Hydraulic Jump
Keith bechtol october 31, 2007, (submitted as coursework for physics 210, stanford university, fall 2007), introduction.
A hydraulic jump is a fluid shockwave created at the transition between laminar and turbulent flow. One common example of a hydraulic jump can be seen in the water radiating outward when the stream of tap water strikes the horizontal surface of a sink. The water initially flows in a smooth sheet with consistent current patterns. In this region, the speed of the water exceeds the local wave speed. Friction against the sink surface slows the flow until an abrupt change occurs. At this point, the depth increases as water piles up in the transition region and flow becomes turbulent [1]. The motion of individual water molecules becomes erratic and unpredictable. The interruption of flow patterns also reduces the kinetic energy of the water. In addition to the kitchen sink example, hydraulic jumps are also typical features of river rapids where the water swirls and foams around rocks and logs.
Basic Theory
Although the hydraulic jump effect is common to everyday experience and has been studied experimentally for many years, the underlying theory describing the phenomena is surprisingly complex. Rayleigh first described the problem in 1914 [2]. He calculated of the change in fluid depth associated with the shockwave and introduced the concepts of continuity and fluid momentum conservation in his derivation. A schematic of fluid flow at a hydraulic jump is depicted in Figure 1.
Diagram of classical hydraulic jump. |
Let region 1 represent section of fast laminar flow preceding the hydraulic jump and region 2 denote the section of slow turbulent flow after the transition. In the present analysis, assume a uniform hydrostatic pressure distribution and a uniform velocity distribution. Consider vertical slices of fluid representing unit areas of the flow. Continuity of fluid flow implies that that the discharge, q=hv , must be equal before and after the hydraulic jump. h denotes the water depth and v is the water velocity.
A useful quantity to define is the Froude number F R =v/(qh) 1/2 . The Froude number is the ratio of flow velocity to wave celerity and marks the boundary between critical and subcritical flow [1]. For conditions in which the Froude number is greater than one, flow velocity exceedes the wave celerity (the speed of an individual wave crest) and the fluid motion is smooth. Subcritical currents with Froude numbers less than exhibit turbulent flow.
Next, call the ratio of fluid depths D=h 1 /h 2 . Solving the two equations above yields the following differential equation in terms of the ratio of fluid depths and Froude number:
The solution of this differential equation provides the ratio of final depth following a hydraulic jump to the initial depth before the jump.
Another important property of the hydraulic jump is the energy dissapated by the transition to turbulent flow. The energy loss is usually measured in terms of change in hydraulic head, H=h+(q 2 /2gA 2 ) where A is the cross sectional area of the flow. Efficiency is written as η=ΔH/H 1 .
Both the change in fluid depth and the energy dissipated can be quantified in terms of the Froude number. As the Froude number increases, the change in fluid depth grows and the energy dissipated by the jump rises.
Diagram of radial hydraulic jump |
Radial Hydraulic Jump
The theoretical framework developed by Rayleigh can be also used to predict the location of a radial hydraulic jump. Consider the arrangement in Figure 2 for a radial hydraulic jump. Following Rayleigh's theoretical approach, one can derive the position of a radial hydraulic jump as
Here, R j is the radial position of the jump, d represents the post-jump depth, a is the jet radius, g is gravitational acceleration, and q is jet discharge. Notice that fluid viscosity does not explicitly enter into Rayleigh's calculation of the jump position in this formula. Indeed, the effect of fluid viscosity on shockwave formation is one of the outstanding theoretical questions concerning hydraulic jumps. In the 1960s, Watson developed a new model to account for the change to turbulent flow following the hydraulic jump [3]. Using a series of approximation techniques, Watson proposed
Radial Jump for Superfluid Helium
Contemporary experiments have put fluid dynamics models to an extreme test by measuring the radii of hydraulic jumps using liquid helium. Above the lambda point critical temperature, liquid helium behaves as a normal fluid with conventional viscosity properties. However, as the temperature drops below 2.17 K, the liquid helium experiences a phase change and becomes a superfluid with effectively zero viscosity. The vast majority of hydraulic jump models consider conventional fluids and until recently, the ability of these models to describe superfluids was untested. The importance of the hydraulic jumps in the study of fluid dynamics motivates further investigation into the superfluid scenario.
In 2007, Rolley, Guthmann, and Pettersen performed an experiment to analyze radial hydraulic jumps in liquid helium [4]. A jet of liquid helium falls vertically downward onto the surface of a horizontal mirror. The depth of the liquid helium is measured by a CCD camera positioned at a shallow viewing angle. Additional mirrors are positioned at angles to observe the radial position of the jump from above. Next the jet of liquid helium is gradually cooled from an initial temperature of 4.2 K through the superfluid transition point to a final temperature of 1.5 K. The jump radius is measured at liquid helium temperatures above and below the superfluid transition to gauge the effect of fluid viscosity.
Above the critical temperature, the position of the hydraulic jump position is well described by theory as demonstrated by a close correspondence with experimental values. Models not accounting for surface tension tend to overestimate jump radii while models incorporating surface tension tend to predict jump radii slightly too low. More surprisingly though, the model proposed by Watson continued to accurately predict shockwave conditions of liquid helium below 2.17 K. Rolley, Guthmann, and Pettersen explain this effect by observing that the liquid helium below 2.17 K is actually a mixture of normal and superfluid components [4]. Additionally, superfluidity is disrupted above a critical fluid velocity. Therefore, even at a temperature of 1.5 K, the effective viscosity of the liquid helium was certainly non-zero. However, the experimenters were able to confirm the presence of a superfluid component by observing ripple patterns uncharacteristic of normal fluids.
Despite experimental limitations, the accuracy of theoretical predictions of shockwave conditions over a wide range in fluid viscosities indicates that current models may be applicable to a larger range of fluid behavior than previously expected.
Industrial Application as Energy Dissipator
Hydraulic jumps remain a topic of continued scientific and technological interest in part due to their industrial utility as energy dissipators. One of the most important applications of the hydraulic jump is to reduce the impact of dams downstream. Rapid outflow from a spillway erodes the channel and can undermine the structural strength of the reservoir if left unchecked. However, hydraulic jump stilling basins can reduce the discharge energy by up seventy percent [5]. Internal friction and mixing high velocity flow into the larger water volume lowers the speed of outflow. Depending on the type of design, changes in slope, channel width, or obstacles positioned along the spillway trigger the transition to turbulent flow. Forces from hydrostatic uplift, cavitation, vibration, and abrasion create unique engineering challenges for each class of spillway. Consequently, the design of a practical energy dissipator must balance efficiency with durability.
Termination Shock Analog
The termination shock of the Solar System is a more exotic analog of the hydraulic jump. The Sun emits a flow of charged particles traveling outward at speeds of 420 km/s. Along the way, plasma interactions with the interstellar medium slow the particles. The point at which the solar wind drops to a subsonic speed of about 100 km/s in the interstellar medium is called the termination shock. In this region, the solar wind experiences compressional heating, pressure fluctuations, and a sudden magnetic field changes. Beyond the termination shock, the solar wind is effectively stopped by the interstellar medium in a region called the heliosphere. The bow shock enveloping the heliosphere marks the outer edge of the Solar System.
The outer reaches of the Solar System remained largely unexplored until the last decade. Two Voyager Mission spacecraft launched in 1977 offered the best hope of directly probing this region. By 1993, Belcher, Lazarus, McNutt, and Gordon were able to use early observations of the Voyager 1 spacecraft to predict the location of the termination shock [6]. Then in December 2004, Voyager 1 entered a region with high intensities of low-energy (~1 MeV) solar wind particles at a distance of 94 AU from the Sun [7]. The encounter marked the first direct observation of the termination shock. This discovery required the efforts of both Voyager spacecraft simultaneously exploring separate regions of the Solar System. Since the solar wind pressure varies over time due to changing solar activity levels, Voyager 2 was needed as a calibration instrument to distinguish the termination shock from pressure fluctions due to normal solar events. Voyager 2 lags Voyager 1 by about 20 AU in its outward trajectory and the two spacecraft travel in different directions to probe distinct regions of the Solar System boundary. Subsequent observations have revealed that the position of the termination shock shifts by about 10 AU during an eleven-year solar acitivity cycle. Although the plasma interactions involved in the termination shock differ substantially from hydraulic jumps in water, the abrupt transition of fluid behavoir when dropping to subcritical flow velocities is shared by both phenomena.
© 2007 Keith Bechtol. The author grants permission to copy, distribute and display this work in unaltered form, with attributation to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] W. H. Hager, Energy Dissipators and Hydraulic Jump , (Kluwer Academic Publishers, 1992).
[2] L. Rayleigh, "On the Theory of Long Waves and Bores," Proc. Roy. Soc. Lond. A 90 , 324 (1914).
[3] E. J. Watson, "The spread of a Liquid Jet Over a Horizontal Plane", J. Fluid Mech. 20 , 481 (1964).
[4] E. Rolley, C. Guthmann and M. S. Pettersen, "The Hydraulic Jump and Ripples in Liquid Helium", Physica B: Cond. Mat. 394 , 46 (2007).
[5] R. M. Khatsuria, Hydraulics of Spillways and Energy Dissipators , (Marcel Decker, 2005).
[6] J. W. Belcher, A. J. Lazarus, R. L. McNutt Jr., and G. S. Gordon, "Solar Wind Conditions in the Outer Heliosphere and the Distance to the Termination Shock", J. Geophys. Res. 98 , 177 (1993).
[7] W. R. Webber, "An Empirical Estimate of the Heliospheric Termination Shock Location with Time with Application to the Intensity Increases of MeV Protons Seen at Voyager 1 in 2002-2005", J. Geophys. Res. 110 , 209 (2005).
Fluid flow: Froude and Reynolds numbers
- November 12, 2021
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19 th century experiments that helped quantify the nature of fluid flow, surface waves, and bedforms .
It all depends on inertia , like the reluctance to get out of bed on a cold winter’s morning. But rather than feeling guilty, acknowledge that by sleeping in you are adhering to the mechanical Laws that prevent the universe from collapsing – the inertial forces that keep planets in orbit around their suns, and suns in motion through their galaxies.
Inertia is loosely defined as a force that resists the change in motion of a body; here motion refers to a vector that describes velocity and direction, and ‘body’ refers to pretty well anything composed of matter, including a body of fluid. The term was coined by astronomer Johannes Kepler (17 th century); his erstwhile colleague Galileo demonstrated its qualities by experimenting with balls rolling along sloping surfaces.
However, it was Isaac Newton who codified the properties of inertia in his three Laws of Motion – apparently Newton credits Galileo with the discovery. The 1 st Law, also called the Law of Inertia , states that the motion of a body will not change unless an external force acts on it (i.e., to accelerate, decelerate, or change its direction). The 2 nd Law quantifies the relationship between an external force F, mass (m) and acceleration (a) as F = ma . And the 3 rd Law states that when an external force is applied, there will be an equal and opposite force that resists the change in motion, i.e., an inertial force – also called the Action-Reaction Law .
Inertial forces depend on the mass of a body – the larger the mass, the greater the force. In fact, the concept of mass itself is based on inertia.
Inertia and fluid flow
Inertial forces are central to the quantification of fluid mechanics. We have William Froude (1810-1879) and Osborne Reynolds (1842-1912) to thank for their eponymous numbers ( Froude number and Reynolds number ) that describe the characteristic states of flow. And because these numbers are dimensionless, they allow experiments with models (e.g., wind tunnels, sediment flumes) that can be scaled to real-world fluid flow phenomena. Scaling can be applied to almost anything related to fluid flow – from the motion of a boat through water, to quantifying the formation of sediment bedforms or sediment gravity flows from small-scale sediment flume experiments. The importance of Froude and Reynolds numbers cannot be overstated.
Froude number
Froude’s influential paper of 1861 was published by the Institute of Naval Architects (PDF available). Froude had surmised that, to predict the behaviour of a ship moving through water, he would need to experiment with much smaller versions of ships, or models , that could be scaled to the behaviour of much larger vessels. Thus, Froude’s number was derived from experiments with model boats, a few metres long.
The number expresses the characteristics of flow, including surface waves and bedforms, as the ratio between inertial forces and gravitational forces:
Fr = V/√(g.D)
Where V is bulk flow velocity (having dimensional units L.T -1 ) that reflects the dominant effect of inertia on surface flows, and the component √(g.D) where g is the gravitational constant (units of L.T -2 ), and D is water depth (units of L). The denominator represents the speed of a surface gravity wave relative to the bulk flow velocity (√(g.D) simplifies to units of velocity). Whether the surface wave is faster, slower or the same speed as the bulk flow will depend on its resistance to move, or its inertia. Fr is dimensionless.
The numerical value of Fr is used to define three conditions of flow. If Fr = 1 (numerator = denominator), then any surface wave will remain stationary – it will not move upstream or downstream. This condition occurs when both the velocities and water depth are at critical values. Not surprisingly, this condition is called critical flow . A common manifestation of critical flow is the formation of stationary waves (or standing waves) above and usually in phase with antidune bedforms (i.e., upper flow regime ).
When Fr < 1, inertial forces dominate, and the result is a subcritical condition – tranquil flow . This corresponds to lower flow regime bedforms such as ripples and larger dune structures.
When Fr > 1, gravitational forces dominate resulting in supercritical flow conditions. The corresponding stream flow surface conditions manifest as an acceleration of flow such that stationary waves break upstream (chutes – upper flow regime), commonly followed by a rapid decrease in flow and formation of a hydraulic jump where Fr < 1 ( chute and pool conditions). A hydraulic jump is the region of turbulence that represents the transition from supercritical (laminar) flow to tranquil flow – as shown in the kitchen sink example below. Supercritical flow is also common in pyroclastic density currents .
The complexity of flow transitions in a small natural system is shown in this video clip of supercritical and subcritical (tranquil) domains in a small, shallow stream. The standing waves (left) represent critical conditions where the speed of the waves matches the stream flow velocity. Supercritical conditions downstream produce chutes. Downstream migrating ripples in the foreground indicate subcritical flow.
Reynolds number
Unlike Froude who was more concerned with the surface configurations of a flowing medium, Reynolds experiments in glass pipes were concerned with the bulk structure of flow, in particular the transition from laminar to turbulent flow ( Reynolds, 1883 , PDF available). To picture this, think of a flowing fluid as a set of flow lines. In laminar flow, the flow lines are parallel, or approximately so, and relatively straight. The flow velocity will be the same across each flow line. By contrast, turbulence is described by flow lines that constantly change direction and velocity. In a flowing stream this is manifested as eddies, boils, and breaking waves. In sedimentary systems, turbulence is an erosive process, and an important mechanism for maintenance of sediment suspension through water columns and in sediment gravity flows.
The video below shows the abrupt transition from laminar flow in the slightly sinuous trail of smoke, to turbulent flow above.
To understand the nature of the laminar-turbulent flow transition, Reynolds considered four variables:
- Fluid density ρ (units of M.L -3 ).
- Fluid viscosity ( μ ) that measures the resistance to shear and is strongly temperature-dependent. μ has units of M.(L.T) -1
- Mean velocity of flow V , that reflects shear rate and inertia forces (units of L.T -1 ), and
- Tube diameter D that influences the degree of turbulence (units of L).
Reynold’s number is written as:
Re = ρVD/μ
that expresses the ratio of inertial (resistance) forces to viscous (resistance) forces . Re is dimensionless.
In his glass tube experiments, Reynolds systematically varied μ , V , and D ( μ was varied by heating the water). For each combination he discovered that the transition from laminar to turbulent flow in water was abrupt, and consistently had Re values of about 12000. Reversing the experiment gave values of about 2000 for the transition from turbulent to laminar flow.
Re can be used to determine the kind of flow in large and small fluid systems. As a general rule:
- Re values <2000 indicate laminar flow ,
- Re >4000 turbulent flow , and
- the region in between these two extremes reflects transitional flow .
Flow in most open-surface geological and geomorphic systems tends to be turbulent, with familiar examples including channelized flow (river, tidal and submarine channels) and more open flow across broad expanses such as continental shelves. It also includes volcaniclastic systems like pyroclastic flows and surges. Experimental flow in flumes produces a variety of bedforms at Re values that range from about 4000 to >100,000.
Laminar flow at low velocities is probably responsible for deposition of lower flow-regime plane beds; Allen (1992) has suggested that laminar flow at higher velocities may be restricted to thin sheet floods. Fluids having high viscosity, such as glacial ice and lava, commonly exhibit laminar flow. The Re value in microscopic rock-fluid systems, such as intercrystal boundaries in diagenetic environments, will also be low because fluid viscosity will dominate in such confined spaces.
Comparing Froude and Reynolds numbers
Froude numbers express a relationship between the free-surface of a flow and the various waves and ruffles that form there, and bedforms at the sediment-water interface. Reynolds numbers deal to the bulk characteristics of flow – whether it has laminar or turbulent structure.
The numbers Fr and Re are like chalk and cheese – they are not comparable. Both are dimensionless ratios, but that’s where the similarity ends. Both functions depend on inertial forces (the resistance to do anything), but for Fr the inertial component is in the denominator, and for Re in the numerator. Thus, if inertial forces become dominant, the numerical value of Fr decreases and that for Re increases.
Both numbers have application well beyond the relatively narrow field of sedimentology. Both are used extensively in scaled models – Fr for elucidating the efficacy of movement through a fluid – boats through water, airplanes through air. Re is used extensively to describe fluid flow in biological systems.
Allen (1992) has given sedimentologists a diagram that generalizes the relationship between Fr and Re in terms of mean flow velocity and flow depth. The boundaries of the 4 domains correspond to critical flow transitions; subcritical (tranquil) to supercritical for Fr , and laminar to turbulent for Re . I have added the most common bedforms to these domains.
There are many publications on this topic, but I highly recommend two publications that provide greater detail of theory and practice on this and other topics in fluid flow and sedimentation:
John Southard’s excellent (open access), online Introduction to Fluid Motion and Sediment Transport .
J.R.L. Allen 1992 (and later editions) Principles of Physical Sedimentology (that no sedimentologist should be without).
Other posts in this series
Identifying paleocurrent indicators
Measuring and representing paleocurrents
Crossbedding – some common terminology
Sediment transport: Bedload and suspension load
The hydraulics of sedimentation: Flow regime
Fluid flow: Shields and Hjulström diagrams
Fluid flow: Stokes Law and particle settling
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William Froude: the father of hydrodynamics
by RINA Editorial | 20th January 2024 | Maritime History & Heritage , RINA News
Froude's combination of calculations and model-based experimentation became the blueprint for measuring water resistance
RINA Historian Mark Barton looks at the life and career of one of the Institution’s, and naval architecture’s, most influential figures
William Froude (1810-1879) was the first person to formulate reliable laws for the resistance that water offers to ships (such as the hull speed equation) and thus enable ship designers to predict their stability and performance. Having worked on railways, Froude was invited by Brunel to turn his attention to the stability of ships in a seaway. He was an early contributor to the then-Institution of Naval Architects and his paper On the Rolling of Ships , was presented at the Institution’s second session in March 1861. Froude would prove hugely influential, and with his associate Henry Brunel, obtained funds from the Admiralty to identify the most efficient hull shape.
He undertook this research by building scale models and established a formula (now known as the Froude number) by which the results of small-scale tests could be used to predict the behaviour of full-sized hulls. He built a sequence of 3, 6 and 12 foot scale models and used them in towing trials. These took place on the River Dart and enabled him to establish their hull resistance and scaling laws, known as the ‘Law of Comparison’. His experiments were vindicated in full-scale trials conducted by the Admiralty. However, he recognised that exposure to wind and waves was impacting adversely on his results.
William Froude’s Torquay home. Source: Mark Barton
As a result, the first ship model towing tank or ‘Ship Tank’ was built, for a sum of £2,000, from the Admiralty. This allowed Froude to build and operate an enclosed experiment facility with a carriage, or ‘truck’ as Froude called it, to tow models at steady speed from one end of the tank to the other whilst measuring their drag under controlled conditions.
Froude’s original test tank circa 1872.
The Admiralty approved the plan but increased the scope of supply to include tests on the rolling of ships and highlighted that any cost overrun would have to be borne by Froude himself. This 85 m long tank was built at his home in Torquay in 1872 and the first model tested being the sloop HMS Greyhound . Here he was able to combine mathematical expertise with practical experimentation to such good effect that his methods are still followed today.
Although Froude died in 1879 at the age of 68, his son Robert Edmund (Eddie) Froude continued the research and established a new experimental facility on a redundant plot adjacent to the naval gunboat yard at Haslar. The last test at Torquay was completed on 5th January 1886 and the new tank at the Admiralty Experiment Works, Haslar opened a month later on 6th February. The new tank was larger and 122 m long.
The new facility was used intensively and tested models of all of the major classes of British warship that fought in World War One. The first submarine models were tested in early 1902, just a couple of months before HMS Holland I commenced her sea trials. Capacity limitations during the early inter-war years led to the Admiralty approving a second, larger tank built at right angles to the first.
The new tank (known as No.2 Ship Tank with the original tank becoming No.1) was 271 m long and had an adjustable false floor to allow experiments to be conducted regarding the impact of shallow water. It remains in use today and has been operated by QinetiQ since 2001.
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Froude number
Our editors will review what you’ve submitted and determine whether to revise the article.
Froude number (Fr) , in hydrology and fluid mechanics , dimensionless quantity used to indicate the influence of gravity on fluid motion. It is generally expressed as Fr = v /( gd ) 1 / 2 , in which d is depth of flow, g is the gravitational acceleration (equal to the specific weight of the water divided by its density , in fluid mechanics), v is the celerity of a small surface (or gravity) wave, and Fr is the Froude number. When Fr is less than 1, small surface waves can move upstream; when Fr is greater than 1, they will be carried downstream; and when Fr = 1 (said to be the critical Froude number), the velocity of flow is just equal to the velocity of surface waves. The Froude number enters into formulations of the hydraulic jump (rise in water surface elevation) that occurs under certain conditions, and, together with the Reynolds number , it serves to delineate the boundary between laminar and turbulent flow conditions in open channels.
Elin Darelius & Team
Elin Darelius & Team's Scientific Adventures
Who is faster, the currents or the waves? The Froude number
A very convenient way to describe a flow system is by looking at its Froude number. The Froude number gives the ratio between the speed a fluid is moving at, and the phase velocity of waves travelling on that fluid. And if we want to represent some real world situation at a smaller scale in a tank, we need to have the same Froude numbers in the same regions of the flow.
For a very strong example of where a Froude number helps you to describe a flow, look at the picture below: We use a hose to fill a tank. The water shoots away from the point of impact, flowing so much faster than waves can travel that the surface there is flat. This means that the Froude number, defined as flow velocity devided by phase velocity, is larger than 1 close to the point of impact.
At some point away from the point of impact, you see the flow changing quite drastically: the water level is a lot higher all of a sudden, and you see waves and other disturbances on it. This is where the phase velocity of waves becomes faster than the flow velocity, so disturbances don’t just get flushed away with the flow, but can actually exist and propagate whichever way they want. That’s where the Froude number changes from larger than 1 to smaller than 1, in what is called a hydraulic jump . This line is marked in red below, where waves are trapped and you see a marked jump in surface height. Do you see how useful the Froude number is to describe the two regimes on either side of the hydraulic jump?
Obviously, this is a very extreme example. But you also see them out in nature everywhere. Can you spot some in the picture below?
But still, all those examples are a little more drastic than what we would imagine is happening in the ocean. But there is one little detail that we didn’t talk about yet: Until now we have looked at Froude numbers and waves at the surface of whatever water we looked at. But the same thing can also happen inside the water, if there is a density stratification and we look at waves on the interface between water of different densities. Waves running on a density interface, however, move much more slowly than those on a free surface. If you are interested, you can have a look at that phenomenon here . But with waves running a lot slower, it’s easy to imagine that there are places in the ocean where the currents are actually moving faster than the waves on a density interface, isn’t it?
For an example of the explanatory power of the Froude number, you see a tank experiment we did a couple of years ago with Rolf Käse and Martin Vogt ( link ). There is actually a little too much going on in that tank for our purposes right now, but the ridge on the right can be interpreted as, for example, the Greenland-Scotland-Ridge, making the blue reservoir the deep waters of the Nordic Seas, and the blue water spilling over the ridge into the clear water the Denmark Strait Overflow. And in the tank you see that there is a laminar flow directly on top of the ridge and a little way down. And then, all of a sudden, the overflow plume starts mixing with the surrounding water in a turbulent flow. And the point in between those is the hydraulic jump, where the Froude number changes from below 1 to above 1.
Nifty thing, this Froude number, isn’t it? And I hope you’ll start spotting hydraulic jumps every time you do the dishes or wash your hands now! 🙂
All pictures in this post are taken from my blog “ Adventures in Oceanography and Teaching “. Check it out if you like this kind of stuff — I do! 🙂
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The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow . The Froude number is a ratio of inertial and gravitational forces.
· Gravity (numerator) - moves water downhill
· Inertia (denominator) - reflects its willingness to do so.
V = Water velocity
D = Hydraulic depth (cross sectional area of flow / top width)
g = Gravity
Fr = 1, critical flow,
Fr > 1, supercritical flow (fast rapid flow),
Fr < 1, subcritical flow (slow / tranquil flow)
The Froude number is a measurement of bulk flow characteristics such as waves, sand bedforms, flow/depth interactions at a cross section or between boulders.
The denominator represents the speed of a small wave on the water surface relative to the speed of the water, called wave celerity. At critical flow celerity equals flow velocity. Any disturbance to the surface will remain stationary. In subcritical flow the flow is controlled from a downstream point and information is transmitted upstream. This condition leads to backwater effects. Supercritical flow is controlled upstream and disturbances are transmitted downstream.
Wave propagation can be used to illustrate these flow states: A stick placed in the water will create a V pattern of waves downstream. If flow is subcritical waves will appear in front of the stick. If flow is at critical waves will have a 45 o angle. If flow is supercritical no upstream waves will appear and the wave angle will be less than 45 o .
Note: Critical flow is unstable and often sets up standing waves between super and subcritical flow. When the actual water depth is below critical depth it is called supercritical because it is in a higher energy state. Likewise actual depth above critical depth is called subcritical because it is in a lower energy state.
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I will answer anything from the world of physics.
Froude number
What is froude number.
The Froude number is a dimensionless parameter used to determine the similarity between the flow of fluids at different scales. It is named after William Froude, a British engineer who studied the movement of ships and waves in water. The Froude number is defined as the ratio of the inertia forces to the gravitational forces acting on a fluid flow system. In simpler terms, it is a measure of how fast the fluid is moving relative to the depth of the fluid.
Calculation of Froude Number
The Froude number is calculated using the equation F = V/√(gL), where F is the Froude number, V is the velocity of the fluid, g is the acceleration due to gravity, and L is the characteristic length of the fluid flow. The characteristic length can be the depth of the fluid in a channel, the radius of a pipe, or the length of a ship. The Froude number is a dimensionless quantity, meaning that it has no units.
Applications of Froude Number
The Froude number is used in a variety of applications, including ship design, hydraulic engineering, and fluid mechanics. In ship design, the Froude number is used to determine the optimal speed of the ship and to ensure that the ship operates in a stable manner in different sea conditions. In hydraulic engineering, the Froude number is used to design channels and culverts for water flow. In fluid mechanics, the Froude number is used to study the behavior of fluids in different scenarios, such as the flow of water over a dam or the flow of air over an airplane wing.
Example of Froude Number in Action
One example of the use of the Froude number is in the design of a hydroelectric power plant. The Froude number is used to determine the optimal speed of the water flowing through the turbines. If the Froude number is too high, the water will not be able to flow smoothly through the turbines, resulting in inefficiency and damage to the equipment. If the Froude number is too low, the turbines will not be able to generate enough power. By calculating the Froude number, engineers can design the plant to operate at the most efficient speed and generate the maximum amount of power.
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Froude number
One of the criteria for similarity of the motion of a liquid or gas, applicable in cases when the influence of gravitational forces is significant. The Froude number characterizes the relationship between the inertial and gravitational forces acting on an elementary volume of the liquid or gas. The Froude number is
$$\mathrm{Fr}=\frac{v^2}{g \ell},$$
where $v$ is the speed of the flow (or the speed of the moving body), $g$ is the gravitational acceleration and $\ell$ is the typical length of the flow or the body.
The Froude number was introduced by W. Froude (1870).
[a1] | L.I. Sedov, "Similarity and dimensional methods in mechanics", Infosearch (1959) (Translated from Russian) |
[a2] | G. Birkhoff, "Hydrodynamics, a study in logic, fact and similitude", Princeton Univ. Press (1960) |
- This page was last edited on 4 January 2024, at 19:56.
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- Froude Number
Explore the intricate concepts of Engineering Fluid Mechanics , starting with a comprehensive analysis of the Froude Number. This article provides in-depth explanations from understanding the theory, its importance, through to practical examples using the Froude Number Equation. It offers detailed insight into flow states, scrutinising both subcritical and critical flow. Additionally, readers can delve into advanced applications using a densimetric perspective and the relationship between the Froude Number and Dimensional Analysis . Unearth valuable knowledge and enhance your engineering proficiency with these real-world examples and case studies.
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What does the Froude Number represent in Engineering Fluid Mechanics?
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Understanding the Froude Number in Engineering Fluid Mechanics
In the field of Engineering Fluid Mechanics , the Froude Number is a significant non-dimensional parameter that plays a crucial role in analyzing various fluid flow scenarios. You'll come across this term quite often as you delve deeper into fluid mechanics, and understanding it is a stepping stone towards mastering this area of engineering.
The Froude Number, represented by \(Fr\), is a dimensionless number defining the ratio of the inertia force to the gravitational force acting on a fluid in motion. It is named after the British engineer and naval architect William Froude.
Definition of Froude Number
You might be curious to know more about the mathematics behind the definition of the Froude Number. For a body or surface moving in a fluid, or a flow past a stationary body or surface, the Froude Number is given by the formula:
- \(v\) is the velocity of the object,
- \(\sqrt{gL}\) is the speed of a gravity wave through the fluid,
- \(g\) is the acceleration due to gravity,
- and \(L\) is a characteristic length (for instance, the depth of flow or height of an object).
A Froude Number less than 1 indicates a subcritical flow condition where gravitational forces dominate. If the Froude Number is equal to 1, the flow is critical, and gravity and inertia forces are balanced. A Froude Number greater than 1 indicates a supercritical flow, dominated by inertia forces. This is quite important to understand as it helps in analyzing the types of flow in different situations.
Importance of Froude Number in Engineering Fluid Mechanics
The Froude Number carries a significant weight in Engineering Fluid Mechanics due to several reasons:
- It plays a pivotal role in predicting the flow regime , whether the flow is laminar, transitional or turbulent.
- In hydrodynamics, the Froude Number is used in the study of the stability of ships and boats, wave generation, and wave resistance.
- The Froude Number is used for scaling fluid flow problems that involve a free surface. This implies that the Froude Number is highly valuable in designing flow models for rivers, channels, and hydraulic structures.
- It’s also employed extensively in the field of environmental engineering while modelling wastewater and stormwater treatment systems.
Suppose there's a river with a flow velocity of 2 m/s and depth of 1.5 m. Considering the acceleration due to gravity as 9.81 m/s², the Froude Number would be \(\frac{2}{\sqrt{9.81 * 1.5}}\) = 0.65. The subcritical (or tranquil) flow state denotes that gravitational forces are the dominant factor in the flow regime of this river.
Comprehending the Froude Number Equation
Validation of the Froude Number and its incorporation into the broader subject of Engineering Fluid Mechanics is largely due to the mathematical equation it is represented by. This equation is pivotal in understanding the dynamics of fluid flow and its interaction with gravitational forces. The equation for the Froude Number is:
Here, \(Fr\) is the Froude Number, \(v\) is the velocity of the fluid, \(g\) is the acceleration due to gravity and \(L\) is a significant length associated with the problem, such as depth of flow.
Guidance on the Froude Number Derivation
The derivation of the Froude Number is based on the principle of dimensional analysis , which preferably involves Buckingham's Pi Theory. This method is widely used in fluid mechanics to generate dimensionless numbers.
To start with, consider the dimensions of velocity \([LT^{-1}]\), gravitational acceleration \([LT^{-2}]\) and length \([L]\). The aim here is to establish a dimensionless quantity by appropriately combining these variables.
Applying Buckingham's theorem with velocity, acceleration, and length as repeating variables, we find that there is one dimensionless group which could be formed. This leads to the combination \(v/\sqrt{gL}\), which is recognised as the Froude Number.
Indicative of its derivation, the Froude Number signifies the importance of inertial to gravitational forces in scaling fluid flow problems involving a free surface.
Applying the Froude Number Equation: Practical Examples
Being a versatile and effective tool, the Froude Number is used ubiquitously within fluid dynamics and engineering applications. Let's look at some practical scenarios and derive the Froude Number in each of them.
Example 1: Consider a water flow in a channel with a velocity of 3 m/s and a depth of 2 m. Given that the acceleration due to gravity is 9.81 m/s², the Froude Number can be calculated as follows:
In this case, as the Froude Number is less than 1, it denotes a subcritical flow condition, which signifies that gravitational forces dominate over inertial forces.
Example 2: Now consider a ship moving through the water with a velocity of 7.5 m/s. Let the significant length \(L\) in this case be the length of the ship's hull submerged in water, let's say 30 m. The Froude Number for this example would be:
This Froude Number much less than 1 indicates a highly subcritical flow, which implies that hydraulic jumps or sudden changes in the water elevation are less likely to occur around this ship.
In both these practical applications, and indeed in numerous others, the Froude Number plays a crucial role in helping us understand and predict the behaviour of fluid flows under the action of gravity and inertial forces.
Insight into Different Flow States: Subcritical and Critical Flow Froude Number
In the study of fluid mechanics, different flow states hold significant meaning. The Froude Number plays a critical role in defining these various states of flow including subcritical and critical flows. It helps in understanding whether the flow regime is dominated by inertial or gravitational forces.
Explanation of Froude Number Subcritical Flow
When addressing the states of flow, it’s essential to commence with the concept of subcritical flow. A subcritical flow happens when the Froude Number is less than one (\(Fr < 1\)). The name 'subcritical' signifies that the flow is under the critical point and is slow or tranquil. In this state, the flow is dominated by gravitational forces more than the inertial forces.
This feature of subcritical flow allows for small distortions or perturbations to propagate both downstream and upstream, which means that any change in the flow’s downstream conditions can influence the upstream flow. In subcritical flow, water profiles are often smooth and gentle showing no abrupt changes unless acted upon by an external force.
We can often see examples of subcritical flow in nature in slower-moving rivers or streams. Also, open channels that carry water at a steady and slow pace usually exhibit subcritical flow.
Consider a canal with a water velocity of 1 m/s and a flow depth of 1.5 m. Given the acceleration due to gravity as 9.81 m/s², the Froude Number for this canal would be calculated as \(Fr = \frac{1}{\sqrt{(9.81*1.5)}}\), which equals 0.26. This value of Froude Number less than 1 indicates a subcritical flow state, inferred as a slow, tranquil flow dominated by gravitational forces.
Scrutinising the Critical Flow Froude Number
Moving further along the spectrum of flow states, you reach critical flow , which occurs when the Froude Number equals one (\(Fr = 1\)). This is considered the dividing point between subcritical and supercritical flows. The critical flow is a state of balance where the inertial and gravitational forces acting on the fluid are equal.
Critical flow condition serves as the transition between the subcritical and supercritical states. Understandably, it is less common in natural or man-made flows given that any slight disturbance will cause the flow to shift into either the subcritical or supercritical state.
However, in practice, the critical flow is seen in certain fluid mechanics phenomena such as hydraulic jumps, where water abruptly transitions from supercritical to subcritical flow, or when analysing the maximum discharge capacity of a run-of-river hydropower plant or a spillway of a dam.
Consider a scenario where a waterfall has a velocity of 10 m/s just before it drops over a cliff edge which is 5 m high. Here, the Froude Number is given by \(Fr = \frac{10}{\sqrt{9.81*5}}\), which equals 1.41. As this value is greater than 1, the flow of the waterfall before the drop is in a supercritical state. However, at the very edge of the cliff where the waterfall begins to drop, the flow condition becomes critical before transitioning into a free fall, essentially a supercritical flow state.
In conclusion, the comprehension of Froude Number resulting in subcritical or critical flows is crucial in several fields of practical engineering, such as hydrology for designing channels, spillways and predicting flood levels, or naval architecture for designing ship hulls to minimise wave resistance.
Advanced Applications of Froude Number: The Densimetric Perspective
The traditional version of the Froude Number plays a significant role in various engineering applications where fluid flow and gravitational forces interact. However, when it comes to situations involving density differences within fluids, such as layered fluids or multiphase flows, a more advanced concept comes into play: the Densimetric Froude Number. This variant of the Froude Number significantly expands its usefulness, taking into account the density contrast in fluid flows, making it especially crucial in environmental and industrial applications.
Conceptualising the Densimetric Froude Number
The Densimetric Froude Number, often denoted as \(Fr_d\), pulls in the element of density difference between two fluids or between regions within a single fluid. The role of density variations becomes notable when we investigate stratified flows or multiphase flows, where lighter and heavier fluid layers or phases intermingle.
The Densimetric Froude Number is defined as: \(Fr_d = \frac{v}{\sqrt{g' \cdot L}}\).
In the above formula:
- \(v\) is the characteristic velocity of the fluid,
- \(L\) is a characteristic length,
- and \(g'\) represents a reduced gravitational acceleration that introduces the density difference between the two layers or phases of the fluid and is given by \(g' = g \cdot \frac{\Delta \rho}{\rho_0}\), where \(\Delta \rho\) is the change in density and \(\rho_0\) is the reference density (often the density of the lighter fluid).
As with the original Froude Number, the Densimetric Froude Number is a dimensionless quantity. The interpretations linked to different values of \(Fr_d\) are similar to those for the traditional Froude Number. Yet, the inclusion of density contrast into the equation makes the Densimetric Froude Number significantly more relevant in scenarios of density stratified flows or multiphase flows.
It's fascinating to note that the Densimetric Froude Number has shown to be an essential parameter in the study of geophysical flows, particularly those linked to atmospheric and oceanographic phenomena. Here, density differences caused by temperature and salinity variations strongly impact the flow behaviour and dynamics, and the Densimetric Froude Number becomes a crucial tool for analysis and modelling.
Real-world Engineering Examples Utilising Densimetric Froude Number
The Densimetric Froude Number finds its applications in an array of real-world scenarios where fluid layers of different densities interact. Here are a couple of engineering cases in which it plays a key role:
Example 1: Atmospheric and Oceanic Flows: Probably the most widespread use of the Densimetric Froude Number is in geophysical fluid dynamics . Both atmospheric and oceanic flows often exhibit stratification due to temperature or salinity-induced density differences. By taking into account these density contrasts, the Densimetric Froude Number aids in appropriately scaling and investigating such phenomena. This can help in predicting weather patterns or ocean currents more accurately.
Example 2: Industrial Multiphase Flows: In industries, multiphase flows are quite common. Whether it's the oil and gas industry dealing with the simultaneous flow of oil, water, and gas in pipelines, or the food and chemical industries handling mixtures of liquids, solids, and gases, the Densimetric Froude Number becomes useful. It aids in characterising the flow regime and predicting phase distribution and pressure drop, thereby optimising the process performance.
Consider a pipeline in an oilfield carrying a mixture of crude oil (density = 850 kg/m³) and natural gas (density = 20 kg/m³). Let's say the mixture's velocity is 3 m/s and the pipeline diameter (characteristic length) is 0.1 m. Given the standard gravity as 9.81 m/s², we calculate the reduced gravity as \(g' = 9.81 \cdot \frac{(850 - 20)}{850}= 10.38\) m/s². The Densimetric Froude Number in this case can then be estimated as \(Fr_d = \frac{3}{\sqrt{10.38*0.1}}= 2.94\), which is greater than 1, indicating that the gas-oil flow in this pipeline is in a supercritical condition and dominated by inertia forces.
The value of the Densimetric Froude Number, in this situation and many like it, allows engineers to anticipate flow behaviour accurately and design effective operational strategies.
Froude Number and Dimensional Analysis
The study of fluid dynamics would be incomplete without the concept of dimensional analysis and the utilisation of dimensionless numbers, a chief one among them being the Froude Number. This section delves into the relationship between the Froude Number and dimensional analysis in the context of engineering fluid mechanics.
Relationship Between Froude Number and Dimensional Analysis
The practice of dimensional analysis is a powerful tool within physics and engineering disciplines, aiding not only in verifying equations and formulas but also in reducing complex physical phenomena to a simpler, more comprehensible form through dimensionless numbers. The Froude Number holds stature as one of these significant dimensionless numbers, primarily in studies involving gravity-driven fluid flows such as waves in oceans, rivers, and channels, where gravitational and inertia forces interact.
The Froude Number is defined as the ratio of inertial forces to gravitational forces: \(Fr = \frac{V}{\sqrt{gL}}\), where \(V\) is the characteristic velocity of the fluid, \(g\) is the acceleration due to gravity, and \(L\) represents a characteristic length.
This dimensionless number signifies the relative influence of these two forces on the flow behaviours. The fact that it's dimensionless makes it particularly useful when studying similar flow situations in differently scaled systems. For instance, water waves in a small laboratory tank or in an extensive ocean can be compared using the Froude Number, provided the flow is dynamically similar.
It is essential to realise that the construction of the Froude Number involves bringing together physical quantities of different dimensions (speed, length, gravity) using a square root operation. This is a classic example of how the process of dimensional analysis helps synthesise dimensionless quantities from dimensional ones.
Looking deeper into the subject, we find that fluid flow scenarios often involve more complexities beyond just inertia and gravity forces. For instance, viscosity and surface tension forces may become influential at smaller scales. Hence, in those situations, other dimensionless numbers like the Reynolds Number for inertial-viscous forces or the Weber Number for inertial-surface tension forces become significant. However, let it be noted that the Froude Number remains the go-to dimensionless number for large scale flows dominated by inertia and gravity forces.
Case Studies of Froude Number Dimensional Analysis in Engineering Fluid Mechanics
Understanding how the Froude Number and dimensional analysis work together can be best realised through practical case studies from the field of engineering fluid mechanics.
Case Study 1: Design of Ship Hulls: In naval architecture - the science of ship design - the hull shape plays a crucial role in a ship's resistance movement through water. The Froude Number is used as a significant parameter to ensure dynamic similarity between model tests in laboratories and real-world scenarios. For similar flows, if the ratios of inertial to gravitational forces (i.e., the Froude Numbers) of the model and the actual ship are equal, the wave patterns, wave resistances, and other hull performance characteristics will correspond. Therefore, making the accurate assessments using a small-scale model possible.
Case Study 2: River Modelling and Flood Prediction: Flood prediction and river management often rely on the construction of physical scale models of river segments. Here, the Froude Number enables the transfer of insights drawn from the scale models to the actual rivers. By ensuring that the Froude Number is the same in the model and reality, engineers can observe how changes in river flow characteristics like velocity, depth, and channel shape affect flood levels and consequently devise effective flood control measures.
For example, consider a situation where a large river is prone to flooding and engineers are designing a levee system to control it. Suppose they create a 1:100 scaled-down physical model of the river in a laboratory. In the model, if a particular flow velocity of 0.1 m/s results in safe water levels, they can use the Froude Number to determine the equivalent safe flow velocity in the actual river. If the model river depth (L) is 0.05 m, then the Froude Number in the model is \(Fr = \frac{0.1}{\sqrt{9.81*0.05}} = 0.45\). Assuming the same Froude Number in the actual river, with a depth of 5 m (100 times the model depth), the safe flow velocity can be calculated as \(V = Fr*\sqrt{9.81*5} = 0.45*\sqrt{9.81*5} = 1 m/s\). Hence, ensuring that the actual river flow velocity is maintained around this value will help achieve the desired safety against flooding, as indicated by the model study.
The above examples illustrate, not just the importance of the Froude Number and dimensional analysis in engineering, but also their real-world implications in mitigating risks and optimising system performance.
Froude Number - Key takeaways
- The Froude Number is a dimensionless quantity which represents the ratio of the inertial force to gravitational force in free surface flow problems, calculated using the formula: \(Fr = \frac{v}{\sqrt{g \cdot L}}\).
- Subcritical flow occurs when the Froude number is less than one (\(Fr < 1\)), suggesting that gravitational forces dominate over inertial forces. This flow state is typically slow and tranquil, with smooth water profiles and both upstream and downstream propagation of disturbances.
- Critical flow is defined by a Froude number of one (\(Fr = 1\)), indicating a balance of inertial and gravitational forces on the fluid. It serves as the transition point between subcritical and supercritical flow states.
- The Densimetric Froude Number accounts for density differences within fluids in scenarios like layered or multiphase flows. It is calculated using the formula: \(Fr_d = \frac{v}{\sqrt{g' \cdot L}}\) where \(g'\) represents a reduced gravitational acceleration considering density difference.
- The Froude Number is derived through the process of dimensional analysis, specifically using Buckingham's Pi theorem. This helps establish the Froude Number as a significant dimensionless group, helping to scale and analyze fluid flow problems involving a free surface.
Flashcards in Froude Number 15
Critical flow is a balance state when the Froude Number equals one. This implies that inertial and gravitational forces acting on the fluid are equal. This condition is the transition point between subcritical and supercritical flows.
The Froude Number equation represents the ratio of inertial to gravitational forces in fluid flow problems involving a free surface. The variables are velocity of the fluid, acceleration due to gravity and significant length related to the problem.
The understanding of subcritical and critical flows is essential in practical engineering fields like hydrology for designing channels, spillways, flood level prediction or naval architecture for designing ship hulls to minimise wave resistance.
The Froude Number is a powerful tool in dimensional analysis as it helps to represent the interaction of gravitational and inertia forces in fluid flows. Its dimensionless nature allows for comparing similar flow situations in differently scaled systems. It's particularly useful in studying large-scale flows dominated by inertia and gravity.
The Froude Number in fluid mechanics defines different states of flow such as subcritical and critical. A flow is considered subcritical when the Froude Number is less than one, meaning the flow regime is dominated by gravitational forces more than the inertial forces.
The Froude Number, symbolized by \(Fr\), is a non-dimensional parameter in Engineering Fluid Mechanics that defines the ratio of the inertia force to the gravitational force acting on a fluid in motion.
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COMMENTS
In continuum mechanics, the Froude number (Fr, after William Froude, / ˈ f r uː d / [1]) is a dimensionless number defined as the ratio of the flow inertia to the external force field (the latter in many applications simply due to gravity).The Froude number is based on the speed-length ratio which he defined as: [2] [3] = where u is the local flow velocity (in m/s), g is the local gravity ...
Froude's experiments were validated in full-scale trials, and his methods of ship scale modeling are still used today. Froude's work on free-surface flows was acknowledged by naming the ratio of inertial and gravitational forces as the "Froude number." It is one of two numbers in fluid mechanics that dominate the scaling of all flows.
William Froude (/ ˈ f r uː d /; [1] 28 November 1810 in Devon [2] - 4 May 1879 in Simonstown, Cape Colony) was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships (such as the hull speed equation) and for predicting their stability.
The Froude number is the ratio of flow velocity to wave celerity and marks the boundary between critical and subcritical flow [1]. For conditions in which the Froude number is greater than one, flow velocity exceedes the wave celerity (the speed of an individual wave crest) and the fluid motion is smooth. ... Contemporary experiments have put ...
Froude number. Froude's influential paper of 1861 was published by the Institute of Naval Architects (PDF available). Froude had surmised that, to predict the behaviour of a ship moving through water, he would need to experiment with much smaller versions of ships, or models, that could be scaled to the behaviour of much larger vessels. Thus ...
Although Froude died in 1879 at the age of 68, his son Robert Edmund (Eddie) Froude continued the research and established a new experimental facility on a redundant plot adjacent to the naval gunboat yard at Haslar. The last test at Torquay was completed on 5th January 1886 and the new tank at the Admiralty Experiment Works, Haslar opened a ...
Froude number (Fr), in hydrology and fluid mechanics, dimensionless quantity used to indicate the influence of gravity on fluid motion. It is generally expressed as Fr = v /( gd ) 1 / 2 , in which d is depth of flow, g is the gravitational acceleration (equal to the specific weight of the water divided by its density , in fluid mechanics), v is ...
REYNOLDS NUMBERS, AND FROUDE NUMBERS INTRODUCTION 1 Steady flow past a solid sphere is important in many situations, both in the natural environment and in the world of technology, and it serves as a good ... obtain the function by experiment, as is commonly the case in problems of flow of real fluids. With flow past the sphere as an example we ...
The towing carriage met Froude's desire for something more scientific for his new experiment tank, something that would pull the models at a steady and measurable rate. It utilized a 3-foot, 3-inch gauge railway for the towing equipment and screw dynamometer, "carried by the roof principals and extended the full length of the waterway.
The Froude number, an important parameter for the study of liquids moving in a free surface (e.g., surface wave motion), is defined as F r = v 2 / gL or F r = ρv 2 L 2 /ρgL 3.Because ρL 3 = m (mass), one can also have F r = ρv 2 L 2 /mg.Since ρv 2 is the inertial force per unit area, the numerator of this last formula represents the total inertial force of the fluid.
Froude similarity considers besides inertia the gravity force, which is dominant in most free surface flows, especially if friction effects are negligible or for highly turbulent phenomena such as wave breaking. The Froude similarity requires identical Froude numbers between model and its prototype for each selected experiment.
For an example of the explanatory power of the Froude number, you see a tank experiment we did a couple of years ago with Rolf Käse and Martin Vogt . There is actually a little too much going on in that tank for our purposes right now, but the ridge on the right can be interpreted as, for example, the Greenland-Scotland-Ridge, making the blue ...
Froude Number and Flow States. The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow. The Froude number is a ratio of inertial and gravitational forces. · Gravity (numerator) - moves water downhill. · Inertia (denominator) - reflects its willingness to do so. Where: V = Water velocity.
The Froude number is a dimensionless parameter used to determine the similarity between the flow of fluids at different scales. It is named after William Froude, a British engineer who studied the movement of ships and waves in water. The Froude number is defined as the ratio of the inertia forces to the gravitational forces acting on a fluid ...
The first term in Equation 2.7.2 2.7.2 is then proportional to μVL/ρL2V2 μ V L / ρ L 2 V 2, or μ/ρLV μ / ρ L V. This is simply the inverse of a Reynolds number. The Reynolds number in any fluid problem is therefore inversely proportional to the ratio of a viscous force and a quantity with the dimensions of a force, the rate of change of ...
Froude number . Fn • The ratio between inertia and gravity: gL • Dynamic similarity requirement between model and full scale: • Equality in . Fn. in model and full scale will ensure that gravity forces are correctly scaled • Surface waves are gravity -driven ⇒ equality in . Fn. will ensure that wave resistance and other wave forces are
Froude number. One of the criteria for similarity of the motion of a liquid or gas, applicable in cases when the influence of gravitational forces is significant. The Froude number characterizes the relationship between the inertial and gravitational forces acting on an elementary volume of the liquid or gas. The Froude number is.
Froude's writings include "Experiments on the Surface-friction Experienced by a Plane Moving Through Water," in British Association for the Advancement of Science Report, 42nd Meeting, 1872; and "On Experiments with H.M.S. Greyhound," in Transactions of the Institution of Naval Architects, 16 (1874), 36-73.
The Froude Number is often depth-averaged or applied as a densiometric function to compare experiments of sediment gravity flows in laboratory tanks, such as Euro Tank (sensu Pohl, 2019). Lower stage plane beds and ripples are common bedforms in fine-grained deepwater environments that are formed by subcritical flows, developed beneath a near ...
C. The Froude Number equation signifies the importance of gravitational forces in fluid mechanics. The variables are mass of the fluid, acceleration due to gravity and the depth of flow. D. The Froude Number equation is based on Buckingham's Pi theory and it represents the force applied on a fluid under various conditions of flow rates and ...
A Froude number of 2.5 to 4.5 is the worst design range, as the jump in this range will create large waves that could cause structural damage. Theory: The performance of a hydraulic jump depends mainly on the value of the Froude number. The Froude number is defined as ... In this experiment, the velocity is varied by varying the speed of the ...
Froude approach. In 1978 the International Towing Tank Congress (ITTC) adopted a updated standardized procedure for performing ship resistance experiments and calculations. The ITTC-78 method considers the total model resistance as the sum of a viscous component and a wave making component (see reference material).
It is expressed as, Froude No = v √gd. Here, 'd' represents the depth of flow, 'g' is the acceleration due to gravity, 'v' is the celerity or gravity of the small surface wave. When Froude No. is at a critical point that is Fr = 1, then the velocity of the flow is equal to the velocity of surface waves, which is also known as the ...
The study utilizes laboratory experiments conducted in a flume measuring 12 m in length, 1.2 m in width, and 0.8 m in depth, with a chute angle of 26.6°. The cascade of spillways consists of 10 steps, each 0.12 m long and 0.06 m high. ... which proved helpful in energy dissipation. The downstream velocity and Froude number for case 4 were ...
Extending the work of Yang-Zumbrun for the hydrodynamically stable case of Froude number \(F<2\), we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow.Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by ...