10.1 Hydraulic jumps

Hydraulic jumps are fascinating phenomena in fluid dynamics where fast, shallow flow suddenly transitions to slower, deeper flow. This abrupt change dissipates energy and creates turbulence, making hydraulic jumps crucial for energy dissipation in structures like spillways and stilling basins.

Understanding hydraulic jumps involves key concepts like Froude number, specific energy, and conjugate depths. These principles help engineers predict jump behavior, design effective energy dissipaters, and control flow in rivers and channels. Hydraulic jumps also find applications in wastewater treatment and recreational activities.

Hydraulic jump fundamentals

• Hydraulic jumps occur when supercritical flow transitions to subcritical flow, resulting in a sudden rise in the water surface elevation and significant energy dissipation
• The Froude number, defined as the ratio of inertial forces to gravitational forces, determines the flow regime and the occurrence of hydraulic jumps
• Supercritical flow has a Froude number greater than 1, characterized by high velocities and shallow depths, while subcritical flow has a Froude number less than 1, with lower velocities and greater depths

Supercritical vs subcritical flow

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• Supercritical flow is characterized by high velocities, shallow depths, and a Froude number greater than 1
  • In supercritical flow, disturbances cannot propagate upstream, and the flow is controlled by downstream conditions
• Subcritical flow has lower velocities, greater depths, and a Froude number less than 1
  • In subcritical flow, disturbances can propagate both upstream and downstream, and the flow is controlled by upstream conditions
• The transition between supercritical and subcritical flow occurs at a Froude number of 1, known as critical flow

Froude number significance

• The Froude number ($Fr$) is a dimensionless parameter that represents the ratio of inertial forces to gravitational forces in open channel flow
  • $Fr = \frac{v}{\sqrt{gy}}$, where $v$ is the flow velocity, $g$ is the gravitational acceleration, and $y$ is the flow depth
• The Froude number determines the flow regime and the occurrence of hydraulic jumps
  • $Fr > 1$: Supercritical flow
  • $Fr < 1$: Subcritical flow
  • $Fr = 1$: Critical flow
• The Froude number is crucial in designing hydraulic structures, such as spillways and stilling basins, to ensure proper energy dissipation and flow control

Specific energy concept

• Specific energy ($E$) is the sum of the potential energy (elevation) and the kinetic energy (velocity head) at a given cross-section in open channel flow
  • $E = y + \frac{v^2}{2g}$, where $y$ is the flow depth and $v$ is the flow velocity
• For a given discharge and channel geometry, there are two possible flow depths for each specific energy value, known as the alternate depths
• The minimum specific energy occurs at critical flow, where the Froude number is equal to 1
• The specific energy concept helps analyze the flow transitions and the formation of hydraulic jumps

Conjugate depth relationship

• In a hydraulic jump, the depths immediately upstream (supercritical) and downstream (subcritical) of the jump are known as conjugate depths
• The conjugate depth relationship is derived from the conservation of momentum principle and relates the upstream and downstream depths based on the upstream Froude number
• The Belanger equation expresses the conjugate depth relationship:
  • $\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$, where $y_1$ and $y_2$ are the upstream and downstream conjugate depths, respectively, and $Fr_1$ is the upstream Froude number
• The conjugate depth relationship is essential for predicting the characteristics of hydraulic jumps and designing hydraulic structures

Types of hydraulic jumps

• Hydraulic jumps can be classified into different types based on their characteristics, such as the upstream Froude number, the jump length, and the surface profile
• The classical hydraulic jump is the most common type, characterized by a distinct roller and a sudden rise in the water surface elevation
• Other types of hydraulic jumps include undular jumps, weak jumps, oscillating jumps, and steady vs. moving jumps

Classical hydraulic jump

• The classical hydraulic jump occurs when the upstream Froude number is between 1.7 and 9
• It is characterized by a well-defined roller, significant energy dissipation, and a distinct rise in the water surface elevation
• The flow downstream of the jump is subcritical, with a lower velocity and greater depth compared to the upstream supercritical flow
• Classical hydraulic jumps are commonly observed in natural streams and are utilized in hydraulic structures for energy dissipation

Undular jump

• An undular jump occurs when the upstream Froude number is between 1 and 1.7
• It is characterized by a series of stationary surface undulations downstream of the jump, with minimal energy dissipation
• The flow remains subcritical throughout the jump, and the water surface rise is gradual compared to the classical hydraulic jump
• Undular jumps are less effective in energy dissipation and are generally avoided in hydraulic structure design

Weak jump

• A weak jump occurs when the upstream Froude number is slightly greater than 1 (typically between 1 and 1.5)
• It is characterized by a small rise in the water surface elevation and minimal energy dissipation
• The flow downstream of the jump remains subcritical, with a slight increase in depth and decrease in velocity compared to the upstream flow
• Weak jumps are less prominent and have limited practical applications in hydraulic engineering

Oscillating jump

• An oscillating jump occurs when the jump is not stable and oscillates back and forth within a certain range
• It is characterized by periodic fluctuations in the jump position and the water surface profile
• Oscillating jumps can occur due to various factors, such as upstream flow conditions, channel geometry, and downstream boundary conditions
• Oscillating jumps can cause undesirable flow conditions and should be avoided in hydraulic structure design

Steady vs moving jumps

• Steady jumps are stationary and maintain a fixed position in the channel, while moving jumps propagate upstream or downstream
• Steady jumps occur when the upstream and downstream flow conditions are in equilibrium, and the jump length remains constant
• Moving jumps can be caused by changes in the upstream or downstream flow conditions, such as variations in the discharge or tailwater depth
• Moving jumps can lead to unsteady flow conditions and should be considered in the design of hydraulic structures

Hydraulic jump characteristics

• Hydraulic jumps exhibit various characteristics that influence their behavior, energy dissipation, and flow properties
• These characteristics include energy dissipation mechanisms, aeration and air entrainment, pressure distribution, velocity profiles and turbulence, and surface roughness effects
• Understanding these characteristics is crucial for the design and analysis of hydraulic structures and the prediction of jump behavior in open channel flows

Energy dissipation mechanisms

• Hydraulic jumps are highly effective in dissipating energy due to the significant turbulence and mixing that occurs within the jump
• The primary energy dissipation mechanisms in hydraulic jumps include:
  • Turbulent mixing and eddy formation in the roller region
  • Aeration and air entrainment, which increases the effective fluid density and enhances energy dissipation
  • Boundary layer development and interaction with the channel bed and walls
• The energy dissipation efficiency of a hydraulic jump depends on factors such as the upstream Froude number, the jump length, and the channel geometry

Aeration and air entrainment

• Hydraulic jumps are characterized by significant aeration and air entrainment, which occur due to the turbulent mixing and the high shear stresses at the water surface
• Air entrainment increases the effective fluid density and the bulk flow volume, leading to enhanced energy dissipation and reduced flow velocities
• The air concentration within the jump varies spatially and depends on factors such as the upstream Froude number and the jump length
• Aeration and air entrainment also influence the pressure distribution, velocity profiles, and the overall flow structure within the jump

Pressure distribution in jumps

• The pressure distribution within a hydraulic jump is non-hydrostatic due to the turbulent nature of the flow and the presence of the roller region
• The pressure distribution varies along the jump length and depends on factors such as the upstream Froude number and the jump geometry
• In the roller region, the pressure distribution is characterized by a significant pressure gradient, with higher pressures near the bed and lower pressures near the water surface
• The non-hydrostatic pressure distribution affects the flow structure, the velocity profiles, and the energy dissipation within the jump

Velocity profiles and turbulence

• Hydraulic jumps exhibit complex velocity profiles and turbulence characteristics due to the sudden change in flow regime and the presence of the roller region
• The velocity profiles within the jump vary along the jump length and across the channel cross-section
• In the roller region, the velocity profiles are characterized by reverse flow near the water surface and high turbulence intensities
• Downstream of the jump, the velocity profiles gradually recover towards a more uniform distribution, with reduced turbulence intensities
• The turbulence characteristics, such as the turbulent kinetic energy and the Reynolds stresses, play a significant role in the energy dissipation and the mixing processes within the jump

Surface roughness effects

• The surface roughness of the channel bed and walls influences the characteristics of hydraulic jumps and their energy dissipation efficiency
• Rough surfaces enhance the turbulence generation and the boundary layer development, leading to increased energy dissipation and shorter jump lengths compared to smooth surfaces
• The surface roughness effects are more pronounced for lower upstream Froude numbers and smaller jump lengths
• In practice, the surface roughness can be manipulated using various techniques, such as the use of baffle blocks, end sills, or roughness elements, to improve the energy dissipation efficiency and the stability of hydraulic jumps

Hydraulic jump applications

• Hydraulic jumps have numerous applications in hydraulic engineering, river and channel flow control, and recreational activities
• The primary application of hydraulic jumps is in the design of energy dissipaters in hydraulic structures, such as spillways and stilling basins
• Hydraulic jumps are also used for river and channel flow control, wastewater treatment, and recreational purposes, such as whitewater kayaking

Energy dissipaters in hydraulic structures

• Hydraulic jumps are widely used as energy dissipaters in hydraulic structures, such as spillways, outlet works, and culverts
• The purpose of energy dissipation is to reduce the high velocities and turbulence downstream of the structure to prevent erosion and ensure safe downstream conditions
• Hydraulic jumps are created by designing a stilling basin or an apron downstream of the structure, which induces the transition from supercritical to subcritical flow
• The design of energy dissipaters involves the selection of the appropriate basin geometry, the determination of the required tailwater depth, and the consideration of factors such as the upstream Froude number and the discharge range

Stilling basins design considerations

• Stilling basins are hydraulic structures designed to create and contain hydraulic jumps for energy dissipation purposes
• The design of stilling basins involves various considerations, such as:
  • The basin length and width, which should be sufficient to accommodate the jump and prevent downstream erosion
  • The basin depth, which should be designed to maintain the required tailwater depth and ensure the stability of the jump
  • The use of appurtenances, such as baffle blocks, end sills, or chute blocks, to enhance energy dissipation and improve the jump stability
• Stilling basin design also requires the consideration of factors such as the range of discharges, the sediment transport characteristics, and the downstream channel conditions

River and channel flow control

• Hydraulic jumps can be used for flow control in rivers and channels to regulate the flow depth, velocity, and energy dissipation
• By creating a hydraulic jump at a specific location, it is possible to raise the water level upstream of the jump, reduce the flow velocity, and dissipate excess energy
• Flow control structures, such as grade control structures or low-head dams, can be designed to induce hydraulic jumps and achieve the desired flow conditions
• The use of hydraulic jumps for river and channel flow control requires the consideration of factors such as the channel geometry, the sediment transport, and the ecological impacts

Whitewater recreation and kayaking

• Hydraulic jumps are a popular feature in whitewater recreation and kayaking, providing challenging and exciting conditions for paddlers
• Whitewater parks and artificial river courses often incorporate hydraulic jumps to create various flow features, such as holes, waves, and eddies
• The design of whitewater features involves the manipulation of the channel geometry and the flow conditions to create the desired hydraulic jump characteristics
• Safety considerations, such as the provision of adequate rescue areas and the management of swimmer and boat passage, are crucial in the design of whitewater recreational facilities

Industrial and wastewater treatment

• Hydraulic jumps can be utilized in industrial and wastewater treatment applications for mixing, aeration, and energy dissipation purposes
• In wastewater treatment plants, hydraulic jumps can be used in aeration basins or mixing chambers to promote the mixing of wastewater and the transfer of oxygen
• Hydraulic jumps can also be used in industrial processes, such as the mixing of chemicals or the dissipation of energy in pipeline systems
• The design of hydraulic jumps for industrial and wastewater treatment applications requires the consideration of factors such as the flow rates, the water quality, and the specific treatment objectives

Hydraulic jump equations and calculations

• The analysis and design of hydraulic jumps involve various equations and calculations based on the principles of conservation of mass, momentum, and energy
• The key equations and calculations include the momentum conservation analysis, the Belanger equation derivation, the sequent depth ratio estimation, the length of jump predictions, and the energy loss computations
• These equations and calculations are essential for predicting the characteristics of hydraulic jumps, designing hydraulic structures, and analyzing the performance of energy dissipaters

Momentum conservation analysis

• The momentum conservation principle is the fundamental basis for the analysis of hydraulic jumps
• The momentum equation states that the net force acting on a control volume is equal to the rate of change of momentum within the control volume
• For a hydraulic jump, the momentum equation can be written as:
  • $\rho Q(v_2 - v_1) = P_1 - P_2 + F_f$
  • where $\rho$ is the fluid density, $Q$ is the discharge, $v_1$ and $v_2$ are the upstream and downstream velocities, $P_1$ and $P_2$ are the hydrostatic pressure forces, and $F_f$ is the friction force
• The momentum conservation analysis allows the determination of the conjugate depths and the jump characteristics based on the upstream flow conditions and the channel geometry

Belanger equation derivation

• The Belanger equation is a fundamental relationship that relates the conjugate depths in a hydraulic jump based on the upstream Froude number
• The equation is derived from the momentum conservation principle, assuming a horizontal rectangular channel and neglecting the friction forces
• The Belanger equation is given by:
  • $\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$
  • where $y_1$ and $y_2$ are the upstream and downstream conjugate depths, respectively, and $Fr_1$ is the upstream Froude number
• The Belanger equation is widely used in the design and analysis of hydraulic jumps, as it provides a simple and accurate estimation of the conjugate depth ratio based on the upstream flow conditions

Sequent depth ratio estimation

• The sequent depth ratio is the ratio of the downstream conjugate depth to the upstream conjugate depth in a hydraulic jump
• The sequent depth ratio can be estimated using the Belanger equation or other empirical relationships based on the upstream Froude number
• For a given upstream Froude number, the sequent depth ratio determines the required downstream depth to form a stable hydraulic jump
• The estimation of the sequent depth ratio is crucial for the design of stilling basins and energy dissipaters, as it determines the required tailwater depth and the basin geometry

Length of jump predictions

• The length of a hydraulic jump is the distance from the toe of the jump (where the supercritical flow transitions to subcritical flow) to the point where the flow becomes fully developed and the water surface profile becomes parallel to the channel bed
• The length of the jump can be predicted using empirical relationships based on the upstream Froude number and the conjugate depth ratio
• One commonly used equation for the length of the jump is:
  • $L_j = 6(y_2 - y_1)$
  • where $L_j$ is the jump length, and $y_1$ and $y_2$ are the upstream and downstream conjugate depths, respectively
• The prediction of the jump length is important for the design of stilling basins and the determination of the required basin dimensions

Energy loss computations

• Hydraulic jumps are highly effective in dissipating energy, and the energy loss within the jump can be computed using the energy conservation principle
• The energy loss in a hydraulic jump is the difference between the specific energy upstream and downstream of the jump
• The specific energy at a given cross-section is given by:
  • $E = y + \frac{v^2}{2g}$
  • where $E$ is the specific energy, $y$ is the flow depth, $v$ is the flow velocity, and $g$ is the gravitational acceleration
• The energy loss in a hydraulic jump can be computed as:
  • $\Delta E = E_1 - E_2$
  • where $\Delta E$ is the energy loss, and $E_1$ and $E_2$ are the specific energies upstream and downstream of the jump, respectively
• The computation of energy loss is essential for evaluating the performance of hydraulic jum