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% 表題: 2 次元非静力学モデル -- 離散モデル  乱流エネルギーの時間方向の離散化
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% 履歴: 2005-02-04 杉山耕一朗
%       2005/04/13  小高正嗣
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\section{乱流運動エネルギーの式}

Klemp and Wilhelmson (1978) および CReSS (坪木と榊原篤志, 2001) と同様
に, 1.5 次のクロージャーを用いることで, 乱流エネルギーの時間発展方程式
は以下ように書ける.
%
\begin{eqnarray}
  \DP{K_{m}}{t} 
  &=& 
   - \left( 
      u \DP{K_{m}}{x} + w \DP{K_{m}}{z}
     \right)
   - \frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} 
      \left(\DP{\theta}{z} \right)
\nonumber \\
  && 
  + \left( C_{m}^{2} l^{2} \right) \left\{ 
       \left( \DP{u}{x} \right)^{2}
     + \left( \DP{w}{z} \right)^{2}
    \right\}
\nonumber \\
  &&
  +  \frac{ C_{m}^{2} l^{2} }{2}
     \left( \DP{u}{z} + \DP{w}{x}\right)^{2}
  - \frac{K_{m}}{3}
     \left( \DP{u}{x} + \DP{w}{z} \right) 
\nonumber \\
   && 
   + \Dinv{2}
       \left(\DP[2]{K_{m}^{2}}{x}
               + \DP[2]{K_{m}^{2}}{z}
       \right)
   + \left(\DP{K_{m}}{x}\right)^{2}
   + \left(\DP{K_{m}}{z}\right)^{2}
\nonumber \\
  && 
   - \Dinv{2 l^{2}} K_{m}^{2}
\Deqlab{kiso:TurbE}
\end{eqnarray}
%
ここで $C_{\varepsilon} = C_{m} = 0.2$, 
混合距離 $l = \left(\Delta x \Delta z \right)^{1/2}$ とする.


\Deqref{kiso:TurbE} 式を離散化する. CReSS にならい, 移流項を $t$ で, 
移流項以外を $t - \Delta t$ で評価する. 
%
\begin{eqnarray}
 \DP{K_{m}}{t} 
  &=& 
   - \left\{
        u_{i(u),k} \left( \DP{K_{m}}{x} \right)_{i(u), k} 
     \right\}_{i,u}^{t} 
   - \left\{
        w_{i,k(w)} \left( \DP{K_{m}}{z} \right)_{i, k(w)} 
     \right\}_{i,u}^{t}
\nonumber \\
  && 
   - \left\{
      \frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}} 
      \left(\DP{\theta}{z} \right)_{i,k(w)}
     \right\}_{i,k}^{t-\Delta t}
\nonumber \\
  && 
  + \left( C_{m}^{2} l^{2} \right)_{i,k}
    \left[
    \left\{ 
       \left( \DP{u}{x} \right)^{2} 
    \right\}_{i(u),k}
    \right]_{i,k}^{t - \Delta t}
  + \left( C_{m}^{2} l^{2} \right)_{i,k}
    \left[
    \left\{ 
       \left( \DP{w}{z} \right)^{2}
    \right\}_{i,k(w)}
    \right]_{i,k}^{t - \Delta t}
\nonumber \\
  &&
  + \left(  \frac{ C_{m}^{2} l^{2} }{2} \right)_{i,k}
    \left[
     \left\{ 
       \left( \DP{u}{z} \right)_{i(u),k(w)}
     \right\}_{i,k}^{t-\Delta t}     
    +
     \left\{ 
       \left( \DP{w}{x} \right)_{i(u),k(w)} 
     \right\}_{i,k}^{t - \Delta t}
    \right]^{2}
\nonumber \\
  &&
  - \left( \frac{K_{m}}{3} \right)_{i,k}^{t - \Delta t}
    \left\{
     \left( \DP{u}{x} \right)_{i,k}^{t-\Delta t}
  + 
     \left( \DP{w}{z} \right)_{i,k}^{t-\Delta t}
    \right\}
\nonumber \\
   && 
   + \Dinv{2}
   \left\{
       \left( 
          \DP[2]{K_{m}^{2}}{x}
       \right)_{i,k}^{t-\Delta t}
   +
       \left(
          \DP[2]{K_{m}^{2}}{z}
       \right)_{i,k}^{t-\Delta t}
   \right\}
\nonumber \\
   && 
   + 
   \left[
   \left\{
     \left(\DP{K_{m}}{x}\right)^{2}
   \right\}_{i(u),k}
   \right]_{i,k}^{t - \Delta t}
   + 
   \left[
   \left\{
     \left(\DP{K_{m}}{z}\right)^{2}
   \right\}_{i,k(w)}
   \right]_{i,k}^{t - \Delta t}
\nonumber \\
   && 
   - \Dinv{2 l^{2}} \left( K_{m}^{2} \right)_{i,k}^{t - \Delta t}
\Deqlab{TurbE_risan}
\end{eqnarray}
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