空气动力学英文课件Chapter_04

nullChapter 4Chapter 4Incompressible Flow Over AirfoilsOf the many problems now engaging attention, the following are considered of immediate importance and will be considered by the committee as rapidly as funds can be secured for the purpose…. The evolution of the more efficient wing sections of practical form, embodying suitable dimension for an economical structure, with moderate travel of the center-of-pressure and still affording a large range of angle-of-attack combined with efficient action. From the first annual report of the NACA, 19154.1 Introduction4.1 IntroductionLudiwig Prandtl and his colleagues at Göttingen, Germany, showed that the aerodynamic consideration of wings could be split into two parts. (1) the study of the section of a wing—an airfoil. And (2) the modification of such airfoil properties to account for the complete, finite wing.nullDefinition of airfoilnullThe purpose of this chapter:Present theoretical methods for the calculation of airfoil aerodynamic properties.Road map of this chapternull4.2 Airfoil nomenclatureLeading edge: 前缘 trailing edge: 后缘 Chord line: 弦线 chord length: 弦长 Thickness: 厚度 camber: 弯度 Mean chamber line: 中弧线 nullNACA “four digit” series airfoilNACA2412: The first digit: maximum camber in hundredths; The second digit: the location of maximum camber along the chord from leading edge in tenth of the chord; The last two digits: maximum thickness in hundredth of the chord.NACA0012: symmetrical airfoilnull4.3 Airfoil characteristics(experiment)nullSpecial definitions lift coefficientangle of attacklift slopeMaximum lift coefficientzero-lift attack angleConsequence of the flow separationIt is impossible for we to calculate with inviscid flow approximation!!nullExperiment results for NACA2412 nullSource of dragprofile dragAerodynamic centernull4.4 Philosophy of theoretical solutions for low-speed flow over airfoils: The Vortex SheetSchematic figure of a point vortexnullSchematic figure of a vortex filamentPoint vortex is simply a section of a straight vortex filamentnullConstruction of a vortex sheetTry to remember the analogous situation for the construction of a source sheet.nullInfinite number of vortex filaments,The strength of each vortex filament is infinitesimally small.nullVelocity induced by at point P Definition of the strength of vortex sheetVelocity potential induced by at point P Difference of the superposition between the velocity vectors and velocity potential.nullThe circulation around the vortex sheet is the sum of the strengths of the elemental vortices.There is a discontinuity change in the tangential component of velocity across the sheet.Let to be the circulation along the dashed line.nullornullassoAs the top and bottom of the dashed line approach the vortex sheet, , become the velocity components tangential to the vortex sheet immediately above and below the sheet.orThe local jump in the tangential velocity across the vortex sheet is equal to the local sheet strength.nullPhilosophy of airfoil theory for inviscid, incompressible flows.Step 1. Replace the airfoil surface with a vortex sheet of strength Step 2. Find a suitable distribution of such that the wall boundary condition can be satisfied. That is, the combination of the free stream flow and the vortex sheet will make the vortex sheet(the surface of the airfoil) a streamline of the flow.nullStep 3. Calculate the circulation around the airfoil, and then get the lift by Kutta-Joukowski theorem Note 1. There are no general analytical solution for an airfoil with arbitrary shape and thickness. This should be solved numerically with suitable digital computers. Vortex panel method (Sec. 4.9)Note 2. Physical significance of the vortex sheet which has been used to replace the surface of the airfoil surface. Boundary layer is a highly viscous region, the vorticity inside the boundary layer is finite. nullStep 4. Approximation for a thin airfoil, shift the vortex sheet from the airfoil surface to the camber line of the airfoil. The upper and lower part of the vortex sheet are coincide together.This time, Find a suitable distribution of such that the wall boundary condition can be satisfied. That is, the combination of the free stream flow and the vortex sheet will make the vortex sheet(camber line of the airfoil) a streamline of the flow.nullNote 3. After the thin airfoil approximation, it is possible to give a closed-form analytical solution of . 4.5 The Kutta ConditionFor potential flows, different choice of gives different lifting flow around circular cylinder. And it is the same to the situation of airfoils.Two different flows around a same airfoil at the same attack anglenullnullThe nature knows how to pick a right solution. We need an additional condition that fixes for a given airfoil at a given attack angle.nullExperimental results for the development of the flow field around an airfoil which is set into motion from an initial state of rest.(a)null(b)null(c)Experimental results demonstrate that the flow is smoothly leaving the top and bottom surface of the airfoil at the trailing edge.nullIf the flow is smoothly leaving the top and bottom surface of the airfoil at the trailing edge, then the circulation is the value the nature adopts.nullThe condition, that the flow is smoothly leaving the top and bottom surface of the airfoil at the trailing edge, which is a physical observation, is called as : Kutta ConditionKutta condition used in theoretical analysisnullThe pressure at both the top and bottom surface immediately adjacent to point a (trailing edge).nullFor a given airfoil at a given angle of attack, the value of circulation around the airfoil is such that the flow leaves the trailing edge smoothly If the trailing-edge angle is finite, then the trailing edge is a stagnation point. If the trailing edge is cusped, then the velocities leaving the top and bottom surfaces at the trailing edge are finite and equal in magnitude and direction.nullAt the trailing edge (TE), we have.For finite-angle trailing edge.For cusped trailing edge.For trailing edge.null4.6 Kelvin’s circulation theorem and the starting vortex Examination of Kutta condition in a detailed way. Kelvin’s Circulation Theoremnull Explanation for the generation of the circulation around an airfoil with Kelvin’s theorem. nullFrom kelvin’s theorem.The circulation around the airfoil is equal and opposite to the circulation around the starting vortex.null4.7 Classical thin airfoil theory: the symmetrical airfoil Where are we in the road map of this chapter? Under the assumption of thin airfoil, the vortex is distributed along the mean camber line. null what will be the condition for the variation of ?The mean camber line should be a streamline of the combined flow and Kutta condition is satisfied at the trailing edge, i.e., further approximation of the placement of the vortex sheetnull what will be the condition for the variation of ?The mean camber line should be a streamline of the combined flow and Kutta condition is satisfied at the trailing edge, i.e., condition expressed by velocity componentsComponent of the free stream velocity normal to the camber line.Component of the velocity induced by vortex sheet normal to the camber line.nullnullFor small angle of attack and small camber.so thatnullComponent of the velocity induced by vortex sheet normal to the camber line.Component of the velocity induced by vortex sheet normal to the chord line.For thin airfoil, the approximation bellow is consistent with the thin airfoil theorynull incremental normal velocity induced by the vortex sheet placed on the camber linenullWhere, if the shape of the camber line is given, are functions of x .For small cambered airfoil, are small values, that meansnullSimplification introduced above is equivalent to satisfying the boundary conditions on the x axis instead of the mean camber line.Conclusion:nullVelocity dw at ponit x induced by the elemental vortex segment at point ξ nullVelocity w at ponit x induced by all elemental vortex segments at along the chord line is obtained by integrating dw from ξ=0 to ξ=c The mean camber line should be a streamline of the combined flow and Kutta condition is satisfied at the trailing edge, i.e., nullnullFundamental equation of thin airfoil theoryIt is simply a statement that the camber line is a streamline of the flow.Note: referring to the textbook(page 270).null Analysis for a symmetric airfoilStep 1. Fundamental equation for a symmetric airfoilStep 2. Transform ξ into θ Note: ξ is a dummy variable, x is a fixed pointnullnullStep 3. Rewritten the fundamental equation of thin airfoil theory in the arguments of Note: pay attention to the limits of integralnullStep 4. An rigorous solution of the equation above for can be obtained from complex variable analysis, but it is beyond the scope of this textbook.The solution is:After the solution been given, what should we do now? Is it a true for our problem? How to prove it? What is the principle to be based to prove?nullStep 5. Verification for the solutionThere are two conditions which the solution must satisfy, they are wall condition and Kutta condition.Wall condition………Substituting the solution into the wall condition given abovenullNow we have to prove thatFortunately, there is a standard integral given asnullSo, the solution satisfies the fundamental equation of thin airfoil theorynullKutta condition… … …the value is undeterminedWith L’Hospital’s rule(罗毕塔法则）So, the solution satisfies Kutta conditionnullStep 6. Calculation of the characteristics for thin airfoilsnullLift per unit span, (Kutta-joukovski)The lift coefficientLift slope =nullMoment coefficient about the leading edgeIncremental lift contributed by the elemental vortex segment dξ is : nullThis incremental lift creates a moment about the leading edgeThe total moment about the leading edge(per unit span) due to the entire vortex sheet is(Problem 4.4)nullMoment coefficientassonullThe moment coefficient about the quarter-chord point isThe center of pressure is at the quarter-chord point for a symmetric airfoilnullAs the moment coefficient about the quarter-chord point of a symmetric airfoil is independent of the attack angle, that is always equal to zero, then, the quarter-chord point of a symmetric airfoil is called as aerodynamic center.Summary: referring back to the textbookExperiment results for a symmetric airfoilnullnull1. 2. Lift slope 3. The center of pressure and the aerodynamic center are both located at the quarter-chord point.Theoretical results for a symmetric airfoilnull4.8 The cambered airfoilThin airfoil theory for a cambered airfoil is a generalization of the method for a symmetric airfoil. That means, the result for a symmetrical airfoil is a special case of the cambered airfoil. To keep the mean camber line of a cambered airfoil be a streamline of the flow, the condition isnullas the value of the camber is not zero, it makes the analysis more difficult than in the case of symmetric airfoil.After we use the same transformWe obtainnull find a solution for from the equation above, at the same time, the solution of must satisfy the Kutta conditionStep 1. Introduction of a rigorous solutionAt the very first, let us make a closed comparison between the styles of the solution of symmetric airfoil and the solution of cambered airfoil.nullSymmetric airfoilCambered airfoilnullThe form of the first term in the solution of a cambered airfoil is nearly the same to the solution of the symmetric airfoil. This term can be looked as the skeleton for the solutions for both symmetrical or cambered airfoil. The Fourier sine series can be looked as a fine tuning of the solutions, so that the camber can be taken in to account.nullStep 2. To find the solution of is equal to find the specific values of all the coefficients, Substituting the solution into the fundamental equation.nullfromWe can havenulland as then Reduced tonullor It is the transformed version of the fundamental equation of thin airfoil theoryNote: the equation is evaluated at a given point x along the chordnullhence the equation is also evaluated at a given point x along the chord, here, and correspond to the same point x along the chord. is a function of . AndnullStep 3. Investigation of Fourier cosine series expansionThe general form of the Fourier cosine series expansion representing a function of f(θ) over an interval 0≤ θ ≤π is given byIf an integration is taken on the both side over the interval 0≤ θ ≤π , thennullnullIf we multiplies cos(θ) on the both sides, and an integration is taken on the both side over the interval 0≤ θ ≤π , thennullnullIn the same way, we can proveConclusion: if a function of f(θ) over an interval 0≤ θ ≤π is given byThe coefficients and should be nullStep 4. Solution of and asnullWith the use of the results for the investigation of Fourier series expansion, we can get a direct expression of and orandnullWhat shall we keep in mind? and what we have to think about at this moment? Please back to our textbook at page 276.null after we get the solution of , then we are ready to obtain expressions for the aerodynamic coefficients for a cambered airfoilStep 1. Calculation of the total circulation of the vortex sheetnullnullStep 2. After the total circulation being evaluated, the lift per unit span isStep 3. The lift coefficient isnullFrom thin airfoil theory for any shape airfoilnullStep 4. Definition of zero lift anglenullComparison between two expressions of the lift coefficientWe have For symmetric airfoilnullStep 5. Moment coefficientFor symmetric airfoilnullMoment coefficient about the quarter-chord pointIt is independent to the attack angle, it depends on the shape of the camber line of the airfoil. Thus, the quarter-chord point is the theoretical location of the aerodynamic center for a cambered airfoilnullStep 6. Location of the pressure centernull4.9 Lifting flows over arbitrary bodies: the vortex panel numerical method Advantages for thin airfoil theory, and its limit for applications. Method suited to calculated the aerodynamic characteristics of bodies of arbitrary shape, thickness and orientation: the vortex panel method Comparison between the vortex panel method and source panel method: lifting body and nonlifting body.null Philosophy of the vortex panel methodCovering the body surface with a vortex sheet. Find the strength distribution to make the surface a streamline of the flow.nullApproximate the vortex sheet by a series of straight panels. Let the vortex strength per unit length be constant over a given panel.nullIf the number of vortex panels is n, there will be n unknowns to be solved, that is Solve the n unknowns, such that the body surface becomes a streamline of the flow and that the Kutta condition is satisfied.null Algorithm of the vortex panel methodVelocity potential introduced at P due to the jth panelnullwhere is a constant over the jth panelandVelocity potential introduced at P due to all the panelsnullPut P at the control point of the ith panelPut P at the control point of the ith panel， that means point P is located at , thennullandAt the control points, the normal component of the velocity is zero, that isnullnullIt represents n equations with n unknownsKutta condition should be satisfied at the trailing edge,