Resonant sloshing in an upright annular tank

Abstract

Resonant sloshing in an upright annular tank is studied by using a new nonlinear modal theory, which is complete within the framework of the Narimanov–Moiseev asymptotics. The applicability is justified for a fairly deep liquid (the liquid-depth-to-outer-tank-radius ratio $1.5\lesssim h=\bar{h}/\bar{r}_{2}$) and away from the non-dimensional inner radii $r_{1}=\bar{r}_{1}/\bar{r}_{2}=0.08546$, 0.17618, 0.27826, 0.31323, 0.31855, 0.43444, 0.46015, 0.48434, 0.68655, 0.70118. The theory is used to describe steady-state (stable and unstable) resonant waves due to a harmonic excitation with the forcing frequency close to the lowest natural sloshing frequency. We show that the surge-sway-pitch-roll excitation is always of either longitudinal or elliptic type. Existing experimental results on the horizontally excited steady-state wave regimes in an upright circular tank ($r_{1}=0$) are utilised for validation. Inserting an inner pole with the radii $r_{1}\approx 0.25$ and 0.35 ($1.5\lesssim h$) causes that no stable swirling and/or irregular waves exist. The response curves for an elliptic-type excitation are examined versus the minor-axis forcing-amplitude component. Stable swirling is then expected being co- and counter-directed to the angular forcing direction. Passage to the rotary (circular) excitation keeps the co-directed swirling stable for all resonant forcing frequencies but the stable counter-directed swirling disappears.

The damped sloshing in an upright circular tank due to an orbital forcing

“…3 shows that the latter branch disappears, as δ increases. This is a very interesting fact, which contradicts the steady-state analysis of the undamped sloshing in [2], where both the co-and counter-directed angular progressive waves exist and can be stable in certain frequency ranges for any 0 1 < δ . In summary, the linear viscous damping matters for the orbitally-excited sloshing in bioreactors of an upright circular cylindrical shape.…”

“…makes it possible to derive an approximate system of ordinary differential equations (nonlinear modal equations [2]) with respect to the free-surface generalized coordinates ( ) [3], which provides a rather accurate theoretical prediction of the logarithmic decrements of the natural sloshing modes due to the boundary layer and the bulk viscosity. The 2π-periodic solutions of the modified modal system describe the resonant steady-state sloshing.…”

“…The 2π-periodic solutions of the modified modal system describe the resonant steady-state sloshing. To find the asymptotic steady-state solutions, we use the Bubnov-Galerkin procedure [2,4] by posing the lowest-order components of the primary resonantly excited modes as…”

“…The Bubnov-Galerkin procedure leads to a necessary solvability condition with respect of a , a , b , and b appearing as a system of nonlinear algebraic equations [2,4,5]. To describe the steady-state sloshing, we should solve the system for any 11 11 / σ = σ σ close to 1.…”

“…The first Lyapunov method can be used to study the stability. The algebraic system is rederived in terms of the integral amplitudes A, B (the main wave elevation components in the Ox and Oy directions, respectively) and the phase-lags , ψ ϕ : ε > ε = ξ = was analyzed in [2,4]. A planar standing wave and the swirling are identified.…”