Hydraulic jump

A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise (a step or standing wave) occurs in the liquid surface. The rapidly flowing liquid is abruptly slowed and increases in height converting some of the flow's initial kinetic energy into an increase in potential energy, with some energy irreversibly lost through turbulence to heat. In an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how a shockwave forms.

The phenomenon is dependent upon the initial fluid speed. If the initial speed of the fluid is below the critical speed then no jump is possible. For initial flow speeds which are not significantly above the critical speed, the transition appears as an undulating wave. As the initial flow speed increases further, the transition becomes more abrupt, until at high enough speeds, the transition front will break and curl back upon itself. When this happens, the jump can be accompanied by violent turbulence, eddying, air entrainment, and surface undulations (waves).

There are two main manifestations of hydraulic jumps and historically different terminology has been used for each. However, the mechanisms behind them are similar because they are simply variations of each other seen from different frames of reference and so the physics and analysis techniques can be used for both types.

The different manifestations are:

* The stationary hydraulic jump - rapidly flowing water transitions in a stationary jump to slowly moving water as shown in Figures 1 and 2.

* The tidal bore - a wall or undulating wave of water moves upstream against water flowing downstream as shown in Figures 3 and 4. If considered from a frame of reference which moves with the wave front, you can see that this case is physically similar to a stationary jump.A related case is a cascade - a wall or undulating wave of water moves downstream overtaking a shallower downstream flow of water as shown in Figure 5. If considered from a frame of reference which moves with the wave front, this is amenable to the same analysis as a stationary jump.

These phenomena are addressed in an extensive literature from a number of technical viewpoints. cite book|author=JF Douglas, JM Gasiorek & JA Swaffield |title= Fluid Mechanics |publisher=Prentice Hall, Essex |year=2001 (4th ed) |id=ISBN 0-582-41476-8 ] cite book|author= Faber, T.E. |title= Fluid Dynamics for Physicists |publisher=Cambridge University Press, Cambridge|year=1995|id= ISBN 0-521-42969-2 ] cite book|author=Faulkner, L.L. |title= Practical Fluid Mechanics for Engineering Applications |publisher=Marcel Dekker AG, Basil, Switzerland |year=2000 |id=ISBN 0-8247-9575-X ] cite book|author=R.W. Fox & A.T. McDonald |title= Introduction to Fluid Mechanics |publisher=John Wiley & Sons |year=1985 |id=ISBN 0-471-88598-3 ] cite book|author=Hager, Willi H.|title=Energy Dissipaters and Hydraulic Jump |publisher=Kluwer Academic Publishers, Dordrecht |year=1995|id=ISBN 90-5410-198-9 ] cite book|author=Khatsuria, R.M. |title=Hydraulics of Spillways and Energy Dissipaters |publisher=Marcel Dekker, New York |year= 2005|id= ISBN 0-8247-5789-0 ] cite book|author=Lighthill, James|title=Waves in Fluids |publisher=Cambridge University Press, Cambridge|year=1978|id=ISBN 0-521-29233-6 ] cite book|author= Roberson, J.A. & Crowe, C.T |title=Engineering Fluid Mechanics |publisher= Houghton Mifflin Company, Boston |year=1990 |id=ISBN 0-395-38124-X ] cite book|author=V.L. Streeter and E.B. Wylie|title=Fluid Mechanics |publisher=McGraw-Hill Book Company, New York |year=1979|id=ISBN 0-07-062232-9 ] cite book|author=Vennard, John K. |title=Elementary Fluid Mechanics |publisher= John Wiley & Sons, New York |year=1963 (4th edition) |id= ] cite book|author=Vischer, D.L. & Hager, W.H.|title=Energy Dissipaters |publisher=A.A. Balkema, Rotterdam |year=1995|id=ISBN 0-8247-5789-0 ] cite book|author= White, Frank M. |title= Fluid Mechanics |publisher= McGraw Hill, Inc. |year=1986 |id=ISBN 0-07-069673-X ] cite book|author=H Chanson |title=The Hydraulic of Open Channel Flow: an Introduction |publisher=Butterworth-Heinemann |year=2004 (2nd ed) |id=ISBN 978 0 7506 5978 9 ]

Classes of hydraulic jumps

Hydraulic jumps can be seen in both a stationary form, called a hydraulic jump and a dynamic or moving form, called a positive surge or "hydraulic jump in translation". They can be described using the same analytic approaches and are simply variants of a single phenomena.

Moving hydraulic jump

A tidal bore (or bore) is a hydraulic jump which occurs when the incoming tide forms a wave (or waves) of water that travel up a river or narrow bay against the direction of the current. As is true for hydraulic jumps in general, bores take on various forms depending upon the difference in the waterlevel upstream and down, ranging from an undular wavefront to a shock-wave-like wall of water. Figure 3 shows a tidal bore with the characteristics common to shallow upstream water - a large elevation difference is observed. Figure 4 shows a tidal bore with the characteristics common to deep upstream water - a small elevation difference is observed and the wavefront undulates. In both cases the tidal wave moves at the speed characteristic of waves in water of the depth found immediately behind the wave front.

Another variation of the moving hydraulic jump is the cascade. In the cascade (an example of which is found in Figure 5), a series of roll waves or undulating waves of water moves downstream overtaking a shallower downstream flow of water.

tationary hydraulic jump

The stationary hydraulic jump is most frequently seen on rivers and on engineered features such as outfalls of dams and irrigation works. They occur when a flow of liquid at high velocity discharges into a zone of the river or engineered structure which can only sustain a lower velocity. When this occurs, the water slows in a rather abrupt rise (a step or standing wave) on the liquid surface.

Comparing the characteristics before and after, one finds:

The other stationary hydraulic jump is when a rapid flow encounters a submerged object which throws the water upwards. The mathematics behind this form is more complex and will need to take into account the shape of the object and the flow characteristics of the fluid around it.

Analysis of the hydraulic jump on a liquid surface

In spite of the apparent complexity of the flow transition, application of simple analytic tools to a two dimensional analysis are effective in providing analytic results which closely parallel both field and laboratory results. Analyses show
* Height of the jump: the relationship between the depths before and after the jump as a function of flow rate.
* Energy loss in the jump
* Location of the jump on a natural or an engineered structure
* Character of the jump: undular or abrupt

Height of the jump

There are several methods of predicting the height of a hydraulic jump. [This section outlines the approaches at an overview level only.] They all reach common conclusions that:
* The ratio of the water depth before and after the jump depend solely on the ratio of velocity of the water entering the jump to the speed of the wave over-running the moving water.
* The height of the jump can be many times the initial depth of the water.

; Applying the energy principleAssuming a two-dimensional situation with flow rate (q) as shown by Figure 7 below, the energy principle yield an expression of the energy loss in the hydraulic jump. Indeed hydraulic jumps are commonly used as energy dissipaters downstream of dam spillways ().

; Applying the Continuity PrincipleIn fluid dynamics, the equation of continuity is effectively an equation of conservation of mass. Considering any fixed closed surface within an incompressible moving fluid, the fluid flows into a given volume at some points and flows out at other points along the surface with no net change in mass within the space since the density is constant. We will assume a rectangular channel. The differential continuity equation, in Cartesian coordinates:

:$\{partial\; ho\; over\; partial\; t\}\; +\; abla\; cdot\; (\; ho\; mathbf\{v\})\; =\; 0$

where ρ is density, t is time, and v is fluid velocity.

Since the density is constant and we are considering only a 2-dimensional case, this integrates to:

: $v\_0\; imes\; h\_0\; =\; v\_1\; imes\; h\_1$

: or $v\_1\; =\; v\_0\; imes\; \{h\_0\; over\; h\_1\}$

Substituting yields a cubic equation which can be solved using Cardano’s method to determine that:

: $\{h\_1\; over\; h\_0\}\; =frac\{-1\; pm\{sqrt\{1+\{frac\{8v\_0^2\}\{gh\_0\}\{2\}$

Additionally, the same result can be achieved by considering conservation of momentum across the jump. Assuming no losses and constant density, conservation of momentum for the fluid can be expressed as:

: $\{d\; over\; dt\}\; int\_\{z\_0\}^\{z\_1\}\; holanglemathbf\{v\}\; angle\; A,ds\; =\; ho\; langlemathbf\{v\}v\; angle\_0A\_0\; -\; ho\; langlemathbf\{v\}v\; angle\_1A\_1\; +\; p\_0mathbf\{A\}\_0\; -\; p\_1mathbf\{A\}\_1\; -\; mathbf\{F\}\; +\; homathbf\{g\}int\_\{z\_0\}^\{z\_1\}A,dz=0$

Where $mathbf\{v\}$ is the velocity field, and $v$ is the component of velocity perpendicular to the control volume surface.

Essentially, the time variation of momentum in the control volume bounded by $z\_0$ and $z\_1$ can be expressed as the difference of momentum fluxes entering and leaving the control volume. For a flow field that is everywhere parallel to z, $mathbf\{v\}$ has the same direction and magnitude as $v$. Thus, $langlemathbf\{v\}v\; angle\; =\; langle\; v^2\; angle$. Also, for turbulent flow, langle v^2 angle} over {langle v angle}^2} = 1 . Additionally, we will only consider changes in x-momentum, so the last integral, which has only a z-component, need not be included in subsequent equations. With these simplifications, the expression for momentum conservation becomes:

:$\{\; ho\; langle\; v\; angle\_0^2\}A\_0\; -\{\; ho\; langle\; v\; angle\_1^2\}A\_1\; +\; p\_0\; mathbf\{A\}\_0-p\_1\; mathbf\{A\}\_1\; =0$

Assuming a uniform velocity over the flow at $z\_0$ and $z\_1$, which implies a surface normal vector $mathbf\{A\}$, parallel to the flow,

:$ho\; v\_0^2A\_0\; -\; ho\; v\_1^2A\_1\; +\; p\_0\; A\_0\; -\; p\_1\; A\_1\; =\; 0$

The static pressure in the flow is simply the hydrostatic pressure, $p=p\_a\; +\; ho\; gh$, where $p\_a$ is the atmospheric pressure. The force caused by the atmospheric pressure will cancel across any boundary, so it need not be considered. The net force caused by the pressure acting on the control volume before and after the jump is:

:$pA\; =\; int\_0^h\; ho\; gh\text{'},dh\text{'}\; =\; \{1\; over\; 2\}\; ho\; gh^2$

The expression for conservation of momentum can now be written as:

:$ho\; v\_0^2h\_0\; -\; ho\; v\_1^2h\_1\; +\; \{1\; over\; 2\}\; ho\; gh\_0^2\; -\; \{1\; over\; 2\}\; ho\; gh\_1^2\; =\; 0$

Dividing by constant $ho$ and introducing the result from continuity gives

:$v\_0^2\; (h\_0-\{h\_0^2\; over\; h\_1\})\; +\; \{g\; over\; 2\}\; (h\_0^2\; -\; h\_1^2)=0$

Which, after some algebra, simplifies to:

:$\{1\; over\; 2\}\; \{h\_1\; over\; h\_0\}(\{h\_1\; over\; h\_0\}\; +\; 1)\; -\; Fr^2\; =\; 0$

Where $Fr^2=\{v\_0^2\; over\; gh\_0\}$. Solving this quadratic yields the same equation for $\{h\_1\; over\; h\_0\}$ as stated above.

Negative answers do not yield meaningful physical solutions, so this reduces to:

: $\{h\_1\; over\; h\_0\}\; =frac\{-1\; +\{sqrt\{1+\{frac\{8v\_0^2\}\{gh\_0$}{2}

This produces three solutions:
* When $frac\{8v\_0^2\}\{gh\_0\}\; =\; 8$, then $\{h\_1\; over\; h\_0\}\; =\; 1$ (i.e., there is no jump)
* When , then (i.e., there is a negative jump - this can be shown as not conserving energy and is only physically possible if some force were to accelerate the fluid at that point)
* When $frac\{8v\_0^2\}\{gh\_0\}\; >\; 8$ or $frac\{v\_0^2\}\{gh\_0\}\; >\; 1$, then $\{h\_1\; over\; h\_0\}\; >\; 1$ (i.e., there is a positive jump)

Since $frac\{v\_0^2\}\{gh\_0\}\; =\; Fr^2$, where $Fr$ is the dimensionless Froude number, this is equivalent to the condition that $Fr\; >\; 1$. Since the $sqrt\{gh\_0\}$ is the speed of a shallow gravity wave, the condition that $Fr\; >\; 1$ is equivalent to stating that the initial velocity represents supercritical flow (Froude number > 1) while the final velocity represents subcritical flow (Froude number < 1).

;Jump height in terms of flow

The ratio of the flow height before the jump and after the jump can be simply expressed in terms of the Froude number of the incoming flow. The greater that the flow is supercritical, the more pronounced the jump will be.

: $\{h\_1\; over\; h\_0\}\; =frac$sqrt{1+8Fr^2} -1{2} known as Bélanger equation.

Practically this means that water accelerated by large drops can create stronger standing waves in the form of hydraulic jumps as it decelerates at the base of the drop. Such standing waves, when found downstream of a weir or natural rock ledge, can form an extremely dangerous "keeper" with a water wall that "keeps" floating objects (e.g., logs, kayaks, or kayakers) recirculating in the standing wave for extended periods.

; Alternate but equivalent approach applying the Impulse-Momentum Principle

A similar analysis, reaching exactly the same results, derives the same results starting with the impulse-momentum principle.

: $Net\; Impulse\; =\; Change\; in\; momentum$

: $ho\; (gh\_0\; -\; gh\_1)t=\; ho\; left\; (\; \{v\_1^2\; over\; 2\}\; -\; \{v\_0^2\; over\; 2\}\; ight\; )t$

: $\{v\_0^2\; over\; 2\}+gh\_0=\{v\_1^2\; over\; 2\}+gh\_1$

This equation yields the same overall relationship between jump height and Froude number.

Energy dissipation by a hydraulic jump

One of the most important engineering applications of the hydraulic jump is to dissipate energy in channels, dam spillways, and similar structures so that the excess kinetic energy does not damage these structures. The energy dissipation or head loss across a hydraulic jump is a function of the magnitude of the jump. The larger the jump as expressed in the fraction of final height to initial height, the greater the head loss.

Analytically (using the model developed by R.W. Fox & A.T. McDonald), the fractional energy loss (FEL) can be expressed in terms of the Froude number ($Fr\_0$) for the incident flow as:

: $FEL\; =\; frac\; \{(sqrt\{1+8Fr\_0^2\}\; -\; 3)^3\}\; \{8\; \{(sqrt\{1+8Fr\_0^2\}\; -\; 1)(sqrt\{Fr\_0^2\; +\; 2\})$

Since $Fr\_0\; =\; sqrt\{frac\{v\_0^2\}\{gh\_0$ this is equivalent to concluding the energy loss can be predicted by predicting or measuring the speed and depth of the entering water.

Location of hydraulic jump in a streambed or an engineered structure

In the design of a dam the energy of the fast-flowing stream over a spillway must be partially dissipated to prevent erosion of the streambed downstream of the spillway, which could ultimately lead to failure of the dam. This can be done by arranging for the formation of a hydraulic jump to dissipate energy. To limit damage, this hydraulic jump normally occurs on an apron engineered to withstand hydraulic forces and to prevent local cavitation and other phenomena which accelerate erosion.

In the design of a spillway and apron, the engineers select the point at which a hydraulic jump will occur. Obstructions (such as a lip) or slope changes are routinely designed into the apron to force a jump at a specific location. Obstructions are unnecessary as the slope change alone is normally sufficient. To trigger the hydraulic jump without obstacles, an apron is designed such that the flat slope of the apron retards the rapidly flowing water from the spillway. If the apron slope is insufficient to maintain the original high velocity, a jump will occur.

Two methods of designing an induced jump are common:

* If the downstream flow is restricted by the down-stream channel such that water backs up onto the foot of the spillway, that downstream water level can be used to identify the location of the jump.

* If the spillway continues to drop for some distance, but the slope changes such that it will no longer support supercritical flow, the depth in the lower subcritical flow region is sufficient to determine the location of the jump.

In both cases, the final depth of the water is determined by the downstream characteristics. The jump will occur if and only if the level of inflowing (supercritical) water level ($h\_0$) satisfies the condition:

: $h\_0\; =\{h\_1over\; 2\}\; left\; (\; \{-1\; +\; sqrt\; \{1\; +\; 8F\_\{rate\}^2h\_1/g\; ight\; )$

: "$F\_\{rate\}$" = fluid flow rate : "g" = acceleration due to gravity (essentially constant for this case): "h" = height of the fluid ($h\_0$ = initial height while $h\_1$ = final downstream height)

Applying wave theory to the hydraulic jump

In fluid dynamics, gravity waves are waves generated in a fluid which has as the restoring force, gravity. Gravity waves on an air-water interface are called surface gravity waves or surface waves. Hydraulic jumps, ocean waves and tsunamis can all be treated as examples of gravity waves.

The wave speed or celerity (speed of individual waves, as opposed to the speed of a group of waves) of gravity waves in shallow water is given by:
* $v\; =\; sqrt\{gh\}sqrt\{frac\{\; anh\{(kh)\{kh$ which approaches $sqrt\{gh\}$ for small h;In which:
* v = wave speed or celerity (m/s)
* g = gravitational acceleration (9.8 m/s² on Earth)
* h = water depth (m)
* $k\; =\; frac\{2pi\}\{lambda\}$ wave number where $lambda$ is the wavelength.

The constraints on the approximation for the speed of a gravity wave as $sqrt\{gh\}$ for shallow depths are:
* For wavelengths close to or less than 1.7 cm the surface tension cannot be neglected so that this approximation is invalid.
* For depths significantly greater than the wavelength, &lambda;, of the wave the speed of the wave is governed only by the wavelength following the equation $speed\; =\; \{g\; lambda\; over\; \{2pi$ where g is the acceleration of gravity.

A hydraulic jump can be viewed as discontinuous waves of all frequencies (wavelengths), which are generated and propagate from a point near the jump. The waves propagate both upstream and downstream. Since a large fraction of the waves fall in a wavelength range where they are shallow water gravity waves that move at the same speed for a given depth, they move upstream at the same rate; however as the water shallows upstream, their speed drops quickly, limiting the rate at which they can propagate upstream to $sqrt\{gh\}$. Shorter wavelengths, which propagate more slowly than the speed of the wave in the deeper downstream water, are swept away downstream. A fairly wide range of wavelengths and frequencies are still present, so Fourier analysis would suggest that a relatively abrupt wave front can be formed and this is indeed observed in practice.

Viewing the hydraulic jump from a wave perspective provides another insight into the phenomena. When the incoming water speed is slow enough, a number of the longer wavelength waves propagate faster than the incoming flow, and can disperse upstream as well as downstream. The deeper the incoming water is the more pronounced the dispersion effect will be. Only a small subset of frequencies (wavelengths) will match the speed of the flow. This truncation of the Fourier spectrum results in a hydraulic jump characterized by undulating waves rather than an abrupt jump. When visible undulations are present, the wavelength of the visible undulations provide a direct indication of the speed of the water upstream of the hydraulic jump.

This characteristic behavior allows one to estimate the pre-jump water depth and water speed simply by observing the height of the jump, the characteristics of the jump, and correlating them as tabulated below. Such an “eyeball” estimate is routinely used by river runners while judging rapids; their conclusions are generally based on an intuitive sense rather than an analytic approach.

Tabular summary of the analytic conclusions

NB: the above classification is very rough. Undular hydraulic jumps have been observed with inflow/prejump Froude numbers up to 3.5 to 4 ] .

Hydraulic jump variations

Although the previous discussion has focused on the straight-forward simple channel approximation, a number of variations are amenable to similar analyses as well. They also serve the important function of allowing the student to perform simple experiments with everyday objects.

hallow fluid hydraulic jumps

;The hydraulic jump in your sinkFigure 2 above illustrates a daily example of a hydraulic jump can be seen in the sink. Around the place where the tap water hits the sink, you will see a smooth looking flow pattern. A little further away, you will see a sudden "jump" in the water level. This is a hydraulic jump.

The nature of this jump differs from those previously discussed in the following ways:
* The water is flowing radially. As a result it continuously grows shallower and slows due to friction (the Froude number drops) up to the point where the jump occurs.
* The flow depth is thin enough that the surface tension can no longer be neglected, changing the wave solution conclusions. The higher speed of the surface tension waves bleed off the high frequency component, making an undular jump the dominant form. [Surface tension effects can be seen by looking closely at the region inside the hydraulic jump. There you will observe thin waves radiating radially and axially from the point of water impact.]

Changes in the behavior of the jump can be observed by changing the flow rate.

Internal wave hydraulic jumps

Hydraulic jumps in abyssal fan formation

Turbidity currents can result in internal hydraulic jumps (i.e., hydraulic jumps as internal waves in fluids of different density) in abyssal fan formation. The internal hydraulic jumps have been associated with salinity or temperature induced stratification as well as with density differences due to suspended materials. When the bed slope over which the turbidity current flattens, the slower rate of flow is mirrored by increased sediment deposition below the flow, producing a gradual backward slope (i.e., a slope which rises against the current). Where a hydraulic jump occurs, the signature is an abrupt backward slope, corresponding to the rapid reduction in the flow rate at the point of the jump. [cite web|url=http://cee.uiuc.edu/people/parkerg/_private/Preprints/TurbJumpJHR.pdf|title=The Response of Turbidity Currents to a Canyon-Fan Transition: Internal Hydraulic Jumps and Depositional Signatures|date=2004|author=Svetlana Kostic, Gary Parker]

Atmospheric hydraulic jumps

Industrial and recreational applications for hydraulic jumps

Industrial

The hydraulic jump is the most commonly used choice of design engineers for energy dissipation below spillways and outlets. A properly designed hydraulic jump can provide for 60-70% energy dissipation of the energy in the basin itself, limiting the damage to structures and the streambed. Even with such efficient energy dissipation, stilling basins must be carefully designed to avoid serious damage due to uplift, vibration, cavitation, and abrasion. An extensive literature has been developed for this type of engineering.

Recreational

While travelling down river, kayaking and canoeing paddlers will often stop and playboat in standing waves and hydraulic jumps. The standing waves and shock fronts of hydraulic jumps make for popular locations for such recreation.

Similarly, kayakers and surfers have been known to ride bores up rivers.

ee also

*Shock wave
* Tidal bore
* Turbulence
* Laminar flow
* Undular bore

References and notes

External links

* [http://espace.library.uq.edu.au/list.php?browse=author&author_id=193 More than 50, freely available, published research articles on hydraulic jumps, energy dissipation and tidal bores by Professor Hubert Chanson, Department of Civil Engineering, The University of Queensland]
* [http://espace.library.uq.edu.au/view.php?pid=UQ:120303 Chanson, H. (2007). "Hydraulic Jumps: Bubbles and Bores."] Proceedings of 16th Australasian Fluid Mechanics Conference AFMC, Gold Coast, Australia, 2-7 December, Plenary Address, pp. 39-53 (CD-ROM) (ISBN 978-1-864998-94-8)