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First, mathematical formulation of linear stability analysis in fluid mechanics is given. Next, several examples in which the linear stability analysis was successfully applied are presented. Then, bifurcation diagrams are shown, and the basic equations by which fluid motions are calculated for specific boundary conditions are explained. Finally, numerical methods to obtain quasi‐analytic solutions in high accuracy are addressed, on which the perturbation is superimposed and thus the linear stability analysis can be applied, in the case that an exact solution is difficult to obtain., "Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

Formation and instability of flow between two corotating disks in an enclosure is investigated numerically, in which the origin of polygonal flow pattern is explored utilizing numerical simulation and linear stability analyses. The outer cylindrical boundary of the flow field is assumed to be stationary, whereas the inner cylinder rotates together with the two disks. Polygonal flow patterns are known to appear due to instability of the axisymmetric flow with respect to the axis of rotation, which is a unique solution at small Reynolds numbers. The origin of the flow structure of so called shift-and-reflect flow pattern is clarified from the solution of the linear stability analyses and occurs due to Hopf bifurcation. In addition, we found that the most unstable mode of disturbance has an unexpected wavenumber as large as 20 to 30 for very narrow disk spacing., 資料番号: AA0063608007, レポート番号: JAXA-SP-07-009

We are concerned with the carbuncle phenomenon, a kind of instability that happens in numerical computation for compressible Euler equations. Through numerical experiments we show that the occurrence of instability is closely related with some linearized stability of discrete temporal evolution., 資料番号: AA0049212030, レポート番号: JAXA-SP-05-017

The thermocapillary flow in liquid bridges is investigated numerically by means of a linear stability analysis. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, and aspect ratio is investigated for intermediate Prandtl numbers. The effect of gravity on the flow stability is compared for small (Pr = 0.02) and large Prandtl numbers. Since the instabilities for small and large Prandtl numbers are markedly different, the gravity forces lead to a different behavior. While the hydrostatic shape of the bridge is the most important gravity effect on the critical conditions for low-Prandtl-number flows, buoyancy is the dominating factor for the flow stability when the Prandtl number is high. In addition to these investigations, the linear stability for acetone and methanol half-zones including the effect of evaporative cooling is calculated and compared with experimental data., 資料番号: AA0032601007, レポート番号: NASDA-TMR-010015E

超音速翼上の境界層の遷移位置を予測するための効率的な数値計算法を提案した。この計算システムは3次元境界層、その線形安定性およびN因子の計算から構成されている。3種類の計算の各々は大量の計算時間と各計算間の複雑な干渉手続きを必要とする。従って、計算時間を節約し計算システムをできるだけ簡素なものにするために、境界層の計算と線形安定性の計算に対して同じ計算アルゴリズムを用いた。, An efficient numerical calculation method for predicting the transition location of the boundary layer over a supersonic wing was proposed. This numerical system consists of the calculations of Three Dimensional (3-D) boundary layer, its linear stability characteristics, and N factor. Each of the three calculations requires a lot of computing time and a rather complex interference procedure between each calculation. Therefore, the same computing algorithm for both boundary layer and linear stability calculations was used to save computing time and to make the calculation system as simple as possible., 資料番号: AA0000440003, レポート番号: NAL SP-31

無限進展円筒上の圧縮性境界層流をグリッド密度または時間進行法をもつナビエ・ストークスコードにより計算した。グリッドに対する境界層流の依存性はグリッドBのような稠密なグリッドを用いることにより消去される様である。対角化ADI(交互方向陰的差分法)スキームとLU-SGS(上下対称Symmetric Gauss Seidel)スキームにより収束された境界層流の間の差は極めて小さかった。付着線での境界層の安定性解析の結果はMalikの得た結果とかなり一致している。, The compressible boundary layer flow on an infinite swept cylinder was computed by the Navier-Stokes code with the grid density or with the time marching method. The dependence of the boundary layer flow on the grid appears to be eliminated by employing a grid as dense as Grid B. There was very little difference between the boundary layer flow converged by the diagonalized ADI (Alternating Direction Implicit) scheme and that converged by the LU-SGS (Lower-Upper Symmetric Gauss-Seidel) scheme. The results of the stability analysis of the boundary layer at the attachment line agree fairly well with those obtained by Malik., 資料番号: AA0000440004, レポート番号: NAL SP-31

超音速層流制御(SLFC)を応用した空力設計システムを開発するために、遷移予測法の数値的研究とSLFCの影響についての実験的研究を行った。第1段階で、線形安定性の解析システムを開発し、いくつかの代表例において評価された増幅率について良い確認を得た。第2段階で、経験的e{N}法を用いて遷移予測システムを開発し、2代表例において評価N因子について良い確認を得た。第3段階で、いくつかの遷移データと超音速流におけるSLFCの影響とを得るために、独自に設計した、たわみデルタ翼モデルを用いて風洞試験を実施した。最後に、モデルの遷移特性の解析を試み、マッハ1.4の遷移においてNが約7の有用な結果を得た。, To develop an aerodynamic design system applying Supersonic Laminar Flow Control (SLFC), numerical study on a transition prediction method and experimental study on the effect of SLFC were performed. In the first phase of study, an analysis system of linear stability was developed and good validation for the estimated amplification rate in several typical cases was obtained. In the second phase, a transition prediction system was developed by applying an empirical e(sup N) method and good validation for the estimated N-factor was obtained in two typical cases. In the third phase, wind tunnel tests using an originally designed warped delta wing model were carried out to obtain some data of transition and the effect of SLFC under supersonic flow condition. And finally the transition characteristics analysis of the model was tried and the useful result of N approximately = 7 at transition at Mach 1.4 was obtained., 資料番号: AA0000440005, レポート番号: NAL SP-31

航空宇宙技術研究所 Sep. 1994 (15th). Mar. 1995 (16th) 東京 日本, National Aerospace Laboratory Sep. 1994 (15th). Mar. 1995 (16th) Tokyo Japan, 下面加熱矩形ダクト内混合対流の線形不安定性を、コロケーション法で数値的に検討した。チェビシェフ多項式で2重拡大法を用いて得られた固有値問題と、それによって発生した一般化固有値問題を数値解析した。進行横ロールはレイノルズ数が、ある臨界値よりも小さい時に、最も不安定な攪乱であることが分かった。一方、縦ロールは、レイノルズ数がこの臨界値以上の場合に、最も危険になる。, The linear stability of mixed convection in a rectangular duct heated from below is investigated numerically by a collocation method. The eigenvalue problem obtained by making use in the double expansion method in Chebyshev polynomials and the resulting generalized eigenvalue problem is solved numerically. It is found that the traveling transverse roll is the most unstable disturbance when the Reynolds number is less than a certain critical value, while the longitudinal roll is most dangerous when the Reynolds number is larger the critical value., 資料番号: AA0000109016, レポート番号: NAL SP-28

線形安定理論は、3次元境界層における横流れ不安定に対して時間依存型モードは定在モードに比べてより不安定であると予測している。3次元境界層の典型的な流れ場である静止流体中で回転する円盤流では定在モードが進行波モードに比べて優勢的に観察されているけれども、その理由については合理的な説明がなされていない。本論文では、これまでの実験的観察と理論的研究のリビューに加えて、壁面曲率項、流線曲率項および非平行性を含めた安定計算を実施して、その選択機構について議論している。この結果、壁面曲率項は横流れ不安定のモード選択機構に重要な役割を演じる、と同時に撹乱の初期振幅も重要であることが明らかにされている。, Linear stability theory predicts that the most unstable disturbance of crossflow instability in three-dimensional boundary layers is a time dependent mode rather than a stationary mode. Stationary rather than traveling mode has been predominantly observed in natural disk flow, spinning in still air, which is known to be a typical example of three-dimensional boundary layers. The present paper discusses this mode selection mechanism of crossflow instability on the basis of reviewing both experimental and theoretical research and making linear stability calculations inclusive of a wall-curvature term as well as streamline-curvature and non-parallel terms. This reveals that the wall curvature plays a key role in the mode selection mechanism of three-dimensional boundary layers and that initial amplitudes of unstable disturbances are of principal importance., ISSNの誤記: 1349-1148, Incorrect ISSN: 1349-1148, 資料番号: AA0047414000, レポート番号: JAXA-RR-03-022E

3次元境界層中に発達する横流れ不安定波のレイノルズ数依存性を、複素特性曲線法を用いて遷移予測の観点から数値的に調べた。複素特性曲線法は、従来の手法では特定できなかった不安定撹乱の波数ベクトルを与えることを可能にするもので、これを用いることで境界層の遷移を引き起こす主要な撹乱波の振幅の大きさや振動数、および撹乱波の波角などのレイノルズ数依存性を定量的に示すことに成功した。流れ場は50度の後退角を持つ無限円柱まわりの境界層とし、主流速度とその方向のコード長に基づくレイノルズ数を0.5×10(exp 6)から1.5×10(exp 6)の範囲で解析を行った。解析に用いる境界層速度分布はN-Sコードにより数値的に求めた。このとき境界層内の計算格子点を50〜70点とし、計算精度の確保に努めた。安定性解析には非圧縮性Navier-Stokes方程式に線形近以と平行流の仮定を施して得られるOrr-Sommerfeld方程式を用いた。数値解析の結果、最も大きく増幅する撹乱波の振幅および振動数はレイノルズ数の変化に対して線形的な依存性を示すことがわかった。ただし波角については常に一定の値をとり、レイノルズ数依存性を全く持たないという結果を得た。, Reynolds number dependence of the cross-flow instability in a three-dimensional boundary layer is numerically investigated using a complex ray theory, which can determine not only the N factor of the e(sup N) method but also the wave angle of the disturbance wave. The flow field is an infinite yawed circular cylinder with a sweep angle of 50 degrees, and the Reynolds number based on the mean flow velocity and the chord length ranges from 0.5 x 10(exp 6) to 1.5 x 10(exp 6). The velocity profiles used in the stability analysis are obtained using an N-S code. The grid points in the boundary layer are set at about 50 to 70 to retain a high degree of precision. The Orr-Sommerfeld equation, which is deduced from the incompressible Navier-Stokes equation with linear approximation and parallel flow assumption, is used to analyze stability. The present study found that amplitude and frequency depend linearly on the Reynolds number. On the other hand, the study also showed that the wave angle of the disturbance wave does not depend on the Reynolds number., 資料番号: AA0047387000, レポート番号: NAL TR-1467