Because compressibility effects arise in real applications of propellers and turbines, the Actuator Disk Theory or Froude’s Momentum Theory was established for compressible, subsonic flow using the three laws of conservation and isentropic thermodynamics. The compressible Actuator Disk Theory was established for the unducted (bare) and ducted cases in which the disk was treated as the only assembly within the flow stream in the bare case and enclosed by a duct having a constant cross-sectional area equal to the disk area in the ducted case. The primary motivation of the current thesis was to predict the ideal performance of a small ram-air turbine (microRAT), operating at high subsonic Mach numbers, that would power an autonomous Boundary Layer Data System during test flights. The compressible-flow governing equations were applied to a propeller and a turbine for both the bare and ducted cases. The solutions to the resulting system of coupled, non-linear, algebraic equations were obtained using an iterative approach. The results showed that the power extraction efficiency and the total drag coefficient of the bare turbine are slightly higher for compressible flow than for incompressible flow. As the free-stream Mach increases, the Betz limit of the compressible bare turbine slightly increases from the incompressible value of 0.593 and occurs at a velocity ratio between the far downstream and the free-stream that is lower than the incompressible value of 0.333. From incompressible to a free-stream Mach number of 0.8, the Betz limit increases by 0.021 while its corresponding velocity ratio decreases by 0.036. The Betz limit and its corresponding velocity ratio for the ducted turbine are not affected by the free-stream Mach and are the same for both incompressible and compressible flow. The total drag coefficient of the ducted turbine is also the same regardless of the free-stream Mach number and the compressibility of the flow; but, the individual contributions of the turbine drag and the lip thrust to the total drag differs between compressible and incompressible flow and between varying free-stream Mach numbers. It was concluded that overall compressibility has little influence on the ideal performance of an actuator disk.

The primary motivation for this work was to predict the conditions that would yield minimum drag for a small Ram-Air Turbine used to provide a specified power requirement for a small flight test instrument called the Boundary Layer Data System. Actuator Disk Theory was used to provide an analytical model for this work. Classic Actuator Disk Theory (CADT) or Froude’s Momentum Theory was initially established for quasi-one-dimensional flows and inviscid fluids to predict the power output, drag, and efficiency of energy-extracting devices as a function of wake and freestream velocities using the laws of Conservations of Mass, Momentum, and Energy. Because swirl and losses due to the effects of viscosity have real and significant impacts on existing turbines, there is a strong motivation to develop models which can provide generalized results about the performance of an energy-extractor, such as a turbine, with the inclusion of these effects. A model with swirl and a model with losses due to the effects of viscosity were incorporated into CADT which yielded equations that predicted the performance of an energy-extractor for both un-ducted and ducted cases. In both of these models, for this application, additional performance parameters were analyzed including the drag, drag coefficient, power output, power coefficient, force coefficient, and relative efficiency. For the un-ducted CADT, it is well known that the wake-to-freestream velocity ratio of 1/3 will give the maximum power extraction efficiency of 59.3%; this result is called the Betz limit. However, the present analysis shows that reduced drag for a desired power extraction will occur for wake-to-freestream velocity ratios higher than the value of 1/3 which results in maximum power extraction efficiency. This in turn means that a turbine with a larger area than the smallest possible turbine for a specified power extraction will actually experience a lower drag. The model with the inclusion of swirl made use of the Moment of Momentum Theorem applied to a single-rotor actuator disk with no stators, in addition to the laws of Conservation of Mass, Momentum, and Energy from the CADT. The results from the model w/swirl showed that drag remains unchanged while power extracted decreases with the addition of swirl, with swirl effects becoming more severe for tip speed ratios below about 5. As for CADT, reduced drag for a specified power extraction can be achieved when the wake-to-freestream velocity ratio is higher that than which provides maximum power extraction efficiency. The model w/losses due to viscosity incorporated the losses into the Conservation of Energy relationship. The results from the model w/losses showed that there is a distinct wake-to-freestream velocity ratio at which minimum drag for a specified power output is achieved, and that this velocity ratio is usually—but not always—higher than that for which the power extraction efficiency is a maximum. It was concluded that a lower drag for a specified power output of an energy-extractor can usually be achieved at a wake-to-freestream velocity ratio higher than that which produces the v maximum power extraction efficiency. The latter condition, known as the Betz limit for CADT, and which defines the minimum size for a turbine to provide a specified power extraction, is therefore not the correct target design condition to achieve lowest drag for a small Ram-Air Turbine to power BLDS.