12.4 Hydraulic Jumps and Rapidly Varied Flow

Hydraulic jumps occur when fast, shallow water suddenly transitions to slower, deeper flow. This abrupt change produces intense turbulence and significant energy loss. The jump's intensity depends on the upstream Froude number, which characterizes how "supercritical" the incoming flow is.

Rapidly varied flow refers to situations where water depth changes dramatically over a short distance. You'll encounter this at spillways, weirs, sluice gates, and other hydraulic structures. Designing these structures safely requires predicting both the flow behavior and the energy losses involved.

Hydraulic Jumps

Formation of hydraulic jumps

A hydraulic jump forms when supercritical flow ($Fr > 1$, high velocity, shallow depth) is forced to transition into subcritical flow ($Fr < 1$, low velocity, greater depth). The flow can't make this transition smoothly, so instead it "jumps" through a violent, turbulent region.

What you'll observe in a hydraulic jump:

• A sudden rise in the water surface elevation
• Intense turbulence and mixing within the jump itself
• A roller near the surface, which is a recirculating zone where water curls back upstream, along with eddies that entrain air into the flow

Several factors control whether and where a jump forms:

• Channel slope: steeper slopes generate faster flow, promoting supercritical conditions upstream of the jump
• Upstream and downstream boundary conditions: the discharge and the downstream depth (often set by a control structure or tailwater level) determine whether the depths are compatible with a jump
• Channel geometry: width, roughness, and cross-sectional shape all influence the flow regime

Classification by Froude numbers

Hydraulic jumps are classified by the upstream Froude number ($Fr_1$). Each type has a distinct appearance and dissipates a different fraction of the flow's energy.

• Undular jump ($1 < Fr_1 < 1.7$): The water surface rises through a series of smooth, standing waves rather than a sharp front. Energy dissipation is minimal.
• Weak jump ($1.7 < Fr_1 < 2.5$): A small roller develops at the surface. The jump is well-defined but energy loss is still relatively low.
• Oscillating jump ($2.5 < Fr_1 < 4.5$): The jump becomes unstable, with the toe oscillating back and forth. Irregular waves propagate downstream, making this type problematic for design because the unsteady forces can damage channel linings.
• Steady jump ($4.5 < Fr_1 < 9.0$): The most well-behaved and commonly designed-for jump. It's stable, clearly defined, and dissipates a significant amount of energy through strong turbulence. This is the range most stilling basins are designed to operate in.
• Strong jump ($Fr_1 > 9.0$): Extremely violent turbulence with large rollers and heavy air entrainment. Energy dissipation is very high, but the forces involved can be destructive if the structure isn't designed to handle them.

Energy dissipation in hydraulic jumps

The energy lost in a hydraulic jump comes from the turbulent mixing inside the jump itself. You quantify this loss as a head loss ($\Delta H$) between the upstream and downstream sections.

The head loss is given by:

$\Delta H = \frac{(y_2 - y_1)^3}{4 y_1 y_2}$

where $y_1$ is the upstream (supercritical) depth and $y_2$ is the downstream (subcritical) depth. Notice that this depends only on the two conjugate depths, not on velocity directly.

To find the relationship between those two depths, you use the sequent depth (Bélanger) equation, derived from conservation of momentum (not energy, since energy is lost):

$\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8\,Fr_1^2} - 1\right)$

The pair $y_1$ and $y_2$ are called conjugate depths. For a given $Fr_1$, there's exactly one conjugate depth ratio. As $Fr_1$ increases, the ratio $y_2 / y_1$ increases, meaning the jump produces a larger depth change and dissipates more energy.

Here's a practical workflow for solving hydraulic jump problems:

1. Determine the upstream conditions: depth $y_1$, velocity $v_1$, and discharge $q$.
2. Calculate the upstream Froude number: $Fr_1 = \frac{v_1}{\sqrt{g\,y_1}}$.
3. Use the sequent depth equation to find $y_2$.
4. Compute the head loss using the formula above.
5. Classify the jump type based on $Fr_1$ to understand its stability and character.

Rapidly Varied Flow

Flow over spillways and weirs

Rapidly varied flow occurs wherever the depth changes significantly over a short distance. Common examples include flow over spillways, flow over weirs, and flow through sluice gates and culverts.

Specific energy ties these situations together. It's defined as:

$E = y + \frac{v^2}{2g}$

This represents the total energy per unit weight measured relative to the channel bottom. For a given discharge, there's a minimum value of $E$ that corresponds to critical depth ($y_c$), where $Fr = 1$. Flow transitions between subcritical and supercritical regimes pass through this critical depth, which is why it shows up repeatedly in rapidly varied flow problems.

For a rectangular channel, critical depth is:

$y_c = \sqrt[3]{\frac{q^2}{g}}$

where $q$ is the discharge per unit width and $g$ is gravitational acceleration.

Broad-crested weirs produce nearly parallel streamlines over the crest, so the flow approaches critical depth on top of the weir. The discharge equation is:

$Q = C_d\,L\,\sqrt{2g}\;H^{3/2}$

where $C_d$ is the discharge coefficient (dependent on weir geometry), $L$ is the crest length, and $H$ is the upstream head measured above the weir crest.

Ogee spillways are shaped to match the trajectory of the lower nappe of a freely falling sheet of water (a ventilated nappe). This design keeps the water in contact with the spillway surface, minimizing sub-atmospheric pressures and reducing cavitation risk.

Sharp-crested weirs are thinner structures where the nappe springs free from the crest. The discharge formulas depend on the weir shape:

• Rectangular weir:

$Q = \frac{2}{3}\,C_d\,\sqrt{2g}\;L\;H^{3/2}$

where $C_d$ typically ranges from 0.6 to 0.65 for standard installations (values up to about 0.8 can occur with specific approach conditions and suppressed contractions).

• V-notch (triangular) weir:

$Q = \frac{8}{15}\,C_d\,\sqrt{2g}\;\tan\!\left(\frac{\theta}{2}\right)\;H^{5/2}$

where $\theta$ is the notch angle. Common angles are 90°, 60°, and 45°. V-notch weirs are particularly useful for measuring low discharges because the $H^{5/2}$ relationship makes them more sensitive to small changes in head than rectangular weirs.