Theory

The following theory described is meant to be minimal, with mainly the equations presented for reference when reading the code.

Geometry

Wing Parametrization

A half-wing is defined in terms of a nested trapezoid. A single trapezoid is called a section. A section consists of two foil profiles, and their associated chord lengths and twist angles. Between them is their span length with associated leading-edge dihedral and sweep angles. So a general half-wing consisting of $n$ sections will have $n$ entries for foils, chords, and twists, and $n - 1$ entries for spans, dihedrals, sweeps for some $n \in \mathbb N$. The following illustration should help visualize the concept.

Aerodynamics

The aerodynamic analyses in AeroFuse mainly utilize potential flow theory and solve problems using a boundary element method. This essentially is the following Laplace equation problem with the following Robin (?) boundary conditions:

\[\nabla^2 \phi = 0, \quad \mathbf V \equiv \nabla \phi \cdot \hat{\mathbf n} = 0, \quad \lim_{\mathbf r \to \infty} \phi(\mathbf r) \to 0\]

Doublet-Source Panel Method

The doublet-source panel method predicts inviscid, incompressible, irrotational, isentropic external flow over surfaces in 2 dimensions.

Source and doublet singularities are placed on the surface, and boundary conditions are imposed on their induced velocity to obtain a well-posed problem. The induced velocity is evaluated by the corresponding free-field Green function for each singularity.

\[\]

The velocities are added to obtain the total induced velocity at a point $\mathbf r$.

\[\]

Vortex Lattice Method

The vortex lattice method predicts inviscid, incompressible, irrotational, isentropic external flow over "thin" surfaces in 3 dimensions.

Vortex filaments are placed on the surface, and boundary conditions are imposed on their induced velocity to obtain a well-posed problem. The induced velocity is evaluated by the Biot-Savart integral for a vortex line of length $\ell$ with a constant circulation strength $\Gamma$.

\[\mathbf V(\mathbf r, \mathbf r') = \frac{\Gamma}{4\pi} \int_0^\ell \frac{d\boldsymbol\ell' \times (\mathbf r - \mathbf r')}{|\mathbf r - \mathbf r'|^3}\]

The vortices can be set up in various configurations consisting of bound or semi-infinite filaments, commonly in the form of horseshoes or vortex rings.

1. Horseshoe elements: These are defined by a finite bound leg and two semi-infinite trailing legs. AeroFuse encodes this information in the Horseshoe type.
2. Vortex rings: These are defined by four bound legs. AeroFuse encodes this information in the VortexRing type.

A quasi-steady freestream condition with velocity $\mathbf U$ and rotation $\boldsymbol\Omega$ (in the body's frame) defines an external flow. The induced velocity at a point is given by:

\[\mathbf V_{\infty}(\mathbf r) = - (\mathbf U + \boldsymbol\Omega \times \mathbf r)\]

The velocities are added to obtain the total induced velocity at a point $\mathbf r$.

\[\mathbf V(\mathbf r) = \sum_i \frac{\Gamma_i}{4\pi} \int_0^{\ell_i} \frac{d\boldsymbol\ell_i' \times (\mathbf r - \mathbf r_i')}{|\mathbf r - \mathbf r_i'|^3} + \mathbf V_\infty(\mathbf r)\]

Imposing the Neumann boundary condition $\mathbf V \cdot \hat{\mathbf n} = 0$ defines the problem. The construction, in essence, fundamentally results in the following linear system to be solved:

\[\mathbf A \boldsymbol\Gamma = -V_{\infty} \cdot [\hat{\mathbf n}_i ]_{i = 1, \ldots, N}\]

Compressibility Corrections

The following Prandtl-Glauert equation, represented in wind axes, is applied for a weakly compressible flow problem ($0.3 \leq M_\infty \leq 0.7$).

\[\beta_{PG}^2\frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} + \frac{\partial^2\phi}{\partial z^2} = 0, \quad \beta_{PG}^2 = \left(1 - M_\infty^2\right)\]

The Prandtl-Glauert transformation $\phi(x,y,z; \beta_{PG}) \to \bar\phi(\bar x, \bar y, \bar z)$ converts this equation into an equivalent incompressible flow problem in a transformed geometric space. This "bar" map scales the coordinates $(\bar x,\bar y, \bar z) = (x,\beta_{PG} y, \beta_{PG} z)$ and the potential $\bar\phi = \beta_{PG}^2 \phi$ in sequence. Hence the transformed equation satisfies the Laplace equation with the Neumann boundary condition:

\[\begin{aligned} \bar\nabla^2 \bar\phi & = 0, \\ \bar{\mathbf V} \cdot \hat{\bar{\mathbf n}} & = 0, \end{aligned}\]

where $\bar\nabla$ is differentiation with respect to the transformed coordinates and $\hat{\bar{\mathbf n}} = (\beta_{PG} \hat n_x, \hat n_y, \hat n_z)$ which can be proved by computing the appropriate cross product.

As the circulation is a scalar (hence invariant of the coordinate transformation but not the potential scaling), the inverse is also readily derived.

\[\begin{aligned} \Gamma & = \int \mathbf V \cdot d\boldsymbol\ell = \int \nabla\phi \cdot d\boldsymbol\ell, \\ \bar{\Gamma} & = \int \bar{\mathbf V} \cdot d\bar {\boldsymbol\ell} = \int\bar\nabla\bar\phi \cdot d\bar{\boldsymbol\ell}, \\ \implies \Gamma & = \bar{\Gamma} / \beta_{PG}^2 \end{aligned}\]

Hence the solution of the resultant incompressible system in transformed coordinates provides the necessary quantities of interest for calculating the dynamics.

Structures

The structural analyses in AeroFuse utilize linear finite-element methods.

Particularly, a $2$-dimensional beam element model has been implemented following the standard formulation using cubic Hermite shape functions based on Euler-Bernoulli beam theory. These are embedded into a $3$-dimensional local coordinate system in the vortex lattice method without loss of generality.

The linear system consisting of the stiffness matrix $\mathbf K$ and load vector $\mathbf f$ are solved to obtain the displacement vector $\boldsymbol\delta$.

\[\mathbf K \boldsymbol\delta = \mathbf f\]

Aeroelasticity

The vortex lattice method and beam element model are combined into a coupled system to perform static aeroelastic analyses. The analysis is made nonlinear via promotion of the angle of attack $\alpha$ to a variable by specifying the load factor $n$ with a given weight $W$ at fixed sideslip angle $\beta$.

Define $\mathbf x = [\boldsymbol\Gamma, \boldsymbol\delta, \alpha]$ as the state vector satisfying the residual equations:

\[\begin{aligned} \mathcal R_A(\mathbf x) & = \mathbf A(\boldsymbol\delta) \boldsymbol\Gamma - \mathbf V_\infty(\alpha) \cdot [\mathbf n_i(\boldsymbol\delta)]_{i = 1,\ldots, N} \\ \mathcal R_S(\mathbf x) & = \mathbf K \boldsymbol\delta - \mathbf f(\boldsymbol\delta, \boldsymbol\Gamma, \alpha) \\ \mathcal R_L(\mathbf x) & = L(\boldsymbol\delta, \boldsymbol\Gamma, \alpha) - n W \\ \end{aligned}\]

where the lift $L$ is obtained by transforming forces computed via the Kutta-Jowkowski theorem into wind axes.

\[[D, Y, L] = \mathbf R_B^W(\alpha, \beta)\left(\sum_{i = 1}^N \rho \mathbf V_i \times \boldsymbol\Gamma_i \boldsymbol \ell_i(\boldsymbol\delta)\right)\]

and the structural load vector $\mathbf f$ is obtained from conservative and consistent load averaging of the Kutta-Jowkowski forces in geometric axes.

\[\]

References

1. Mark Drela. Flight Vehicle Aerodynamics. MIT Press, 2014.
2. Joseph Katz and Allen Plotkin. Low-Speed Aerodynamics, Second Edition. Cambridge University Press, 2001.

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