Modified numerical methods for analyzing transonic flows enclosing a
two-dimensional or threedimensional wing with embedding shock waves have
been developed. In the present approach, the linearized solution and local
linearized solution are used to construct an approximation pressrue
distribution on the airfoil under the effect of embedding shock waves. A
numerical algorithm combing the proposed approximation solution with a
modified finite-difference scheme forms a fastconvergence non-conservative
relaxation method for the two-dimensional flow. The locations of the sonic
line and the shock wave, as well as the pressure distribution on various
airfoils have been calculated by using the present method and compared
with experimental data from other sources. The agreement is excellent, and
the convergence rate is 2 to 8 times faster than other numerical methods
up to date. For flows over a three-dimensional wing, the matched
asymptotic expansion method is employed to introduce a transonic
lifting-line theory; and then governing equations and boundary conditions
of the first order and second order inner flow solutions are established.
The first-ouder solution represents a two-dimensional nonlinear flow over
wing sections and the second-order solution represents a two-dimensional
effect. Numerical results from the present method have been compared with
those from the three-dimensional finite-difference method. It hav been
shown that for the same accuracy, the present method needs.only about one
tenth of the computer storage. Results obtained by using this method have
shown that the smaller is aspect ratio, the weaker the three-dimensional
transonic effect will be. This conclusion can be used as a criterion to
justify the performance of a shock-free supercritical wing.
本文以相配漸近展開法（Matched asymptotic expansion method） 處理有嵌埋 式震
波（Embedding shock wave）存在時的非線性穿音速流場。對於含有嵌埋式震波 之二
維翼剖面流，本文提出一快速收歛連續線鬆弛法（A fast convergence successive
line relaxation method）。新法結合局部線性法（Local linearization method）
、線性化的流異元素法（Singularity-element method）及改進的有限差分法（ Fini
te difference method），並利用遠場擾動流勢解析解為邊界條件來求解。使用 本法
分析所得之流場音速線（Sonic line）、震波線（Shock line）以及震波效應下 的翼
面壓力係數分佈等結果，較現有之其他數值方法更為接近實驗值；且本法之收歛 速度
遠較現有的方法為快。對於三維翼流場之分析，本文利用相配漸近展開法來討論 穿音
速昇力線理論（Transonic lifting line theory） ，建立第一階和第二階內場 流勢
統制方程式及其邊界條件。第一階內場流勢代表翼剖面之二維非線性流場。第 二階流
勢則代表三維的展弦比效應修正流場，對於不同翼形之三維翼翼面壓力係數分佈 則利
用解出的第一階和第二階壓力係數配合翼形函數來計算。本法所得之翼面壓力分 佈與
使用三維有限差分法所得者相當吻合。但本法可節省十倍以上的計算機儲存量（ Comp
uter storage）。利用本法進行三維翼展弦比效塵分析之結果顯示，三維的非線 性穿
音速效應係隨展弦比減小而減弱。此結論提供一研判除震（Shock free）超臨界 翼（
Supercritical wing）功能的重要考慮因素。