%deffont "thick" xfont "helvetica-bold-r", tfont "/usr/share/fonts/truetype/Arial_Bold.ttf", tmfont "/usr/share/fonts/truetype/kochi/kochi-gothic.ttf", vfont "goth" %deffont "standard" xfont "helvetica-medium-r", tfont "/usr/share/fonts/truetype/Times_New_Roman.ttf", tmfont "/usr/share/fonts/kochi/kochi-mincho.ttf", vfont "min" %deffont "typewriter" xfont "courier-medium-r", tfont "/usr/share/fonts/truetype/Courier_New.ttf", tmfont "goth.ttf" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Default settings per each line numbers. %% %default 1 area 90 90, leftfill, size 2, fore "saddlebrown", back "white", font "standard", hgap 0 %default 2 size 5, hgap 20, vgap 10, prefix " ", font "thick", ccolor "black" %default 3 size 3, hgap 10, bar "gray70", vgap 10 %default 4 size 5, hgap 10, fore "gray20", vgap 30, prefix " ", font "standard" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Default settings that are applied to TAB-indented lines. %% %tab 1 size 5, vgap 40, prefix " ", icon box "green" 50 %tab 2 size 4, vgap 40, prefix " ", icon arc "yellow" 50 %tab 3 size 3, vgap 40, prefix " ", icon delta3 "white" 40 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %nodefault %fore "red", size 8, back "darkblue", font "standard", vgap 0, ccolor "gray" %bgrad 0 0 128 0 1 "black" "black" "blue" "black" "black" "black" "black" "black" %center, fore "yellow", font "thick" シアー不安定の基礎 %font "standard" %size 5, fore "coral", font "standard" 九州大学 応用力学研究所 伊賀 啓太 %size 4 iga@riam.kyushu-u.ac.jp %size 2 %right, fore "green", font "thick" (2003年夏 GFDセミナー, 2003/09/09 奈井江) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" %pcache 1 1 0 30 目次 シアー不安定の数学的な取り扱い(固有値問題) シアー不安定の定性的な説明・擬運動量 波の共鳴による不安定の理解(通常のモードどうし) Linの定理・連続モード 通常のモードと連続モードの共鳴による不安定 連続モードに隠されたモードの共鳴による不安定 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %nodefault %fore "red", size 8, back "darkblue", font "standard", vgap 0, ccolor "gray" %bgrad 0 0 128 0 1 "black" "black" "blue" "black" "black" "black" "black" "black" %center, fore "yellow", font "thick" シアー不安定の数学的な取り扱い (固有値問題) %font "standard" %size 5, fore "coral", font "standard" 九州大学 応用力学研究所 伊賀 啓太 %size 4 iga@riam.kyushu-u.ac.jp %size 2 %right, fore "green", font "thick" (2003年夏 GFDセミナー, 2003/09/09 奈井江) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 流れの安定・不安定 %font "standard", fore "black" %left, fore "black" 流れを記述する方程式である\ Euler 方程式 (あるいは Navier-Stokes 方程式)\ のある厳密解 →これは実際に実現されるものであるか? その厳密解に小さな擾乱を加えて、\ その後の振舞いを調べる。 →時間が経っても元の状態から離れないなら安定、\ 時間とともに元の状態から離れていくなら不安定。 不安定な状態は実際には実現されにくい。 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" シアー不安定の実例 %font "standard", fore "black" %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-basic-jump eqn" % \begin{displaymath} U = \left\{ \begin{array}{cl} U_0 & (y>0) \\ - U_0 & (y<0) \end{array} \right. \end{displaymath} %endfilter %image "./eqn/1-basic-jump.eps" 0 350 350 1 %left, fore "black" を基本場とする流れの時間変化 %center %image "./anim/anim_shear_init.gif" 0 100 100 1 %pause %system "xanim -Zr ./anim/anim_shear.gif -geometry %75x150+38+31" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 流れの不安定の数学的な取り扱い %font "standard", fore "black" %left, fore "black" 基本方程式(Euler方程式・連続の式) %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-Euler-continuity eqn" % \begin{eqnarray*} && \PD{}{\Vectm{u}}{t} + (\Vectm{u} \cdot \nabla ) \Vectm{u} + f \hat{\Vectm{k}}\times \Vectm{u} = - \frac{1}{\rho} \nabla p \\ && \nabla \cdot \Vectm{u} = 0 \end{eqnarray*} %endfilter %image "./eqn/1-Euler-continuity.eps" 0 350 350 1 %left, fore "black" 基本場の定常解(厳密解) %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-exact-solution eqn" % \begin{eqnarray*} && (\Vectm{U} \cdot \nabla ) \Vectm{U} + f \hat{\Vectm{k}}\times \Vectm{U} = - \frac{1}{\rho} \nabla P \\ && \nabla \cdot \Vectm{U} = 0 \end{eqnarray*} %endfilter %image "./eqn/1-exact-solution.eps" 0 350 350 1 %left, fore "black" 諸量を基本場とそれからのずれ(擾乱)に分ける %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-disturbance-def eqn" % \begin{eqnarray*} && \Vectm{u}(\Vectm{r}, t) = \Vectm{U}(\Vectm{r}) + \Vectm{u}'(\Vectm{r}, t) \\ && p(\Vectm{r}, t) = P(\Vectm{r}) + p'(\Vectm{r}, t) \end{eqnarray*} %endfilter %image "./eqn/1-disturbance-def.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" 基本方程式に代入して基本場の式を差し引く: %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-disturbance-eq eqn" % \begin{eqnarray*} && \PD{}{\Vectm{u}'}{t} + (\Vectm{U} \cdot \nabla ) \Vectm{u}' + (\Vectm{u}' \cdot \nabla ) \Vectm{U} + (\Vectm{u}' \cdot \nabla ) \Vectm{u}' + f \hat{\Vectm{k}}\times \Vectm{u}' = - \frac{1}{\rho} \nabla p' \\ && \nabla \cdot \Vectm{u}' = 0 \end{eqnarray*} %endfilter %image "./eqn/1-disturbance-eq.eps" 0 350 350 1 %left, fore "black" 擾乱の2次の項を無視して線形方程式: %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-linear eqn" % \begin{eqnarray*} && \PD{}{\Vectm{u}'}{t} + (\Vectm{U} \cdot \nabla ) \Vectm{u}' + (\Vectm{u}' \cdot \nabla ) \Vectm{U} + f \hat{\Vectm{k}}\times \Vectm{u}' = - \frac{1}{\rho} \nabla p' \\ && \nabla \cdot \Vectm{u}' = 0 \end{eqnarray*} %endfilter %image "./eqn/1-linear.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" 2次元運動を考え、基本場が平行流なら: %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-linear-2D eqn" % \begin{eqnarray*} && \PD{}{u'}{t} + U \PD{}{u'}{x} + v' \D{}{U}{y} - fv' = - \frac{1}{\rho} \PD{}{p'}{x} \\ && \PD{}{v'}{t} + U \PD{}{v'}{x} + fu' = - \frac{1}{\rho} \PD{}{p'}{y} \\ && \PD{}{u'}{x} + \PD{}{v'}{y} = 0 \end{eqnarray*} %endfilter %image "./eqn/1-linear-2D.eps" 0 350 350 1 %left, fore "black" 1変数の方程式にすると: %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-vorticity-eq eqn" % \begin{eqnarray*} && \PD{}{}{t} \nabla^2 \phi + U \PD{}{}{x} \nabla^2 \phi + \left( \beta - \D{2}{U}{y} \right) \PD{}{\phi}{x} = 0 \\ && \mbox{where} \ \ u' \equiv - \PD{}{\phi}{y} , \ \ v' \equiv \PD{}{\phi}{x} \end{eqnarray*} %endfilter %image "./eqn/1-vorticity-eq.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" これは線形化した渦位(渦度)方程式であることに注意 %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-basic-2D eqn" % \begin{eqnarray*} && \PD{}{u}{t} + u \PD{}{u}{x} - fv = - \frac{1}{\rho} \PD{}{p}{x} \\ && \PD{}{v}{t} + u \PD{}{v}{x} + fu = - \frac{1}{\rho} \PD{}{p}{y} \\ && \PD{}{u}{x} + \PD{}{v}{y} = 0 \end{eqnarray*} %endfilter %image "./eqn/1-basic-2D.eps" 0 350 350 1 %font "standard", fore "black" %left, fore "black" の∂/∂x(第2式)-∂/∂y(第1式)に、第3式を考慮して %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-vorticity-eq-nonlinear eqn" % \begin{eqnarray*} && \hspace*{30mm} \D{}{q}{t} = 0 \\ && \mbox{where} \ \ \D{}{}{t} \equiv \PD{}{}{t} + u \PD{}{}{x} + v \PD{}{}{y}, \ \ q \equiv \PD{}{v}{x} - \PD{}{u}{y} + f \end{eqnarray*} %endfilter %image "./eqn/1-vorticity-eq-nonlinear.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" この渦位方程式を線形化したものは %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-vorticity-eq-linear eqn" % \begin{eqnarray*} && \PD{}{q'}{t} + U \PD{}{q'}{x} + v' \D{}{Q}{y} = 0 \\ && \mbox{where} \ \ q' = \PD{}{v'}{x} - \PD{}{u'}{y} = \nabla^2 \phi, \\ && \ \ \ \ \ \ \ \ \ \D{}{Q}{y} = \D{}{}{y} \left( f - \D{}{U}{y} \right) = \beta - \D{2}{U}{y} \end{eqnarray*} %endfilter %image "./eqn/1-vorticity-eq-linear.eps" 0 350 350 1 %font "standard", fore "black" %left, fore "black" 浅水系ではラグランジュ的に保存するのは %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-shallow-PV-def eqn" % \begin{displaymath} q \equiv \frac{\displaystyle \PD{}{v}{x} - \PD{}{u}{y} + f}{H} \end{displaymath} %endfilter %image "./eqn/1-shallow-PV-def.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" Fourier 成分に分けて考える: %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-sinusoidal eqn" % \begin{displaymath} \phi(x, y, t) = \hat{\phi}(y) \exp \left[ i (kx - \omega t) \right] \end{displaymath} %endfilter %image "./eqn/1-sinusoidal.eps" 0 350 350 1 %left, fore "black" → 形式的には、方程式の中で %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-replace eqn" % \begin{displaymath} \PD{}{}{x} \rightarrow ik , \ \ \PD{}{}{t} \rightarrow -i\omega \end{displaymath} %endfilter %image "./eqn/1-replace.eps" 0 350 350 1 %left, fore "black" と置き換えればよい。 %left, fore "black" 置き換えた結果(Rayleigh 方程式): %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-Rayleigh-eq eqn" % \begin{eqnarray*} && \left( U - c \right) \left( \D{2}{\phi}{y} - k^2 \phi \right) + \D{}{Q}{y} \phi = 0 \\ && \mathrm{where} \ \ c \equiv \frac{\omega}{k} \end{eqnarray*} %endfilter %image "./eqn/1-Rayleigh-eq.eps" 0 350 350 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %left, fore "black" Rayleigh方程式を、適当な境界条件 \ (例えばφ=0 at y=±1 など)で解く。 →一般には(適当なcを与えたのでは)、両方の境界条件を\ 同時に満たすことができず、\ 境界条件を双方ともに満たすのは\ 特定のcの値(飛び飛びに無限個ある時もある)の時のみになる。 (cを固有値とする固有値問題) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定性の数学的な取り扱い %font "standard", fore "black" %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-real-imaginary-c eqn" % \begin{eqnarray*} \exp \left[ i (kx - \omega t) \right] &=& \exp \left[ i (kx - \mathrm{Re}(\omega) t) + \mathrm{Im}(\omega) t \right] \\ &=& \underbrace{ \exp \left[ i (kx - \mathrm{Re}(\omega) t) \right] }_{\mbox{oscillate with amplitude 1}\ \ \ } \times \underbrace{ \exp \left[ \mathrm{Im}(\omega) t \right] }_{\mbox{increase or decrease}} \end{eqnarray*} %endfilter %image "./eqn/1-real-imaginary-c.eps" 0 300 300 1 %left, fore "black" → Im(c)の符号によって振舞いが異なってくる Im(c) = 0: 時間が経っても振幅は変わらない。(中立) Im(c) > 0: 時間とともに振幅が増加する。(不安定) Im(c) < 0: 時間とともに振幅が減少する。(安定) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定解析の計算例 %font "standard", fore "black" %left, fore "black" 基本場 %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %%%filter "./bin/latex2eps-with-ext-dir.sh 1-basic-jump eqn" %%% %%\begin{displaymath} %% U = \left\{ \begin{array}{cl} %% U_0 & (y>0) \\ %% - U_0 & (y<0) %% \end{array} \right. %%\end{displaymath} %%%endfilter %image "./eqn/1-basic-jump.eps" 0 350 350 1 %left, fore "black" に対する計算結果 %area 0 0 5 40 %image "ncarg/jump-c.eps" 0 75 75 1 %area 0 0 40 40 %image "ncarg/jump-phi.eps" 0 125 125 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %page %bgrad 0 100 256 0 0 "white" "lightblue" 線形安定解析の計算例 %font "standard", fore "black" %left, fore "black" 基本場 %center, fore "black" %% 以下, eps ファイルを作る TeX ファイル %% %filter "./bin/latex2eps-with-ext-dir.sh 1-basic-zone eqn" % \begin{displaymath} U = \left\{ \begin{array}{cl} U_0 & (y>L) \\ \displaystyle \frac{U_0}{L} y & (-L