Bernoulli loss
• 07 Feb 2023
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# Bernoulli loss

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Article Summary

The Bernoulli Loss uses Bernoulli's equation to model head losses, such as those caused by changes in cross section across open channel constrictions or expansions.

Discrete energy losses which cannot be accounted for in the friction term of the Saint-Venant equations can be modelled using the Bernoulli equation. Applications include energy losses generated by a sudden expansion or contraction in channel geometry, such as entry and exit losses associated with culverts, bridges or severe bends in a channel. ## Theory and Guidance

### Equations

The Bernoulli Equation:

where
= Area at upstream/downstream node of Bernoulli loss
= Energy loss coefficient for forward/backward flow
= Flow at both upstream and downstream nodes
= Water surface elevation at upstream node
= Water surface elevation at downstream node
= Acceleration due to gravity

### General

Using the Bernoulli equation, changes to channel geometry can also be modelled without energy loss simply by having K factors as zero and appropriate values for the upstream and downstream areas.

To model a bridge, the upstream and downstream areas are usually set to be identical, being the area of the bridge opening. These areas are used with the flow to define a reference velocity to which the K factors are related. The processed results of a model run at the Bernoulli Loss nodes take the velocities from the units upstream and downstream of the Bernoulli Loss and not from the areas defined in the Bernoulli Loss itself.

Standard values of loss coefficients taken from the literature are not necessarily directly applicable to the Bernoulli Loss. To derive K values for use in Flood Modeller it is recommended that you base your values on one or more of the following:

1. head loss coefficients from the literature for similar equations (for example Chow, VT (1959) or Ritzema (ed) (1994)) which may need adjusting to account for the differences in the equations
2. experience of using Flood Modeller and other software on previous structures
3. calibration data

You must ensure that the highest water level specified for the Bernoulli Loss is greater than the maximum water level expected to be encountered. This cannot be checked by Flood Modeller when reading in data. The same applies to the minimum water level.

The principle of superposition may be applicable in some cases, so that two losses could be grouped together in one Bernoulli Loss.

Bernoulli Loss units can be used in conjunction with the Junction where the assumptions of equal water levels on the Junction branches is not a sufficiently accurate approximation.

If Data Interpolation is set to SPLINE then a natural cubic spline is fitted to the area and head loss coefficient tables. Otherwise the program interpolates linearly between each point. A spline should only be specified if the data sets are reasonably smooth. If there is noise in the data it may be amplified by the spline. Similarly, data with large changes in gradient can lead to oscillations.

It should be noted that the equations used by the Bernoulli Loss will be less accurate at high flow velocities. This is for two reasons:

• the turbulent losses that are not modelled by this unit become more significant at higher velocities
• the Bernoulli Loss will use the upstream head value to look up areas and loss coefficients from the user-defined table. However, downstream area values should really be obtained using the downstream head value.

At low flow velocities this will not be much of a problem, as the head loss through the Bernoulli Loss will be small. The discrepancy will become larger at higher flow velocities.

## Data

Field in data entry formDescriptionName in datafile
Upstream labelUpstream node labelLabel1
Downstream labelDownstream node labelLabel2
ElevationElevation of water surface above datum (m AD)hi
Upstream areaUpstream area corresponding to hi (m2)A1,i
Downstream areaDownstream area corresponding to hi (m2)A2,i
K - forward flowHead loss coefficient for forward flow corresponding to hi (typically in the range 0.1 to 2.0)K12,i
K - reverse flowHead loss coefficient for backward flow corresponding to hi (typically in the range 0.1 to 2.0)K21,i
Data interpolationSPLINE if a spline is to be fitted to the data, or LINEAR to use linear interpolation. If the field is blank then linear interpolation is used.smooth

## Datafile format

### General

Line 1 : keyword 'BERNOULLI' [comment]

Line 2 : Label1, Label2

Line 3 : n1 [smooth]

Line 4 to Line 3+n1 : hi, A1,i, A2,i, K12,i, K21,i

where n1 is the number of data sets following, items in square brackets are optional, and other parameters are as defined in the data section.

### Example

BERNOULLI
UNIT061     UNIT062
3    LINEAR
0.000     0.000     0.000     1.000     0.950
1.000     2.000     1.000     0.900     0.850
2.000     4.000     3.000     0.800     0.750
Chow V.T. (1959)
Open channel hydraulics, McGraw-Hill

Ritzema (ed) (1994)
Drainage Principles and Applications, ILRI Publication 16