2. Pitt–Peters dynamic inflow model

In dynamic inflow models, the goal is to derive an expression for the axial-induced velocity at the rotor disk. This is done by relating a set of inflow variables (states) to the rotor aerodynamic loads via a set of first-order differential equations.

Dynamic inflow models are applicable to edgewise AND axial flow conditions. However, for axial (or mostly axial) flow, we recommend BEM theory due to its simplicity, robustness and reasonable accuracy.

In the simplest formulation, there are three flow states, one constant (\(\lambda_{0}\)) and two linear (\(\lambda_{c},\lambda_{s}\)), which represent the perturbations in the wake-induced downwash. The normalized induced velocity (\(\lambda=\frac{u_x^i}{\Omega R}\)) is approximated by a first order Fourier expansion, in which the coefficients are the flow states, such that

\[ \lambda(\overline{r}, \psi) = \lambda_{0} + \overline{r}\lambda_{c}\cos{\psi} + \overline{r}\lambda_{s}\sin{\psi}, \]

where \(\overline{r}\) is the normalized radius and \(\psi\) is the azimuth angle in the rotor plane. The response of the inflow states to aerodynamic perturbations is described a set of linear, first order differential equations as

\[\begin{split} LM \begin{bmatrix} \dot{\lambda}_{0} \\ \dot{\lambda}_{c} \\ \dot{\lambda}_{s} \end{bmatrix} + \begin{bmatrix} \lambda_{0} \\ \lambda_{c} \\ \lambda_{s} \end{bmatrix} = L \begin{bmatrix} C_T \\ -C_{M_{y}} \\ C_{M_{x}} \end{bmatrix}, \end{split}\]

where the aerodynamic loading coefficient are in terms of the integrated thrust and moment coefficients, which are given by

\[ C_T = \frac{T}{\rho A \left(\Omega R\right)^2} \]

\[ C_{M_{x}} = \frac{M_x}{\rho A \left(\Omega R\right)^2 R} \]

\[ C_{M_{y}} = \frac{M_y}{\rho A \left(\Omega R\right)^2 R}, \]

where \(M_x\) is the rotor hub roll moment, \(M_y\) is the rotor hub pitching moment, and \(A\) is the rotor area. It is important to note that to compute the aerodynamic loads, we require an airfoil model just like in BEM theory.

Pitt and Peters [PP80] developed expression for the stability matrix \(L\) and the mass matrix \(M\), which are given as

\[\begin{split} L = \frac{1}{V_{\textrm{eff}}}\begin{bmatrix} \frac{1}{2} & \frac{-15\pi}{64}\sqrt{\frac{1-\cos\chi}{1+\cos\chi}} & 0\\ \frac{15\pi}{64}\sqrt{\frac{1-\cos\chi}{1+\cos\chi}} & \frac{4\cos\chi}{1 + \cos\chi}& 0 \\ 0 & 0 & \frac{4}{1 + \cos\chi} \end{bmatrix}, \end{split}\]

and

\[\begin{split} M = \begin{bmatrix} \frac{128}{75\pi} & 0 & 0 \\ 0 & \frac{64}{45\pi} & 0 \\ 0 & 0 & \frac{64}{45\pi} \\ \end{bmatrix}, \end{split}\]

where \(\chi\) is the wake skew angle and \(V_{\text{eff}}\) is the effective velocity given as

\[ V_{\text{eff}} = \frac{\mu^2 + \lambda\left(\lambda + \lambda_i\right)}{\sqrt{\mu^2 + \lambda^2}}. \]

In the expression for the effective velocity, \(\mu\) is the normalized in-plane component of inflow velocity, i.e., \(\mu=\frac{V_{\infty}\cos i}{\Omega R}\). This quantity is also know as the rotor advance ratio where \(i\) is the rotor disk incidence angle or angle of attack (positive for forward tilt). The other unknown the expression for \(V_{\text{eff}}\) is \(\lambda_i\), which is the mean (axial) induced velocity and is related to the thrust coefficient via momentum theory as

\[ \lambda_i = \frac{C_T}{2\sqrt{\mu^2 + \left(\lambda_i+\mu_z\right)^2}}, \]

where \(\mu_z=\frac{V_{\infty}\sin i}{\Omega R}\) is the axial (i.e., normal) component of the rotor inflow and is also referred to as axial velocity ratio. Solving for \(\lambda_i\) is typically done via a Newton-Raphson algorithm.

Lastly, we note that in the current version of BladeAD, we solve for steady state and neglect transient effects

For a more detailed derivation of the dynamic inflow model in general, we refer the reader to a comprehensive text book on rotorcraft aero-mechanics by Johnson [Joh13], specifically (Ch. 11). Chen [Che89] provides a comprehensive, high-level summary of various nonlinear inflow models that have been developed.

2.1. Bibliography

Che89

Robert TN Chen. A survey of nonuniform inflow models for rotorcraft flight dynamics and control applications. In European Rotorcraft Forum, number A-89220. 1989.

Joh13

Wayne Johnson. Rotorcraft aeromechanics. Volume 36. Cambridge university press, 2013.

PP80

Dale M Pitt and David A Peters. Theoretical prediction of dynamic-inflow derivatives. SIXTH EUROPEAN ROTORCRAFT AND POWERED LIFT AIRCRAFT FORUM, 1980.