C 13 HAPTER C OMPRESSIBLE THIN AIRFOIL THEORY 13.1 C OMPRESSIBLE POTENTIAL FLOW 13.1.1 THE FULL POTENTIAL EQUATION In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy from an inviscid, irrotational model of the flow. Recall the equations developed in Chapter 6 governing steady, irrotational, homentropic ((cid:162)s = 0) flow in the absence of body forces. (cid:162)((cid:108)U) = 0 U •• U (cid:162)P (cid:163) (cid:165) (cid:162) --------------- + -------- = 0 (cid:164) 2 (cid:166) (cid:108) . (13.1) P (cid:108) (cid:97) (cid:163) (cid:165) ------ = ------ P (cid:164)(cid:108) (cid:166) 0 0 The gradient of the isentropic relation is 2 (cid:162)P = a (cid:162)(cid:108). (13.2) Recall from the development in Chapter 6 that P (cid:97)– 1 (cid:162)P (cid:163) (cid:165) (cid:163) (cid:165) (cid:162) --- = ------------ -------- (13.3) (cid:164)(cid:108)(cid:166) (cid:164) (cid:97) (cid:166) (cid:108) Using (13.3) the momentum equation becomes (cid:97) P U ••U (cid:163)(cid:163) (cid:165) (cid:165) (cid:162) ------------ --- + --------------- = 0 (13.4) (cid:164)(cid:164)(cid:97)– 1(cid:166)(cid:108) 2 (cid:166) Substitute (13.2) into the continuity equation and use (13.3). The continuity equa- tion becomes 2 2 U ••(cid:162)a + ((cid:97)– 1)a (cid:162)••U = 0. (13.5) Equate the Bernoulli integral to free stream conditions. bjc 13.1 5/31/13 Compressible potential flow 2 2 2 2 a U a a U •• U (cid:39) (cid:39) (cid:39) (cid:163) (cid:97)– 1 2(cid:165) ------------ + --------------- = ------------ + -------- = ------------ 1 + ------------M = C T (13.6) (cid:97)– 1 2 (cid:97)– 1 2 (cid:97)– 1(cid:164) 2 (cid:39)(cid:166) p t Note that the momentum equation is essentially equivalent to the statement that the stagnation temperature T is constant throughout the flow. Using (13.6) we t can write 2 a U ••U (cid:163) (cid:165) ------------ = h – --------------- (13.7) (cid:164)(cid:97)– 1(cid:166) t 2 The continuity equation finally becomes U ••U U ••U (cid:163) (cid:165) (cid:163) (cid:165) ((cid:97)– 1) h – --------------- (cid:162)••U – U ••(cid:162) --------------- = 0 (13.8) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) The equations governing compressible, steady, inviscid, irrotational motion reduce to a single equation for the velocity vector U. The irrotationality condition (cid:162)×U = 0 permits the introduction of a velocity potential. U = (cid:162)(cid:92) (13.9) and (13.8) becomes (cid:163) (cid:162)(cid:92)••(cid:162)(cid:92)(cid:165) 2 (cid:163)(cid:162)(cid:92)••(cid:162)(cid:92)(cid:165) ((cid:97)– 1) h – ------------------------ (cid:162) (cid:92)–(cid:162)(cid:92)••(cid:162) ------------------------ = 0. (13.10) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) For complex body shapes numerical methods are normally used to solve for (cid:92). However the equation is of relatively limited applicability. If the flow is over a thick airfoil or a bluff body for instance then the equation only applies to the sub- sonic Mach number regime at Mach numbers below the range where shocks begin to appear on the body. At high subsonic and supersonic Mach numbers where there are shocks then the homentropic assumption (13.2) breaks down. Equation (13.8) also applies to internal flows without shocks such as fully expanded nozzle flow. 13.1.2 THE NONLINEAR SMALL DISTURBANCE APPROXIMATION In the case of a thin airfoil that only slightly disturbs the flow, equation (13.8) can be simplified using small disturbance theory. 5/31/13 13.2 bjc Compressible potential flow Consider the flow past a thin 3-D airfoil shown below. v y=f(x,z) U + u (cid:39) U (cid:39) y x z The velocity field consists of a freestream flow plus a small disturbance U = U + u (cid:39) V = v (13.11) W = w where u  U « 1 , v  U « 1 , w  U « 1. (13.12) (cid:39) (cid:39) (cid:39) Similarly the state variables deviate only slightly from freestream values. P = P + P' (cid:39) T = T + T' (13.13) (cid:39) (cid:108) = (cid:108) +(cid:108)' (cid:39) and a = a + a'. (13.14) (cid:39) This decomposition of variables is substituted into equation (13.8). Various terms are 2 U 2 2 2 U ••U (cid:39) u v w --------------- = -------- + uU + ----- + ----- + ------ 2 2 (cid:39) 2 2 2 (13.15) (cid:162)••U = u + v + w x y z and bjc 13.3 5/31/13 Compressible potential flow U ••U (cid:163) (cid:165) (cid:162) --------------- = (u U + uu + vv + ww , (cid:164) 2 (cid:166) x (cid:39) x x x (13.16) u U + uu + vv + ww , y (cid:39) y y y u U + uu + vv + ww ) z (cid:39) z z z as well as (cid:163) U ••U(cid:165) ((cid:97)– 1) h – --------------- (cid:162) •• U = (cid:164) t 2 (cid:166) 2 (cid:163) (cid:163)U(cid:39) u2 v2 w2(cid:165)(cid:165) ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU + ----- + ----- + ------(cid:180)(cid:180)u + t 2 (cid:39) 2 2 2 x (cid:164) (cid:164) (cid:166)(cid:166) (13.17) 2 (cid:163) (cid:163)U(cid:39) u2 v2 w2(cid:165)(cid:165) ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU + ----- + ----- + ------(cid:180)(cid:180)v + t 2 (cid:39) 2 2 2 y (cid:164) (cid:164) (cid:166)(cid:166) 2 (cid:163) (cid:163)U(cid:39) u2 v2 w2(cid:165)(cid:165) ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU + ----- + ----- + ------(cid:180)(cid:180)w t 2 (cid:39) 2 2 2 z (cid:164) (cid:164) (cid:166)(cid:166) and, finally (cid:163)U ••U(cid:165) 2 U ••(cid:162) --------------- = u U + uu U + vv U + ww U + (cid:164) 2 (cid:166) x (cid:39) x (cid:39) x (cid:39) x (cid:39) 2 uu U + u u + uvv + uww + x (cid:39) x x x . (13.18) 2 vu U + vuu + v v + vww + y (cid:39) y y y 2 wu U + wuu + wvv + w w z (cid:39) z z z Neglect terms in (13.17) and (13.18) that are of third order in the disturbance velocities. Now 5/31/13 13.4 bjc Compressible potential flow U ••U U ••U (cid:163) (cid:165) (cid:163) (cid:165) ((cid:97)– 1) h – --------------- (cid:162)••U – U ••(cid:162) --------------- (cid:129) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) 2 (cid:163) (cid:163)U (cid:165)(cid:165) (cid:39) ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU (cid:180)(cid:180)u + t 2 (cid:39) x (cid:164) (cid:164) (cid:166)(cid:166) 2 (cid:163) (cid:163)U (cid:165)(cid:165) (13.19) (cid:39) ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU (cid:180)(cid:180)v + t 2 (cid:39) y (cid:164) (cid:164) (cid:166)(cid:166) 2 (cid:163) (cid:163)U (cid:165)(cid:165) (cid:39) 2 ((cid:97)– 1)(cid:178)h – (cid:178)-------- + uU (cid:180)(cid:180)w – (u U + uu U + vv U + ww U ) – t 2 (cid:39) z x (cid:39) x (cid:39) x (cid:39) x (cid:39) (cid:164) (cid:164) (cid:166)(cid:166) uu U – vu U – wu U x (cid:39) y (cid:39) z (cid:39) or U ••U U ••U (cid:163) (cid:165) (cid:163) (cid:165) ((cid:97)– 1) h – --------------- (cid:162)••U – U ••(cid:162) --------------- (cid:129) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) 2 (13.20) ((cid:97)– 1)(h – uU )(u + v + w ) – (u U + uu U + vv U + ww U ) (cid:39) (cid:39) x y z x (cid:39) x (cid:39) x (cid:39) x (cid:39) –uu U – vu U – wu U x (cid:39) y (cid:39) z (cid:39) 2 Recall that ((cid:97)– 1)h = a . Equation (13.20) can be rearranged to read (cid:39) (cid:39) U ••U U ••U (cid:163) (cid:165) (cid:163) (cid:165) ((cid:97)– 1) h – --------------- (cid:162)••U – U ••(cid:162) --------------- (cid:129) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) 2 2 (13.21) a (u + v + w ) – U u – ((cid:97)+ 1)uu U – (vv U + ww U ) (cid:39) x y z (cid:39) x x (cid:39) x (cid:39) x (cid:39) ((cid:97)– 1)(uv U + uw U ) – vu U – wu U y (cid:39) z (cid:39) y (cid:39) z (cid:39) 2 Divide through by a . (cid:39) bjc 13.5 5/31/13 Compressible potential flow U ••U U ••U (cid:163) (cid:165) (cid:163) (cid:165) ((cid:97)– 1) h – --------------- (cid:162)••U – U ••(cid:162) --------------- (cid:129) (cid:164) t 2 (cid:166) (cid:164) 2 (cid:166) ((cid:97)+ 1)M 2 (cid:39) (1 – M )u + v + w – ---------------------------uu – (cid:39) x y z a x (13.22) (cid:39) M (cid:39) ---------(((cid:97)– 1)(uv + uw ) + vu + wu + vv + ww ) a y z y z x x (cid:39) This equation contains both linear and quadratic terms in the velocity disturbances and one might expect to be able to neglect the quadratic terms. But note that the first term becomes very small near M = 1. Thus in order to maintain the small (cid:39) disturbance approximation at transonic Mach numbers the uu term must be x retained. The remaining quadratic terms are small at all Mach numbers and can be dropped. Finally the small disturbance equation is ((cid:97)+ 1)M 2 (cid:39) (1 – M )u + v + w – ---------------------------uu = 0. (13.23) (cid:39) x y z a x (cid:39) The velocity potential is written in terms of a freestream potential and a distur- bance potential (cid:92) = U x +(cid:113)(x,y,z). (13.24) (cid:39) The small disturbance equation in terms of the disturbance potential becomes M 2 (cid:39) (1 – M )(cid:113) +(cid:113) +(cid:113) = ((cid:97)+ 1)---------(cid:113) (cid:113) (13.25) (cid:39) xx yy zz a x xx (cid:39) Equation (13.25) is valid over the whole range of subsonic, transonic and super- sonic Mach numbers. 13.1.3 LINEARIZED POTENTIAL FLOW If we restrict our attention to subsonic and supersonic flow, staying away from Mach numbers close to one, the nonlinear term on the right side of (13.25) can be dropped and the small disturbance potential equation reduces to the linear wave equation. 5/31/13 13.6 bjc Compressible potential flow 2 .(cid:96) (cid:113)xx – ((cid:113) +(cid:113) ) = 0 (13.26) yy zz 2 where (cid:96) = M – 1. In two dimensions (cid:39) 2 (cid:96) (cid:113)xx –(cid:113) = 0. (13.27) yy The general solution of (13.27) can be expressed as a sum of two arbitrary functions (cid:113)(x,y) = F(x –(cid:96)y) + G(x +(cid:96)y). (13.28) 2 Note that if M < 1 the 2-D linearized potential equation (13.27) is an elliptic (cid:39) equation that can be rescaled to form Laplace’s equation and (13.28) expresses the solution in terms of conjugate complex variables. In this case the subsonic flow can be analyzed using the methods of complex analysis. Presently we will restrict our attention to the supersonic case. The subsonic case is treated later in the chapter. 2 If M > 1 then (13.27) is the 2-D wave equation and has solutions of hyperbolic (cid:39) type. Supersonic flow is analyzed using the fact that the properties of the flow are constant along the characteristic lines x ±(cid:96)y = constant. The figure below illustrates supersonic flow past a thin airfoil with several characteristics shown. Notice that in the linear approximation the characteristics are all parallel to one another and lie at the Mach angle µ of the free stream. Information about the (cid:39) flow is carried in the value of the potential assigned to a given characteristic and in the spacing between characteristics for a given flow change. Right-leaning characteristics carry the information about the flow on the upper surface of the wing and left-leaning characteristics carry information about the flow on the lower surface. bjc 13.7 5/31/13 Compressible potential flow x –(cid:96)y = constant y µ (cid:39) x M > 1 (cid:39) µ = Sin-1(1/M ) (cid:39) (cid:39) x +(cid:96)y = constant All properties of the flow, velocity, pressure, temperature, etc. are constant along the characteristics. Since disturbances only propagate along downstream running characteristics we can write the velocity potential for the upper and lower surfaces as (cid:113)(x,y) = F(x –(cid:96)y) y>0 . (13.29) (cid:113)(x,y) = G(x +(cid:96)y) y<0 Let y = f(x) define the coordinates of the upper surface of the wing and y = g(x) define the lower surface. The full nonlinear boundary condition on the upper surface is v df ---- = ------. (13.30) U dx y = f In the spirit of the thin airfoil approximation this boundary condition can be approximated by the linearized form v df -------- = ------ (13.31) U dx (cid:39) y = 0 which we can write as 5/31/13 13.8 bjc Compressible potential flow (cid:44)(cid:113)(x,y) df (cid:163) (cid:165) -------------------- = U ------ (13.32) (cid:44)y (cid:39)(cid:164)dx(cid:166) y = 0 or U (cid:39)(cid:163)df(cid:165) F'(x) = –-------- ------ . (13.33) (cid:96) (cid:164)dx(cid:166) On the lower surface the boundary condition is U (cid:39)(cid:163)dg(cid:165) G'(x) = -------- ------ . (13.34) (cid:96) (cid:164)dx(cid:166) In the thin airfoil approximation the airfoil itself is, in effect, collapsed to a line along the x-axis, the velocity potential is extended to the line y = 0 and the sur- face boundary condition is applied at y = 0 instead of at the physical airfoil surface. The entire effect of the airfoil on the flow is accounted for by the vertical velocity perturbation generated by the local slope of the wing. The linearized boundary condition is valid on 2-D thin wings and on 3-D wings of that are of “thin planar form”. Recall that (13.26) is only valid for subsonic and supersonic flow and not for tran- 2 sonic flow where (1 – M ) « 1. (cid:39) 13.1.4 THE PRESSURE COEFFICIENT Let’s work out the linearized pressure coefficient. The pressure coefficient is P – P (cid:39) 2 (cid:163) P (cid:165) C = ------------------- = ------------ ------- – 1 . (13.35) P 1 2 2 (cid:164)P (cid:166) ---(cid:108) U (cid:97)M (cid:39) 2 (cid:39) (cid:39) (cid:39) The stagnation enthalpy is constant throughout the flow thus T 1 2 2 2 2 ------- = 1 + ------------------(U – (U + v + w )). (13.36) T 2C T (cid:39) (cid:39) p (cid:39) Similarly the entropy is constant and thus the pressure and temperature are related by bjc 13.9 5/31/13 Compressible potential flow (cid:97) ------------ P (cid:163) 1 2 2 2 2 (cid:165)(cid:97)– 1 ------- = 1 + -------------------(U – (U + v + w )) (13.37) P (cid:164) 2C T (cid:39) (cid:166) (cid:39) P (cid:39) and the pressure coefficient is (cid:97) (cid:168) ------------ (cid:172) 2 (cid:171)(cid:163) 1 2 2 2 2 (cid:165)(cid:97)– 1 (cid:171) C = ------------(cid:169) 1 + ------------------(U – (U + v + w )) – 1(cid:173). (13.38) P 2 (cid:164) 2C T (cid:39) (cid:166) (cid:97)M (cid:171) p (cid:39) (cid:171) (cid:39) (cid:170) (cid:174) The velocity term in (13.37) is small 2 2 2 2 2 2 2 U – (U + v + w ) = –(2uU + u + v + w ). (13.39) (cid:39) (cid:39) n 2 Now use the binomial expansion (1 –(cid:161)) (cid:129)1 – n(cid:161)+ n(n – 1)(cid:161)  2 to expand the term in parentheses in (13.39). Note that the expansion has to be carried out to second order. The pressure coefficient is approximately 2 2 2 (cid:163) 2u 2 u v + w (cid:165) C (cid:129) –(cid:178)-------- + (1 – M )-------- + -------------------(cid:180) . (13.40) P U (cid:39) 2 2 (cid:164) (cid:39) U U (cid:166) (cid:39) (cid:39) Equation (13.40) is a valid approximation for small perturbations in subsonic or supersonic flow. For 2-D flows over planar bodies it is sufficient to retain only the first term in (13.40) and we use the expression u C (cid:129) –2--------. (13.41) P U (cid:39) For 3-D flows over slender approximately axisymmetric bodies we must retain the last term and so 2 2 (cid:163) 2u v + w (cid:165) C (cid:129) –(cid:178)-------- + -------------------(cid:180) . (13.42) P U 2 (cid:164) (cid:39) U (cid:166) (cid:39) 5/31/13 13.10 bjc