potential-based panel methods -...

Potential-Based Panel Methods - Mathematics

AA210b

Lecture 5

January 20, 2009

AA210b - Fundamentals of Compressible Flow II 1

AA200b - Applied Aerodynamics II Lecture 5

Mathematical Foundations

In the previous lecture we derived the Hess-Smith panel method mainlybased on physical intuition. Certain assumptions were made regardingthe constancy of the vortex strength distribution and the variation of thesource/sink strength within a panel. At the time, we mentioned that thiswas just one of the many possibilities to satisfy the boundary conditionsof our problem. There are other ways to solve the problem which can bemore easily seen if a more formal mathematical approach is followed in thederivation of potential methods.

It must be said, however, that Hess and Smith’s method was the firsttruly usable (in practice) panel method, which, together with a boundarylayer formulation, was responsible for a variety of airfoil designs at theformer Douglas Aircraft Company.

In this lecture, we will pursue the consequences of the Divergence

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Theorem to find an expression for the potential field created by a bodyimmersed in a free stream. This potential field will be generated by adistribution of sources and doublets of varying strengths on the surface ofthe body of interest.

In contrast with more advanced (and more computationally demanding)CFD models, the discretization process will occur only at the surface of thebody, not within the volume of interest. This property of panel methodsallows them to be solved with far more modest computational resourcesthan a typical Euler or Reynolds Averaged Navier-Stokes solution.

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Divergence Theorem

Most of our derivation relies on the divergence theorem and itsinterpretation in the context of a vector field that we will define below.Firstly, the divergence theorem can be stated as follows:

Theorem 1. [Divergence Theorem] The integral of the divergence of avector field U over an arbitrary volume V is equal to the flux integral of thesame vector field U over the bounding surface of the volume, S.∫

V∇ ·UdV = −

∫Sn̂ ·UdS (1)

In this statement, n̂ is the unit normal vector defined at every point of Sthat points into the domain. This definition of the direction of n̂ forces us

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to place a negative sign in front of the right hand side of Equation 1. Thewell-known divergence theorem is nothing but a multidimensional statementof the fundamental theorem of calculus

∫ b

a

df

dxdx = f(b)− f(a)

where the divergence operator has taken the place of the derivative d/dx,and the surface integral in the right hand side plays the role of the functionevaluation at the limits of the one-dimensional interval f(b)− f(a).

The vector field in question, U, must be continuous inside of V. Thedomain of interest and the notation used throughout this lecture can beseen in the Figure below.

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Figure 1: Control Volume for the Application of the Divergence Theorem

In our derivations, we will be applying the divergence theorem to thefollowing vector field

U = φ∇φs − φs∇φ (2)

where φ is the potential of the flow we are interested in (generated by

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the airfoil under examination), and φs is the potential of a source of unitstrength located at some arbitrary point P within V. The potential of asource of unit strength in two- and three dimensions can be expressed as

φs =1

2πln r in two dimensions (3)

φs =−14π

1r

in three dimensions

As can be seen in the Figure below

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Figure 2: Source Potential Nomenclature

r is simply the scalar distance between the location of the source (at pointP ) and the point at which we are evaluating the potential.

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Continuity Requirements and Branch Cuts

Notice that the vector field U must be continuous in V, and therefore,so must, in general, its components φ, ∇φ, φs, and ∇φs. This continuityrequirement will certainly present a problem at the location of the source,P , since the source potential is singular there and the derivatives of φsbecome infinite. Therefore, a valid approach is to carve out a sphericalvolume of radius ε centered at P . Later we will take the limit of variousexpressions as the radius of this spherical volume tends to zero, ε→ 0, thusrecovering the initial control volume. This new control volume, Vε, beingcarved out of the original control volume V can be seen in the Figure below.

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Figure 3: New Control Volume Used to Avoid Discontinuities of U.

For similar reasons, we must exclude flow entities such as the vortexsheet that emanates from the trailing edge of a three-dimensional wing offinite span. The tangential component of velocity is discontinuous acrossthis vortex sheet, thus violating the assumptions of the divergence theorem.In two-dimensions, although a well-defined vortex sheet does not exist andthe velocity is continuous off the body, the potential φ is not. This can

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be illustrated by the simple example of the potential created by a circularcylinder of radius a with circulation, Γ, given by

φ = V∞r cos θ(

1 +a2

r2

)− Γ

2πθ

Since θ is not a single valued function of position (by continuing to go aroundand around the cylinder/airfoil θ can take values such as 0, 2π, 4π, 6π, . . .,etc, effectively generating an arbitrary amount of lift. Although not explicitlymentioned in introductory texts, the assumption is usually made that θ liesin the interval [0, 2π].

For both two- and three-dimensional flows, this problem can be takencare of by making a cut in the domain of interest. Since the cut nowbecomes part of the boundary of the volume of interest, both ∇φ and φ canbe allowed to be discontinuous across this cut. This concept is usually calleda Riemmann cut / branch cut in complex variable jargon. The need for

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this cut can be further emphasized by examining the concept of circulationaround a closed contour, C, around the cylinder / airfoil, Γ:

Γ =∮CV · dl

Looking at the Figure below, one can see that the potentials at two differentpoints on the curve C are related by:

φ(P2)− φ(P1) =∫ P2

P1

∇φ · dl

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Figure 4: Multivaluedness of Potential Function

If we continue from P2 back onto P1, we have that

φ(P1)− φ(P1) =∮CV · dl = Γ

and therefore, φ is multi-valued at P1. The insertion of a branch cut orRiemmann cut is the solution to this problem.

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With these modifications to the original volume of interest, V, ourdomain is now simply connected. The bounding surface S is now composedof three different parts

• SB, the surface of the body of interest

• S∞, the far-field bounding surface

• SC, the branch cut surface that sandwiches the discontinuities in eitherφ or ∇φ

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Divergence Theorem - Volume Integral

Let’s now turn our attention back to the volume integral portion of thedivergence theorem. With our definition of U given in Equation 2, we havethat, applying the chain rule,

∇ ·U = ∇φ · ∇φs + φ∇ · ∇φs −∇φs · ∇φ− φs∇ · ∇φ (4)

The first and third terms of the equation cancel out leaving us with

∇ ·U = φ∇2φs − φs∇2φ = 0 in Vε

since both the total potential and the source potential satisfy Laplace’sequation in the domain of interest. Therefore, in the divergence theorem,

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we are left with the right hand side (surface integral portion):

0 = −∫S+Sε

n̂ · (φ∇φs − φs∇φ)dS (5)

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Divergence Theorem - Surface Integral

This surface integral can be split into two parts∫Sε

n̂ · (φ∇φs − φs∇φ)dS = −∫Sn̂ · (φ∇φs − φs∇φ)dS

Let’s first look at the left hand side of this equation as we allow the radius ofthe control volume Sε to shrink to zero. The values of both the potential φand its gradient ∇φ approach their values at P , φP and VP, and therefore,∫

Sεn̂ · (φ∇φs − φs∇φ)dS ≈ φP

∫Sε

n̂ · ∇φsdS −VP ·∫Sε

n̂φsdS

Notice that the first integral on the right hand side is simply equal to thevolume flux out of the small spherical control volume. This will be unity,

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since φs was constructed to be a unit-strength source at point P . Notealso that this is also true in the limit ε → 0. The second integral on theright hand side can be simplified since the control volume is spherical. Assuch, all points on its surface are equidistant from the point P . Since thepotential of the source φs located at P only depends on the distance fromP , φs can be considered to be a constant in this integral, and can thereforebe taken out from under the integral sign, leaving us with

φs

∫Sε

n̂dS

which can be easily seen to be zero by symmetry. This is true, in fact,regardless of the shape of the control volume.

Equation 5 then leaves us with an expression for the potential of theflow at point P (something that we are after but that will require further

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interpretation):

φP =∫S

[(n̂ · ∇φ)φs − φ(n̂ · ∇φs)]dS (6)

which is an expression for the potential of the flow at an arbitrary point Pin terms of information on the surface S of the domain of interest. Thisis the type of expression we were after. In fact this is the expression thatpermits us to call this approach a panel method. What is the meaning ofthe two terms in this integral? Let’s look at them in more detail.

The first term of the integral is given by∫S

(n̂ · ∇φ)φsdS

According to our derivation, φs is the potential generated by a source of unit

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strength located at P at the specific location on the surface S that is underconsideration in the integral. However, as we saw before, this potentialdepends only on the distance between the point P and the portion dS ofthe surface under consideration. Here comes the critical trick

φs can just as well be considered to be the potential created by asource of unit strength located at dS as measured at point P .

This role-reversal allows us to interpret the first term of the integral asthe potential of a distribution of sources of strength per unit area equal ton̂ · ∇φ (the component normal to S of the fluid velocity) on the surface ofthe volume of interest V measured at an arbitrary point in the domain P .This should sound familiar, since it is quite similar to the approach takenby Hess and Smith in the early 1960’s.

Similarly, the second term of the integral can be interpreted as the

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potential generated at point P by a distribution of doublets of a givenstrength. Let’s look at

∫Sφ(n̂ · ∇φs)dS

The term n̂ · ∇φs can be interpreted as as the rate of change of φs inthe direction normal to the differential element of area dS. Figure 5 belowshows a better picture of this explanation

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Figure 5: Interpretation of Second Term of Integral of Equation 6

Let Q1 and Q2 be two points located a distance δ apart so that

~Q2Q1 = δn̂

Let φ1 and φ2 be the values of the potential at Q1 and Q2 generated by a

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unit source at P . Then

n̂ · ∇φs = limδ→0

φ1 − φ2

δ

However, φ1 and φ2 can also be interpreted as the values of the potentialsof unit sources located at Q1 and Q2 as measured at P . Then, φ1/δ−φ2/δis the sum of the potentials of a source and a sink of strength ±1/δ atQ1 and Q2. As δ → 0, they coalesce into a doublet whose strength is theproduct of the source/sink strength and the distance between the sourceand the sink, i.e. δ/δ = 1.

Therefore, the second term of Equation 6 can be regarded as thepotential measured at P of a doublet distribution on S of strength φ, thelocal velocity potential.

In sum,

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The velocity potential of any irrotational flow can be representedby a distribution of sources and doublets over the bounding surfacesof the domain of interest. The strength of the source and doubletdistributions per unit area are, respectively, the boundary values ofthe normal derivative of φ and of φ itself.

Since we know the form of the potential in the far-field, we can separatethe various components of the surface integral in Equation 6. In particular,on S∞,

φ ≈ V∞(x cosα+ y sinα)which yields the final expression for φP

φP = V∞(xP cosα+ yP sinα) +∫SB+SC

[(n̂ · ∇φ)φs− φ(n̂ · ∇φs)]dS (7)

Using this expression we can start concocting a variety of approaches to

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solve for the unknown in this equation, φ.

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