Panel Method

Surface Panel Method (aerodynamics/cfd/panel-method)

Use when the task is a surface panel method for incompressible potential flow: discretizing an airfoil or body into source or doublet panels, applying the Neumann or Dirichlet boundary condition, enforcing the Kutta condition, or computing the pressure distribution and force coefficients.

Domain quick reference

• Panel methods replace the body surface with flat panels carrying constant-strength singularities. Source panels model thickness and non-lifting bodies; doublet panels (or vortex panels) model lifting surfaces; the singularity strengths are the unknowns of a linear system assembled from influence coefficients.
• Neumann boundary condition (source panel method): the total velocity normal to the surface is zero at each control point (panel midpoint), sum_j A_ij sigma_j = -(V_inf dot n_i), with the source self-influence A_ii = 1/2. The source panel method reproduces the analytic cylinder pressure coefficient Cp = 1 - 4 sin^2(theta) to machine precision for an inscribed polygon.
• Dirichlet boundary condition (doublet panel method): the interior velocity potential is zero at each control point, sum_j D_ij mu_j = -phi_inf_i with D_ii = 1/2. The doublet strength equals the surface velocity potential, so the surface velocity is the surface derivative of the doublet distribution.
• Kutta condition: for a lifting body with a sharp trailing edge, the tangential velocities on the upper and lower trailing-edge panels must be equal, which fixes the circulation. Thin-airfoil theory gives the flat plate result Gamma = pi c V_inf sin(alpha), so c_l = 2 pi sin(alpha); the Kutta-Joukowski relation converts circulation to lift with c_l = 2 Gamma / (c V_inf).
• Pressure: the Bernoulli relation for incompressible potential flow gives Cp = 1 - (V/V_inf)^2. Integrating Cp times the panel outward normal over the closed surface yields the lift and drag coefficients; d'Alembert's paradox states the drag is zero in inviscid potential flow, so a symmetric body at zero angle of attack must show zero lift and near-zero drag.
• 3D bodies: the sphere is the standard validation anchor for 3D panel codes. Potential flow over a sphere gives Cp = 1 - (9/4) sin^2(theta) and the exact source strength 2 V_inf cos(theta). Quadrilateral surface panels need area, centroid, and unit normal checks.

Workflow

1. Build the panel geometry with build_panels from a closed, counter-clockwise point list (first point equals last). Each panel gets its midpoint, length, outward normal, and tangent.
2. For a non-lifting body, assemble the influence matrix with source_influence_matrix and solve for the source strengths with neumann_source_solution, which applies the Neumann boundary condition at the control points.
3. For the Dirichlet form, assemble the doublet influence matrix with dirichlet_doublet_matrix and solve with dirichlet_doublet_solution, then recover the surface velocity with surface_velocity_from_doublets.
4. Evaluate the surface velocity and pressure coefficient with surface_velocity_and_cp, then integrate with force_coefficients for the lift and drag coefficients.
5. For lifting cases, apply the Kutta condition: check trailing-edge velocity equality with kutta_condition_check and estimate the circulation with flat_plate_circulation and cl_from_circulation.
6. For 3D bodies, check the panel bookkeeping with quad_panel_properties and validate against the sphere solution with sphere_potential_flow_cp and sphere_source_strength.
7. Solve any assembled system with solve_linear_system (Gaussian elimination with partial pivoting).

Pitfalls

• Wrong boundary condition: the Neumann form works on velocities (zero normal velocity), the Dirichlet form works on potentials (zero interior potential); mixing the two assemblies produces a wrong system.
• Forgetting the doublet self-influence: the half-space solid angle D_ii = 1/2 is required for the Dirichlet matrix, and the source self-influence A_ii = 1/2 for the Neumann matrix.
• Computing the subtended panel angle from principal-value atan2 differences: endpoint directions wrap at 180 degrees and break the influence matrix; use the signed cross and dot product angle instead.
• Ignoring the Kutta condition on a lifting airfoil: without it the circulation is undetermined and the lift is wrong.
• Treating potential-flow drag as physical: the panel method returns zero inviscid drag by construction (d'Alembert); viscous drag needs a boundary-layer or CFD treatment.
• Clockwise point ordering flips the normals inward and inverts the boundary condition signs; require a positive signed area.

Behavior contract (gate 3)

The panel geometry, influence, boundary condition, Kutta, pressure, and 3D body logic is exercised by the gate 3 contract test: scripts/test_panel_method.py against scripts/panel_method_logic.py (stdlib unittest, offline). Run: python3 scripts/test_panel_method.py

Compliance

• NACA Report 824 is US government work (public domain); summary and reference data only, per standards-map.yaml. The panel method methodology here is common potential-flow practice, not reproduced text.
• compliance: STANDARDS-REF, gated: false.