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1、PART 3Inviscid, Compressible Flow无粘可压缩流邓 磊E-mail: *Department of Fluid Mechanics, School of AeronauticsNorthwestern Polytechnical UniversityCHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS: LINEAR THEORY绕翼型的可压缩亚音速流: 线化理论问题：为什么需要把微分方程线性化？ 线性化有什么好处？ 当微分方程为线性方程，边界条件也是线性时，方程的解满足叠加原理。即，可以将一个复杂的问题，分解成若
2、干简单的问题，分别求解，然后将解叠加。这样问题大大简化。 如： 绕翼型的流动=有攻角的平板+无厚度无攻角的弯度+无攻角无弯度的厚度+=Figure 11.1 Road Map for Chap.11.velocity potential equation Linearized velocity potential equation Prandtl-GlauetCompressibilty correction Improved compressibilty Correction Critical Mach umber The area rule for transonic flow Super
3、critical airfoils Drag-Divergence Mach number: Sound Barrier 速度势方程 线性化的速度势方程 Prandtl-Glauet压缩性修正改进的压缩性修正 临界马赫数跨音速面积律 超临界翼型 Figure 11.1 11章路线图阻力发散马赫数:音障亚音速气动特性跨音速气动特性REVIEWContinuity EquationTrue for all flows:Steady or Unsteady,Viscous or Inviscid,Rotational or Irrotational2-D Incompressible Flows(S
4、teady, Inviscid and Irrotational)2-D Compressible Flows(Steady, Inviscid and Irrotational)steadyirrotationalLaplaces Equation(linear equation)Does a similar expression exist for compressible flows?Yes, but it is non-linear11.2 The Velocity Potential Equation(速度势方程)STEP 1: VELOCITY POTENTIAL CONTINUI
5、TYFlow is irrotationalx-component y-componentContinuity for 2-Dcompressible flowSubstitute velocityinto continuity equationGrouping like termsExpressions for dr?STEP 2: MOMENTUM + ENERGYEulers (Momentum) EquationSubstitute velocity potentialFlow is isentropic:Change in pressure, dp, is relatedto cha
6、nge in density, dr, via a2Substitute into momentum equationChanges in x-directionChanges in y-directionRESULTVelocity Potential Equation: Nonlinear EquationCompressible, Steady, Inviscid and Irrotational FlowsNote: This is one equation, with one unknown, fa0 (as well as T0, P0, r0, h0) are known con
7、stants of the flowReview: Incompressible, Steady, Inviscid and Irrotational FlowsVelocity Potential Equation: Linear EquationIn this equation , the speed of sound is also the function of (from 8.34) :结论：1）速度势方程是只有一个未知变量的偏微分方程（PDE）；2）11.12式是连续方程、动量方程和能量方程的综合。3）理论上，给出远场边界条件和物面边界条件，就可以通过上式求解出绕二维外形的流动参数
8、。infinite boundary condition：wall boundary condition ：4） How to use?Once is known, all the other value flow variables are directly obtained as follows: (a0, T0, P0, r0, h0 are known quantities) 1. Calculate u and v: and2.Calculate a: 4. Calculate T, p, : 3.Calculate M: WHAT DOES THIS MEAN, WHAT DO W
9、E DO NOW?线性偏微分方程: 偏微分方程分为线性和非线性 线性偏微分方程: 方程未知数 以及未知数的所有导数只以线性形式存在，不存在交叉乘及平方等等可压缩流动非线性速度势的偏微分方程不存在解析解 借助于数值求解方法是否可以将非线性方程在一定的条件下，简化为线性方程(easy to solve)?1. Slender bodies 细长体2. Small angles of attack 小攻角 如果可以，就可以应用于翼型的研究中，并提供在亚音速可压缩流中的定性和定量的特性Next steps: 介绍小扰动理论 (finite and small) 在1、2的条件下线化速度势方程。11.3
10、 THE LINEARIZED VELOCITY POTENTIAL EQUATION 线化速度势方程 对二维、无旋、等熵流动：perturbation velocity potential equation(扰动速度势方程).(11.14)Perturbation velocity potential: same equation, still nonlinear(11.14a)(11.15)为了加深理解,我们将(11.14)用扰动速度表示:用扰动速度表示的能量方程为:将(11.15a)代入到(11.14a), 并重新整理可得:即:(11.15a)(11.16)方程(11.16)仍然是无旋、
11、等熵流动的精确方程。这时扰动速度 、 的值可大、可小，即对于大扰动、小扰动都成立。线性非线性assume that the body in Fig.11.2 is a slender body at small angle of attack (假设物体是细长的,迎角为小迎角). 在这种情况下,有:small perturbation (小扰动)situation: 同时 、 与它们的导数也非常小。Compare terms (coefficients of like derivatives) across equal signCompare C and A: If 0 M 0.8 or M
12、1.2 C A Neglect CCompare D and B: If M 5 D 5 (or so) terms C, D and E may be large even if perturbations are smallA ABCDEHOW TO LINEARIZERESULTAfter order of magnitude analysis, we have following resultsMay also be written in terms of perturbation velocity potentialEquation is a linear PDE and is ra
13、ther easy to solveRecall: Equation is no longer exact Valid situation: Slender bodies Small angles of attack Subsonic and Supersonic Mach numbers Keeping in mind these assumptions equation is good approximation(11.18)(11.17)Summary of commonly-used equations and the correspondingassumption (常用控制方程及其
14、相应假设小结):求解速度势方程的目的在于得到物体表面的压强分布，进而得到气动力。下面我们推导用速度势表示的压强系数的表达式：(11.19)(11.21)(11.22)回忆：(11.27)(11.27)仍然是一个精确表达式。忽略(11.32)式（11.32)是亚音速或超音速小扰动线化压力系数公式，只适用于小扰动情况；压强系数只依赖于x方向的扰动速度。远场边界条件:物面:(11.34)VVuv(11.34)物面流动相切条件的近似表达式(11.18)(11.32)小结：本节推导的三个重要公式亚音速或超音速小扰动速度势方程亚音速或超音速小扰动线化压力系数公式 速度势方程 线性化的速度势方程 Prand
15、tl-Glauet压缩性修正改进的压缩性修正 临界马赫数跨音速面积律 超临界翼型 Figure 11.1 11章路线图阻力发散马赫数:音障(11.18)HOW DO WE USE EQUATION (11.18)？11.4 PRANDTL-GLAUERT COMPRESSIBILITY CORRECTION (PRANDTL-GLAUERT压缩性修正)通过修正不可压缩流的结果来近似考虑压缩性影响的方法称为压缩性修正。 我们考虑绕某翼型的无粘、亚音速流动问题：HOW DO WE SOLVE EQUATION Note behavior of sign of leading term for su
16、bsonic and supersonic flows Equation is almost Laplaces equation, if we could get rid of b coefficient Strategy Coordinate transformation Transform into new space governed by and In transformed space, new velocity potential may be writtenTRANSFORMED VARIABLES (1/2) Definition of new variables (determining a useful transformation is done) Perform chain rule to express in terms of transformed variablesTRANSFORMED VARIABLES (2/2) Differentiate with respect to x a second time Differentiate with resp
