Vulcan Wind Models User Guide

This guide covers the wind modeling utilities in Vulcan, providing atmospheric wind profiles and turbulence models for flight simulation and trajectory optimization.

Quick Start

#include <vulcan/wind/WindShear.hpp>
#include <vulcan/wind/DrydenTurbulence.hpp>
 
using namespace vulcan;
 
// Create a power-law wind shear profile
double alt = 100.0;  // meters
double wind = wind_shear::power_law(alt, 10.0, 10.0);  // 10 m/s at 10m ref
 
// Get MIL-spec turbulence parameters
auto params = wind::mil_spec_params(500.0, wind::TurbulenceSeverity::Moderate);
// params.sigma_u, params.L_u, etc.

Wind Types

WindVector - 3D Wind in NED Frame

#include <vulcan/wind/WindTypes.hpp>
 
wind::WindVector<double> w{.north = 10.0, .east = 5.0, .down = 0.0};
 
double speed = w.speed();              // Total magnitude [m/s]
double horiz = w.horizontal_speed();   // Horizontal component
double dir = w.direction_from();       // Meteorological convention [rad]

Turbulence Parameters (MIL-F-8785C)

// Get MIL-spec parameters for given altitude and severity
auto params = wind::mil_spec_params(
    altitude,                           // AGL [m]
    wind::TurbulenceSeverity::Moderate  // Light, Moderate, or Severe
);
 
// Access σ (intensity) and L (scale length) for each axis
std::cout << "σ_u = " << params.sigma_u << " m/s\n";
std::cout << "L_u = " << params.L_u << " m\n";

Constant Wind Field

Simple uniform wind - useful as baseline and for testing.

#include <vulcan/wind/ConstantWind.hpp>
 
// From NED components
auto w = constant_wind::from_ned(10.0, 5.0, 0.0);
 
// From speed and direction (meteorological convention)
auto w = constant_wind::from_speed_direction(
    12.0,           // speed [m/s]
    M_PI / 4.0      // direction FROM [rad], clockwise from North
);

Wind Shear Profiles

Altitude-dependent wind models for boundary layer effects.

Power-Law Profile (Most Common)

#include <vulcan/wind/WindShear.hpp>
 
// V(h) = V_ref * (h / h_ref)^α
double wind = wind_shear::power_law(
    altitude,       // Current altitude [m]
    10.0,           // Reference wind at h_ref [m/s]
    10.0,           // Reference height h_ref [m]
    wind_shear::exponent::NEUTRAL  // α = 0.25
);
 
// Available exponents:
// - exponent::UNSTABLE = 1/7 (daytime, strong heating)
// - exponent::NEUTRAL  = 1/4 (overcast, moderate wind)
// - exponent::STABLE   = 1/3 (nighttime, light wind)

Logarithmic Profile (Boundary Layer Theory)

// V(h) = (u* / κ) * ln((h - d) / z₀)
double wind = wind_shear::logarithmic(
    altitude,           // [m]
    0.5,                // friction velocity u* [m/s]
    roughness::OPEN_TERRAIN,  // z₀ = 0.03 m
    0.0                 // displacement height d [m]
);
 
// Available roughness lengths:
// - roughness::OPEN_WATER   = 0.0002 m
// - roughness::OPEN_TERRAIN = 0.03 m
// - roughness::RURAL        = 0.1 m
// - roughness::SUBURBAN     = 0.5 m
// - roughness::URBAN        = 1.0 m

Linear Shear

// V(h) = V_base + shear_rate * (h - h_base)
double wind = wind_shear::linear(
    altitude,           // Current altitude [m]
    10.0,               // Base wind [m/s]
    0.0,                // Base altitude [m]
    0.01                // Shear rate [1/s]
);

Dryden Turbulence Model

The Dryden model uses rational transfer functions to filter white noise into realistic turbulence. Easier to implement than von Kármán.

Power Spectral Density

#include <vulcan/wind/DrydenTurbulence.hpp>
 
// Longitudinal PSD: Φ_u(Ω) = σ²·(2L/π) / [1 + (L·Ω)²]
double psd = dryden::psd_longitudinal(omega, sigma_u, L_u);
 
// Lateral/vertical PSD (different shape)
double psd_v = dryden::psd_lateral(omega, sigma_v, L_v);

Forming Filter (Time-Domain Simulation)

// 1. Initialize filter state
auto state = dryden::init_state<double>();
 
// 2. Compute filter coefficients for given conditions
auto coeffs = dryden::mil_spec_coeffs(
    altitude,                           // AGL [m]
    wind::TurbulenceSeverity::Moderate, // severity
    100.0,                              // airspeed [m/s]
    0.01                                // time step [s]
);
 
// 3. Step filter with white noise input
std::mt19937 rng(12345);
std::normal_distribution<double> noise(0.0, 1.0);
 
for (int i = 0; i < 1000; ++i) {
    auto gust = dryden::step(
        state, coeffs,
        noise(rng),  // longitudinal noise
        noise(rng),  // lateral noise  
        noise(rng)   // vertical noise
    );
    // Use gust.u_g, gust.v_g, gust.w_g
}

von Kármán Turbulence Model

More accurate spectral match to atmospheric data, but uses irrational exponents (5/6, 11/6) requiring higher-order filter approximation.

Power Spectral Density

#include <vulcan/wind/VonKarmanTurbulence.hpp>
 
// Φ_u(Ω) = σ²·(2L/π) / [1 + (1.339·L·Ω)²]^(5/6)
double psd = von_karman::psd_longitudinal(omega, sigma_u, L_u);

Forming Filter

Same API as Dryden, but uses 3rd/4th order filters internally:

auto state = von_karman::init_state<double>();
auto coeffs = von_karman::mil_spec_coeffs(altitude, severity, airspeed, dt);
 
auto gust = von_karman::step(state, coeffs, noise_u, noise_v, noise_w);

Symbolic Computation

All wind functions work with janus::SymbolicScalar for gradient-based optimization:

using Scalar = janus::SymbolicScalar;
 
Scalar alt = janus::sym("altitude");
Scalar wind = wind_shear::power_law(alt, 10.0, 10.0);
 
// Compute gradient dV/dh symbolically
auto dv_dh = janus::jacobian(wind, alt);
 
// Create CasADi function for optimization
janus::Function f("wind_profile", {alt}, {wind, dv_dh});

Optimization Example

Optimize aircraft climb profile considering wind shear:

janus::Opti opti;
 
auto alt1 = opti.variable(500.0);  // First waypoint
auto alt2 = opti.variable(1000.0); // Second waypoint
 
// Objective: minimize fuel cost (function of headwind)
auto cost = mission_cost(alt1, alt2, ref_wind);
opti.minimize(cost);
 
// Constraints
opti.subject_to(alt1 >= 200.0);
opti.subject_to(alt2 >= alt1 + 100.0);
 
auto solution = opti.solve();

Graph Visualization

Export computational graphs as interactive HTML:

auto alt = janus::sym("altitude");
auto wind = wind_shear::power_law(alt, 10.0, 10.0, wind_shear::exponent::NEUTRAL);
 
janus::export_graph_html(wind, "graph_wind_profile", "Wind_Profile");

Remarks

Interactive Examples - Explore the computational graphs:

• 🌬️ Power-Law Wind Profile

API Reference

WindTypes.hpp

Type/Function	Description
WindVector<Scalar>	3D wind in NED frame
GustVelocity<Scalar>	Turbulent gusts in body frame (u_g, v_g, w_g)
TurbulenceParams<Scalar>	σ and L values for each axis
TurbulenceSeverity	Enum: Light, Moderate, Severe
mil_spec_params(alt, severity)	MIL-F-8785C parameter lookup

ConstantWind.hpp

Function	Description
from_ned(n, e, d)	WindVector from NED components
from_speed_direction(spd, dir)	WindVector from speed and meteorological direction

WindShear.hpp

Function	Description
linear(alt, base, base_alt, rate)	Linear profile
power_law(alt, ref, ref_alt, exp)	Power-law profile
logarithmic(alt, u_star, z0, d)	Logarithmic profile
friction_velocity_from_ref(...)	Compute u* from reference wind

DrydenTurbulence.hpp

Function	Description
psd_longitudinal(ω, σ, L)	Longitudinal PSD
psd_lateral(ω, σ, L)	Lateral/vertical PSD
init_state<Scalar>()	Initialize filter state
compute_filter_coeffs(params, V, dt)	Discretize filters
step(state, coeffs, n_u, n_v, n_w)	Step filter, return gusts
mil_spec_coeffs(alt, sev, V, dt)	Convenience: params + discretize

VonKarmanTurbulence.hpp

Function	Description
psd_longitudinal(ω, σ, L)	von Kármán longitudinal PSD
psd_lateral(ω, σ, L)	von Kármán lateral/vertical PSD
init_state<Scalar>()	Initialize higher-order filter state
compute_filter_coeffs(params, V, dt)	Discretize 3rd/4th order filters
step(state, coeffs, n_u, n_v, n_w)	Step filter, return gusts

Example: Wind Optimization Demo

See examples/wind/wind_optimization.cpp for a complete demonstration including:

• Wind shear profile survey
• Fuel cost optimization with janus::Opti
• MIL-spec turbulence parameters
• Interactive graph export

# Build and run the demo
./scripts/build.sh
./build/examples/wind_optimization

The demo outputs:

Optimal climb profile:
  Waypoint 1: 1200.0m
  Waypoint 2: 1300.0m
  Total cost: 3204.38
 
Improvement over baseline: 1.03%