Free Vibration Sloshing Analysis in Axisymmetric Baffled Containers under Low-Gravity Condition

Abstract

The free vibrations analysis of liquid sloshing is carried out for arbitrary axisymmetric containers under low-gravity condition using boundary element method. A potential flow theory is used to model the flow field and the free-surface Laplace-Young equation is used to model the surface tension effect. The obtained governing equations are solved using eigenanalysis techniques to determine the natural frequencies and mode shapes of the sloshing liquid. The results for a circular cylindrical container are compared to the analytical values and very good agreement is achieved for the slipping and anchored contact line assumptions. Furthermore, some baffled containers are also analysed and the effects of baffles on the sloshing frequencies under low and zero gravity conditions are investigated and some conclusions are outlined.

Appendix

The method of calculation of the derivative operators which were used in “Free-Surface Governing Equation” is defined in this section in detail. For instance, consider vector F as follows:

$$ \textbf{F} = \left\{ {\begin{array}{*{20}{c}} {f(x_{-2})}\\ {f(x_{-1})}\\ {f(x_{0})}\\ {f(x_{1})}\\ {f(x_{2})} \end{array}} \right\} $$

(23)

where f is an unknown function and the distances between the points x ₋₂ to x ₂ are equal and named Δx. Using finite difference method, one can approximate the derivative of this function at the mentioned points. This approximation is implemented in each point using the values of the function at that point and the neighbor points based on the central, forward, and backward approximation techniques. To calculate this approximation, one can use different numbers of neighbor points. For example, the value of the first and second derivative of f at the point x ₋₂ can be approximated using the forward method as follows (Fornberg 1988):

$$ {{f^{\prime}}(x_{-2})}\approx \frac{{ - \frac{{11}}{6}f({x_{-2}}) + 3f({x_{-1}}) - \frac{3}{2}f({x_{0}}) + \frac{1}{3}f({x_{1}})}}{{\Delta x}} $$

(24a)

$$ {{f^{\prime\prime}}(x_{-2})}\approx \frac{{2f({x_{-2}}) -5f({x_{-1}})+4f({x_{0}})-f({x_{1}})}}{{({\Delta} x)^{2}}} $$

(24b)

and for the point x ₂ based on the backward method the following approximations can be used

$$ {{f^{\prime}}(x_{2})}\approx \frac{{\frac{{11}}{6}f({x_{2}}) - 3f({x_{1}})+\frac{3}{2}f({x_{0}}) -\frac{1}{3}f({x_{-1}})}}{{\Delta x}} $$

(24c)

$$ {{f^{\prime\prime}}(x_{2})}\approx \frac{{2f({x_{2}}) -5f({x_{1}})+4f({x_{0}})-f({x_{-1}})}}{{({\Delta} x)^{2}}} $$

(24d)

In these equations three neighbor points are used for approximation; it is possible to use more or less number of neighbor points for this purpose. Table 1 presents the coefficients of the forward approximation. The Accuracy column in this table shows the order of accuracy. The coefficients of the backward approximations are achieved by changing the signs of the coefficients only for the odd derivatives.

Table 1 The coefficients of the forward approximation (Fornberg 1988)

For the medial points one may use the central approximation. For instance, the value of the first and second derivative of f at the point x ₀ can be approximated as follows:

$$ {{f^{\prime}}(x_{0})}\approx \frac{{ - \frac{{1}}{2}f({x_{-1}}) + \frac{1}{2}f({x_{1}})}}{{\Delta x}} $$

(24e)

$$ {{f^{\prime\prime}}(x_{0})}\approx \frac{{ f({x_{-1}}) -2f({x_{0}}) + f({x_{1}})}}{{({\Delta} x)^{2}}} $$

(24f)

The coefficients of the central approximation technique are presented in Table 2.

Table 2 The coefficients of the central approximation (Fornberg 1988)

To calculate the first and second derivatives for all five points presented in Eq. 23, there would be five equations which can be presented in the following matrix form set of equations (with second-order accuracy):

$$ \left\{ {\begin{array}{*{20}{c}} {f^{\prime}({x_{- 2}})}\\ {f^{\prime}({x_{- 1}})}\\ {f^{\prime}({x_{0}})}\\ {f^{\prime}({x_{1}})}\\ {f^{\prime}({x_{2}})} \end{array}} \right\} = \frac{1}{2{\Delta x}}\left[ {\begin{array}{*{20}{c}} {-3}&4&{-1}&0&0\\ {-1}&0&1&0&0\\ 0&{ - 1}&0&1&0\\ 0&0&{ - 1}&0&1\\ 0&0&1&{ - 4}&3 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {f({x_{-2}})}\\ {f({x_{-1}})}\\ {f({x_{0}})}\\ {f({x_{1}})}\\ {f({x_{2}})} \end{array}} \right\}={\textbf{D}^{(1)}}\textbf{F} $$

(25a)

$$ \left\{ {\begin{array}{*{20}{c}} {f^{\prime\prime}({x_{- 2}})}\\ {f{\prime\prime}({x_{- 1}})}\\ {f{\prime\prime}({x_{0}})}\\ {f{\prime\prime}({x_{1}})}\\ {f{\prime\prime}({x_{2}})} \end{array}} \right\} = \frac{1}{{{{({\Delta} x)}^{2}}}}\left[ {\begin{array}{*{20}{c}} 1&{ - 2}&1&0&0\\ 1&{ - 2}&1&0&0\\ 0&1&{ - 2}&1&0\\ 0&0&1&{ - 2}&1\\ 0&0&1&{ - 2}&1 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {f({x_{- 2}})}\\ {f({x_{- 1}})}\\ {f({x_{0}})}\\ {f({x_{1}})}\\ {f({x_{2}})} \end{array}} \right\} = {\textbf{D}^{(2)}}\textbf{F} $$

(25b)

Although the matrices D ⁽¹⁾ and D ⁽²⁾ are determined with second-order accuracy in this section and this accuracy is appropriate for the problem which is studied in this paper, but these matrices may be calculated with higher accuracy based on the coefficients presented in Tables 1 and 2.