隱式逆擴散加權基本不震盪算則的發展及其應用

THE DEVELOPMENT AND APPLICATIONS OF IMPLICIT ANTI-DIFFUSIVE WEIGHTED ESSENTIALLY NON-OSCILLATORY SCHEMES

謝蒼仁(Tsang-Jen Hsieh)

指導教授 ： 王興華

共同指導教授 ： 楊照彥

國立臺灣大學/工學院/機械工程學研究所/博士(2009年)

https://doi.org/10.6342/NTU.2009.00992

摘要

本文發展應用於可壓縮流之逆擴散加權基本不震盪算則，在時間離散方面，採用LU-SGS(Lower-Upper Symmetric-Gauss-Seidel)隱式解法以增強數值之穩定性及加速程式之收歛，同時亦採用多區塊有限體積網格法以彈性處理複雜幾何邊界問題。在空間離散方面，無黏性通量採用原始WENO (Weighted Essentially Non-Oscillatory)與逆擴散通量修正之WENO (Anti-diffusive WENO)高解析算則，黏性通量則採用中央差分法。紊流流場計算採用Spalart-Allmaras單方程式紊流模式。
本文採用數種不同之算則進行一系列可壓縮流的測試及驗證，算例包含了震波撞擊楔型體之繞射、第四型(type IV)的多震波干涉模擬、穿音速流經NACA 0012 及 RAE 2822 翼剖面、穿音速紊流流經ONERA M6 翼､超音速紊流流經三角翼之分析以及F16戰機在超音速流場之分析等，除了詳細比較原始WENO算則與逆擴散通量修正之WENO、逆擴散通量修正之Mapped WENO、逆擴散通量之修正平滑指示器(Modified Smoothness Indicator ; MSI) WENO算則等計算結果外，並與實驗值及其他數值分析結果比較。
一般可壓縮流數值分析方法應用於自由流馬赫數低於0.3時，數值收斂困難。因此，本文另一主要目的為擴展前述已發展成熟之可壓縮流程式，參考Weiss 及 Smith預調矩陣法，將原始WENO算則以及Anti-diffusive WENO算則轉化應用於預調矩陣系統，使得程式也可以有效的處理低速不可壓縮流的流場問題，期望所建構的分析工具可初步滿足未來氣動力設計需求。分析案例包含了二維低速拖曳方穴流場、三維低速高攻角(0.069馬赫、攻角27度)三角翼流場分析以及三維低速高攻角(0.1馬赫、攻角40度)三維戰機流場分析。
經由一系列的測試及驗證，顯示本文所採用之Anti-diffusive WENO算則除了具有良好的收斂性能與高階精度外，在相同的網格下對不連續接觸面的解析比原始WENO算則更為陡峭。在所有的測試案例中，不論是二維或三維，無黏性及紊流流場的模擬均能獲得精確的結果，並且與相關流場問題的數值解及實驗結果的比較也相當的吻合。

關鍵字

逆擴散WENO算則 ； 預調矩陣 ； 高解析算則

並列摘要

A class of lower-upper symmetric-Gauss-Seidel (LU-SGS) implicit anti-diffusive WENO schemes for solving the two- and three-dimensional compressible Navier-Stokes equations with Spalart-Allmaras one equation turbulence model is presented. Weighted essentially non-oscillatory spatial operator with and without anti-diffusive flux is employed for inviscid fluxes and central differencing for viscous fluxes. A numerical flux of WENO scheme in flux limiter form is adopted, which consists of first-order and high-order fluxes and allows for a more flexible choice of first order dissipative methods. Numerical experiments with several variants of the original WENO schemes, including anti-diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator WENO scheme are tested for the two- and three-dimensional inviscid/viscous flows. Computations of unsteady oblique shock wave diffraction over a wedge and steady flows over NACA 0012, RAE 2822 airfoils, ONERA M6 wing, delta wing and F16 fighter are presented to test and compare the methods.
It is known that most of the numerical algorithms developed for compressible flows are often inefficient or even inaccurate at low Mach numbers. Therefore, the other objective of this study is to extend the developed compressible flow solver for imcompressible flows computation. The Weiss-Smith preconditioned scheme is adopted and three computations are performed for validation, one is the two dimensional cavity flow, the second one is the three dimensional low subsonic flow over delta wing with high angle of attack and the other one is the three dimensional low subsonic flow over F16 fighter with high angle of attack.
By using the WENO scheme with anti-diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high order accuracy is maintained and in particular the contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. The present solutions are also compared with experimental data and other computational results and exhibit good agreement.

並列關鍵字

anti-diffusive WENO schemes ； pre-conditioned matrix ； high resolutuin scheme

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