The flow inside a fluid damper where a piston reciprocates sinusoidally inside an outer casing containing high-viscosity silicone oil is simulated using a finite volume method, at various excitation frequencies. The oil is modeled by the Carreau-Yasuda (CY) and Phan-Thien and Tanner (PTT) constitutive equations. Both models account for shear-thinning, but only the PTT model accounts for elasticity. The CY and other generalised Newtonian models have been previously used in theoretical studies of fluid dampers, but the present study is the first to perform full two-dimensional (axisymmetric) simulations employing a viscoelastic constitutive equation. It is found that the CY and PTT predictions are similar when the excitation frequency is low, but at medium and higher frequencies, the CY model fails to describe important phenomena that are predicted by the PTT model and observed in experimental studies found in the literature, such as the hysteresis of the force-displacement and force-velocity loops. Elastic effects are quantified by applying a decomposition of the damper force into elastic and viscous components, inspired from large amplitude oscillatory shear theory. The CY model also overestimates the damper force relative to the PTT model because it underpredicts the flow development length inside the piston-cylinder gap. It is thus concluded that (a) fluid elasticity must be accounted for and (b) theoretical approaches that rely on the assumption of one-dimensional flow in the piston-cylinder gap are of limited accuracy, even if they account for fluid viscoelasticity. The consequences of using lower-viscosity silicone oil are also briefly examined.
REFERENCES
The temperature rise of the fluid due to viscous dissipation can be important and deserves a separate study. It is investigated theoretically in Refs. 29 and 30 and experimentally in Ref. 13. In Ref. 13, experiments performed with a damper of much larger size than the present model showed that if the amplitude of oscillation is significantly greater than the piston diameter then the oil temperature in the neighbourhood of the piston can rise by 50 or more degrees Celsius during 6 oscillation cycles (the temperature rise is smaller farther away from the piston), which leads to a 10% drop in the maximum damper force during the same period. A force drop is expected because such a temperature rise would reduce the oil viscosity to half its room temperature value.10,31 However, other experiments (again in Ref. 13) with a smaller damper showed a negligible effect of the temperature rise on the force magnitude, so this issue requires further investigation.
The “microscopic” fluid element is still much larger than the molecular dimensions so that the fluid can be regarded as a continuum.
The instantaneous energy balance for the damper fluid includes the rate of work of the force Ff l, the rate of viscous dissipation of energy into heat, and the rate of change of energy stored in the fluid in the form of elastic energy. However, considering a full period of oscillation, if the flow has reached the periodic state, the stored elastic energy at time t is exactly equal to that at time t + T. Therefore, the work of Ff l during a complete cycle is equal to the amount of energy dissipated to heat by viscous action during the same period.
Note, however, that the lPTT-10 geometry differs slightly from that of the lPTT-100 and lPTT-500 simulations, and comparing Rec, De, and Wi values between different geometries is not a completely valid way of assessing the relative importance of the effects that these numbers quantify (see our previous publication, Ref. 22).
For the f = 2 Hz simulation, we reduced the time step to Δt = T/32 000 to ensure that it is quite smaller than the relaxation time; this results in λ/Δt ≈ 45.
The Lambert W function is double-valued on the interval (−1/e, 0), but here 2ab2 > 0 and there is only one branch to follow.
