Formulation of the general momentum equations for the actuator disc with a rotating wake

Based on the actuator disc representation of a rotor (or propeller), many simplified momentum theories have been developed and are usually coupled to blade element forces to provide a blade element momentum theory (BEM). The general momentum theory of Glauert, unlike most simplified theories, accounts for the sub-atmospheric pressure variation in the wake vortex but has lacked closure in an analytic formulation of the system of equations that would relate the flow field induction factors and wake expansion. The specific problem has been in expressing the balance in axial force (usually loosely described as the momentum equation) in an explicit way that accounts for the influence of a radial pressure gradient. Without approximation or CFD based empirical corrections, the effect of the radial variation of suction pressure in the wake as influencing the axial force balance is accounted. This enables formulation of a complete equation system relating the induction factors and wake expansion.