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Chapter 12 Wings of Finite Span

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Chapter 12 Wings of Finite Span Equation Chapter 12 Section 1 CHAPTER 12 WINGS OF FINITE SPAN ___________________________________________________________________________________________________________ 12.1 FLOW OVER A THREE-DIMENSIONAL WING The images below from Van Dyke’s Album of Fluid Motion depict low speed flow over a lifting wing of finite span. The images include a view from the side, a plan view from above and a series of images showing the vortex wake in a series of planes normal to the wing wake at increasing downstream distances from the wing. The span of the wing is b and the chord is called C . (a) (b) (c) Figure 12.1 Images of the flow past a finite span wing at low speed. From An Album of Fluid Motion by M. Van Dyke. 1 The wings are not all the same. Figure 12.1a shows a wing in water visualized by colored dye. Figure 12.1b shows smoke lines in the flow over a wing in air. And figure 12.1c shows the wake of a wing towed in water visualized by very small hydrogen bubbles emitted by a fine wire just downstream of the wing trailing edge. These images beautifully illustrate the three-dimensional flow over the wing. In each case the wing has lift leading to reduced pressure above the wing and increased pressure below compared to the free stream pressure. The consequences of this pressure difference are illustrated in the figures. The dyelines in Figure 12.1a emanate from the boundary layer on the lower surface of the wing. At the wing tip, vorticity from the lower and upper surface of the wing separates and rolls up to form a trailing vortex. Figure 12.1b illustrates the same process as viewed from above looking down on the suction surface of the wing. Smoke lines introduced ahead of the wing are positioned so that the lines pass just below the wing. The upstream influence of the elevated pressure below the wing leads to a divergence of the smoke lines ahead of the wing. If the smoke lines had been positioned to pass above the wing one would instead see a convergence driven by the low pressure on the suction surface. This very important effect can also be clearly seen in the smoke lines that leave the trailing edge of the wing and diverge outward to join the vortex rollup from the wing tips. Figure 12.2 schematically shows the flow on the centerline of a three-dimensional wing. Figure 12.2 – Velocity field normal to a wing comprising a transverse bound vortex of circulation Γ plus downwash generated by a semi-infinite system of free vortices in the wake. 2 Near the wing the bound circulation due to lift leads to an up-wash ahead of the wing and downwash behind the wing similar to the flow produced by a two-dimensional lifting wing of infinite span. A very important effect is generated by the flow due to the vortex pair that comprises the wake. The semi-infinite sheet of vorticity distributed in the wake produces a downward velocity component in the free-stream ahead of the wing, at the wing and far downstream as illustrated in Figure 12.3. Figure 12.3 Upstream and downstream effect of the wake of a finite span lifting wing. The downwash by the wake leads to a reduction in the angle-of-attack of the wing relative to the free stream, reducing the lift. In addition the downwash rotates the oncoming flow vector at the wing leading to a component of drag as shown in Figure in 12.3. The change in angle of attack due to the downwash generated by the wake is ⎛ Uz (0,0,0)⎞ (12.1) αi = ArcTan⎜ ⎟ < 0 ⎝ U∞ ⎠ where Uz (0,0,0) is the z component of velocity induced by the vortex system of the wake at the mid span of the wing. The downwash produced by the circulation of the bound vortex of the wing is zero at the wing and therefore does not contribute to the induced drag. The span-wise divergence and convergence of the flow at the wing trailing edge seen in Figure 12.1b is illustrated in Figure 12.4. Figure 12.4 Span-wise flow in a plane perpendicular to the wing trailing edge 3 The general direction of the flow about the wing is from high pressure to low pressure. At the wing trailing edge where the pressures above and below the wing are equal the stream-wise vorticity produced by the spanwise flow over the wing convects into the wake where it feeds the rolling up vortices as can be seen in figure 12.1b. The connection between the bound vorticity on the wing and the free vorticity in the wake can be determined using the contour shown in Figure 12.5. (a) (b) Figure 12.5 Contour used to connect the circulation bound to a lifting wing, the span- wise flow at the wing trailing edge and free vorticity in the wake. The idea behind Figure 12.5 is that the circulation about a contour that lies entirely in a potential flow is zero since the potential is continuous over such a contour. UicˆdC = ∇ΦicˆdC = dΦ = Φ final − Φinitial = 0 (12.2) !∫ !∫ !∫ Follow the arrows closely in Figure 12.5a. The contour CA passes over the wing in a downstream direction and, before reaching the vortex sheet, connects to contour CB . Contour CB then turns 90 degrees and upward almost reversing direction. From just above the vortex sheet CB passes around and below the wake vortex sheet approaching the sheet from below. Contour CB connects to CA then turns back 90 degrees, nearly 4 reversing direction again and passes under the wing where CA closes. At no point does the contour penetrate the vortex sheet that is discontinuous in the potential because of the discontinuity in span-wise velocity across the sheet. Using (12.2) we can write !∫ UicˆdC = ∫ UicˆdC + ∫ UicˆdC = 0 (12.3) CA CB To clarify this idea, in Figure 12.5b the contours are imagined to be shrunk-wrapped about the wing and the trailing edge vortex sheet while remaining imbedded entirely in potential flow. Equation (12.3) implies ΓTrailingEdge (y) = −ΓWing (y) (12.4) By the way it is not necessary to shrink-wrap the contour to get to this result. Figure 12.4a is enough, either contour gets to the result (12.4). 12.2 CIRCULATION AND PRESSURE Figure 12.6 shows a cross-section of a finite-span wing moving at a constant velocity U∞ in an incompressible flow. Figure 12.6 Airfoil cross-section The dashed line with length C connects the leading edge of the wing to the trailing edge and is called the chord line. The section lift (lift per unit span) is ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ dL C(y) ⎜ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎟ = ⎜ PLower − PUpper ⎟dx (12.5) ∫0 ⎜ 2 1/2 ⎟ ⎜ 2 1/2 ⎟ dy ⎜ ⎛ ⎛ dz ⎞ ⎞ ⎜ ⎛ ⎛ dz ⎞ ⎞ ⎟ ⎟ ⎜ 1+ Lower ⎟ 1+ Upper ⎜ ⎜ ⎜ ⎝⎜ dx ⎠⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ ⎝⎜ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎝ dx ⎠ ⎠ ⎠ ⎠⎟ where zLower (x, y) and zUpper (x, y) are the z coordinates of the lower and upper surfaces of the wing at the spanwise position y . The Bernoulli equation is 5 1 2 1 2 1 2 P∞ + ρU∞ = PLower + ρ(ULower ) = PUpper + ρ UUpper (12.6) 2 2 2 ( ) If the airfoil is relatively thin we can approximate the pressure by 1 P + ρU 2 ≅ P + ρU u = P + ρU u (12.7) ∞ 2 ∞ Lower ∞ Lower Upper ∞ Upper The lift per unit span is then dL C = ρU uUpper − uLower dx = ρU Γ(y) (12.8) dy ∞ ∫0 ( ) ∞ where Γ(y) is the circulation about the airfoil at the section y . Γ = UicˆdC (12.9) !∫C 12.3 FORCES AND MOMENTS ON A WING Figure 12.7 shows the relationship between the various forces that act on a differential slice dy at some spanwise location on a three-dimensional wing with span b . Fig 12.7 Differential forces on a section of a 3-D wing The differential normal force at a given spanwise location is dF⊥ (y) = ρUR (y)Γ(y)dy (12.10) The differential lift and induced drag at the position y are U∞ dL(y) = dF⊥ (y)Cos(αi ) = dF⊥ (y) = ρU∞Γ(y)dy (12.11) UR (0, y,0) 6 and Uz (0, y,0) dDi (y) = −dF⊥ (y)Sin(αi ) = −dF⊥ (y) = −ρUz (0, y,0)Γ(y)dy (12.12) UR (0, y,0) The induced flow angle is generally very small so we can apply the approximation αi ≅ Uz (0, y,0) /U∞ . The differential induced drag is related to the differential lift by dDi (y) = −αi (y)dL(y) (12.13) The lift and drag are found by integrating the above differential relations b/2 L = ρU∞ Γ(y)dy (12.14) ∫−b/2 b/2 Di = −ρ Uz (0, y,0)Γ(y)dy (12.15) ∫−b/2 The rolling moment vector is aligned with the positive x-axis and also involves integrating the circulation. b/2 b/2 M x = M Roll = ydL(y) = ρU∞ yΓ(y)dy (12.16) ∫−b/2 ∫−b/2 The yawing moment is aligned with the positive z-axis and, like drag, its integral involves the circulation and downwash velocity. b/2 b/2 M z = MYaw = − ydDi (y) = ρ yUz (0, y,0)Γ(y)dy (12.17) ∫−b/2 ∫−b/2 12.4 LIFTING LINE THEORY Assume that the flow over a finite wing can be treated as locally two-dimensional with the flow geometry depicted in Figure 12.8 applied at each position y along the span of the wing.
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