Panel Methods : Mini-Lecture. David Willis

Transcription

1 Pael Methods : Mii-Lecture David Willis

2 3D - Pael Method Examples

3 Other Applicatios of Pael Methods

4 Boudary Elemet Methods Pael methods belog to a broader class of methods called Boudary Elemet Methods (BEM). Used i Capacitace extractio Structural aalysis (some) Drug desig MEMs Stokes flow

5 Start with the Basic Idea A iitial thought experimet with vortices.

6 Let s Start by Lookig At This Problem Flow tagecy: The equatio we wat to satisfy at each cotrol poit j, o the airfoil:

7 Example of the Flow Aroud A Airfoil With 10 Poit ortices Full Image Elargemet/Zoom Flow tagecy is satisfied at the cotrol poit

8 Example of the Flow Aroud A Airfoil With 50 Poit ortices Full Image Elargemet/Zoom

9 Example of the Flow Aroud A Airfoil With 100 Poit ortices Full Image Elargemet/Zoom

10 I search of a more elegat solutio NACA 4412 Pael Γ 1 Γ 2 Γ 3 Γ 4 Γ 5 Rather tha discrete vortices, use a collectio of vortex sheets (aka. paels) o the foil.

11 Poit ortex vs. orticity Distributios Costat orticity Distr. Poit ortex Liear orticity Distr.

12 Liear ortex Pael Method

13 Geometry! Paels Step 1: Get the geometry (airfoil, sphere, etc). Step 2: Discretize the geometry ito paels (or computatioally discrete elemets) Reality NACA 4412 Pael ormal Pael Cetroid = Evaluatio Pt. Pael/Elemet. Computatioal Represetatio of the geometry ertex/node

14 Small Note O Mesh Geeratio May umerical methods distribute discrete elemets ear regios of large chages (either i solutio value or geometry). Pael Methods are o differet. We usually wat to cluster paels ear regios where the flow chages rapidly or geometry has sigificat curvature. 40 Equal Sized Paels L.E. Clusterig L.E. ad T.E. Clusterig

15 Step 3 : Place a liearly varyig distributio of vorticity o each pael γ 8 γ 7 γ 6 γ 5 γ 4 γ 3 γ 2 γ 1 γ NP γ NP+1 Usig a pael method, we are tryig to determie the vorticity distributio aroud the airfoil. By determiig γ 1! γ N+1 we have solved the problem. The value of the orticity sheet stregth varies liearly betwee the values o each vertex of the pael.

16 Step 4 : Buildig the Ifluece Matrix 5,1 4,1 3,1 2,1 1,1 N1 C1 C2 C3 C4 C5 C6 C7 C8 N1 1,1 2,1 3,1 4,1 + = 5,1 if 0 : : : : γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 CN N,1 γ N γ Ν+1

17 Step 4: Buildig the Ifluece Matrix 5,2 4,2 3,2 N2 1,2 if C1 C2 C3 C4 C5 C6 C7 C8 1,1 2,1 3,1 4,1 + = 5,1 1,2 2,2 3,2 4,2 5,2 γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 0 CN N,2 N,1 N,2 γ N γ Ν+1

18 Step 5: Addig the Freestream elocity! RHS 1 if if if * 1 if * 2 if * 3 if * 4 if * 5 if * N 1,1 2,1 3,1 + = 4,1 5,1 1,2 2,2 3,2 4,2 5,2 N,2 N,1 N,2 1,3 2,3 3,3 4,3 5,3 N,3 1,N+1 2,N+1 3,N+1 4,N+1 5,N+1 N,N+1 γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ N γ Ν+1

19 Step 5: Addig the Freestream elocity! RHS 1,1 5,1 = 1,2 5,2 1,3 5,3 1,N+1 if * 0 1 2,1 2,2 2,3 2,N+1 if * 2 0 if * 3 3,1 3,2 3,3 3,N+1 if * 4 4,1 4,2 4,3 4,N+1 if * 5 + 5,N+1 γ 1 γ 2 γ 3 γ 4 γ 5 γ if * N N,2 N,1 N,2 N,3 N,N+1 γ N 0 γ Ν+1

20 Step 5: Addig the Freestream elocity! RHS 1,1 2,1 3,1 4,1 5,1 = 1,2 2,2 3,2 4,2 5,2 1,3 2,3 3,3 4,3 5,3 1,N+1 2,N+1 3,N+1 4,N+1 5,N+1 γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 - if * 1 - if * 2 - if * 3 - if * 4 - if * 5 N,2 N,1 N,2 N,3 N,N+1 γ N - if * N γ Ν+1

21 Some Possible Solutios: No Net Circulatio Net Negative Circulatio So, which solutio is the correct oe, ad how do we esure we get that solutio? Net Positive Circulatio Net Positive Circulatio

22 Smooth T.E. Flow

23 What Should Happe at the Trailig Edge? We ca eforce pressure of upper surface ad lower surface flows to be the same at the trailig edge. For steady flows, pressure equality at trailig edge implies velocity equality at trailig edge (From Beroulli Equatio). ½ ρ 2 = ½ ρ 2 2 = 2 =

24 Added Commet o elocities at the Trailig Edge For a fiite agled trailig edge: v U v L Implies v U v L For a cusped trailig edge: Stagatio Poit v U v L See Aderso for details.

25 Coditio which is applied i the pael method : The Kutta Coditio We should implemet a Kutta Coditio ito the liear vortex pael method! esure there is o et vorticity at the T.E.

26 Step 6: Implemetig the Kutta Coditio i the Liear orticity Pael Method γ 1 γ NP+1 1,1 2,1 3,1 4,1 5,1 = 1,2 2,2 3,2 4,2 5,2 1,3 2,3 3,3 4,3 5,3 1,N+1 2,N+1 3,N+1 4,N+1 5,N+1 γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 - if * 1 - if * 2 - if * 3 - if * 4 - if * 5 N,2 N,3 N,N+1 γ N γ Ν+1 0 N,1 N,2 - if * N γ 1 + γ NP+1 = 0

27 Step 7: Solve the System of Equatios 1,1 2,1 3,1 4,1 5,1 = 1,2 2,2 3,2 4,2 5,2 N,2 N,1 N,2 1,3 2,3 3,3 4,3 5,3 N,3 1,N+1 2,N+1 3,N+1 4,N+1 5,N+1 N,N+1 γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ N γ Ν+1 - if * 1 - if * 2 - if * 3 - if * 4 - if * 5 - if * N

28 Step 8: Postprocessig the Results Circulatio: Lift: (Kutta Joukowski) Pressure Coefficiet:

29 Summary of Key Steps (These steps apply to most pael methods) Get the geometry Place paels oto the geometry Choose a pael type (source, vortex, source-doublet, etc.) Setup the Aerodyamic Ifluece Coefficiet matrix Geerate a RHS Add a Kutta coditio Solve the system of equatios Post-process the result

30 Pressure Coefficiet Upper Surface -C P Lower Surface

31 The Source Pael Method Source Pael Method: NOTE: The source-oly-pael method ca ot be used to compute the flow aroud a liftig airfoil. Uless modified, the source pael method will always solve for the zero-circulatio flow aroud a body.

32 Source-Doublet Pael Method Source-Doublet Pael methods Wake Elemets represetig vorticity Source doublet pael methods are commoly used ad ca be used to solve for the flow aroud liftig bodies.

33 Other Optios for Pael Methods Source-Doublet Pael methods Source doublet pael methods are commoly used i 3D They ca be used to solve for the flow aroud liftig bodies.

34 Applyig Pael Methods to Moder 3D Problems

35 Trailig ortices

36 Trailig ortices" Cessa airplae. Photo by Paul Bowe.

37 From a Spitfire to Liftig Lie Model.

38 Determiig the prescribed bat Wid Tuel Side view geometry Typical flight coditios: Forward velocity 2-7 m/s 7 cm chord 40 cm wigspa Flappig frequecy 8-10 Hz Kiematics: Sychro-Stereo imagig at 1000 fps Wake velocities: Future PI at 200Hz

39 Wigbeat Re-Costructios Frot iew Side iew Top iew

40 Example of Similar Methods i Actio

41 Develop a mesh/discretizatio ad Apply a ortex Distributio O each of the paels we place a distributio of vorticity

42 Setup a Liear Matrix System The matrix system is N x N i size: For large umber of paels or elemets it is costly to solve this matrix system. We apply techiques which approximate the matrix system, allowig us to rapidly solve the problem.

43 Usteady Effects : From a Spitfire to a flappig wig. Spitfire% jpg Flappig wig Usteady motios Usteady wake structures

44 Flapper Desig : 3-D Structural Tailorig Typical Level Flight Result Reducig Stiffess B B A A Dowstroke Upstroke Outboard Sectio Iboard Sectio 1 MOIE 2 Lower Stiffess A-A Higher Stiffess B-B

45 Wig Desig Example Select Wig Plaform Determie Feathered 3D Wig Shape Determie Shape Modificatio

46 Results