【正文】 PART 3 Inviscid, Compressible Flow 無粘可壓縮流 鄧 磊 Email: 2022年 6月 2日星期四 Department of Fluid Mechanics, School of Aeronautics Northwestern Polytechnical University CHAPTER 11 SUBSONIC COMPRESSIBLE FLOW OVER AIRFOILS: LINEAR THEORY 繞翼型的可壓縮亞音速流 : 線化理論 問題：為什么需要把微分方程線性化？ 線性化有什么好處？ 當(dāng)微分方程為線性方程，邊界條件也是線性時(shí)，方程的解滿足疊加原理。 即，可以將一個(gè)復(fù)雜的問題，分解成若干簡單的問題，分別求解，然后將解疊加。這樣問題大大簡化。 如： 繞翼型的流動(dòng) =有攻角的平板 +無厚度無攻角的彎度 +無攻角無彎度的厚度 α + + α = Introduction 第四章學(xué)習(xí)了低速不可壓流動(dòng)流過翼型的問題。 1）如果高亞音速流動(dòng)流過翼型會(huì)發(fā)生什么 ? 2）壓縮性如何影響翼型的氣動(dòng)特性？ 3）如何分析和計(jì)算壓縮性的影響？ 本章的目的是研究 M1時(shí)二維翼型的流動(dòng)特性 ,這時(shí)不可壓假設(shè)不再成立 . Figure Road Map for . velocity potential equation Linearized velocity potential equation PrandtlGlauet Compressibilty correction Improved pressibilty Correction Critical Mach umber The area rule for transonic flow Supercritical airfoils DragDivergence Mach number: Sound Barrier 速度勢方程 線性化的速度勢方程 PrandtlGlauet壓縮性修正 改進(jìn)的壓縮性修正 臨界馬赫數(shù) 跨音速面積律 超臨界翼型 Figure 11章路線圖 阻力發(fā)散馬赫數(shù) :音障 亞音速氣動(dòng)特性 跨音速氣動(dòng)特性 REVIEW Continuity Equation True for all flows: Steady or Unsteady, Viscous or Inviscid, Rotational or Irrotational 2D Inpressible Flows (Steady, Inviscid and Irrotational) 2D Compressible Flows (Steady, Inviscid and Irrotational) steady irrotational Laplace’s Equation (linear equation) Does a similar expression exist for pressible flows? Yes, but it is nonlinear ? ? 0?????? Vt ???? ?? ? 0000?????????????????????????????????yvyvxuxuVVVVVVt????????????????? ?000002?????????????????????????VVVVVt????? The Velocity Potential Equation(速度勢方程 ) ? ?000 22222222???????????????????????????????????????????????????????????????????????yyxxyxyyyxxxyvyvxuxuVyvxujyixjviuV??????????????????????????????STEP 1: VELOCITY POTENTIAL → CONTINUITY Flow is irrotational xponent yponent Continuity for 2D pressible flow Substitute velocity into continuity equation Grouping like terms Expressions for d?? STEP 2: MOMENTUM + ENERGY Euler’s (Momentum) Equation Substitute velocity potential Flow is isentropic: Change in pressure, dp, is related to change in density, d?, via a2 Substitute into momentum equation Changes in xdirection Changes in ydirection ? ? ? ???????????????????????????????????????????????????????????????????????????????????????????????????????????????222222222222222222222yyyxxayyxyxxaxyxdaddadpyxddpvudVddpV d Vdp???????????????????????RESULT 0))((2)(11)(112222222222 ????????????????????????????????yxyxayyaxxa???????Velocity Potential Equation: Nonlinear Equation Compressible, Steady, Inviscid and Irrotational Flows Note: This is one equation, with one unknown, ? a0 (as well as T0, P0, ?0, h0) are known constants of the flow Review: Inpressible, Steady, Inviscid and Irrotational Flows Velocity Potential Equation: Linear Equation 02 ?? ?2220 21????? Vaa ?In this equation , the speed of sound is also the function of φ (from ) : ?????????????? 22202 )()(21yxaa???結(jié)論： 1）速度勢方程是只有一個(gè)未知變量的偏微分方程（ PDE）； 2） 、動(dòng)量方程和能量方程的綜合。 3）理論上，給出遠(yuǎn)場邊界條件和物面邊界條件，就可以通過上式求解出繞二維外形的流動(dòng)參數(shù)。 0???n?????? Vxu? 0????yv?infinite boundary condition： wall boundary condition ： 4） How to use? Once φ 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: and xu ??? ?yv ??? ? a: ? ?100020222220221121??????????????????????????????????????????????????????????????TTppMTTavuaVMyxaa4. Calculate T, p, ρ: M: WHAT DOES THIS MEAN, WHAT DO WE DO NOW? ? 線性偏微分方程 : 偏微分方程分為線性和非線性 – 線性偏微分方程 : 方程未知數(shù) φ 以及未知數(shù)的所有導(dǎo)數(shù)只以線性形式存在，不存在交叉乘及平方等等 ? 可壓縮流動(dòng)非線性速度勢的偏微分方程不存在解析解 – 借助于數(shù)值求解方法 ? 是否可以將非線性方程在一定的條件下，簡化為線性方程 (easy to solve)? 1. Slender bodies 細(xì)長體 2. Small angles of attack 小攻角 – 如果可以，就可以應(yīng)用于翼型的研究中，并提供在亞音速可壓縮流中的定性和定量的特性 ? Next steps: – 介紹小擾動(dòng)理論 (finite and small) – 在 2的條件下線化速度勢方程。 THE LINEARIZED VELOCITY POTENTIAL EQUATION 線化速度勢方程 對(duì)二維、無旋、等熵流動(dòng)： vyuxvvuVu???????????????????? ??? ? xVyyxVx??????????????????yxyxyyxx??????????????????????????22222222220?)?)(?(2?)?(?)?(222222222 ???????????????????????????????????? yxyxVyyaxxVa???????perturbation velocity potential equation(擾動(dòng)速度勢方程 ). () Perturbation velocity potential: same equation, still nonlinear