OYE Model

In QBlade, dynamic stall may be modeled in unsteady Lifting Line Free Vortex Wake or Blade Element Momentum Method simulations by using the dynamic stall model proposed by Oye 1. It should be noted that this model only captures the dynamics of separated flow. The additional attached-flow dynamics due to airfoil wake memory effects are captured intrinsically by the Lifting Line Free Vortex Wake model. In its implementation in QBlade, the model contribution is smoothly faded out toward the static polar near ±50°.

In Oye’s work, dynamic stall is modeled with the help of a separation function \(f\). It is used to calculate the dynamic lift \(Cl_{dyn}\) as:

\[\begin{align} Cl_{dyn} = f \cdot Cl_{att} + (1-f) \cdot Cl_{sep}. \end{align}\]

Here, \(Cl_{att}\) is the fully attached inviscid lift contribution and \(Cl_{sep}\) is the fully separated lift contribution. Applying the same relation to steady conditions gives:

\[\begin{align} Cl^{st} = f^{st} \cdot Cl_{att}^{st} + (1-f^{st}) \cdot Cl_{sep}^{st}. \end{align}\]

The superscript \(st\) refers to steady conditions. \(Cl^{st}\) is obtained from the static polar data, while \(Cl_{att}^{st}\) is obtained by extrapolating the linear part of the lift curve to the required angle of attack. Following Hansen 2, the steady separation function \(f^{st}\) is calculated as:

\[\begin{align} f^{st} = \left(2\sqrt{\frac{Cl^{st}}{Cl_{att}^{st}}}-1\right)^2. \end{align}\]

The ratio \(Cl^{st}/Cl_{att}^{st}\) is evaluated with its sign. For negative angles of attack, this ratio remains positive as long as \(Cl^{st}\) and \(Cl_{att}^{st}\) have the same sign. If \(Cl_{att}^{st}\) is close to zero, \(f^{st}\) is set to 1. The resulting value of \(f^{st}\) is limited to the range \(0 \le f^{st} \le 1\).

The dynamic separation function \(f\) is assumed to return to the static value \(f^{st}\) according to:

\[\begin{align} \frac{df}{dt} = \frac{f^{st} - f}{\tau}. \end{align}\]

Integrating this equation yields:

\[\begin{align} f(t) = f^{st}(t) + \left(f(t-\Delta t) - f^{st}(t)\right)e^{-\Delta t/\tau}. \end{align}\]

The time constant \(\tau\) is defined as:

\[\begin{align} \tau = \frac{A c}{2 V_{rel}}, \end{align}\]

where \(A\) is a model parameter, typically around 8, \(c\) is the airfoil chord, and \(V_{rel}\) is the relative velocity at the airfoil section.

After \(f\) has been evaluated, \(Cl_{sep}\) is obtained from:

\[\begin{align} Cl_{sep} = \frac{Cl^{st}-f^{st}Cl_{att}}{1-f^{st}}. \end{align}\]

For values of \(f^{st}\) close to 1, this expression is regularized in the implementation to avoid division by a very small number. \(Cl_{att}\) is obtained by extrapolating the linear lift curve. With these quantities known, \(Cl_{dyn}\) can be computed.

In QBlade, the Oye dynamic stall model also determines a dynamically changing drag coefficient \(Cd_{dyn}\). Following Bergami, the dynamic drag is evaluated as:

\[\begin{align} Cd_{dyn} = Cd^{st} + \left(Cd^{st}-Cd^{st}_0\right) \left[0.5\left(\sqrt{f^{st}}-\sqrt{f}\right) - 0.25\left(f-f^{st}\right) \right]. \end{align}\]

Here, \(Cd^{st}_{0}\) is the drag coefficient at zero angle of attack.

1

Stig Oye. Dynamic stall simulated as time lag of separation. 1991.

2

Morten Hartvig Hansen, Mac Gaunaa, and Helge Aagaard Madsen. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulation. Technical Report, DTU, 2004.