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In [[fluid dynamics]], '''aerodynamic potential flow codes''' or '''panel codes''' are used to determine the fluid velocity, and subsequently the pressure distribution, on an object.  This may be a simple two-dimensional object, such as a circle or wing, or it may be a three-dimensional vehicle.

A series of singularities as sources, sinks, vortex points and [[doublet (potential flow)|doublets]] are used to model the panels and wakes.  These codes may be valid at subsonic and supersonic speeds.

== History ==
Early panel codes were developed in the late 1960s to early 1970s.  Advanced panel codes, such as Panair (developed by [[Boeing]]), were first introduced in the late 1970s, and gained popularity as computing speed increased.  Over time, panel codes were replaced with higher order panel methods and subsequently CFD ([[Computational Fluid Dynamics]]).  However, panel codes are still used for preliminary aerodynamic analysis as the time required for an analysis run is significantly less due to a decreased number of elements.

== Assumptions ==
These are the various assumptions that go into developing potential flow panel methods:
* [[Inviscid flow|Inviscid]]
* [[Incompressible flow|Incompressible]] <math> \nabla \cdot V=0 </math>
* Irrotational <math>\nabla \times V=0</math>
* [[Steady state|Steady]] <math> \frac{\partial}{\partial t}=0 </math>

However, the incompressible flow assumption may be removed from the potential flow derivation leaving: 
* Potential flow (inviscid, irrotational, steady) <math>\nabla^2 \phi=0</math>

== Derivation of panel method solution to potential flow problem ==
* From Small Disturbances
:<math> (1-M_\infty^2) \phi_{xx} + \phi_{yy} + \phi_{zz} = 0 </math> (subsonic)

* From [[Divergence theorem|Divergence Theorem]]
:<math>\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{S}\mathbf{F}\cdot\mathbf{n}\, dS</math>

* Let Velocity U be a twice continuously differentiable function in a region of volume V in space.  This function is the stream function <math> \phi </math>.
* Let P be a point in the volume V
* Let S be the surface boundary of the volume V.
* Let Q be a point on the surface S, and <math> R = |P-Q|</math>.

As Q goes from inside V to the surface of V, 
* Therefore:
:<math>U_p= -\frac{1} {4 \pi} \iiint\limits_V\left(\frac{\nabla^2\cdot\mathbf{U}}{R}\right) dV_Q</math>
:<math>   -\frac{1} {4 \pi} \iint\limits_S\left(\frac{\mathbf{n}\cdot \nabla \mathbf{U}  }{R}\right) dS_Q</math>
:<math>   +\frac{1} {4 \pi} \iint\limits_S\left(\mathbf{U}\mathbf{n} \cdot\nabla \frac{1}{R}\right) dS_Q</math>

For :<math>\nabla^2 \phi=0</math>, where the surface normal points inwards.
:<math>\phi_p = -\frac{1} {4 \pi} \iint\limits_S\left(\mathbf{n} \frac{  \nabla \phi_{U} - \nabla \phi_{L}}{R} - \mathbf{n} \left( \phi_{U} - \phi_{L}\right) \nabla \frac{1}{R} \right) dS_Q</math>

This equation can be broken down into both a source term and a doublet term.

The Source Strength at an arbitrary point Q is:
:<math> \sigma = \nabla \mathbf{n} (\nabla \phi_U-\nabla \phi_L )</math>

The Doublet Strength at an arbitrary point Q is:
:<math> \mu =\phi_U - \phi_L </math>

The simplified potential flow equation is:
:<math>\phi_p = -\frac{1} {4 \pi} \iint\limits_S\left(\frac{\sigma}{R} - \mu \cdot \mathbf{n}  \cdot \nabla \frac{1}{R} \right) dS</math>

With this equation, along with applicable boundary conditions, the potential flow problem may be solved.

== Required boundary conditions ==
The velocity potential on the internal surface and all points inside V (or on the lower surface S) is 0.
:<math> \phi_L = 0 </math>

The Doublet Strength is:
:<math> \mu =\phi_U - \phi_L </math>
:<math> \mu = \phi_U  </math>

The velocity potential on the outer surface is normal to the surface and is equal to the freestream velocity.
:<math> \phi_U = -V_\infty \cdot \mathbf{n} </math>

These basic equations are satisfied when the geometry is a 'watertight' geometry.  If it is watertight, it is a well-posed problem.  If it is not, it is an ill-posed problem.

== Discretization of potential flow equation ==
The potential flow equation with well-posed boundary conditions applied is:
:<math>\mu_P = \frac{1} {4 \pi} \iint\limits_S\left(\frac{V_\infty \cdot \mathbf{n}}{R}  \right) dS_U + \frac{1} {4 \pi} \iint\limits_S\left(\mu \cdot \mathbf{n}  \cdot \nabla \frac{1}{R} \right) dS</math>

*Note that the <math> dS_U </math> integration term is evaluated only on the upper surface, while th <math>dS</math> integral term is evaluated on the upper and lower surfaces.

The continuous surface S may now be discretized into discrete panels.  These panels will approximate the shape of the actual surface.  This value of the various source and doublet terms may be evaluated at a convenient point (such as the centroid of the panel).  Some assumed distribution of the source and doublet strengths (typically constant or linear) are used at points other than the centroid.  A single source term s of unknown strength <math>\lambda</math> and a single doublet term m of unknown strength <math>\lambda</math> are defined at a given point.

:<math>\sigma_Q = \sum_{i=1}^n \lambda_i s_i(Q)=0</math>
:<math>\mu_Q = \sum_{i=1}^n \lambda_i m_i(Q)</math>

where:
:<math>s_i = ln(r)</math>
:<math>m_i = </math>

These terms can be used to create a system of linear equations which can be solved for all the unknown values of <math>\lambda</math>.

== Methods for discretizing panels ==
* constant strength - simple, large number of panels required
* linear varying strength - reasonable answer, little difficulty in creating well-posed problems
* quadratic varying strength - accurate, more difficult to create a well-posed problem

Some techniques are commonly used to model surfaces.<ref>Section 7.6</ref>
* Body Thickness by line sources
* Body Lift by line doublets
* Wing Thickness by constant source panels
* Wing Lift by constant pressure panels
* Wing-Body Interface by constant pressure panels

== Methods of determining pressure ==
Once the Velocity at every point is determined, the pressure can be determined by using one of the following formulas.  All various [[Pressure coefficient]] methods produce results that are similar and are commonly used to identify regions where the results are invalid.

Pressure Coefficient is defined as:
:<math>C_p = \frac{p-p_\infty}{q_\infty}=\frac{p-p_\infty}{\frac{1}{2} \rho_\infty V_\infty^2} = \frac{p-p_\infty}{\frac{\gamma}{2}  p_\infty M_\infty^2 }</math>

The Isentropic Pressure Coefficient is:
:<math>C_p = \frac{2} {\gamma M_\infty^2} \left( \left(1+\frac{\gamma-1} {2} M_\infty^2 \left[\frac{1-|\vec{V}|^2}{|\vec{V_\infty}|^2}\right]\right)^{ \frac{\gamma}{\gamma-1} } -1   \right) </math>

The Incompressible Pressure Coefficient is:
:<math>C_p = 1 - \frac{|\vec{V}|^2}{|\vec{V_\infty}|^2} </math>

The Second Order Pressure Coefficient is:
:<math>C_p = 1-|\vec{V}|^2 + M_\infty^2 u^2 </math>

The Slender Body Theory Pressure Coefficient is:
:<math>C_p = -(2u +v^2 +w^2) </math>

The Linear Theory Pressure Coefficient is:
:<math>C_p = -2u </math>

The Reduced Second Order Pressure Coefficient is:
:<math>C_p = 1-|\vec{V}|^2 </math>

== What panel methods cannot do ==

*Panel methods are inviscid solutions. You will not capture viscous effects except via user "modeling" by changing the geometry.
*Solutions are invalid as soon as the flow changes locally from subsonic to supersonic (i.e. the critical Mach number has been exceeded) or vice versa.

== Potential flow software ==
{{See also|Computational fluid dynamics#Background and history}}

{| class="wikitable" style="font-size:90%;" 
! rowspan=2 | Name !! rowspan=2 | License !! rowspan=2 | Lan !! colspan=3 | Operating system !! rowspan=2 | Geometry import !! colspan=3 | Meshing !! rowspan=2 | Body Representation !! rowspan=2 | Wake model !! rowspan=2 | Developer
|-
| [[Linux]] || [[OS X]] || [[Microsoft Windows]] || Structured || Unstructured || Hybrid
|-
| [http://www.aeolus-aero.com Aeolus ASP]|| {{proprietary}} || Java / [[Fortran]] || {{yes}} || || {{yes}} || || {{yes}} || || || Quadrilaterals|| || [http://www.aeolus-aero.com Aeolus Aero Sketch Pad]
|-
| CMARC      || {{proprietary}} || [[C (programming language)|C]] || {{yes}} || || {{yes}} || || || || || || || {{URL|http://www.aerologic.com/cmarc.html|Homepage}}, AeroLogic, based on PMARC-12
|-
| DesignFOIL || {{proprietary}} || || {{no|[[Wine (software)|Wine]]}} || || {{yes}} || || || || || || || {{URL|http://www.dreesecode.com/}}, DreeseCode Software LLC
|-
| [http://www.researchinflight.com/products.html FlightStream] || {{proprietary}} || Fortran / C++ || || || {{yes}} || CAD, Discrete || || {{Yes}} || {{Yes}} || Solids || || [http://www.researchinflight.com Research in Flight Company] 
|-
| HESS       || {{proprietary}} || || || || || || || || || || || [[Douglas Aircraft Company]]
|-
| LinAir    || {{proprietary}} || || {{yes}} || || || || || || || || || [[Desktop Aeronautics]]
|-
| MACAERO   || {{proprietary}} || || || || || || || || || || || [[McDonnell Aircraft]]
|-
| [http://www.flowsol.co.uk/products/newpan NEWPAN] || {{proprietary}} || [[C++]] || {{yes}} || || || || {{yes}} || {{yes}} || || || || [http://www.flowsol.co.uk Flow Solutions Ltd.]
|-
| [http://www.openvogel.org Tucan] || {{free|[[GPLv3]]}} || [[VB.NET]]/[[C Sharp (programming language)|C#.NET]] || {{yes}} (Console) || || {{yes}} || STL || {{yes}}|| || || Quadrilaterals & triangles || Free || G. Hazebrouck & contributors
|-
| [[QBlade]] || {{free|[[GPLv2]]}} || [[C (programming language)|C]]/[[C++]] || {{yes}} || ||   || || || || || || || [[Technical University of Berlin|TU Berlin]]
|- 
| Quadpan    || {{proprietary}} || |||| || || || || || || || || [[Lockheed Corporation|Lockheed]]
|-
| PanAir a502 || {{free|[[Public domain software]]}} || [[Fortran]] || {{yes}} || {{yes}} || {{yes}} || || || || || || || {{URL|http://www.pdas.com/panair.html|Homepage}}, [[Boeing]]?
|-
| [https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/PANUKL PANUKL] || {{proprietary|[[freeware]]}} || [[C++]]/[[Fortran]] || {{yes}} || || {{yes}} || NX - partially || {{yes}} || || || Quadrilaterals || || [[Warsaw University of Technology]], {{URL|https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/PANUKL|PANUKL}} exports data to {{URL|https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/SDSA|SDSA}} and to [[Calculix]] 
|-
| PMARC       || {{URL|https://software.nasa.gov/software/ARC-14407-1|Free On Request}} || [[Fortran 77]] || {{yes|UNIX}} || {{yes}} || {{yes}} || || || || || || || [[NASA]], descendant of VSAERO
|-
| [[VSAero]]    || {{proprietary}} || || {{yes|UNIX}} || || || || || || || || || {{URL|https://starkaerospace.com/products-services/ami/software/|Homepage}}
|-
| [[Vortexje]] || {{free|[[GPLv2]]}} || [[C++]] || {{yes}} || || || || || || || || || Baayen & Heinz GmbH
|-
| [[XFOIL]]   || {{free|[[GPLv2]]}} || [[Fortran]] || {{yes}} || {{yes}} || {{yes}} || || || || || || || {{URL|http://web.mit.edu/drela/Public/web/xfoil/}}
|-
| [[XFLR5]]   || {{free|[[GPLv2]]}} || [[C (programming language)|C]]/[[C++]] || {{yes}} || ||  || || || || || || || {{URL|http://www.xflr5.com}}
|-
| VSPAERO Packaged with [[OpenVSP]]   || {{free|[[NASA Open Source Agreement|NOSA]]}} || [[C++]] || {{yes}} || {{yes}} || {{yes}} || || || {{yes}} || || Polygons, typically quad & tri dominated || Free & rigid || {{URL|http://openvsp.org}}
|-
| MachLine   || {{free|[[MIT License]]}} || [[Fortran]] || {{yes}} || untested || untested || STL, VTK, TRI || || {{yes}} || || Solid bodies using surface tris || Rigid || Utah State University AeroLab {{URL|aerolab.usu.edu}} {{URL|http://github.com/usuaero/MachLine}}
|}

== See also ==
*[[Stream function]]
*[[Conformal mapping]]
*[[Velocity potential]]
*[[Divergence theorem]]
*[[Joukowsky transform]]
*[[Potential flow]]
*[[Circulation (fluid dynamics)|Circulation]]
*[[Biot–Savart law#Aerodynamics applications|Biot–Savart law]]

==Notes==
{{Reflist}}

==References==
* [http://www.pdas.com/ Public Domain Aerodynamic Software], A Panair Distribution Source, Ralph Carmichael
* [http://hdl.handle.net/2060/19920013405 Panair Volume I, Theory Manual, Version 3.0], Michael Epton, Alfred Magnus, 1990 [[Boeing]]
* [http://hdl.handle.net/2060/19920013622 Panair Volume II, Theory Manual, Version 3.0], Michael Epton, Alfred Magnus, 1990 [[Boeing]]
* [http://hdl.handle.net/2060/19850002614 Panair Volume III, Case Manual, Version 1.0], Michael Epton, Kenneth Sidewell, Alfred Magnus, 1981 [[Boeing]]
* [http://hdl.handle.net/2060/19920013399 Panair Volume IV, Maintenance Document, Version 3.0], Michael Epton, Kenneth Sidewell, Alfred Magnus, 1991 [[Boeing]]
* [https://wayback.archive-it.org/all/20100424140724/http://hdl.handle.net/2060/19710009882 Recent Experience in Using Finite Element Methods For The Solution Of Problems In Aerodynamic Interference], Ralph Carmichael, 1971 [[NASA Ames Research Center]]
* [https://ntrs.nasa.gov/citations/19910009745 NASA-TP-2995 Panel methods: An introduction], Larry L. Erickson, 1990 NASA

[[Category:Fluid dynamics]]