.. sectionauthor:: 清水康行 .. _xy2sn: ================================================ Appendix II (s-n 座標の2次元流れの基礎式) ================================================ 2次元流れ基礎方程式のs-n座標への変換 ====================================== :math:`s-n` 座標を直交曲線座標とします. :math:`s-n` 座標において, :math:`s` 軸は任意の曲線, :math:`n` 軸は `s` 軸に直交する直線座標軸とします． .. _xy_sn: .. figure:: images/0C/xy_sn.png :width: 80% : :math:`x-y` 座標と, :math:`s-n` 座標 :numref:`xy_sn` に示すように, :math:`x-y` 座標における :math:`x` 軸方向の流速を `u`, :math:`y` 軸方向の流速を :math:`v`, :math:`s-n` 座標における :math:`s` 軸方向の流速を `u_s`, :math:`n` 軸方向の流速を :math:`u_n` とします. また, :math:`s` 軸と :math:`x` 軸の角度を :math:`\theta`, :math:`s` 軸の曲率半径を :math:`r` とします． :numref:`xy_sn` に示すように, :math:`s` 軸のプラス方向に向かって :math:`\theta` が増えていくような曲がりの時の 曲率半径で :math:`r` が正になるような定義とします． 曲率の定義より, .. math:: :label: sn1 r d\theta = ds, \; \; \cfrac{\partial \theta}{\partial s}= \cfrac{1}{r}, \; \; \cfrac{\partial \theta}{\partial n}=0 :math:`u,v` と :math:`u, v` は下記の関係となります． .. math:: :label: sn2 u=u_s \cos \theta - u_n \sin \theta \\ v=u_s \sin \theta + u_n \cos \theta .. math:: :label: sn3 u_s=u \cos \theta + v \sin \theta \\ u_n=-u \sin \theta + v \cos \theta また，各偏微分は下記のような関係になります． .. math:: :label: sn4 \left( \begin{array}{c} \cfrac{\partial}{\partial x} \\ \\ \cfrac{\partial}{\partial y} \end{array} \right) = \left( \begin{array}{cc} \cos{\theta} & -\sin{\theta} \\ \\ \sin{\theta} & \cos{\theta} \end{array} \right) \left( \begin{array}{c} \cfrac{\partial}{\partial s} \\ \\ \cfrac{\partial}{\partial n} \end{array} \right) .. math:: :label: sn5 \left( \begin{array}{c} \displaystyle{{\partial}\over{\partial s}} \\ \\ \displaystyle{{\partial}\over{\partial n}} \end{array} \right) = \left( \begin{array}{cc} \cos{\theta} & \sin{\theta} \\ \\ -\sin{\theta} & \cos{\theta} \end{array} \right) \left( \begin{array}{c} \displaystyle{{\partial}\over{\partial x}} \\ \\ \displaystyle{{\partial}\over{\partial y}} \end{array} \right) また, .. math:: :label: sn6 \cfrac{\partial s}{\partial x}=\cos \theta, \; \; \cfrac{\partial s}{\partial y}=\sin \theta, \; \; \cfrac{\partial n}{\partial x}=-\sin \theta, \; \; \cfrac{\partial n}{\partial y}=\cos \theta 連続式 -------------- .. math:: :label: sn7 \cfrac{\partial h}{\partial t} +\cfrac{\partial (hu)}{\partial x} +\cfrac{\partial (hv)}{\partial y}=0 .. math:: :label: sn8 \cfrac{\partial h}{\partial t} +\cfrac{\partial}{\partial x} \left\{ h(u_s \cos \theta - u_n \sin \theta) \right\} +\cfrac{\partial}{\partial y} \left\{ h(u_s \sin \theta + u_n \cos \theta) \right\}=0 .. math:: :label: sn9 \cfrac{\partial h}{\partial t} +\cos \theta \cfrac{\partial}{\partial s} \left\{ h(u_s \cos \theta - u_n \sin \theta) \right\} -\sin \theta \cfrac{\partial}{\partial n} \left\{ h(u_s \cos \theta - u_n \sin \theta) \right\} \\ +\sin \theta \cfrac{\partial}{\partial s} \left\{ h(u_s \sin \theta + u_n \cos \theta) \right\} +\cos \theta \cfrac{\partial}{\partial n} \left\{ h(u_s \sin \theta + u_n \cos \theta) \right\}=0 .. math:: :label: sn10 \cfrac{\partial h}{\partial t} +\cos \theta \left\{ \cos\theta \cfrac{\partial(h u_s)}{\partial s}+h u_s \cfrac{\partial \cos \theta}{\partial s} -\sin\theta \cfrac{\partial(h u_n)}{\partial s}-h u_n \cfrac{\partial \sin \theta}{\partial s} \right\} \\ -\sin \theta \left\{ \cos\theta \cfrac{\partial(h u_s)}{\partial n}+h u_s \cancelto{0}{\cfrac{\partial \cos \theta}{\partial n}} -\sin\theta \cfrac{\partial(h u_n)}{\partial n}-h u_n \cancelto{0}{\cfrac{\partial \sin \theta}{\partial n}} \right\} \\ +\sin \theta \left\{ \sin\theta \cfrac{\partial(h u_s)}{\partial s}+h u_s \cfrac{\partial \sin \theta}{\partial s} +\cos\theta \cfrac{\partial(h u_n)}{\partial s}+h u_n \cfrac{\partial \cos \theta}{\partial s} \right\} \\ +\cos \theta \left\{ \sin\theta \cfrac{\partial(h u_s)}{\partial n}+h u_s \cancelto{0}{\cfrac{\partial \sin \theta}{\partial n}} +\cos\theta \cfrac{\partial(h u_n)}{\partial n}+h u_n \cancelto{0}{\cfrac{\partial \cos \theta}{\partial n}} \right\} .. math:: :label: sn11 \cfrac{\partial h}{\partial t} +\cos^2\theta\cfrac{\partial(hu_s)}{\partial s} \cancel{-\cos\theta\sin\theta hu_s\cfrac{\partial\theta}{\partial s}} \cancel{-\cos\theta\sin\theta\cfrac{\partial(hu_n)}{\partial s}}-\cos^2\theta hu_n\cfrac{\partial\theta}{\partial s} \\ \cancel{-\sin\theta\cos\theta\cfrac{\partial(hu_s)}{\partial n}} +\sin^2\theta\cfrac{\partial(hu_n)}{\partial n} \\ +\sin^2\theta\cfrac{\partial(hu_s)}{\partial s} +\cancel{\sin\theta\cos\theta hu_s\cfrac{\partial\theta}{\partial s}} \cancel{+\sin\theta\cos\theta\cfrac{\partial(hu_n)}{\partial s}}-\sin^2\theta hu_n\cfrac{\partial\theta}{\partial s} \\ \cancel{+\cos\theta\sin\theta\cfrac{\partial(hu_s)}{\partial n}} +\cos^2\theta\cfrac{\partial(hu_n)}{\partial n} .. math:: :label: sn12 \cfrac{\partial h}{\partial t} +(\cos^2\theta+\sin^2\theta)\cfrac{\partial(hu_s)}{\partial s} -(\cos^2\theta+\sin^2\theta)hu_n\cfrac{\partial\theta}{\partial s} \\ +(\cos^2\theta+\sin^2\theta)\cfrac{\partial(hu_n)}{\partial n} =0 .. math:: :label: sn13 \cfrac{\partial h}{\partial t} +\cfrac{\partial(hu_s)}{\partial s}-\cfrac{hu_n}{r} +\cfrac{\partial(hu_n)}{\partial n} =0 ここで, .. math:: :label: sn14 \cfrac{\partial(rhu_n)}{\partial n}=r\cfrac{\partial(hu_n)}{\partial n} +hu_n\cfrac{\partial r}{\partial n} \\ = r\cfrac{\partial(hu_n)}{\partial n}-hu_n なので，最終的に :math:`s-n` 座標における連続式は, .. math:: :label: sn15 \cfrac{\partial h}{\partial t} +\cfrac{\partial(hu_s)}{\partial s} +\cfrac{1}{r}\cfrac{\partial(rhu_n)}{\partial n} =0 となります． 運動方程式 ---------------------- :math:`x-y` 座標における運動方程式を再記します． .. math:: :label: sn_a1 \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+ v \frac{\partial u}{\partial y}= -g \frac{\partial H}{\partial x} -\frac{\tau_x}{\rho h}+D_x .. math:: :label: sn_a2 \frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+ v \frac{\partial v}{\partial y}= -g \frac{\partial H}{\partial y} -\frac{\tau_y}{\rho h}+D_y ただし, :math:`x, y` は互いに直交する平面座標軸, :math:`t` は時間, :math:`u, v` は :math:`x, y` 方向の水深平均流速, :math:`h` は水深, :math:`H` は水位, :math:`g` は重力加速度, :math:`\tau_x, \tau_y` は :math:`x, y` 方向の河床せん断力, :math:`\rho` は水の密度, :math:`D_x, D_y` は :math:`x, y` 方向の拡散項です. :eq:`sn_a1` 式, :eq:`sn_a2` 式を以下のように表します． .. math:: :label: sn_a3 A_x = P_x + F_x + D_x\\ A_y = P_y + F_y + D_x ただし, :math:`A_x, A_y, P_x, P_y, F_x, F_y, D_x, D_y` はそれぞれ, :math:`x, y` 方向の移流項(加速度項), 圧力勾配項, 摩擦項および拡散項です． また, これらの添え字を :math:`s,n` に変えたものを :math:`s,n` 軸方向のそれぞれ 各項とします． :math:`x, y` 軸方向の加速度(移流項) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: :label: adv1 A_x = {{\partial u}\over{\partial t}} +u{{\partial u}\over{\partial x}} +v{{\partial u}\over{\partial y}} ={{\partial u}\over{\partial t}} +u\left( \cos\theta{{\partial u}\over{\partial s}} -\sin\theta{{\partial u}\over{\partial n}} \right) +v\left( \sin\theta{{\partial u}\over{\partial s}} +\cos\theta{{\partial u}\over{\partial n}} \right) \\ ={{\partial u}\over{\partial t}}+ {{\partial u}\over{\partial s}} \underbrace{(u\cos\theta+v\sin\theta)}_{u_s} +{{\partial u}\over{\partial n}} \underbrace{(-u\sin\theta+v\cos\theta)}_{u_n} ={{\partial u}\over{\partial t}}+ u_s{{\partial u}\over{\partial s}} +u_n{{\partial u}\over{\partial n}} .. math:: :label: adv_2 A_y = {{\partial v}\over{\partial t}} +u{{\partial v}\over{\partial x}} +v{{\partial v}\over{\partial y}} ={{\partial u}\over{\partial t}} +u\left( \cos\theta{{\partial v}\over{\partial s}} -\sin\theta{{\partial v}\over{\partial n}} \right) +v\left( \sin\theta{{\partial v}\over{\partial s}} +\cos\theta{{\partial v}\over{\partial n}} \right) \\ ={{\partial v}\over{\partial t}}+ {{\partial v}\over{\partial s}} \underbrace{(u\cos\theta+v\sin\theta)}_{u_s} +{{\partial v}\over{\partial n}} \underbrace{(-u\sin\theta+v\cos\theta)}_{u_n} ={{\partial u}\over{\partial t}}+ u_s{{\partial v}\over{\partial s}} +u_n{{\partial v}\over{\partial n}} :math:`s` 軸方向の加速度(移流項) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: :label: sn_a4 A_s = & A_x\cos\theta+A_y\sin\theta =\left( {{\partial u}\over{\partial t}}+ u_s{{\partial u}\over{\partial s}} +u_n{{\partial u}\over{\partial n}} \right)\cos\theta+ \left( {{\partial v}\over{\partial t}}+ u_s{{\partial v}\over{\partial s}} +u_n{{\partial v}\over{\partial n}} \right)\sin\theta = &\cos\theta{{\partial u}\over{\partial t}} + \cos\theta u_s {{\partial}\over{\partial s}} (u_s\cos\theta-u_n\sin\theta) +\cos\theta u_n{{\partial}\over{\partial n}} (u_s\cos\theta-u_n\sin\theta) & +\sin\theta{{\partial v}\over{\partial t}} +\sin\theta u_s {{\partial}\over{\partial s}} (u_s\sin\theta+u_n\cos\theta) +\sin\theta u_n {{\partial}\over{\partial n}} (u_s\sin\theta+u_n\cos\theta) = & {{\partial u_s}\over{\partial t}} +u_s\cos^2\theta {{\partial u_s}\over{\partial s}} +u_s^2\cos\theta{{\partial(\cos\theta)}\over{\partial s}} -u_s\sin\theta \cos\theta{{\partial u_n}\over{\partial s}} -u_s u_n\cos\theta{{\partial(\sin\theta)}\over{\partial s}} & +u_n\cos^2\theta {{\partial u_s}\over{\partial n}} +u_s u_n\cos\theta{{\partial(\cos\theta)}\over{\partial n}} -u_n\sin\theta \cos\theta{{\partial u_n}\over{\partial n}} -u_n^2\cos\theta{{\partial(\sin\theta)}\over{\partial n}} & +u_s\sin^2\theta {{\partial u_s}\over{\partial s}} +u_s^2\sin\theta{{\partial(\sin\theta)}\over{\partial s}} +u_s\sin\theta \cos\theta{{\partial u_n}\over{\partial s}} +u_s u_n\sin\theta{{\partial(\cos\theta)}\over{\partial s}} & +u_n\sin^2\theta {{\partial u_s}\over{\partial n}} +u_s u_n\sin\theta{{\partial(\sin\theta)}\over{\partial n}} +u_n\sin\theta \cos\theta{{\partial u_n}\over{\partial n}} +u_n^2\sin\theta{{\partial(\cos\theta)}\over{\partial n}} = & {{\partial u_s}\over{\partial t}} +u_s{{\partial u_s}\over{\partial s}} +u_n{{\partial u_s}\over{\partial n}} & +u_s^2\cos\theta{{\partial(\cos\theta)}\over{\partial s}} -u_s u_n\cos\theta{{\partial(\sin\theta)}\over{\partial s}} +u_s u_n\cos\theta{{\partial(\cos\theta)}\over{\partial n}} -u_n^2\cos\theta{{\partial(\sin\theta)}\over{\partial n}} & +u_s^2\sin\theta{{\partial(\sin\theta)}\over{\partial s}} +u_s u_n\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +u_s u_n\sin\theta{{\partial(\sin\theta)}\over{\partial n}} +u_n^2\sin\theta{{\partial(\cos\theta)}\over{\partial n}} = & {{\partial u_s}\over{\partial t}} +u_s{{\partial u_s}\over{\partial s}} +u_n{{\partial u_s}\over{\partial n}} +u_s^2\left\{\sin\theta{{\partial(\sin\theta)}\over{\partial s}} +\cos\theta{{\partial(\cos\theta)}\over{\partial s}}\right\} & +u_s u_n\left\{ \cos\theta{{\partial(\cos\theta)}\over{\partial n}} -\cos\theta{{\partial(\sin\theta)}\over{\partial s}} +\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +\sin\theta{{\partial(\sin\theta)}\over{\partial n}} \right\} & +u_n^2\left\{ \sin\theta{{\partial(\cos\theta)}\over{\partial n}} -\cos\theta{{\partial(\sin\theta)}\over{\partial n}} \right\} ここで, .. math:: :label: sn_a5 \sin\theta{{\partial(\sin\theta)}\over{\partial s}} +\cos\theta{{\partial(\cos\theta)}\over{\partial s}} =\sin\theta\cos\theta{{\partial \theta}\over{\partial s}} -\cos\theta\sin\theta{{\partial \theta}\over{\partial s}}=0 .. math:: :label: sn_a6 & \cos\theta{{\partial(\cos\theta)}\over{\partial n}} -\cos\theta{{\partial(\sin\theta)}\over{\partial s}} +\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +\sin\theta{{\partial(\sin\theta)}\over{\partial n}} = & \cancel{-\cos\theta\sin\theta{{\partial \theta}\over{\partial n}}} -\cos^2\theta{{\partial \theta}\over{\partial s}} -\sin^2\theta{{\partial \theta}\over{\partial s}} \cancel{+\sin\theta\cos\theta{{\partial \theta}\over{\partial n}}} = &-\left(\cos^2\theta+\sin^2\theta\right){{\partial \theta}\over{\partial s}} = &-{{\partial \theta}\over{\partial s}}=-{1 \over r} また, .. math:: :label: sn_a7 \sin\theta{{\partial(\cos\theta)}\over{\partial n}} -\cos\theta{{\partial(\sin\theta)}\over{\partial n}}= -\sin^2\theta{{\partial \theta}\over{\partial n}} -\cos^2\theta{{\partial \theta}\over{\partial n}} =-{{\partial \theta}\over{\partial n}}=0 なので, :math:`s` 軸方向の移流項は次式のようになります． .. math:: :label: sn_a8 A_s= {{\partial u_s}\over{\partial t}} +u_s{{\partial u_s}\over{\partial s}} +u_n{{\partial u_s}\over{\partial n}} -{{u_s u_n}\over{r}} :math:`n` 軸方向の加速度(移流項) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: :label: nn_a4 A_n = & -A_x\sin\theta+A_y\cos\theta \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = & -\left( {{\partial u_x}\over{\partial t}}+ u_s{{\partial u_x}\over{\partial s}} +u_n{{\partial u_x}\over{\partial n}} \right)\sin\theta+ \left( {{\partial u_y}\over{\partial t}}+ u_s{{\partial u_y}\over{\partial s}} +u_n{{\partial u_y}\over{\partial n}} \right)\cos\theta = & {{\partial u_n}\over{\partial t}} -\sin\theta u_s {{\partial}\over{\partial s}} (u_s\cos\theta-u_n\sin\theta) -\sin\theta u_n{{\partial}\over{\partial n}} (u_s\cos\theta-u_n\sin\theta) & +\cos\theta u_s {{\partial}\over{\partial s}} (u_s\sin\theta+u_n\cos\theta) +\cos\theta u_n {{\partial}\over{\partial n}} (u_s\sin\theta+u_n\cos\theta) = & {{\partial u_n}\over{\partial t}} -u_s\sin\theta\cos\theta {{\partial u_s}\over{\partial s}} -u_s^2\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +u_s\sin^2\theta {{\partial u_n}\over{\partial s}} +u_s u_n\sin\theta{{\partial(\sin\theta)}\over{\partial s}} & -u_n\sin\theta\cos\theta {{\partial u_s}\over{\partial n}} -u_s u_n\sin\theta{{\partial(\cos\theta)}\over{\partial n}} +u_n\sin^2\theta {{\partial u_n}\over{\partial n}} +u_n^2\sin\theta{{\partial(\sin\theta)}\over{\partial n}} & +u_s\cos\theta\sin\theta {{\partial u_s}\over{\partial s}} +u_s^2\cos\theta{{\partial(\sin\theta)}\over{\partial s}} +u_s\cos^2\theta{{\partial u_n}\over{\partial s}} +u_s u_n\cos\theta{{\partial(\cos\theta)}\over{\partial s}} & +u_n\sin\theta\cos\theta {{\partial u_s}\over{\partial n}} +u_s u_n\cos\theta{{\partial(\sin\theta)}\over{\partial n}} +u_n\cos^2\theta{{\partial u_n}\over{\partial n}} +u_n^2\cos\theta{{\partial(\cos\theta)}\over{\partial n}} = & {{\partial u_n}\over{\partial t}} +u_s{{\partial u_n}\over{\partial s}} +u_n{{\partial u_n}\over{\partial n}} & -u_s^2\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +u_s u_n\sin\theta{{\partial(\sin\theta)}\over{\partial s}} -u_s u_n\sin\theta{{\partial(\cos\theta)}\over{\partial n}} +u_n^2\sin\theta{{\partial(\sin\theta)}\over{\partial n}} & +u_s^2\cos\theta{{\partial(\sin\theta)}\over{\partial s}} +u_s u_n\cos\theta{{\partial(\cos\theta)}\over{\partial s}} +u_s u_n\cos\theta{{\partial(\sin\theta)}\over{\partial n}} +u_n^2\cos\theta{{\partial(\cos\theta)}\over{\partial n}} = & {{\partial u_n}\over{\partial t}} +u_s{{\partial u_n}\over{\partial s}} +u_n{{\partial u_n}\over{\partial n}} +u_s^2\left\{-\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +\cos\theta{{\partial(\sin\theta)}\over{\partial s}}\right\} & +u_s u_n\left\{ \sin\theta{{\partial(\sin\theta)}\over{\partial s}} -\sin\theta{{\partial(\cos\theta)}\over{\partial n}} +\cos\theta{{\partial(\cos\theta)}\over{\partial s}} +\cos\theta{{\partial(\sin\theta)}\over{\partial n}} \right\} & +u_n^2\left\{ \sin\theta{{\partial(\sin\theta)}\over{\partial n}} +\cos\theta{{\partial(\cos\theta)}\over{\partial n}} \right\} ここで, .. math:: :label: nn_a5 -\sin\theta{{\partial(\cos\theta)}\over{\partial s}} +\cos\theta{{\partial(\sin\theta)}\over{\partial s}} =\sin^2\theta{{\partial \theta}\over{\partial s}} +\cos^2\theta{{\partial \theta}\over{\partial s}}={1 \over r} .. math:: :label: nn_a6 & \sin\theta{{\partial(\sin\theta)}\over{\partial s}} -\sin\theta{{\partial(\cos\theta)}\over{\partial n}} +\cos\theta{{\partial(\cos\theta)}\over{\partial s}} +\cos\theta{{\partial(\sin\theta)}\over{\partial n}} = & \cancel{\sin\theta\cos\theta{{\partial \theta}\over{\partial s}}} +\sin^2\theta{{\partial \theta}\over{\partial n}} \cancel{-\sin\theta\cos\theta{{\partial \theta}\over{\partial s}}} +\cos^2\theta{{\partial \theta}\over{\partial n}} = & {{\partial \theta}\over{\partial n}}=0 .. math:: :label: nn_a7 \sin\theta{{\partial(\sin\theta)}\over{\partial n}} +\cos\theta{{\partial(\cos\theta)}\over{\partial n}}= \sin\theta\cos\theta{{\partial \theta}\over{\partial n}} -\sin\theta\cos\theta{{\partial \theta}\over{\partial n}} =0 なので, :math:`n` 軸方向の移流項は次式のようになります． .. math:: :label: nn_a8 A_n= {{\partial u_n}\over{\partial t}} +u_s{{\partial u_n}\over{\partial s}} +u_n{{\partial u_n}\over{\partial n}} +{{u_s^2}\over{r}} 圧力項 ------------- :math:`x,y` 軸方向の圧力勾配は, .. math:: :label: ps_1 P_x= & -g\cfrac{\partial H}{\partial x} =-g\left(\cos\theta\cfrac{\partial H}{\partial s} -\sin\theta\cfrac{\partial H}{\partial n} \right) P_y= & -g\cfrac{\partial H}{\partial y} =-g\left(\sin\theta\cfrac{\partial H}{\partial s} +\cos\theta\cfrac{\partial H}{\partial n} \right) これを, :math:`s, n` 軸方向の圧力勾配, :math:`P_s, P_n` に変換します． .. math:: :label: ps_2 P_s= & P_x \cos\theta+P_y \sin\theta \\ = & \cos\theta\left\{ -g\left(\cos\theta\cfrac{\partial H}{\partial s} -\sin\theta\cfrac{\partial H}{\partial n} \right) \right\} +\sin\theta\left\{ -g\left(\sin\theta\cfrac{\partial H}{\partial s} +\cos\theta\cfrac{\partial H}{\partial n} \right) \right\} \\ = & g\left( -\cos^2\theta\cfrac{\partial H}{\partial s} \cancel{+\cos\theta\sin\theta\cfrac{\partial H}{\partial n}} -\sin^2\theta\cfrac{\partial H}{\partial s} \cancel{-\cos\theta\sin\theta\cfrac{\partial H}{\partial n}} \right) \\ = & -g\cfrac{\partial H}{\partial s} .. math:: :label: ps_3 P_n= & -P_x \sin\theta+P_y \cos\theta \\ = & -\sin\theta\left\{ -g\left(\cos\theta\cfrac{\partial H}{\partial s} -\sin\theta\cfrac{\partial H}{\partial n} \right) \right\} +\cos\theta\left\{ -g\left(\sin\theta\cfrac{\partial H}{\partial s} +\cos\theta\cfrac{\partial H}{\partial n} \right) \right\} \\ = & g\left( \cancel{\cos\theta\sin\theta\cfrac{\partial H}{\partial s}} -\sin^2\theta\cfrac{\partial H}{\partial n} \cancel{-\cos\theta\sin\theta\cfrac{\partial H}{\partial s}} -\cos^2\theta\cfrac{\partial H}{\partial s} \right) \\ =& -g\cfrac{\partial H}{\partial n} 摩擦項 ------------- :math:`x,y` 軸方向の摩擦項目はマニング則を用いた場合, :eq:`tauxy` より, 以下のように表されます． .. math:: :label: frc_1 F_x = & -\cfrac{\tau_x}{\rho h} = -{{g n_m^2 u \sqrt{u^2+v^2}}\over h^{4/3}} =-u V_\tau F_y = & -\cfrac{\tau_y}{\rho h} = -{{g n_m^2 u \sqrt{u^2+v^2}}\over h^{4/3}} =-v V_\tau ただし, .. math:: :label: frc_2 V_\tau = \cfrac{g n_m^2 \sqrt{u^2+v^2}}{h^{4/3}} = \cfrac{g n_m^2 \sqrt{u_s^2+u_n^2}}{h^{4/3}} :eq:`frc_1` 式を, :math:`s,n` 軸方向に変換します. :math:`F_s, F_n` を :math:`s, n` 方向の摩擦項とします． .. math:: :label: frc_3 F_s & = F_x\cos\theta+F_y\sin\theta = -u V_\tau\cos\theta-v V_\tau\sin\theta & = -(u_s \cos \theta - u_n \sin \theta)V_\tau\cos\theta -(u_s \sin \theta + u_n \cos \theta) V_\tau\sin\theta & = -u_s V_\tau \cos^2 \theta \cancel{- u_n V_\tau \sin\theta\cos\theta} -u_s V_\tau \sin^2\theta +\cancel{u_n V_\tau \sin\theta\cos\theta} & = -u_s(\sin^2\theta+\cos^2 \theta)V_\tau = -u_s V_\tau =-{{g n_m^2 u_s \sqrt{u_s^2+u_n^2}}\over h^{4/3}} .. math:: :label: frc_4 F_n & = -F_x\sin\theta+F_y\cos\theta =u V_\tau\sin\theta-v V_\tau\cos\theta & = (u_s \cos\theta - u_n \sin \theta)V_\tau\sin\theta -(u_s \sin \theta + u_n \cos \theta)V_\tau\cos\theta & = \cancel{u_s V_\tau\cos\theta\sin\theta}-u_nV_\tau\sin^2\theta -\cancel{u_s V_\tau\sin\theta\cos\theta}-u_nV_\tau\cos^2\theta & = -u_n(\sin^2\theta+\cos^2 \theta)V_\tau = -u_n V_\tau =-{{g n_m^2 u_n \sqrt{u_s^2+u_n^2}}\over h^{4/3}} 拡散項 ------------- :math:`s-n` 座標における濃度拡散方程式( :eq:`sn_dif5` 式)と同様の考え方で運動量の拡散については 以下のように表されます． .. math:: :label: frc_5 D_s & = \cfrac{\partial}{\partial s}\left(\nu_t\cfrac{\partial u_s}{\partial s}\right) +\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_s}{\partial n}\right) +\cfrac{\nu_t}{r}\cfrac{\partial u_s}{\partial n} D_n & =\cfrac{\partial}{\partial s}\left(\nu_t \cfrac{\partial u_n}{\partial s}\right) +\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_n}{\partial n}\right) +\cfrac{\nu_t}{r}\cfrac{\partial u_n}{\partial n} まとめ ------------------- 以上より, :math:`s-n` 座標における2次元自由水面流れの連続式および運動方程式は以下となります． .. math:: :label: sn_cont \cfrac{\partial h}{\partial t} +\cfrac{\partial(hu_s)}{\partial s} +\cfrac{1}{r}\cfrac{\partial(rhu_n)}{\partial n} =0 .. math:: :label: sn_mom1 {{\partial u_s}\over{\partial t}} +u_s{{\partial u_s}\over{\partial s}} +u_n{{\partial u_s}\over{\partial n}} -{{u_s u_n}\over{r}}= \hspace{4.5cm} -g\cfrac{\partial H}{\partial s} -{{g n_m^2 u_s \sqrt{u_s^2+u_n^2}}\over h^{4/3}} +\cfrac{\partial}{\partial s}\left(\nu_t\cfrac{\partial u_s}{\partial s}\right) +\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_s}{\partial n}\right) +\cfrac{\nu_t}{r}\cfrac{\partial u_s}{\partial n} .. math:: :label: sn_mom2 {{\partial u_n}\over{\partial t}} +u_s{{\partial u_n}\over{\partial s}} +u_n{{\partial u_n}\over{\partial n}} +{{u_s^2}\over{r}}= \hspace{4.5cm} -g\cfrac{\partial H}{\partial n} -{{g n_m^2 u_n \sqrt{u_s^2+u_n^2}}\over h^{4/3}} +\cfrac{\partial}{\partial s}\left(\nu_t \cfrac{\partial u_n}{\partial s}\right) +\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_n}{\partial n}\right) +\cfrac{\nu_t}{r}\cfrac{\partial u_n}{\partial n}