10.1 Hydraulic jumps

Hydraulic jump fundamentals

A hydraulic jump is the abrupt transition from fast, shallow (supercritical) flow to slower, deeper (subcritical) flow in an open channel. This sudden change creates intense turbulence and dissipates a large fraction of the flow's kinetic energy. Engineers rely on hydraulic jumps to protect channels and structures from erosion downstream of spillways, dams, and other high-velocity outlets.

Three concepts are central to understanding hydraulic jumps: the Froude number (which tells you the flow regime), specific energy (which links depth and velocity at a cross-section), and conjugate depths (the paired depths on either side of the jump).

Supercritical vs subcritical flow

The Froude number divides open-channel flow into two regimes:

• Supercritical flow ($Fr > 1$): High velocity, shallow depth. Disturbances (like a wave from a tossed pebble) cannot travel upstream because the flow outruns them. The flow is controlled by downstream conditions.
• Subcritical flow ($Fr < 1$): Lower velocity, greater depth. Disturbances can propagate both upstream and downstream. The flow is controlled by upstream conditions.
• Critical flow ($Fr = 1$): The boundary between the two regimes. This is the condition of minimum specific energy for a given discharge.

A hydraulic jump forms when supercritical flow is forced to transition to subcritical flow, for example when fast flow from a spillway meets a deeper, slower tailwater pool.

Froude number significance

The Froude number is a dimensionless ratio of inertial forces to gravitational forces in open-channel flow:

$Fr = \frac{v}{\sqrt{gy}}$

where $v$ is the flow velocity, $g$ is gravitational acceleration, and $y$ is the flow depth.

Why does this single number matter so much? It determines whether a hydraulic jump can occur, what type of jump forms, and how much energy the jump dissipates. A higher upstream $Fr$ means a stronger, more turbulent jump with greater energy loss. Designers of spillways and stilling basins use the Froude number as the primary input for sizing energy-dissipation structures.

Specific energy concept

Specific energy ($E$) is the total mechanical energy per unit weight of fluid measured relative to the channel bed:

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

For a fixed discharge in a given channel, you can plot $E$ against depth $y$ to get the classic specific energy curve. Two key features of this curve:

• For every value of $E$ above the minimum, there are two possible depths called alternate depths: one supercritical (shallow, fast) and one subcritical (deep, slow).
• The minimum specific energy corresponds to critical flow ($Fr = 1$).

A hydraulic jump moves the flow from the supercritical alternate depth to a different, deeper depth. The downstream depth is not the subcritical alternate depth (which would conserve energy); instead it's the conjugate depth, found from momentum conservation. The difference in specific energy between the two sides of the jump is the energy lost to turbulence.

Conjugate depth relationship

The depths immediately upstream ($y_1$, supercritical) and downstream ($y_2$, subcritical) of a hydraulic jump are called conjugate depths. They are linked by momentum conservation, not energy conservation, because energy is lost in the jump.

For a horizontal, rectangular channel with negligible bed friction, the relationship is given by the Bélanger equation:

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

where $Fr_1$ is the upstream Froude number.

For example, if $Fr_1 = 3$, the equation gives $y_2/y_1 \approx 3.77$, meaning the downstream depth is nearly four times the upstream depth. This ratio is essential for sizing stilling basins: you need to know $y_2$ to set the correct tailwater level that will hold the jump in place.

Types of hydraulic jumps

Hydraulic jumps are classified primarily by the upstream Froude number, which controls the jump's strength, appearance, and energy dissipation. The boundaries between types are approximate and vary slightly across references.

Classical hydraulic jump

The classical (or "steady") hydraulic jump occurs for upstream Froude numbers roughly between 4.5 and 9. It features:

• A well-defined surface roller with strong recirculation
• Significant energy dissipation (can exceed 50–70% of upstream kinetic energy)
• A distinct, abrupt rise in the water surface

This is the type most commonly exploited in stilling basin design because it is stable, predictable, and highly effective at removing energy.

Undular jump

When $Fr_1$ is between about 1.0 and 1.7, the jump produces a train of standing waves (undulations) on the surface rather than a single abrupt rise. Energy dissipation is small, and the surface remains relatively smooth. Because undular jumps dissipate little energy, they are generally not useful for engineering energy dissipation.

Weak jump

For $Fr_1$ between roughly 1.7 and 2.5, the jump is classified as weak. The surface rise is modest, some small rollers form, and energy loss is limited. The downstream flow may show mild surface irregularities but nothing dramatic.

Oscillating jump

In the range of $Fr_1$ from about 2.5 to 4.5, the jump becomes oscillating. The toe of the jump shifts back and forth, and irregular waves can propagate downstream, potentially causing bank erosion or structural vibration. Oscillating jumps are problematic in design because their instability makes tailwater conditions unpredictable. Engineers typically try to avoid operating in this Froude number range.

Steady vs moving jumps

• A steady jump stays at a fixed location when upstream and downstream conditions are in equilibrium.
• A moving jump propagates upstream or downstream in response to changing discharge or tailwater level. For instance, a sudden increase in tailwater depth can push the jump upstream.

Moving jumps create unsteady flow and must be accounted for in design, especially for structures that experience a wide range of operating discharges.

Hydraulic jump characteristics

Beyond the basic depth and velocity changes, several physical processes inside a hydraulic jump affect its performance and the loads it imposes on structures.

Energy dissipation mechanisms

Most of the energy entering a hydraulic jump is converted to heat through three main mechanisms:

• Turbulent mixing and roller formation: The large-scale recirculating roller at the surface generates intense shear and small-scale eddies that break down kinetic energy.
• Air entrainment: Turbulence at the surface pulls air into the flow, increasing the bulk density of the air-water mixture and enhancing dissipation.
• Bed and wall friction: Boundary layers along the channel surfaces contribute additional shear losses, though these are usually secondary to the roller-driven dissipation.

Dissipation efficiency increases with the upstream Froude number. A jump with $Fr_1 = 9$ can dissipate over 70% of the incoming energy, while a jump with $Fr_1 = 2$ dissipates only about 7%.

Aeration and air entrainment

The violent surface turbulence in a hydraulic jump entrains large volumes of air. Air concentration is highest near the toe of the jump and decreases downstream as bubbles rise to the surface. The entrained air:

• Increases the bulk volume of the flow (important for freeboard design)
• Reduces cavitation risk on the channel bed by cushioning pressure fluctuations
• Enhances oxygen transfer, which is why hydraulic jumps are sometimes used in water treatment

Pressure distribution in jumps

Inside the jump, the pressure distribution is non-hydrostatic. Near the toe, strong vertical accelerations in the roller create pressures that deviate significantly from the simple $\rho g y$ hydrostatic assumption. Pressures tend to be higher near the bed and lower near the surface in the roller zone. Downstream of the roller, the pressure gradually returns to hydrostatic. Designers must account for these pressure fluctuations when specifying the structural thickness of stilling basin floors.

Velocity profiles and turbulence

Velocity profiles through a hydraulic jump are far from uniform:

• Near the bed, a high-speed forward jet persists from the incoming supercritical flow.
• Near the surface in the roller region, flow actually reverses direction (moves upstream).
• Turbulence intensity peaks in the shear layer between the forward jet and the surface roller.

Downstream of the jump, the velocity profile gradually becomes more uniform and turbulence decays, but full recovery can take 10–20 times the downstream depth.

Surface roughness effects

Roughening the channel bed (with baffle blocks, sills, or textured concrete) has several effects:

• Increases turbulence production near the bed, boosting energy dissipation
• Shortens the jump length, allowing a more compact stilling basin
• Stabilizes the jump position, reducing the tendency to oscillate

These effects are more pronounced at lower Froude numbers. At very high Froude numbers, the jump is already so turbulent that added roughness provides diminishing returns.

Hydraulic jump applications

Energy dissipaters in hydraulic structures

The most common engineering use of hydraulic jumps is downstream of spillways, outlet works, and culverts. High-velocity flow exiting these structures would scour the downstream channel if left unchecked. By designing conditions that force a hydraulic jump, engineers convert much of that kinetic energy into turbulence and heat before the flow continues downstream.

The design process involves:

1. Determine the range of discharges the structure must handle.
2. Calculate the upstream Froude number and conjugate depth for each discharge.
3. Set the tailwater elevation (using a stilling basin floor elevation or an end sill) so that the actual tailwater depth matches the required conjugate depth.
4. Size the basin length to fully contain the jump.

Stilling basins design considerations

A stilling basin is a reinforced concrete apron built to contain the hydraulic jump. Key design factors include:

• Basin length: Must be long enough to contain the full jump. A common estimate is $L_j \approx 6(y_2 - y_1)$, but specific basin types (USBR Type I through IV) have their own empirical length guidelines.
• Basin floor elevation: Set so the tailwater depth equals the required conjugate depth across the operating range.
• Appurtenances: Chute blocks at the basin entrance break up the incoming jet. Baffle blocks on the basin floor increase turbulence and shorten the jump. End sills at the downstream edge help stabilize the jump and prevent it from sweeping out.
• Discharge range: The basin must perform adequately not just at the design flood but across the full range of expected flows.

River and channel flow control

Hydraulic jumps can be induced at specific locations in rivers and channels using grade-control structures or low-head weirs. This raises the upstream water level (useful for irrigation diversions or habitat maintenance), reduces downstream velocity, and dissipates energy that would otherwise cause bed erosion. Design must consider sediment transport and ecological impacts, since a permanent jump can block fish passage.

Whitewater recreation and kayaking

Whitewater parks deliberately create hydraulic jumps and related features (holes, waves, eddies) by shaping the channel bed with concrete or boulders. The Froude number and channel geometry are tuned to produce features that are challenging but safe for paddlers. Safety design includes adequate pool depth downstream, accessible eddies for rescue, and avoidance of strongly recirculating "keeper" hydraulics that can trap swimmers.

Industrial and wastewater treatment

Hydraulic jumps promote rapid mixing and oxygen transfer, making them useful in:

• Aeration basins in wastewater treatment plants, where the entrained air supplies dissolved oxygen for biological processes
• Chemical mixing chambers, where the intense turbulence ensures rapid and uniform blending of treatment chemicals
• Pipeline energy dissipation, where a jump in an open section downstream of a pressurized pipe reduces velocity before discharge into a receiving channel

Hydraulic jump equations and calculations

Momentum conservation analysis

The hydraulic jump is analyzed using the momentum equation (not the energy equation) because energy is lost to turbulence. For a control volume enclosing the jump in a horizontal, rectangular channel:

$\frac{1}{2}\rho g y_1^2 + \rho q^2 / y_1 = \frac{1}{2}\rho g y_2^2 + \rho q^2 / y_2$

where $q$ is the discharge per unit width. The left side represents the sum of hydrostatic pressure force and momentum flux at the upstream section; the right side is the same at the downstream section. Friction along the short jump length is typically neglected.

This equation can be rearranged into the momentum function (also called the specific force):

$M = \frac{q^2}{gy} + \frac{y^2}{2}$

At the conjugate depths, $M_1 = M_2$. Setting the momentum functions equal and solving yields the Bélanger equation.

Belanger equation derivation

Starting from $M_1 = M_2$ for a rectangular channel:

1. Write: $\frac{q^2}{gy_1} + \frac{y_1^2}{2} = \frac{q^2}{gy_2} + \frac{y_2^2}{2}$
2. Substitute $q = v_1 y_1$ and $Fr_1 = v_1 / \sqrt{gy_1}$, so $q^2 = Fr_1^2 \, g \, y_1^3$.
3. Rearrange into a quadratic in $y_2/y_1$.
4. Solve the quadratic (discarding the negative root):

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

This is the Bélanger equation. It assumes a horizontal, rectangular, frictionless channel. For trapezoidal or circular channels, the momentum balance must be solved numerically.

Sequent depth ratio estimation

The sequent depth ratio $y_2/y_1$ from the Bélanger equation grows with $Fr_1$:

Knowing this ratio is critical for stilling basin design: you need the downstream depth $y_2$ to set the basin floor elevation so that the available tailwater matches the required conjugate depth.

Length of jump predictions

The jump length $L_j$ (from the toe to where the surface profile levels out) is not given by a single theoretical equation. The most widely used approximation for a classical jump in a rectangular channel is:

$L_j \approx 6(y_2 - y_1)$

More refined estimates from USBR experiments relate $L_j/y_2$ to $Fr_1$, typically giving $L_j/y_2$ values between 5 and 7 for $Fr_1$ in the range of 4 to 12. The basin must be at least this long to fully contain the jump and prevent scour downstream.

Energy loss computations

The energy lost in a hydraulic jump equals the drop in specific energy across the jump:

$\Delta E = E_1 - E_2 = \left(y_1 + \frac{v_1^2}{2g}\right) - \left(y_2 + \frac{v_2^2}{2g}\right)$

For a rectangular channel, this can be expressed purely in terms of the conjugate depths:

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

This compact formula shows that energy loss increases sharply with the difference between conjugate depths. As a quick check: for $Fr_1 = 5$ with $y_1 = 0.5$ m, you get $y_2 \approx 3.23$ m and $\Delta E \approx 2.88$ m of head lost, which is roughly 50% of the upstream specific energy. This kind of calculation is the basis for evaluating whether a stilling basin provides adequate energy dissipation.