On the initial configurations of collapsible channel flow

This paper studies the effect of the initial configurations of the governing equations on flows in a collapsible channel where the upper elastic wall is replaced by a pre-stretched beam. The aim is to check the existence of a “tongue” shaped neutral stability curve in the Reynolds number–tension space from a fluid-beam model [Luo XY, Cai ZX. Effects of wall stiffness on the linear stability of flow in an elastic channel. In: de Langre E, Axisa F, editors. Proceedings of the eighth international conference on flow-induced vibrations, FIV2004, vol. II. Paris, France: 2004. p. 167–70], in a properly formulated initial strain configuration. It was found that, for a given Reynolds number, as the tension is lowered to a critical value, the system becomes unstable, which is to be expected. However, a further decrease of the tension re-stabilizes the system before it becomes unstable again. It was possible that this puzzling finding was an artefact since the elastic equations used in the model were not properly derived from the zero initial stress configuration (Ogden, private communication). To check this, in this paper, a range of steady solutions are studied with both zero and non-zero initial wall tension. These are compared with the results using the finite element package Adina 8.3 using both the initial strain and initial stress configurations. As expected, the fluid-beam model agrees with Adina when using the initial stress configuration, but not when using the initial strain configuration. For cases with a small initial tension or small deformation (very large initial tension), both initial stress and initial strain configurations lead to very similar results, however, when the initial tension is comparable with the stretching induced tension, there are obvious differences in these two configurations. The “tongue” stability curve is then re-calculated with a zero initial tension, and re-plotted in the Reynolds number–effective tension space. It is interesting to see that though slightly different in shape, the “tongue” stable zone appears again when the zero initial tension is used. Thus it is highly likely that the puzzling “tongue” in the neutral stability curve is not due to the modelling approximation, but indicating a real, interesting physical phenomenon.