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 x d x d t = x 0 ( α − τ 2 ˙ α ) − x τ 1 x x x x x x x x x x d x d t = x 0 ( α − τ 2 ˙ α ) − x τ 1 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 x τ 2 = k 2 c V x x x x x x x x x x τ 2 = k 2 c V is the instantaneous separation point position, x 0 x 0 x 0 x 0 = 0.5 [ 1 − tanh ( K S ⋅ ( α a d j − ϕ ) ) ] x 0 ) x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 x 0 = 0.5 [ 1 − tanh ( K S ⋅ ( α a d j − ϕ ) ) ] is the steady-state separation point for a given angle of attack α α α α a d j = α − τ 2 ˙ α α α α α α α α α α a d j = α − τ 2 ˙ α , ˙ α ˙ α ˙ α K S ˙ α [ ˙ α ˙ α ˙ α ˙ α τ 1 = k 1 c V ˙ α ˙ α ˙ α ˙ α K S is the rate of change of angle of attack, and τ 1 τ 1 τ 1 ϕ τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 2 = k 2 c V τ 1 τ 1 τ 1 τ 1 ϕ and x 0 = { 1.0 if  | α a d j | < α s t a l l 1.0 − 0.5 | α a d j | − α s t a l l 20.0 if  | α a d j | ≥ α s t a l l τ 2 τ 2 x 0 − s t = 0.5 [ 1 − tanh ( K S ⋅ ( α − ϕ ) ) ] are time constants.

3 Time Constants

The time constants τ 1 τ 1 τ 1 x ( t + Δ t ) = x ( t ) + d x d t Δ t τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 x 0 ( α ) τ 1 τ 1 τ 1 τ 1 x ( t + Δ t ) = x ( t ) + d x d t Δ t and τ 2 τ 2 τ 2 d x d t τ 2 τ 2 τ 2 τ 2 τ 2 τ 2 x ( t + Δ t ) = x ( t ) + d x d t Δ t τ 2 τ 2 τ 2 τ 2 d x d t 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 c C L ( α , x ) = C L − s t ( 0 ) + d C L − s t ( 0 ) d α sin ( α ) ( 1 + √ x 2 ) 2 c c c c c c x ( t + Δ t ) = x ( t ) + x 0 − x ( t ) τ 1 Δ t c c c c C L ( α , x ) = C L − s t ( 0 ) + d C L − s t ( 0 ) d α sin ( α ) ( 1 + √ x 2 ) 2 is the chord length, V V V C L − s t V V V V V V Δ t V V Δ t V V Δ t C L − s t is the relative velocity, and k 1 k 1 k 1 C D ( α , x ) = C D − s t ( α ) + ( C D − s t ( α ) − C D − s t ( 0 ) ) [ 0.5 ( √ x 0 − s t − √ x ) − 0.25 ( x − x 0 − s t ) ] k 1 k 1 k 1 k 1 k 1 k 1 x k 1 k 1 x k 1 k 1 x C D ( α , x ) = C D − s t ( α ) + ( C D − s t ( α ) − C D − s t ( 0 ) ) [ 0.5 ( √ x 0 − s t − √ x ) − 0.25 ( x − x 0 − s t ) ] and k 2 k 2 k 2 C D − s t k 2 k 2 k 2 k 2 k 2 k 2 C L ( α , x ) = π 2 sin α ( 1 + √ x ) 2 k 2 k 2 k 2 k 2 C D − s t 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 x 0 C m ( α , x ) = C L ( α , x ) ⋅ 5 ( 1 − √ x ) 2 + 4 √ x 16 x 0 x 0 x 0 x 0 x 0 x 0 C m ( α , x ) = C L ( α , x ) ⋅ 5 ( 1 − √ x ) 2 + 4 √ x 16 x 0 x 0 x 0 x 0 C m ( α , x ) = C L ( α , x ) ⋅ 5 ( 1 − √ x ) 2 + 4 √ x 16 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 
$$\frac{dx}{dt}$$
x d x d t d x d t d x d t d x d t x ( t + Δ t ) = x ( t ) + x 0 − x ( t ) τ 1 Δ t d x d t x x d x d t x x d x d t x x d x d t d x d t d x d t d x d t d ( d t d x d t d x d t d x d t d x d t , we get:
$$x(t + \Delta t) = x(t) + \frac{x_0 - x(t)}{\tau_1} \Delta t$$
(6)
where Δ t C L Δ t C L Δ t C L Δ t Δ t Δ t Δ t Δ t Δ t d x d t Δ t d x d t Δ t d x d t x is the simulation time step.

6 Lift, drag and Moment Coefficients

Once the separation point x x x C m x x C m x x C m x x x 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 τ 1 x x x x x τ 1 x τ 1 x τ 1 x x x x x .

7 Implementation Flow

The implementation of the GK model in the code follows these steps:
  1. Calculate the rate of change of angle of attack τ 2 ˙ α ˙ α ˙ α ˙ α τ 2 ˙ α τ 2 ˙ α τ 2 ˙ α ˙ α ˙ α ˙ α ˙ α ˙ α ˙ α ˙ α
  2. Calculate the time constants x 0 τ 1 τ 1 τ 1 τ 1 x 0 τ 1 x 0 τ 1 x 0 τ 1 τ 1 τ 1 τ 1 τ 1 and d x d t τ 2 τ 2 τ 2 τ 2 τ 2 d x d t τ 2 d x d t τ 2 d x d t τ 2 τ 2 τ 2 τ 2 τ 2
  3. Determine the appropriate stall angle based on Reynolds number
  4. Calculate the steady-state separation point x x 0 x 0 x 0 x 0 x 0 x x 0 x x 0 x x 0 x 0 x 0 x 0 x 0
  5. Calculate x d x d t x d x d t x d x d t x d x d t d x d t d x d t d x d t d x d t using equation (1)
  6. Update the separation point C L x C L x C L x C L x x x x x using equations (5) and (6)
  7. Ensure C m x x x x C m x C m x C m x x x x x stays within the valid range [0,1]
  8. Calculate the lift coefficient C L C L C L C L C L C L C L C L using equation (7)
  9. Calculate the moment coefficient C m C m C m C m C m C m C m C m using equation (8)
  10. Update the aerodynamic coefficients