2.1 Liquid vibrations in cylindrical tanks with flexible membranes covering free surfaces

In this section free vibrations of a liquid in a rigid cylindrical tank with a flexible membrane covering the free surface are considered, Fig. 1a). Suppose that the liquid in the container is an ideal and incompressible one, and its motion is irrotational. Then the relative liquid velocity V has a potential $\overline{\mathrm{\Phi }}=\overline{\mathrm{\Phi }}\left(r,\theta ,z,t\right)$ , so that $\mathit{V}=\mathrm{\nabla }\overline{\mathrm{\Phi }}$ .

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Fig. 1

Cylindrical tank with flexible membranes and levels of membrane installation.

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In this paper the linear water wave theory is in use. In above formulated suppositions the velocity potential satisfies the Laplace equation

The following boundary value problem (BVP) is formulated for the velocity potential $\overline{\mathrm{\Phi }}$ . In the fluid domain it satisfies the Laplace equation (1). At the wall $r=R$ and at the bottom $z=0$ of the cylindrical tank the impermeability conditions are formulated as follows:

To formulate the boundary condition on the free surface, consider at first the following equation of the membrane motion:

where w is a membrane deflection, T is a tension per unit length, μ is mass per unit area of the membrane, p is the liquid pressure on the membrane.

The following boundary conditions are formulated for membrane deflections, at r = R:

On the free liquid surface at $z=h$ the equality of normal components of liquid and membrane velocities have to be satisfied

as well as the dynamic boundary condition

So, we have coupled boundary value problem (1)-(6) for evaluating the unknown functions Φ and w.

Consider harmonic oscillations and assume that $$\overline{\mathrm{\Phi }}(r\text{,}\theta \text{,}z,t\text{)}=\stackrel{\sim }{\mathrm{\Phi }}(\text{r,}\theta \text{,}z{\text{)e}}^{i\overline{\omega }t}$$

Using the separation of variables method, we get

where $k_{\mathit{mn}}$ are zeros of the first derivative of Bessel’s function $J_m^{\prime }\left(\mathit{kr}\right)=0$ at $r=R$ . From (6) we have the following expression for pressure: $$p=e^{i\overline{\omega }t}\left[-\rho i\overline{\omega }\sum _{m=0}^{\infty }\sum _{n=1}^{\infty }{A}_{\mathit{mn}}{J}_m\left(\frac{k_{\mathit{mn}}r}{R}\right)cos\left(m\theta \right)cosh\left(\frac{k_{\mathit{mn}}h}{R}\right)\right]-\rho gw.$$

For flexible membrane we suppose that

It would be noted that membrane thickness is negligible. Using above equations for p and w in membrane equation (4), we get

which is a non-homogeneous differential equation. Solution for membrane deflections of Eq. (10) is following:

Condition (5) at $z=h$ gives

Using condition w = 0 at r = R from (5), we get

So, we have equations (12) and (13) to determine unknown coefficients $A_{\mathit{mn}},B_m$ and unknown frequency $\overline{\omega }$ .

For a fixed mode m the truncated series are received as

There are constant coefficients before unknowns $A_{\mathit{mn}},B_m,n=1,..,N$ in equation (14), and variable ones in equation (15), so we have to choose N points including end points in $0⩽r⩽R$ namely,

r_i=(Ri)/(N −1), I = 0,1,…N

at the free surface for satisfying equation (15). Then we obtain the following N + 1 homogeneous algebraic equations in the coefficients $\mathit{X}=A_{m1},A_{m2},...,A_{\mathit{mN}},B_m$ :

In (17) we suppose that n₁ = 0,1,…N-1.

In matrix form system (16)-(17) can be written as

The condition of a non-trivial solution of system (18) gives the non-linear frequency equation (the equality of the system determinant to zero)

for evaluating the frequencies $\overline{\omega }$ .

2.2 Liquid vibrations in cylindrical tanks with internal flexible membranes

Free vibrations of the liquid in the cylindrical tank with an internal flexible membrane are considered, Fig. 1b). The incompressible and inviscid liquid is supposed to perform irrotational motion in the fluid domain divided into two sub-domains by internal flexible membrane that is installed at the height h₁, Fig. 1c). The first fluid sub-domain is confined by bottom, the lower part of the cylindrical wall, and the flexible membrane. The liquid velocity in this domain is described by potential ${\overline{\mathrm{\Phi }}}_1$ . The second fluid sub-domain is bounded by the flexible membrane, the upper part of the cylindrical wall, and the free surface. The liquid velocity in this domain is described by potential ${\overline{\mathrm{\Phi }}}_2$ .

Two BVP are formulated to determine these potentials. So, for potential ${\overline{\mathrm{\Phi }}}_1$ we have the Laplace equation

with wall and bottom impermeability conditions as follows:

$\frac{\partial {\overline{\mathrm{\Phi }}}_1}{\partial r}=0$ , r = R, 0 < z < h₁, $\frac{\partial {\overline{\mathrm{\Phi }}}_1}{\partial z}=0$ , z = 0. (21)

For potential ${\overline{\mathrm{\Phi }}}_2$ we have the analogical BVP with Laplace’s equation

and the next impermeability conditions at the cylindrical wall:

At the free surface $z=h_{\text{2}}$ we have

To formulate the boundary conditions on the flexible membrane at $z=h_1$ , consider the equation of membrane motion as follows:

The impermeability conditions at $z=h_1$ are following:

The additional boundary conditions are formulated for membrane deflections, at r = R

For pressure components at z =  $h_1$ we consider

Assuming that

insert expressions (28) for $p_k\left(k=1,2\right)$ into membrane equation (25). Taking into account equations (20), (22), we get at z =  $h_1$

Assuming that

and using the separation of variables method, we get

and considering at z =  $h_1$ the following boundary condition $$\frac{\partial {\overline{\mathrm{\Phi }}}_1}{\partial z}=\frac{\partial {\overline{\mathrm{\Phi }}}_2}{\partial z}$$

we get truncated series for the fixed mode m $$\sum _{n=1}^N\frac{k_{\mathit{mn}}}{R}g\left(B_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)+C_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}r}{R}\right)-$$ $$-\sum _{n=1}^N{\overline{\omega }}^2\left(B_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)+C_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}r}{R}\right)=0$$ $$\sum _{n=1}^N{A}_{\mathit{mn}}\frac{k_{\mathit{mn}}}{R}{J}_m\left(\frac{k_{\mathit{mn}}r}{R}\right)sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)=\sum _{n=1}^N\frac{k_{\mathit{mn}}}{R}\left(B_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+C_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}r}{R}\right),$$ $$\sum _{n=1}^N{A}_{\mathit{mn}}\left[\left({\left(\frac{k_{\mathit{mn}}}{R}\right)}^3+\frac{{\overline{\omega }}^2}{T}\frac{k_{\mathit{mn}}\mu }{R}\right)sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+\frac{\rho {\overline{\omega }}^2}{T}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right]J_m\left(\frac{k_{\mathit{mn}}r}{R}\right)=$$ $$=\sum _{n=1}^N\frac{\rho {\overline{\omega }}^2}{T}\left(B_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+C_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}r}{R}\right).$$

Then we choose N points including end points in $0⩽r⩽R$ at the free surface, namely r_i = (Ri)/(N-1), i = 0,1,…N and obtain the following 3 N homogeneous algebraic equations in the coefficients $\mathit{Y}=A_{m1},A_{m2},...,A_{\mathit{mN}},B_{m1},B_{m2},...,B_{\mathit{mN}},C_{m1},C_{m2},...,C_{\mathit{mN}}$ : $$\sum _{n=1}^N\frac{k_{\mathit{mn}}}{R}g\left(B_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)+C_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}{n}_1}{N-1}\right)-$$ $$-\sum _{n=1}^N{\overline{\omega }}^2\left(B_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)+C_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_2}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}{n}_1^{}}{N-1}\right)=0$$

for $n_1=0,1,...,N-1$ , $$\begin{array}{c}\sum _{n=1}^N{A}_{\mathit{mn}}\frac{k_{\mathit{mn}}}{R}{J}_m\left(\frac{k_{\mathit{mn}}{n}_1}{N-1}\right)sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)=\\ =\sum _{n=1}^N\frac{k_{\mathit{mn}}}{R}\left(B_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+C_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}{n}_1}{N-1}\right),\end{array}$$

for $n_1=0,1,...,N-1$ , $$\sum _{n=1}^N{A}_{\mathit{mn}}\left[\left({\left(\frac{k_{\mathit{mn}}}{R}\right)}^3+\frac{{\overline{\omega }}^2}{T}\frac{k_{\mathit{mn}}\mu }{R}\right)sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+\frac{\rho {\overline{\omega }}^2}{T}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right]J_m\left(\frac{k_{\mathit{mn}}{n}_1}{N-1}\right)=$$ $$=\sum _{n=1}^N\frac{\rho {\overline{\omega }}^2}{T}\left(B_{\mathit{mn}}cosh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)+C_{\mathit{mn}}sinh\left(\frac{k_{\mathit{mn}}{h}_1}{R}\right)\right)J_m\left(\frac{k_{\mathit{mn}}{n}_1^{}}{N-1}\right),$$

for $n_1=0,1,...,N-1$ .

In matrix form this system can be written as

The condition of a non-trivial solution of system (34) gives the non-linear frequency equation (the equality of the system determinant to zero)

for evaluating the frequencies $\overline{\omega }$ .

3.1 Validation study

Numerical results are obtained by two methods. First, we obtain frequencies $\overline{\omega }$ for both considered problems by solving the non-linear equations (19), (35) using combination of Newton and parabolic interpolation methods (Choudhary and Bora, 2017).

To validate our results, the methods based on coupled FEM and BEM methods (Gnitko et al, 2017; 2019; Jamalabadi, 2020), are in use.

Consider the rigid partially filled cylindrical tank with radius R = 0.5 m and filling level h = 1 m. Let the liquid density be ρ = 998 kg/m³.

To testify our numerical results, we compare data for the first non-axisymmetric frequency, obtained by boundary (BEM) and finite (FEM) elements methods with analytical values of Ibrahim (2005), obtained by the following formula $${\omega }_k=\sqrt{g\frac{{\mu }_k}{R}tanh\left(\frac{{\mu }_k}{R}H\right)}/2\pi$$ where ${\mu }_k$ are roots of the equation $J_1^{\prime }\left(x\right)=0$ , J₁(x) is the Bessel function of the first kind.

The comparison of numerical results obtained by using coupled BEM and FEM methods with analytical ones is presented in Table 1. The results demonstrate good agreement. In BEM the one-dimensional elements with constant approximation of densities are applied. It would be noted that considered frequency is the lowest one of liquid vibrations in the tank.

Fundamental sloshing modes of fluid vibrations are shown in Fig. 2.

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Fig. 2

Fundamental sloshing modes and frequencies of liquid in the cylindrical tank,

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Next, we consider vibrations of flexible membranes of different materials, without interaction with liquids. Consider the clamped silicon membrane with radius R = 0.5 m, thickness h_m = 0.001 m, material density ρ_m = 2800 kg/m³, Young modulus E = 50Mpa, Poison’s ratio ν = 0.49 and the Eva plastic membrane with radius R = 0.5 m, thickness h_m = 0.001 m, material density ρ_m = 950 kg/m³, Young modulus E = 24.5 MPa, and Poison’s ratio ν = 0.48.

The BVP is solved for the following equation:

with boundary conditions (5). First modes of silicon and Eva plastic membranes that correspond to axisymmetric and non-axisymmetric vibrations are shown in Fig. 3 and Fig. 4, respectively.

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Fig. 3

Two first modes and frequencies of plastic membranes axisymmetric vibrations.

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Fig. 4

Two first modes and frequencies of plastic membranes vibrations of first harmonic.

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For validation study the present BEM-FEM approach is also employed to simulate the hydro-elastic frequencies in an upright cylindrical tank of radius R = 1 m with an elastic free-surface membrane.

Frequencies ω²_ijR/g, associated with the first three vibration modes corresponding to the axisymmetric circumferential mode, i = 0;j = 1,2,3, and to non-axisymmetric mode i = 1 and j = 1,2,3 at different filling levels, h/R, ranging from 0.1 to 0.5 for liquid density equal to ρ = 1000 kg/m³ were calculated for comparison with data (Kolaei and Rakhej, 2019). The results were obtained for μ = 1 kg/m² and T = 10 N/m. Tables 2 shows the first normalized frequencies for different numbers of circumferential mode at different filling levels and relative difference δ between solutions obtained by proposed method and data from (Kolaei and Rakhej, 2019).

3.2 Coupled membrane and liquid vibrations in cylindrical tanks

Suppose that the flexible membrane is installed into cylindrical tank, or covered the free surface, Fig. 1. Both silicon and Eva plastic membranes are examined. Fig. 5 demonstrates changing in values of frequencies for axisymmetric vibrations of Eva plastic membrane and liquid in dependence of the installation level h₁, Fig. 1b) and Fig. 1c). The system “Liquid- Membrane” performs coupled vibrations, and mutual influence of both components is sufficient.

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Fig. 5

First natural frequencies of the Eva plastic membrane and the liquid free surface (ω) via level (h₁).

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As follows from numerical simulations, if the membrane is installed inside the cylinder, then the most important parameter affecting the result, is the height h₁ of the membrane installation. If the membrane is installed at a considerable distance from the free surface, then the sloshing frequency practically does not change, and more precisely, it slightly increases. This phenomenon is clearly demonstrated in Fig. 5, between 0 and 0.85 value of membrane installation level. Solid lines correspond to calculations made by using BEM-FEM software, points correspond to calculations made by using analytical approach, described above, with N = 40. The results are in good agreement. The difference between analytical and numerical results is near 0.001. When the membrane is near the free surface, the sloshing frequency drops significantly, approaching the membrane frequencies. That is clearly demonstrated in Fig. 5, in the interval from 0.85 to 0.99 values of the membrane installation level.

Also, depending on the vertical arrangement of the membrane, the sloshing modes are also changed.

At low levels of the membrane installation both sloshing modes and frequencies resemble the modes and frequencies of the tank without membranes. When the membrane is close to the free surface, the modes of the latter are exposed to the modes of the membrane, and look different, see Fig. 6a).

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Fig. 6

First modes and frequencies: a) first sloshing mode at h₁ = 0,99 m, b) first mode and frequency of membrane covered the liquid free surface.

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When the membrane is placed on the liquid free surface, the modes and frequencies of the membrane change drastically. Instead of the axisymmetric mode, the mode of first harmonic becomes corresponding to the lowest frequency, Fig. 6b).

The frequencies of the membrane installed in the fluid are significantly reduced due to the added mass of fluid. The lowest frequency of the membrane is obtained when the membrane is near the bottom of the cylinder. With increasing the level of the membrane installation its frequency increases up to the level of installation of 0.5 m. Further, the membrane frequency decreases up to the level of its installation on the free surface of the liquid, where a sharp decrease in the frequency of the membrane occurs. When installing the membrane on the surface of the liquid, the sloshing frequency and the membrane frequency tend to be equal.

The silicon membrane installed in liquid-filled cylindrical tank shows the similar characteristics.

Fig. 7 demonstrates changing values of frequencies of silicon membrane and liquid in dependence of the installation level h₁.

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Fig. 7

First natural frequencies of the silicon plastic membrane and the liquid free surface (ω) via level (h₁).

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In case of placing the membrane directly on the free surface, the frequencies and modes of the sloshing practically coincide with ones of the membrane, Fig. 8 and present modes and frequencies of this coupled “Liquid- Membrane” system. As a result, the lowest frequencies of the system with silicon roof on waterline are equal to ω₁ = 0.0763 Hz, ω₂ = 0.3081 Hz, ω₃ = 0.7645 Hz. The lowest frequencies of the system with Eva plastic silicon roof on waterline (without free surface), are equal to ω₁ = 0.0527 Hz, ω₂ = 0.214 Hz, ω₃ = 0.523 Hz.

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Fig. 8

First modes and frequencies of “Liquid-Membrane system”with silicon plastic roof on waterline.

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These results show the significant decrease in the frequency of sloshing with membrane on the waterline, compared with installing the membrane at the level of 0.99 m, and the significant increasing in the frequency of the membrane on the waterline compared with internal membrane at the same installation level h₁ = 0.99 m.