direction is always CW. When the rotation direction of the cylindrical container is reversed, each swirling direction with respect to forcing frequency f is also reversed. In both cases, the relative swirling direction changes near the resonance frequency were predicted by linear theory ( f = 2.97 Hz). Fig. 5 shows the results of FFT analysis of the pressure change at each forcing frequency f. Fig. 6 shows the frequency response of three main frequency components detected by FFT analysis. Table 1 summarizes our frequency component results as a function of relative swirling direction (i.e., in comparison to container rotation). Because the pressure sensor is fixed to the cylindrical container, it rotates at together with the cylindrical container. The rotation speed of the swirling is the forcing frequency f. Therefore, the pressure sensor detects swirling in the direction opposite to the rotation of the cylindrical container as the frequency component f + and detects swirling in the same direction to the rotation of the cylindrical container as the frequency component f − . Thus, swirling in a rotating container includes waves that rotate at f in the same direction and the opposite direction relative to the rotation of the cylindrical container. The apparent direction of the swirling is the rotation direction of the larger wave. With respect to the frequency component 2f, we could not identify the wave type. Because we detected the wave
94
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
Fig. 5. Power spectra of P.
5. Concluding remarks
Fig. 6. Experimental results for the frequency response of the frequency components.
at frequency 2f irrespective of the rotation direction and rotation speed of the cylindrical container, it is possible that the swirling includes a wave that has no node in the circumferential direction, i.e., nodes only in the radial direction. However, when we detected the frequency component 2f, the swirling appeared in the direction opposite to the rotation of the cylindrical container, precluding visual confirmation.
We examined the frequency component of the dynamic pressure change of water in an upright, rotating, laterally oscillating cylindrical container. Our goal was to understand the relation between sloshing and swirling of water. The wave that appears when applying rotational motion to the lateral oscillation of the container contains three frequency components. Two of these components are waves that rotate in the same direction and opposite direction relative to the rotation of the container. The apparent swirling direction depends on the relative magnitude of both frequency components. We experimentally elucidated the approximate physical mechanism of the free-surface behavior in an axially symmetric rotating container oscillating in the lateral direction through measurements of time-dependent dynamic pressure. We have yet to clarify the wave type that corresponds to the third frequency component. A unifying theoretical framework remains undeveloped. We are continuing our efforts in these directions.
Acknowledgments We would like to thank Mr. T. Seki of Tokushu Keisoku Co. Ltd. for his assistance in the experiments. We also wish to acknowledge the contribution of Mr. R. Maejima, who was a student of Keio University.
Y. Saito, T. Sawada / Journal of Ocean Engineering and Science 3 (2018) 91–95
References [1] R.E. Hutton, An investigation of resonant, nonlinear, non-planar free surface oscillations of a fluid, NASA Tech Note d-1870, (1963). [2] R.A. Ibrahim, Liquid Sloshing Dynamics, Part I, Cambridge University Press, Cambridge, 2005. [3] Y. Yam, D.L. Mingori, D. Halsmer, J. Guid. Control Dyn. 20 (1997) 306–312.
95
[4] H.F. Bauer, W. Eidel, Forschung Im Ingenieurwesen 67 (2002) 93–99. [5] E. Zhang, J. Tang, M. Tao, J. Appl. Mech. 63 (1996) 101–105. [6] M. Ohaba, T. Sawada, S. Sudo, T. Tanahashi, in: Proceedings of JSME Fluids Engineering Conference, 1995, pp. 203–204. (in Japanese). [7] T. Ishiyama, S. Kaneko, S. Takemoto, T. Sawada, Mater. Sci. Forum 792 (2014) 33–38.