# E5 Relevant Equations of Steady Open Channel Flow

The Chezy equation for uniform flow can be written:

The Chezy Equation

 $\displaystyle Q$ $\displaystyle=AC\sqrt{RS_{o}}$ $\displaystyle=ACR^{1/2}S_{o}^{1/2}$ (Eq-1)

Using the Manning’s n and Chezy C relationship $C=\frac{R^{1/6}}{n}$ we obtain the Manning’s equation for uniform flow:

The Manning’s Equation

 $\displaystyle Q$ $\displaystyle=\frac{1}{n}AR^{1/6}\sqrt{RS_{o}}$ $\displaystyle=\frac{1}{n}AR^{2/3}S_{o}^{1/2}$ (Eq-2)

Where $R=A/P$ and these are calculated depending on the shape of the channel cross-section e.g. Rectangular, Trapezoidal, etc. Table 3, below, may help.

Wide Channels
Note that when a channel is said to be wide then it is assumed to be approximately rectangular and that the width $B$, is much greater than depth $y$, i.e. $B\gg y$. This results in:

 $\displaystyle R$ $\displaystyle=\frac{A}{P}=\frac{By}{B+2y}\approx\frac{By}{B}$ $\displaystyle R$ $\displaystyle\approx y$

Also, for wide channels we often specify flow per unit width i.e. $q$ in units of $m^{3}/s/m$.

The Manning’s Equation - wide channel

$Q=\frac{1}{n}by_{n}\;y_{n}^{2/3}S_{o}^{1/2}=\frac{1}{n}by_{n}^{5/3}S_{o}^{1/2}$ (Eq-3)

In terms of $q$

$q=\frac{1}{n}y_{n}^{5/3}S_{o}^{1/2}$ (Eq-4)

This has the consequence that a normal depth calculation becomes explicit i.e.

$y_{n}=\left(\frac{Qn}{bS_{o}^{1/2}}\right)^{3/5}$ (Eq-5)
$y_{n}=\left(\frac{qn}{S_{o}^{1/2}}\right)^{3/5}$ (Eq-6)

The Chezy Equation - wide channel

$Q=bCy_{n}^{3/2}S_{o}^{1/2}$ (Eq-7)

In terms of $q$

$q=Cy_{n}^{3/2}S_{o}^{1/2}$ (Eq-8)

And the explicit normal depth calculation becomes

$y_{n}=\left(\frac{Q}{bCS_{o}^{1/2}}\right)^{2/3}$ (Eq-9)

In terms of $q$

$y_{n}=\left(\frac{q}{CS_{o}^{1/2}}\right)^{2/3}$ (Eq-10)

Froude Number $Fr$

 $\displaystyle Fr$ $\displaystyle=\frac{V}{\sqrt{gD_{m}}}=\frac{V}{\sqrt{g\frac{A}{B}}}$ (Eq-11) $\displaystyle=\frac{Q}{A\sqrt{g\frac{A}{B}}}$ (Eq-12) $\displaystyle=\left(\frac{Q^{2}B}{gA^{3}}\right)^{1/2}$ (Eq-13)

and

$Fr^{2}=\frac{Q^{2}B}{gA^{3}}$ (Eq-14)

For a rectangular channel $A=By$ and $q=Q/B$, so

 $\displaystyle Fr$ $\displaystyle=\frac{V}{\sqrt{gy}}$ (Eq-15) $\displaystyle=\frac{Q}{By\sqrt{gy}}$ (Eq-16)

and

$Fr^{2}=\frac{Q^{2}}{B^{2}gy^{3}}$ (Eq-17)

For a wide channel

$Fr=\frac{q}{y\sqrt{gy}}$ (Eq-18)

Critical depth, $y_{c}$

At critical depth $y_{c}$, $Fr=1$, so equation Eq-13 becomes

$Fr=1=\left(\frac{Q^{2}B}{gA^{3}}\right)^{1/2}$ (Eq-19)

and similarly equation Eq-14 become

$Fr^{2}=\frac{Q^{2}B}{gA^{3}}$ (Eq-20)

Either of these can be solved, for $y_{c}$. You must use the appropriate expressions for geometry for $A$ and $B$.

For a rectangular channel, $A=By$ and $B=b$ so

 $\displaystyle 1$ $\displaystyle=\frac{Q}{By\sqrt{gy_{c}}}$ $\displaystyle 1$ $\displaystyle=\frac{Q^{2}}{B^{2}gy_{c}^{3}}$ (Eq-21)

And the explicit expression for critical depth $y_{c}$ is obtained:

$y_{c}=\sqrt[3]{\frac{Q^{2}}{B^{2}g}}=\left(\frac{Q^{2}}{B^{2}g}\right)^{1/3}$ (Eq-22)

In terms of $q$

$y_{c}=\sqrt[3]{\frac{q^{2}}{g}}=\left(\frac{q^{2}}{g}\right)^{1/3}$ (Eq-23)

Critical Slope, $So_{c}$

Critical slope, $So_{c}$ is the slope $S_{o}$ in the uniform flow equation (Manning’s or Chezy) at which depth is equal to critical depth, $y_{c}$.

Rearranging the Manning’s equation Eq-2 and combining with the Froude number equated to 1, equation Eq-13 or equation Eq-14 (and using the subscript c to indicate parameters evaluated at critical depth) results in:

$So_{c}=\frac{gn^{2}P_{c}}{B_{c}R_{c}^{1/3}}=\frac{gn^{2}A_{c}}{B_{c}R_{c}^{4}/3}$ (Eq-24)

Critical Slope - wide channel
For a wide rectangular channel then as $P\approx B$ and thus $R=y$ this results in:

$So_{c}=\frac{gn^{2}}{y_{c}^{1/3}}$ (Eq-25)

Bernoulli and Specific Energy

$y+\frac{\alpha V^{2}}{2g}+z=H$ (Eq-26)
$E_{s}=y+\frac{\alpha V^{2}}{2g}$ (Eq-27)

Where $\alpha=1$ in most cases, but can be calculated using:

$\alpha=\frac{\int\rho u^{3}\;dA}{\rho V^{3}A}$ (Eq-28)

where $V\;(=Q/A)$ is the mean velocity.

Conservation of Specific Energy We nearly always assume that specific energy is conserved so from point 1 to point 2 downstream in a flow, where point 2 may be raised a small height $\Delta z$, then

$Es_{1}=E_{s}2+\Delta z$ (Eq-29)
$y_{1}+\frac{V_{1}^{2}}{2g}=y_{2}+\frac{V_{2}^{2}}{2g}+\Delta z$ (Eq-30)
$y_{1}+\frac{Q^{2}}{(B_{1}y_{1})^{2}2g}=y_{2}+\frac{Q^{2}}{(B_{2}y_{2})^{2}2g}+\Delta z$ (Eq-31)

Note that for a converging or narrowing channel the with changes and $B_{1}\neq B_{2}$ and thus $q_{1}\neq q_{2}$. In terms of $q$ the specific energy equation is written between points 1 and 2 as.
This is the most common form.

$y_{1}+\frac{q_{1}^{2}}{y_{1}^{2}2g}=y_{2}+\frac{q_{2}^{2}}{y_{2}^{2}2g}+\Delta z$ (Eq-32)

Relationship between critical depth and specific energy

$y_{c}=\frac{2}{3}E_{sc}$ (Eq-33)

Specific Force

$F=\frac{Q^{2}}{gA}+\bar{z}A$ (Eq-34)

Momentum force $F$

Momentum force (from Newton’s 2nd law), is given by

$F=\rho\>Q\;\beta(V_{2}-V_{1})$ (Eq-35)

Where $beta$ is a momentum correction factor, often taken as equal to 1, but should the flow across a section, or along a reach be very non uniform then this expression can be used:

$\beta=\frac{\int\rho u^{2}\;dA}{\rho V^{2}A}$ (Eq-36)

Pressure Force $P$

Pressure force $P$ is the force due to changes in pressure, given by:

$P=\rho g(\overline{z}_{1}A_{1}-\overline{z}_{2}A_{2})$ (Eq-37)

For a rectangular channel

$P=\rho gb\frac{1}{2}(y^{2}_{1}-y^{2}_{2})$ (Eq-38)

Hydraulic Jump

For a hydraulic jump in a rectangular channel the following relate the the depths on the upstream $y_{1}$, and downstream $y_{2}$, sides of the jump. These depths are know as conjugate depths and are depths of equal specific force

$y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8F_{r1}^{2}}-1\right)$ (Eq-39)

And

$y_{1}=\frac{y_{2}}{2}\left(\sqrt{1+8F_{r2}^{2}}-1\right)$ (Eq-40)

The equations can be manipulated to give this expression for energy loss in a jump

$\Delta E=\frac{(y_{2}-y_{1})^{3}}{4y_{1}y_{2}}$ (Eq-41)

$\frac{dy}{dx}=\frac{S_{o}-S_{f}}{1-Fr^{2}}$ (Eq-42)
$\frac{dE}{dx}=S_{o}-S_{f}$ (Eq-43)
$\frac{dH}{dx}=-S_{f}$ (Eq-44)

$S_{f}$ is calculated from the uniform flow equations (Chezy or Manning’s) where it is equated to the slope of the channel $S_{o}$. From Eq-1 and Eq-2 we get:

$S_{f}=S_{o}=\left(\frac{Q}{ACR^{1/2}}\right)^{2}=\left(\frac{Qn}{AR^{2/3}}% \right)^{2}$ (Eq-45)

And for a wide channel

$S_{f}=\left(\frac{Q}{bCy^{3/2}}\right)^{2}=\left(\frac{q}{Cy^{3/2}}\right)^{2}$ (Eq-46)

and

$S_{f}=\left(\frac{Qn}{by^{5/3}}\right)^{2}=\left(\frac{qn}{y^{5/3}}\right)^{2}$ (Eq-47)

Direct step method

$\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{mean}$ (Eq-48)

(Where the method to obtain the mean value is flexible, see below.)

Standard step method

$\Delta E_{s}=\Delta x(S_{o}-S_{f})_{mean}$ (Eq-49)

(Where the method to obtain the mean value is flexible, see below.)

Numerical Integration There are a great many of numerical methods for integrating first-order numerical method, such as the Gradually Varied Flow equation. Here three are identified that are used in this course.

For distance from depth integrations we have:

1. i)

As in the first-order Euler method (where the subscript ${}_{o}$ indicates at the known, or initial, point), i.e. $y_{0}$:

$\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{o}$ (Eq-50)
2. ii)

At the averaged depth $y_{1/2}=(y_{0}+y_{1})/2$

$\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{mean}=\Delta y% \left(\frac{1-Fr^{2}_{y{{}_{1/2}}}}{S_{o}-S_{f_{y_{1/2}}}}\right)=\Delta y% \left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{{1/2}}$ (Eq-51)
3. iii)

or the whole function averaged between the initial and subsequent point:

 $\displaystyle\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{mean}$ $\displaystyle=\Delta y\left.\left[\left(\frac{1-Fr_{0}^{2}}{S_{o}-S_{f_{y_{0}}% }}\right)+\left(\frac{1-Fr_{1}^{2}}{S_{o}-S_{f_{y_{1}}}}\right)\right]\middle/% 2\right.$ $\displaystyle=\Delta y\left.\left[\left(\frac{1-Fr}{S_{o}-S_{f}}\right)_{0}+% \left(\frac{1-Fr}{S_{o}-S_{f}}\right)_{1}\right]\middle/2\right.$ (Eq-52)

For the depth from distance formulation we solve

$\Delta y=\Delta x\left(\frac{S_{o}-S_{f}}{1-Fr^{2}}\right)_{0}\\$ (Eq-53)

This is usually applied using an Euler approach similar to i) above.

Common Channel Section geometric Formulae

All of the above expressions must be adapted for the appropriate channel geometry. Table 3 of three very common channel shapes may help: