Flow Instability due to Pipeline Leakage

Leakage in a pipeline is modelled by presence of one small hole on pipeline wall. The hole sets up a localized asymmetric pressure difference which alters the steady state velocity of the fluid flowing inside the pipeline. We use Reynolds's averaging technique and Prandtl's mixing length to study the turbulent flow perturbation to an initially laminar steady state flow. The resulting equations are two-dimensional by assuming negligible velocity gradient along the pipeline length. We propose a potential function expression to account for the angular dependence from the asymmetry. Results from the analysis give the leakage flow rate as a function of leakage radius and pressure difference between the pipe inside and the ambient.