C-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory

Transcription

1 ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos? How? Source Pael Method Vortex Pael Method

2 Page 1 of 12 Ut C-3C Problem Solutos? How? PANEL METHODS / THIN AIRFOIL THEORY: PURE THEORETICAL SCHEMES Pael methods ad th arfol theory are bult upo potetal flow aalyss. I other words, these are pure theoretcal schemes. Smlar to the ut B-3, as we deal wth the pure theory, oe must be very carefully recogze (ad keep track of) the followgs: (1) What are the assumptos made to smplfy the equato? Steady-state? Ivscd? No body forces? Icompressble? Irrotatoal? (2) Because of the assumptos made, how your theoretcal soluto dfferet (or apart) from the actual (or real) flow feld pheomea?

3 Page 2 of 12 Ut C-3C Source Pael Method (1) SOURCE PANEL METHOD A source sheet ca be defed as fte umber of le sources (sde by sde) where the stregth of each le source s ftesmally small. Source stregth per ut legth ca be defed as: () s. Cosder a arbtrary pot P( x, y )... a small secto of the source sheet of stregth ds duces a ftesmally small velocty potetal d : ds d l r 2 Thus, the complete velocty potetal at pot P duced by the etre source sheet s: b ds ( x, y) l r 2 a

4 Page 3 of 12 Ut C-3C Source Pael Method (2) Let us approxmate the source sheet by a seres of straght paels, moreover, let the source stregth per ut legth be costat over a gve pael. For the total umber of paels, the source stregths per ut legth for each pael ca be represeted,,,,, by: The, solve:, 1,2,3, such that the body surface becomes a streamle of the flow: meas that the boudary codto s mposed umercally by s such that the ormal compoet of the flow velocty s zero at the mdpot of each pael (called, the cotrol pot ). Aga, cosder a pot P (x,y) the flow feld, ad let r p be the dstace from ay pot o the th pael to P. The velocty potetal duced at P due to the th pael s: l rpds 2

5 Page 4 of 12 Ut C-3C Source Pael Method (3) The potetal at P due to all paels s: P l rpds where, the dstace r ( x x ) ( y y ) 2 2 p Next, let us put P at the cotrol pot of the th pael P( x, y ): x y, l r ds 1 2 r ( x x ) ( y y ) 2 2 Ths s the cotrbuto of all paels to the potetal at the cotrol pot of the th pael. The free stream compoet ormal to the pael s: V ˆ, V V cos The ormal compoet of velocty duced at ( x, y ) by a pael s: V ( x, y )

6 Page 5 of 12 Ut C-3C Source Pael Method (4) Combg ths wth the potetal ( x, y), whch s cotrbuto of all the paels to the potetal at the cotrol pot of the th pael: V l r ds ( 1) Applyg the boudary codto, V, V 0, yelds: l r ds V cos ( 1) Let: I, (l r, ) ds => I, V cos Ths s a lear algebrac equato wth ukows ( 1, 2, 3,,, ) => these ukows ca be solved ( equatos wth ukows) Oce s are all solved (, 1,2,3, )... The compoet of free stream velocty taget to the surface s: V, s Vs The tagetal velocty V s at the cotrol pot of the th pael duced by all the paels s: Vs l r ds s 2 s 1 The total surface velocty at the th cotrol pot V s the sum of the cotrbuto from the free stream ad from the source paels: V V, s Vs Vs l r ds 2 s 1 Fally, the pressure coeffcet at the th pael ca be gve as: Cp, 1 ( V V ) 2

7 Page 6 of 12 Class Example Problem C-3-1C Related Subects... Source Pael Method Ut C-3C There s a theoretcal arfol, called a Joukowsk arfol. The shape of ths arfol ca be obtaed from a crcle by a mathematcal trasformato betwee two domas. Sce ths arfol s a theoretcally derved shape, the surface pressure coeffcet dstrbuto (C p ) ca also be obtaed theoretcally. Usg source pael method, calculate the pressure coeffcet dstrbuto (C p ) of symmetc Joukowsk arfol at AOA = 0. Compare the theoretcal dstrbuto of C p (Joukowsk arfol: kow values) agast C p computed by the source pael method. Symmetrc Joukowsk Arfol Source pael code has two fudametal problems: Source pael does ot smulate crculato. Therefore, the source pael code s applcato s heretly lmted to a olftg flow aroud a body oly. Source pael caot eforce the Kutta codto at the tralg edge. Therefore, the stagato pot s formed at a arbtrary locato as the agle of attack s creased, ad ths causes the ophyscal solutos.

8 Page 7 of 12 Ut C-3C Vortex Pael Method (1) VORTEX SHEET A vortex flamet ca be defed as a straght le perpedcular to the page, gog through pot O (exted to fty both sdes). A vortex sheet ca be defed as fte umber of vortex flamets sde by sde, wth the stregth of each flamet s ftesmally small. Cosder a arbtrary pot P (x, z). The small secto of vortex sheet of stregth ds duces a small velocty potetal ad velocty dv : ds d ad 2 dv ds 2 r The velocty potetal at P due to etre vortex sheet from a to b s, therefore: b b 1 ( x, z) ds 2 where, the crculato s: ds a a

9 Page 8 of 12 Ut C-3C Vortex Pael Method (2) u u 1 2 ds Cosder a dashed path eclosg a vortex sheet of stregth ds : V ds ( v2d u1ds v1d u2ds) ( u1 u2) ds ( v1 v2) d C Let the top ad bottom of the dashed le approach the vortex sheet: d 0, ad u1 u2 Importat cocluso: the local ump tagetal velocty across the vortex sheet s equal to the local sheet stregth. Ths s the fudametal cocept of th arfol theory. The vortex pael method s bult upo the dea that, f we replace the 2-D arfol shape by a vortex sheet, t s possble to obta the lft usg the Kutta-Joukowsk theorem. Th arfol theory ad vortex pael method are bult upo the same fudametal models of vortex sheet. If arfol s th => th arfol theory If arfol s ot th => vortex pael method

10 Page 9 of 12 Ut C-3C Vortex Pael Method (3) VORTEX PANEL METHOD Very smlar approach (as dscussed source pael method). A seres of vortex sheets approxmates the body of arbtrary shape (vortex paels): Total umber of paels wth the vortex stregths per ut legth of: 1, 2, 3,,, At the cotrol pots, the ormal compoet of the velocty s zero: V, V 0 V cos ds 0 2 or 1 V cos I Note that the Kutta codto ca be appled precsely at the tralg edge ad s gve by (TE) 0, meas at tralg edge: 1 Ths Kutta codto forces the flow feld to have a proper stagato pot at (or ear) the tralg edge of the flow feld.

11 Page 10 of 12 Ut C-3C Vortex Pael Method (4) Oce the ukow vortex stregth, 1, 2, 3,,,, are obtaed, the total crculato for the gve flow feld ca be calculated as: s 1 Therefore, the lft per ut spa s (from the Kutta-Joukowsk theorem): L' V s 1

12 Page 11 of 12 Ut C-3C Vortex Pael Method (5) VARIETY OF SOURCE / VORTEX PANEL METHOD SCHEMES 1st order vortex pael method: the stregth of vortex pael s costat for a gve pael. 2d order vortex pael method: the stregth of vortex pael vares learly over a gve pael. Combato of source + vortex paels: use source pael to smulate the arfol thckess, use vortex pael to troduce crculato (lft).

13 Secto Lft Coeffcet Page 12 of 12 Class Example Problem C-3-2C Related Subects... Vortex Pael Method Ut C-3C Usg Vrga Tech vortex pael method (Java Applet), determe the lft coeffcet of NACA 0009 arfol over a rage of agles of attack from 0 to 16 degrees. Plot a lft curve (c l v.s. ) ad compare t agast NACA arfol data. There are a few mportat restrctos of usg Vortex Pael Java Applet Arfol coordate data must be arraged that: Tralg Edge (1,0) => Lower Surface Data => Leadg Edge (0,0) => Tralg Edge Too much paels clustered ear the tralg edge wll make t dffcult to properly apply Kutta codto: you may wat to maually remove some paels. Total umber of paels should be the rage of 30 to 100. Usg too may paels wll slow dow computatos ad wll ot ecessarly mprove the accuracy. Lft Coeffcet v.s. AOA 2 AOA Lft Coeffcet (degrees) Vortex Pael Method NACA 0009 Data AOA (degrees) Vortex Pael Method NACA 0009 Data