Application of Panel Methods

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Session 02

Application of Panel Methods

Session delivered by:Session delivered by:

. . .. . .

102 M.S. Ramaiah School of Advanced Studies, Bengaluru


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Session Objectives

-- At the end of this session the dele ate would have

understood

How to test a anel code for accurac

How to validate and interpret results from a panel code

How to interpret airfoil results

The meaning of a higher order panel method

How to extend the method to 3-D cases



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ess on op cs

1. Revision of the Panel Methods2. Verification of the quality of Panel

e o esu s

3. Validation of the Panel Method Results

4. Airfoil Characteristics- Lift PitchinMoment, Drag

5. Special Airfoils, Inverse Design

. -



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INTRODUCTION

n ncompress e ow, e e ec s o v scos y can e

neglected, satisfies the Laplace equation. The continuity equation and conditions of irrotationality lead

to the Laplace equation.

The Laplace equation is a linear PDE and hence a large body

found accurately and efficiently. We are solving only one PDE.

Flows with Mach numberM< 0.3 can be approximated as

ncompress e ows. ence, e v scous e ec s arenegligible, a lowMflow can be approximated by the Laplace

equation and solved conveniently.



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A summary of the Panel method is given below by listing the steps

a en n e r o xamp e.

1. The geometry is to be represented by N panels. It is a closed

surface.

2. Boundary condition

representing a closed surface at infinity

3. Geometrical quantities are calculated. They include: areas,

slopes i , unit vectors normal and tangential to the panels

n i, i , no a or co oca on po n s a ong w e rcoordinates.

4. On each of theNpanels assume uniform source strength q i and


the vortex strength , assumed to be same on each panel.


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5. Now at any point in the field can be expressed in terms of (N

va ues o s ngu ar es an n n y va ue

:

s s v v

relate velocities to singularities. These influence coefficients are

geometric relations and can be evaluated.



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7. The velocities are related to the singularities by the relations:

8. Apply the tangency boundary condition for velocities on theN

pane s. ese are res equa ons or s ngu ar es q i an :



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9. Apply the Kutta condition at the trailing edge to get one more

equa on. s equa on s

10. Thus we have generated (N+1) equations to close the system.

This linear system of equations can be solved for (qi

and ).

can be evaluated at each panel (Bernoullis theorem) and then

lift, drag & moment coefficients can be calculated.



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Variation of pressure coefficient on the top and bottom surfaces is

s own or a r o a ong w e exac resu s o a ne

by conformal mapping.

= 6 1. Notice max & min values.

2. How to get CL, CD, CM?.

4. What do you mean by

exact results here?

. omment on agreement.6. The method can be

extended to 3-D.



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Sensitivity of the Results to the Number of Panels (1)

1. The discussion here is the same as grid independence study in

CFD.

2. T e resu ts we get s ou not epen on t e num er o pane s.

3. The number of panels is not a part of the original flow problem. It

is a numerical artefact we have introduced to solve the roblem.

Hence if the results depend on the number of panels, they are not

the solution of the original problem, i.e., they are not the results.. ,

better.

5. Also, an increase inN, results in better representation of the

s ngu ar ty str ut ons. emem er q i s un orm on a pane .6. The integral representations we use are exact. But they are

restricted to irrotational flows onl .



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1. We increase the number of panelsN, and observe if the results

change. If they change we have not reached the desired stage.

2. Q: I t ey o not c ange w t respect to an ncrease n , w at

can we conclude?

3. When we increase , it is uite im ortant to have a strate . Also,

keep in mind that coarseness ofNmeans part of the flow geometry

is not represented properly. The strategy should be such that asN .

4. The effect of an increase inN, should be concluded only if this

increase is sufficiently large. (Not changingNfrom 200 to 210

an t en conc u e t at t e resu ts are nsens t ve to t s c ange5. What is the level of sensitivity of the results to a change inNthat

is acce table? It de ends on what uantities we are ins ectin and


the purpose of our study and also the resources available.


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1. Sensitivity of the drag coefficient to the number of panelsN .

2. What should be the asymptotic value asN becomes large?



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1. Sensitivity of the lift coefficient to the number of panelsN .

2. Keep in mind the expanded scale used.

3. Comment on t e non-monoton c e av our o t e t coe c ent

to the number of panelsN .



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Based on the sensitivity of the results toN as seen in the previous

two figures consider:

1. Comment on the choice of N = 20. See that the lift values are

the same for N = 20 & 100 !

2. Is the value N = 80 or 100 a satisfactory choice?

3. Suppose we decidedN = 100 to be a satisfactory choice (last,

choice for = 2 ? Is it likely to be a satisfactory choice for = 12 ?

4. If we increase the number of panels sufficiently, can we resolvethe boundary layer and make the results more realistic?


.

viscous flow?


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1. The same results are plotted now as a function of ( 1/n ) which is

proportional to (or a measure of average) panel size.

2. Sens t v ty o t e rag coe c ent to ( 1 n ).



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2. Sens t v ty o t e t coe c ent to ( 1 n ).



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2. Sens t v ty o t e moment coe c ent to ( 1 n ).



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Based on the sensitivity of the results to ( 1 / n ) as seen in the

previous three figures consider:

1. As the number of panels becomes very large ( 1 / n ) 0 and

we should et exact results.

2. If we plot these graphs on log scale it may be possible to

evaluate the accuracy of the numerical scheme.. D

4. Is it possible to extrapolate CL curve to ( 1 / n ) 0 ?

5. Is it possible to extrapolate Cm curve to ( 1 / n ) 0 ?

. e cons er ng t e top t ree quest ons, ta e a oo at t eprevious two figures also.



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The quantities we used ( CD , CL & Cm ) for comparison are integral

values. We will compare now CP distribution. This is a more

eta e an sens t ve test. Argue w y we nee more pane s near

the leading edge.



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1. In the previous figure we see that with 20 panels we do not

recoverCP = 1 at the leading edge. But results of 60 & 100

pane s are n st ngu s a egrap ica y.



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Validation of the Results (1)

1. Step 1: In the previous comparisons we have assumed that the

inviscid model is valid and verifiedthe computational procedure

to ma e sure t at our approx mate so ut on s n ee c ose to t e

exact solution of the assumed model.

2. Ste 2: Once we are convinced of this, it is ossible to check

(validate) how close is the assumed theoretical model to physical

reality.

. .

4. In the validation step 2 here we usually make comparison for a

wider range of parameters of practical relevance.

. t s o ten cu t to ver y our computat ona proce ure orevery possible range of parameters. We, of course, select the

ran e of arameters of ractical interest and then make


verification tests to more stringent cases.


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made here withthe experimental

data.

See next slide

for comments.



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NACA 0012 and NANA 4412. The fist airfoil is symmetrical

and both of them have 12 % thickness.

. e pane met o g ves t e correct s ope o , pre cte y

the thin airfoil theory (Verify).3. For NACA 0012 the curve passes through the origin but for

NACA 4412 the zero lift angle is about Z L = - 4 .

4. Disagreement with the experimental data is gradual as ncreases, ut t s g er or t e a r o .

5. Q: Is it possible to argue that by shifting one set of curves that

the disagreement is roughly the same in both the sets? Keep

thin airfoil theory in mind.6. For the cambered airfoil the flow separates first at the trailing


.

separates first at the leading edge and stall is abrupt.


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Predicting the Pitching Moment (1)

1. Pitching moment plotted is about the quarter chord point.



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Predicting the Pitching Moment (2)

1. For the NACA 0012 airfoil experimental data indicate that the

quarter chord point is the aerodynamic centre (recall the

.

indicating that the quarter chord point is not the aerodynamiccentre.

2. Once the flow separates disagreement is large.

3. For the cambered NACA 4412 airfoil experimental data indicate

- .

4. Recall that for this airfoil flow separates at the trailing edge and

disagreement is gradual as is increased. Pitching moment is

large and there is a qualitative difference between the two setsof results.

5. Com arison made here is ver exactin and also of ractical


importance for stability. But failure is not of computations.


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Discussion of the Airfoil Results by Panel Methods (1)

1. The panel methods solve for an irrotational flow. Hence the results

may be OK for thin airfoils at low Mach number (M< 0.3) and small

.

2. Results may be acceptable for pressure distribution and lift but notfor drag. See the next figure.

3. The 2-D drag value predicted by the panel methods should be strictly

zero but there may be a small value indicating numerical error.

4. The skin friction dra of an airfoil cannot be redicted b the anel

methods. Because of viscous effects the rear stagnation pressure will

not be fully recovered and will contribute for a small value of

. .

be predicted by an inviscid model.

5. Since the panel methods are accurate and fast, they can be used to


study the effect of geometry keeping in mind viscous effects.


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Discussion of the Airfoil Results by Panel Methods (2)

1. Key areas of interest in airfoil pressure distribution.



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Airfoil Characteristics (1)

Effect of Angle of Attack :NACA 0012 Airfoil

CP minimum

value decreases

with increase in

A larger

deceleration with



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ec o r o c ness:

= 0 NACA 0012 Airfoil

CP minimum

value decreases

with increase in

thickness

A larger

increase in

thickness


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ec o r o c ness:

= 4 NACA 0012 Airfoil

The thinnest airfoil

shows a large expansion /

recom ression due to the

stagnation point being

below the L.E.

The thicker airfoil

results in milder expansion

and subsequentrecompression.


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Effect of on Cambered Airfoil CP:

= 0 & 4 NACA 4412 Airfoil

Due to camber lift is

generated without a large

. .

subsequent recompression.

T s re uces t e

possibility of L.E.

se aration. See next

figure for comparison.


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am er ec s on r o P:

CL = 0.48 for both the airfoils Due to camber lift is

generated without a

lar e L.E. ex ansion and

subsequent

recompression.

Next two figures

indicate it for CL = 0.96

& for CL = 1.43.


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CL = 0.96 for both the airfoils


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CL = 1.43 for both the airfoils


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Even though the panel methods do not give drag values, it is

nstruct ve to oo at t e rag va ues or t ese two a r o s anappreciate the effectiveness of the camber.


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Airfoil Characteristics (8-A)

ec s o am er on r o P:NACA 6712 Airfoil

E ect o extreme a t cam er was use y R c ar W tcom n

the development of the NASA supercritical airfoils. It o ens u the ressure distribution near the leadin ed e. See

next figure.


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Airfoil Characteristics (8-B)

ec s o am er on r o P:NACA 6712 Airfoil Aft camber opens

up the pressure

distribution near the

leading edge.

But it has led to a

large zero lift pitching

.

It has also led to

delayed and thenrapid pressure

recover and ossible


early B.L. separation.

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Airfoil Characteristics (9-A)

ec s o am er on r o P:GA(W)-1 also known as NASA LS(1)-0417 Airfoil

T e a r o s s own ear er were eve ope n t e 1930s an t e

geometry was given by simple formulas. Modern airfoils developedin the 1970s are defined b tables of coordinates.

The figure below shows 17 % thick GA(W)-1 airfoil (General

Aviation) .


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Airfoil Characteristics (9-B)

ec s o am er on r o P:GA(W)-1 Airfoil, 17 % thick It has better

maximum lift and

stall characteristics.

The ressure

recovery rate is

constant and hence

.

If the camber is too

steep, flow mayseparate first at the

bottom surface .


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Laminar Rooftop Airfoil by Liebeck for High Lift

Note the high lift curve in the next figure.

Design and tests from R.H. Liebeck,Re . MDC-J5667/01, Au ust 1972

From: R.T. Jones, Wing Theory.


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for High Lift

Note the pressure recovery in the

.

Design and tests from R.H.e ec ,

Rep. MDC-J5667/01, August

1972

From: R.T. Jones, Wing Theory.


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ACD2506Drag value for Liebeck s High Lift Airfoil

Note the low drag value even at very high lift!


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Inverse Problem of Airfoil Design

1. The pressure distribution is specified. Of course, it has to be

something possible.

.

distribution.3. This is the inverse problem different from the analysis problem

where we obtain the pressure distribution corresponding to a

known geometry.

4. If we succeed in the inverse desi n it is ossible to enerate anideal airfoil section to meet an specific needs, provided it meets

other requirements, for example from the structural viewpoint.

.

does not exist. Hence the restriction in item 1 above.

6. Panel method being accurate and fast comes in a handy, specially


in 2-D problems. But we have to suitably modify them.

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A E l f I Ai f il D i


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An Example of Inverse Airfoil Design


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EXMP: Wing-body-tail configuration with wakes


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EXMP: Details of the wake model required.


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EXMP: The space shuttle mounted on a Boeing 747.


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It is tempting to apply the panel methods to 3-D flows.

A 3-D lifting body has induced drag even in a steady, irrotational

flow. Panel methods can be used to calculate the induced drag. But.

Application of the Kutta condition needs a careful consideration.

It applies to distinct edges. It can lead to difficulties.

ow o mo e e wa es a so nee s care u cons era on.

Since the 3-D bodies considered are usually complex, the wake

from one part may interfere with some other parts and hence the flow

may not be strictly irrotational. Usually quadrilateral panels are used to define a surface. It


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Hi her Order Panel Methods

Panel methods are approximate methods.

,

we approximate the geometry, (2) how well we approximate theequations and (3) the round-off errors in the arithmetic. The last part

can be controlled and we will not bother about it here.

We need a larger number of panels for an accurate representation

of the eometr . Often non- lanar anels are used. In higher order panel methods singularity distributions are not

constant on the panel. This improves the formal accuracy of the

.

A smaller number of panels can be used in higher order methods.

However, in practice the need to resolve geometric details dictates


the number of panels and we cannot reduce the number of panels.

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Summary

The following topics were dealt in this session

1. Revision of the Panel Methods

2. Verification of the quality of Panel Method Results

.

4. Airfoil Characteristics- Lift, Pitching Moment, Drag

5. Special Airfoils, Inverse Design

6. Higher order and 3-D Panel Methods


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Documents

Application of Panel Methods