Hydraulic Jump Open Channel Flow & Energy Dissipation Calculator

1. The Hydraulic Jump Phenomenon in Civil & Water Engineering

In civil engineering hydraulics, flood control, and dam spillway design, a hydraulic jump represents the rapid transition from a supercritical shooting flow regime (characterized by high velocity, shallow depth, and Froude number $Fr > 1$) to a subcritical tranquil flow regime ($Fr < 1$, deep depth, slow velocity).

Uncontrolled high-velocity supercritical jets from dam spillways can travel at 20 to 40 meters per second, possessing immense kinetic energy capable of destroying downstream river channels, undermining bridge piers, and causing catastrophic dam toe scour. Engineers intentionally force a hydraulic jump inside a concrete stilling basin to safely dissipate up to 70–80% of excess kinetic energy into harmless heat and turbulence.

2. The Froude Number and Supercritical Flow Criteria

The Froude number ($Fr$) is the dimensionless ratio of fluid inertial forces to gravitational forces, representing the ratio of flow velocity to the propagation speed of surface gravity waves ($c = sqrt{g y}$):

• Subcritical Flow ($Fr < 1$): Flow velocity is slower than gravity wave speed. Disturbances can travel upstream against the flow; water surface responds smoothly to downstream control.
• Critical Flow ($Fr = 1$): Specific energy is at its absolute thermodynamic minimum for a given unit discharge $q = Q / B$. Critical depth is $y_c = sqrt[3]{q^2 / g}$.
• Supercritical Flow ($Fr > 1$): Flow velocity exceeds gravity wave propagation speed. Downstream disturbances cannot propagate upstream; flow is shooting and unstable.

3. The Bélanger Momentum Equation Derivation

Because intense turbulence inside the jump produces unknown viscous energy dissipation, the energy equation cannot be solved directly. Instead, applying Newton's second law (conservation of linear momentum) between section 1 (upstream) and section 2 (downstream) across a short horizontal reach where boundary wall friction is negligible:

For a rectangular channel of width $B$, hydrostatic pressure forces are $P_1 = frac{1}{2} rho g B y_1^2$ and $P_2 = frac{1}{2} rho g B y_2^2$. Substituting unit discharge $q = V_1 y_1 = V_2 y_2$:

Rearranging and solving the quadratic equation for the depth ratio yields the famous Bélanger equation (1828):

The depths $y_1$ and $y_2$ are known as conjugate (sequent) depths.

4. Energy Dissipation & Head Loss Formulation

The specific energy $E$ at any cross section is:

The head loss $Delta E = E_1 - E_2$ across the jump simplifies algebraically to:

The energy dissipation efficiency is:

5. USBR Classification of Hydraulic Jumps

The US Bureau of Reclamation (USBR) categorizes hydraulic jumps into five distinct behavioral regimes based on upstream Froude number $Fr_1$:

1. Undular Jump ($1.0 < Fr_1 le 1.7$): The water surface shows slight standing undulations or ripples with negligible surface rollers. Head loss is practically zero ($eta_d < 5%$).
2. Weak Jump ($1.7 < Fr_1 le 2.5$): A series of small surface rollers develop. Velocity distribution remains relatively uniform. Energy dissipation is low ($eta_d approx 5% - 15%$).
3. Oscillating Jump ($2.5 < Fr_1 le 4.5$): An unstable, pulsating jet oscillates periodically from channel bottom to surface. Each oscillation sends large destructive shock waves downstream that can travel for miles, damaging earthen canal banks. Engineers avoid designing stilling basins in this range.
4. Steady Jump ($4.5 < Fr_1 le 9.0$): The ideal engineering regime. A stable, well-formed turbulent roller forms whose position remains locked. Energy dissipation reaches $45% - 70%$, providing optimal quiet water exit.
5. Strong Jump ($Fr_1 > 9.0$): A violent, rugged jump with intense air entrainment and spray. Highly effective dissipation ($eta_d > 70% - 85%$), but requires heavy reinforced concrete basins to resist severe cavitation and dynamic uplift pressures.

6. Step-by-Step Worked Numerical Example: Dam Spillway Stilling Basin

Consider water discharging down a concrete spillway chute into a horizontal rectangular stilling basin:

• Upstream shooting depth $y_1 = 0.80text{ m}$
• Upstream flow velocity $V_1 = 12.0text{ m/s}$
• Basin width $B = 10.0text{ m}$ ($g = 9.80665text{ m/s}^2$)

1. Calculate Upstream Froude Number ($Fr_1$): $$Fr_1 = frac{V_1}{sqrt{g y_1}} = frac{12.0}{sqrt{9.80665 times 0.80}} = frac{12.0}{sqrt{7.8453}} = frac{12.0}{2.801} approx 4.284$$ This falls on the boundary of an Oscillating and Steady jump regime.
2. Calculate Downstream Sequent Depth ($y_2$): $$frac{y_2}{y_1} = frac{1}{2} left( sqrt{1 + 8(4.284)^2} - 1 right) = frac{1}{2} left( sqrt{1 + 8 times 18.354} - 1 right) = frac{1}{2} (sqrt{147.83} - 1) = frac{1}{2} (12.158 - 1) = 5.579$$ $$y_2 = 5.579 times 0.80text{ m} approx 4.463text{ meters} quad (14.64text{ ft})$$
3. Calculate Downstream Velocity and Froude Number ($V_2, Fr_2$): $$q = V_1 y_1 = 12.0 times 0.80 = 9.60text{ m}^2/text{s}$$ $$V_2 = frac{q}{y_2} = frac{9.60}{4.463} approx 2.151text{ m/s}$$ $$Fr_2 = frac{2.151}{sqrt{9.80665 times 4.463}} = frac{2.151}{sqrt{43.767}} = frac{2.151}{6.616} approx 0.325 quad (< 1.0 text{ Subcritical})$$
4. Calculate Head Loss ($Delta E$) and Energy Dissipation Efficiency: $$y_2 - y_1 = 4.463 - 0.80 = 3.663text{ m}$$ $$Delta E = frac{(3.663)^3}{4 times 0.80 times 4.463} = frac{49.15}{14.28} approx 3.442text{ meters of head}$$ Upstream total specific energy: $$E_1 = y_1 + frac{V_1^2}{2g} = 0.80 + frac{144.0}{19.6133} = 0.80 + 7.342 = 8.142text{ m}$$ $$eta_d = frac{Delta E}{E_1} times 100% = frac{3.442}{8.142} times 100% approx 42.27%$$ The jump safely removes $42.3%$ of the destructive mechanical kinetic energy.
5. Estimate Stilling Basin Length ($L_j$): $$L_j approx 6.0 times y_2 approx 6.0 times 4.463 approx 26.8text{ meters}$$

7. Stilling Basin Appurtenances & Scour Control Guidelines

• Chute Blocks: Placed at the entrance of the basin to serrate the incoming jet into smaller interweaving streams, lifting the jet off the floor and shortening jump length.
• Baffle Piers / Baffle Blocks: Reinforced concrete impact blocks anchored to the basin floor that absorb kinetic momentum directly through drag forces, reducing required basin length by up to $35–45%$ (e.g., USBR Type III Basins for $Fr_1 > 4.5$).
• Dentated End Sills: Positioned at the basin exit threshold to deflect bottom currents upward toward the surface, generating a reverse bottom roller that deposits sediment against the basin lip rather than scouring the natural riverbed.

8. Tailwater Rating Curve Matching: Drowned vs Repelled Jumps

In practical hydraulic design, the actual water depth in the downstream river channel ($y_t$, known as the tailwater depth) is governed by downstream channel roughness and slope:

• Jump at Basin Toe ($y_t = y_2$): The design ideal. The jump forms exactly at the toe of the spillway chute, maximizing energy dissipation within the reinforced concrete apron.
• Drowned / Submerged Jump ($y_t > y_2$): Occurs when tailwater exceeds the sequent depth. The jump is forced up the spillway face, creating a submerged surface roller with a high-velocity bottom current that can bypass the surface dissipation mechanism.
• Repelled Jump ($y_t < y_2$): The most hazardous operating condition. The tailwater is insufficient to support conjugate depth $y_2$, causing the high-velocity supercritical shooting jet to shoot completely out of the protected stilling basin into the unprotected natural alluvial riverbed, triggering catastrophic scouring and undermining.

9. Sloping Channel Bottom Modifications

When stilling basins are constructed on sloping floors (slope angle $theta > 0$), gravity adds a positive component to the downstream momentum balance:

Sloping basins (USBR Type V) are employed when topographical bedrock excavations are expensive, allowing the jump to stabilize along the inclined apron across a wide range of tailwater stage fluctuations.