(This note is in response to Ron Kilmartin's recent post entitled "High Velocity Flows in Extran - 2")
Your May 20 posting was interesting and stimulated thought on the subject. However, there are a couple of assumptions that you made in the posting that need to be clarified or corrected. First, EXTRAN always reports hydraulic grade line information. Energy grade line information is only reported coincidentally when the velocity head is zero. Whether or not a junction is in surcharge does not affect this reporting convention.
Next, form losses in EXTRAN are always controlled by the user. The options for controlling form losses are equivalent pipes and discrete local losses (new in Version 4.4). So, EXTRAN does not assume anything in terms of energy conversion due to form losses. Rather, this is left to the control of the user. What we recommend is the following:
1. Use the discrete local loss option (NEQUAL = 4 or 5 in SWMM 4.4). This option was developed by XP Software and incorporates local losses into an additional term in the momentum equation.
2. Select an entrance loss coefficient based upon entrance geometry (ENTK on the C1 line).
3. Select an exit loss coefficient based upon in-pipe and exit velocities (EXITK).
4. If applicable, select an appropriate loss coefficient for bends, etc. (OTHERK).
5. Incorporate an additional loss coefficient into OTHERK based upon approach, in-pipe and exit velocities to ensure the conservation of energy between conduits.
Whether or not the "energy is distributed along the entire conduit" (as can be thought of using the equivalent pipe method) or lost at a point is unimportant. Both yield the same results; they are simply equivalent ways of expressing the same thing. Regardless of where the energy is lost conceptually, it is still unavailable for the rest of the system. An analogy is solving a set of simultaneous linear equations. Regardless of which variable is solved first, the answer will be the same.
The discussions of "total" conversion of KE to PE and hydraulic jumps within junctions were a bit confusing. A total conversion of KE to PE implies a discharge into a conduit/storage area with a velocity head of zero. However, this is not the case involved in your discussion. The discussion of hydraulic jumps seemed to imply that jumps should be allowed to occur within junctions. However, junctions have no length, and hydraulic jumps do. We would be interested in further clarification from you on these subjects.
Brett and Mike and others interested in this problem:
Thanks much for your critique. I agree with much of what you wrote. I am trying to resolve in my own mind how to interpret the output in high-velocity situations. What I am finding out is that there are a number of different situations that can arise in which high velocities raise questions.
Fundamentally there are differences between the conservation of momentum and the conservation of energy under high velocity conditions. EXTRAN was formulated on the basis of conservation of momentum and continuity. However, the two approaches should be more or less consistent with each other. There is apparently no guarantee that the principle of conservation of energy will not be violated when velocity head is a major component of the total energy in a pipe. Bob Dickenson once mentioned on the net that there is an upper limit on velocity in the program of 50 fps. Still that is a velocity head of 39 feet - really roaring!
For steep slopes with high open channel velocities it may be found that the energy required for the velocity determined with the EXTRAN average velocity is a lot more than is locally available in the system. When this occurs EXTRAN is apparently reporting a flow and depth that is not physically possible from the energy point of view; here there is a real problem of interpretation of program results.
Where it appears in profile that the system can provide the KE required in the conduits, the computed free surface Qs and depths are probably ok from the energy point of view. This also applies to surcharged conduits; however, the drop in pressure within surcharged conduits due to subtracting Hv from the EGL could become a problem if the resulting HGL plots below the conduit crown, which would indicate local negative pressure. If the negative pressure becomes great enough it could suck in air from the many openings in the system, and the real world conduit would convert to open channel while EXTRAN considers it surcharged and flowing over the full section. The open channel condition in the real world would probably be very unstable due to making and breaking of the semi-vacuum condition.
I Agree that EXTRAN reports the HGL at the node ; this is fairly clear from the manual for the open channel case. I think you are also agreeing that in a surcharged node that since there is no velocity, hence, by default, the HGL in the node = the EGL. In the free-surface node however, there is something of a conflict between definitions. The water surface is the HGL, but Hv is significant in the upstream conduit exit and the downstream conduit entrance, hence there is an implied total head at the node equal to the HGL+ Hv for the downstream conduit +/- form loss effects. This could be partly piezometric and partly KE in the real conduit junction.
For the free surface condition, the velocity head in the conduit + the depth must equal the head in the upstream node. While not a physical assumption of the equations, it would seem that for a simple node it will indeed have an implied velocity head or PE equivalent, since at the conduit entrance the specific energy must be equal to the flow depth + the Hv head.
But consider a large storage node such as a pond with a channel coming in and one going out, and EXTRAN computes a high velocity on both incoming and outgoing channels; free surface everywhere and in general the channels could be supercritical or just above critical depth with a relatively high velocity.
If the pond is big enough, the KE of incoming conduit is likely spent in dissipation. Cross flow across pond is at low velocity with negligible velocity head. Whether to model the pond in cross-section with its own node s is a separate question - depends how big. The outflow conduit still requires full specific energy appropriate to the high velocity, at the pipe entrance. EXTRAN computes a high velocity because of the steep slope to the next node. What appears to be missing here is that in real world the velocity does not jump from zero to a maximum value upon entrance to the conduit; there is a transition to the reach of maximum velocity.
In the EXTRAN momentum equation, if the conduit is too long there may be an entirely different hydraulic environment at the downstream end as compared to the upstream end. The conduit hydraulics as modeled by the momentum equation are based on averages of V, R, and H at the two nodes. Maybe this averaging over a significant drop in Z is a problem with respect to interpreting results. Maybe modeling long steep conduits in shorter segments would result in a better EGL definition. I plan a little experimenting along this line and will report the results.
Also, part of the problem is that EXTRAN output only reports averages and not the up and downstream depths and velocities at both ends of the conduits. It uses these latter values in computing the averages used in the momentum equation, but only reports the averages. Thus, my initial inclination to plot the average velocity head as a uniform drop from the surcharged nodes' "hydraulic head" as an EGL line is an oversimplification of the situation. While EXTRAN assumes no variation of Q within the conduit over a timestep, velocity and depth and hence velocity head do vary from one end to the other.
The average EXTRAN computed velocity head is not really right to use at either node; it is larger at one and smaller at the other (I am speaking of the conduit velocity entering/leaving the node). For low velocities it makes little difference. For high velocities the squaring effect in getting the head value can be very signficant. Using the average velocity head at the node with low real velocity can result in an indicated need for specific energy well above the invert, and often well above the system's total energy level in the vicinity; this leads the user to an inaccurate estimate of the required nodal local total energy level.
I agree 100% with all your recommendations on recognition of form losses, although I have not yet examined the new 4.4 features in this regard.
There was recently some traffic on the net discussing the possibility of variable losses as a function of depth or discharge. That might be something to consider where concern is for both low flows and high flows. As it is now one would have to have two data sets of n - one for low flows and one for high flows. But that is another subject.
Getting back to High velocity, Providing a priori for losses equivalent to a hydraulic jump is a problem. You are quite right that the jump will normally be in the conduit, forced by an adverse pressure gradient (in the real conduit). Test runs will show profiles where hydraulic jumps are likely. Then one needs to assign losses to that conduit corresponding to expected jump losses, like you described. This would be a situation incidentally where variable n as a function of depth would be doubly valuable, since jump characteristics vary importantly with Froude number.
One program enhancement that would be of great assistance in evaluating supercritical velocity cases would be time histories of some of the data that EXTRAN uses but does not presently print out: up and downstream end conduit depths and velocities, critical depths, Froude Numbers and velocity heads; only the velocity heads would be something not presently calculated. These could be handled like flows and heads are handled on the B6 and B7 cards. If Wayne and Chuck are listening, I would like to throw that on the wish list pile (which seems to have gotten rather deep in the past few weeks! ah, priorities...; admittedly enhancements for high-velocity interpretation are probably not on everybody's wish list).
I feel like I am rambling here. Hope your concerns are at least partially answered somewhere above. I do not think we have yet reached a bonified set of recommended steps for evaluating high velocities in 4.4 , but I hope we can gradually reach that point. I feel there is room for more input here Brett, Mike, and if anyone else would like to chime in.
Apologies again for length.
Ron Kilmartin
Consulting Engineer
This posting is in response to Ron Kilmartin's comments discussing high velocities in EXTRAN (Re: Conservation of Energy in EXTRAN, June 6, 1998). We believe that this is an important topic and offer our dis- cussion to clarify any misunderstanding. We invite additional comments from Ron and others on this issue.
Conservation of energy and conservation of momentum are two different ways of solving for flow. In theory, the two are not all that closely related. Energy is a scalar quantity; momentum is a vector quantity. The energy equation measures losses due to INTERNAL energy dissipation; the momentum equation measures losses due to EXTERNAL forces (friction, gravity, pressure, and added force) exerted on the flow. One could only expect these two equations to yield identical results under uniform flow conditions. In practice, however, we do not generally encounter situations where there are noticeable disparities in total energy in the system. Granted, this could be due in part to the methodologies that we employ.
In your last posting, you stated that "for steep slopes with high open channel velocities it may be found that the energy required for the velocity determined with the EXTRAN average velocity is a lot more than is locally available in the system." We have tested a variety of steep systems, and we are unable to duplicate this finding. Would it be possible for you to post a simple example of where this occurs? Also, we caution you on the use of average velocity in making energy comp- utations. Each conduit in EXTRAN uses three velocities (upstream, downstream, and middle/average), and it is imperative under non-uniform flow conditions to use the appropriate velocity when applying the energy equation.
Your discussion of local negative pressures in a closed conduit is also intriguing. Would it be possible to post a simple data set where this occurs?
There still seems to be an apparent misunderstanding as to what EXTRAN is reporting at a node. EXTRAN ALWAYS reports the HGL at a node. It DOES NOT matter if the node is surcharged or under free surface cond- itions. Also, surcharging at a node does not imply that there is no velocity, as your last posting seemed to imply. It is important that this point is understood. It applies to many items raised in the middle of your last posting.
We do not see the problem that you do with the "big pond" example. The incoming conduit has a high incoming velocity and loses its kinetic energy upon entrance to the pond. This loss is (must be) handled with an exit loss coefficient of 1.0 (one of the few cases where an exit loss of 1.0 is appropriate). You stated that there appears to be some- thing missing at the other end of the pond where "in the real world the velocity does not jump from zero to a maximum value upon entrance to the conduit." You are correct that this is not instantaneous in the real world, but there is nothing missing in the model unless you have not accounted for an entrance loss. Most entrances to restrictive culverts constitute rapidly varied flow. Since this is extremely diff- icult to deal with on a theoretical basis, it is almost universally dealt with by employing an empirical entrance loss coefficient coupled with the velocity head in the culvert. EXTRAN employs this method (with proper input, of course). It is advisable not to utilize the energy equation at this particular location, because the underlying assumption of hydrostatic pressure distribution is violated. In terms of there being "an entirely different hydraulic environment at the downstream end as compared to the upstream end," that should not be a problem. Conduit hydraulics are NOT, as you stated, "based on aver- ages." Conduit hydraulics are based on upstream, downstream, and middle/average characteristics. They are only REPORTED for the average, which you recognized in your next paragraph. We reaffirm your caveats about using the average velocity at the upstream and downstream end of conduits. This would only be appropriate for uniform flow situations.
In closing, your wish for additional time histories has largely been fulfilled already in the latest release (version 4.4) of SWMM. There is an option to output a binary file that contains upstream, down- stream, and average conduit characteristics. This file is used by the new product MIKE SWMM, and can also be used by other means. Also, you may be interested in the new options for treatment of S2 and M2 curves in the latest release.
Thanks much for your reply. I cannot address all issues raised at the moment but I will as soon as I can . Here are some thoughts on some of the issues you raised. In general, I think you have helped me to formulate a better line of reasoning on the high velocity issue, at least in my own mind.
1. First let me stress that the basic purpose I intended for this thread was to search out how to review 4.4 program results for reasonableness when it is indicating very high velocities, e.g., 25 to 50 feet per second.
2. I think your concise comparative discussion of Momentum and Energy methods was excellent. I would only add that friction would also be considered in an energy approach.
3. Equation 5-4, as given in the manual ( p 140), is the basis for my conclusion that EXTRAN calculates Q in the next time step based on averaged values of R, V, and A, from the previous time step, as stated in the explanation of that equation, and not based on the end values themselves.
4. Since the only measure of friction loss known to EXTRAN is Manning's n, I think for the purpose of this discussion, we can probably ignore form losses as clouding the main issue of concern. With the understanding that form losses are a matter of user due diligence to take care of in specifying Manning's n, we can focus on just how to interpret basic EXTRAN output, for whatever Manning's n is specified. This also applies to areas with non-linear pressure distributions - EXTRAN is assuming hydrostatic everywhere.
5. None of the equations in Section 5 of the manual indicate any dependence on velocity in the nodes (other than vertical velocity which would have negligible velocity head). I guess my fundamental question is, what should the user assume for KE or velocity head in the nodes, when reviewing the program output for reasonableness? I think you have clarified one matter for me, and that is that in general, EXTRAN is giving us the HGL in the nodes, and not the EGL, except by possible coincidence.
6. Based on the EXTRAN intermediate output of Q and nodal depth, one can back calculate the depth, area, and hydraulic radius at each end for open channel cases. Or if the conduit is surcharged, then the conduit geometry becomes the area, hydraulic radius, etc. The constant Q assumption within the time step enables the computation of the end velocities and the corresponding velocity heads.
7. Now if EXTRAN is projecting a free surface at a node (defined as the junction + half the length of the connecting conduits), it seems clear that the velocity head in each adjacent conduit plots directly above its water surface and this carries over to the junction itself, as illustrated in Figure 5-2 of the manual.
8. If the node is surcharged, entirely different arithmetic is brought to bear. The horizontal (+/-) area of the connecting pipes is no longer in the equations; instead, the free-surface equations are replaced by Eq. 5-16, 17, and 18. There is thus a discontinuity in the mathematical methodology between the open channel and the surcharged case.
9. It is apparent that these equations are still solving for H and that H is the depth at the node. It is also apparent that no account is taken of velocity within the nodal junction other than vertical velocity, again negligible.
10. It is true that this depth in the surcharged junction corresponds to the HGL. The question is, does it also correspond to the EGL for the typical junction, or does a velocity head of kinetic energy plot above the HGL to yield an EGL higher than the HGL?
11. More directly, if the velocity head is say 15 feet in the upstream end of the outflow conduit, is there a "velocity head" in the node of 15 feet that is additive to the water depth H to get the EGL? Or is the HGL in fact equal to the EGL within the confined junction?
12. If this H reaches ground level, it floods. The flood waters obviously have no velocity head that would be additive to the HGL of the node. Therefore, it is concluded that for the surcharged flooding condition, the HGL necessarily defines the EGL, and, the HGLs in the adjacent conduits are equal to that EGL, less the conduit's local velocity head.
13. It seems clear that this reasoning can be extended to cover the entire surcharge zone between ground level and top of conduit, since there is no difference in program logic other than spilling at the ground surface.
14. Physical reasoning would lead to the conclusion that in the general case, the surcharged zone within the typical junction above the conduits might be fairly turbulent, but whether a discernible velocity exists in the manhole, out of one conduit and into the next, will depend greatly on the local manhole geometry. This of course is where form loss considerations need to be brought to bear. However, based on the above reasoning, it seems clear that EXTRAN is not implying a velocity head in surcharged nodes, and therefore that its calculated value of H, by default, is also the EGL within the node.
15. This reasoning leads then to the conclusion that between surcharged nodes, the EGL is defined as the straight line between the HGL=EGL in the upper node to the HGL=EGL in the lower node. To get the local HGL in the conduit, one must deduct the velocity head at the upper node from the EGL=HGL in the upper node and similarly at the downstream node. The hydraulic gradient in the conduit is then the line connecting these two points.
16. If this hydraulic gradient plots below the top of conduit, negative pressures are induced. For low velocities this is generally not a problem. However, for high velocities and corresponding velocity heads, negative pressures can become significant. For example, going back to item 11 above, with a velocity head of 15 feet, if there is a surcharge depth in the munction on the top of the conduit of 5 feet, then there is a negative pressure of 10 feet just inside the conduit.
17. This reasoning thus indicates that in the open channel junction case the velocity head is added to the EXTRAN HGL at the node to get the EGL, whereas in the surcharged case, to get the HGL in the conduit, the velocity head is subtracted from the HGL=EGL at the node.
Well, all of this reasoning is based on my personal interpretation of the manual and to some extent, on some numeric experiments. So this epistle is not exactly founded on rock. I, would really like to get more discussion on this matter from you and from other members of the group.
An interesting related paper recently came to my attention on the hydraulic jump problem. This is in the IAHR Journal No. 5, 1997, By H. Capart, et al, "Numerical and Experimental Water Transients in Sewer Pipes". Apparently this logic could be added to EXTRAN. for example as a callable routine for pre-specified conduits that the user feels will contain a jump. This would be a bit of work, of course.
I am glad to read there is a 4.4 option to output a binary file with some of the desired values. How is this option invoked? Can you tell me what routine writes it and whether there is available a utility program to read these data?
Ron has started a very interesting discussion on solving high velocity conditions. I believe that this is a key issue of solving unsteady flow problem with stability while representing reality. Ron has tackled two very tough problems for numerical models: surcharge condition and high velicity or supercritical flow (leads to hydraulic jump). Ron, thanks for mentioning my paper in one of your early posting.
I agree with Brett and Mike in their posting that conservation of energy and conservation of momentum are two different ways of solving flow. Energy equation is applicable under a special condition of steady flow. In fact, energy equation can be derived, by dropping the unsteady term of the momentum equation and integrates all terms over a distance. In other words, momentum equation is a general representation of flow movement and should be used in all situations in computation of unsteady flow problems.
In my opion, confusion comes when applying the concept in energy equation to the solving process of momentum equation. When solving momentum equation coupled with continuity equation, we are basically dealing with flow (Q or V) and water surface elevation (h or H). As long as the combination of Q and H satisfies the continuity and momentum conservation equation at a particular time, we called it the solution of the problem for that time. If we want to analyse the results using the concept of kinetic and potential energy, we can use the results and calculate the energy. But it won't change the solution for the flow problem.
I have struggled for sometime about the high velocity and hydraulic jump with specific energy approach. My question was: is there a need for special treatment of supercritical flow and hydraulic jump when dealing with the computation of one-dimension flow problem? With analysis of literatures of hydraulic jumps, sub/super critical flow, specific energy, etc. I got a conclusion that there is no need to have a special treatment of supercritical flow and hydraulic jump in solving one-dimensional flow problems. The law of conservation of continuity and conservation of momentum is general enough to cover the phenomena.
To make it simple, picture a channel section of high velocity flow and analyse the momentum or force balance. To simplify the discussion, assume the flow is in steady state. If the slope is steep, the gravitational force component in the flow direction will be high. If the difference between the pressure applied to the upstream and downstream of the section and the bottom and side friction cannot balance the gravitational force component, the flow would either accelerate (that would increase the friction by increasing the velocity) or reduce the depth (which would reduce the gravitational force) or both. The equilibrium state would be a normal supercritical flow and should be described by the momentum equation. If there is a downstream restriction of flow, the pressure difference becomes large enough to balance the gravitational force. The equilibrium state will be a hydraulic jump in the section. In either of the above conditions, momentum euqation is applicable to solve the problem.
When surchaged, H becomes the representation of pressure which determines the upstream and downstream pressure forces. The overall problem of solving the flow and pressure (or the head) is still the same.
A few comments to Ron Kilmartin's posting on June 16th:
1. EXTRAN uses upstream, downstream, and average/mid-pipe character- istics to solve for flow. Please refer to subroutines ROUTE and HEAD for more details.
2. EXTRAN does not assume hydrostatic pressure distribution everywhere, since it substitutes imposed solutions in areas where a hydrostatic pressure distribution cannot be assumed (for example, weirs and orifices).
3. In a link-node model, nodes do not have properties of flow. Nodes are used to solve for continuity and HGL.
4. Be careful of applying the reported head as the conduit depth. There are several special cases where this is inappropriate. These cases are covered in the EXTRAN manual. An example of one case is a drop manhole where there is the (higher) critical depth in the incoming pipe and the lower elevation in the manhole/outgoing pipe.
5. Free surfaces are not defined in EXTRAN as the junction plus half the length of the connecting conduits. This referenced definition has to do with how surface areas are assigned to nodes. See page 145 (and Figure 5-4) of the EXTRAN manual for the five cases of how surface area is distributed.
6. The EGL can (and always will if there is a velocity head) plot above the ground elevation for flooded nodes. The energy in the water flowing overland should not be equated to the energy of the water in the pipes. The HGL and EGL are never equal unless there is zero velocity.
Brett and Mike - thanks again for your critique and I think you are right with every one of your points except maybe on point 6; my intent was that if EXTRAN finds flooding it calculates a flow rate out the node in the intermediate output but not a depth and not a surface velocity- it cant since it has no geometry to base it on...unless you physically provide a channel at the surface on the Cx cards, in which case it is not flooding until the channel is overtopped and that brings you back to square one: when EXTRAN floods at a node as given in the intermediate output the EGL=HGL at the ground surface, as far as EXTRAN is concerned.
I am glad you mentioned figure 5-4. It appears to me that there is a case missing here, and that is with critical depth upstream ...actually just inside the conduit... and flow in the downstream direction. This can occur when the upstream node is above critical depth as a free surface condition, and also continued as a surcharged condition in a range up to a velocity head equal to D/2, with critical depth just inside the conduit. This may change from critical depth to surcharged if drowned out from downstream conditions. This is a fairly common culvert design condition but as far as I can see it does not fit into any of the five cases. Can you throw any light on this?
Linear Mode
