E5 Relevant Equations of Steady Open Channel Flow

The Chezy equation for uniform flow can be written:

The Chezy Equation

Q\displaystyle Q =ACRSo\displaystyle=AC\sqrt{RS_{o}}
=ACR1/2So1/2\displaystyle=ACR^{1/2}S_{o}^{1/2} (Eq-1)

Using the Manning’s n and Chezy C relationship C=R1/6nC=\frac{R^{1/6}}{n} we obtain the Manning’s equation for uniform flow:

The Manning’s Equation

Q\displaystyle Q =1nAR1/6RSo\displaystyle=\frac{1}{n}AR^{1/6}\sqrt{RS_{o}}
=1nAR2/3So1/2\displaystyle=\frac{1}{n}AR^{2/3}S_{o}^{1/2} (Eq-2)

Where R=A/PR=A/P and these are calculated depending on the shape of the channel cross-section e.g. Rectangular, Trapezoidal, etc. Table 1, 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 BB, is much greater than depth yy, i.e. ByB\gg y. This results in:

R\displaystyle R =AP=ByB+2yByB\displaystyle=\frac{A}{P}=\frac{By}{B+2y}\approx\frac{By}{B}
R\displaystyle R y\displaystyle\approx y

Also, for wide channels we often specify flow per unit width i.e. qq in units of m3/s/mm^{3}/s/m.

The Manning’s Equation - wide channel

Q=1nbynyn2/3So1/2=1nbyn5/3So1/2Q=\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 qq

q=1nyn5/3So1/2q=\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.

yn=(QnbSo1/2)3/5y_{n}=\left(\frac{Qn}{bS_{o}^{1/2}}\right)^{3/5} (Eq-5)
yn=(qnSo1/2)3/5y_{n}=\left(\frac{qn}{S_{o}^{1/2}}\right)^{3/5} (Eq-6)

The Chezy Equation - wide channel

Q=bCyn3/2So1/2Q=bCy_{n}^{3/2}S_{o}^{1/2} (Eq-7)

In terms of qq

q=Cyn3/2So1/2q=Cy_{n}^{3/2}S_{o}^{1/2} (Eq-8)

And the explicit normal depth calculation becomes

yn=(QbCSo1/2)2/3y_{n}=\left(\frac{Q}{bCS_{o}^{1/2}}\right)^{2/3} (Eq-9)

In terms of qq

yn=(qCSo1/2)2/3y_{n}=\left(\frac{q}{CS_{o}^{1/2}}\right)^{2/3} (Eq-10)

Froude Number FrFr

Fr\displaystyle Fr =VgDm=VgAB\displaystyle=\frac{V}{\sqrt{gD_{m}}}=\frac{V}{\sqrt{g\frac{A}{B}}} (Eq-11)
=QAgAB\displaystyle=\frac{Q}{A\sqrt{g\frac{A}{B}}} (Eq-12)
=(Q2BgA3)1/2\displaystyle=\left(\frac{Q^{2}B}{gA^{3}}\right)^{1/2} (Eq-13)


Fr2=Q2BgA3Fr^{2}=\frac{Q^{2}B}{gA^{3}} (Eq-14)

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

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


Fr2=Q2B2gy3Fr^{2}=\frac{Q^{2}}{B^{2}gy^{3}} (Eq-17)

For a wide channel

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

Critical depth, ycy_{c}

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

Fr=1=(Q2BgA3)1/2Fr=1=\left(\frac{Q^{2}B}{gA^{3}}\right)^{1/2} (Eq-19)

and similarly equation Eq-14 become

Fr2=Q2BgA3Fr^{2}=\frac{Q^{2}B}{gA^{3}} (Eq-20)

Either of these can be solved, for ycy_{c}. You must use the appropriate expressions for geometry for AA and BB.

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

1\displaystyle 1 =QBygyc\displaystyle=\frac{Q}{By\sqrt{gy_{c}}}
1\displaystyle 1 =Q2B2gyc3\displaystyle=\frac{Q^{2}}{B^{2}gy_{c}^{3}} (Eq-21)

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

yc=Q2B2g3=(Q2B2g)1/3y_{c}=\sqrt[3]{\frac{Q^{2}}{B^{2}g}}=\left(\frac{Q^{2}}{B^{2}g}\right)^{1/3} (Eq-22)

In terms of qq

yc=q2g3=(q2g)1/3y_{c}=\sqrt[3]{\frac{q^{2}}{g}}=\left(\frac{q^{2}}{g}\right)^{1/3} (Eq-23)

Critical Slope, SocSo_{c}

Critical slope, SocSo_{c} is the slope SoS_{o} in the uniform flow equation (Manning’s or Chezy) at which depth is equal to critical depth, ycy_{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:

Soc=gn2PcBcRc1/3=gn2AcBcRc4/3So_{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 PBP\approx B and thus R=yR=y this results in:

Soc=gn2yc1/3So_{c}=\frac{gn^{2}}{y_{c}^{1/3}} (Eq-25)

Bernoulli and Specific Energy

y+αV22g+z=Hy+\frac{\alpha V^{2}}{2g}+z=H (Eq-26)
Es=y+αV22gE_{s}=y+\frac{\alpha V^{2}}{2g} (Eq-27)

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

α=ρu3𝑑AρV3A\alpha=\frac{\int\rho u^{3}\;dA}{\rho V^{3}A} (Eq-28)

where V(=Q/A)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 Δz\Delta z, then

Es1=Es2+ΔzEs_{1}=E_{s}2+\Delta z (Eq-29)
y1+V122g=y2+V222g+Δzy_{1}+\frac{V_{1}^{2}}{2g}=y_{2}+\frac{V_{2}^{2}}{2g}+\Delta z (Eq-30)
y1+Q2(B1y1)22g=y2+Q2(B2y2)22g+Δzy_{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 B1B2B_{1}\neq B_{2} and thus q1q2q_{1}\neq q_{2}. In terms of qq the specific energy equation is written between points 1 and 2 as.
This is the most common form.

y1+q12y122g=y2+q22y222g+Δzy_{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

yc=23Escy_{c}=\frac{2}{3}E_{sc} (Eq-33)

Specific Force

F=Q2gA+z¯AF=\frac{Q^{2}}{gA}+\bar{z}A (Eq-34)

Momentum force FF

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

F=ρQβ(V2V1)F=\rho\>Q\;\beta(V_{2}-V_{1}) (Eq-35)

Where betabeta 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:

β=ρu2𝑑AρV2A\beta=\frac{\int\rho u^{2}\;dA}{\rho V^{2}A} (Eq-36)

Pressure Force PP

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

P=ρg(z¯1A1z¯2A2)P=\rho g(\overline{z}_{1}A_{1}-\overline{z}_{2}A_{2}) (Eq-37)

For a rectangular channel

P=ρgb12(y12y22)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 y1y_{1}, and downstream y2y_{2}, sides of the jump. These depths are know as conjugate depths and are depths of equal specific force

y2=y12(1+8Fr121)y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8F_{r1}^{2}}-1\right) (Eq-39)


y1=y22(1+8Fr221)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

ΔE=(y2y1)34y1y2\Delta E=\frac{(y_{2}-y_{1})^{3}}{4y_{1}y_{2}} (Eq-41)

Gradually Varied Flow Equation

dydx=SoSf1Fr2\frac{dy}{dx}=\frac{S_{o}-S_{f}}{1-Fr^{2}} (Eq-42)
dEdx=SoSf\frac{dE}{dx}=S_{o}-S_{f} (Eq-43)
dHdx=Sf\frac{dH}{dx}=-S_{f} (Eq-44)

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

Sf=So=(QACR1/2)2=(QnAR2/3)2S_{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

Sf=(QbCy3/2)2=(qCy3/2)2S_{f}=\left(\frac{Q}{bCy^{3/2}}\right)^{2}=\left(\frac{q}{Cy^{3/2}}\right)^{2} (Eq-46)


Sf=(Qnby5/3)2=(qny5/3)2S_{f}=\left(\frac{Qn}{by^{5/3}}\right)^{2}=\left(\frac{qn}{y^{5/3}}\right)^{2} (Eq-47)

Direct step method

Δx=Δy(1Fr2SoSf)mean\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

ΔEs=Δx(SoSf)mean\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{}_{o} indicates at the known, or initial, point), i.e. y0y_{0}:

    Δx=Δy(1Fr2SoSf)o\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{o} (Eq-50)
  2. ii)

    At the averaged depth y1/2=(y0+y1)/2y_{1/2}=(y_{0}+y_{1})/2

    Δx=Δy(1Fr2SoSf)mean=Δy(1Fry1/22SoSfy1/2)=Δy(1Fr2SoSf)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:

    Δx=Δy(1Fr2SoSf)mean\displaystyle\Delta x=\Delta y\left(\frac{1-Fr^{2}}{S_{o}-S_{f}}\right)_{mean} =Δy[(1Fr02SoSfy0)+(1Fr12SoSfy1)]/2\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.
    =Δy[(1FrSoSf)0+(1FrSoSf)1]/2\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

Δy=Δx(SoSf1Fr2)0\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 1 of three very common channel shapes may help:

Rectangle Trapezoid Circle
[Uncaptioned image]
[Uncaptioned image]
[Uncaptioned image]
(ϕ\phi in radians)
Area, AA byby (b+sy)y(b+sy)y 18(ϕsinϕ)D2\frac{1}{8}(\phi-\sin\phi)D^{2}
Wetted perimeter, PP b+2yb+2y b+2y1+s2b+2y\sqrt{1+s^{2}} 12ϕD\frac{1}{2}\phi D
Top width, BB bb b+2syb+2sy (sin(ϕ/2))D(\sin(\phi/2))D
Hydraulic Radius, R=APR=\frac{A}{P} byb+2y\frac{by}{b+2y} (b+sy)yb+2y1+x2\frac{(b+sy)y}{b+2y\sqrt{1+x^{2}}} 14(1sinϕϕ)D\frac{1}{4}\left(1-\frac{\sin\phi}{\phi}\right)D
Hydraulic mean depth, Dm=ABD_{m}=\frac{A}{B} yy (b+sy)yb+2sy\frac{(b+sy)y}{b+2sy} 18(ϕsinϕsin(ϕ/2))D\frac{1}{8}\left(\frac{\phi-\sin\phi}{\sin(\phi/2)}\right)D
Table 1: Equations of Section Properties for Rectangular, Trapezoidal and Circular sections