Advancing Fluid Dynamics Stability Analysis: Construction of Lyapunov Functions via the Generalized Kinetic Energy Approach

Abstract: The energy method, also known as the Reynolds-Orr equation, is widely utilized in predicting the unconditional stability threshold of shear flows owing to the zero contribution of nonlinear terms to the time derivative of perturbation kinetic energy. However, it often underestimates the critical Reynolds numbers compared to experimental measurements. On the other hand, linear stability analysis tends to yield impractically high limits due to the occurrence of subcritical transitions. A novel methodology is introduced to enhance and validate the generalized kinetic energy formulation, aiming to provide a more accurate estimation of transition. This method considers the influence of nonlinear terms in calculating the threshold amplitude. The efficacy of this approach is showcased through the utilization of basic low-order turbulence models and the Poiseuille flow as illustrative examples. Through the proposed technique, the objective is to bridge the disparity between theoretically predicted critical Reynolds numbers and experimental observations, thus providing a more precise evaluation of shear flow stability. This research contributes to the advancement of stability analysis methods, offering practical implications for diverse fluid flow scenarios.