Stability of a rotating cylindrical shell containing axial viscous flow

The present thesis studies the stability of a rotating cylindrical shell containing a co-rotating axial viscous flow. The system can be thought of as a long thin-walled pipe carrying an internal axial flow while the whole is in a frame of reference rotating at a prescribed rate. The equations of the previously solved inviscid model are rederived and the problem is studied further. The results obtained for purely axial flow are reproduced, but as expected from literature, it is impossible to obtain satisfactory results for the system subjected to rotation due to the presence of singularities in the flow pressure solution. A hypothetical physical explanation for these singularities is put forward and has similarities with the phenomenon of atmospheric flow blocking. Considering the unsuccessful results obtained with the inviscid theory, it is believed that the added realism brought in by the introduction of viscosity in the theory can lead to a successful model. Assuming a travelling-wave perturbation scheme, the linear Donnell-Mushtari thin shell equations are coupled with the fluid stresses obtained by solving numerically the incompressible Navier-Stokes equation for a laminar or turbulent flow. A novel triple-perturbation approach is established to consider the interaction between the fluid and the structure. This triple-perturbation approach is in essence a superposition of three fluid fields caused by the three components of the shell deformation for a given oscillation mode. It is found that the usual technique for linear aeroelasticity studies consisting of applying the fluid boundary conditions at the undeformed position of the wall instead of the instantaneous deformed position greatly alters the stability of the system. To remedy to this problem, three different corrections are applied and tested on the carefully derived model. The dynamics of the system subjected to purely axial flow with no rotation is successfully studied with the viscous model for both laminar and turbulent flow conditions. Because no experimental or previous theoretical data is available, it is impossible to validate the results obtained in the laminar regime. For the turbulent regime, as the Reynolds number is increased, the results tend more and more towards those obtained with the inviscid theory. The results obtained for small rates of rotation show that both in the laminar and in the turbulent regime, the system tends to be stabilised when subjected to a small rate of rotation. On the other hand, this tendency should be reversed for higher rates of rotation, but it is impossible to show this due to the limitations of the root-finding method employed.

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