Classic energy problem in open-channel flow - Wikiversity
Introduction; Channel Flow; Flow Velocity Formulas; Specific Energy . Then, if you measure the depth of the river at good points in a cross Now, if we know y, R and S we can just substitute in the equation and find q. Energy, Specific Energy, and Gradually Varied Flow. Channel Geometry Characteristics. • Depth, y. • Area, A. • Wetted perimeter, P. • Top width, T Flow in Open Channels: Manning. Equation. Manning's equation is used to relate the average channel . Hydraulic energy head measured with respect to the local. The critical depth value mentioned in the E–y diagram corresponds to the minimum energy a flow can possess for a.
For solid bodies, the frictional force is about independent of velocity, so they just go on accelerating if they move at all, and never reach a steady state, ideally. The shear force between the liquid and its bed depends on the square of its velocity, which is suggested by dimensional analysis.
There is no physics in this; we have just used dimensional analysis and the wonder is that it turns out to be almost right. Therefore, we are pretty sure water will reach a steady velocity in flowing down an incline. In surveying, slope is usually referred to horizontal distance. In fact, the slopes of water channels are usually so small that the difference is negligible. A slope of 0. The frictional force is exerted only along this perimeter.
Note that the coefficient C' has dimensions, and so differs when different units are used. Then, C' is on the order of The Swiss engineers E.
Kutter showed that much better results could be obtained if the constant C' depended on R, S and a constant n that was characteristic of the roughness of the channel as follows: This, of course, is for metres. For feet, it is multiplied by 1. This will give some idea of the variation of n.
Better values can be found in engineering references. Kutter's formula works well, but is very difficult to use in iterative and theoretical calculations because it cannot be solved for R or S easily.
Manning found a formula that gives results very close to Kutter's, but is much simpler, and which is now used almost exclusively. It is absolutely remarkable that such a simple formula gives such good results, and that it uses the same roughness parameter n. Manning's formula is easily solved for R or S, a great advantage.
If you use feet, multiply by 1. Let us use a simple rectangular channel as an example. Many channels are actually rectangular, or close to it, so this is a useful example. We will show the modifications necessary when the channel is not rectangular, and will note that the results are pretty much the same qualitatively. Let y be the depth of the water above the bottom of the channel, and b the width of the channel.
Now, if we know y, R and S we can just substitute in the equation and find q. In many problems, however, we are given some desired discharge, and want to find the corresponding y and V. This was once an arduous process, involving an iterative solution.
Now, with a tool like the HPG, it is straightforward. All you have to do is express the equation in terms of numbers and the variable y, type it in, and press SOLVE.
Even in Daugherty's excellent and accurate text, on p. The HP not only saves work, but more importantly it is less subject to error. Many of our illustrations will be for an even simpler example, the wide rectangular channel in which only bottom friction is present.
Results for the wide rectangular channel will be similar to those for a channel of any shape. If we do not have a rectangular channel, but one of arbitrary shape, we must work with Q instead of q, and y is usually the depth from the lowest point in the channel. Then the area A occupied by fluid and the wetted perimeter P are functions of y, which may be expressed analytically for channels of a geometric shape. Then, R can be expressed as a function of y, and used in the Manning equation.
We have, in fact, done this for the rectangular channel. All we have to do is replace the square bracket above by R y. If R is expressed as, say, a table of values, the best way to proceed is to plot or tabulate q as a function of y, and then enter the table with q instead of y.
You should feel comfortable with these calculations after a little practice. They can solve many problems all by themselves. The value of y corresponding to a given Q and S is called the normal depth for those conditions. The three energy components are elevation, pressure, and velocity. All play a role in open-channel flow. For any flow, there is an energy grade line that can be imagined above the flow, and its slope is S'. Below this is the bed grade line, with slope S, and usually below that is the horizontal datum, the reference surface.
The streamlines of the flow are parallel. Therefore, C, the energy per unit weight, has the same value at any depth. When a closed channel runs full, then the depth can no longer vary to accommodate the discharge, and the pressure becomes different from the atmospheric pressure, and must be taken into account in using Bernoulli's theorem.
This is the fundamental difference between open channel flow and pipe flow. The curve of q as a function of y for a fixed E is plotted at the right. We notice that q is a double-valued function of y, and has a maximum possible value qm.
Energy–depth relationship in a rectangular channel
The corresponding depth y can be found by differentiating q with respect to y and setting the derivative equal to zero. For depths greater than the critical depth, the velocity is smaller than the critical velocity. Flow in this region is called subcritical. For depths smaller than the critical depth, the velocity is greater than the critical velocity.
Flow in this region is called supercritical. Note that the sub- and super- refer to the velocity of flow. The same discharge q is possible with given E in either region. In the upper region, we have greater flow area, in the lower region greater flow velocity. Because the frictional resistance varies rapidly with velocity, subcritical uniform flow is associated with gentle slopes, supercritical uniform flow with steep slopes. Note that the curve is plotted with respect to dimensionless variables, so the same curve can be used for any E or qm.
Consider flows described by points a and c. Since they are on the same vertical line, the discharge is the same for each.
The same holds for point c, but here the static part is much smaller and the dynamic part larger. At the point of maximum discharge for this value of E, point e, the static energy is twice the dynamic energy. Assume there is no head loss, the specific energy does not change, so the flows in the constriction are represented by points b and d.
We note that in subcritical flow, the depth of flow decreases, while in supercritical flow the depth increases. Again, this is for a rectangular channel but diagrams for other channel shapes are similar. The point a corresponds to an upper-stage or tranquil flow.
Note that the specific energy is a minimum at the critical depth.
This minimum value will, of course, depend on the discharge. Let us suppose there is a hump in the bed of the channel that decreases the specific energy from E1 to E2. The height of the hump will be the decrease in the specific energy. If the flow is subcritical, we see that depth will decrease slightly to point b. If the flow is supercritical, the depth will, on the other hand, increase slightly to point e.
This is exactly the same as the response to a lateral constriction. If the hump is high enough, the flow may become critical at it. For any larger hump, the specific energy cannot decrease further, and instead the upstream depth must increase to keep the flow critical over the hump. For this reason, such a point may be called a control section, since it controls the upstream depth.
The vertical tangent at critical depth means that small changes in E will cause large changes in y, so the surface may appear disturbed. On any declining slope S, uniform flow will be established if possible at a depth y called the normal depth for a given Q. If the normal depth is greater than the critical depth, then the slope S is called mild and the flow tranquil or upper-stage. If it is smaller, then the slope S is called steep and the flow is called rapid or lower-stage.
Whether a slope is mild or steep depends on Q and n, the discharge and roughness. A slope giving exactly the critical velocity is called, unsurprisingly, critical, and is not often found. Maximum discharge occurs for critical flow.
It expresses the relative strength of inertial and gravitational forces, and was first used in ship modelling in the estimation of wave drag effects. If the Froude numbers of two flows are the same, then these effects will be similar. It expresses the ratio of inertial forces to viscous forces. Therefore, for any depth of water we can find the frictional head loss.
These ideas are summarized in the diagram at the right, which shows a reach between 1 and 2 in which the flow is uniform. The direction of flow is, of course, the direction in which the energy line EL falls. Note that the slope of the EL s is the same as the bed slope s'.
The average velocity V has adjusted itself to make this so, determining the depth y from the known specific energy E or discharge Q. This depth is the standard depth, and the velocity is the standard velocity, for this flow. The critical depth yc is the lower boundary of the region for which the specific energy decreases with increasing depth subcritical flow and the upper boundary of the region where the specific energy increases with a decrease in depth supercritical flow.
Nonuniform Flow The slope s' of the channel may be different in different reaches; the channel may change width or shape, there may be humps and hollows in the channel, or weirs and sudden drops, and other factors that change the flow conditions. The resulting flow will be steady, although the elements of the water will experience acceleration from point to point.
Classic energy problem in open-channel flow
We can usually make a good approximation to the flow by using Bernoulli's theorem and dividing the problem into lengths of approximately uniform conditions. We will want to know how the depth y and the velocity V vary with position, as well as the other characteristics of the flow. These problems are quite interesting, have many practical applications, and show the power of engineering hydraulics. Consider a hump in the water surface, a surface wave travelling down the channel with some velocity V.
That such waves exist is an experimental fact. Now suppose the water in the channel is moving with velocity V in the other direction; the wave will appear to stand still. In deep water or short wavelengthsthe velocity is proportional to the square root of the wavelength, as in the ocean, where waves of different periods separate themselves from each other when propagating long distances.
The group velocity in this case is half the phase velocity. The propagation of gravity waves on a water surface is said to be dispersive.
We observe both kinds of waves on the surface of the water in our channels. The most important thing to us is that the speed of long surface waves is exactly the critical velocity! If the water is moving faster than the critical velocity, as in rapid flow, then wave disturbances cannot propagate upstream.
On the other hand, in tranquil flow waves from a disturbance can propagate upstream as well as down. This is one way to determine whether an observed flow is supercritical or subcritical, by simple observation.
Suppose that the slope increases from mild to steep at a certain point. The upstream normal depth y is greater than yc, while the downstream normal depth y' is less than yc.
The depth decreases as E decreases on the upper-stage curve, then continues to decrease as E goes through a minimum at the critical depth near the break point of the profile, and then E increases again until the normal depth for the steep slope is reached. Because of this, the point at which critical depth is reached is called a control section, because it controls the upstream depth. If the slope decreases from steep to mild, something very different takes place.
As y increases toward the critical depth, a flow instability occurs at some point, and the flow becomes turbulent until the new normal depth is attained downstream in tranquil flow. This is called a hydraulic jump, which will be analyzed below. This equation tells us whether the water surface is rising or falling in the direction of motion. The dimensionless parameter F is the Froude Number. It is analogous to the Mach Number in compressible fluid flow. It was originally defined by Froude as the speed of a ship divided by the square root of its water level length not dimensionless.
The relative wave resistances of hulls of different sizes and speeds are the the same if the Froude numbers are equal, permitting resistance tests on model ships. When water is flowing at critical depth, the surface is typically disturbed. However, the equation cannot account for the details of flow in this region, and, of course, the infinte slopes are not observed.
Since the flow is subcritical, this means that the depth decreases while the velocity increases. The flow is no longer uniform, but is still steady, and the discharge is constant at any cross-section. It is easy to get an expression for the change in specific energy by equating the vertical distances at 1 and 2: This relation is usually rearranged to give the distance L, when the depths at each end of the reach are assumed: Suppose we have uniform subcritical flow on a mild slope, and let us modify the downstream end of the channel.
One modification would be to create a dam that would raise the depth in front of it. The discharge must get by the dam somehow, either by flowing over the top or through a gate, for example. A backwater deeper than the normal depth would form that would slowly approach the normal depth as we pass upstream. Or, we could allow a free discharge from the end of the channel that would then fall as a free jet. In this case, we would have a drawdown water surface with a depth less than the normal depth, approaching normal depth as we go upstream.
These surface profiles are denoted M1 and M2, respectively, by Bakhmeteff. If the water discharges through a gate at the bottom of a dam with water behind it at greater than critical depth, its velocity will be greater than critical. The depth will then rise as the water decelerates because of the large resistance.
Before it reaches critical depth, a hydraulic jump will occur, making the transition to subcritical flow. The resulting profile is denoted M3.
The water profile can be calculated by the formula just derived, starting from some point where the depth is known and finding the distances to points where the depth takes a series of increments approaching the normal depth.
On a steep slope, the normal depth is less than the critical depth, so the water profiles are different from those on a mild slope.
If the depth is greater than the normal depth but less than the critical depthit will approach the normal depth as it accelerates downstream, while if the depth is less than normal depth, the water will decelerate with increasing depth, approaching normal depth asymptotically. These are the profiles S2 and S3, respectively. A supercritical flow approaching a dam will undergo a hydraulic jump and become subcritical, then rise on the steep slope with a backwater curve designated S1.
Downstream influences do not affect upstream flows when the velocity is supercritical. In particular, there will be no drawdown curve approaching a free exit. These six flow regimes cover most applications, but special profiles can be identified for critical slopes normal and critical depths equaland for horizontal and adverse slopes normal depth infinite. The profiles for horizontal and adverse slopes are similar to those for a mild slope with an infinite depth, but of course normal uniform flow is not possible.
Diagrams of these profiles are shown in Daugherty and Franzini p. When the channel slope changes from mild to steep, the initially uniform subcritical flow must change to the finally uniform supercritical flow, while the flow velocity accelerates from its initial value to its final value. This transition is smooth and efficient, with little additional loss of head, like converging flow in a pipe.
In the region of acceleration on the mild slope, the velocity is higher than its normal value, so the rate of head loss is greater. The energy line now approaches the profile line, so that the specific energy which is the vertical distance between them decreases. From the specific energy diagram, we see that the flow depth decreases as a result.
At the break in bed slope, the flow velocity is less than the normal value, so the resistance is less, and the energy line and profile line now diverge, and the specific energy increases. The point of the break, then, is the point of minimum specific energy, and the corresponding depth of flow is the critical depth for the given discharge. Because of the rapid change of depth with specific energy vertical tangent to the specific energy curve there may be surface disturbances at this point.
As the specific energy increases further on, the depth of flow decreases until it finally approaches the normal depth for the profile gradient. Similar flows occur at the outflow of reservoirs and spillways, but the location of critical depth may be somewhat further upstream. This transition is always characterized by a decrease in depth of flow in the direction of flow. At a free outfall from a mild slope, the specific energy decreases up to the end of the channel.
Nevertheless, critical depth is reached before this point, typically a distance 4yc from the lip. The depth at the lip is about 0. The reason for this is in the change in the flow pattern near the end. In the free jet, the velocity may well be uniform across the jet, very different from the velocity distribution in the channel. It is usually accurate enough to assume that critical depth is reached near a crest of the flow.
Water Profiles Let's use the relation we have just obtained to find the water levels in a practical problem. The depth of steady flow yo can be found from Manning's formula. Since this is less than the actual depth, the flow is tranquil or upper-stage, and the slope is mild.
It is easy to repeat the calculation for any Q. Now suppose we have placed a dam with a spillway height of 8 m above datum at the lower end of the channel. We assume that the width is the same, 6 m, for simplicity, and that the height of the water over the spillway sill will be the critical depth at that point.
Now we can estimate conditions at the end of the channel at the dam. This is not quite correct, but the error will not be large, so a more detailed estimate is not worth the effort. For this station, the numbers are Instead of trying to find the conditions at some distance L upstream, it is much easier to assume a new depth, and then find the distance L that corresponds.
The columns of the table can be filled in with little trouble. The specific energy has decreased from Fluvial Design Guide - Chapter 7 Hydraulic analysis and design Three significant concepts or principles form the building blocks for all types of hydraulic analysis: Many textbooks provide detail on the theory and how the fundamental principles are developed to account for unsteady flow. Potential energy refers to the energy a fluid has due to its elevation: W and z are respectively the weight of water and the distance the water is located above the reference point or datum.
Kinetic energy refers to the energy possessed by fluid flow due to its velocity where: V and g are respectively the velocity of the water and the acceleration due to gravity. There is also a pressure head term, but this is usually neglected. The units for H are metres m. The energy grade line should always drop in the direction of flow except if energy is added to the system by a mechanical device.
Mass and continuity The principles of conservation of mass are used to develop the equation of continuity. For steady flow in open channels, this is expressed in terms of discharge or flow rate.
A and V are respectively the cross-sectional area and average velocity at locations 1 and 2. Momentum has both magnitude and direction.