Enhancements and Modifications to the Full Equations (FEQ) Model, March 1995 to August 1999
Note: This document is separate from the U.S. Geological Survey report by Franz and Melching (1997). This description of enhancements and modifications to the Full Equations Utilities Model has not been approved by the Director of the U.S. Geological Survey.

Update for section 8.1.2.1.2.7, Conservation of Momentum/Energy: Code 13, Franz and Melching (1997), p. 80

Section 8.1.2.1.2.7 Conservation of Momentum/Energy: Code 13

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Conservation of momentum/energy: Code 13.- Code 13 represents a junction between what we call a main channel and up to two tributary channels. The flow in the main channel does not change direction, that is, the flow in the main channel is approximately in a constant direction as it moves from upstream to downstream through the junction. If we assume that the flows from the tributary channels, called side channels, enter at right angles to the flow, these flows would not contribute any momentum flux to the main channel. Thus the momentum balance, ignoring friction, bottom slope, and assuming that the cross section on the main channel upstream and downstream of the entry of the side channels was the same, was simple. Conservation of momentum under these idealized situations was merely equality of the sum of momentum flux and hydrostatic pressure force upstream and downstream of the junction. These assumptions were used in all previous versions.

However, the flow picture is more complex than this idealized case. A cross slope exists in the side channel for some distance upstream of its junction with the main channel. This cross slope forms as the water begins to change direction some distance upstream of the point of entry of the water into the main channel. This cross slope grows in magnitude, reaching its maximum value as the side channel flow enters the main channel. The slope of the water surface across the side channel is such that there is a net force in the downstream direction of the flow in the main channel. That is, the water surface on the left bank is higher than on the right bank when we face downstream. Code 13 did not include this net downstream force. Consequently the estimate of losses at junctions using Code 13 were too large.

There are two ways to improve the representation of the losses at a junction using Code 13. In the first, we move the face of the control volume on the side channels far enough upstream so that the cross slope at that point in these channels is negligible. Then we estimate the net force on the side channels in the direction of the flow in the main channel and add these forces to the momentum balance. The major problems with this approach are that we have no experimental data on which to base the location of the face of the control volume in the side channels nor do we have any good means of estimating the cross slope in the side channels. Thus we must seek another approach.

In the second approach, we avoid these force terms by moving the face of the control volume in the side channels to the edge of the main channel such that the downstream force arising from the cross slope of the water surface in the side channel no longer applies to the control volume. However, we have traded one difficulty for another, because now the water in the side channel enters the junction at an angle that differs from the angle of the side channel. Furthermore the angle of the flow as well as the velocity of the flow entering the main channel from the side channel varies across this face of the control volume. For side channels entering the main channel at an angle of 90 degrees or less, the average entry angle of the water is less than the angle of entry of the walls of the side channel. Therefore, the flow from the side channel does contribute a momentum flux in the downstream direction of the main channel even if the walls of the side channel are at right angles to the walls of the main channel. Fortunately in this case, some experiments are available to help us estimate the angle of entry of the water given the angle between the side channel and the main channel.

Experiments with combining flows at junctions have been performed in the laboratory at various times over the last 50 years. However, the variety of junctions is such that no systematic study has yet been done on anything approaching natural channel junctions. Rectangular channels with junction edges not rounded have been studied most extensively. In most cases the main channel and the side channel have the same width. Best and Reid (1984) report some results on the measurement of the separation zone at an equal-width rectangular junction with subcritical flows with most Froude numbers being less than 0.5. Hager (1987), in a discussion of their article, found that a simple model for predicting the separation zone gave improved results if he allowed for a difference between the direction of entry of the water and the direction of entry of the side channel. Best and Reid (1987), in their response to the discussion, estimated the angle of entry of the water at the center of the side channel as the flow entered the main channel. They found that the relationship between the two angles varied with the ratio of the side-channel flow to the main-channel flow.

Best and Reid (1987) presented results for three values of side-channel flow relative to the flow downstream of the junction: 0.35, 0.51, and 0.60. By fitting their estimated flow angle results with a least-squares linear relationship forced to pass through the origin, I found that the reduction factors for the side-channel entry angle were: 0.60, 0.71, and 0.73 respectively. This is a small sample of values but these values apply approximately to side-channel angles varying from 15 to 90 degrees. The main conclusion is that the reduction factor changes more slowly as the relative side-channel flow increases. These results were developed from rectangular channels of constant width with sharp edges at the junction.

In natural channels and in non-rectangular channels it seems reasonable to assume that the angle of entry of the water would be relatively smaller than in sharp-cornered junctions in rectangular channels. The size of the separation zone would be smaller because the walls of the side channel near the junction would be in closer agreement with the direction of flow in the side channel. This smoothing of the transition between the two channels would make the angle of entry of the flows smaller than the abrupt transition formed by sharp-edged boundaries in rectangular channels. This means that using the factors from the laboratory experiments will still make the angle of entry too large and therefore the estimated downstream component of momentum flux from the side channel will still be too small. However, the error committed will be smaller than if the flux is neglected completely. Further refinement of the representation must await measurements at junctions with different forms than sharp-edged rectangular.

Another limitation of the Best and Reid results is the constant width channel used. It is more typical that a side channel is smaller than the main channel. The effect of main-channel width on the entry angle is not known from measurements but it seems reasonable to assume that the reduction coefficient would approach 1.0 as the width of the main channel increases and the flows and the width of the side channel are held constant. However, the rate of approach to this limit is unknown. Thus we are left with some tantalizing possibilities combined with remaining imponderables.

The following seems reasonable:

1. Using the angle of the side channel as the angle of entry for the flow and using the mean velocity to compute the momentum flux will result in overestimating the loss of mechanical energy at the junction. Both factors lead to an underestimate of the momentum flux entering the main channel from the side channel. The true angle of flow varies across the control volume face but at each point the flow angle will be less than the side channel entry angle. Also the momentum flux computed from an average velocity for a cross section is always less than the true momentum flux. If the flux contribution from the side channel is too small then the mechanical energy losses will be too large.

2. The values derived from the Best and Reid results appear to yield a factor that is too large in general if the two channels are of similar size. No measurements for the effect of dissimilarity in size are available. However, Best and Reid (1984, p. 1589) state that the momentum flux ratio is a measure of the interaction between two merging flows that applies to channels of varying widths. The values of this ratio for the above relative flow values were approximately: 0.24, 1.1, and 2.3 respectively. A least squares line through the three points of momentum ratio and reduction factor for the entry angle fits them with a maximum relative error of about 5 per cent. The slope also has the correct sign. Let  reduction factor, and  the momentum flux in the side channel relative to the momentum flux in the main channel upstream of the junction. The fitted relationship is then . If the flow ratios are held constant but the width of the main channel is doubled, the momentum ratio will also double and the values of the reduction factor will increase by ratios of 1.02, 1.1, and 1.19 respectively. Thus to the degree that the momentum ratio can correctly reflect the effect of channel width, doubling the main channel width increases the reduction factor by at most 19 per cent. Extrapolating this crude relationship to the limit when  gives a main channel width of about 6.5 relative to the side channel. In truth the channel width at which  will be larger than this value derived by extrapolation of a linear fit to a non-linear relationship.

The changes to FEQ for Code 13 still assumes that the main-channel cross section upstream and downstream of the side channel is the same. However, up to two exterior nodes can be specified to represent the momentum flux entering from side channels. The angles of entry of the flow, NOT the angle of entry of the side channel, must also be given. Because the data for the relationship between the angle of entry of the channel and the angle of entry of the flow is limited, a fixed value for angle of flow entry is used. The above discussion can guide the choice for this angle. The exterior nodes representing the side channel flows must be on a branch because we need a flow area as well as a flow rate to compute a momentum flux. This excludes dummy branch and level-pool reservoir nodes because free nodes have no area.

With these changes the momentum balance for Code 13 becomes:

where average entry angle for flow, and  number of side channels. Note that if  degrees, then the momentum relationship in earlier versions is obtained. If no angle of entry is given, FEQ assumes that the water from the side channel enters at right angles to the main channel flow.

The energy balance relationship is

where mechanical energy loss coefficient with . No checking for possible critical depth is done and therefore the loss coefficient should not be much different than zero. In most cases the loss in mechanical energy in the channel engendered by flow leaving the channel is small.

Flow over a side weir is often approximated by assuming that the specific energy is unaffected by the outflow of water because the outflow takes place smoothly. Therefore, this option, in conjunction with others, models flow over a side weir. Flows over a side weir tend to remove the higher velocity portions of the flow in the channel. Thus energy is removed from the channel via this route. Using a small value of  can approximate this effect but there is no data on the value of  to use. Field measurements or physical model studies would be needed to define the value of  to use.