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Aerodynamic potential flow code

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

A series of sources and doublets are used to model the panels and wakes respectively. 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 methods. However, panel codes are still used for preliminary aerodynamic analysis.

Assumptions

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

Inviscid $nabla^2 phi=0$
Incompressible $nabla cdot V=0$
Irrotational $nabla times V=0$
Steady $frac{d}{dt}=0$

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

Potential Flow (inviscid, irrotational, steady) $nabla^2 phi=0$

Derivation of Panel Method Solution to Potential Flow Problem

From Small Disturbances

(1-M_infty^2) phi_{xx} + phi_{yy} + phi_{zz} = 0

(subsonic)

From Diverence Theorem

iiintlimits_Vleft(nablacdotmathbf{F}right)dV=iintlimits_{S}mathbf{F}cdotmathbf{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 $phi$ .
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= -frac{1} {4 pi} iiintlimits_Vleft(frac{nabla^2cdotmathbf{U}}{R}right) dV_Q

-frac{1} {4 pi} iintlimits_Sleft(frac{mathbf{n}cdot nabla mathbf{U} }{R}right) dS_Q

+frac{1} {4 pi} iintlimits_Sleft(mathbf{U}mathbf{n} cdotnabla frac{1}{R}right) dS_Q

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

phi_p = -frac{1} {4 pi} iintlimits_Sleft(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 the a both a source term and a doublet term.

The Source Strength at an arbitrary point Q is:

sigma = nabla mathbf{n} (nabla phi_U-nabla phi_L )

The Doublet Strength at an arbitrary point Q is:

mu =phi_U - phi_L

The simplified potential flow equation is:

phi_p = -frac{1} {4 pi} iintlimits_Sleft(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.

phi_L = 0

The Doublet Strength is:

mu =phi_U - phi_L

mu = phi_U

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

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:

mu_P = frac{1} {4 pi} iintlimits_Sleft(frac{V_infty cdot mathbf{n}}{R} right) dS_U + frac{1} {4 pi} iintlimits_Sleft(mu cdot mathbf{n} cdot nabla frac{1}{R} 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 $lambda$ and a single doublet term m of unknown strength $lambda$ are defined at a given point.

sigma_Q = sum_{i=1}^n lambda_i s_i(Q)=0

mu_Q = sum_{i=1}^n 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 $lambda$ .

Methods for Discritizing 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.

- 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: