Potential-Based Panel Methods - Variations AA210b Lecture 6

Potential-Based Panel Methods - Variations

AA210b

Lecture 6

January 22, 2009

AA210b - Fundamentals of Compressible Flow II 1

AA210b - Fundamentals of Compressible Flow II Lecture 6

Integral Equation for the Surface Potential

In the last lecture we arrived at an integral equation for the potentialfunction on the surface of the configuration of interest. This expression wasshown to be:

φP = V∞(xP cosα+ yP sinα) +∫SB+SC

[(n · ∇φ)φs− φ(n · ∇φs)]dS (1)

Now, the surface integral portion of this Equation can be simplified further bymaking a few observations. Firstly, the flow tangency boundary conditionestablishes that, on the surface of the body SB, n · V = n · ∇φ = 0.Consequently, the first term of the surface integral will be zero for theportion of the integral that includes SB.

Furthermore, the portion of the first term of the integral that includesSC can also be eliminated using the following rationale. The velocity

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component tangential to SC can be discontinuous in certain situations (atrailing vortex sheet, for example). However, the component of velocitynormal to SC is continuous. Since the unit normals on both sides of SChave directly opposite directions, their contributions cancel out exactly, thuseliminating the whole first part of the integral in Equation 1 leaving us with

φP = V∞(xP cosα+ yP sinα)−∫SB+SC

φ(n · ∇φs)dS (2)

Hmm...according to our earlier results, we could obtain the potential of theflow by using a distribution of sources and doublets on the surface of theconfiguration, but we have now completely wiped off the portion relating tothe source distribution, (n · ∇φ)φs. What is going on? Well, we can seelater (and you can read about it in Moran, pp. 282–285) that an alternativevector field U∗ can be used in conjunction with the divergence theoremto recover the source distribution in the expression for the potential, thus

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explaining in a mathematically rigorous way the approach taken by Hessand Smith. We will not discuss this topic during lecture.

Now, the potential φ itself can be discontinuous across SC (and indeedit is in all cases of lifting flows). Using the concept of the potential jumpacross SC defined as:

∆φ = φ+ − φ−

we can rewrite our expression for φP as follows

φP = V∞(xP cosα+yP sinα)−∫SBφ(n·∇φs)dS−

∫SC

∆φ(n·∇φs)dS (3)

As shown in the Figure below, φ+ is the value of the potential on the side ofSC for which n goes into the fluid, while φ− is the value of the potential onthe other side. It is, of course, understood that the integral is now carriedout over a single side of SC.

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Figure 1: Potential Jump Across SC.

The potential jump ∆φ can be easily related to the values of thepotential on SB. If P1 and P2 are the intersections of both sides of SC withSB, then

φ|P16= φ|P2

Instead

∆φ = φ|P2− φ|P1

=∮ P2

P1

∇φ · dl = Γ (4)

This is not only true at the trailing edge of the airfoil, but also all along the

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wake cut SC. This can be easily seen by repeating the circulation calculationabove for closed contour that begin and end at matching points on SC.Since the value of the circulation integral is always the same, regardless ofthe location of the intersection of the contour with SC, the potential jump∆φ = Γ everywhere on SC. Please refer to the Figure below.

Figure 2: Potential Jump Across SC

Finally we are left with an equation that can be used in practice for a

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variety of potential-based panel methods:

φP = V∞(xP cosα+ yP sinα)−∫SBφ(n · ∇φs)dS −Γ

∫SC

n · ∇φsdS (5)

with Γ being related to the values of φ on SB according to Equation 4.

Notice, once more, that this Equation is an integral equation for φ onSB which can be solved either analytically (difficult) or numerically in amanner similar to that followed in the Hess-Smith method. In other words,for the numerical solution of Equation 5, we will discretize the surface of theconfiguration into a series of panels [1, . . . , N ], we will assume a particularvariation of the potential φ on each panel (constant, linear, quadratic) andparameterize this variation with a series of coefficients (the unknowns ofour equations), and we will obtain a number of independent equations equalto the number of unknowns by allowing the point P and its associatedpotential φP to approach points on the surface of the body SB where the

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values of the potential are known according to the chosen parameterization.Again, this will result in a set of simultaneous linear equations Ax = bwhich can be easily solved using standard procedures.

As a reminder, the setup of panels and nodes, together with theirnumbering for a typical airfoil analysis problem can be found in Figure 3below.

Figure 3: Airfoil Surface Panelization Conventions

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Constant-Potential Method

This method is also known as the two-dimensional version of Morinoand Kuo. In its simplest incarnation, we choose the potential to be constanton each panel of the configuration, but the values of these constants areallowed to differ from panel to panel.

φ = φj

on panel j, and, according to Equation 4,

Γ = φN − φ1

Therefore, we have N unknowns (the values of the potential on each panel),and we need N independent equations to solve for these unknowns. These

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equations can be obtained by evaluating Equation 5 at the midpoint of eachpanel. Defining

xi =12

(xi + xi+1) yi =12

(yi + yi+1)

we have

φi = V∞(xi cosα+ yi sinα)−N∑j=1

φj

∫panel j

n · ∇φsdS (6)

− (φN − φ1)∫SC

n · ∇φsdS for i = 1, . . . , N

Once again, as we saw in the Hess-Smith method, the integrals in thisequation need to be evaluated and they are most easily performed in

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a coordinate system aligned with the panel that is contributing to thepotential. Let (x∗, y∗) be the coordinates of the midpoint of panel i on acoordinate system fixed on panel j such as the one in the Figure below.

Figure 4: Local Panel-fixed Coordinates

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After all integrations are carried out, one arrives at an equation like

N∑j=1

Aijφj = bi for i = 1, . . . , N

which can be solved through the usual methods. See handout from Moranfor details about the coefficients.

Finally, the velocity can be calculated in a variety of ways. The simplestof all is to compute the velocity at the nodes of the discretization by using

Vi =φi − φi−1

d(7)

The various elements of this calculation are defined in the Figure below.

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Figure 5: Computation of Nodal Velocities in Constant Potential Method

Finally, with the velocities already calculated, one can continue as usualand compute the pressure distribution using Bernoulli’s equation, whoseresult can then be used to integrate the forces on the body.

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Linear-Potential Method

A more accurate alternative to the constant potential method is toassume a linear variation of the value of the potential within each panel.For example,

φ = φj +ξ

lj(φj+1 − φj) (8)

on panel j, where ξ is simply the coordinate along the length of the panel(which varies from 0 to 1 from one end of the panel to the other), andwhere φj is the potential at the jth node. See Figure 4 for more details.Note that the potentials are now described at the nodes of the panelizationand not at the midpoints of each panel. Because of this situation, we willnow have a number of unknowns equal to N + 1, instead of N .

By allowing the point P in Equation 5 to become each and every oneof the N + 1 nodes of the surface panelization, we obtain a set of linear

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equations for the unknowns φj. This has to be done quite carefully. Seepp. 274-275 of Moran’s book for more details. Effectively,

φP = V∞(xP cosα+ yP sinα)

+1

2π

N∑j=1

[φj +

x∗

lj(φj+1 − φj)

]βPj +

y∗

lj(φj+1 − φj) ln

rPj+1

rPj

+1

2π(φN+1 − φ1)βPN+1 (9)

with especially careful definitions of βPj and ln rPj+1

rPj. Unfortunately,

the equations that result for φP = φ1 and φP = φN+1 are identical, andtherefore, we need to enforce the Kutta condition explicitly to obtain asolvable system. This can be done in a manner similar to that used in the

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Hess-Smith panel method:

φN+1 − φNlN

=φ1 − φ2

l1

Once the linear system is solved, the tangential velocity component at themidpoints of the panels can be easily calculated as

Vt(xi, yi) ≈φj+1 − φj

lj

and the pressure distribution can be calculated with the help of the Bernoulliequation. From the pressure distribution, the force and moment coefficientscan be computed in the usual way.

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Equivalent Vortex Distributions

Without going into much detail, the following can be stated:

For every doublet distribution, there corresponds an equivalent vortexdistribution.

This statement allows the rigorous mathematical development of vortex-based panel methods (such as the Hess-Smith method).

Integrals of the type

12π

∫panelj

φy∗dξ

(x∗ − ξ)2 + y∗2=

12πφθ

∣∣∣∣ξ=ljξ=0

− 12π

∫ lj

0

∂φ

∂ξθdξ

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which show up in the development of doublet-based panel methods can beshown to be equivalent to a distribution of vortices of strength ∂φ

∂ξ withineach panel, plus two concentrated vortices at the end of each panel whosestrength is equal to the value of the potential at the end of the panel.

This equivalence between doublet and vortex distributions can help shedsome light into the development of vortex-based panel methods (see Moranpp. 280–282) for more details.

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Source-Based Panel Methods

In our mathematical development of panel methods, we had obtainedan expression for the potential at any point in the flow field, P , in termsof surface source and doublet distributions. The strength of the sourcedistribution was found to be equal to the normal component of velocity atthe surface. Because of the flow tangency boundary condition, this sourcestrength turned out to be zero, and the source distribution dropped out ofour equation.

However, had we defined our vector field U in a different way, we couldhave arrived at slightly different results. In particular, if the vector field ofinterest U∗ is defined as follows

U∗ = φ∗∇φs − φs∇φ∗

where φ∗ is the solution to Laplace’s equation inside the body of interest,

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while φS is still the potential of a unit-strength source located at a pointP outside of the body of interest, we would have arrived (using both thedivergence theorem and the result of the earlier definition of U) at thefollowing conclusion:

φP = V∞(x cosα+ y sinα) +∫SB+SC

(σφs − µn · ∇φs)dS

where

σ = n · ∇(φ− φ∗)µ = φ− φ∗

This is the same conclusion that we had arrived at earlier, only now thesource and doublet strength distributions are not as simply related to thenormal component of the velocity and the potential values themselves.

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They are related to the difference between φ and φ∗ and its gradient, whosevalues at the surface of the body can be quite arbitrary. This arbitrarinesscan be easily exploited to make sure that the source distribution is noteliminated by the flow tangency boundary condition.

Using the equivalence between doublet and vortex distributions, theexpressions obtained with this approach can be made to look very similarto the Hess-Smith method discussed earlier in class.

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Potential-Based Panel Methods - Variations AA210b Lecture 6