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improvements to theodorsen
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we are close as the shape is good but we have some sort of negative offset with acceleration of a plate
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@samayala22
samayala22 committed Apr 12, 2024
1 parent 1bbd65f commit 7f92456
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Showing 3 changed files with 134 additions and 13 deletions.
45 changes: 34 additions & 11 deletions data/theodorsen.py
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Original file line number Diff line number Diff line change
Expand Up @@ -14,6 +14,9 @@ def derivative2(f, x):
def solve_ivp(x0: float, s0: float, sf: float, f: callable):
return spi.solve_ivp(f, [s0, sf], [x0]).y[-1] # return only the result at t=sf

def pade_approximation(k: float):
return (0.5177*k*k + 0.2752*k + 0.01576) / (k*k + 0.3414*k + 0.01582)

# Some info in Katz Plotkin p414 (eq 13.73a)
# Jone's approximation of Wagner function
b0 = 1
Expand All @@ -25,9 +28,13 @@ def solve_ivp(x0: float, s0: float, sf: float, f: callable):
# UVLM parameters
rho = 1 # fluid density
u_inf = 1 # freestream
ar = 4 # aspect ratio
ar = 500 # aspect ratio
b = 0.5 # half chord
a = ar / (2*b) # full span
c = 2*b # chord
a = ar / c # full span
pitch_axis = -1 # leading edge

def atime(t: float): return 2. * u_inf * t / c

amplitudes = [0.1, 0.1, 0.1]
reduced_frequencies = [0.5, 0.75, 1.5]
Expand All @@ -46,26 +53,42 @@ def solve_ivp(x0: float, s0: float, sf: float, f: callable):
amplitude = amp / (2*b)
omega = k * u_inf / (2*b) # pitch frequency

def pitch(t): return 0
def heave(t): return amplitude * np.sin(omega * t)
# sudden acceleration
def pitch(t): return np.radians(5)
def heave(t): return 0

def w(s: float): return u_inf * pitch(s) + derivative(heave, s) + b * (0.5 - a) * derivative(pitch, s)
# pure heaving
# def pitch(t): return 0
# def heave(t): return amplitude * np.sin(omega * t)

def w(s: float):
return u_inf * pitch(s) + derivative(heave, s) + b * (0.5 - pitch_axis) * derivative(pitch, s)

def dx1ds(s: float, x1: float): return b1 * beta_1 * w(s) - beta_1 * x1
def dx2ds(s: float, x2: float): return b2 * beta_2 * w(s) - beta_2 * x2

def cl_theodorsen(t: float):
L_m = rho * b * b * np.pi * (u_inf * derivative(pitch, t) + derivative2(heave, t) - b * a * derivative2(pitch, t))
L_c = -2 * np.pi * rho * u_inf * b * ((b0 + b1 + b2) * w(t) + solve_ivp(0, 0, t, dx1ds)[-1] + solve_ivp(0, 0, t, dx2ds)[-1])
return (L_m + L_c) / (0.5 * rho * u_inf * u_inf * a * b)

x1_solution = spi.solve_ivp(dx1ds, [0, t_final], [0], t_eval=t)
x2_solution = spi.solve_ivp(dx2ds, [0, t_final], [0], t_eval=t)

def x1(s: float): return np.interp(s, x1_solution.t, x1_solution.y[0])
def x2(s: float): return np.interp(s, x2_solution.t, x2_solution.y[0])

def cl_theodorsen(t: float): # using Wagner functions and Kholodar formulation
L_m = rho * b * b * np.pi * (u_inf * derivative(pitch, t) + derivative2(heave, t) - b * pitch_axis * derivative2(pitch, t))
L_c = -2 * np.pi * rho * u_inf * b * ((b0 + b1 + b2) * w(t) + x1(t) + x2(t))
return (L_m + L_c) / (0.5 * rho * u_inf * u_inf * c)

# def cl_theodorsen(t: float): # using Pade approximation
# return 0.5 * np.pi * (derivative2(heave, t) + derivative(pitch, t) - 0.5 * pitch_axis * derivative2(pitch, t)) + 2.0 * np.pi * (pitch(t) + derivative(heave, t) + 0.5 * derivative(pitch, t) * (0.5 - pitch_axis)) * pade_approximation(k)
# L = rho * b * b * np.pi * (u_inf * derivative(pitch, t) + derivative2(heave, t) - b * pitch_axis * derivative2(pitch, t)) + 2 * np.pi * rho * u_inf * b * (u_inf * pitch(t) + derivative(heave, t) + b * (0.5 - pitch_axis) * derivative(pitch, t)) * pade_approximation(k)
# return L / (0.5 * rho * u_inf * u_inf * a * (2*b))

cl = np.array([cl_theodorsen(ti) for ti in t])
coord_z = np.array([heave(ti) / (2*b) for ti in t])

axs["time"].plot(t, cl, label=f"k={k}")
axs["heave"].plot(coord_z[len(cl)//2:], cl[len(cl)//2:], label=f"k={k}")


uvlm_cl = []
uvlm_t = []
uvlm_z = []
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98 changes: 98 additions & 0 deletions mesh/infinite_rectangular_5x20.x
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4 changes: 2 additions & 2 deletions tests/test_uvlm_theodorsen.cpp
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Expand Up @@ -64,7 +64,7 @@ void dump_buffer(std::ofstream& stream, T* start, T* end) {

int main() {
// const std::vector<std::string> meshes = {"../../../../mesh/infinite_rectangular_2x8.x"};
const std::vector<std::string> meshes = {"../../../../mesh/rectangular_5x10.x"};
const std::vector<std::string> meshes = {"../../../../mesh/infinite_rectangular_5x20.x"};

const std::vector<std::string> backends = get_available_backends();

Expand All @@ -77,7 +77,7 @@ int main() {
const f32 t_final = 30.0f;
const f32 u_inf = 1.0f; // freestream velocity
const f32 amplitude = 0.1f; // amplitude of the wing motion
const f32 k = 1.5f; // reduced frequency
const f32 k = 0.5f; // reduced frequency

const f32 omega = k * u_inf / (2*b);

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