Linear forced sloshing for a baffled rectangular fuel tank

Abstract

Sloshing in a fuel tank of a commercial vehicle, which is subdivided by a perforated plate as baffle, is investigated to predict occuring forces. Using linear potential theory and a two-dimensional modal description, the eigenmodes and eigenfrequencies are determined for the baffled tank and compared to experimental results. Damping due to viscous boundary layers and the pressure drop at the baffle is examined and discussed with regard to experimental findings. MotivationSloshing, that is the dynamic behaviour of a fluid with a free surface, has to be considered in different applications like spacecrafts [1], liquid cargo carrying ships [2, Chap. 1] and tuned liquid dampers [3], for example. Regarding fuel tanks of commercial vehicles, special attention has to be paid to the three-dimensional excitation due to street unevenness and to the perforated plates used as baffles ( Fig. 1). Knowledge of the sloshing forces is indispensable for an accurate description of the dynamic behavior of the entire commercial vehicle. For an horizontal excitation of the fuel tank, free-surface modes and corresponding eigenfrequencies can be observed whereas vertical excitation leads to parameter excited vibrations [4].2 Linear forced sloshing Fig. 1: View inside a truck fuel tank [Daimler AG].x1 x A 2 A x B 2 B x3 h x z y z 217 188 Fig. 2: Rectangular tank with dimensions 317 mm×217 mm×188 mm (left) and baffle positions A and B, examined perforated plate (middle) and simplified plate with slits (right) [5].To describe the sloshing behaviour, linear potential theory is applied assuming an incompressible, inviscid, homogeneous fluid, irrotational fluid motion, neglectable surface tension and small excitation amplitudes and the fuel tank is approximated as rectangular (Fig. 2). Using a two-dimensional modal description, the motion of each mode is given by [4] β m (t) + 2ξ m ω mβm (t) + ω 2 m 1 +zin the case of linear viscous damping with eigenmode ϕ m (x, z), eigenfrequency ω m and free-surface elevation ζ(x, t) = ∞ m=1 β m (t)ϕ m (x, 0).ẍ 0 (t) andz 0 (t) denote the horizontal and vertical excitation, respectively. The hydrodynamic coefficients λ 1m = ρ ∂V f ϕ m (x, 0)xdx, µ m = ρ κm ∂V f ϕ m (x, 0) 2 dx and α m = ρ x ϕ m (x, 0)dx are integrals over the fluid surface ∂V f . The last (nonlinear) term on the left side represents the pressure drop ∆p = 1 2 ρK|u|u with approach velocity u and pressure drop coefficient K at the perforated plate [6]. The resulting horizontal sloshing force can be calculated by ResultsSloshing eigenfrequencies As can be seen from the given formulas, eigenfrequencies and eigenmodes as well as corresponding damping values have to be determined to predict the free-surface shape and occurring sloshing forces. Fig. 3 shows the measured amplitude reduction and frequency shift depending on the baffle geometry for a rectangular tank filled with a diesel