The Venturi flume

The broad-crested weir is a device that combines the functions of measuring a flow and dissipating the potential energy where a reduction in level of a channel is required. There are many applications where the flow must be measured with a minimum of loss of energy and the Venturi flume has evolved for this purpose. The broad-crested weir depends for its success as a measuring device on a contraction in the flow produced by raising the bed of a channel and so producing critical flow conditions on the crest. This permits the evolution of a calibration expression. A transition to critical flow can just as easily be produced in a horizontal channel by a contraction in width or by a combination of a contraction in width and a small hump in the bed. This is the basis of design of the Venturi flume.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 10-26 represents the plan and side elevation of a convergent-divergent fitting in a channel. It is likely that it could be a Venturi flume.  The best shape for a contraction would be one that produced no oblique waves on the surface whilst producing the desired change from tranquil flow to critical flow. There seems to be no shape that does this so, in the interests of simplicity I have let the contraction have straight sides and given it a contraction ratio of 2.5 to 1. I have shown a short parallel throat and then a relatively long divergence back to the original width. The flow throughout is the same but, as the channel narrows, the flow per unit width increases and in this case by 2.5 times. Inevitably the level drops to produce a profile like that shown in the side elevation of the flume in figure 10-16 but the surface will be criss-crossed with oblique waves.

 

I have re-drafted graph 10-3 to become graph 10-7 and supposed that the flume to be carrying a flow of  at the full width of the channel. This means that, at the throat, the flow per unit width will be . Now the loss in a contraction is always small so this change in flow per unit width takes place at sensibly constant specific energy. Once the throat is passed the depth goes on falling and the flow becomes rapid and, provided that the downstream flow pattern  is suitable, reaches a new depth that is the alternate depth to the upstream depth. Then there will be a hydraulic jump in the flow just down-stream of the flume.

 

This can be represented on graph 10-7 as the line 1 to 2 which represents, on the graph, the flow through the contraction to the throat where the depth must be critical and line 2 to 3 which takes place in the divergence. Then, in the hydraulic jump, the level rises to 4 and a depth lower than .

 

The new feature is that whilst the flow is unchanged in the contraction the flow per unit width changes. In addition, because the flow is in a contraction the loss will be very small and the specific energy the same throughout. It is evident that, providing that rapid flow can take place in the divergence, the flow will go though critical conditions somewhere in the throat. It makes sense to apply our specific energy equation to section 1 in the channel just before the contraction and to the section in the throat 2 where somewhere conditions are critical. We get:-

                                               .

Given critical conditions at section 2 we know that . Then we can put:-

                                              .

If the flow is  we have  where suffix  indicates throat and  is the flow in the throat. Now we can substitute and : -                 .

From this we get :-        and     .  If we simplify again we get   and multiplying through by  the final form is :-

                                              .

On examination it becomes clear that, as the ratio of the widths of the convergence is fixed for a given flume, the ratio of the depth of the approach flow to the critical depth in the throat is also fixed. It means that we can find the flow from a knowledge of the upstream depth if the water is free to flow with rapid flow in some or all of the divergence.

 

We already have                                       .        

So                                                            

And then                                               

This is the final form except for evaluating the constant when  in SI units.

 

This is an interesting outcome because it says that the flow is dependent on the upstream level and the width of the throat. The width of the channel is not relevant.

 

This is the physics of a Venturi flume and an engineer would be expected to carry on from here to design a Venturi flume for some application. No doubt the channel to which the flume is to be fitted exists or is planned and the expected range of flow known. If the flume is to be a convergent-divergent fitting the most obvious dimension that is required is the width of the throat. There must be some link between the ratio of the width of the throat and the width of the channel to the ratio of the depth upstream of the flume and the depth at the throat where the flow is critical. We can explore that link.

 

We have for a Venturi flume . This can be rearranged to give :-

 and if  this becomes .

Then

.

A graph can be drawn of r versus k and this is shown in graph 10-8.

 

Be careful with this plot because I have suppressed the zero and started at a ratio of 1 because lower ratios are invalid. Note the way that high values of the width ratio lead to quite impractical flumes and that the upper limit of the depth ratio is 1.5 as we might expect. In the range of practical ratios for the depth are really from about 0.5 to about 2.5.

 

Keep in mind that detritus of all sorts get into open channels and the wider the throat the less likelihood there is of a rubbish jam. 

 

These figures may give a ratio of depths but they do not give us any idea of the relationship between flow and upstream depth. For this we need another equation.

 

We can go back to the continuity equation and if critical conditions exist somewhere in the throat, and write         .

We know that  and this can be substituted to give, .

Once more we have this awkward critical depth term but we can accept a minor approximation and put  and then   and .  This can also be plotted to give graph 10-9. Once more I have suppressed the zero because the depths before the throat for widths less than 1 metre are not really practical. This is not a graph that can be plotted in a general way and I have used a flow per unit width of  and let the channel width vary from 2 to 5 metres. The graphs have no meaning for an upstream depth below 1.11 metres when the width of the throat equals that of the channel and the black line shows the boundary.

 

The implication of this graph is again that big flows take place in big channels.

 

In my view these graphs and the physics above are sufficient to design a simple convergent-divergent Venturi flume and the use of a mathematics package makes exploration of possible designs very easy. It is well to remember that the simplification of ignoring the kinetic energy of the approach flow when compared with the sum of the pressure and potential energies, i.e. the depth, and this introduces a small error of 1% or 2% in the calibration for practical designs and losses in the convergent were also ignored and these my give another error of about the same magnitude. We can expect a coefficient of discharge of between 0.95 and 0.97.

 

There is the question of the net loss of energy caused by the fitting of the Venturi flume. The major loss occurs in the hydraulic jump and that must be present if the flume is to be of any practical value. Then the loss depends on the size of the jump and that depends on the down stream depth. If the downstream depth is low the jump will occur as I showed it in figure 10-6 and the loss will be the maximum. As the depth increases the jump will move into the divergence becoming smaller in height as it does so. Eventually it reaches the throat and, whilst this gives the smallest loss, it also starts to interfere with our requirement that critical conditions shall exist in the throat. So it is best if the jump is as close to the throat as possible consistent with it not interfering with calibration.

 

It is possible to buy Venturi-flumes made in fibreglass up to about the physical proportions of a motor car. Beyond this size each design is unique and will probably be made in concrete. In use the upstream depth can be measured using a float gauge that might well be in a side chamber. It can be monitored with a recorder. It may well be that the Venturi meter will be too dependent on the down stream conditions and then a hump is incorporated on the flume with its maximum height at the throat. Then the calibration expression involves the difference in level between the upstream surface and the top of the hump.