ME 425: Aerodynamics

Transcription

1 ME 45: Aeroynamis Dr. A.B.M. Toufique Hasan Professor Department of Mehanial Engineering Banglaesh University of Engineering & Tehnology BUET, Dhaka Leture-6 /5/8 teaher.buet.a.b/toufiquehasan/ Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 Airfoil Nomenlature The straight line onneting the leaing L.E. an trailing eges T.E. is the hor line. Mean amber line is the lous of points halfway between the upper an lower surfaes as measure perpeniular to the mean amber line itself. The amber is the maimum istane between the mean amber line an the hor line, measure perpeniular to the hor line. Wing Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8

2 Airfoil Nomenlature NACA airfoils National Avisory Committee for Aeronautis NASA National Aeronautis an Spae Aministration NACA X X X X maimum thikness in hunreths of hor loation of maimum amber from LE in tenths of hor maimum amber in hunreths of hor NACA maimum thikness% zero amber symmetriairfoil NACA 4 maimum thikness% maimumamberis.4 from LE maimumamber % ambee airfoil Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 3 Low Spee Flow Over Airfoils Infinite no. of point vorties along a straight line etening to infinity + to Infinite no. of vorte filaments sie by sie, where strength of eah filament is infinitesimally small. Inue veloity, ue to infinitesimal vorte filament irulation aroun theashe path is s v n u s v n u s u u s s u u s loal C u u jump in tengential veloity aross the vorte sheet is equal to the loal sheet strength. ; n ; is theloalsheetstrength per unit length Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 4

3 Low Spee Flow Over Airfoils The onept of vorte sheet is instrumental in the theoretial aeroynamis of low-spee airfoil. A philosophy of airfoil theory of invisi, inompressible flow is as follows: Consier an airfoil of arbitrary shape an thikness in a free stream with veloity as skethe in the figure. Replae the airfoil surfae with a vorte sheet of variable strength γs. Calulate the variation of γ as a funtion of s suh that the inue veloity fiel from the vorte sheet when ae to the uniform veloity of magnitue will make the vorte sheet hene the airfoil surfae a streamline of the flow. In turn, the irulation aroun the airfoil will given by s where the integral is taken aroun the omplete surfae of Finally, lift is alulate by the Kutta - Joukowski theorem : L the airfoil. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 5 Low Spee Flow Over Airfoils The onept of replaing the airfoil surfae with a vorte sheet is more than just a mathematial evie; it also has physial signifiane. In real ase, there is a thin bounary layer on the surfae, ue to the ation of frition between the surfae an the air flow. This bounary layer is highly visous region in whih large veloity graients proue substantial vortiity url. Hene, in real life, there is a istribution of vortiity along the airfoil surfae ue to visous effet an the philosophy of replaing the airfoil surfae with a vorte sheet an be onstrute as a way of moeling this effet in an invisi flow. Moeling of bounary layer as vorties vorte sheet: url Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan

4 Low Spee Flow Over Airfoils Imagine that the airfoil is mae very thin. If we were to stan bak an look at suh a thin airfoil from a istane, the portion of the vorte sheet on the top an bottom surfae of the airfoil woul almost oinie. This give rise to a metho of approimating a thin airfoil be replaing it with a single vorte sheet istribute over the amber line of the airfoil, as shown in figure. The strength of this vorte sheet is alulate 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. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 7 Consier a vorte sheet plae on the amber line of an airfoil, as shown in Fig. a. The free-stream veloity is, an the airfoil is at an angle of attak AOA, α. The istane measure along the amber line is enote by s. Theshapeoftheamberlineisgivenbyz = z. The hor length is. w' is the omponent of veloity normal to the amber line inue by the vorte sheet; w' = w' s If the airfoil is thin, the amber line is lose to the hor line, an viewe from a istane, the vorte sheet appears to fall approimately on the hor line as shown in Fig. b. Here γ = γ an γ = γ is alulate to satisfy that the amber line not the hor line is a streamline. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan

5 For the amberline to be a streamline, the omponent veloity normal to the amber line must be zero at all points along the amber line. The the flow is the sum of uniform veloity an the veloity inue by the vorte sheet. Let,n normal to the amber line. Thus, for the amberline to be a streamline,,n be amber line is the omponent of the free stream veloity,n z w s at every point along the amber line. At any point, P on the amber line, where the slope of the geometry of Figure yiels z z sin tan,n z z attak an tan for a thin airfoil at small angle of, of veloity at any point in the Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 9 If the airfoil is thin, the amber line is lose to the hor line, an it is onsistant with thin airfoil theory to make the approimation that w s w an epression for w in terms of the strength of the vorte sheet is easily obtainable as follows : Consier an elemental vorte of istane from the origin along the hor line, as shown in figure. The strength of the vorte sheet varies with the istane along the hor line; that is.the veloity wat point inue by the elemental vorte at point is : w strength γ loate at a reall: r γ = orte strength per unit length ; ue to vorte lokwise Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 5

6 6 Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 s w w : to the trailingege leaingege vortiesalong the hor line is obtaine by integrating from the elemental winueat point by all In turn,the veloity for the amberline to be a streamline,,n z s w z Funamental equation of thin airfoil theory. To be solve for symmetri airfoil an ambere airfoil. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 z Symmetri airfoil A symmetri airfoil has no amber; the amber line is oinient with the hor line. To eal with the integral in above equation; aopt the following transformation: os it orrespons as fie point in the bove equation, is a sine Further, ; T.E. at ; L.E. at sin os z

7 Substitute the above transformation gives: sin os os A rigorous solution of the above equation an be obtaine from the mathematial theory of integral equations whih is beyon the sope of ME 45: Aeroynamis os sin Distribution of irulation aroun a symmetri airfoil Now, the total irulation aroun the symmetri airfoil is sin ; os sin sin os transformation, os Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan. 8 3 In ase of symmetri airfoil; Using Kutta-Joukowski theorem; L L L The setionallift oeffiient lift per unit span of airfoil/setional lift l l Theoretial lift oeffiient is linearly proportional to the angle of attak AOA. lift slope l Theoretial lift slope is π per raian, whih is. per egree of AOA. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan

8 theory l ep. Theory an t preit the stalling phenomena. Theory an t ifferential the effet of Re number. Theory an t aommoate the thikness of the airfoil. Dr. A.B.M. Toufique Hasan BUET L-4 T-, Dept. of ME ME 45: Aeroynamis Jan