Aerodynamic potential-flow code

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.

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

• Inviscid
• Incompressible $\backslash nabla\; \backslash cdot\; V=0$
• Irrotational $\backslash nabla\; \backslash times\; V=0$
• Steady $\backslash frac\{\backslash partial\}\{\backslash partial\; t\}=0$

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

• Potential flow (inviscid, irrotational, steady) $\backslash nabla^2\; \backslash phi=0$

• From Small Disturbances

:$(1-M\_\backslash infty^2)\; \backslash phi\_\{xx\}\; +\; \backslash phi\_\{yy\}\; +\; \backslash phi\_\{zz\}\; =\; 0$ (subsonic)

• From Divergence Theorem

:$\backslash iiint\backslash limits\_V\backslash left(\backslash nabla\backslash cdot\backslash mathbf\{F\}\backslash right)dV=\backslash iint\backslash limits\_\{S\}\backslash mathbf\{F\}\backslash cdot\backslash mathbf\{n\}\backslash ,\; dS$

• Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function $\backslash phi$.
• 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 $R\; =\; |P-Q|$.

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

• Therefore:

:$U\_p=\; -\backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iiint\backslash limits\_V\backslash left(\backslash frac\{\backslash nabla^2\backslash cdot\backslash mathbf\{U\}\}\{R\}\backslash right)\; dV\_Q$

:$-\backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash frac\{\backslash mathbf\{n\}\backslash cdot\; \backslash nabla\; \backslash mathbf\{U\}\; \}\{R\}\backslash right)\; dS\_Q$

:$+\backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash mathbf\{U\}\backslash mathbf\{n\}\; \backslash cdot\backslash nabla\; \backslash frac\{1\}\{R\}\backslash right)\; dS\_Q$

For :$\backslash nabla^2\; \backslash phi=0$, where the surface normal points inwards.

:$\backslash phi\_p\; =\; -\backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash mathbf\{n\}\; \backslash frac\{\; \backslash nabla\; \backslash phi\_\{U\}\; -\; \backslash nabla\; \backslash phi\_\{L\}\}\{R\}\; -\; \backslash mathbf\{n\}\; \backslash left(\; \backslash phi\_\{U\}\; -\; \backslash phi\_\{L\}\backslash right)\; \backslash nabla\; \backslash frac\{1\}\{R\}\; \backslash right)\; dS\_Q$

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

The Source Strength at an arbitrary point Q is:

:$\backslash sigma\; =\; \backslash nabla\; \backslash mathbf\{n\}\; (\backslash nabla\; \backslash phi\_U-\backslash nabla\; \backslash phi\_L\; )$

The Doublet Strength at an arbitrary point Q is:

:$\backslash mu\; =\backslash phi\_U\; -\; \backslash phi\_L$

The simplified potential flow equation is:

:$\backslash phi\_p\; =\; -\backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash frac\{\backslash sigma\}\{R\}\; -\; \backslash mu\; \backslash cdot\; \backslash mathbf\{n\}\; \backslash cdot\; \backslash nabla\; \backslash frac\{1\}\{R\}\; \backslash right)\; dS$

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

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

:$\backslash phi\_L\; =\; 0$

The Doublet Strength is:

:$\backslash mu\; =\backslash phi\_U\; -\; \backslash phi\_L$

:$\backslash mu\; =\; \backslash phi\_U$

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

:$\backslash phi\_U\; =\; -V\_\backslash infty\; \backslash cdot\; \backslash mathbf\{n\}$

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.

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

:$\backslash mu\_P\; =\; \backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash frac\{V\_\backslash infty\; \backslash cdot\; \backslash mathbf\{n\}\}\{R\}\; \backslash right)\; dS\_U\; +\; \backslash frac\{1\}\; \{4\; \backslash pi\}\; \backslash iint\backslash limits\_S\backslash left(\backslash mu\; \backslash cdot\; \backslash mathbf\{n\}\; \backslash cdot\; \backslash nabla\; \backslash frac\{1\}\{R\}\; \backslash right)\; dS$

• Note that the $dS\_U$ integration term is evaluated only on the upper surface, while th $dS$ 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 $\backslash lambda$ and a single doublet term m of unknown strength $\backslash lambda$ are defined at a given point.

:$\backslash sigma\_Q\; =\; \backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; s\_i(Q)=0$

:$\backslash mu\_Q\; =\; \backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; m\_i(Q)$

where:

:$s\_i\; =\; ln(r)$

:$m\_i\; =$

These terms can be used to create a system of linear equations which can be solved for all the unknown values of $\backslash lambda$.

• 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.Section 7.6

• 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

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:

:$C\_p\; =\; \backslash frac\{p-p\_\backslash infty\}\{q\_\backslash infty\}=\backslash frac\{p-p\_\backslash infty\}\{\backslash frac\{1\}\{2\}\; \backslash rho\_\backslash infty\; V\_\backslash infty^2\}\; =\; \backslash frac\{p-p\_\backslash infty\}\{\backslash frac\{\backslash gamma\}\{2\}\; p\_\backslash infty\; M\_\backslash infty^2\; \}$

The Isentropic Pressure Coefficient is:

:$C\_p\; =\; \backslash frac\{2\}\; \{\backslash gamma\; M\_\backslash infty^2\}\; \backslash left(\; \backslash left(1+\backslash frac\{\backslash gamma-1\}\; \{2\}\; M\_\backslash infty^2\; \backslash left[\backslash frac\{1-|\backslash vec\{V\}|^2\}\{|\backslash vec\{V\_\backslash infty\}|^2\}\backslash right]\backslash right)^\{\; \backslash frac\{\backslash gamma\}\{\backslash gamma-1\}\; \}\; -1\; \backslash right)$

The Incompressible Pressure Coefficient is:

:$C\_p\; =\; 1\; -\; \backslash frac\{|\backslash vec\{V\}|^2\}\{|\backslash vec\{V\_\backslash infty\}|^2\}$

The Second Order Pressure Coefficient is:

:$C\_p\; =\; 1-|\backslash vec\{V\}|^2\; +\; M\_\backslash infty^2\; u^2$

The Slender Body Theory Pressure Coefficient is:

:$C\_p\; =\; -(2u\; +v^2\; +w^2)$

The Linear Theory Pressure Coefficient is:

:$C\_p\; =\; -2u$

The Reduced Second Order Pressure Coefficient is:

:$C\_p\; =\; 1-|\backslash vec\{V\}|^2$

• 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.

{{See also|Computational fluid dynamics#Background and history}}

• Stream function
• Conformal mapping
• Velocity potential
• Divergence theorem
• Joukowsky transform
• Potential flow
• Circulation
• Biot–Savart law

{{Reflist}}

• [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/archive/nasa/casi.ntrs.nasa.gov/19910009745_1991009745.pdf]

Category:Fluid dynamics