Equation of motion for vibrating pipes with laminar and turbulent flow profiles

Non-uniform flow velocity distributions occur in real fluids in fluid-conveying pipes.The widely used equation of motion for fluid-conveying pipes,based on ideal flow was modified to account for non-ideal effects.For laminar flow in a circular pipe,the parabolic profile gives a flow profile modification factor for the centrifugal force term of 1.333.For turbulent flow in a circular pipe,both the exponential and logarithmic velocity profiles give similar results with the modification factors decreasing with increasing Reynolds number and lying in 1.018—1.053 for Re=103—105.The critical flow velocities for divergence in these cases were found to be 13.4%,0.9%—2.5% lower than for ideal fluid flow,while those for flutter are even lower,36% for the laminar flow profile.An equivalent flow velocity and an equivalent mass were developed for vibrating pipe flow problems with different flow profiles for solutions using the equation of motion for ideal fluid flow.