Applied Aerodynamics

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1 Applied Aeodynamics Def: Mach Numbe (M), M a atio of flow velocity to the speed of sound Compessibility Effects Def: eynolds Numbe (e), e ρ c µ atio of inetial foces to viscous foces iscous Effects If is a eaction foce acting on a body and C is its dimensionless coefficient: ρ C then dimensional analysis (Buckingham s PI theoem) tells us S C f (, M e ) which implies C C C L D M f (, M e f (, M 3 e f (, M ) ) 4 e )

2 You Fluid Mechanics Couses: Basic Fluid Mechanics Application of Fma to fluids Imcompessible Flows Mostly Inviscid (some pipe flow if you wee lucky) Fundamental Aeodynamics Incompessible Flow Theoy (no compessibility, no viscousity) Souces, ks and votices Panel Methods (D only) Gas Dynamics Mach Numbe effects (mostly shock/expansion theoy, supesonic flows) Possible coections fo high subsonic flow Applied Aeodynamics 3D Bodies iscous Flows Compessibility Effects Aeodynamic Design Issues Inviscid 3D Wing Theoy iscous Aifoil Codes Xfoil fom MIT

3 Class Plan: eview D Aifoil Theoy (Chaptes 3&4 of Andeson) Intoduce 3D Wing Theoy (Chaptes 5&6) Navie-Stokes and Bounday Laye Equations XFOIL Applied Aeodynamics Design Concepts (along the way) 3

4 4 eview of Aifoil Theoy and D Aeodynamics Elemental Solutions Flows about D aifoils ae built fom fou elemental solutions: unifom flow, souce/k, doublet and votex. These elemental solutions ae solutions to the govening equations of incompessible flow, Laplace s equation. 0 ϕ 0 (3.) which elies on the flow being iotational Equations (3.) ae solved fo 0 (3.) N - the velocity potential - the steam function The velocity potential is such that, i.e., u, v (3.3) x y wheeas the steam function is always othogonal (pependicula) to the velocity potential, u ϕ ϕ, v (3.4) y x

5 5 Convenient Coodinate Systems Catesian: ),, ( z y x z y x (3.5) Cylindical: ),, ( z z (3.6) Spheical: ),, ( Φ 0 Φ Φ + + (3.7) Big Advantage: Laplace s Equation is LINEA Supeposition is possible

6 6 Bounday Conditions Solution about a specific body equies that the body be a steamline, i.e., flow tangency condition. nˆ 0 (3.8) o 0 n ϕ 0 s whee n- suface nomal, s suface tangent Anothe way to say it is that the suface has to be a steamline: dy b v steamline equation dx u

7 7 Unifom Flow C x + C C y + C3 ϕ (3.9) u x C ϕ v 0 y x ϕ y In cylindical coodinates cos ϕ (3.0)

8 8 Souce/Sink c, 0 (3.) v Souce Stength Λ & (3.) l m& Whee v& is the volume flow ate (sometimes called Q) v & ρ Λ So (3.3) Λ ln Λ 0 Def: Ciculatio n ': ds ( ) ds C Λ ϕ ϕ Λ ϕ 0 (3.4) (3.5) if S is an aea that includes the souce, 0 souce / k

9 9 Supeposition of Souce and Unifom Flow Λ ϕ + ϕ Λ cos + ϕ (3.6) Stagnation Point Set 0 Λ, (3.7) The stagnation point can now be used to define the body steamline, i.e., inset the stagnation point coodinates into the steam function equation to get its constant value. That new equation defines the body steamline.

0 Supeposition of Souce, Sink and Unifom Flow Λ Λ ϕ + (3.8) Doublet Flow The souce and k of the last section wee sepaated by a distance l and they had identical stengths.

10 0 Supeposition of Souce, Sink and Unifom Flow Λ Λ ϕ + (3.8) Doublet Flow The souce and k of the last section wee sepaated by a distance l and they had identical stengths. If one imagines that thei steam function is vaiable and thei distance can be alteed a new condition would esult if they ae moved togethe while holding constant the poduct l7. In the limit, as l 0 a doublet is fomed. K cos K ϕ (3.9) whee K l7. Steamlines ae found by choog constant so that we get K d c (3.0)

11 Doublet + Unifom Flow When a doublet is supeposed togethe with a unifom flow we get the flow ove a cicula cylinde. K ϕ (3.) o if we let K ϕ (3.) elocities become: ϕ ϕ cos cos ( )

Stagnation Points ae again found by setting the velocities to zeo. cos 0 + 0 (3.

12 Stagnation Points ae again found by setting the velocities to zeo. cos (3.5) Which gives (,0) and (,B). The stagnation steamline then becomes ϕ 0, i.e.,, a cicle (3.6)

13 3 Pessue Coefficient Fom the Benoulli Equation we get: C p (3.7) Which becomes on the cylinde C p 4 (3.8) Fom this we see that the axial and nomal foce coefficients become: c Cn ( C p, l C p, u ) dx c C a c 0 TE LE ( C C )dy p, f p, b (3.9) Both tun out to be zeo ce the pessue coefficient is symmetic., i.e., D Alembet s paadox!

14 4 otex Flow ϕ ln (3.30) 0 Ciculation ds C But we also know that ( ) ds (3.3) (3.3) How does this squae if the flow is eveywhee iotational? Answe, it doesn t, the flow is iotational eveywhee except at the cente of the votex. The angula velocity becomes infinite at that gulaity.

15 5 Doublet + Unifom Flow + otex ( ) ln ϕ + (3.33) ϕ ϕ cos + (3.34) This time the stagnation points depend upon '. > < ( ) ± 3, 4, 3 4,

16 6 + + SC q L C C C l d l p ρ 0 4 (3.35) Lift is diectly popotional to ciculation

7 Panel Methods The pimay technique fo low speed analysis (incompessible flow) is the panel method. We will discuss two, the souce panel method and the votex panel method.

17 7 Panel Methods The pimay technique fo low speed analysis (incompessible flow) is the panel method. We will discuss two, the souce panel method and the votex panel method. Souce Panel Method To begin conside what would happen if you stetched a point souce/k out ove some cuve in space Defined by the equation λds d ln (3.36) whee λ λ(s), s being the diection along the souce sheet. To detemine the potential at some point P(x,y) induced by the souce sheet, one integates Eq along the sheet. b λds ( x, y) ln (3.37) a

8 When combined with a unifom flow this yields the flow ove an abitay body if λ λ(s) is pescibed in such a way that the desied body shape becomes a steamline.

The equation fo the potential induced by the th panel is then λ ln ids (3.38) Which gives the contibution to N at point P fom panel.

18 8 When combined with a unifom flow this yields the flow ove an abitay body if λ λ(s) is pescibed in such a way that the desied body shape becomes a steamline. In geneal, it is difficult to find an analytical expession fo λ λ(s), so it is appoximated eithe by a seies of constant souce panels o by linealy vaying souce panels. The equation fo the potential induced by the th panel is then λ ln ids (3.38) Which gives the contibution to N at point P fom panel. The potential fo the entie flow field then comes by summing the contibutions fom all the panels. At the cente of each panel is a contol point at which the basic equation fo the suface nomal velocity can be found fom

19 λ λ ds (3.39) n n i [ ( xi, yi )] + ( ln i ) n i ni i This is the nomal velocity contibution at the contol point fom only the souce sheet. Howeve, what we want is the body to be a steamline, so we must have: i (3.40) n n i This leads to the equation that can be solved fo the λ unknowns λ n λ i + I i i + cos β i 0 (3.4) whee I is the integal in Eq The suface velocity on the body at the contol point can then be found fom λ ds (3.4) s n [ ( xi, yi )] ( ln i ) s i s once the λ ae found. Lastly, the integal can be evaluated as on page 56 of the Andeson text and C found fom the Benoulli equation. p Unfotunately, ce we ae dealing only with souces and ks the lift and dag ae both zeo!!!

0 Lifting Aifoils The poblem of lift in geneal is bound up in the idea of ciculation and hence equies votices in the panel method, which we showed ealie to be the only elemental solution to exhibit

20 0 Lifting Aifoils The poblem of lift in geneal is bound up in the idea of ciculation and hence equies votices in the panel method, which we showed ealie to be the only elemental solution to exhibit ciculation. To accommodate this we use an idea simila to the souce sheet, the votex sheet Whose potential is given by the equation whee γ γ (s). b, γds (4.) ( x z) a Ciculation comes about fom the votices and is elated diectly to the distibution of γ, via: b γ ds (4.) a Fom this point the entie thin aifoil theoy can be developed (which should be eviewed) o one can follow the same appoach

used to develop the souce panel method ug the votex sheet equation to poduce a votex panel method fo abitay lifting bodies. Howeve, one thing misg is the specification of stagnation points.

Kutta Condition The basic idea is elated to the fact that an infinite numbe of stagnation point locations can exist on a lifting aifoil if only the ealie theoy is applied, and because of that,

21 used to develop the souce panel method ug the votex sheet equation to poduce a votex panel method fo abitay lifting bodies. Howeve, one thing misg is the specification of stagnation points. Which equies the Kutta Condition. Kutta Condition The basic idea is elated to the fact that an infinite numbe of stagnation point locations can exist on a lifting aifoil if only the ealie theoy is applied, and because of that, vitually any ciculation (lift) is possible. Howeve, Wilhelm Kutta obseved that the flow leaves a pointed tailing edge aifoil smoothly fom the tailing edge, placing the stagnation point at the tailing edge. This is the so-called Kutta Condition and it povides the additional infomation we need to detemine the ciculation and hence lift of an abitay aifoil.

If it stats out zeo aound an aifoil at est, it must emain zeo along that fluid element.

22 Stat-Up otex Anothe inteesting, yet appaently contadictoy, idea associated with aifoils comes about because of Kelvin s Theoem, which says ciculation does not change along a fixed fluid element. If it stats out zeo aound an aifoil at est, it must emain zeo along that fluid element. Fo ciculation to exist about the aifoil (and hence lift) it must coexist with an equal stength negative ciculation shed at aifoil stat-up and convecting downsteam bounded by the oiginal fluid element.

3 otex Panel Method So ust as in the souce panel method, the votex sheet can be appoximated by a seies of discete votex panels of constant o vaying stength.

23 3 otex Panel Method So ust as in the souce panel method, the votex sheet can be appoximated by a seies of discete votex panels of constant o vaying stength. Again the body must be a steamline and hence the flow tangency condition can be applied at the contol points. The esulting equation fo votex stength is n γ i cos β i ds 0 (4.3) n This poduces a system of n unknowns and n equations. We think good but not so, because it still doesn t incopoate the Kutta Condition. A way to do this is to make the tailing edge panels vey small and enfoce i γ i i γ (4.4) as illustated in the Figue below. Unfotunately, now the system is ove detemined. To obtain a deteminate system one must eliminate one of the panels fom the linea equation set. Then, once the system is solved, the lift can be detemined diectly fom the known coefficients, i.e., Next up Wing Theoy!!!! n L ρ γ s (4.5)

ME 425: Aerodynamics

ME 425: Aerodynamics ME 5: Aeodynamics D ABM Toufique Hasan Pofesso Depatment of Mechanical Engineeing, BUET Lectue- 8 Apil 7 teachebuetacbd/toufiquehasan/ toufiquehasan@mebuetacbd ME5: Aeodynamics (Jan 7) Flow ove a stationay