If you're interested in hydraulic engineering, this hydraulic jump calculator will be right up your alley. With its help, you can determine all characteristics of a hydraulic jump, including the flow velocity up- and downstream, the height and length of the jump, and the total head loss. We will also provide you with a Froude number equation and teach how to use it to identify the type of hydraulic jump.
Make sure to take a look at the Reynolds number calculator, too!
What is a hydraulic jump?
By definition, hydraulic jump happens when the flow of a fluid changes from supercritical to subcritical. The abrupt change in flow characteristics is accompanied by substantial energy losses, as well as turbulence in the flow.
The video below illustrates an example of a hydraulic jump (which occurs at the end of the video). You can observe a change of water level, as well as a turbulent transition zone.
What exactly are the supercritical and subcritical flows, then? To understand these terms, you first need to know what is a critical depth - depth of flow in which the energy is at a minimum for a given discharge
Q. Once you know the critical depth, you can instantly distinguish between
Supercritical flow (also called a rapid flow) - flow depth is lower than the critical depth;
Subcritical flow (also called a slow flow) - flow depth is higher than the critical depth.
The Froude number equation
Instead of analyzing the energy of the flow, hydraulic engineers use the Froude number to determine whether a flow is supercritical or subcritical. For any open channel flow, it can be calculated with the following formula:
Fr = v / √(g * D)
Fris the Froude number of the flow,
vis the flow velocity,
gis the gravitational acceleration,
Dis the flow depth.
If the value of the Froude number is greater than one, the flow is supercritical. If it's smaller than one, the flow is supercritical.
Let's examine the Froude number equation more closely. As it is a dimensionless number, the denominator must be expressed in the units of speed - meters per second or feet per second. But does this value have any physical interpretation?
In fact, it does. The value in the denominator,
√(g * D), is the velocity of wave propagation, also called the wave celerity. For example, if you threw a pebble into the water, it would create ripples on the surface that would propagate with this celerity.
If the flow is supercritical (Fr > 1), the velocity of the flow is higher than the celerity. It means that any waves or disturbances in the flow are transmitted downstream.
On the other hand, if the flow is subcritical (Fr < 1), the velocity of the flow is smaller than the celerity. Waves and disturbances in the flow will propagate upstream.
What happens in the case of a critical flow? As Froude number is equal to 1, velocity will be equal to celerity. In this scenario, any disturbance in the flow will stay in one place, moving neither upstream nor downstream.
Calculating the hydraulic jump properties
Our hydraulic jump calculator analyzes the phenomenon of a hydraulic jump in detail. It finds multiple characteristics, even including jump efficiency.
Remember that these formulas are only valid for the following assumptions:
- We're considering an open channel flow;
- The channel is rectangular and horizontal (without a slope);
- The jump occurs from a supercritical to subcritical flow.
1. Flow rate formula
For both upstream and downstream flows, the discharge
Q is equal to flow velocity
v multiplied by the surface area of the channel cross-section. As we assume a non-varying rectangular channel, this formula can be written down as
Q = v * y * B
Qis the discharge (in m³/s or cu ft/s),
vis the flow velocity (in m/s or ft/s),
yis the flow depth (in m or ft),
Bis the channel width (in m or ft).
2. Conjugate depth equation
The momentum functions of the flow depths upstream and downstream are equal. Thanks to this property, we can say that the depths
y₁ (upstream) and
y₂ (downstream) are conjugate depths, and use the formula below:
y₂/y₁ = 0.5 * [√(1 + 8 * Fr₁²) - 1]
y₂/y₁is the depth ratio (visible in the advanced mode of the hydraulic jump calculator),
Fr₁is the Froude number of the flow upstream.
3. Head loss formula
In any hydraulic jump, although momentum is conserved (what allows us to use the conjugate depth equation), some energy is lost. We can call this an energy loss or a head loss. It can be expressed with the following formula:
ΔE = (y₂ - y₁)³ / (4 * y₁ * y₂)
4. Hydraulic jump length equation
The equation determining the length of a hydraulic jump has been found in numerous field and laboratory experiments. You should remember, though, that this is only an approximation - the real length of the hydraulic jump might vary depending on many additional factors such as the flow turbulence.
L = 220 * y₁ * tanh[(Fr₁ - 1) / 22]
5. Hydraulic jump height formula
The height of the hydraulic jump is simply the difference between the flow depth downstream and upstream. As the flow upstream is supercritical, it will always be at a lower flow depth:
h = y₂ - y₁
6. Jump efficiency equation
You can also estimate what percentage of the original energy is lost during a hydraulic jump. This value is dependent mostly on the Froude number of the upstream flow
Fr₁ and is governed by the following equation:
En = [[√(8 * Fr₁² + 1)]³ - 4 * Fr₁² + 1] / [8 * Fr₁² * (2 + Fr₁²)] * 100%
Types of hydraulic jumps
The way a hydraulic jump looks like depends strongly on the Froude number upstream
Fr₁. The ranges below show different types of jumps that differ in the flow patterns.
Fr₁ < 1.7: undular jump. The jump is minimal, which results in small undulations on the surface and very low energy loss.
1.7 < Fr₁ < 2.5: weak jump. the jump is still considered small, and the energy dissipation is quite low.
2.5 < Fr₁ < 4.5: oscillating jump. Irregular waves are generated during this jump. Energy losses start to be considerable.
4.5 < Fr₁ < 9: steady jump. The jump is steady and constricted to one location. Energy losses can reach 70%.
Fr₁ > 9: strong jump. In such jumps, supercritical water jets are formed, the difference in velocities up- and downstream is high, and the energy dissipation can reach 85%.