Dynamic fluid sloshing in a one-dimensional array of coupled vessels

This paper investigates the coupled motion between the dynamics of $N$ vessels coupled together in a one-dimensional array by springs and the motion of the inviscid fluid sloshing within each vessel. We develop a fully nonlinear model for the system relative to a moving frame such that the fluid in each vessel is governed by the Euler equations and the motion of each vessel is modeled by a forced spring equation. By considering a linearization of the model, the characteristic equation for the natural frequencies of the system is derived and analyzed for a variety of nondimensional parameter regimes. It is found that the problem can exhibit a variety of resonance situations from the $1\text{:}1$ resonance to $(N+1)$-fold $1\text{:}\cdots \text{:}1$ resonance, as well as more general $r\text{:}s\text{:}\cdots \text{:}t$ resonances for natural numbers $r,s,t$. This paper focuses in particular on determining the existence of regions of parameter space where the $(N+1)$-fold $1\text{:}\cdots \text{:}1$ resonance can be found.

Figure 1

Schematic diagram of the one-dimensional array of $N$ rectangular vessels coupled together by springs.

Figure 2

Plot of $D_2\left(s\right)$, for the identical, symmetric system for $R_1=0.5,G_1=1.0,$ and (a) $({\delta }_1,G_2)=(0.1,0.0)$, (b) $(0.1,1.0)$, (c) $(0.1,3.0),$ and (d) $(0.01,1.0)$. The circles indicate the roots of $D_2\left(s\right)=0$.

Figure 3

Plot of the first four roots of (4.2), for the identical, symmetric system as a function of $G_2$ for $R_1=0.5,G_1=1.0,$ and ${\delta }_1=0.1$. The dashed lines give the asymptotic form of $s_n$ from (4.3). The asymptotic and numerical results for $s_1$ and $s_3$ are indistinguishable.

Figure 4

Contour plot of the value of $G_{1\max }\ge 0$ from (4.4) as a function of $(R_1,{\delta }_1)$, for (a) $m=1$ and (b) $m=2$.

Figure 5

Plot of $D_2\left(s\right)$, for the nonsymmetric system for $R_1=R_2=0.5,G_1=1.0,{\widehat{L}}_2=M_2=1,$ and (a) ${\delta }_1={\delta }_2=0.1$ with $(G_2,G_3)=(0.0,3.0)$, (b) ${\delta }_1={\delta }_2=0.1$ with $(G_2,G_3)=(3.0,3.0)$, (c) ${\delta }_1=2{\delta }_2=0.1$ with $(G_2,G_3)=(1.0,1.0),$ and (d) ${\delta }_1=4{\delta }_2=0.1$ with $(G_2,G_3)=(1.0,1.0)$. The circles indicate the roots of $D_2\left(s\right)=0$.

Figure 6

Plot of the first four roots of $D_2\left(s\right)=0$, for the nonsymmetric system as a function of $G_2$ for $R_1=R_2=0.5,G_1=1.0,G_3=3.0,$ and ${\delta }_1={\delta }_2=0.1$. The dashed lines give the asymptotic form of $s_n$ from (4.3).

Figure 7

Plot of (a) the resonance contour (4.5) for the nonsymmetric, two-vessel system in the $(G_1,G_2)$ plane for $G_3=3.0,{\widehat{L}}_2=M_2=1,m=1,{\delta }_1={\delta }_2=0.1$ with $R_1=R_2=1.0$ (solid line) and $R_1=R_2=0.5$ (dashed line). The remaining panels show contour plots of $G_2$ for (b) the case shown in panel (a), (c) the case shown in panel (a) but with $m=2,$ and (d) the case $G_3=3.0,{\widehat{L}}_2=M_2=1,m=1,$ and $R_1=R_2=1.0$. The arrows indicate the 1:1:1 resonance value. The “in” and “out” in each region signifies whether the 1:1 resonance is an in-phase or out-of-phase mode, respectively.

Figure 8

Plot of (a) $D_3\left(s\right)$ and (b) $D_8\left(s\right)$ for the symmetric system when $R_i=0.5,{\delta }_i=0.1,{\widehat{L}}_i=M_i=1$ for $i=1,...,N$ and $G_i=1.0$ for $i=1,...,N+1$.

Figure 9

Contour plot of $G_2$ showing the regions of the $(R_1,G_1)$ parameter space where a possible $(N+1)$-fold $1:\cdots :1$ resonance can exist for ${\delta }_1=0.1,m=1$ with (a) $N=3$ and (b) $N=8$. Here $R_i=R_1,{\delta }_i={\delta }_1,{\widehat{L}}_i=M_i=1\forall i\ge 2$ and $G_i=G_1,\forall i\ge 3$. Panels (c) and (d) plot contours of ${\widehat{c}}_2$ and ${\widehat{c}}_3$, respectively, for the $N=3$ result in panel (a).