ME 425: AERODYNAMICS

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1 ME 45: AERODYNAMICS - Dr. A.B.M. Toufique Hasan Professor Department of Mehanial Engineering, BUET Leture # 3 4 Otober, 6 ME45: Aerodynamis Airfoil Nomenlature Mean amber line is the lous of points halfway between the upper and lower surfae as measured perpendiular to the mean amber line itself. The straight line onneting the leading and trailing edges is the hord line. The amber is the maximum distane between the mean amber line and the hord line, measured perpendiular to the hord line. ME45: Aerodynamis

2 Airfoil Nomenlature NACA airfoils (National Advisory Committee for Aeronautis ontd NASA (National Aeronautis and Spae Administration NACA X X X X maximum thikness in hundreths of hord loation of maximum amber from LE in tenths of hord maximum amber in hundreths of hord NACA maximum thikness% zero amber (symmetri airfoil NACA 4 maximum thikness% maximum amber is.4 from LE maximum amber % (ambeed airfoil ME45: Aerodynamis 3 Low-speed flow over airfoils Infinite no. of point vorties along a straight line extending to infinity (+ to - irulation around thedashedpathis r r Γ = ds = v dn + u ds v dn u ds Γ = ( u u ds γ ds = ( u u ds loal C γ = u u the loal sheet strength. ; dn Infinite no. of vortex filaments side by side, where strength of eah filament is infinitesimallysmall. ; γ is theloalsheet strength per unit length jump in tengential veloity aross the vortex sheet is equal to Indued veloity, d due to infinitesimalvortex filament ME45: Aerodynamis 4

3 Low-speed flow over airfoils ontd The onept of vortex sheet is instrumental in the theoretial aerodynamis of lowspeed airfoil. A philosophy of airfoil theory of invisid, inompressible flow is as follows: Consider an airfoil of arbitrary shape and thikness in a free stream with veloity as skethed in the figure. Replae the airfoil surfae with a vortex sheet of variable strength γ(s. Calulate the variation of γ as a funtion of s suh that the indued veloity field from he vortex sheet when added to the uniform veloity of magnitude will make the vortex sheet (hene the airfoil surfae a streamline of the flow. In turn, the irulation around the airfoil will given by Γ = γ ds where the integral is taken around the omplete surfae of Finally, lift is alulated by the Kutta - Joukowski theorem : L = ρ Γ ME45: Aerodynamis the airfoil. 5 Low-speed flow over airfoils ontd The onept of replaing the airfoil surfae with a vortex sheet is more than just a mathematial devie; it also has physial signifiane. In real ase, there is a thin boundary layer on the surfae, due to the ation of frition between the surfae and the air flow. This boundary layer is highly visous region in whih large veloity gradients produe substantial vortiity. r (urlof veloityfieldis finite Hene, in real life, there is a distribution of vortiity along the airfoil surfae due to visous effet and the philosophy of replaing the airfoil surfae with a vortex sheet an be onstruted as a way of modeling this effet in an invisid flow. Modeling of boundary layer as vorties r ME45: Aerodynamis 6 3

4 Low-speed flow over airfoils ontd Imagine that the airfoil is made very thin. If we were to stand bak and look at suh a thin airfoil from a distane, the portion of the vortex sheet on the top and bottom surfae of the airfoil would almost oinide. This give rise to a method of approximating a thin airfoil be replaing it with a single vortex sheet distributed over the amber line of the airfoil, as shown in figure. The strength of this vortex sheet is alulated suh that, in ombination with free stream, the amber line beomes a streamline of the flow. This philosophy is known as lassial thin airfoil theory. ME45: Aerodynamis 7 Thin Airfoil Theory Consider a vortex sheet plaed on the amber line of airfoil, as shown in figure (a.the free - stream veloity is and the airfoil is at angle of attak (AOA α. The distane measured along the amber line is denoted by s.the shape of the amber line is given by z = z( x.the hord length is. an, w is the omponent of veloity normal to amber line indued by the vortex sheet; w = w ( s. If the airfoil is thin, the amber line is lose to the hord line, and viewed from a distane, the vortex sheet appears to fall approximately on the hord line as shown in figure (b. Hereγ = γ ( xandγ = γ ( x is alulated to satisfy that the amber line (not the hord line is a streamline. ME45: Aerodynamis 8 4

5 Thin Airfoil Theory For the amberline to be a streamline, the omponent of veloity normal to the amber line must be zero at all points along the amber line. The veloity at any point in the flow is the sum of indued by the vortex sheet. Let,n normal to the amber line. Thus, for the amberline to be a streamline, be the omponent of the free stream veloity,n + w ( s = at every point along theamber line. At any point, P on the amber line, where the slope of dz amber line is, thegeometry of Figure yields dx = sin + dz,n α tan,n = dx for a thin airfoil at small angle of uniform veloity and the veloity attak α and the dz α dx dz dx ME45: Aerodynamis ontd 9 Thin Airfoil Theory ontd If theairfoil is thin, the amber line is lose to the hord line, and it is onsistant with thin airfoil theory to make the approximation that w ( s w( x an expression for w(x in terms of the strength of the vortex sheet is easily obtainable as follows : Consider an elemental vortex of distaneξ from the origin along the hord line, as shown in figure. The strength of distane along the hord line; that is γ = γ ( ξ.the veloity dwat point x indued by the elemental vortex at point ξ is : γ ( ξ dξ dw= π ( x ξ strength γ dξ loated at a the vortex sheet γ varies with the reall: Γ θ = π r ; due to vortex ME45: Aerodynamis 5

6 Thin Airfoil Theory ontd In turn, the veloity w(x indued at point x by all the elemental vorties along the hord line is obtained by integrating from leading edge ( ξ = to the trailing edge ( ξ = : w( x = w ( s = γ ( ξ dξ π ( x ξ γ ( ξ dξ π ( x ξ for the amberline to be a streamline, + w ( s =,n dz α dx γ ( ξ dξ = π ( x ξ γ ( ξ dξ = π ( x ξ dz α dx Fundamental equation of thin airfoil theory. To be solved for symmetri airfoil and ambered airfoil. ME45: Aerodynamis Thin Airfoil Theory ontd Symmetri airfoil A symmetri airfoil has no amber; the amber line is oinident with the hord line. dz = dx γ ( ξ dξ = α π ( x ξ To deal with the integral in above equation; adopt the following transformation ξ = ( osθ dξ = sinθ dθ at LE ξ = ; θ = at TE ξ = Further, sine x is a x = ; θ = π ( osθ fixed point in the bove equation, it orresponds as ME45: Aerodynamis 6

7 Thin Airfoil Theory ontd Substitute the above transformation gives: π π γ ( θ sinθ dθ osθ osθ = α A rigorous solution of the above equation an be obtained from the mathematial theory of integral equations (whih is beyond the sope of ME 45 Aerodynamis γ ( θ = α ( + osθ sinθ Now, the total irulation around the symmetri airfoil is Γ = Γ = Γ = π π Γ = α Γ = π α γ ( ξ dξ γ ( θ sinθ dθ ; α π ( + osθ sinθ dθ sinθ ( + osθ dθ transformation, ξ = ( osθ ME45: Aerodynamis 3 Thin Airfoil Theory In ase of symmetri airfoil; ontd Γ = π α Using Kutta-Joukowski theorem; L = ρ Γ L = π α ρ The lift oeffiient l L = ρ (lift per unit span of airfoil ( Q Γ = π α π α ρ = ρ l = πα Theoretial lift oeffiient is linearly proportional to the angle of attak (AOA. lift slope = d l dα = π Theoretial lift slope is π per radian, whih is. per degree of AOA. ME45: Aerodynamis 4 7

8 Thin Airfoil Theory ontd theory exp. ME45: Aerodynamis 5 8