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### Theme 3

1. 1. SECTION 1. AERODYNAMICS OF LIFTING SURFACES THEME 3. FEATURES OF WINGS FLOW The features of wings flow are connected with interaction of flows on the lowerand upper surfaces. The features of flow depend on Mach numbers M ∞ ( M ∞ < 1 orM ∞ > 1 ), sweep angles, angles of attack and other parameters. Lets consider an influence of these parameters onto process of wings flow. 3.1. Subsonic speeds of wings flow M ∞ < 1 . 3.1.1. Unswept high-aspect-ratio wings The features of unswept high-aspect-ratio wings flow are determined by overflowfrom the lower surface to the upper surface at the wing tips. The appearance of wingspanwise flow is due to the fact that the pressure on the upper surface is less thanpressure on the lower surface. So, we shall consider a finite-span wing which is streamlined by a straight-line flow having a constant velocity. If the wing creates the lift force, then there is a zone of reduced pressure above a wing, and under a wing is a zone of increased pressure (Fig. 3.1). Under influence of pressure difference there is an air overflow passing through wing tip edges from area of the increased pressure into area of reduced pressure. A flow parallel to wing Fig. 3.1. Flow lines curvature span appears. The cross flow along wing span caused by cross flow is more intensive at the wing tips andattenuates to central cross-section. This flow caused a curvature of flow lines: on theupper surface towards the plane of symmetry, and on the lower - to the wing tips. 21
2. 2. Lets note, that the flow of a finite-span wing is not two dimensional parallel flow,but three-dimensional, especially near its tips. At that wing tips effect the whole surface,therefore aerodynamic characteristics of a finite-span wing differ from the aerodynamiccharacteristics of an airfoil While calculating the aerodynamic characteristics of high-aspect-ratio wings( λ > 6 ) it is possible, in a whole, to neglect the mentioned above flow curvature and touse a hypothesis of flat cross-sections. The hypothesis of flat cross-sections assumes,that each cross-section of a wing is streamlined by its own two-dimensional parallelflow. In particular, the wing lifting force in this case is determined by summation of liftof wing cross-sections, which is calculated under the Zhukovsky formula: l 2 Ya = ρ∞V∞ ∫ Г ( z )dz , −l (3.1) 2Where Г ( z ) is the circulation of speed along the contour covering a wing in cross-section of a chosen element. The vortex sheet is formed as a result of interaction of the upper and lower flowsbehind the wing. The vortex sheet consists of vortex threads, which occur due to variousdirection of speeds on the trailing edge at the approach to the upper and lower surfaces,however, values of these speeds are equal (postulate by Zhukovsky-Chaplygin) (Fig.3.2,a). The vorticity value is decreased with approaching to a plane of symmetry. In aninviscid flow the vortex sheet behind a wing reaches infinity and at small angles ofattack (in the linear theory) is directed along speed of incoming flow. In real conditions the vortex sheet is unstable and is turned off in two high-powervortex cores (Fig. 3.2,b), which for transport airplanes stretch for tens kilometers. Whenlight airplanes happen to be in a bundle track behind a heavy airplane it can result in itscrash (failure). 22
3. 3. Fig. 3.2. Formation of a vortex sheet behind a wing. The vortex sheet induces behind a wing a velocity field Vi (Fig. 3.3,a), whichdeflects an undisturbed flow on an angle ε , which is called as downwash angle Vi ε ≈ tg ε = (3.2) V∞And the angle of attack of section (cross-section) receives the following true value α real = α − ε . (3.3) If it may be assumed , that in an inviscid flow the total aerodynamic force whicheffecting wing cross-section is perpendicular to the true flow velocityVreal = V∞ + Vi2 , and the lift force is perpendicular to V∞ , then a force component 2along the direction of incoming flow velocity appears (Fig. 3.3,b). Fig. 3.3. Occurrence of a downwash behind a wing. 23
4. 4. This force is called as force of induced drag l 2 X i = ρ∞ ∫ Vi ( z ) Г ( z )dz . −l (3.4) 2 The occurrence of induced drag is the important point in the course of studyingthe features of finite-aspect-ratio wings flow. Naturally the question of obtaining minimum value of induced drag X i atspecified values of lift force Ya and wing span l emerges. This problem is ofvariational type, which solution is got in case of a constancy of induced speed Vispanwise. Thus the distribution of speed circulation Г ( z ) should have the elliptical law(for a wing in a boundless flow). What should an optimum wing be? The answer to this question is an ambiguous: • chords of a flat wing should change under the elliptical law (elliptical wing plan form); • twist spanwise should vary under the elliptical law for the rectangular wing plan form; • the minimum value of induced drag is also reached by twist application which law depends on the wing plan form for a wing of any form. 3.1.2. Swept high-aspect-ratio wings There is an effect of slipping on swept wing in addition to the considered abovefeatures: an additional curvature of flow lines caused by spanwise flow from centre tothe wing tips (for sweepback) appears. Lets consider a slipping wing. We shall write down undisturbed speed ofincoming flow as: V∞ = Vn + Vτ2 , where Vn is the velocity component normal to the 2leading edge; Vτ is the tangent component (fig. 3.4). 24
5. 5. The wing disturbs a flow. Lets designate tangent Vτ∗ and normal Vn velocity ∗ ∗components of a perturbed flow at a wing surface and we shall assume that Vnanywhere does not reach value of sound velocity. Tangent component is constant: Vτ∗ = Vτ = const and Vn in normal cross- ∗ section varies in a flow in the same way as the flow above airfoil in this cross-section. ∗ Summing Vτ and Vn , we receive total speed V and we find flow lines. The effect of slipping on swept wing is realized in less degree in the middle and tip area because of Vτ ≠ const (smaller values in central and tip areas). Finite span and the existence of a plane of symmetry result in redistribution of pressure on wing airfoils in various cross-sections. At the wing tips the peak of rarefaction on the upper surface isFig. 3.4. Flow lines on a slipping wing: increased and displaces forward as a result ofa) - the angle of attack is equal to zero; slipping effect and overflow influence b) - positive angle of attack (fig. 3.5, b). Simultaneously positive pressuregradient considerably increases behind peak of rarefaction (along flow). Pressuredifference decreases near a plane of symmetry at the leading edge (fig. 3.5, b). As aresult the peak of rarefaction displaces back. The positive pressure gradient behind peakof rarefaction is reduced. On a swept-forward wing (fig. 3.5, c) the peak of rarefaction grows and displacesforward in a plane of symmetry. The positive pressure gradient behind it increases alsowith the peak of rarefaction growth. 25
6. 6. Fig. 3.5. Sweep influence on load distribution chordwise: a) - unswept wing; b) - sweepback wing; c) - swept-forward wing. Existence of the spanwise wing flow to the wing tip and the displacement ofrarefaction peak to the leading edge results in probability of flow stalling on the swept-back wing tips. Aerodynamic fences or leading edges fractures are used to prevent flowstall at the wing tips (Fig. 3.6). Fig. 3.6. The design solutions for prevention of flow stall: a) - aerodynamic fence; b) - aerodynamic “dog-tooth”; c) - “slit ((saw)kerf)” 26
7. 7. 3.1.3. Small-aspect-ratio wings Additional effects are essential influence of cross-sectional flow spanwise ontolongitudinal flow (curvature of flow lines) and occurrence of vortex structures above awing descending from lateral and leading edges. The nonlinearity of the characteristicsC ya = f (α ) and m z = f (α ) is the result of effects mentioned above. The narrowing of flow lines on the upper surface ( causes occurrence of additional speeds + ΔVupper and ) pressure decreasing; the divergence of flow lines on the lower surface reduces flow rates ( − ΔVlower ) and increases pressure (Fig. 3.7). An additional lift appears. It is caused by influence of cross-sectional flow onto longitudinal flow. Cross-sectional flow does not create lift itself, its influence has an effect on friction drag. The additional contribution to the aerodynamic Fig. 3.7. characteristics is introduced by vortex structures abovethe wing (Fig. 3.8), in particular, playing a role of washers interfering pressurecompensation between the lower and upper surfaces. At angles of attack increasingdependencies C ya = f (α ) and m z = f (α ) also become non-linear (Fig. 3.9). In ageneral the lift coefficient value is possible to represent as the sum of two items: C ya = C ya line + ΔC ya , (3.5)Where C ya line the lift coefficient is determined without the account of effects of small-aspect-ratio wing; ΔC ya is non-linear additive; 27
8. 8. Fig. 3.8. Fig. 3.9. It is possible to achieve the increasing of non-linear additive caused by formationof vortex structures above the wing by application of curvilinear edges, saws etc. 3.2. Supersonic velocities of wings flow M ∞ > 1 . The features of wings flow are determined by basic property of supersonic flows -existence of influence areas limited by Mach cones (Fig. 3.10). Fig. 3.10. A Mach cone. The areas of influence divide wing edges into subsonic and supersonic ones,with various flow features. Leading edges. Subsonic edges. Velocity component (Mach number M ∞ ) perpendicular to theleading edge is subsonic one ( M ∞n = M ∞ cos χ l .e . < 1 ). Suction force appears at edgeflow, so it is necessary to apply rounded edges (subsonic airfoils) to the greater 28
9. 9. realization of suction force and smooth flow or flap sharp edges providing shock-freeflow (Fig. 3.11). Lets define parameter n describing character of edge flow. The leading edge will be subsonic if the following conditions are satisfiedχ l .e . > 90 °− μ ∞ , or tgχ l .e . > tg(90 °− μ ∞ ) and 2 tgχ l .e . > M ∞ − 1 . We have 2n = M ∞ − 1 ctgχ l .e . < 1 . For example, for a triangular wing the aspect ratio is equal to λ = 4ctgχ l .e . , so we 1 2receive n = λ M∞ − 1 < 1 . 4 Supersonic edges. Velocity component perpendicular to the leading edge issupersonic one and M ∞ cos χ l .e . > 1 . Edges should be sharp (supersonic airfoil) forwave drag decreasing (Fig. 3.12). 2 We have for a supersonic leading edge n = M ∞ − 1 ⋅ ctgχ l .e . > 1 . Fig. 3.11. A subsonic leading edge Fig. 3.12. A supersonic leading edge The concepts of subsonic, sound and supersonic lateral and trailing edges offinite-span wing are analogous ones. 29
10. 10. Lateral edges. Subsonic edge. There is an overflow and pressure compensation between the lower and upper surfaces (analogously to subsonic flow). Supersonic edge. The upper and lower surfaces are separately streamlined without Fig. 3.13. A subsonic and supersonic mutual influence. lateral edge. Trailing edges. Subsonic edge. The postulate Chaplygin-Zhukovsky about flow stall from thetrailing edge and relations Vupper = Vlower and ΔC р = C р lower − C р upper = 0 shouldbe executed that is conditioned by mutual influence of the upper and lower surfaces ofthe trailing edge (Fig. 3.14). Supersonic. The upper and lower surfaces are separately streamlined. The flowdepartures from the trailing edge without the requirement of fulfillment of Zhukovsky-Chaplygin postulate (Fig. 3.15). Fig. 3.14. A subsonic trailing edge Fig. 3.15. A supersonic trailing edge It is necessary to note, that the concepts of subsonic and supersonic edges areconnected both with wing plan form, and with Mach number M ∞ . The edge can besubsonic or supersonic, depending on Mach number M ∞ at specified wing geometry. 30