Analytical solution for earthquake induced hydrodynamic pressure on a circular cylinder in a two-layer fluid ＊

Pi-guang Wang , Mi Zhao , Xiu-li Du , Jing-bo Liu

1. Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing 100124, China

2. College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 1000124, China

3. Department of Civil Engineering, Tsinghua University, Beijing 100084, China

Abstract: The analytical method is adopted to investigate the earthquake-induced hydrodynamic pressure on a circular cylinder surrounded by a two-layer fluid. Based on the radiation theory, the analytical expressions of the hydrodynamic pressure and the hydrodynamic force on a rigid circular cylinder are obtained. Then, the dispersion relation for the radiation problem in a two-layer fluid, and the added mass and the damping coefficients on the circular cylinder are analyzed. The earthquake-induced hydrodynamic forces on the circular cylinder in a two-layer fluid are furtherly investigated. It is founded that the hydrodynamic pressures on the circular cylinder include not only the surface-wave and perturbing wave modes, but also the internal-wave mode. This is quite different from the case in a single-layer fluid. However, the results indicate that the internal-wave mode has little influence on the hydrodynamic forces on the circular cylinder in a two-layer fluid. Lastly, the seismic responses of the flexible cylinder in single-layer and two-layer fluids are studied. Some examples show that the seismic responses of the cylinder in a two-layer fluid are much larger than those in a single-layer fluid.

Key words: Circular cylinder, earthquake, hydrodynamic force, surface wave, two-layer fluid

Introduction

It is well known that the seismic response of offshore structures surrounded by water involves the water-structure interaction and the hydrodynamic forces due to the water-structure interaction may have significant effects on the dynamic responses and the dynamic properties of the offshore structures[1-4].Therefore, the hydrodynamic forces on the offshore structures should be important issues in their safety design.

The existing studies for the earthquake-induced hydrodynamic forces on offshore structures mainly focused on the case of a single-layer fluid. Liaw and Chopra[4]investigated the effects of the hydrodynamic forces on the seismic response of cantilever circular cylinders. It was pointed out that the water-structure interaction can be conveniently modeled as an “added mass” when the surface waves and the fluid compressibility are ignored. The dynamic responses of the circular cylinders subjected to a horizontal ground excitation were furtherly investigated by Wang et al.[5]The expressions of the hydrodynamic forces on circular cylinders were simplified by Li and Yang[6],Du et al.[7], Jiang et al.[8], and Wang et al.[9-10].

In addition to the circular cylinder surrounding by a single-layer fluid mentioned above, some other water-cylinder interaction models were also proposed.For axisymmetric cylinders, Wang et al.[11]analyzed the earthquake-induced hydrodynamic pressure on circular tapered cylinders in water; and Avilés and Li[12]adopted a boundary method to evaluate the hydrodynamic forces on rigid axisymmetric cylinders with consideration of the seabed flexibility. Simplified models were proposed to evaluate the added mass for the intake tower with a cross section having two-symmetric-axes[13]and uniform cylinders of an arbitrary cross section[14]. The simplified formulas for evaluating the hydrodynamic forces on elliptical,rectangular and round-ended cylinders and multiple circular cylinders were obtained by Wang et al.[9,10,15-16].

However, the variation of the water temperature and salinity with depth may cause the density stratification in the ocean, which is often simply modeled by a two-layer fluid, where the density in each layer is constant. The interaction of waves with a cylinder in a two-layer fluid was investigated by You et al.[17], Linton and Mclver[18]and Cadby and Linton[19]. It should be pointed out that the internal waves will be generated on the interface between the upper and lower layers when there are disturbing sources in a two-layer fluid. The wave load on the offshore structures can sometimes be quite different from that in a single-layer fluid because of the interaction of the internal waves. In this paper, we investigate the earthquake-induced hydrodynamic forces on a circular cylinder and the seismic responses of the flexible cylinder in a two-layer fluid.

1. Mathematical formulation

The interaction between the vertical circular cylinder and the fluid is shown in Fig. 1. The fluid is a two-layer one and it is assumed to be incompressible and inviscid. It is also assumed that the fluid domain extends to infinity. The water is treated as in the calm state before the action of the earthquake. The circular cylinder with the radiusais fixed on the rigid ground.The cylinder is assumed to be rigid in the present study. Denote the density and the depth of the upper and lower fluid layers by1ρ,h1and2ρ,h2,respectively. Leth=h1+h2be the total depth andthe density ratio. A harmonic wave of frequencyωis considered here as the seismic excitation and the displacement of the ground motion is expressed as. HereUgis the amplitude,tis the time. A cylindrical coordinate systemO-rθzis adopted with the originOat the bottom of the fluid and thez-axis pointing upwards.

It should be noted that the steady-state response is also a simple harmonic motion of the same frequency when the excitation is a simple harmonic motion, as a property of linear time invariant systems[1]. Therefore, the hydrodynamic pressure of the fluid can also be expressed asin the cylindrical coordinate system,where the superscriptm=1,2 denotes the upper and lower layers, respectively. The governing equation for the fluid domain in the cylindrical coordinate system can be expressed as

The boundary conditions are as follows:

(1) The hydrodynamic pressure at the free surface

(2) No vertical motion at the boundaryz=0,i.e.

(3) The boundary conditions at the interface of the upper and lower layers

(4) The condition of symmetry about theθ=0 plan

(5) The radial component of the motion of the fluid atr=ais the same as the radial motion of the outer surface of the cylinder

(6) The waves propagate away from the structure and decay with the traveling distance, and the hydrodynamic pressure at infinity is zero

2. Exact solution

The method of separation of variables is used to solve this problem andPmcan be expressed as

Then Eq. (9) yields three ordinary differential equations

wherek2andn2are constants.

According to the boundary conditions (6) and (7),the solution of Eq. (11) contains terms involving cosθonly, which meansn=1. According to the boundary condition Eq. (8), the solution of Eq. (12)contains terms involvingonly, whereis the Hankel function of the first kind of order one. The solution of Eq. (10) can be written as

whereAm,jandBm,jare undetermined coefficients.

Taking the boundary conditions (2) through (4)into consideration, the vertical eigenfunctions in Eq.(13) can be written as:

wherejAis an undetermined coefficient, and the constantsαandβare expressed as

Substituting Eqs. (14) and (15) into Eq. (5), after some manipulations, we obtain

wheren=1,2.

For a homogenous fluid, the dispersion relation Eq. (19) can be expressed as

It can be seen from Eqs. (19) and (20) that,unlike the case of a homogenous fluid, the waves of the frequencyωin a two-layer fluid can propagate with two kinds of different wave numbersk1,jand. This means that not only the radiation waves but also the perturbing potentials in the surface-wave mode and the internal-wave mode will be generated due to the oscillation of the bottom-mounted cylinder in a two-layer fluid[17]. If, whereκn,jis a positive real number. The wave numberk=kn,jcan be obtained from Eq. (19).

Neglecting the surface waves, the boundary condition is equivalent to assigningg=0[1]. Then,the dispersion relation in Eq. (19) can be rewritten as

In conclusion, the hydrodynamic pressurePcan be written as

Substituting Eq. (22) into Eq. (7), after some manipulations, we obtain

Correspondingly, the horizontal hydrodynamic force on the circular cylinder at the heightzcan be expressed as

whereJis the mode number.

Then, the total horizontal hydrodynamic force on the circular cylinder can be obtained as

whereCmandCdare the added mass and the damping coefficient, respectively.

h h,γ=0.80 anda=0.5h, whereCm,1andCm,2are the added mass coefficients in the surface-wave mode and in the internal-wave mode,respectively. It can be seen that the present solution agrees well with the solution given by You et al.[17].

Figure 3 shows the added mass coefficients against the dimensionless frequency0ωfor different mode numbersJwithh1/h2=1,γ=0.80 anda= 0.1h. It can be seen that the accuracy of the solution is sufficient whenJ=6.

3. Numerical results

Fig. 2 Comparison of the added mass coefficients between the present solution and the solution given by You et al.[17]with h1/h2 = 1,γ = 0.80 and a = 0.5h

Fig. 3 (Color online) The added mass coefficients obtained by different mode numbers J with h1/ h2=1, γ=0.80 and a=0.1h

The fluid density of the lower layer is assumed to be larger than that of the upper layer, and the fluid density of the single-layer is the same as that of the upper layer. The dispersion relation, the added mass and the damping coefficients, and the hydrodynamic force are discussed in the following sections.

3.1 Dispersion relation

Fig. 4 (Color online) The wave numbers against the wave frequency ω0 with h1 /h2 =1, γ=0.50

Figure 4 presents the dimensionless wave number withj=0, 1 andh1/h2=1 against the dimensionless frequency0ω, when the density ratioγ= 1.0, 0.5. The dimensionless wave number is found to be the same for different values ofhfor a fixed dimensionless frequency0ωwith the same values ofγandh1/h2. Compared with the single-layer fluid,there is an additional traveling wave propagating with the wave numberk2,0in a two-layer fluid, which is said to be in the internal-wave mode. The traveling wave propagating with the wave numberk1,0is in the surface-wave mode and the other traveling waves are in the perturbing wave modes. The wave number in the internal-wave and surface-wave modes increases with the increase of the frequencyω0, whereas the wave number in the perturbing wave modes decreases with the increase of the frequency0ω. It also can be seen that the wave numberk1,0in the two-layer fluid and in the single-layer fluid is essentially the same,whereas the wave numbersk1,jandk2,jwithj≥1 in the two-layer fluid and in the single-layer fluid may be quite different.

Fig. 5 (Color online) The wave numbers against the density ratio γ with h1/h2=1 for different values of ω0

Fig. 6 (Color online) The wave numbers against the depth ratio h1/ h2 with γ=0.80 for different values of ω0

Figure 5 shows the dimensionless wave numbers in the internal-wave mode and in the perturbing wave modes against the density ratioγfor different values ofω0, whenh1/h2=1. It can be seen that the wave number in the internal-wave mode dramatically increases with the increase of the density ratioγ. The wave number in the perturbing wave modes increases or decreases with the increase of the density ratioγ.It also can be seen that the frequency has little influence on the wave number in the perturbing wave modes when the dimensionless frequency is large.Figure 6 shows the dimensionless wave numbers in the internal-wave mode and in the perturbing wave modes against the depth ratioh1/h2for different values of0ω, whenγ=0.80. The wave number in the internal-wave mode decreases significantly with the increase of the depth ratioh1/h2. The depth ratioh1/h2also has a significant influence on the wave number in the perturbing wave modes for a small frequency0ω.

3.2 Added mass and damping coefficients

Figure 7 shows the added mass and the damping coefficients against the dimensionless frequency0ωfor different values of the density ratioγwhena= 10 m ,h=20 m andh1/h2=1. It can be seen that the density ratioγhas significant effects on the added mass and the damping coefficients. The added mass and the damping coefficients increase with the increase of the density ratioγ. In addition, as the frequency increases, the added mass coefficient is found to increase first, then sharply decrease and gradually increase to a certain value, and the damping coefficient is found to increase first and then decrease to zero.

Fig. 7 (Color online) The added mass and the damping coefficients against the wave frequency ω0 with h1/ h2 =1,a = 10m and h=20m for different values of γ

Fig. 8 (Color online) The added mass and the damping coefficients against the wave frequency ω0 with γ=0.97,a = 2 m and h=20 m for different values of h1/ h2

Figures 8 and 9 show the added mass and the damping coefficients against the dimensionless frequency0ωfor different values of the depth ratiowithγ=0.97, 0.70, respectively, whena=2 m ,h=20 m. It is observed that the depth ratioh1/h2has little effects on the added mass and the damping coefficients when the density difference of the two-layer fluid is small (γ= 0.97), whereas it has significant effects when the density difference of the two-layer fluid is large (γ= 0.70). The added mass and the damping coefficients are also found to decrease with the increase of the depth ratioh1/h2.

Figure 10 shows the added mass and the damping coefficients against the dimensionless frequency0ωin the surface-wave, internal-wave and perturbing wave modes, whenh1/h2=1,γ=0.80,a=2 m andh=20 m. It is observed that the added mass is mainly caused by the surface-wave and perturbing wave modes, while the damping is mainly caused by the surface-wave mode. It should be noted that effects of the internal-wave mode can be neglected. In addition, it can be seen that the added mass coefficient due to the surface-wave mode first increases and then sharply decreases to zero near a certain frequency,whereas the added mass coefficient due to the perturbing wave mode sharply increases from zero to a certain value. Figure 11 shows the added mass

Fig. 9 (Color online) The added mass and the damping coefficients against the wave frequency ω0 with γ = 0.70,a =2m and h=20m for different values of h1/ h2

coefficient due to the surface-wave and perturbing wave modes against the dimensionless frequencyω0for different values of the density ratioγand the depth ratioh1/h2, whena=2 m,h=20 m. It can be seen that the added mass coefficients due to the surface-wave and perturbing wave modes decrease with the increase of the density ratioγand the depth ratioh1/h2.

3.3 Hydrodynamic force

Figure 12 shows the total hydrodynamic forces against the dimensionless frequency0ωfor different values of the density ratioγ, the depth ratio1/2hhand the width-depth ratio 2a/h. It is interesting to note that the variation trend of the dimensionless hydrodynamic forces with0ωis basically consistent with the added mass coefficient. Obviously, the dimensionless hydrodynamic forces decrease with the increase of the density ratioγand the depth ratioh1/h2. In addition, it can be seen that the width-depth ratio 2a/hhas significant effects on the dimensionless hydrodynamic forces. For large frequencies (ω0＞ 2), the dimensionless hydrodyna-mic forces are found to decrease with the increase of the width-depth ratio 2a/h.

Fig. 10 (Color online) The added mass and the damping coefficients in different wave modes against the wave frequency 0ω with γ=0.80, a=2m and h=20m

Figure 13 shows the proportions of different wave modes in the hydrodynamic forces against the dimensionless frequency0ωfor different values of the density ratioγin the case of 2a/h= 0.2,h1/h2=1, whereR1denotes the ratio of the hydrodynamic forces caused by the surface-wave modes to the total wave forces andR2denotes the ratio of the hydrodynamic forces caused by the perturbing modes to the total wave forces caused by the perturbing wave modes. It can be seen that the proportion of the surface-wave modes decreases nearly from 100% to 0% while the proportion of the perturbing modes increases nearly from 0% to 100%when0ωincreases from 1 to 6.

It should be noted that the free surface boundary condition will be of little consequence in the earthquake response of the cylinders surrounded by the water subjected to the high frequency ground motion[1]. In this case the fluid-cylinder dynamic interaction can be replaced by an added massmm,0,, as in the following expression

Fig. 11 (Color online) The added mass coefficients in different wave modes against the wave frequency ω0 with h = 20m for different values of γ and 2a/ h

Figure 14 shows the added mass along the height of the cylinder for different values of the density ratioγand the width-depth ratio 2a/h, whenh1/h2=1.It can be seen that the added mass of the cylinder in the lower layer of the two-layer fluid is much larger than that in the single-layer fluid.

4. Application

Fig. 12 (Color online) The total hydrodynamic forces against the wave frequency 0ω with h=20 m for different values of γ, h1/ h2 and 2a/ h

In this section, the effect of the hydrodynamic force on the seismic responses of the flexible cylinder is investigated in the case when the peak acceleration(amax) is equal to 0.2 g.The ground motions,including the El-Centro, Christchuch and Sanfernado waves are shown in Fig. 15. The added mass for the hydrodynamic force is calculated by Eq. (27). The material parameters of the cylinder including the Young’s modulus and the density areEs=30000MPa,ρs=2500kg/m3, respectively. The damping ratioξof the structure under the earthquake action is about 0.03-0.07, and the damping ratio for the reinforced concrete structure is aboutξ=0.05[20]. Therefore,ξ=0.05 is considered in the present study. The height of the cylinder isHs=40m. The mass of the superstructure isMs=δmHs, whereδis the ratio of the mass of the superstructure to the mass of the cylinder andm=ρsπa2is the mass of the cylinder per unit of height.The fluid density of the upper layer is 1 000 kg/m3.The simplified model of the fluid-structure system is shown in Fig. 16.

Fig. 13 (Color online) The proportions of different wave modes in the hydrodynamic forces against the wave frequency 0ω for different values of γ

The Euler beam model is used for the elastic cylinder. Based on the assumed-mode method, the deformation of the cylinder can be approximately expressed as[1]

Fig. 14 (Color online) The added mass along the height withh1/ h2=1 for different values of γ and 2a/ h

Fig. 15 (Color online) Acceleration curves of seismic waves

Fig. 16 Simplified model of fluid-cylinder system

The effects of the hydrodynamic forces on the seismic responses of the cylinder surrounded by a single-layer fluid for different values of the water depth are investigated in the case ofδ=2,a=4 m.Figure 17 shows the peak displacement response of the cylinder. It can be seen that the seismic response of the cylinder can be significantly increased by the surrounded water, particularly when the water depth is equal to the height of the cylinder. In general, the effects of the hydrodynamic forces on the seismic responses of the cylinder decrease with the increase of the water depth.

Fig. 17 (Color online) Peak displacement of the cylinder in single-layer fluid with δ = 2 , γ =1.0 , a = 4 m and Hs=40 m for different values of h

The seismic responses of the cylinder surrounded by a two-layer fluid withh=40 m are investigated in the case of different values ofδanda. A dimensionless parameter is defined asR1=, whereu1maxandu2maxdenote the peak displacement on the top of the cylinder in a single-layer and in a two-layer fluid respectively.Figure 18 shows the parameterR1againsth2withγ= 0.85. Figure 19 shows the parameterR1againstγwitha=10 m,h2=10 m, 20 m and 30 m. It can be seen from Figs. 18 and 19 that the seismic response of the cylinder in a two-layer fluid may be much larger than that in a single-layer fluid in some cases. For instance, the seismic response of the cylinder in a two-layer fluid is 13% larger than that in a single-layer fluid in the case ofh2=30 m,γ= 0.85,a=10 m,δ=10 and subjected to the Sanfernado wave. In general, the effects of the hydrodynamic forces in a two-layer fluid on the seismic response of the cylinder increase with the increase ofh2and the decrease ofγ.

Fig. 18 (Color online) Peak displacement of the cylinder in single-layer fluid with δ=2, a=4m and Hs=40m for different values of h

5. Conclusions

In this study, the analytical method is adopted to analyze the wave radiation problem due to a vertical circular cylinder surrounded by a fluid under the earthquake action. Analytical expressions for the hydrodynamic force, as well as the added mass and the damping coefficients are obtained. For a circular cylinder in a two-layer fluid, there are waves in three different modes, that is, the surface-wave, internalwave and perturbing wave modes. However, there are only the surface-wave and perturbing wave modes for a circular cylinder in a single-layer fluid. The wave number in the surface-wave mode in a two-layer fluid is essentially the same as that in a single-layer fluid,but the wave numbers in the perturbing wave modes can be quite different. The density ratio and the depth ratio can have significant effects on the wave numbers in the surface-wave and perturbing wave modes when the frequency is small.

Fig. 19 (Color online) The parameter R1 against γ with a = 10m , h = H s =40 m for different values of δ and h2

Numerical results indicate that the internal-wave mode has little effect on the added mass and the damping coefficients. The added mass is mainly caused by the surface-wave and perturbing wave modes, while the damping is mainly caused by the surface-wave mode. The density ratio and the load frequency can have a great effect on the added mass and the damping coefficients. If the density ratio is small, the effect of the depth ratio on the damping coefficient is small. The hydrodynamic force on a circular cylinder is deeply influenced by the density ratio, the depth ratio, and the width-depth ratio. It is also found that the free surface boundary condition has little influence on the earthquake-induced hydrodynamic forces for the high frequency ground motion. In this case the fluid-cylinder dynamic interaction can be replaced by an added mass, which does not depend on the earthquake frequency. The added mass in the lower layer can be much larger than that in a single-layer fluid. Moreover, the seismic responses of the flexible cylinder in the single-layer and two-layer fluids may be quite different when the density ratio and the depth of the lower layer are large.