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Photograph and CrossSection of Trapezoidal Channel:
Gradually Varied Flow Profiles:
Units in gradually varied flow calculation: cm=centimeter, cfs=cubic feet per second, ft=feet, gal=US gallon, gpm=US gallons per minute,
gph=US gallons per hour, gpd=US gallons per day, km=kilometer, m=meter, MGD=Millions of US
gallons per day, min=minute, s=second
Links on this page: Equations Variables Manning n coefficients
Error messages References
Introduction
In long prismatic (constant crosssectional geometry) channels, flowing water will attempt
to reach the "normal depth" (also known as the "uniform flow
depth"). Normal depth is the water depth determined using Manning's equation (please see our other web page for design of trapezoidal channels using Manning's equation). A
gradually varied flow (GVF) profile is a plot of water depth versus distance along the
channel as the water depth gradually achieves normal depth. A GVF computation in a
trapezoidal channel involves starting at a known depth Y_{s} and making
successive water depth computations at small distance intervals. The method involves
the continuity equation and energy slope equations. The LMNO Engineering
calculation initially computes normal depth, critical depth, and GVF profile type.
Then, it computes the water depth profile and plots it. The calculation also
displays flow properties (depth, velocity, Froude number, etc.) at a specific location X_{p}
entered by the user. A GVF profile is also known as a water depth profile, backwater
calculation, and nonuniform flow computation. It is for steady state flows
(discharge remains constant).
The LMNO Engineering calculation plots GVF profiles for M1, M2, S2,
S3, C1, and C3 curves. M3 and S1 curves cross over the critical depth in order
to achieve normal depth. Flows crossing the critical depth are called "rapidly
varied flows" and cannot be computed using GVF methods.
Equations and Methodology
Fundamental flow equations are first presented, followed by equations for computing the
critical depth Y_{c} and normal depth Y_{n}. Then,
using the input value of Y_{s}, the GVF profile type is determined and
the GVF profile is computed using the Improved Euler method. References for the
equations are shown alongside the equations. Manning's equation for Y_{n}
and the equation for the friction slope S_{f} are empirical; they are
shown in the form that uses meters and seconds for units. Units for all other
equations can be from any consistent set of units.
Fundamental equations
The following equations are always valid for trapezoidal channels (Chanson, 1999; Chow,
1959; Simon and Korom, 1997):
Critical depth computation
To compute critical depth Y_{c} the Froude number F is set to
1.0. Then, we use the Newton method (Kahaner, Moler, and Nash, 1989; Rao, 1985)
along with the fundamental equations above to solve for Y_{c}.
Normal depth computation
To compute normal depth Y_{n} a cubic solution technique (Rao, 1985) is
used to solve the fundamental equations above in conjunction with the Manning Equation
(Chanson, 1999; Chaudhry, 1993; Chow, 1959; Simon and Korom, 1997):
Gradually varied flow profile determination (Chanson,
1999; Chaudhry, 1993; Chow, 1959; Simon and Korom, 1997):
If Y_{n}>Y_{c}, then the channel is considered to have a
mild (M) slope. If Y_{n}<Y_{c}, the slope is steep
(S). If Y_{n}=Y_{c}, then the slope is termed critical (C).
The slopes are further classified by a number (1, 2, or 3) as follows:
For mild slopes (Y_{n}>Y_{c}):
If Y_{s}>Y_{n}, then the slope is an M1. The GVF
calculation starts downstream at X_{max} at a depth of Y_{s}
and proceeds upstream to X=0. The water depth gets closer to Y_{n}
as the calculation proceeds further and further upstream.
If Y_{n}>Y_{s} >Y_{c}, then the slope is an
M2. The GVF calculation starts downstream at X_{max} at a depth of Y_{s}
and proceeds upstream to X=0. The water depth gets closer to Y_{n}
as the calculation proceeds further and further upstream.
If Y_{c}>Y_{s}, then the slope is an M3. This is an
unstable GVF calculation since the water depth begins below both Y_{n}
and Y_{c}. Since the slope is mild, an hydraulic jump will
occur. Hydraulic jumps are rapidly varied flow situations that cannot be modeled by
a GVF calculator. Therefore, the message, "Cannot plot S1 or M3", will be
shown.
For steep slopes (Y_{c}>Y_{n}):
If Y_{s}>Y_{c}, then the slope is an S1. This is an
unstable GVF calculation since the water depth begins above both Y_{c}
and Y_{n}. Since the slope is steep, the water depth will have to
pass through the critical depth in order to reach the normal depth. Passing through
the critical depth is a rapidly varied flow situation that cannot be modeled by a GVF
calculator. Therefore, the message, "Cannot plot S1 or M3", will be shown.
If Y_{c}>Y_{s}>Y_{n}, then the slope is an
S2. The GVF calculation starts upstream at X=0 at a depth of Y_{s}
and proceeds downstream to X_{max}. The water depth gets closer to Y_{n}
as the calculation proceeds further and further downstream.
If Y_{n}>Y_{s}, then the slope is an S3. The GVF
calculation starts upstream at X=0 at a depth of Y_{s} and
proceeds downstream to X_{max}. The water depth gets closer to Y_{n}
as the calculation proceeds further and further downstream.
For critical slopes (Y_{c}=Y_{n}):
If Y_{s}>Y_{c}, then the slope is a C1. The GVF
calculation starts downstream at X_{max} at a depth of Y_{s}
and proceeds upstream to X=0. The water depth gets closer to Y_{n}
as the calculation proceeds further and further upstream.
If Y_{c}>Y_{s}, then the slope is a C3. The GVF
calculation starts upstream at X=0 at a depth of Y_{s} and
proceeds downstream to X_{max}. The water depth gets closer to Y_{n}
as the calculation proceeds further and further downstream.
There is no such thing as a C2 slope  since Y_{c}=Y_{n}, Y_{s}
cannot be between Y_{c} and Y_{n}.
Gradually varied flow profile (graph)
computation
To compute the gradually varied flow profile (graph), the Improved Euler
method (Chaudhry, 1993) is used:
At control section, i=1 and Y_{i}=Y_{s}.
Repeat for i=2 to n in increments of distance dX where dX
is negative for downstream control and dX is positive for upstream control.
Compute T_{i}, A_{i}, and P_{i} using the fundamental equations shown above using Y=Y_{i}.
Compute the friction slope, depth increment, and intermediate depth (note: for the
friction slope equation shown, the friction slope variables must be in meters and
seconds):
Compute T_{2}, A_{2}, and P_{2}
using the fundamental equations shown above with Y=Y_{2}.
Then, compute the friction slope based on T_{2}, A_{2},
and P_{2} followed by computation of a second depth increment.
Finally, compute the water depth, Y_{i+1} by using the average of the two
differential depth increments (this is the basis of the Improved Euler method).
Then repeat the loop by incrementing i.
The LMNO Engineering calculation uses an unequal node spacing so that more nodes are
used at the beginning of the calculation to improve accuracy. The first node spacing
is approximately 10^{10} m, and there are 4500 distance increments. The
results have been checked against hand calculations, spreadsheets, and results shown in
Chaudhry (1993), Chow (1959), French (1985), Henderson (1966), and Simon and Korom (1997).
Variables
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to calculation
Variables are shown below in SI units (metric). If you work through the above
equations by hand, use the SI units shown  since many of the equations are empirical and
are valid only with the indicated units. (The calculation
performs internal unit conversions which allow you to select a variety of different
units.)
A = Channel crosssectional area [m^{2}].
A_{i} = Area computed at successive i intervals in Improved Euler method [m^{2}].
A_{p} = Area at X_{p} [m^{2}].
A_{2} = Area for intermediate computation in Improved Euler method [m^{2}].
dX = Distance increment for Improved Euler method [m]. Negative for M1, M2,
and C1 since computation proceeds upstream. Positive for S2, S3, and C3 since
computation proceeds downstream
(dY/dX)_{1} = First depth increment for Improved Euler method [m].
(dY/dX)_{2} = Second depth increment for Improved Euler method [m].
B = Channel bottom width [m].
E = Elevation [m]. The calculation automatically sets the channel invert
elevation to 0.0 at X_{max}.
E_{pi} = Elevation of channel invert at X_{p} [m].
Invert means bottom of the channel.
E_{py} = Elevation of water surface at X_{p} [m].
F = Froude number [dimensionless].
F_{p} = Froude number at X_{p} [dimensionless].
g = Acceleration due to gravity, 9.8066 m/s^{2}.
i = Loop index for computing GVF profile.
n = Manning's n value [dimensionless]. See table below for
values.
P = Channel wetted perimeter [m].
P_{i} = Wetted perimeter computed at successive i intervals in Improved
Euler method [m].
P_{2} = Second wetted perimeter computed in Improved Euler method [m].
Q = Discharge (flowrate) of water in the channel [m^{3}/s].
S_{o} = Slope of bottom of channel (vertical to horizontal ratio) [m/m].
S_{f1} = First energy slope for Improved Euler method [dimensionless].
S_{f2} = Second energy slope for Improved Euler method [dimensionless].
T = Top width of water in channel [m].
T_{i} = Top width computed at successive i intervals in Improved Euler
method [m].
T_{2} = Second top width computed in Improved Euler method [m].
T_{p} = Top width at X_{p} [m].
V = Average velocity of water [m/s].
V_{p} = Velocity at X_{p} [m/s].
X = Distance along channel [m].
X_{max} = Maximum distance for computing GVF profile [m]. Profile is
always plotted from X=0 to X_{max}. For M1, M2, and C1
profiles, Y_{s} is at X=X_{max}. For S2, S3, and
C3 profiles, Y_{s} is at X=0.
X_{p} = Distance entered by user for showing channel properties [m].
Cannot exceed X_{max}. If user enters X_{p}>X_{max},
the calculation will automatically set X_{p} to X_{max}.
Y = Water depth [m].
Y_{c} = Critical depth [m].
Y_{i} = Water depth computed at successive i intervals in Improved Euler
method [m].
Y_{n} = Normal depth [m].
Y_{p} = Depth at X_{p} [m].
Y_{s} = Starting depth [m]. This is also known as the depth at the
control section. It is the depth that GVF calculations start at.
Y_{2} = Second depth computed in Improved Euler method [m].
Z_{1} = One channel side slope (horizontal to vertical ratio) [m/m].
Z_{2} = The other channel side slope (horizontal to vertical ratio) [m/m].
Manning n Coefficients
Back to calculation
The Manning's n coefficients were compiled from Chaudhry (1993), Chow (1959),
French (1985), and Mays (1999).
Material 
Manning n 
Material 
Manning n 
Natural Streams 
Excavated Earth Channels 
Clean and Straight 
0.030 
Clean 
0.022 
Major Rivers 
0.035 
Gravelly 
0.025 
Sluggish with Deep Pools 
0.040 
Weedy 
0.030 


Stony, Cobbles 
0.035 

Metals 
Floodplains 
Brass 
0.011 
Pasture, Farmland 
0.035 
Cast Iron 
0.013 
Light Brush 
0.050 
Smooth Steel 
0.012 
Heavy Brush 
0.075 
Corrugated Metal 
0.022 
Trees 
0.15 

NonMetals 
Glass 
0.010 
Finished Concrete 
0.012 
Clay Tile 
0.014 
Unfinished Concrete 
0.014 
Brickwork 
0.015 
Gravel 
0.029 
Asphalt 
0.016 
Earth 
0.025 
Masonry 
0.025 
Planed Wood 
0.012 


Unplaned Wood 
0.013 
Error Messages
Back to calculation
Initial input checks. The following messages are generated
from improper input values:
"Need 1e20<Q<1e50 m^{3}/s", "Need 1e20<B<1e6
m", "Need Z_{1}, Z_{2} ≥0", "Z_{1}, Z_{2}
cannot both be 0", "Need 1e9<n<20", "Need 1e20<S_{o}<1e99",
"Need 0.001<X_{max}<1e6 m", "Need 1e20<Y_{s}<100
m", "Need X_{p}≥0".
Runtime messages. The following messages may be
generated during execution:
"Infeasible input". Inputs are unusually large or small causing
the program to have trouble computing Y_{n} or Y_{c}.
"Cannot plot S1 or M3". As discussed
above, these two GVF profiles encounter rapidly varied flow where the water depth
crosses through critical depth.
"No graph. Y_{s}=Y_{n}". This is a uniform flow
situation, not a GVF calculation. Water depth will remain at normal depth, so the
GVF profile is not computed. One would expect this message to occur if Y_{n} is copied into the Y_{s} field. However, the message may not appear since Y_{n} is shown to only 8 significant figures but is stored internally to 18 significant figures.
"Y_{n} at x=874.231 m". This is the distance where the
water depth is within 0.01% of the normal depth.
References
Back to calculation
Chanson, H. 1999. The Hydraulics of Open Channel Flow. John Wiley and
Sons, Inc.
Chaudhry, M. H. 1993. OpenChannel Flow. PrenticeHall, Inc.
Chow, V. T. 1959. OpenChannel Hydraulics. McGrawHill, Inc. (the
classic text)
French, R. H. 1985. OpenChannel Hydraulics. McGrawHill Book Co.
Henderson, F. M. 1966. Open Channel Flow. MacMillan Publishing Co.
Kahaner, D, C. Moler, and S. Nash. 1989. Numerical Methods and Software.
PrenticeHall, Inc. 2ed.
Mays, L. W. editor. 1999. Hydraulic design handbook. McGrawHill Book
Co.
Rao, S. 1985. Optimization: Theory and Applications. Wiley Eastern
Limited. 2ed.
Simon, A. and S. Korom. 1997. Hydraulics. PrenticeHall, Inc.
4ed.
© 20012022 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or questions about gradually varied flow.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com https://www.LMNOeng.com

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