OYE Dynamic Stall 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. Currently, QBlades does not have an adapted dynamic stall model that captures the attached flow dynamics for simulations using the Blade Element Momentum Method.

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

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

Where \(Cl_{att}\) is the fully attached inviscid lift contribution and \(Cl_{sep}\) the fully separated lift contribution. To solve the equation above, it is applied to steady conditions. The result is:

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

where the superscript \(st\) refers to steady conditions. Comparing the equations above, \(Cl_{att} = Cl^{st}_{att}\) and \(Cl_{sep} = Cl^{st}_{sep}\). \(Cl^{st}\) is obtained by reading in static polar data and \(Cl_{att}^{st}\) is obtained by extrapolating the linear part of the lift curve to the required angle of attack. Following Hansen 2, \(f^{st}\) can be calculated in the following way:

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

The value of \(f^{st}\) is limited to be between 0 and 1. Now \(f\) is assumed to return to the static value \(f^{st}\) as follows:

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

Integrating the equation above allows the determination of the dynamic behavior of \(f(t)\):

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

\(\tau\) is a time constant, defined as:

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

where \(A\) is a parameter (typically around 8), \(c\) is the airfoil chord and \(V_{rel}\) the relative velocity at the airfoil section. After \(f\) has been evaluated, \(Cl_{sep}\) can be obtained from:

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

\(Cl_{att}\) is gained by extrapolation of the linear lift curve.

Now, all variables have been determined and the dynamic lift \(Cl_{dyn}\) can be computed.

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

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

In this equation \(Cd^{st}_{0}\) is the drag at 0 degree 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.