Effective Force Interaction Area and Energy Derivation of Flow Dynamic Pressure Force at an Arbitrary Angle of Attack

This paper presents a consistent force and energy analysis of the dynamic pressure of an incompressible liquid and resolves a fundamental methodological error that has persisted in classical hydrodynamics for several centuries and is reproduced in virtually all physics textbooks. The essence of this error lies in automatically equating the geometric cross-sectional area of a free jet with the effective area of its force interaction with a surface or vessel, which leads to a direct twofold contradiction between calculations based on Bernoulli's dynamic pressure and the law of conservation of momentum. Based on an analysis of pressure force work and flow kinetic energy, it is rigorously proven that when a liquid discharges from an orifice or decelerates normally against a barrier, the effective pressure distribution area is exactly twice the cross-sectional area of the flow itself. Within an inertial reference frame, a direct energy derivation of the jet impact force on a moving blade is performed for both normal incidence and an arbitrary angle of attack. The resulting generalized relationship accurately describes all flow-surface interaction regimes—from grazing flow along a plane and normal impact to complete jet reversal in Pelton-type turbines—without relying on artificial empirical corrections. this is an English version of the original article