Momentum Theory Rotor
Velocity Pressure (Red is high, blue is low)
\(w_0, p_\infty\)
...
\(w, p_\infty\)
...
\(w, p_\infty\)
\(p_+\)
Rotor
\(p_-\)
Rotor
\(p_-\)
Equations
Conservation of Momentum: \[ T = \dot m (w -w_0) \] Conservation of Energy: \[ T v = \frac12 \dot m (w^2 -w_0^2) \] Derivations: \[ v = \frac12 \left(\frac{w^2 -w_0^2}{w-w_0}\right) \] Pressure distributions can be found by simply applying Bernoulli's Theorem. Note that the Theorem does not hold across the rotor disk, only within each region (upstream, downstream), since energy is being added by the rotor disk. Definition of Terms:\(A\): area
\(\dot m\): mass flow rate
\(T\): thrust
\(\rho\): density
\(v\): velocity at rotor disk
\(w\): final velocity (far downstream)
\(w_0\): initial velocity (far upstream)
\(p\): pressure
\(p_\infty\): ambient pressure
\(p_+\): pressure just above rotor disk
\(p_-\): pressure just below rotor disk
Hover Performance Calculator
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