【正文】 0 . 8 50 . 80 . 7 50 . 70 . 6 50 . 60 . 5 50 . 50 . 4 50 . 40 . 3 50 . 30 . 2 50 . 20 . 1 50 . 10 . 0 50M∞= α=0度 NACA0012 翼型 等馬赫數(shù)云圖 XCP0 0 . 2 5 0 . 5 0 . 7 5 1 0 . 6 0 . 4 0 . 200 . 20 . 40 . 60 . 81M∞= α=0度 NACA0012 翼型表面壓力系數(shù)分布 XCP0 0 . 2 5 0 . 5 0 . 7 5 1 0 . 500 . 51M∞==0度 NACA0012 翼型表面壓力系數(shù)分布 臨界馬赫數(shù)： 在翼型表面速度最大點剛好達到聲速時對應的 自由來流馬赫數(shù)，稱為翼型的 臨界馬赫數(shù) 。 We deal with several aspects of transonic flow from a qualitative point of view. 在本節(jié)我們定性地討論一下跨音速流動的特征。 ??? )(121 220, ??????Mcc ll??2?lc IMPROVED COMPRESSIBILITYCORRECTIONS 改進的壓縮性修正公式 ? PrandtlGlauret – Shortest expression – Tends to underpredict experimental results ? To account for some of nonlinear aspects of flow field KarmanTsien – Most widely used Laitone – Most recent 20,1 ???MCC PP2111 0,2220,PPPCMMMCC????????????????0,22220,122111 PPPCMMMMCC??????????? ??????三種修正公式的比較 PrandtlGlauert壓縮性修正：基于線性理論，因此適用于薄物體、小迎角、亞音速、不適合高亞音速。 )(11 220, ????????MCC pp例 由第四章，我們得出繞對稱、薄翼型的不可壓流動的理論升力系數(shù)為 。 例 在翼型表面一給定點，已知在繞流速度極低時的壓強系數(shù)為 。 翼型相似 ?ddquVv ???FINAL RESULTS ? Insert transformation results into linearized CP ? PrandtlGlauert rule: If we know the inpressible pressure distribution over an airfoil, the pressible pressure distribution over the same airfoil may be obtained ? Lift and moment coefficients are integrals of pressure distribution (inviscid flows only) 20,20,20,0,1 1 1 211212 ?2?2???????????????????????????????????????MccMccMCCCVuCVxVxVVuCmmllPPPPP????????連續(xù)方程 動量方程 能量方程 φ 速度勢方程 （非線性） 擾動速度 擾動速度勢方程 （非線性） 小擾動假設 小擾動速度勢方程 （線性） 轉換空間 0??)1( 22222 ??????? ? yxM ???? ta n? ???? Vy??? VuC p ?2拉普拉斯方程 （線性） 02222 ?????? ??????? ta n???? V(ξ， η) 20,1 ??? MCC PP結論：滿足小擾動假設條件的可壓縮流動的壓力系數(shù)可以通過繞相同外形的不可壓縮流動的壓力系數(shù)，修正而得到。 我們考慮繞某翼型的無粘、亞音速流動問題： HOW DO WE SOLVE EQUATION ? Note behavior of sign of leading term for subsonic and supersonic flows ? Equation is almost Laplace’s equation, if we could get rid of b coefficient ? Strategy – Coordinate transformation – Transform into new space governed by ξ and η ? In transformed space, new velocity potential may be written ? ?? ? ? ?yxyxyxMyxM,?,0??10??1222222222222??????????????????????????????????TRANSFORMED VARIABLES (1/2) ? Definition of new variables (determining a useful transformation is done) ? Perform chain rule to express in terms of transformed variables ??????????????????????????????????????????????????????????????????????????????????????????????????????????????1??? ,0 ,0 ,1??????yxyxyxyyyxxxyx? ? ? ?yx ,?, ????? ?TRANSFORMED VARIABLES (2/2) ? Differentiate with respect to x a second time ? Differentiate with respect to y a second time ? Substitute in results and arrive at a Laplace equation for transformed variables ? Recall that Laplace’s equation governs behavior of inpressible flows 0?1?222222222222????????????????????????????yx? Transformation relates pressible flow over an airfoil in (x, y) space to inpressible flow in (ξ,η) space over same airfoil 變換將 (x, y) 空間的翼型上的可壓縮流動和 (ξ,η) 空間內相同翼型上的不可壓流動聯(lián)系起來 翼型外形 dxdfuVv ??? ??小擾動邊界條件： ydxdfV??????精確： 物面： )(xfy ? )(?? q???? ???? ddqV????????y? （ ） dxdfddq ??（ ） 在轉換空間的翼型形狀與物理空間的翼型形狀相同 。 ,1? ???Vu,1? 22????Vu 1? 22 ????Vv? ? ????????? ???? ?? ??1111pp)1(2 2 ???? ppMC p ?222 ???2??????VvuVuCp???VuC p?2忽略 xVVuCp ??????????2 ?2 () 式（ )是亞音速或超音速小擾動線化壓力系數(shù)公式， 只適用于小擾動情況；壓強系數(shù)只依賴于 x方向的擾動速度 。 v?u?)]??)(?1(?[?]?21?21?)1[(?]?21?21?)1[(??)1(222222222222xvyuVuVvMyvVuVvVuMxuVvVuVuMyvxuM???????????????????????????????????????????????? Compare terms (coefficients of like derivatives) across equal sign ? Compare C and A: – If 0 ≤ M∞ ≤ or M∞ ≥ – C A – Neglect C ? Compare D and B: – If M∞ ≤ 5 – D B – Neglect D ? Examine E – If M∞ ≤ 5 ,E ~ 0 – Neglect E ? Note that if M∞ 5 (or so) terms C, D and E may be large even if perturbations are small A B C D E HOW TO LINEARIZE RESULT ? After order of magnitude analysis, we have following results ? May also be written in terms of perturbation velocity potential ? Equation is a linear PDE and is rather easy to solve ? Recall: – Equation is no longer exact – Valid situation: ? Slender bodies ? Small angles of attack