Category Archives: Non Uniform & Critical Open Channel Flow

Backwater Curve Calculations Spreadsheet

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Where to Find a Backwater Curve Calculations Spreadsheet

To obtain a backwater curve calculations spreadsheet to calculate surface profiles for non uniform open channel flow, click here to visit our spreadsheet store.  Obtain a convenient, easy to use backwater curve calculations spreadsheet at a reasonable price.  Read on for information about the use of an Excel spreadsheet for non uniform flow open channel surface profile step wise calculations.

Background on Non Uniform and Uniform Open Channel Flow

uniform nonuniform flow diagram - backwater curve calculations spreadsheetsThe diagram at the right illustrates uniform and nonuniform open channel flow.  Uniform flow in an open channel consists of a constant volumetric flow of liquid through a reach of channel with a constant bottom slope, surface roughness, and hydraulic radius (that is constant channel size and shape).  For those constant channel conditions, the water will flow at a constant depth, called the normal depth, for the  particular channel conditions and volumetric flow rate. The diagram shows a reach of uniform open channel flow, followed by a change in bottom slope that causes non-uniform flow, ending with another reach of uniform open channel flow.  This article is about means of calculating the surface profile (depth vs distance down the channel) for a reach of non uniform flow.


Classifications of Non Uniform Open Channel Flow for a Backwater Curve Calculations Spreadsheet

backwater curve calculations spreadsheet - uniform surface profile typesClassifications of Non Uniform Open Channel Flow (Mild or Steep Channel Slope)

The diagram above shows the three possible non uniform flow patterns for a mild slope (channel slope less than the critical slope) and the three for a steep slope (channel slope greater than the critical slope).  The three mild slope classifications are M1, M2, and M3.  The “M” indicates mild slope and the number shows the relationship among depth of flow, y, critical depth, yc, and normal depth, yo , as shown in the diagram.  Similarly the three steep slope classifications are S1, S2, and S3, with the numbers having the same meaning.  The diagram shows a typical physical situation that will give rise to each of these six types of non uniform open channel flow.

The Energy Equation for a Backwater Curve Calculations Spreadsheet

The energy equation (the first law of thermodynamics applied to a flowing fluid), which has many applications in fluid mechanics, can be used for non uniform open channel flow surface profile stepwise calculations.  The diagram below shows the parameters that will be used at each end of a reach of channel with non uniform flow.

Backwater curve calculations spreadsheet - non uniform flow parametersA Reach of Open Channel with Non Uniform Flow

The energy equation written across a reach of channel is illustrated graphically in the diagram above.  The sum of the three items on the upstream end of the channel reach must equal the sum of the three items on the downstream end of the channel reach, giving the equation:

Where the parameters in the equation are as follows:

  • y1 =  the upstream depth of flow in ft (m for S.I. units)
  • y2 =  the downstream depth of flow in ft (m for S.I. units)
  • V1 =  the upstream average velocity in ft/sec (m/s for S.I. units)
  • V2 =  the downstream average velocity in ft/sec (m/s for S.I. units)
  • g  =  the acceleration due to gravity  =  32.17 ft/sec2 (9.81 m/s2 for S.I. units)
  • ΔL  =  the horizontal length of the channel reach in ft (m for S.I. units)
  • So =  the bottom slope of the channel, which is dimensionless
  • Sf =  the slope of the energy grade line (thus head loss is hL = SfΔL)

For specified flow rate, Q, channel bottom slope, So , Manning roughness coefficient, n, and channel width for a rectangular channel, the energy equation can be used to calculate the length, ΔL, for transition from a known upstream depth, y1 , to a selected downstream depth, y2 .  This process can be repeated as many times as necessary to determine the total distance to a specified downstream depth.

The energy equation can be rearranged to give the following equation for ΔL:

The Manning equation is typically used to calculate the slope of the energy grade line, Sf .  Although the Manning equation only applies for uniform flow, the use of mean cross-sectional area and mean hydraulic radius with a relatively small step for the calculation gives a good approximation.  The equation for Sf is as follows:

Sf =  {Qn/[1.49Am(Rhm2/3)]]}2, where  Am is the mean area and Rhmis the mean hydraulic radius between sections 1 and 2.  For S.I. units, the 1.49 constant in this equation becomes 1.00.

Screenshot of a Backwater Curve Calculations Spreadsheet

Consider a 20 ft wide rectangular channel with bottom slope equal to 0.0003, carrying 1006 cfs.  The normal depth for this flow is 10 ft.   An M1 backwater curve is generated due to a downstream obstruction.  Calculate the channel length for the transition from a depth of 12 ft to a depth of 12.5 ft in this backwater curve.

Solution: The spreadsheet shown in the screenshot below shows the solution.  It actually has the entire M1 curve from a depth of 10 ft to a depth of 16 ft.  It shows DL for the transition from 12 ft depth to 12.5 ft depth to be 3853 ft.

The Excel spreadsheet template shown above can be used to calculate an M1 surface profile for a rectangular channel with specified flow rate, bottom width, bottom slope, and Manning roughness coefficient.  Why bother to make these calculations by hand?  This backwater curve calculations spreadsheet and others with similar calculations for a trapezoidal channel, and for any of the six mild or steep nonuniform flow surface profiles are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

References

1. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

2. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959.

3. Bengtson, Harlan H., Open Channel Flow II – Hydraulic Jumps and Supercritical and Nonuniform Flow – An online, continuing education course for PDH credit.

4.  Bengtson, Harlan H., “Non Uniform Flow in Open Channels“, an online blog article

Hydraulic Jump Calculator Excel Spreadsheets

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Where to Find Hydraulic Jump Calculator Excel Spreadsheets

For an Excel spreadsheets to use as an open channel flow, hydraulic jump calculator, click here to visit our spreadsheet store.  Obtain a convenient, easy to use rectangular channel hydraulic jump calculator spreadsheet for only $14.95. Read on for information about the use of an Excel spreadsheet as a horizontal, rectangular channel hydraulic jump calculator.

Background for Hydraulic Jump Calculator

In order to discuss hydraulic jumps it’s necessary to talk about subcritical and supercritical flow.  In general subcritical flow takes place at low velocities and high flow depths, while supercritical flow occurs at high velocities and low flow depths.  For more details about critical, subcritical, and supercritical flow, see the article, “Open Channel Flow Spreadsheets – Critical Depth and Critical Slope.”  The diagram above shows supercritical flow on a steep slope, changing to subcritical flow on a mild slope.  As shown, the transition from supercritical flow to subcritical flow takes place with a hydraulic jump.  Whenever supercritical flow takes place on a slope that isn’t steep enough to maintain supercritical flow, the transition to subcritical flow will take place through the mechanism of a hydraulic jump as illustrated in the diagram.

Hydraulic Jump Calculator Parameters

Hydraulic jump calculations center on relationships among the supercritical conditions before the jump (upstream or initial conditions) and the subcritical conditions after the jump (downstream or sequent conditions).  The diagram at the left shows initial supercritical parameters and sequent subcritical parameters for a hydraulic jump.  The parameters and their typical units are summarized below:

  • y1 = the initial (upstream) depth of flow in ft for U.S. or m for S.I. units
  • V1 = the initial (upstream) liquid velocity in ft/sec for U.S. or m/s for S.I. units
  • E1 = the initial (upstream) head in ft for U.S. or m for S.I. units
  • y2 = the sequent (downstream) depth of flow in ft for U.S. or m for S.I. units
  • V2 = the sequent (downstream) liquid velocity in ft/sec for U.S. or m/s for S.I. units
  • E2 = the sequent (downstream) head in ft for U.S. or m for S.I. units
  • Q = the flow rate through the hydraulic jump in cfs for U.S. or m3/s for S.I. units
  • ΔE = the head loss across the hydraulic jump in ft for U.S. or m for S.I. units

An Excel Spreadsheet as a Hydraulic Jump Calculator

The Excel spreadsheet template shown below can be used to carry out hydraulic jump calculations.   Why bother to make these calculations by hand?  This Excel spreadsheet can calculate the sequent depth, sequent velocity, jump length, head loss across the jump, and hydraulic jump efficiency for specified initial depth, flow rate and channel width.  These spreadsheets are available in either U.S. or S.I. units at a very low cost (only $14.95 in our spreadsheet store.  These spreadsheets also have a tab for calculation of flow rate under a sluice gate and all of the equations used in the spreadsheet calculations are shown on the spreadsheets.

Note that some of the equations used in the spreadsheet calculations apply only for rectangular, horizontal channels, so the spreadsheets should be used only for channels that are at least approximately rectangular in cross-section and have a zero or very small slope.

References

1. Harlan H. Bengtson, “Hydraulic Jumps and Supercritical and Nonuniform Open Channel Flow,”  an online continuing education course for Professional Engineers.

2.  U.S. Department of Transportation, FHWA, Hydraulic Design of Energy Dissipators for Culverts and Channels, Hydraulic Engineering Circular No. 14, 3rd Ed, Chapter 6: Hydraulic Jump.

Critical Depth Open Channel Flow Spreadsheet

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Where to Find a Critical Depth Open Channel Flow Spreadsheet

To obtain a critical depth open channel flow spreadsheet for calculating critical depth and/or critical slope for open channel flow, click here to visit our spreadsheet store.  Read on for information about the use of a critical depth open channel flow spreadsheet for critical depth and critical slope calculations.

The Froude Number and Critical, Subcritical and Supercritical Flow

Any particular example of open channel flow will be critical, subcritical, or supercritical flow.  In general, supercritical flow is characterized by high liquid velocity and shallow flow, while subcritical flow is characterized by low liquid velocity and relatively deep flow.  Critical flow is the dividing line flow condition between subcritical and supercritical flow.

The Froude number is a dimensionless number for open channel flow that provides information on whether a given flow is subcritical, supercritical or critical flow.  The Froude number is defined to be:  Fr = V/(gL)1/2 , where V is the average velocity, g is the acceleration due to gravity, and L is a characteristic length for the particular type of open channel flow.  For flow in a rectangular channel:  Fr = V/(gy)1/2 ,   where y is the depth of flow.  For flow in an open channel with a shape other than rectangular:  Fr = V/[g(A/B)]1/2 , where A is the cross-sectional area of flow, and B is the surface width.

The value of the Froude number for a particular open channel flow situation gives the following information:

  • For Fr < 1, the flow is subcritical
  • For Fr = 1, the flow is critical
  • For Fr > 1, the flow is supercritical

Calculation of Critical Depth

It is sometimes necessary to know the critical depth for a particular open channel flow situation.  This type of calculation can be done using the fact that Fr = 1 for critical flow.  It is quite straightforward for flow in a rectangular channel and a bit more difficult, but still manageable for flow in a non-rectangular channel.

For flow in a rectangular channel (using subscript c for critical flow conditions), Fr = 1 becomes:   Vc/(gyc)1/2 = 1.  Substituting Vc =  Q/Ac =  Q/byc and  q = Q/b  (where b = the width of the rectangular channel), and solving for yc gives the following equation for critical depth: yc =  (q2/g)1/3.   Thus, the critical depth can be calculated for a specified flow rate and rectangular channel width.

For flow in a trapezoidal channel, Fr = 1 becomes:  Vc/[g(A/B)c]1/2 = 1.  Substituting the equation above for Vc together with Ac =  yc(b + zyc)    and   Bc =  b  +  zyc2 leads to the following equation, which can be solved by an iterative process to find the critical depth:

Critical Depth Open Channel Flow Spreadsheet Formula1

Calculation of Critical Slope

After the critical depth, yc ,  has been determined, the critical slope, Sc , can be calculate using the Manning equation if the Manning roughness coefficient, n, is known.  The Manning equation can be rearranged as follows for this calculation:

Critical Depth Open Channel Flow Spreadsheet Formula2Note that Rhc , the critical hydraulic radius, is given by:

Rhc =  Ac/Pc,  where Pc =  b  +  2yc(1 + z2)1/2

Note that calculation of the critical slope is the same for a rectangular channel or a trapezoidal channel, after the critical depth has been determined.  The Manning equation is a dimensional equation, in which the following units must be used:  Q is in cfs, Ac is in ft2, Rhc is in ft, and Sc and n are dimensionless.

Calculations in S.I. Units

The equations for calculation of critical depth are the same for either U.S. or S.I. units.  All of the equations are dimensionally consistent, so it is just necessary to be sure that an internally consistent set of units is used.  For calculation of the critical slope, the S.I. version of the Manning equation must be used, giving:

Critical Depth Open Channel Flow Spreadsheet Formula4In this equation, the following units must be used:  Q is in m3/s, Ac is in m2, Rhc is in m, and Sc and n are dimensionless.

A Critical Depth Open Channel Flow Spreadsheet Screenshot

The critical depth open channel flow spreadsheet template shown below can be used to calculate the critical depth and critical slope for a rectangular channel with specified flow rate, bottom width, and Manning roughness coefficient.  Why bother to make these calculations by hand?  This Excel spreadsheet and others with similar calculations for a trapezoidal channel are available in either U.S. or S.I. units at a very low cost in our spreadsheet store.

Critical Depth Open Channel Flow Spreadsheet Screenshot

References

1. Munson, B. R., Young, D. F., & Okiishi, T. H., Fundamentals of Fluid Mechanics, 4th Ed., New York: John Wiley and Sons, Inc, 2002.

2. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959.

3. Bengtson, Harlan H. Open Channel Flow II – Hydraulic Jumps and Supercritical and Nonuniform Flow – An online, continuing education course for PDH credit.