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Supersonic Thin Airfoil Theory Aa200b Lecture 5 January 20, 2005

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Supersonic Thin Airfoil Theory Aa200b Lecture 5 January 20, 2005 Supersonic Thin Airfoil Theory AA200b Lecture 5 January 20, 2005 AA200b - Applied Aerodynamics II 1 AA200b - Applied Aerodynamics II Lecture 5 Foundations of Supersonic Thin Airfoil Theory The equations that govern the steady subsonic and supersonic flow of an inviscid fluid around thin bodies can be shown to be very similar in appearance to Laplace’s equation @2Á @2Á + = 0; (1) @x2 @y2 which we have used in previous lectures for incompressible flow. In fact, the equations for a free stream Mach number different from zero are given by @2Á @2Á (1 ¡ M 2 ) + = 0: (2) 1 @x2 @y2 Notice that although Eqns. 1 and 2 look very similar, a fundamental change in behavior occurs as the free stream Mach Number, M1, increases past 2 AA200b - Applied Aerodynamics II Lecture 5 the value of 1: the PDE goes from elliptic to hyperbolic and it takes on a wavelike behavior. Eqn. 2 governs the behavior of a supersonic flow where only small disturbances have been introduced. You should already be familiar with a number of results for two- dimensional, isentropic, supersonic flow such as the Prandtl-Meyer function (weak reversible turning analysis, both positive and negative). These notes expand on that knowledge an apply it to the computation of forces and moments on supersonic airfoils. The main difference when compared to incompressible thin airfoil theory is the appearance of drag (in an inviscid flow). The source of this drag is the radiation to infinity of pressure waves created by disturbances at the surface of the airfoil. We will call this drag wave drag. 3 AA200b - Applied Aerodynamics II Lecture 5 Use of 3D Weak-Turning Wave Theory It can be shown that for small angle changes, either compressive or expansive, one can relate flow speed and static pressure changes to the infinitesimal flow angle changes: du dθ = ¡p (3) u M 2 ¡ 1 dp γM 2dθ = +p : p M 2 ¡ 1 The idea of thin airfoil theory is to use these expressions directly for finite- but-small angle changes imposed on the incoming flow by airfoil surface shapes and angle of attack. Thus, if the airfoil is thin enough and the angle of attack is small enough, one may expect to be able to approximate flow 4 AA200b - Applied Aerodynamics II Lecture 5 property variations in the forms ∆w ∆θ = ¡p (4) 2 w M1 ¡ 1 ∆p γM 2 ∆θ = +p 1 ; 2 p M1 ¡ 1 where ∆p ´ p ¡ p1 ∆w ´ w ¡ U1; and where,p if (u; v) are the (x; y)-components of the 2D velocity then w ´ u2 + v2. Also, in these expressions, the local Mach number, M, has been replaced by M1, the incoming Mach number, since the variations 5 AA200b - Applied Aerodynamics II Lecture 5 in M are themselves expected to be small for thin airfoils at low angles of attack. The situation is illustrated in Fig. 1 for a thin biconvex airfoil at zero angle of attack. Addition of a small angle of attack ® of order ¡1 t t tan c » c would not change the figure substantially. 6 AA200b - Applied Aerodynamics II Lecture 5 − dyu ∆θu = dx (x) yu(x) ∆θ(x) t p1 0 c M1 yl(x) dyl ∆θl = dx (x) c Figure 1: Thin Curved Airfoil in Supersonic Flow. 7 AA200b - Applied Aerodynamics II Lecture 5 ¡1 t t In order for the airfoil to be thin we require that tan c » c, i.e. t << 1; c where t and c are illustrated in the figure, and, of course, if there were an angle of attack we would also require that ® << 1 (radians): The condition that ∆θ is everywhere small implies further that, in thin airfoil applications v v tan ∆θ = » << 1; (5) u U1 whereas Eqn. 4, together with Eqn. 5, implies that ∆w » ∆u » ∆u << 1. w w U1 Thus, in thin airfoil applications one need not distinguish between u and w 8 AA200b - Applied Aerodynamics II Lecture 5 as they occur in Eqns. 4-5. It is also true in the same limit that ∆½ M 2 ∆θ = p 1 (6) 2 ½ M1 ¡ 1 ∆c γ ¡ 1 ∆θ = M 2 p ; 1 2 c 2 M1 ¡ 1 and µ ¶ ∆M γ ¡ 1 ∆θ = ¡ 1 + M 2 p : (7) 1 2 M 2 M1 ¡ 1 In all of these expressions, ∆θ is to be taken positive in the compressive sense. Also, because ∆θ must be small for thin airfoil theory to hold, we can write sin ∆θ ¼ tan ∆θ ¼ ∆θ; cos ∆θ ¼ 1 when convenient. Throughout this approximate theory any given quantity is known or can be calculated only to its lowest order in ∆θ. A variation in 9 AA200b - Applied Aerodynamics II Lecture 5 pressure, for example, is linear in ∆θ; terms of order (∆θ)2 and higher can and should be consistently neglected. Similarly, a quantity such as airfoil drag is inherently a product of two linear quantities and thus of order (∆θ)2; for such quantities, terms of order (∆θ)3 and higher are to be neglected. 10 AA200b - Applied Aerodynamics II Lecture 5 Airfoil Performance: Approximate Thin-Airfoil Version As sketched in Fig. 1, the upper and lower surfaces of the airfoil are located at y = yu(x) y = yl(x): On the upper surface, ∆θ is positive (compressive) for yu(x) increasing; thus, on that surface dy ∆θ = u: u dx On the lower surface, by contrast, ∆θ is in the compressive sense when yl(x) becomes increasingly negative. In that case dy ∆θ = ¡ l: l dx 11 AA200b - Applied Aerodynamics II Lecture 5 Using Eqn. 5 together with the expressions above, we have that ∆p p ¡ p γM 2 dy u = u 1 = p 1 u (upper surface) (8) 2 p1 p1 M1 ¡ 1 dx ∆p p ¡ p γM 2 dy l = l 1 = p 1 l (lower surface) 2 p1 p1 M1 ¡ 1 dx In aerodynamic applications it is customary to introduce the dimensionless pressure coefficient, Cp, defined by p ¡ p C ´ 1 : (9) p 1 2 2½1U1 Note that 1 γM 2 ½ U 2 = 1p ; 2 1 1 2 1 12 AA200b - Applied Aerodynamics II Lecture 5 and therefore, for M 6= 0, one can relate ∆p and the pressure coefficient 1 p1 conveniently by 2 ∆p Cp = 2 : γM1 p1 Applied to the pressure variations on the upper and lower surfaces of the airfoil, this gives 2 dyu Cp = p (10) u 2 M1 ¡ 1 dx 2 dyl Cp = ¡p : l 2 M1 ¡ 1 dx Knowing the approximate surface pressures, this puts us in a position to be able to calculate both the lift and the drag of the airfoil, to the same approximation. For example, an element at x on the upper surface, of area 13 AA200b - Applied Aerodynamics II Lecture 5 dx per unit depth into the page, contributes an amount cos(∆θu(x)) µ ¶ dx dlu = ¡pu cos ∆θu = ¡pudx (11) cos ∆θu to the airfoil lift per unit span. Similarly, an element on the lower surface contributes dll = +pldx: Thus, the lift/unit span is Z c l = (pl(x) ¡ pu(x)) dx: (12) 0 In coefficient form, this can be shown to be Z 1 c ¡ ¢ Cl ´ Cpl(x) ¡ Cpu(x) dx (13) c 0 14 AA200b - Applied Aerodynamics II Lecture 5 Using our approximate formulas for Cpu and Cpl we can get an explicit expression for the thin-airfoil lift coefficient. Z c µ ¶ 2 1 dyl dyu Cl = ¡p + dx 2 M1 ¡ 1 c 0 dx dx · ¸ 2 y (c) ¡ y (0) y (c) ¡ y (0) = ¡p l l + u u ; 2 M1 ¡ 1 c c or, since yl(c) = yu(c) and yl(0) = yu(0) for a closed-contour airfoil 4 yu(c) ¡ yu(0) 4 Cl = ¡p = +p ®; (14) 2 2 M1 ¡ 1 c M1 ¡ 1 yu(c)¡yu(0) since c = tan ® ¼ ®. The angle of attack ® is in radians. This is a classical result in aerodynamics and provides a measure, for 15 AA200b - Applied Aerodynamics II Lecture 5 supersonic airfoils, of achievable lift for wings at small angles of attack. This approximation works rather well, even when extended to three dimensions, to predict the lift curve slope of a wing, particularly one where the sections behave in a two-dimensional fashion (issues of subsonic and supersonic leading edges and regions of influence need to be considered here). A review of the derivation clearly shows that the lift coefficient is independent of the airfoil profile shape: the lift coefficient in supersonic flow, for a given angle of attack ® is the same for a flat plate, a diamond airfoil, or a biconvex airfoil. 16 AA200b - Applied Aerodynamics II Lecture 5 Wave Drag Calculation We have just seen that in supersonic thin airfoil theory, the the lift coefficient is independent of airfoil shape. Airfoil drag, however, is another matter; this depends strongly on the shape of the airfoil. To obtain its value, note that an element on the upper surface contributes to the drag amount (per unit span) dx dyu d(drag) = pu sin ∆θu = pu tan ∆θudx = pu dx (upper surface); cos ∆θu dx and, similarly dy d(drag) = ¡p ldx (lower surface): l dx 17 AA200b - Applied Aerodynamics II Lecture 5 Integrating over the chord we obtain Z c µ ¶ dyu dyl drag/unit span = pu ¡ pl dx; 0 dx dx which in coefficient form can be shown to be Z c µ ¶ 1 dyu dyl Cd = Cpu ¡ Cpl dx: (15) c 0 dx dx Using our expressions from the thin airfoil limit we can show " # Z c µ ¶2 µ ¶2 2 1 dyu dyu Cd = p + dx (thin airfoil) (16) 2 M1 ¡ 1 c 0 dx dx 18 AA200b - Applied Aerodynamics II Lecture 5 For the case of a flat plate at an angle of attack ® 4®2 Cd = p ; (17) 2 M1 ¡ 1 while for a diamond airfoil with half angle ¿ 4(®2 + ¿ 2) Cd = p : (18) 2 M1 ¡ 1 For a curved biconvex airfoil, Cd must be calculated for each specified shape.
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