lectures-chapter10

lectures-chapter10 - 650:460 Aerodynamics Chapter 10 Prof....

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Unformatted text preview: 650:460 Aerodynamics Chapter 10 Prof. Doyle Knight Tel: 732 445 4464, Email: doyleknight@gmail.com Office hours: Tues and Thur, 4:30 pm - 6:00 pm and by appointment Fall 2009 1 Supersonic Flows Around Thin Airfoils Linear Theory Assumptions Inviscid Steady Supersonic Thin airfoil Fig. 10.1 2 Supersonic Flows Around Thin Airfoils Linear Theory Atributed to Jakob Ackeret. Diploma in Mechanical Engineering from ETH Zurich (1920) Worked with Ludwig Prandtl in G ottingen (1921-1927) PhD from ETH Zurich (1927) Coined the term Mach number after Ernst Mach Jakob Ackeret (1898-1981) 3 Supersonic Flows Around Thin Airfoils Linear Theory Eulers equation for compressible flow dp =- UdU Assuming the changes dU are small compared to the freestream velocity U , we can integrate holding U U to obtain p- p =- U ( U- U ) From Eqn (8.55) dU U = d M 2- 1 where d is the change in the angle of the velocity vector. 4 Supersonic Flows Around Thin Airfoils Linear Theory Defining the flow angle = 0 ( i.e. , measuring flow angles relative to the flow upstream), U - U U = radicalbig M 2 - 1 where we have used the approximation that the changes in velocity are small compared to U . A positive ( i.e. , a counter-clockwise rotation of the flow) results in a compression wave ( i.e. , a reduction in velocity and increase in pressure). Combining with the equation for the change in pressure yields p- p = U 2 radicalbig M 2 - 1 5 Supersonic Flows Around Thin Airfoils Linear Theory Thus, defining the pressure coefficient c p = p- p 1 2 U 2 then c p = 2 radicalbig M 2 - 1 On the airfoil surface, the tangent (see figure) is u = + dz u dx- l =- dz l dx + where z u ( x ) and z l ( x ) are the upper and lower surface of the airfoil 6 Supersonic Flows Around Thin Airfoils Lift The elemental lift on segment ABCD is dl = p l ds l cos l- p u ds u cos u Since ds cos = dx , dl = ( p l- p u ) dx and thus dc l = ( c p l- c p u ) d parenleftBig x c parenrightBig Fig. 10.4 7 Supersonic Flows Around Thin Airfoils Lift From previous c p l- c p u = 2 radicalbig M 2 - 1 parenleftbigg 2 - dz l dx- dz u dx parenrightbigg Therefore the lift coefficient is c l = integraldisplay 1 o dc l d parenleftBig x c parenrightBig Now integraldisplay 1 o dz l dx d parenleftBig x c parenrightBig = 0 since the leading and trailing edges are on the z = 0 axis. A similar result holds for dz u / dx ....
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This note was uploaded on 01/24/2012 for the course MECHE 343 taught by Professor Professor during the Spring '11 term at Carnegie Mellon.

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lectures-chapter10 - 650:460 Aerodynamics Chapter 10 Prof....

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