【正文】 0))((2)()(222222222 ????????????????????????????????yxyxyyaxxa???????0??)?(2?)?(?])?([ 2222 ????????????? ?? yuvuVyvvaxuuVa2?)?(12122222 vuVaVa ????????????() () 為了加深理解 ,我們將 ()用擾動速度表示 : 用擾動速度表示的能量方程為 : 將 ()代入到 (), 并重新整理可得 : 即 : )???2(2 1 2222 vuuVaa ????? ?? ? () )]??)(?1(?[?]?21?21?)1[(?]?21?21?)1[(??)1(222222222222xvyuVuVvMyvVuVvVuMxuVvVuVuMyvxuM???????????????????????????????????????????????() 方程 ()仍然是無旋、等熵流動的 精確方程 。這時擾動速度 、 的值可大、可小，即 對于大擾動、小擾動都成立 。 v?u?線性 非線性 assume that the body in is a slender body at small angle of attack (假設(shè)物體是細長的 ,迎角為小迎角 ). 在這種情況下 ,有 : small perturbation (小擾動 )situation: 1?,? ,1?,? 2222? ? ??????? VvVuVvVu同時 、 與它們的導(dǎo)數(shù)也非常小。 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 ? Subsonic and Supersonic Mach numbers – Keeping in mind these assumptions equation is good approximation () () 0??)1( 22222 ???????? yxM??0??)1( 2 ??????? ? yvxuM E q u at ion In viscid I r r o t a t i o n a l S m all P e r t u r b a t i o n In c om p r e ssi b le Not e s Navie r S tokes Hom og e n e ou s Re yn old s A ve rage d Navie r S tokes M od e led T u r b u l e n c e E u ler √ F u ll P o te n tial √ √ T r an son ic S m all Pe r turbation √ √ √ S u b so n ic, S u p e r son ic, S m all Pe r turbation √ √ √ Ac ou stic √ √ √ L ap lace √ √ √ Summary of monlyused equations and the corresponding assumption (常用控制方程及其相應(yīng)假設(shè)小結(jié) ): 求解速度勢方程的目的在于得到物體表面的壓強分布，進而得到氣動力。下面我們推導(dǎo)用速度勢表示的壓強系數(shù)的表達式： ????qppCp22 ??? ? Mpq?() () )1(2 2 ???? ppMC p?() pp cVTcVT2222?? ??? 1?? ??Rcp2222221211????????????aVVRTVVTT ???)1/(222???? ?? ?? RVVTT回憶： 222 ?)?( vuVV ??? ?)???2(2 11 222 vuVuaTT ????? ????1)( ???? ??TTpp12222 )???2(211 ??????????? ????????VvuVuMpp () 1222 )???2(211 ????????? ????? ??? vuVua12222 )???2(211 ??????????? ????????VvuVuMpp????????????)???2(21 2222VvuVuMpp ?()仍然是一個精確表達式。 ,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 。 遠場邊界條件 : c o n st a n t?0??c o n st a n t?0??0????????????????????yvxuvu?????? V vuV vuv ???t a n ?物面 : ?? ta n? ???? Vy() V∞ V θ u v ?? t a n????? Vy() 物面流動相切條件的近似表達式 () 0??)1( 22222 ???????? yxM????? V uC p ?2() 小結(jié)：本節(jié)推導(dǎo)的三個重要公式 亞音速或超音速小擾動速度勢方程 亞音速或超音速小擾動線化壓力系數(shù)公式 速度勢方程 線性化的速度勢方程 PrandtlGlauet壓縮性修正 改進的壓縮性修正 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 0??)1( 22222 ???????? yxM?? () HOW DO WE USE EQUATION ()？ PRANDTLGLAUERT COMPRESSIBILITY CORRECTION (PRANDTLGLAUERT壓縮性修正 ) 通過修正不可壓縮流的結(jié)果來 近似考慮壓縮性影響 的方法稱為 壓縮性修正 。 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 ???????????????????????