Subsonic Potential Aerodynamics

A 2D Aerodynamic Potential-Flow Code

Theoretical Model

Velocity Field

• incompressible: $\nabla \cdot \underline{V} = 0$
• conservative: $\underline{V} = \nabla \phi \implies \nabla \times \underline{V} = \nabla \times \nabla \phi = 0$ (irrotational)

$$\nabla \cdot \underline{V} = \nabla \cdot \nabla \phi = \nabla^2 \phi = 0$$

Integral Equation of velocity potential $\phi$

$$\phi(\underline{r}_p) = \iint_S \frac{\sigma}{2\pi} \ln{(\lVert \underline{r} - \underline{r}_p \rVert)} \mathrm{d}S + \iint_{S \cup S_w} - \frac{\mu}{2\pi} (\underline{e}_n \cdot \nabla) \ln{(\lVert \underline{r} - \underline{r}_p \rVert)} \mathrm{d}S + \phi_\infty(\underline{r}_p)$$

where:

• $\mu = \phi - \phi_i$
• $\sigma = (\underline{e}_n \cdot \nabla)(\phi - \phi_i)$
• $\phi_\infty(\underline{r}_p) = \iint_{S_\infty} \left[ (\underline{e}_n \cdot \nabla) \phi \ln{ \left( \frac{\lVert \underline{r} - \underline{r}_p \rVert}{2\pi} \right) } - \phi (\underline{e}_n \cdot \nabla) \ln{ \left( \frac{\lVert \underline{r} - \underline{r}_p \rVert}{2\pi} \right) } \right] \mathrm{d}S$

if $\mu = \phi - \phi_i = \phi - \phi_\infty$ and $\sigma = (\underline{e}_n \cdot \nabla)(\phi - \phi_i) = (\underline{e}_n \cdot \nabla)(\phi - \phi_\infty )$, then for $P \in S^-$ :

$$\iint_S \frac{\sigma}{2\pi} \ln{(\lVert \underline{r} - \underline{r}_p \rVert)} \mathrm{d}S + \iint_{S \cup S_w} - \frac{\mu}{2\pi} (\underline{e}_n \cdot \nabla) \ln{(\lVert \underline{r} - \underline{r}_p \rVert)} \mathrm{d}S = 0, \qquad \forall (x_p, y_p, z_p) \in S: (\underline{r} - \underline{r}_p) \cdot \underline{e}_n \to 0^+$$

note: $\sigma = (\underline{e}_n \cdot \nabla)(\phi - \phi_i) = (\underline{e}_n \cdot \nabla)(\phi - \phi_\infty ) = (\underline{e}_n \cdot \nabla)\phi - (\underline{e}_n \cdot \nabla)\phi_\infty = \underline{V} - \underline{e}_n \cdot \underline{V}_\infty = - \underline{e}_n \cdot \underline{V}_\infty$

Notable Observations:

• A vector field $\underline{V}: U \to \mathbb{R}^n$, where $\mathbb{U}$ is an open subset of $\mathbb{R}^n$, is said to be conservative if there exists a $\mathrm{C}^1$ (continuously differentiable) scalar field $\phi$ on $\mathbb{U}$ such that $\underline{V} = \nabla \phi$.

• According to Poincaré's Lemma, A continuously differentiable ($\mathrm{C}^1$) vector field $\underline{V}$ defined on a simply connected subset $\mathbb{U}$ of $\mathbb{R}^n$ ($\underline{V} \colon \mathbb{U} \subseteq \mathbb{R}^n \to \mathbb{R}^n$), is conservative if and only if it is irrotational throughout its domain ($\nabla \times \underline{V} = 0$, $\forall \underline{x} \in \mathbb{U}$).

• Circulation $\Gamma = \oint_{C} \underline{V} \cdot \mathrm{d} \underline{l} = \iint_S \nabla \times \underline{V} \cdot \mathrm{d}\underline{S}$ In a conservative vector field this integral evaluates to zero for every closed curve. $\Gamma = \oint_{C} \underline{V} \cdot \mathrm{d} \underline{l} = \iint_S \nabla \times \underline{V} \cdot \mathrm{d}\underline{S} = \iint_S \nabla \times \nabla \phi \cdot \mathrm{d}\underline{S} = 0$

• The space exterior to a 2D object (e.g. an airfoil) is not simply connected

Numerical Model (Panel Methods)

$$\sum_{j=0}^{N_s - 1} B_{ij} \sigma_j + \sum_{j=0}^{N_s + N_w - 1} C_{ij} \mu_j = 0 , \qquad 0 \le i < N_s$$

where:

• $B_{ij} = \frac{1}{2\pi} \iint_{S_j} \ln{(\lVert \underline{r} - \underline{r}_{cp_i} \rVert)} \mathrm{d}{S_j} = \frac{1}{2\pi} \iint_{t_1}^{t_2} \ln \left(\sqrt{(t_{cp_i} - t)^2 + n_{cp_i}^2} \right) \mathrm{d}{t_j}$

• $C_{ij} = \frac{1}{2\pi} \iint_{S_j} (\underline{e}_n \cdot \nabla) \ln{(\lVert \underline{r} - \underline{r}_{cp_i} \rVert)} \mathrm{d}{S_j} = \frac{1}{2\pi} \iint_{S_j} \frac{n_{cp_i}}{(t_{cp_i} - t)^2 + n_{cp_i}^2} \mathrm{d}{t_j}$

from Kutta Condition: $\mu_w = const = \mu_U - \mu_L$

$$A_{ij} \mu_j = - B_{ij} \sigma_j , \qquad A_{ij} = \begin{cases} C_{ij} + \sum\limits_{k} C_{ik} & \text{if $j=0$}\\\ C_{ij} & \text{if $0 < j < N_s - 1$}\\\ C_{ij} - \sum\limits_{k} C_{ik} & \text{if $j=N_s-1$} \end{cases}$$ $$0 \le i < N_s \qquad 0 \le j < N_s \qquad N_s \le k < N_s + N_w$$

Features

1. Calculation of Non-Lifting Potential Flow about 2D arbitrarily-shaped rigid bodies

   1. Steady simulations
   2. Unsteady simulations

2. Calculation of Lifting Pseudo-Potential Flow around 2D arbitrarily-shaped rigid bodies

   1. Steady state simulations with flat rigid wake model
   2. Steady state iterative simulations with flexible wake model
   3. Unsteady simulations with a shedding wake model

Simulation Results

Potential Flow around a 2D Circular Object

Potential Flow around an Airfoil

Steady Simulation with rigid wake

Steady Simulation with iterative wake

Unsteady Simulation with wake roll up