By Hermann T. Schlichting, Erich A. Truckenbrodt (transl. by Heinrich J. Ramm)
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The pressures on the upper and lower surfaces of the profile are designated as pu and pl, respectively (see Fig. 2-3), and the difference d p = (p1- pu) is a measure for the normal force dZ = A pb dx acting on the surface element dA = b dx (see Fig. 2-5). bc (2-9b) where cZ is the normal force coefficient from Eq. (1-21) (see Fig. 2-3). bc2 (2-11 b) where nose-up moments are considered as positive. The pitching-moment coefficient is, accordingly, CM=-1 f c dcp(x)dx (2-12) 0 2-2 FUNDAMENTALS OF LIFT THEORY 2-2-1 Kutta-Joukowsky Lift Theorem Treatment of the theory of lift of a body in a fluid flow is considerably less difficult than that of drag because the theory of drag requires incorporation of the viscosity of the fluid.
For small angles of attack, a G< 1, and small camber, the lift coefficient becomes cL = 27r (Cl +2 C (2-36) The lift slope dcL/da is again equal to 27r for small angles of attack, as in the case of the inclined flat plate according to Eq. (2-31). For the zero-lift angle of attack this equation yields ao = -2(h/c). The pitching-moment coefficient about the profile leading edge becomes CM = - 2 (a+4 h) (2-37) resulting in cMo = -ir(h/c) for the zero-moment coefficient when ao = -2(h/c). The velocity distribution on the circular-arc profile is given for small camber and small angles of attack by WC-u'c, 1t4C Y1-4(x)2± (2-38) The + sign applies to the upper profile surface, the - sign to the lower profile surface.
The velocity components in the x and y directions, that is, u and v, are given by AIRFOIL OF INFINITE SPAN IN INCOMPRESSIBLE FLOW (PROFILE THEORY) 37 u a0 d IF c9x 7y V c70 0'l-1 Jy Jx The function F(z) is called a complex stream function. From this function, the velocity field is obtained immediately by differentiation in the complex plane, where dF dz = it - i V = w(z) (2-17) Here, w = u - iv is the conjugate complex number to w = u + iv, which is obtained by reflection of w on the real axis.