SherbyNet - /page/wikibook/entry.mu

Micron Document

`:top

In `F33f`_`[fluid dynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fluid_dynamics]`_`f, `!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 `F33f`_`[doublets`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doublet_(potential_flow)]`_`f are used to model the panels and wakes. These codes may be valid at subsonic and supersonic speeds.

>>Contents

• `F0af`_`[History`#history]`_`f

• `F0af`_`[Assumptions`#assumptions]`_`f

• `F0af`_`[Derivation of panel method solution to potential flow problem`#derivation-of-panel-method-solution-to-potential-flow-problem]`_`f

• `F0af`_`[Required boundary conditions`#required-boundary-conditions]`_`f

• `F0af`_`[Discretization of potential flow equation`#discretization-of-potential-flow-equation]`_`f

• `F0af`_`[Methods for discretizing panels`#methods-for-discretizing-panels]`_`f

• `F0af`_`[Methods of determining pressure`#methods-of-determining-pressure]`_`f

• `F0af`_`[What panel methods cannot do`#what-panel-methods-cannot-do]`_`f

• `F0af`_`[Potential flow software`#potential-flow-software]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>History

Early panel codes were developed in the late 1960s to early 1970s. Advanced panel codes, such as Panair (developed by `F33f`_`[Boeing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boeing]`_`f), 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 (`F33f`_`[Computational Fluid Dynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Computational_Fluid_Dynamics]`_`f). 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:

• `F33f`_`[Inviscid`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inviscid_flow]`_`f

• `F33f`_`[Incompressible`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Incompressible_flow]`_`f ∇ ∇ ⋅ ⋅ V = 0 {\\displaystyle \\nabla \\cdot V=0}

• Irrotational ∇ ∇ × × V = 0 {\\displaystyle \\nabla \\times V=0}

• `F33f`_`[Steady`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Steady_state]`_`f ∂ ∂ ∂ ∂ t = 0 {\\displaystyle {\\frac {\\partial }{\\partial t}}=0}

However, the incompressible flow assumption may be removed from the potential flow derivation leaving:

• Potential flow (inviscid, irrotational, steady) ∇ ∇ 2 ϕ ϕ = 0 {\\displaystyle \\nabla ^{2}\\phi =0}

>>Derivation of panel method solution to potential flow problem

• From Small Disturbances

( 1 − − M ∞ ∞ 2 ) ϕ ϕ x x + ϕ ϕ y y + ϕ ϕ z z = 0 {\\displaystyle (1-M_{\\infty }^{2})\\phi _{xx}+\\phi _{yy}+\\phi _{zz}=0} (subsonic)

• From `F33f`_`[Divergence Theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divergence_theorem]`_`f

∭ ∭ V ( ∇ ∇ ⋅ ⋅ F ) d V = ∬ ∬ S F ⋅ ⋅ n d S {\\displaystyle \\iiint \\limits _{V}\\left(\\nabla \\cdot \\mathbf {F} \\right)dV=\\iint \\limits _{S}\\mathbf {F} \\cdot \\mathbf {n} \\,dS}

• Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function ϕ ϕ {\\displaystyle \\phi } .

• 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 R = | P − − Q | {\\displaystyle R=|P-Q|} .

As Q goes from inside V to the surface of V,

• Therefore:

U p = − − 1 4 π π ∭ ∭ V ( ∇ ∇ 2 ⋅ ⋅ U R ) d V Q {\\displaystyle U_{p}=-{\\frac {1}{4\\pi }}\\iiint \\limits _{V}\\left({\\frac {\\nabla ^{2}\\cdot \\mathbf {U} }{R}}\\right)dV_{Q}}

− − 1 4 π π ∬ ∬ S ( n ⋅ ⋅ ∇ ∇ U R ) d S Q {\\displaystyle -{\\frac {1}{4\\pi }}\\iint \\limits _{S}\\left({\\frac {\\mathbf {n} \\cdot \\nabla \\mathbf {U} }{R}}\\right)dS_{Q}}

+ 1 4 π π ∬ ∬ S ( U n ⋅ ⋅ ∇ ∇ 1 R ) d S Q {\\displaystyle +{\\frac {1}{4\\pi }}\\iint \\limits _{S}\\left(\\mathbf {U} \\mathbf {n} \\cdot \\nabla {\\frac {1}{R}}\\right)dS_{Q}}

For : ∇ ∇ 2 ϕ ϕ = 0 {\\displaystyle \\nabla ^{2}\\phi =0} , where the surface normal points inwards.

ϕ ϕ p = − − 1 4 π π ∬ ∬ S ( n ∇ ∇ ϕ ϕ U − − ∇ ∇ ϕ ϕ L R − − n ( ϕ ϕ U − − ϕ ϕ L ) ∇ ∇ 1 R ) d S Q {\\displaystyle \\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}}

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

The Source Strength at an arbitrary point Q is:

σ σ = ∇ ∇ n ( ∇ ∇ ϕ ϕ U − − ∇ ∇ ϕ ϕ L ) {\\displaystyle \\sigma =\\nabla \\mathbf {n} (\\nabla \\phi _{U}-\\nabla \\phi _{L})}

The Doublet Strength at an arbitrary point Q is:

μ μ = ϕ ϕ U − − ϕ ϕ L {\\displaystyle \\mu =\\phi _{U}-\\phi _{L}}

The simplified potential flow equation is:

ϕ ϕ p = − − 1 4 π π ∬ ∬ S ( σ σ R − − μ μ ⋅ ⋅ n ⋅ ⋅ ∇ ∇ 1 R ) d S {\\displaystyle \\phi _{p}=-{\\frac {1}{4\\pi }}\\iint \\limits _{S}\\left({\\frac {\\sigma }{R}}-\\mu \\cdot \\mathbf {n} \\cdot \\nabla {\\frac {1}{R}}\\right)dS}

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.

ϕ ϕ L = 0 {\\displaystyle \\phi _{L}=0}

The Doublet Strength is:

μ μ = ϕ ϕ U − − ϕ ϕ L {\\displaystyle \\mu =\\phi _{U}-\\phi _{L}}

μ μ = ϕ ϕ U {\\displaystyle \\mu =\\phi _{U}}

The velocity potential on the outer surface is normal to the surface and is equal to the freestream velocity.

ϕ ϕ U = − − V ∞ ∞ ⋅ ⋅ n {\\displaystyle \\phi _{U}=-V_{\\infty }\\cdot \\mathbf {n} }

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:

μ μ P = 1 4 π π ∬ ∬ S ( V ∞ ∞ ⋅ ⋅ n R ) d S U + 1 4 π π ∬ ∬ S ( μ μ ⋅ ⋅ n ⋅ ⋅ ∇ ∇ 1 R ) d S {\\displaystyle \\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}

• Note that the d S U {\\displaystyle dS_{U}} integration term is evaluated only on the upper surface, while th d S {\\displaystyle dS} 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 λ λ {\\displaystyle \\lambda } and a single doublet term m of unknown strength λ λ {\\displaystyle \\lambda } are defined at a given point.

σ σ Q = ∑ ∑ i = 1 n λ λ i s i ( Q ) = 0 {\\displaystyle \\sigma _{Q}=\\sum _{i=1}^{n}\\lambda _{i}s_{i}(Q)=0}

μ μ Q = ∑ ∑ i = 1 n λ λ i m i ( Q ) {\\displaystyle \\mu _{Q}=\\sum _{i=1}^{n}\\lambda _{i}m_{i}(Q)}

where:

s i = l n ( r ) {\\displaystyle s_{i}=ln(r)}

m i = {\\displaystyle m_{i}=}

These terms can be used to create a system of linear equations which can be solved for all the unknown values of λ λ {\\displaystyle \\lambda } .

>>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.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

• 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 `F33f`_`[Pressure coefficient`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pressure_coefficient]`_`f methods produce results that are similar and are commonly used to identify regions where the results are invalid.

Pressure Coefficient is defined as:

C p = p − − p ∞ ∞ q ∞ ∞ = p − − p ∞ ∞ 1 2 ρ ρ ∞ ∞ V ∞ ∞ 2 = p − − p ∞ ∞ γ γ 2 p ∞ ∞ M ∞ ∞ 2 {\\displaystyle 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}}}}

The Isentropic Pressure Coefficient is:

C p = 2 γ γ M ∞ ∞ 2 ( ( 1 + γ γ − − 1 2 M ∞ ∞ 2 [ 1 − − | V → → | 2 | V ∞ ∞ → → | 2 ] ) γ γ γ γ − − 1 − − 1 ) {\\displaystyle 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)}

The Incompressible Pressure Coefficient is:

C p = 1 − − | V → → | 2 | V ∞ ∞ → → | 2 {\\displaystyle C_{p}=1-{\\frac {|{\\vec {V}}|^{2}}{|{\\vec {V_{\\infty }}}|^{2}}}}

The Second Order Pressure Coefficient is:

C p = 1 − − | V → → | 2 + M ∞ ∞ 2 u 2 {\\displaystyle C_{p}=1-|{\\vec {V}}|^{2}+M_{\\infty }^{2}u^{2}}

The Slender Body Theory Pressure Coefficient is:

C p = − − ( 2 u + v 2 + w 2 ) {\\displaystyle C_{p}=-(2u+v^{2}+w^{2})}

The Linear Theory Pressure Coefficient is:

C p = − − 2 u {\\displaystyle C_{p}=-2u}

The Reduced Second Order Pressure Coefficient is:

C p = 1 − − | V → → | 2 {\\displaystyle C_{p}=1-|{\\vec {V}}|^{2}}

>>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

|---|---|---|---|---|---|---|---|---|---|---|---|---|

| CMARC | Proprietary | C | Yes | | Yes | | | | | | | Homepage , AeroLogic, based on PMARC-12 |

| NEWPAN | Proprietary | C++ | Yes | | | | Yes | Yes | | | | Flow Solutions Ltd. |

| PanAir a502 | Public domain software | Fortran | Yes | Yes | Yes | | | | | | | Homepage , Boeing ? |

| XFOIL | GPLv2 | Fortran | Yes | Yes | Yes | | | | | | | web .mit .edu /drela /Public /web /xfoil / |

| VSPAERO Packaged with OpenVSP | NOSA | C++ | Yes | Yes | Yes | | | Yes | | Polygons, typically quad & tri dominated | Free & rigid | openvsp .org |

>>See also

• `F33f`_`[Stream function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stream_function]`_`f

• `F33f`_`[Conformal mapping`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Conformal_mapping]`_`f

• `F33f`_`[Velocity potential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Velocity_potential]`_`f

• `F33f`_`[Divergence theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divergence_theorem]`_`f

• `F33f`_`[Joukowsky transform`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joukowsky_transform]`_`f

• `F33f`_`[Potential flow`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Potential_flow]`_`f

• `F33f`_`[Circulation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Circulation_(fluid_dynamics)]`_`f

• `F33f`_`[Biot–Savart law`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Biot–Savart_law]`_`f

>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f Section 7.6

>>References

• Public Domain Aerodynamic Software, A Panair Distribution Source, Ralph Carmichael

• Panair Volume I, Theory Manual, Version 3.0, Michael Epton, Alfred Magnus, 1990 `F33f`_`[Boeing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boeing]`_`f

• Panair Volume II, Theory Manual, Version 3.0, Michael Epton, Alfred Magnus, 1990 `F33f`_`[Boeing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boeing]`_`f

• Panair Volume III, Case Manual, Version 1.0, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1981 `F33f`_`[Boeing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boeing]`_`f

• Panair Volume IV, Maintenance Document, Version 3.0, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1991 `F33f`_`[Boeing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boeing]`_`f

• Recent Experience in Using Finite Element Methods For The Solution Of Problems In Aerodynamic Interference, Ralph Carmichael, 1971 `F33f`_`[NASA Ames Research Center`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=NASA_Ames_Research_Center]`_`f

• [1]

`c`F0af`_`[↑ Back to top`#top]`_`f`a