Appendix II (Basic equations for 2D flow in the s-n coordinates)
Conversion of the basic 2D flow equations to the s-n coordinates
Let us assume that the \(s-n\) coordinates are Cartesian curvilinear coordinates. On the \(s-n\) coordinates, the \(s\) axis is an arbitrary curve, and the \(n\) axis is a straight coordinate axis perpendicular to the s axis.
Figure 11 : The \(x-y\) coordinate and the \(s-n\) coordinate
As shown in Figure 11, for the \(x-y\) coordinates, let the flow velocity in the \(x\) axis direction be u and the flow velocity in the \(y\) axis direction be \(v\), and for the \(s-n\) coordinates, let the flow velocity in the \(s\) axis direction be u_s and the flow velocity in the \(n\) axis direction be \(u_n\). Also, let the angle between the \(s\) axis and the \(x\) axis be \(\theta\), and the radius of curvature of the \(s\) axis be \(r\). As shown in Figure 11, let us define the radius of curvature such that \(r\) is positive for a curvature whose \(\theta\) increases in the positive direction of the \(s\) axis.
From the definition of the curvature,
(73)\[r d\theta = ds, \; \; \cfrac{\partial \theta}{\partial s}= \cfrac{1}{r}, \; \;
\cfrac{\partial \theta}{\partial n}=0\]
\(u,v\) and \(u, v\) have the following relationships.
(74)\[\begin{split}u=u_s \cos \theta - u_n \sin \theta \\
v=u_s \sin \theta + u_n \cos \theta\end{split}\]
(75)\[\begin{split}u_s=u \cos \theta + v \sin \theta \\
u_n=-u \sin \theta + v \cos \theta\end{split}\]
Each partial derivative has the following relationship.
(76)\[\begin{split}\left( \begin{array}{c}
\cfrac{\partial}{\partial x} \\
\\
\cfrac{\partial}{\partial y}
\end{array} \right)
=
\left( \begin{array}{cc}
\cos{\theta} & -\sin{\theta} \\
\\
\sin{\theta} & \cos{\theta}
\end{array} \right)
\left( \begin{array}{c}
\cfrac{\partial}{\partial s} \\
\\
\cfrac{\partial}{\partial n}
\end{array} \right)\end{split}\]
(77)\[\begin{split}\left( \begin{array}{c}
\displaystyle{{\partial}\over{\partial s}} \\
\\
\displaystyle{{\partial}\over{\partial n}}
\end{array} \right)
=
\left( \begin{array}{cc}
\cos{\theta} & \sin{\theta} \\
\\
-\sin{\theta} &
\cos{\theta}
\end{array} \right)
\left( \begin{array}{c}
\displaystyle{{\partial}\over{\partial x}} \\
\\
\displaystyle{{\partial}\over{\partial y}}
\end{array} \right)\end{split}\]
Also,
(78)\[\cfrac{\partial s}{\partial x}=\cos \theta, \; \;
\cfrac{\partial s}{\partial y}=\sin \theta, \; \;
\cfrac{\partial n}{\partial x}=-\sin \theta, \; \;
\cfrac{\partial n}{\partial y}=\cos \theta\]
Equation of continuity
(79)\[\cfrac{\partial h}{\partial t}
+\cfrac{\partial (hu)}{\partial x}
+\cfrac{\partial (hv)}{\partial y}=0\]
(80)\[\cfrac{\partial h}{\partial t}
+\cfrac{\partial}{\partial x}
\left\{ h(u_s \cos \theta - u_n \sin \theta) \right\}
+\cfrac{\partial}{\partial y}
\left\{ h(u_s \sin \theta + u_n \cos \theta) \right\}=0\]
(81)\[\begin{split}\cfrac{\partial h}{\partial t}
+\cos \theta \cfrac{\partial}{\partial s}
\left\{ h(u_s \cos \theta - u_n \sin \theta) \right\}
-\sin \theta \cfrac{\partial}{\partial n}
\left\{ h(u_s \cos \theta - u_n \sin \theta) \right\} \\
+\sin \theta \cfrac{\partial}{\partial s}
\left\{ h(u_s \sin \theta + u_n \cos \theta) \right\}
+\cos \theta \cfrac{\partial}{\partial n}
\left\{ h(u_s \sin \theta + u_n \cos \theta) \right\}=0\end{split}\]
(82)\[\begin{split}\cfrac{\partial h}{\partial t}
+\cos \theta \left\{
\cos\theta \cfrac{\partial(h u_s)}{\partial s}+h u_s \cfrac{\partial \cos \theta}{\partial s}
-\sin\theta \cfrac{\partial(h u_n)}{\partial s}-h u_n \cfrac{\partial \sin \theta}{\partial s}
\right\} \\
-\sin \theta \left\{
\cos\theta \cfrac{\partial(h u_s)}{\partial n}+h u_s \cancelto{0}{\cfrac{\partial \cos \theta}{\partial n}}
-\sin\theta \cfrac{\partial(h u_n)}{\partial n}-h u_n \cancelto{0}{\cfrac{\partial \sin \theta}{\partial n}}
\right\} \\
+\sin \theta \left\{
\sin\theta \cfrac{\partial(h u_s)}{\partial s}+h u_s \cfrac{\partial \sin \theta}{\partial s}
+\cos\theta \cfrac{\partial(h u_n)}{\partial s}+h u_n \cfrac{\partial \cos \theta}{\partial s}
\right\} \\
+\cos \theta \left\{
\sin\theta \cfrac{\partial(h u_s)}{\partial n}+h u_s \cancelto{0}{\cfrac{\partial \sin \theta}{\partial n}}
+\cos\theta \cfrac{\partial(h u_n)}{\partial n}+h u_n \cancelto{0}{\cfrac{\partial \cos \theta}{\partial n}}
\right\}\end{split}\]
(83)\[\begin{split}\cfrac{\partial h}{\partial t}
+\cos^2\theta\cfrac{\partial(hu_s)}{\partial s} \cancel{-\cos\theta\sin\theta hu_s\cfrac{\partial\theta}{\partial s}}
\cancel{-\cos\theta\sin\theta\cfrac{\partial(hu_n)}{\partial s}}-\cos^2\theta hu_n\cfrac{\partial\theta}{\partial s} \\
\cancel{-\sin\theta\cos\theta\cfrac{\partial(hu_s)}{\partial n}}
+\sin^2\theta\cfrac{\partial(hu_n)}{\partial n} \\
+\sin^2\theta\cfrac{\partial(hu_s)}{\partial s} +\cancel{\sin\theta\cos\theta hu_s\cfrac{\partial\theta}{\partial s}}
\cancel{+\sin\theta\cos\theta\cfrac{\partial(hu_n)}{\partial s}}-\sin^2\theta hu_n\cfrac{\partial\theta}{\partial s} \\
\cancel{+\cos\theta\sin\theta\cfrac{\partial(hu_s)}{\partial n}}
+\cos^2\theta\cfrac{\partial(hu_n)}{\partial n}\end{split}\]
(84)\[\begin{split}\cfrac{\partial h}{\partial t}
+(\cos^2\theta+\sin^2\theta)\cfrac{\partial(hu_s)}{\partial s}
-(\cos^2\theta+\sin^2\theta)hu_n\cfrac{\partial\theta}{\partial s} \\
+(\cos^2\theta+\sin^2\theta)\cfrac{\partial(hu_n)}{\partial n} =0\end{split}\]
(85)\[\cfrac{\partial h}{\partial t}
+\cfrac{\partial(hu_s)}{\partial s}-\cfrac{hu_n}{r}
+\cfrac{\partial(hu_n)}{\partial n} =0\]
Where,
(86)\[\begin{split}\cfrac{\partial(rhu_n)}{\partial n}=r\cfrac{\partial(hu_n)}{\partial n}
+hu_n\cfrac{\partial r}{\partial n} \\
= r\cfrac{\partial(hu_n)}{\partial n}-hu_n\end{split}\]
Finally, the equation of continuity on the \(s-n\) axis is…
(87)\[\cfrac{\partial h}{\partial t}
+\cfrac{\partial(hu_s)}{\partial s}
+\cfrac{1}{r}\cfrac{\partial(rhu_n)}{\partial n} =0\]
Momentum equation
Let us review the equations of motion for the \(x-y\) axis.
(88)\[\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+
v \frac{\partial u}{\partial y}=
-g \frac{\partial H}{\partial x}
-\frac{\tau_x}{\rho h}+D_x\]
(89)\[\frac{\partial v}{\partial t}+u \frac{\partial v}{\partial x}+
v \frac{\partial v}{\partial y}=
-g \frac{\partial H}{\partial y}
-\frac{\tau_y}{\rho h}+D_y\]
Where, \(x, y\) are plane coordinate axes that are mutually perpendicular, \(t\) is time, \(u, v\) are the depth-averaged velocity in the \(x, y\) directions, \(h\) is the water depth, \(H\) is the water level, \(g\) is the gravitational acceleration, \(\tau_x, \tau_y\) are the riverbed shear force in the directions of \(x, y\), \(\rho\) is the density of water and \(D_x, D_y\) are the diffusion term in the \(x, y\) directions.
Then, Equation (88) and Equation (89) are expressed as follows.
(90)\[\begin{split}A_x = P_x + F_x + D_x\\
A_y = P_y + F_y + D_x\end{split}\]
Where, \(A_x, A_y, P_x, P_y, F_x, F_y, D_x, D_y\) respectively express the advection term (acceleration term), the pressure gradient term, the friction term, and the diffusion terms in the directions of \(x, y\). By changing the subscript of each term to \(s,n\), we have each term in the \(s,n\) axis directions.
The advection terms
Acceleration in the \(x, y\) directions can be written as follows:
(91)\[\begin{split}A_x = {{\partial u}\over{\partial t}}
+u{{\partial u}\over{\partial x}}
+v{{\partial u}\over{\partial y}}
={{\partial u}\over{\partial t}}
+u\left(
\cos\theta{{\partial u}\over{\partial s}}
-\sin\theta{{\partial u}\over{\partial n}} \right)
+v\left(
\sin\theta{{\partial u}\over{\partial s}}
+\cos\theta{{\partial u}\over{\partial n}} \right) \\
={{\partial u}\over{\partial t}}+
{{\partial u}\over{\partial s}}
\underbrace{(u\cos\theta+v\sin\theta)}_{u_s}
+{{\partial u}\over{\partial n}}
\underbrace{(-u\sin\theta+v\cos\theta)}_{u_n}
={{\partial u}\over{\partial t}}+
u_s{{\partial u}\over{\partial s}}
+u_n{{\partial u}\over{\partial n}}\end{split}\]
(92)\[\begin{split}A_y = {{\partial v}\over{\partial t}}
+u{{\partial v}\over{\partial x}}
+v{{\partial v}\over{\partial y}}
={{\partial u}\over{\partial t}}
+u\left(
\cos\theta{{\partial v}\over{\partial s}}
-\sin\theta{{\partial v}\over{\partial n}} \right)
+v\left(
\sin\theta{{\partial v}\over{\partial s}}
+\cos\theta{{\partial v}\over{\partial n}} \right) \\
={{\partial v}\over{\partial t}}+
{{\partial v}\over{\partial s}}
\underbrace{(u\cos\theta+v\sin\theta)}_{u_s}
+{{\partial v}\over{\partial n}}
\underbrace{(-u\sin\theta+v\cos\theta)}_{u_n}
={{\partial u}\over{\partial t}}+
u_s{{\partial v}\over{\partial s}}
+u_n{{\partial v}\over{\partial n}}\end{split}\]
Acceleration along the channel downstream direction
Then the acceleration in the \(s\) directions becomes:
(93)\[ \begin{align}\begin{aligned}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\}\end{aligned}\end{align} \]
Where,
(94)\[\sin\theta{{\partial(\sin\theta)}\over{\partial s}}
+\cos\theta{{\partial(\cos\theta)}\over{\partial s}}
=\sin\theta\cos\theta{{\partial \theta}\over{\partial s}}
-\cos\theta\sin\theta{{\partial \theta}\over{\partial s}}=0\]
(95)\[ \begin{align}\begin{aligned}& \cos\theta{{\partial(\cos\theta)}\over{\partial n}}
-\cos\theta{{\partial(\sin\theta)}\over{\partial s}}
+\sin\theta{{\partial(\cos\theta)}\over{\partial s}}
+\sin\theta{{\partial(\sin\theta)}\over{\partial n}}\\= & \cancel{-\cos\theta\sin\theta{{\partial \theta}\over{\partial n}}}
-\cos^2\theta{{\partial \theta}\over{\partial s}}
-\sin^2\theta{{\partial \theta}\over{\partial s}}
\cancel{+\sin\theta\cos\theta{{\partial \theta}\over{\partial n}}}\\= &-\left(\cos^2\theta+\sin^2\theta\right){{\partial \theta}\over{\partial s}}\\= &-{{\partial \theta}\over{\partial s}}=-{1 \over r}\end{aligned}\end{align} \]
Also,
(96)\[\sin\theta{{\partial(\cos\theta)}\over{\partial n}}
-\cos\theta{{\partial(\sin\theta)}\over{\partial n}}=
-\sin^2\theta{{\partial \theta}\over{\partial n}}
-\cos^2\theta{{\partial \theta}\over{\partial n}}
=-{{\partial \theta}\over{\partial n}}=0\]
Therefore, the advection term in the \(s\) axis direction is as follows.
(97)\[A_s=
{{\partial u_s}\over{\partial t}}
+u_s{{\partial u_s}\over{\partial s}}
+u_n{{\partial u_s}\over{\partial n}}
-{{u_s u_n}\over{r}}\]
Acceleration in the transverse direction
Acceleration in the channel transverse direction, \(n\) axis directions can be written as,
(98)\[ \begin{align}\begin{aligned}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\}\end{aligned}\end{align} \]
Where,
(99)\[-\sin\theta{{\partial(\cos\theta)}\over{\partial s}}
+\cos\theta{{\partial(\sin\theta)}\over{\partial s}}
=\sin^2\theta{{\partial \theta}\over{\partial s}}
+\cos^2\theta{{\partial \theta}\over{\partial s}}={1 \over r}\]
(100)\[ \begin{align}\begin{aligned}& \sin\theta{{\partial(\sin\theta)}\over{\partial s}}
-\sin\theta{{\partial(\cos\theta)}\over{\partial n}}
+\cos\theta{{\partial(\cos\theta)}\over{\partial s}}
+\cos\theta{{\partial(\sin\theta)}\over{\partial n}}\\= & \cancel{\sin\theta\cos\theta{{\partial \theta}\over{\partial s}}}
+\sin^2\theta{{\partial \theta}\over{\partial n}}
\cancel{-\sin\theta\cos\theta{{\partial \theta}\over{\partial s}}}
+\cos^2\theta{{\partial \theta}\over{\partial n}}\\= & {{\partial \theta}\over{\partial n}}=0\end{aligned}\end{align} \]
(101)\[\sin\theta{{\partial(\sin\theta)}\over{\partial n}}
+\cos\theta{{\partial(\cos\theta)}\over{\partial n}}=
\sin\theta\cos\theta{{\partial \theta}\over{\partial n}}
-\sin\theta\cos\theta{{\partial \theta}\over{\partial n}}
=0\]
Therefore, the advection term in the \(n\) axis direction is as follows.
(102)\[A_n=
{{\partial u_n}\over{\partial t}}
+u_s{{\partial u_n}\over{\partial s}}
+u_n{{\partial u_n}\over{\partial n}}
+{{u_s^2}\over{r}}\]
Pressure term
The pressure gradient in the \(x,y\) axis directions are…
(103)\[ \begin{align}\begin{aligned}P_x= & -g\cfrac{\partial H}{\partial x}
=-g\left(\cos\theta\cfrac{\partial H}{\partial s}
-\sin\theta\cfrac{\partial H}{\partial n}
\right)\\P_y= & -g\cfrac{\partial H}{\partial y}
=-g\left(\sin\theta\cfrac{\partial H}{\partial s}
+\cos\theta\cfrac{\partial H}{\partial n}
\right)\end{aligned}\end{align} \]
Convert these to the pressure gradients, \(P_s, P_n\), in the \(s, n\) axis directions.
(104)\[\begin{split}P_s= & P_x \cos\theta+P_y \sin\theta \\
= & \cos\theta\left\{
-g\left(\cos\theta\cfrac{\partial H}{\partial s}
-\sin\theta\cfrac{\partial H}{\partial n}
\right)
\right\}
+\sin\theta\left\{
-g\left(\sin\theta\cfrac{\partial H}{\partial s}
+\cos\theta\cfrac{\partial H}{\partial n}
\right)
\right\} \\
= & g\left(
-\cos^2\theta\cfrac{\partial H}{\partial s}
\cancel{+\cos\theta\sin\theta\cfrac{\partial H}{\partial n}}
-\sin^2\theta\cfrac{\partial H}{\partial s}
\cancel{-\cos\theta\sin\theta\cfrac{\partial H}{\partial n}}
\right) \\
= & -g\cfrac{\partial H}{\partial s}\end{split}\]
(105)\[\begin{split}P_n= & -P_x \sin\theta+P_y \cos\theta \\
= & -\sin\theta\left\{
-g\left(\cos\theta\cfrac{\partial H}{\partial s}
-\sin\theta\cfrac{\partial H}{\partial n}
\right)
\right\}
+\cos\theta\left\{
-g\left(\sin\theta\cfrac{\partial H}{\partial s}
+\cos\theta\cfrac{\partial H}{\partial n}
\right)
\right\} \\
= & g\left(
\cancel{\cos\theta\sin\theta\cfrac{\partial H}{\partial s}}
-\sin^2\theta\cfrac{\partial H}{\partial n}
\cancel{-\cos\theta\sin\theta\cfrac{\partial H}{\partial s}}
-\cos^2\theta\cfrac{\partial H}{\partial s}
\right) \\
=& -g\cfrac{\partial H}{\partial n}\end{split}\]
Friction term
From (13), when Manning’s law is adopted, the friction term in the \(x,y\) axis directions are expressed as follows:
(106)\[ \begin{align}\begin{aligned}F_x = & -\cfrac{\tau_x}{\rho h}
= -{{g n_m^2 u \sqrt{u^2+v^2}}\over h^{4/3}} =-u V_\tau\\F_y = & -\cfrac{\tau_y}{\rho h}
= -{{g n_m^2 u \sqrt{u^2+v^2}}\over h^{4/3}} =-v V_\tau\end{aligned}\end{align} \]
Where,
(107)\[V_\tau = \cfrac{g n_m^2 \sqrt{u^2+v^2}}{h^{4/3}}
= \cfrac{g n_m^2 \sqrt{u_s^2+u_n^2}}{h^{4/3}}\]
Convert Equation (106) to the \(s,n\) axis directions. Assume \(F_s, F_n\) are the friction term in the \(s, n\) directions.
(108)\[ \begin{align}\begin{aligned}F_s & = F_x\cos\theta+F_y\sin\theta = -u V_\tau\cos\theta-v V_\tau\sin\theta\\& = -(u_s \cos \theta - u_n \sin \theta)V_\tau\cos\theta
-(u_s \sin \theta + u_n \cos \theta) V_\tau\sin\theta\\& = -u_s V_\tau \cos^2 \theta \cancel{- u_n V_\tau \sin\theta\cos\theta}
-u_s V_\tau \sin^2\theta +\cancel{u_n V_\tau \sin\theta\cos\theta}\\& = -u_s(\sin^2\theta+\cos^2 \theta)V_\tau = -u_s V_\tau
=-{{g n_m^2 u_s \sqrt{u_s^2+u_n^2}}\over h^{4/3}}\end{aligned}\end{align} \]
(109)\[ \begin{align}\begin{aligned}F_n & = -F_x\sin\theta+F_y\cos\theta =u V_\tau\sin\theta-v V_\tau\cos\theta\\& = (u_s \cos\theta - u_n \sin \theta)V_\tau\sin\theta
-(u_s \sin \theta + u_n \cos \theta)V_\tau\cos\theta\\& = \cancel{u_s V_\tau\cos\theta\sin\theta}-u_nV_\tau\sin^2\theta
-\cancel{u_s V_\tau\sin\theta\cos\theta}-u_nV_\tau\cos^2\theta\\& = -u_n(\sin^2\theta+\cos^2 \theta)V_\tau = -u_n V_\tau
=-{{g n_m^2 u_n \sqrt{u_s^2+u_n^2}}\over h^{4/3}}\end{aligned}\end{align} \]
Diffusion term
Similar to the density diffusion equation (Equation (119)) on the \(s-n\) coordinates, the diffusion of momentums are expressed as follows.
(110)\[ \begin{align}\begin{aligned}D_s & =
\cfrac{\partial}{\partial s}\left(\nu_t\cfrac{\partial u_s}{\partial s}\right)
+\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_s}{\partial n}\right)
+\cfrac{\nu_t}{r}\cfrac{\partial u_s}{\partial n}\\D_n & =\cfrac{\partial}{\partial s}\left(\nu_t \cfrac{\partial u_n}{\partial s}\right)
+\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_n}{\partial n}\right)
+\cfrac{\nu_t}{r}\cfrac{\partial u_n}{\partial n}\end{aligned}\end{align} \]
Summary
The equations of motion and the equations of continuity of 2D free surface flow on the \(s-n\) coordinates are as follows.
(111)\[\cfrac{\partial h}{\partial t}
+\cfrac{\partial(hu_s)}{\partial s}
+\cfrac{1}{r}\cfrac{\partial(rhu_n)}{\partial n} =0\]
(112)\[ \begin{align}\begin{aligned}{{\partial u_s}\over{\partial t}}
+u_s{{\partial u_s}\over{\partial s}}
+u_n{{\partial u_s}\over{\partial n}}
-{{u_s u_n}\over{r}}= \hspace{4.5cm}\\-g\cfrac{\partial H}{\partial s}
-{{g n_m^2 u_s \sqrt{u_s^2+u_n^2}}\over h^{4/3}}
+\cfrac{\partial}{\partial s}\left(\nu_t\cfrac{\partial u_s}{\partial s}\right)
+\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_s}{\partial n}\right)
+\cfrac{\nu_t}{r}\cfrac{\partial u_s}{\partial n}\end{aligned}\end{align} \]
(113)\[ \begin{align}\begin{aligned}{{\partial u_n}\over{\partial t}}
+u_s{{\partial u_n}\over{\partial s}}
+u_n{{\partial u_n}\over{\partial n}}
+{{u_s^2}\over{r}}= \hspace{4.5cm}\\-g\cfrac{\partial H}{\partial n}
-{{g n_m^2 u_n \sqrt{u_s^2+u_n^2}}\over h^{4/3}}
+\cfrac{\partial}{\partial s}\left(\nu_t \cfrac{\partial u_n}{\partial s}\right)
+\cfrac{\partial}{\partial n}\left(\nu_t\cfrac{\partial u_n}{\partial n}\right)
+\cfrac{\nu_t}{r}\cfrac{\partial u_n}{\partial n}\end{aligned}\end{align} \]