Solution of Glauert's contraction/expansion equations for wind turbines and powered rotors with swirl

Glauert (1934) developed a set of wake contraction or expansion equations for lifting rotors in either wind turbine or powered mode. Up until now, no one has been able to solve these equations. Goorjian and Wu (1972) proved that, for hover, Glauert's equations have no solution with a bounded wake rotation over the range 0 ≤ r̄ ≤ 1. Goorjian also showed that momentum theory is internally inconsistent and cannot result in a wake contraction solution that matches both the centrifugal force balance and conservation of angular momentum in the far wake, or, in other words, the stream tubes will mix together in the real world. In this work, it is shown that an internally consistent momentum theory can be developed if the pressure balance in the far wake is relaxed, Glauert actually makes this assumption later in his work but does not revisit the wake contraction implications. With this new, internally consistent momentum theory, it is shown that the wake contraction equations can be solved for both wind turbines and helicopters in climb, descent, or hover.