Hydraulic jumps

Determining the strength and location of hydraulic jumps is important for designing energy dissipation structures and assessing the effectiveness of stream barbs or step-pool structures. The following equation is used to estimate energy dissipation at a hydraulic jump:

Energy is expressed in units of length (a head loss). The height of a jump for a channel of small slope can be estimated from:

where:
y₁ = upstream depth
y₂ = downstream flow depth
F₁ = Froude number of the upstream flow. This equation is derived from the specific force formulation for a rectangular channel.

where:

And the Froude number is:

Substituting VA for Q and bd for A, where b is the channel bottom width, as well as making use of the definition of Froude number, this equation can be simplified to:

Knowing the depth of the approaching flow and its Froude number, the flow depth downstream of the jump can be calculated. Froude numbers can also be used to specify different types of jumps as shown in table 1 (Chow 1959).

The length of a well-defined hydraulic jump is the distance from the upstream face of the jump to the point on the surface just downstream of the roller. Chow indicates that it cannot be easily determined theoretically and is best estimated empirically. The U.S. Department of Interior Bureau of Reclamation performed numerous experiments and provides figure 20 for determining jump length based on upstream Froude number and upstream flow depth (Peterka 1984). L is jump length, y1 is upstream depth, and the Froude number is that of the flow coming into the jump.

The location along the channel profile of the upstream beginning of the hydraulic jump can be generally determined from

However, the jump length has a bearing on this estimate. For example, the location of a hydraulic jump formed by a broad-crested weir in the channel can be used to illustrate this situation (fig. 21). Downstream tailwater affects the location of the jump, moving it farther upstream and closer to the weir, as the tailwater is raised. A lower tailwater elevation produces a jump farther downstream. Increasing the height of the weir moves the jump upstream, whereas decreasing it moves the jump downstream.

However, the weir will not cause an hydraulic jump if it is drowned out by downstream tailwater. The downstream depth must be less than critical depth over the weir plus the weir height, or, using the definition of critical depth:

where:
y₁ = depth upstream of the weir
h = weir height
y₃ = tailwater depth downstream