空气动力学英文课件Chapter_05

nullChapter 5Chapter 5Incompressible Flow Finite Wings5.1 Introduction5.1 IntroductionThe properties of airfoils are the same as the properties of a wing with infinite span. However, all real airplanes have wings of finite span.In the present chapter, we will apply our knowledge of airfoil properties to the analysis for finite wings. As we have mentioned in the previous chapter, the analysis for the aerodynamics of wings is separated in two steps. Now, we are going on the second step in Prandtl’s philosophy of wing theory.nullnullnullQuestion : why are the aerodynamic characteristics of a wing any different from the properties of its section? Airfoil can be respected as two-dimensional body, but any finite wing is a three-dimensional body.※Attention should be paid for the flow pattern near the wing tips. And try to understand the reason for such a flow phenomena. ※ For general cases, there is a spanwise velocity component on both top and bottom surface of the wing, but their direction are different. ※ A trailing vortex is created at each wing tip. These wing-tip vortices downstream of the wing induce a small downward velocity in the neighborhood of the wing itself.null※ The two vortices tend to drag the surrounding air with them, and this secondary movment induces a small component is called downwash(下洗）.※ The downwash velocity combines with the freestream velocity to produce a local relative wind which is canted downward in the vicinity of each airfoil section of the wing.※ definition of induced angle of attacknullnullnull※ Two important effects due to the downwash.1 the angle of attack actually seen(or feel) by the local section is the angle between the chord line and the local relative wind. This angle is defined by 2 The local lift vector is in the direction perpendicular to the local relative wind. As a subsequence, there is a drag created by the presence of downwash.nullnullConclusion: the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag ---- the induced drag . And this drag is not produced by viscous friction.Explanation of induced drag in physical sense a wing-tip vortices destroy the net pressure balance b the wing-tip vortices contain large amount of translational and rotational energy, and this energy serves no useful purpose. In effect, the extra power should be provided by the engine to overcome the the induced drag.null※ Road map and purpose of this chapter※ Support points for our analysis 1. Curved vortex filament 2. Biot-Savart Law 3. Helmholtz’s vortex theoremsnull5.2 The Vortex Filament, The Biot-Savart Law, And Helmholtz’s TheoremsThe vortex filamentnullBiot-Savart LawApplication of Biot-Savart Law 1. To a straight vortex filament of infinite length null 2. To a semi-infinite vortex filamentnullHelmholtz’s TheoremsFor inviscid and incompressible flows 1. The strength of a vortex filament is constant along its length. 2. A vortex filament cannot end in a fluid; it must extend to the boundary of the fluid or form a closed path.Concept of lift distribution along the span of a finite wing1. : lift per unit span at location nullnull2. With the different location in the span direction, the chord and attack angle may be different, that means will be different. 3. Concept of geometric twist. washout and washin. has a distribution along the span direction4. Concept of aerodynamic twist. has a distribution along the span direction5. As there is different distribution of the chord of the airfoil, geometric angle of attack, and zero lift angle of attack, the lift per unit span at location , will be different from null6. As the lift per unit span is proportional to the circulation, so, the circulation is also a function of y7. The lift distribution goes to zero at the wing tips.8. The calculation of the lift distribution [or the circulation distribution ] is one of the central problems of the finite-wing theory.null5.3 Prandtl’s Classical Lifting-line TheoryImportance of the Prandtl’s lifting-line theorybound vortex and free vortexReplacement of the finite wing with a bound vortexnullSingle horseshoe vortexnull1. The bound vortex induces no velocity along itself 2. The two trailing vortices both contribute to the induced velocity along the bound vortex, and their contributions are in the downward direction.3. The origin is taken at the center of the bound vortex.4. The induced velocity at any location of y along the bound vortex by the two trailing vortices is :nullThe above equation can be reducedIt approaches to infinite as y approaches the tipsnullSuperposition of large number of horseshoe vortices.1. Problems for the single horseshoe vortex 2. Superposition of a large number of horseshoe vortices, each with a different length of the bound vortex, but all the bound vortices coincident along a single line, called the lifting line. 3. Description for the lifting line and trailing vortices system.4. The strength of each trailing vortices is equal to the change in circulation along the lifting linenullCombination of three horseshoe vortices.nullinfinite number of horseshoe vortices.nullCase for a infinite number of horseshoe vortices.1. Each single horseshoe vortex has a vanishingly small strength 2. For this case, the distribution of is continuous along the lifting line. The value of the circulation at the origin is .3. The collection of the trailing vortices becomes continuous vortex sheet.4. The vortex sheet is parallel to the free stream direction.nullInduced velocity at a given location along the lifting line by an entire semi-infinite trailing vortex located at y.1. The strength of the trailing vortex at y is equal to the change of in circulation along the lifting line.2. Application of Biot-Savart law, induced velocity at along the lifting line by the entire trailing vortex at total change in circulation over the segment dynull3. The total velocity w induced at by the entire trailing vortex sheet. 4. Pause for a while, and referring back to the text book on page 328.nullnullInduced angle of attack at the arbitrary spanwise station For general cases, w is much smaller than , then nullInduced angle of attack in terms of the circulation distribution along the lifting line. It is an important result in our process for the derivation of the finite wing theory. We have to review back to remember based on what tools and approximations this result is achieved.nullRelation between the effective angle of attack and the lift coefficient Lift slope of arbitrary airfoil shapenullnullFundamental equation of Prandtl’s lifting line theory It simply states that the geometric angle of attack is equal to the sum of the effective angle plus the induced angle of attack. It is an integro-differential equation. Why it is called as lifting line theory? What are the known and what is the unknown inside this equation for general cases?nullAfter the solution is obtained from the fundamental equation of the lifting line theory, three aerodynamic characteristics can be achieved1. Lift distribution2. Total liftnullLift coefficient3. Induced drag and induced drag coefficientSection induced dragnullnullTotal induced dragInduced drag coefficient it is obviously to see, in Prandtl’s lifting line theory, the solution of is a key to obtain the aerodynamic characteristics.null FAQs for the lifting-line theory1. Location for the lifting-line relative to the actual wing geometry2. Shape and orientation of the trailing vortex sheet3. Is there any conflict between the boundary condition respect to the airfoil surface and the induced velocity on the lifting-line?null5.3.1 Elliptical Lift distributionConsider a circulation distribution given by That can be expressed in a more familiar form It is an elliptical equation nullSome features related to the above circulation distribution1. is the circulation at the origin, or to say, at the root of the finite wing2. The circulation along the lifting line varies elliptically with the distance along the span, according to Kutta-Joukovski theorem, we have Hence, we are dealing with an elliptical lift distribution null3. As , the lift goes to zero at the wing tips. The lift distribution given above is not obtained from the lifting-line theory of finite wing, it is stipulated as an elliptic distribution. 4. Aerodynamic characteristics of finite wing with such an elliptic distribution※ distribution of induced velocitynullIntroducing a integral variable transformation nullornullThe downwash is constant over the span for an elliptical lift distribution ※ induced angle of attackNote: As the wing span becomes infinite, both downwash and induced angle of attack go to zero.null※ total lift on the wing due to the elliptical lift distr. nullasor※ another expression of the induced angle of attack. nullAspect ratio The induced angle of attack is inversely proportional to the aspect ratio for elliptical lift distribution null※ the induced drag coefficient The induced angle of attack is constantnull※ two important explanations for the induced drag , referring back to our textbook.nullnullnullnull※ elliptical planform wing 1. If the finite wing has neither geometric twist nor aerodynamic twist, as ,and then as the lift coefficient is given byFor thin airfoil, so, must be constant along the spannull2. The lift per unit span is express the distribution of chord with lift distributionIt is clear to see that with constant dynamic pressure and lift coefficient per unit span, and elliptical lift distribution, the chord must vary elliptically along the span, i.e., the wing planform is ellipticalnull5.3.2 General Lift distribution※ with the transformationElliptical distributionnull※ general distributionThe coefficient are unknowns, and they should satisfy the fundamental equation of Prandtl’s lifting-line theory.The differential of the of the circulation distribution isnullnull※ description of the equation above, what are the known quantities and what is the unknown? Strategy for solve the circulation distribution.After the circulation distribution being solved1. Lift coefficientnullThe lift coefficient is only depends on the leading coefficient of the Fourier series expansion. But, should be solved together with the all coefficient.2. Induced drag coefficientnull can be replaced by nullnullnullnullWhere Where The lift distribution which yields minimum induced drag is the elliptical lift distributionnull5.3.3 General Lift distributionEffects of aspect ratio and on the induced drag nullThe induced drag is inversely proportional to AR, but how to verify it with experimental results?Prandtl’s verification※ total drag of a finite wingnullnull※ drag coefficients of two wings with different aspect ratio, assuming that the wings are at the sameif the shape of the airfoil is the same for these two wings, then the profile drag coefficients will be the same. Moreover, the variation of e between thee wings is only a few percent and can be ignored.nullnullnullnullnullnullnullnull