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: B. 音波減衰項について : 2 次元非静力学モデルの離散化 : 4. 参考文献


A. 圧力方程式 (3.9) の左辺の空間微分の書き下し

appendix-a

(3.9) 左辺の変形を行う.

$\displaystyle \Deqref{uwpi:sabun}\mbox{left side}$ $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,k}$  
    $\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{k(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k(w)}
\right\}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...t)_{k-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k-1(w)}
\right\}$  
  $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,k}$  
    $\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...elta \tau}_{i,k+1}
- \pi^{\tau + \Delta \tau}_{i,k}}{\Delta z}
\right)
\right\}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...elta \tau}_{i,k}
- \pi^{\tau + \Delta \tau}_{i,k-1}}{\Delta z}
\right)
\right\}$  
  $\textstyle =$ $\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b...
...} \bar{\theta}_{v}^{2}
\right)_{k(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k+1}$  
    $\displaystyle + \left[
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}...
...theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\right]
\pi^{\tau + \Delta \tau}_{i,k}$  
    $\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k-1}.$ (A.1)

A.1 下部境界

下部境界($k(w) = 0(w)$)について考える. この時 (3.7) 式は,

$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}$ $\textstyle =$ $\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \...
...)
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,0(w)}$  
  $\textstyle \equiv$ $\displaystyle E_{i,0(w)}$ (A.2)

となるので, (3.9) 式の左辺は, $k = 1$ の場合には,
$\displaystyle \Deqref{uwpi:sabun}\mbox{left side}$ $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,1}$  
    $\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,1(w)}
\right\}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{0(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}
\right\}$  
  $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,1}$  
    $\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...\Delta \tau}_{i,2}
- \pi^{\tau + \Delta \tau}_{i,1}}{\Delta z}
\right)
\right\}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{0(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}
\right\}$  
  $\textstyle =$ $\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b...
...ho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,2}$  
    $\displaystyle +
\left\{
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}...
...ho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\} \pi^{\tau + \Delta \tau}_{i,1}$  
    $\displaystyle + \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{0(w)}
E_{i,0(w)}$  

A.2 上部境界

上部境界($k(w) = km(w)$)について考える. (3.9) 式の左辺は,

$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}$ $\textstyle =$ $\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \...
...
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,km(w)}$  
  $\textstyle \equiv$ $\displaystyle E_{i,km(w)}$ (A.3)

となるので, (3.9) 式の左辺は, $k(w) = km(w)$ の場合には,
$\displaystyle \Deqref{uwpi:sabun}\mbox{left side}$ $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,km}$  
    $\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...ght)_{km(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}
\right\}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
..._{km-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km-1(w)}
\right\}$  
  $\textstyle =$ $\displaystyle \pi^{\tau + \Delta \tau}_{i,km}$  
    $\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}$  
    $\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}
{\bar{c_{p...
...ta \tau}_{i,km}
- \pi^{\tau + \Delta \tau}_{i,km-1}}{\Delta z}
\right)
\right\}$  
  $\textstyle =$ $\displaystyle \left\{
1 +
\beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...\bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km}$  
    $\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...ar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km-1}$  
    $\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}$  


next up previous
: B. 音波減衰項について : 2 次元非静力学モデルの離散化 : 4. 参考文献
SUGIYAMA Ko-ichiro 平成22年3月5日