Basics of Open channel flow

Spatially varied flow (SVF) is a steady non-uniform open channel flow in which the discharge changes along the channel length because water is added or removed along the way, either as lateral inflow or outflow. Since $Q$ changes with distance, $dQ/dx\neq0$.

Types

• Increasing discharge (water added along the length): side channel spillway, roadside gutter, drainage channel collecting from a field, wash-water trough.
• Decreasing discharge (water removed along the length): side weir / lateral spillway, irrigation channel with outlets, bottom rack (trench weir), perforated or porous channel.

 Increasing Q               Decreasing Q
  | | |  (inflow)             ~~~~~~~~~~~ ~~~~~~~~~~~~>  Q grows      ~~~~~~~~~~~~~>  Q falls
 ___________________         ____/ side weir ____

The usual momentum or energy equation for gradually varied flow is modified to include the lateral discharge term. In a side-channel spillway, for example, the depth rises along the channel as flow increases.

Gradually varied flow (GVF)

Steady non-uniform flow in which depth changes slowly over a long distance, so streamlines are nearly parallel and pressure is hydrostatic. Example: backwater curve upstream of a dam or weir, drawdown towards a free fall.

Rapidly varied flow (RVF)

Steady non-uniform flow in which depth changes abruptly over a short distance; curvature of streamlines is large and pressure is not hydrostatic. Energy loss is local and large. Example: hydraulic jump, flow over a weir or spillway crest, flow below a sluice gate.

Spatially varied flow (SVF)

Flow in which discharge changes along the channel due to lateral inflow or outflow. Example: side channel spillway, side weir, roadside gutter, drain with lateral inflow.

 GVF (backwater)      RVF (jump)          SVF (side weir)
   ____                 ___                 ~~~~~~~\ 
 ~~~     ~~~~            \_/\___            ~~~~~~~~~~>
 ~~~~~~~~~~~~~~        ~~~~~\_ ____        ____/weir____
  dam |                    jump

GVF: backwater            RVF: hydraulic jump
     ___                   
 ~~~~    ~~~~|  dam          ~~~~\___/\~~~~~
 ___________|                 ___________

SVF: increasing Q            SVF: decreasing Q
   | | |  lateral inflow       ~~~~~~~~~~\
 ~~~~~~~~~~~~~~>                ~~~~~~~~~~~~~> 
 ___________________            _____/weir___

Steady uniform flow

Flow in which depth, velocity and discharge do not change with time or with distance. The water surface, energy line and bed are parallel ($S_w=S_f=S_0$). Examples:

1. Flow in a long straight lined irrigation canal of constant section and slope.
2. Flow in a long, prismatic natural-like drainage channel or a long straight concrete flume.

Steady non-uniform flow

Flow in which discharge is constant with time but depth and velocity change along the length. It is of two kinds: gradually varied (GVF) and rapidly varied (RVF). Examples:

1. Backwater curve upstream of a dam or weir (GVF).
2. Flow through a channel with changing slope or section; drawdown at a free overfall (GVF).
3. Hydraulic jump below a spillway or sluice gate (RVF).

Spatially varied flow (SVF)

Steady flow in which discharge changes along the channel because of lateral inflow or outflow. Examples:

1. Side channel spillway, roadside gutter and drainage ditch receiving rain run-off (increasing discharge).
2. Side weir or lateral intake, irrigation canal with many outlets, trench weir or bottom rack (decreasing discharge).

Energy slope ($S_f$) is the rate of loss of total energy head per unit length of channel. It is the slope of the energy grade line (EGL).

$$S_f=-\frac{dH}{dx}=\frac{h_f}{L}$$

where $H=z+y+\dfrac{\alpha V^2}{2g}$ is the total head and $h_f$ is the friction head loss over length $L$. It is a positive number since energy falls in the flow direction.

   EGL  ------\
                 \----  slope S_f
   WSL  ---------\---   slope S_w
   Bed  -----------\--  slope S_0

For uniform flow, $S_f=S_w=S_0$ (all three lines are parallel). For non-uniform flow, $S_f\neq S_0$. In Manning's equation, $S$ is the energy slope: $S_f=\dfrac{n^2V^2}{R^{4/3}}$.

Hydraulic radius $R=\dfrac{A}{P}$ is the ratio of flow area to wetted perimeter. For a rectangle of bed width $b$ and depth $y$:

$$A=by,\qquad P=b+2y,\qquad R=\frac{by}{b+2y}$$

Variation

• For a given $b$, $R$ increases with $y$ from zero (at $y=0$) and approaches the limit $b/2$ for very large $y$.
• For a given $y$, $R$ increases with $b$ and approaches $y$ for very large $b$.

Wide rectangular channel

Divide numerator and denominator by $b$:

$$R=\frac{y}{1+2y/b}$$

For $b\gg y$, $2y/b\to0$, so

$$R\approx y$$

The side walls contribute negligible wetted perimeter, so $P\approx b$ and $R=by/b=y$.

Deep narrow channel (gorge)

Divide numerator and denominator by $y$:

$$R=\frac{b}{b/y+2}$$

For $y\gg b$, $b/y\to0$, so

$$R\approx\frac{b}{2}$$

Here the bed is small compared with the two deep walls, so $P\approx2y$ and $R=by/2y=b/2$.

 wide: R ~ y            gorge: R ~ b/2
 ~~~~~~~~~~~~~~~        |~~|
 |_____________|        |  |  y >> b

A prismatic channel is an open channel whose cross-sectional shape and size and bed slope do not change along its length. Its area is a function of depth only, $A=f(y)$.

Examples: a long straight rectangular or trapezoidal concrete canal, and a circular culvert of constant diameter and slope.

 same section along the whole length
 ____________________________________
 |  []   |   []   |   []   |   []   |   (constant section, constant slope)

Most theoretical open channel formulas (uniform flow, GVF, critical depth computation) are derived for prismatic channels.

A non-prismatic channel is an open channel whose cross-sectional shape or size, or the bed slope, changes along its length. Its area depends on both depth and distance, $A=f(y,x)$.

Examples: all natural rivers and streams, a canal with transitions (contraction or expansion), a channel with a change of bed slope, or a channel with a change in width at a bridge.

 Plan view of a transition
 ________
         \______   narrower section
 ________/ ______

Analysis (for example critical depth) of non-prismatic channels is done section by section, because properties such as $A$, $T$ and $R$ vary from one section to the next.

Hydraulic depth ($D_h$ or $\bar{D}$) is the ratio of flow area to the top width of the water surface:

$$D_h=\frac{A}{T}$$

It is the depth of a rectangular channel having the same area and top width. For a rectangular channel $D_h=y$, for a triangular channel $D_h=y/2$, and for a trapezoidal channel $D_h=\dfrac{(b+zy)y}{b+2zy}$.

It is used in the Froude number, $Fr=\dfrac{V}{\sqrt{gD_h}}$, and in the critical-flow condition $\dfrac{Q^2T}{gA^3}=1$ for non-rectangular sections. Hydraulic depth should not be confused with hydraulic radius, $R=A/P$.

Conveyance ($K$) of an open channel is a measure of its flow-carrying capacity per unit of the square root of the energy slope. From $Q=\dfrac{1}{n}AR^{2/3}S^{1/2}$ (Manning):

$$Q=K\sqrt{S_f},\qquad K=\frac{1}{n}AR^{2/3}$$

(for Chezy, $K=CAR^{1/2}$). It depends only on channel geometry (depth, shape and size) and roughness, not on the slope. Units are m³/s in SI.

Conveyance is used to compare channel sections, and to find discharge in compound sections where $K=\sum K_i$ and for computing GVF profiles.

Section factor is a geometric property of a channel section used in solving for depth. Two forms are used.

For critical flow

$$Z=A\sqrt{\frac{A}{T}}=\sqrt{\frac{A^3}{T}}$$

Critical flow occurs when $Z=\dfrac{Q}{\sqrt g}$. It depends only on depth and shape and is used to find the critical depth.

For uniform flow

$$AR^{2/3}=\frac{nQ}{\sqrt{S_0}}$$

The right side is known from the data, so the left side is found for a normal depth by trial.

Both section factors are functions of depth only; for a given channel they are tabulated or plotted against $y/D$ or $y/b$.

Gradually varied flow (GVF) is steady non-uniform flow in which the depth changes gradually along the channel length, so that streamlines are almost parallel and the pressure distribution is hydrostatic. The discharge is constant.

Examples: the backwater curve upstream of a weir or dam, drawdown curve approaching a free overfall, and flow in a canal where the slope changes from mild to steep.

        ______ GVF (backwater) 
   ~~~~~~          ~~~~~~|
   _____________________| dam

Its profile is governed by the dynamic equation

$$\frac{dy}{dx}=\frac{S_0-S_f}{1-Fr^2}$$

It is different from rapidly varied flow, in which the depth changes abruptly over a short length (for example, a hydraulic jump).

Pipe flow takes place in a closed conduit running full, driven by pressure difference. Open channel flow has a free surface exposed to atmosphere, driven by gravity.

A pipe flowing partly full behaves as open channel flow.

Gradually varied flow (GVF)

GVF is steady non-uniform flow in which the depth changes slowly from section to section over a long distance. Streamlines are nearly parallel, so the pressure is hydrostatic and the friction loss can be found from a uniform-flow formula (Manning or Chezy) applied locally. Discharge is constant: $dQ/dx=0$.

Governing (dynamic) equation:

$$\frac{dy}{dx}=\frac{S_0-S_f}{1-Fr^2}$$

Example: backwater curve upstream of a dam/weir (the depth rises gradually towards the weir); drawdown curve in a mild channel approaching a free overfall.

    backwater (GVF)
  ~~~~~~~___-----~~~~|
  y_n               ||  weir
  ____________________||

Spatially varied flow (SVF)

SVF is steady flow in which discharge varies along the channel length because water is added or removed along the way. $dQ/dx\neq0$. Momentum, not energy, is usually used since the lateral flow causes unknown energy loss at the junction.

Equation of dynamic type: $\dfrac{dy}{dx}=\dfrac{S_0-S_f-\dfrac{2Q}{gA^2}\dfrac{dQ}{dx}}{1-Fr^2}$ (for lateral inflow with negligible momentum from the inflow in the flow direction).

Example: side channel spillway or a roadside gutter (increasing Q); side weir or a canal with several outlets (decreasing Q).

Hydraulic slope (hydraulic gradient, $S_w$) is the slope of the hydraulic grade line (HGL), which in an open channel coincides with the water surface. It is the fall of the water surface (piezometric head $z+y$) per unit length of the channel:

$$S_w=-\frac{d(z+y)}{dx}$$

For uniform flow the water surface is parallel to the bed, so $S_w=S_0$ (bed slope) and also equals the energy slope $S_f$. For non-uniform flow $S_w$ differs from both $S_0$ and $S_f$.

 EGL ------\
 HGL(WSL)    ------\      S_w = slope of water surface
 Bed             ------\  S_0 = slope of bed

Definitions

• Bed slope ($S_0$): slope of the channel bottom, $S_0=-\dfrac{dz}{dx}=\sin\theta\approx\tan\theta$.
• Hydraulic slope ($S_w$): slope of the hydraulic grade line, which is the free water surface: $S_w=-\dfrac{d(z+y)}{dx}$.
• Energy slope ($S_f$): slope of the energy grade line (EGL), the rate at which total head $H=z+y+\alpha V^2/2g$ is lost: $S_f=-\dfrac{dH}{dx}$.

      EGL  -----\ ____  V1^2/2g
                   \    -----\
      WSL  ----------\        \------
                       \  y1       y2
      Bed  ----------------\--------\---
      datum ......................

Why they are not parallel in non-uniform flow

1. In uniform flow depth $y$ and velocity $V$ are constant, so $\dfrac{dy}{dx}=0$ and $\dfrac{d}{dx}\left(\dfrac{V^2}{2g}\right)=0$. Then the fall of the water surface equals the fall of the bed, and the fall of the EGL equals both: $S_0=S_w=S_f$. All three lines are parallel.
2. In non-uniform flow, depth and velocity change along the channel. Therefore $z$, $z+y$ and $z+y+V^2/2g$ fall at different rates.
3. The relations are $S_w=S_0-\dfrac{dy}{dx}$ and $S_f=S_w-\dfrac{d}{dx}\left(\dfrac{\alpha V^2}{2g}\right)$.
4. For a rising depth (backwater) $dy/dx>0$ so $S_w<S_0$; for a falling depth $S_w>S_0$. Velocity head also changes, so $S_f\neq S_w$.
5. Friction loss $S_f$ is always positive, so the EGL always falls in the flow direction.

Only when $dy/dx=0$ and $dV/dx=0$ do the three slopes become equal.

Non-uniform (varied) flow is open channel flow in which the depth, and hence the velocity, changes from section to section along the channel: $\dfrac{\partial y}{\partial x}\neq0$. In steady non-uniform flow the discharge at a given section does not change with time, but the water surface is not parallel to the bed ($S_w\neq S_0$). It is divided into:

1. Gradually varied flow (GVF) where depth changes slowly. Example: backwater curve upstream of a weir or dam; flow in a canal with a change of bed slope.
2. Rapidly varied flow (RVF) where depth changes abruptly. Example: hydraulic jump, flow over a spillway crest or below a sluice gate.
3. Spatially varied flow (SVF) where discharge also changes along the length. Example: side weir, roadside gutter.

Natural rivers are non-uniform because their section and slope change from place to place. A uniform flow is only an idealised case of constant depth in a long prismatic channel.

Nature of velocity distribution

In an open channel, velocity is zero at the bed and banks (no-slip) and increases away from the boundary. The maximum velocity does not occur at the free surface but a little below it, at about 0.05 to 0.25 of the depth below the surface. This is because of the surface shear of air and the secondary currents. The lines joining points of equal velocity are called isovels.

Mean velocity in a vertical is nearly equal to the velocity at $0.6y$ from the surface (or the average of velocities at $0.2y$ and $0.8y$).

Vertical velocity profile (any shape):

 WS ~~~~~~~~~~~~~
    |    --->
    |     ---->      <- v_max just below surface
    |   --->
    |  ->
 bed|_>  (v = 0)

Rectangular channel

Isovels are roughly parallel to the bed in the middle and curve upward near the sides. The maximum velocity is at the centre line, below the surface. In a wide channel the profile at the centre follows the logarithmic law.

 ~~~~~~~~~~~~~~~~~~~~~~~
 |   ( ( (  v_max ) ) ) |
 |   (  (    )  )  )    |
 |___(______)__________|

Triangular channel

The deepest point is at the centre. Isovels are curved, and the maximum velocity is at the centre line just below the surface. Velocity near the sloping sides is small.

 ~~~~~~~~~~~~~~~~~~
  \    (  v_max )   /
   \   (    )     /
    \   (  )    /
     \  ______/

Trapezoidal channel

Similar to the triangular and rectangular shapes. Isovels follow the sloping sides near the banks and become flatter in the middle. The maximum velocity occurs on the centre line below the surface; the lowest velocities occur in the corners where bed meets sides.

 ~~~~~~~~~~~~~~~~~~~~
  \   ( ( v_max ) )  /
   \  (  (   )  )  /
    \_(____________)/

In all shapes, the velocity distribution is affected by roughness, shape, bends and the presence of vegetation; velocity is smaller at the corners.