Constant-Strength Source

(Adapted fron Hess, J. L., & Smith, A. M. O. (1967), Calculation of potential flow about arbitrary bodies)

Given the following planar polygonal panel

the potential at an arbitrary point $(x,\, y,\, z)$ is calculated in the panel's coordinate system with constant-strength source as

\[ \begin{align*} \phi = -\frac{\sigma}{4\pi} \int\limits_{S} \frac{\mathrm{d}S}{r} ,\end{align*}\]

with $r=\sqrt{(x-\xi)^2 + (y-\eta)^2 + z^2}$.

NOTE: The local coordinate system used by Hess and Smith (and also Katz and Plotkin) follows the opposite to the right-hand rule to define its normal. For this reason, the local coordinate system implemented in all $\phi$ and $\mathbf{u}$ functions in FLOWPanel define $\hat{\boldsymbol\xi} = \hat{\mathbf{o}}$, $\hat{\boldsymbol\eta} = \hat{\mathbf{t}}$, and $\hat{\mathbf{n}}_\mathrm{HS} = -\hat{\mathbf{n}}$.

Integrating over a planar element with $n$ vertices results in

\[\begin{align*} \phi(x,y,z) = -\frac{\sigma}{4\pi} \sum\limits_{i,j\in A} \left[ R_{ij} Q_{ij} + % \vert z \vert J_{ij} \right] ,\end{align*}\]

where $A = \{(1,2),\,\dots,\,(n-1, n),\,(n, 1) \}$ and

\[\begin{align*} \bullet \quad & S_{i,j} = \frac{\eta_j - \eta_i}{d_{i,j}} \\ \bullet \quad & C_{i,j} = \frac{\xi_j - \xi_i}{d_{i,j}} \\ \bullet \quad & Q_{i,j} = \ln{\left(\frac{ r_i+r_j+d_{i,j} }{ r_i+r_j-d_{i,j} }\right)} \\ \bullet \quad & J_{i,j} = \arctan{\left(\frac{ R_{i,j}\lvert z \rvert( r_i s_{i,j}^{(j)} - r_j s_{i,j}^{(i)}) }{ r_i r_j R_{i,j}^2 + z^2 s_{i,j}^{(j)} s_{i,j}^{(i)} }\right)} \\ \bullet \quad & s_{i,j}^{(k)} = (\xi_k - x)C_{i,j} + (\eta_k - y)S_{i,j} \\ \bullet \quad & R_{i,j} = (x - \xi_i)S_{i,j} - (y - \eta_i)C_{i,j} \\ \bullet \quad & d_{i,j} = \sqrt{(\xi_j-\xi_i)^2 + (\eta_j-\eta_i)^2} \\ \bullet \quad & r_i = \sqrt{(x-\xi_i)^2 + (y-\eta_i)^2 + z^2} .\end{align*}\]

NOTE: The $\arctan$ defined in $J_{ij}$ is intended to be evaluated in the range $-\pi$ to $\pi$.

The velocity induced by the panel at $(x,y,z)$ is calculated as

\[\begin{align*} \mathbf{u} = \nabla \phi \end{align*}\]

which results in

\[\begin{align*} \bullet \quad & U_x = \frac{\partial\varphi}{\partial x} = - \frac{\sigma}{4\pi} \sum\limits_{i=1}^n S_i Q_i \\ \bullet \quad & U_y = \frac{\partial\varphi}{\partial y} = \frac{\sigma}{4\pi} \sum\limits_{i=1}^n C_i Q_i \\ \bullet \quad & U_z = \frac{\partial\varphi}{\partial z} = \text{sgn}(z) \frac{\sigma}{4\pi} \left( \Delta\theta - \sum\limits_{i=1}^n J_i \right) ,\end{align*}\]

where $\Delta\theta=2\pi$ if $(x,y,0)$ lies inside the quadrilateral, $\Delta\theta=0$ if not. Hess & Smith mentions that the point lies inside the quadrilateral iff all $R_{i}$ are positive.

NOTE: The $U_z$ velocity poses a discontinuity at the surface since $\lim\limits_{z\rightarrow \pm 0} U_z (0, 0, z) = \pm \frac{\sigma}{2}$. Hess and Smith recommends setting $U_z (0, 0, 0) = + \frac{\sigma}{2}$. In FLOWPanel, however, we let $U_z (0, 0, 0) = 0$, but we also have shifted all control points slightly in the direction of $\hat{\mathbf{n}}$. Remembering that $\hat{\mathbf{n}}_\mathrm{HS} = -\hat{\mathbf{n}}$, the control points are thus shifted in the $-z$ direction, effectively obtaining $\boxed{U_z (\mathbf{x}_\mathrm{cp}) \approx - \frac{\sigma}{2}}$.

NOTE 2: The offset of the control points is controlled through the properties body.CPoffset and body.characteristiclength of the Body type. Here, body.CPoffset is a small non-dimensional number $s$ and body.characteristiclength is a user-defined function of the form (nodes, panel) -> l that returns a characteristic length $\ell$ (which can be either computed based on the panel, or it can be set the same for all panels). Each control point $\mathbf{x}_\mathrm{cp}$ is then computed as $\mathbf{x}_\mathrm{cp} = \mathbf{x}_\mathrm{centroid} + s\ell\hat{\mathbf{n}}$.

NOTE 3: By default, FLOWPanel sets the characteristic length to be the square root of the panel area, but it is strongly recommended that the user provides their own characteristic length, and that this length is the same for all panels.

ASSUMPTIONS

• The panel is a polygon of $n$ number of vertices with $n\ge3$.
• The polygon is planar, i.e., all vertices lay on the same plane.
• Vectors betweens nodes 1 and 2 and nodes 1 and 3 are not collinear.
• The polygon must be concave.

OBSERVATIONS

• The term $Q_{i,j}$ makes this formulation singular at all vertices and edges of a panel; hence, FLOWPanel adds a small epsilon to the denominator of the log argument to avoid the singularity.

NOTE: The small offset added to the denominator of $Q_{i,j}$ corresponds to body.kerneloffset.

The potential and velocity field of a source panel of unitary strength ($\sigma=1$) is shown below

$\nabla \phi$ = $\mathbf{u}$ verification