SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Aerodynamic potential-flow code</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Aerodynamic potential-flow code</span></span>

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, <b>aerodynamic potential flow codes</b> or <b>panel codes</b> 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.

</p><p>A series of singularities as sources, sinks, vortex points and <a href="Doublet_(potential_flow)" class="mw-redirect" title="Doublet (potential flow)">doublets</a> are used to model the panels and wakes. These codes may be valid at subsonic and supersonic speeds.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>Early panel codes were developed in the late 1960s to early 1970s. Advanced panel codes, such as Panair (developed by <a href="Boeing" title="Boeing">Boeing</a>), 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 (<a href="Computational_Fluid_Dynamics" class="mw-redirect" title="Computational Fluid Dynamics">Computational Fluid Dynamics</a>). 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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Assumptions">Assumptions</h2></div>

<p>These are the various assumptions that go into developing potential flow panel methods:

</p>

<ul><li><a href="Inviscid_flow" title="Inviscid flow">Inviscid</a></li>

<li><a href="Incompressible_flow" title="Incompressible flow">Incompressible</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot V=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \cdot V=0}</annotation>

</semantics>

</math></span><img src="./5e630f52a4f810d8d273a3c8063e61470e554608.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.663ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot V=0}" loading="lazy"></span></li>

<li>Irrotational <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times V=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla \times V=0}</annotation>

</semantics>

</math></span><img src="./343c3504401fa1251f4a934e76fcb988b01faf95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.824ex; height:2.176ex;" alt="{\displaystyle \nabla \times V=0}" loading="lazy"></span></li>

<li><a href="Steady_state" title="Steady state">Steady</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t}}=0}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}=0}</annotation>

</semantics>

</math></span><img src="./c3c8d3347d35a6969a730dc4c8b19ba7a98cf8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.255ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}=0}" loading="lazy"></span></li></ul>

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

</p>

<ul><li>Potential flow (inviscid, irrotational, steady) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\phi =0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\phi =0}</annotation>

</semantics>

</math></span><img src="./046b1f213d422442e4710eb5d4927f67d58953e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.636ex; height:3.009ex;" alt="{\displaystyle \nabla ^{2}\phi =0}" loading="lazy"></span></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Derivation_of_panel_method_solution_to_potential_flow_problem">Derivation of panel method solution to potential flow problem</h2></div>

<ul><li>From Small Disturbances</li></ul>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-M_{\infty }^{2})\phi _{xx}+\phi _{yy}+\phi _{zz}=0}</annotation>

</semantics>

</math></span><img src="./192572988928d8de5081d9c33222e0cb38d4756e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.79ex; height:3.176ex;" alt="{\displaystyle (1-M_{\infty }^{2})\phi _{xx}+\phi _{yy}+\phi _{zz}=0}" loading="lazy"></span> (subsonic)</dd></dl>

<ul><li>From <a href="Divergence_theorem" title="Divergence theorem">Divergence Theorem</a></li></ul>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iiint \limits _{V}\left(\nabla \cdot \mathbf {F} \right)dV=\iint \limits _{S}\mathbf {F} \cdot \mathbf {n} \,dS}</annotation>

</semantics>

</math></span><img src="./9608ea6fa450c426020ee88170836d797813d090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.339ex; height:7.343ex;" alt="{\displaystyle \iiint \limits _{V}\left(\nabla \cdot \mathbf {F} \right)dV=\iint \limits _{S}\mathbf {F} \cdot \mathbf {n} \,dS}" loading="lazy"></span></dd></dl>

<ul><li>Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span>.</li>

<li>Let P be a point in the volume V</li>

<li>Let S be the surface boundary of the volume V.</li>

<li>Let Q be a point on the surface S, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=|P-Q|}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=|P-Q|}</annotation>

</semantics>

</math></span><img src="./f6e95df36b4daa3f5ef098fc59d1618288a546b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.58ex; height:2.843ex;" alt="{\displaystyle R=|P-Q|}" loading="lazy"></span>.</li></ul>

<p>As Q goes from inside V to the surface of V,

</p>

<ul><li>Therefore:</li></ul>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{p}=-{\frac {1}{4\pi }}\iiint \limits _{V}\left({\frac {\nabla ^{2}\cdot \mathbf {U} }{R}}\right)dV_{Q}}</annotation>

</semantics>

</math></span><img src="./773da1562ee8576379138305d2d131ccaa08e24e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.731ex; height:7.843ex;" alt="{\displaystyle U_{p}=-{\frac {1}{4\pi }}\iiint \limits _{V}\left({\frac {\nabla ^{2}\cdot \mathbf {U} }{R}}\right)dV_{Q}}" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{4\pi }}\iint \limits _{S}\left({\frac {\mathbf {n} \cdot \nabla \mathbf {U} }{R}}\right)dS_{Q}}</annotation>

</semantics>

</math></span><img src="./033eda8d2bf3b71b1a0df0c72256aa16168e44ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.307ex; height:7.676ex;" alt="{\displaystyle -{\frac {1}{4\pi }}\iint \limits _{S}\left({\frac {\mathbf {n} \cdot \nabla \mathbf {U} }{R}}\right)dS_{Q}}" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +{\frac {1}{4\pi }}\iint \limits _{S}\left(\mathbf {U} \mathbf {n} \cdot \nabla {\frac {1}{R}}\right)dS_{Q}}</annotation>

</semantics>

</math></span><img src="./bf6b5675ed4405fcc6fd28eb72727df62b6a2bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.071ex; height:7.676ex;" alt="{\displaystyle +{\frac {1}{4\pi }}\iint \limits _{S}\left(\mathbf {U} \mathbf {n} \cdot \nabla {\frac {1}{R}}\right)dS_{Q}}" loading="lazy"></span></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\phi =0}</annotation>

</semantics>

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}</annotation>

</semantics>

</math></span><img src="./172e204a5f719a344e190bc2aecf23e8554d6a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:57.407ex; height:7.676ex;" alt="{\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}}" loading="lazy"></span></dd></dl>

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

</p><p>The Source Strength at an arbitrary point Q is:

</p>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma =\nabla \mathbf {n} (\nabla \phi _{U}-\nabla \phi _{L})}</annotation>

</semantics>

</math></span><img src="./2db0aa2e4e815b393117370e42417367db372d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.986ex; height:2.843ex;" alt="{\displaystyle \sigma =\nabla \mathbf {n} (\nabla \phi _{U}-\nabla \phi _{L})}" loading="lazy"></span></dd></dl>

<p>The Doublet Strength at an arbitrary point Q is:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\phi _{U}-\phi _{L}}</annotation>

</semantics>

</math></span><img src="./5be7c78e5391d1c4a22015d6ec9c025017b066b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.956ex; height:2.676ex;" alt="{\displaystyle \mu =\phi _{U}-\phi _{L}}" loading="lazy"></span></dd></dl>

<p>The simplified potential flow equation is:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}</annotation>

</semantics>

</math></span><img src="./a59c38e16ee775e3a8d94584e841d951379b656d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:37.621ex; height:7.676ex;" alt="{\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}" loading="lazy"></span></dd></dl>

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

</p>

<div class="mw-heading mw-heading2"><h2 id="Required_boundary_conditions">Required boundary conditions</h2></div>

<p>The velocity potential on the internal surface and all points inside V (or on the lower surface S) is 0.

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{L}=0}</annotation>

</semantics>

</math></span><img src="./f15d8681d0387b4a7ff11e2d0bf4112d28ea3703.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.998ex; height:2.509ex;" alt="{\displaystyle \phi _{L}=0}" loading="lazy"></span></dd></dl>

<p>The Doublet Strength is:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\phi _{U}-\phi _{L}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\phi _{U}}</annotation>

</semantics>

</math></span><img src="./63351576597fe90a93efeef4f08c154162ddbaba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.378ex; height:2.676ex;" alt="{\displaystyle \mu =\phi _{U}}" loading="lazy"></span></dd></dl>

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

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{U}=-V_{\infty }\cdot \mathbf {n} }</annotation>

</semantics>

</math></span><img src="./ea20865eeb277bfe698b0dbb9619e75d83282315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.18ex; height:2.509ex;" alt="{\displaystyle \phi _{U}=-V_{\infty }\cdot \mathbf {n} }" loading="lazy"></span></dd></dl>

<p>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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Discretization_of_potential_flow_equation">Discretization of potential flow equation</h2></div>

<p>The potential flow equation with well-posed boundary conditions applied is:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}</annotation>

</semantics>

</math></span><img src="./fc7debe6509de9343c777684fd5bee0c0a7703da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:56.334ex; height:7.676ex;" alt="{\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}" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dS_{U}}</annotation>

</semantics>

</math></span><img src="./b2b71277c37997337b8c03baacadd1406532e4ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.134ex; height:2.509ex;" alt="{\displaystyle dS_{U}}" loading="lazy"></span> integration term is evaluated only on the upper surface, while th <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dS}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dS}</annotation>

</semantics>

</math></span><img src="./66f6ab8a2a8b85df78efbea896fcfeb3d4ea39e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle dS}" loading="lazy"></span> integral term is evaluated on the upper and lower surfaces.</li></ul>

<p>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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> and a single doublet term m of unknown strength <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{Q}=\sum _{i=1}^{n}\lambda _{i}s_{i}(Q)=0}</annotation>

</semantics>

</math></span><img src="./809518d9d82f6aac7eccdccf34aef191b05b137d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.654ex; height:6.843ex;" alt="{\displaystyle \sigma _{Q}=\sum _{i=1}^{n}\lambda _{i}s_{i}(Q)=0}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{Q}=\sum _{i=1}^{n}\lambda _{i}m_{i}(Q)}</annotation>

</semantics>

</math></span><img src="./3419af48763f4c9129b1554e983f5a7578ebe314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.417ex; height:6.843ex;" alt="{\displaystyle \mu _{Q}=\sum _{i=1}^{n}\lambda _{i}m_{i}(Q)}" loading="lazy"></span></dd></dl>

<p>where:

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{i}=ln(r)}</annotation>

</semantics>

</math></span><img src="./94901667396de289e3cfd8c18b2a6bc34584cb3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.935ex; height:2.843ex;" alt="{\displaystyle s_{i}=ln(r)}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{i}=}</annotation>

</semantics>

</math></span><img src="./fbde2bbeceaafd8481d78602b91d60c95cf613bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.293ex; height:2.009ex;" alt="{\displaystyle m_{i}=}" loading="lazy"></span></dd></dl>

<p>These terms can be used to create a system of linear equations which can be solved for all the unknown values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading2"><h2 id="Methods_for_discretizing_panels">Methods for discretizing panels</h2></div>

<ul><li>constant strength - simple, large number of panels required</li>

<li>linear varying strength - reasonable answer, little difficulty in creating well-posed problems</li>

<li>quadratic varying strength - accurate, more difficult to create a well-posed problem</li></ul>

<p>Some techniques are commonly used to model surfaces.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li>Body Thickness by line sources</li>

<li>Body Lift by line doublets</li>

<li>Wing Thickness by constant source panels</li>

<li>Wing Lift by constant pressure panels</li>

<li>Wing-Body Interface by constant pressure panels</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Methods_of_determining_pressure">Methods of determining pressure</h2></div>

<p>Once the Velocity at every point is determined, the pressure can be determined by using one of the following formulas. All various <a href="Pressure_coefficient" title="Pressure coefficient">Pressure coefficient</a> methods produce results that are similar and are commonly used to identify regions where the results are invalid.

</p><p>Pressure Coefficient is defined as:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}}</annotation>

</semantics>

</math></span><img src="./5f50f2d87d4688d367d9de48451c4d50331e8db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.449ex; height:6.676ex;" alt="{\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}}}}" loading="lazy"></span></dd></dl>

<p>The Isentropic Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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)}</annotation>

</semantics>

</math></span><img src="./78255320a973bbbae0381f30bd1ae6fb3b8da3a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:54.765ex; height:11.176ex;" alt="{\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)}" loading="lazy"></span></dd></dl>

<p>The Incompressible Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

<mover>

<msub>

</mrow>

</msub>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}=1-{\frac {|{\vec {V}}|^{2}}{|{\vec {V_{\infty }}}|^{2}}}}</annotation>

</semantics>

</math></span><img src="./8d5435f6760a00edbe4e64003ba6303149712318.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:16.237ex; height:8.343ex;" alt="{\displaystyle C_{p}=1-{\frac {|{\vec {V}}|^{2}}{|{\vec {V_{\infty }}}|^{2}}}}" loading="lazy"></span></dd></dl>

<p>The Second Order Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}=1-|{\vec {V}}|^{2}+M_{\infty }^{2}u^{2}}</annotation>

</semantics>

</math></span><img src="./c8dc3f99ffdba96fd1e3d057f9ac7fdde0ab3251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.311ex; height:3.509ex;" alt="{\displaystyle C_{p}=1-|{\vec {V}}|^{2}+M_{\infty }^{2}u^{2}}" loading="lazy"></span></dd></dl>

<p>The Slender Body Theory Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}=-(2u+v^{2}+w^{2})}</annotation>

</semantics>

</math></span><img src="./8d9076ebc5aea4b97ba98afd1558bde594678f5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.51ex; height:3.343ex;" alt="{\displaystyle C_{p}=-(2u+v^{2}+w^{2})}" loading="lazy"></span></dd></dl>

<p>The Linear Theory Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}=-2u}</annotation>

</semantics>

</math></span><img src="./7f9ea7fee9fd669c7ef615b7412f0cdfe574a115.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.12ex; height:2.843ex;" alt="{\displaystyle C_{p}=-2u}" loading="lazy"></span></dd></dl>

<p>The Reduced Second Order Pressure Coefficient is:

</p>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{p}=1-|{\vec {V}}|^{2}}</annotation>

</semantics>

</math></span><img src="./ca3adc3e0d72403b6933108dfe5ad21be586c6ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.957ex; height:3.509ex;" alt="{\displaystyle C_{p}=1-|{\vec {V}}|^{2}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="What_panel_methods_cannot_do">What panel methods cannot do</h2></div>

<ul><li>Panel methods are inviscid solutions. You will not capture viscous effects except via user "modeling" by changing the geometry.</li>

<li>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.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Potential_flow_software">Potential flow software</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Computational_fluid_dynamics#Background_and_history" title="Computational fluid dynamics">Computational fluid dynamics § Background and history</a></div>

<th rowspan="2">License</th>

<th colspan="3">Operating system</th>

<th rowspan="2">Geometry import</th>

<th colspan="3">Meshing</th>

<th rowspan="2">Body Representation</th>

<th rowspan="2">Wake model</th>

<th rowspan="2">Developer

</th></tr>

<tr>

<td><a href="Linux" title="Linux">Linux</a></td>

<td><a href="Microsoft_Windows" title="Microsoft Windows">Microsoft Windows</a></td>

<td>Structured</td>

<td>Unstructured</td>

<td>Hybrid

</td></tr>

<tr>

<td><a rel="nofollow" class="external text" href="http://www.aeolus-aero.com">Aeolus ASP</a></td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td>Java / <a href="Fortran" title="Fortran">Fortran</a></td>

<td>Quadrilaterals</td>

<td><a rel="nofollow" class="external text" href="http://www.aeolus-aero.com">Aeolus Aero Sketch Pad</a>

</td></tr>

<tr>

<td>CMARC</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><span class="url"><a rel="nofollow" class="external text" href="http://www.aerologic.com/cmarc.html">Homepage</a></span>, AeroLogic, based on PMARC-12

</td></tr>

<tr>

<td>DesignFOIL</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><span class="url"><a rel="nofollow" class="external text" href="http://www.dreesecode.com/">www<wbr>.dreesecode<wbr>.com</a></span>, DreeseCode Software LLC

</td></tr>

<tr>

<td><a rel="nofollow" class="external text" href="http://www.researchinflight.com/products.html">FlightStream</a></td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td>Fortran / C++</td>

<td>CAD, Discrete</td>

<td>Solids</td>

<td><a rel="nofollow" class="external text" href="http://www.researchinflight.com">Research in Flight Company</a>

</td></tr>

<tr>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><a href="Douglas_Aircraft_Company" title="Douglas Aircraft Company">Douglas Aircraft Company</a>

</td></tr>

<tr>

<td>LinAir</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td>Desktop Aeronautics

</td></tr>

<tr>

<td>MACAERO</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><a href="McDonnell_Aircraft" class="mw-redirect" title="McDonnell Aircraft">McDonnell Aircraft</a>

</td></tr>

<tr>

<td><a rel="nofollow" class="external text" href="http://www.flowsol.co.uk/products/newpan">NEWPAN</a></td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><a rel="nofollow" class="external text" href="http://www.flowsol.co.uk">Flow Solutions Ltd.</a>

</td></tr>

<tr>

<td><a rel="nofollow" class="external text" href="http://www.openvogel.org">Tucan</a></td>

<td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes (Console)</td>

<td>Quadrilaterals & triangles</td>

<td>G. Hazebrouck & contributors

</td></tr>

<tr>

<td><a href="QBlade" title="QBlade">QBlade</a></td>

<td><a href="Technical_University_of_Berlin" class="mw-redirect" title="Technical University of Berlin">TU Berlin</a>

</td></tr>

<tr>

<td>Quadpan</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><a href="Lockheed_Corporation" title="Lockheed Corporation">Lockheed</a>

</td></tr>

<tr>

<td>PanAir a502</td>

<td style="background: #DFF; color:black; vertical-align: middle; text-align: center;" class="free table-free"><a href="Public_domain_software" class="mw-redirect" title="Public domain software">Public domain software</a></td>

<td><a href="Fortran" title="Fortran">Fortran</a></td>

<td><span class="url"><a rel="nofollow" class="external text" href="http://www.pdas.com/panair.html">Homepage</a></span>, <a href="Boeing" title="Boeing">Boeing</a>?

</td></tr>

<tr>

<td><a rel="nofollow" class="external text" href="https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/PANUKL">PANUKL</a></td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Freeware" title="Freeware">freeware</a></td>

<td><a href="C%2B%2B" title="C++">C++</a>/<a href="Fortran" title="Fortran">Fortran</a></td>

<td>NX - partially</td>

<td>Quadrilaterals</td>

<td><a href="Warsaw_University_of_Technology" title="Warsaw University of Technology">Warsaw University of Technology</a>, <span class="url"><a rel="nofollow" class="external text" href="https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/PANUKL">PANUKL</a></span> exports data to <span class="url"><a rel="nofollow" class="external text" href="https://www.meil.pw.edu.pl/add/ADD/Teaching/Software/SDSA">SDSA</a></span> and to <a href="Calculix" title="Calculix">Calculix</a>

</td></tr>

<tr>

<td>PMARC</td>

<td><span class="url"><a rel="nofollow" class="external text" href="https://software.nasa.gov/software/ARC-14407-1">Free On Request</a></span></td>

<td><a href="Fortran_77" class="mw-redirect" title="Fortran 77">Fortran 77</a></td>

<td><a href="NASA" title="NASA">NASA</a>, descendant of VSAERO

</td></tr>

<tr>

<td>VSAero</td>

<td style="background: #E7E7FF; color:black; vertical-align: middle; text-align: center;" class="table-proprietary"><a href="Proprietary_software" title="Proprietary software">Proprietary</a></td>

<td><span class="url"><a rel="nofollow" class="external text" href="https://starkaerospace.com/products-services/ami/software/">Homepage</a></span>

</td></tr>

<tr>

<td>Vortexje</td>

<td>Baayen & Heinz GmbH

</td></tr>

<tr>

<td><a href="XFOIL" title="XFOIL">XFOIL</a></td>

<td><a href="Fortran" title="Fortran">Fortran</a></td>

<td><span class="url"><a rel="nofollow" class="external text" href="http://web.mit.edu/drela/Public/web/xfoil/">web<wbr>.mit<wbr>.edu<wbr>/drela<wbr>/Public<wbr>/web<wbr>/xfoil<wbr>/</a></span>

</td></tr>

<tr>

</td></tr>

<tr>

<td>VSPAERO Packaged with <a href="OpenVSP" title="OpenVSP">OpenVSP</a></td>

<td>Polygons, typically quad & tri dominated</td>

<td>Free & rigid</td>

<td><span class="url"><a rel="nofollow" class="external text" href="http://openvsp.org">openvsp<wbr>.org</a></span>

</td></tr>

<tr>

<td>MachLine</td>

<td style="background: #DFF; color:black; vertical-align: middle; text-align: center;" class="free table-free"><a href="MIT_License" title="MIT License">MIT License</a></td>

<td><a href="Fortran" title="Fortran">Fortran</a></td>

<td>untested</td>

<td>Solid bodies using surface tris</td>

<td>Rigid</td>

<td>Utah State University AeroLab <span class="url"><a rel="nofollow" class="external text" href="http://aerolab.usu.edu">aerolab<wbr>.usu<wbr>.edu</a></span> <span class="url"><a rel="nofollow" class="external text" href="https://github.com/usuaero/MachLine">github<wbr>.com<wbr>/usuaero<wbr>/MachLine</a></span>

</td></tr></tbody></table>

<ul><li><a href="Stream_function" title="Stream function">Stream function</a></li>

<li><a href="Conformal_mapping" class="mw-redirect" title="Conformal mapping">Conformal mapping</a></li>

<li><a href="Velocity_potential" title="Velocity potential">Velocity potential</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li>

<li><a href="Joukowsky_transform" title="Joukowsky transform">Joukowsky transform</a></li>

<li><a href="Potential_flow" title="Potential flow">Potential flow</a></li>

<li><a href="Circulation_(fluid_dynamics)" class="mw-redirect" title="Circulation (fluid dynamics)">Circulation</a></li>

<li><a href="Biot%E2%80%93Savart_law#Aerodynamics_applications" title="Biot–Savart law">Biot–Savart law</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Section 7.6</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.pdas.com/">Public Domain Aerodynamic Software</a>, A Panair Distribution Source, Ralph Carmichael</li>

<li><a rel="nofollow" class="external text" href="http://hdl.handle.net/2060/19920013405">Panair Volume I, Theory Manual, Version 3.0</a>, Michael Epton, Alfred Magnus, 1990 <a href="Boeing" title="Boeing">Boeing</a></li>

<li><a rel="nofollow" class="external text" href="http://hdl.handle.net/2060/19920013622">Panair Volume II, Theory Manual, Version 3.0</a>, Michael Epton, Alfred Magnus, 1990 <a href="Boeing" title="Boeing">Boeing</a></li>

<li><a rel="nofollow" class="external text" href="http://hdl.handle.net/2060/19850002614">Panair Volume III, Case Manual, Version 1.0</a>, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1981 <a href="Boeing" title="Boeing">Boeing</a></li>

<li><a rel="nofollow" class="external text" href="http://hdl.handle.net/2060/19920013399">Panair Volume IV, Maintenance Document, Version 3.0</a>, Michael Epton, Kenneth Sidewell, Alfred Magnus, 1991 <a href="Boeing" title="Boeing">Boeing</a></li>

<li><a rel="nofollow" class="external text" href="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</a>, Ralph Carmichael, 1971 <a href="NASA_Ames_Research_Center" class="mw-redirect" title="NASA Ames Research Center">NASA Ames Research Center</a></li>

This article is issued from <a class="external text" title="Last edited on 2022-09-13" href="https://en.wikipedia.org/wiki/?title=Aerodynamic_potential-flow_code&oldid=1110144161">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>