This study investigates the fluid-structure interaction of two coaxial cylinders separated by a Newtonian fluid under turbulent axial flow. The theoretical framework treats the inner cylinder as a rigid body mounted on a flexible blade modeled as a Rayleigh beam. The goals of this study are to determine the free vibration modes and frequencies, identify the fluid-elastic instability threshold, and establish an analytical expression for the mean-square displacement of the structure. The approach integrates various fluid forces and torques, such as Archimedean thrust, fluid-elastic forces for a quiescent fluid, fluid-elastic forces due to flow, and the effects of fluid turbulence. The new approach reveals that vibration modes, frequencies, instability thresholds, and meansquare displacement each depend on a different set of dimensionless parameters: 8, 11, and 12, respectively. These parameters include the cylinder aspect ratio and fluid gap radius ratio. By incorporating models from the literature for viscous friction coefficients, turbulent pressure power spectral density, and coherence function, the study demonstrates stability conditions and the scaling of mean-square displacement with Reynolds number squared. The study, presented in a fully dimensionless formulation, aims to assist engineers in constructing small-scale experiments representative of pressure vessel vibrations. To facilitate this, a Python code for system stability determination and mean-square displacement calculation is provided.