def test_softmax(plot=False):
# Test softmax
x = np.linspace(-10, 10, 100)
y1 = x
y2 = -2 * x - 3
hardness = 0.5
y_soft = np.softmax(y1, y2, hardness=hardness)
assert np.softmax(0, 0, hardness=1) == np.log(2)
if plot:
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(font_scale=1)
fig, ax = plt.subplots(1, 1, figsize=(6.4, 4.8), dpi=200)
plt.plot(x, y1, label="y1")
plt.plot(x, y2, label="y2")
plt.plot(x, y_soft, label="softmax")
plt.xlabel(r"x")
plt.ylabel(r"y")
plt.title(r"Softmax")
plt.tight_layout()
plt.legend()
plt.show()
def Cf_flat_plate(Re_L: float, method="hybrid-sharpe-convex") -> float:
"""
Returns the mean skin friction coefficient over a flat plate.
Don't forget to double it (two sides) if you want a drag coefficient.
Args:
Re_L: Reynolds number, normalized to the length of the flat plate.
method: The method of computing the skin friction coefficient. One of:
* "blasius": Uses the Blasius solution. Citing Cengel and Cimbala, "Fluid Mechanics: Fundamentals and
Applications", Table 10-4.
Valid approximately for Re_L <= 5e5.
* "turbulent": Uses turbulent correlations for smooth plates. Citing Cengel and Cimbala,
"Fluid Mechanics: Fundamentals and Applications", Table 10-4.
Valid approximately for 5e5 <= Re_L <= 1e7.
* "hybrid-cengel": Uses turbulent correlations for smooth plates, but accounts for a
non-negligible laminar run at the beginning of the plate. Citing Cengel and Cimbala, "Fluid Mechanics:
Fundamentals and Applications", Table 10-4. Returns: Mean skin friction coefficient over a flat plate.
Valid approximately for 5e5 <= Re_L <= 1e7.
* "hybrid-schlichting": Schlichting's model, that roughly accounts for a non-negligtible laminar run.
Citing "Boundary Layer Theory" 7th Ed., pg. 644
* "hybrid-sharpe-convex": A hybrid model that blends the Blasius and Schlichting models. Convex in
log-log space; however, it may overlook some truly nonconvex behavior near transitional Reynolds numbers.
* "hybrid-sharpe-nonconvex": A hybrid model that blends the Blasius and Cengel models. Nonconvex in
log-log-space; however, it may capture some truly nonconvex behavior near transitional Reynolds numbers.
You can view all of these functions graphically using
`aerosandbox.library.aerodynamics.test_aerodynamics.test_Cf_flat_plate.py`
"""
Re_L = np.abs(Re_L)
if method == "blasius":
return 1.328 / Re_L**0.5
elif method == "turbulent":
return 0.074 / Re_L**(1 / 5)
elif method == "hybrid-cengel":
return 0.074 / Re_L**(1 / 5) - 1742 / Re_L
elif method == "hybrid-schlichting":
return 0.02666 * Re_L**-0.139
elif method == "hybrid-sharpe-convex":
return np.softmax(Cf_flat_plate(Re_L, method="blasius"),
Cf_flat_plate(Re_L, method="hybrid-schlichting"),
hardness=1e3)
elif method == "hybrid-sharpe-nonconvex":
return np.softmax(Cf_flat_plate(Re_L, method="blasius"),
Cf_flat_plate(Re_L, method="hybrid-cengel"),
hardness=1e3)
def mass_hpa_propeller(
diameter,
max_power,
include_variable_pitch_mechanism=False
):
"""
Returns the estimated mass of a propeller assembly for low-disc-loading applications (human powered airplane, paramotor, etc.)
:param diameter: diameter of the propeller [m]
:param max_power: maximum power of the propeller [W]
:param include_variable_pitch_mechanism: boolean, does this propeller have a variable pitch mechanism?
:return: estimated weight [kg]
"""
mass_propeller = (
0.495 *
(diameter / 1.25) ** 1.6 *
np.softmax(0.6, max_power / 14914, hardness=5) ** 2
) # Baselining to a 125cm E-Props Top 80 Propeller for paramotor, with some sketchy scaling assumptions
# Parameters on diameter exponent and min power were chosen such that Daedalus propeller is roughly on the curve.
mass_variable_pitch_mech = 216.8 / 800 * mass_propeller
# correlation to Daedalus data: http://journals.sfu.ca/ts/index.php/ts/article/viewFile/760/718
if include_variable_pitch_mechanism:
mass_propeller += mass_variable_pitch_mech
return mass_propeller
def separation_parameter(alpha, Re=0):
"""
Positive if separated, negative if attached.
This will be an input to a tanh() sigmoid blend via asb.numpy.blend(), so a value of 1 means the flow is
~90% separated, and a value of -1 means the flow is ~90% attached.
"""
return 0.5 * np.softmax(alpha - alpha_stall_positive,
alpha_stall_negative - alpha)
def underlying_function(x): # Softmax of three linear functions
def f1(x):
return -3 * x - 4
def f2(x):
return -0.25 * x + 1
def f3(x):
return 2 * x - 12
return np.softmax(f1(x), f2(x), f3(x), hardness=1)
def induced_drag_ratio_from_ground_effect(h_over_b # type: float
):
"""
Gives the ratio of actual induced drag to free-flight induced drag experienced by a wing in ground effect.
Artificially smoothed below around h/b == 0.05 to retain differentiability and practicality.
Source: W. F. Phillips, D. F. Hunsaker, "Lifting-Line Predictions for Induced Drag and Lift in Ground Effect".
Using Equation 5 from the paper, which is modified from a model from Torenbeek:
Torenbeek, E. "Ground Effects", 1982.
:param h_over_b: (Height above ground) divided by (wingspan).
:return: Ratio of induced drag in ground effect to induced drag out of ground effect [unitless]
"""
h_over_b = np.softmax(h_over_b, 0, hardness=1 / 0.03)
return 1 - np.exp(-4.01 * (2 * h_over_b)**0.717)
def Cd_wave_Korn(Cl, t_over_c, mach, sweep=0, kappa_A=0.95):
"""
Wave drag_force coefficient prediction using the low-fidelity Korn Equation method;
derived in "Configuration Aerodynamics" by W.H. Mason, Sect. 7.5.2, pg. 7-18
:param Cl: Sectional lift coefficient
:param t_over_c: thickness-to-chord ratio
:param sweep: sweep angle, in degrees
:param kappa_A: Airfoil technology factor (0.95 for supercritical section, 0.87 for NACA 6-series)
:return: Wave drag coefficient
"""
smooth_abs_Cl = np.softmax(Cl, -Cl, hardness=10)
mach = np.fmax(mach, 0)
Mdd = kappa_A / np.cosd(sweep) - t_over_c / np.cosd(
sweep)**2 - smooth_abs_Cl / (10 * np.cosd(sweep)**3)
Mcrit = Mdd - (0.1 / 80)**(1 / 3)
Cd_wave = np.where(mach > Mcrit, 20 * (mach - Mcrit)**4, 0)
return Cd_wave
def mach_crit_Korn(
CL,
t_over_c,
sweep=0,
kappa_A=0.95
):
"""
Wave drag_force coefficient prediction using the low-fidelity Korn Equation method;
derived in "Configuration Aerodynamics" by W.H. Mason, Sect. 7.5.2, pg. 7-18
Args:
CL: Sectional lift coefficient
t_over_c: thickness-to-chord ratio
sweep: sweep angle, in degrees
kappa_A: Airfoil technology factor (0.95 for supercritical section, 0.87 for NACA 6-series)
Returns:
"""
smooth_abs_CL = np.softmax(CL, -CL, hardness=10)
M_dd = kappa_A / np.cosd(sweep) - t_over_c / np.cosd(sweep) ** 2 - smooth_abs_CL / (10 * np.cosd(sweep) ** 3)
M_crit = M_dd - (0.1 / 80) ** (1 / 3)
return M_crit
def approximate_CD_wave(
mach,
mach_crit,
CD_wave_at_fully_supersonic,
):
"""
An approximate relation for computing transonic wave drag, based on an object's Mach number.
Considered reasonably valid from Mach 0 up to around Mach 2 or 3-ish.
Methodology is a combination of:
* The methodology described in Raymer, "Aircraft Design: A Conceptual Approach", Section 12.5.10 Transonic Parasite Drag (pg. 449 in Ed. 2)
and
* The methodology described in W.H. Mason's Configuration Aerodynamics, Chapter 7. Transonic Aerodynamics of Airfoils and Wings.
Args:
mach: Mach number at the operating point to be evaluated
mach_crit: Critical mach number, a function of the body geometry
CD_wave_at_fully_supersonic: The wave drag coefficient of the body at the speed that it first goes (
effectively) fully supersonic.
Here, that is taken to mean at the Mach 1.2 case.
This value should probably be derived using something similar to a Sears-Haack relation for the body in
question, with a markup depending on geometry smoothness.
The CD_wave predicted by this function will match this value exactly at M=1.2 and M=1.05.
The peak CD_wave that is predicted is ~1.23 * this value, which occurs at M=1.10.
In the high-Mach limit, this function asymptotes at 0.80 * this value, as empirically stated by Raymer.
However, this model is only approximate and is likely not valid for high-supersonic flows.
Returns: The approximate wave drag coefficient at the specified Mach number.
The reference area is whatever the reference area used in the `CD_wave_at_fully_supersonic` parameter is.
"""
mach_crit_max = 1 - (0.1 / 80)**(1 / 3)
mach_crit = -np.softmax(-mach_crit, -mach_crit_max, hardness=50)
### The following approximate relation is derived in W.H. Mason, "Configuration Aerodynamics", Chapter 7. Transonic Aerodynamics of Airfoils and Wings.
### Equation 7-8 on Page 7-19.
### This is in turn based on Lock's proposed empirically-derived shape of the drag rise, from Hilton, W.F., High Speed Aerodynamics, Longmans, Green & Co., London, 1952, pp. 47-49
mach_dd = mach_crit + (0.1 / 80)**(1 / 3)
### Model drag sections and cutoffs:
return CD_wave_at_fully_supersonic * np.where(
mach < mach_crit,
0,
np.where(
mach < mach_dd,
20 * (mach - mach_crit)**4,
np.where(
mach < 1.05,
cubic_hermite_patch(mach,
x_a=mach_dd,
x_b=1.05,
f_a=20 * (0.1 / 80)**(4 / 3),
f_b=1,
dfdx_a=0.1,
dfdx_b=10),
np.where(mach < 1.2,
cubic_hermite_patch(mach,
x_a=1.05,
x_b=1.2,
f_a=1,
f_b=1,
dfdx_a=10,
dfdx_b=-4),
np.blend(
switch=4 * 2 * (mach - 1.2) / (1.2 - 0.8),
value_switch_high=0.8,
value_switch_low=1.2,
)
# 0.8 + 0.2 * np.exp(20 * (1.2 - mach))
))))
def CL_over_Cl(
aspect_ratio: float,
mach: float = 0.,
sweep: float = 0.,
Cl_is_compressible: bool = True
) -> float:
"""
Returns the ratio of 3D lift coefficient (with compressibility) to the 2D lift coefficient.
Specifically: CL_3D / CL_2D
Args:
aspect_ratio: The aspect ratio of the wing.
mach: The freestream Mach number.
sweep: The sweep of the wing, in degrees. To be most accurate, this should be the sweep at the locus of
thickest points along the wing.
Cl_is_compressible: This flag indicates whether the 2D airfoil data already has compressibility effects
modeled.
For example:
* If this flag is True, this function returns: CL_3D / CL_2D, where CL_2D is the sectional lift
coefficient based on the local profile at the freestream mach number.
* If this flag is False, this function returns: CL_3D / CL_2D_at_mach_zero, where CL_2D_... is the
sectional lift coefficient based on the local profile at mach zero.
For most accurate results, set this flag to True, and then model profile characteristics separately.
"""
prandtl_glauert_beta_squared_ideal = 1 - mach ** 2
# beta_squared = 1 - mach ** 2
beta_squared = np.softmax(
prandtl_glauert_beta_squared_ideal,
-prandtl_glauert_beta_squared_ideal,
hardness=3.0
)
### Alternate formulations
# CL_ratio = aspect_ratio / (aspect_ratio + 2) # Equivalent to equation in Drela's FVA in incompressible, 2*pi*alpha limit.
# CL_ratio = aspect_ratio / (2 + np.sqrt(4 + aspect_ratio ** 2)) # more theoretically sound at low aspect_ratio
### Formulation from Raymer, Sect. 12.4.1; citing DATCOM.
# Comparison to experiment suggests this is the most accurate.
# Symbolically simplified to remove the PG singularity.
eta = 0.95
CL_ratio = aspect_ratio / (
2 + (
4 + (aspect_ratio ** 2 * beta_squared / eta ** 2) + (np.tand(sweep) * aspect_ratio / eta) ** 2
) ** 0.5
)
if Cl_is_compressible:
CL_ratio = CL_ratio * beta_squared ** 0.5
return CL_ratio