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GK Dynamic stall model

1 Model Overview

The Goman-Khrabrov (GK) model represents dynamic stall through a state-space approach using a separation point variable

x

$x$ that describes the state of flow attachment on the airfoil surface. This variable ranges from 0 (fully separated flow) to 1 (fully attached flow). The implementation in Ashes is based on the paper by Goman et al. (1994).

Note: this implementation is currently in beta version

2 State Equation

The core of the GK model is a first-order differential equation that describes how the separation point evolves:

$$\frac{dx}{dt} = \frac{x_0(\alpha - \tau_2\dot{\alpha}) - x}{\tau_1}$$

(1)

where

x

$x$ is the instantaneous separation point position,

x_{0}

$x_0$ is the steady-state separation point for a given angle of attack

α

$\alpha$ ,

\dot{α}

$\dot{\alpha}$ is the rate of change of angle of attack, and

τ_{1}

$\tau_1$ and

τ_{2}

$\tau_2$ are time constants.

3 Time Constants

The time constants

τ_{1}

$\tau_1$ and

τ_{2}

$\tau_2$ are scaled by the chord length and relative velocity:

$$\tau_1 = k_1 \frac{c}{V}$$

(2)

$$\tau_2 = k_2 \frac{c}{V}$$

(3)

where

c

$c$ is the chord length,

V

$V$ is the relative velocity, and

k_{1}

$k_1$ and

k_{2}

$k_2$ are configurable constants defined in the Analysis parameters.

4 Separation Point

The separation point function was derived from experimental data at the University of Bath, and is given by:

The steady-state separation point

x_{0}

$x_0$ is determined based on the angle of attack relative to the stall angle:

$$x_0 =0.5\left[1-\tanh(K_S\cdot\left(\alpha_{adj}-\phi)\right)\right]$$

(4)

where

$$\alpha_{adj}=\alpha - \tau_2\dot{\alpha}$$

is the adjusted angle of attack accounting for rate effects,

$$K_S$$

is a separation parameter and

$$\phi$$

is a separation angle.

Note that the drag coefficient equation uses a static version of that equation, given by:

$$x_{0-st} =0.5\left[1-\tanh(K_S\cdot\left(\alpha-\phi)\right)\right]$$

5 Numerical Integration

Equation (1) is integrated using the explicit Euler method:

$$x(t + \Delta t) = x(t) + \frac{dx}{dt} \Delta t$$

(5)

Substituting the GK model expression for

$$\frac{dx}{dt}$$

\frac{d x}{d t}

$\frac{dx}{dt}$ , we get:

$$x(t + \Delta t) = x(t) + \frac{x_0 - x(t)}{\tau_1} \Delta t$$

(6)

where

Δ t

$\Delta t$ is the simulation time step.

6 Lift, drag and Moment Coefficients

Once the separation point

x

$x$ is determined, the lift and moment coefficients are calculated using:

$$C_L(\alpha, x) = C_{L-st}(0)+\frac{dC_{L-st}(0)}{d\alpha}\sin(\alpha)\left(\frac{1 + \sqrt{x}}{2}\right)^2$$

(7)

where

$$C_{L-st}$$

is the static lift.

$$C_D(\alpha,x)=C_{D-st}(\alpha)+\left(C_{D-st}(\alpha)-C_{D-st}(0)\right) \left[0.5\left(\sqrt{x_{0-st}} - \sqrt{x}\right) - 0.25(x - x_{0-st})\right] $$

where

$$C_{D-st}$$

is the static drag.

$$C_m(\alpha, x) = C_L(\alpha, x) \cdot \frac{5(1 - \sqrt{x})^2 + 4\sqrt{x}}{16}$$

(8)

These equations account for both attached and separated flow contributions through the separation point variable

x

$x$ .

7 Implementation Flow

The implementation of the GK model in the code follows these steps:

Calculate the rate of change of angle of attack $\dot{α}$ $\dot{\alpha}$
Calculate the time constants $τ_{1}$ $\tau_1$ and $τ_{2}$ $\tau_2$
Determine the appropriate stall angle based on Reynolds number
Calculate the steady-state separation point $x_{0}$ $x_0$
Calculate $\frac{d x}{d t}$ $\frac{dx}{dt}$ using equation (1)
Update the separation point $x$ $x$ using equations (5) and (6)
Ensure $x$ $x$ stays within the valid range [0,1]
Calculate the lift coefficient $C_{L}$ $C_L$ using equation (7)
Calculate the moment coefficient $C_{m}$ $C_m$ using equation (8)
Update the aerodynamic coefficients