Dynamic equation of GVF, wide rectangular form

Dynamic equation of GVF

Assumptions: steady flow; hydrostatic pressure distribution (small slope); the channel is prismatic; the energy loss is given by the uniform-flow formula using the local depth and velocity; $\alpha = 1$.

Take a short reach $dx$ of an open channel. The total head above a datum is

$$H = z + y + \frac{V^2}{2g}$$

where $z$ is the bed level, $y$ the depth and $V = Q/A$. The bed slope is $S_0 = -dz/dx$ and the energy slope is $S_f = -dH/dx$. Differentiating with respect to $x$:

$$\frac{dH}{dx} = \frac{dz}{dx} + \frac{dy}{dx} + \frac{d}{dx}\left(\frac{V^2}{2g}\right)$$ $$-S_f = -S_0 + \frac{dy}{dx} + \frac{d}{dy}\left(\frac{Q^2}{2gA^2}\right)\frac{dy}{dx}$$

Since $dA = T\,dy$:

$$\frac{d}{dy}\left(\frac{Q^2}{2gA^2}\right) = -\frac{Q^2}{gA^3}\frac{dA}{dy} = -\frac{Q^2 T}{gA^3} = -Fr^2$$

Therefore

$$-S_f = -S_0 + \frac{dy}{dx}\left(1 - Fr^2\right)$$ $$\boxed{\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}}\qquad Fr^2 = \frac{Q^2 T}{gA^3}$$

Physical meaning: the numerator is the difference between the bed slope and the slope of the energy line. If $S_0 > S_f$, depth tends to increase in the flow direction (for subcritical flow); the denominator decides the sign by flow regime ($1 - Fr^2 > 0$ subcritical, $< 0$ supercritical). When $y = y_n$, $S_f = S_0$ and $dy/dx = 0$ (uniform flow). When $y = y_c$, $Fr = 1$ and $dy/dx \to \infty$ (the surface meets the critical depth line steeply).

Wide rectangular form using Manning's equation

For a wide rectangular channel, $R \approx y$, $A = by$ and $q = Q/b$.

Friction slope from Manning's equation $V = \frac{1}{n}R^{2/3}S_f^{1/2}$:

$$S_f = \frac{n^2 V^2}{R^{4/3}} = \frac{n^2 q^2}{y^{10/3}}$$

At normal depth $S_f = S_0$ and $y = y_n$:

$$S_0 = \frac{n^2 q^2}{y_n^{10/3}} \;\Rightarrow\; \frac{S_f}{S_0} = \left(\frac{y_n}{y}\right)^{10/3}$$

Froude number: at critical depth $q^2 = g\,y_c^3$, so

$$Fr^2 = \frac{q^2}{g y^3} = \left(\frac{y_c}{y}\right)^3$$

Substituting into the dynamic equation:

$$\boxed{\frac{dy}{dx} = \frac{S_0\left[1 - \left(\dfrac{y_n}{y}\right)^{10/3}\right]}{1 - \left(\dfrac{y_c}{y}\right)^3}}$$