EFFECT OF CROSSSECTIONAL SHAPEON FREESURFACE CHANNEL INSTABILITY
Victor M. Ponce and Andrea C. Scott
San Diego State University, San Diego,
California
ABSTRACT
A study of the effect of crosssectional shape on freesurface channel
hydrodynamic
instability is accomplished. At the outset, the rating exponent β, Froude number
F, and Vedernikov number V are identified
as the controlling variables.
A steep, lined channel is specified for the analysis.
The selected design discharge is Q = 100 m^{3}/s, with bottom slope
S = 0.06 and Manning's n = 0.025, closely resembling the flow conditions of the
Huayñajahuira river, in La Paz, Bolivia, where roll waves have been shown
to recur with disturbing regularity. The testing program considers
the variation of the bottom width b in the range 5 ≥ b ≥ 1, at 1m
intervals,
and the side slope z in the range 0.25 ≥ z ≥ 0, at 0.05 intervals.
The online calculator onlinechannel15b
is used to calculate the relevant hydraulic variables, culminating in the
values of rating exponent β, Froude number F, and
Vedernikov number V for each of thirty (30) cases.
The results show conclusively that as the channel width b is reduced from 5 to 1 m,
and the side slope z reduced from 0.25 to 0,
that β, F,
and V are reduced, first gradually, and then sharply as z →
0, with the asympotic value z = 0 corresponding to a rectangular channel.
For a given design application,
these findings may be used to determine optimal geometric crosssectional values b and
z
in order to assure that V < 1 and, therefore,
avoid flow hydrodynamic instability and the associated roll waves.

1. INTRODUCTION
Freesurface instability in openchannel flow
is generally manifested by the development of roll waves.
These are unsteady flow features associated with steep, lined channels,
when the Vedernikov number V is greater than or equal to 1 (V ≥ 1).
(Ponce, 2014: Section 11.4).
However, it can be shown that the actual development of a roll wave
depends primarily on the shape of the cross section, whether
it is trapezoidal, rectangular, or triangular.
For a given channel cross section, there is a
unique relationship between the exponent β of the rating
(the discharge Q vs flow area A equilibrium rating),
and
the ratio V/F, in which F = Froude number
(Ponce and Choque Guzmán, 2019).
In some circumstances,
roll wave phenomena may be of such magnitude
as to actually place at risk
life and property.
This fact is confirmed by the
roll waves that occur with worrisome regularity in
the Huayñajahuira river, in La Paz, Bolivia, as shown in
Fig. 1 and the accompanying video.
[Click on top of photo to watch video] 
Fig. 1 Roll wave event on a channelized reach
of the Huayñajahuira river, La Paz, Bolivia, on December 11, 2021.

We posit that the design of a lined channel for the control of
roll waves may be
accomplished by a judicious choice of crosssectional shape.
The way to accomplish this is to choose, at the design stage,
a channel shape
that effectively reduces the Vedernikov number below the thresholed value of
1 (V < 1).
Therefore, the design focus centers on the value of
β, the exponent of the rating, a parameter
defined in terms of
V/F.
Herein we use the online calculator
onlinechannel15b,
which determines values of
F, V, and β for a prismatic channel
(Ponce and Boulomytis, 2021). We run the calculator
for a series of crosssectional shapes,
including trapezoidal and rectangular, keeping constant
the following variables: (1)
discharge Q, (2) Manning's n,
and (3) bottom slope S.
The effect of the crosssectional shape is tested by running
the calculator for several suitable values of side slope z (z H: 1 V),
with the flow depth y set to correspond with the
selected discharge Q.
The aim is
to examine the
behavior and sensitivity of the flow variables to the Froude F and
Vedernikov V numbers, and to the concomitant value of β. In practice, it may be shown that
channel stability is attained for values of
β close to but clearly greater than 1.
Therefore, the optimal cross section, from the
standpoint of channel stability, corresponds to the
lowest value of β, greater than 1,
that is
compatible with project cost, optimal footprint dimensions,
and other relevant considerations.
2. BACKGROUND
The theory of hydrodynamic stability of openchannel channel flow
is due to Vedernikov (1945). Several years later, Craya clarified the Vedernikov criterion
by stating it in terms of the wave celerities (Craya, 1952).
The VedernikovCraya criterion states
that occurrence of
roll waves will form when the Seddon celerity
equals or exceeds the
Lagrange celerity, that is, when the kinematic wave celerity, governed by gravitational and
frictional forces, equals or exceeds the dynamic wave celerity, governed by inertial and pressuregradient forces.
In this case, the Vedernikov number exceeds 1: V ≥ 1.
Otherwise, V < 1, i.e., dynamic waves travel faster than kinematic waves and,
consequently, the flow is stable.
In 1907, Cornish showed, apparently for the first time, a photograph of the fascinating phenomenon
in a paper published in the Journal of the Royal Geographical Society (Fig. 2)
(Cornish, 1907). In 1948, Powell christened the concept, by stating, to wit: "This criterion,
which I am calling the Vedernikov number..." (Powell, 1948).
Later, Ven Te Chow referred to the phenomenon as the "Instability of Uniform Flow,"
implying that under certain conditions, the flow could become unstable
and break into a train of roll waves (Chow, 1959: Section 8.8).
Fig. 2 Roll waves observed in a canal
in the Swiss Alps at the turn of the 20th century
(Cornish, 1907).

The roles of mass and energy are seen to be central to the development
of roll waves.
While kinematic waves transport mass,
dynamic waves transport energy (Lighthill and Whitham, 1955).
Therefore, the occurrence of roll waves is seen to be related
to the unsteady transport of mass overcoming the unsteady
transport of energy. In this light, roll waves are a curious
physical manifestation of the preponderance of mass transport
over energy transport in unsteady openchannel flow in steep channels
(Ponce and Choque Guzmán, 2019).
3. RELATIONSHIP BETWEEN β AND V/F
There are three characteristic velocities in openchannel hydraulics
(Ponce, 1991):
 The mean velocity u of the normal, steady flow, expressed by the Manning or Chezy formulas;
 The relative velocity v of
kinematic waves, expressed by the Seddon celerity formula;
and
 The relative velocity w of
dynamic waves, expressed by the Lagrange celerity formula.
These three velocities
can only define two independent, dimensionless ratios, or numbers, to wit:
the Froude and Vedernikov numbers
(Ponce, 2014: Section 1.3).
The Froude number is the ratio of the velocity of the normal, steady flow u
to the relative celerity of dynamic waves w:
u u
F _{ = } ____ _{ = } ________
w ( g D )^{1/2}
 (1) 

in which D = hydraulic depth (D = A /T ); A = flow area; T =
top width; g = gravitational acceleration (Ponce and Choque Guzmán, 2019;
Ponce and Boulomytis, 2021).
The Vedernikov number is the ratio of the relative celerity of kinematic waves v to the relative celerity of dynamic waves w:
v v
V _{ = } ____ _{ = } ________
w ( g D )^{1/2}
 (2) 

The third ratio, a function of the other two,
is the dimensionless relative kinematic wave celerity v/u,
expressed as follows (Ponce and Choque, 2019):
v V
_____ _{ = } _{β  1} _{ = } _____
u F
 (3) 

The neutralstability Froude number F_{ns} is that which
corresponds to the Vedernikov number V = 1.
Therefore, the neutralstability Froude number is a function only of β, the exponent of the rating:
1
_{ Fns } _{ = } ______
β  1

 (4) 
Table 1 shows corresponding values of β and F_{ns} for three asymptotic
crosssectional shapes and two types of friction.
The shape of the inherently stable channel has been documented. first
by Liggett (1975), and later by
Ponce and Porras (1995) (Fig. 3).
Table 1. Values of β and F_{ns}
corresponding to three asymptotic crosssectional shapes.

Crosssectional shape 
Friction 
β 
F_{ns} 
Hydraulically wide 
Manning 
5/3 
3/2 
Chezy 
3/2 
2 
Triangular 
Manning 
4/3 
3 
Chezy 
5/4 
4 
Inherently stable 
Manning or Chezy 
1 
∞ 
Fig. 3 Shape of the inherently stable channel. 
Equation 4 shows that as β ⇒ 1,
the neutralstability Froude number F_{ns}
⇒ ∞.
In practice, however, the Froude number is bounded by the demonstrably finite amount of friction,
and maximum Froude numbers do not realistically exceed a value in the range 2530.
Therefore, the inherently stable channel must be considered a theoretical construct.
More importantly, however, certain crosssectional shapes featuring
values of β close to but greater than 1 result in an actual
increase in the value of the neutral stability
Froude number F_{ns}, effectively reducing the probability that the flow will become unstable.
This line of reasoning is pursued in this paper: To find the optimal shape of cross section, typically trapezoidal,
that will show to be both practical and stable.
4. TESTING PROGRAM
4.1 Rationale
The online calculator onlinechannel15b
calculates the value of β, the exponent of the rating,
corresponding to a rectangular, trapezoidal, or triangular crosssectional shape.
The calculator requires the following input (Fig. 4):
Input to onlinechannel15b:

Bottom width b

Flow depth y

Side slope z_{1}

Side slope z_{2}

Manning's n

Bottom slope S.

Fig. 4 Definition sketch for a cross section of trapezoidal, rectangular, or triangular shape.

At the outset, for each design application, determine the applicable values of
Manning's n and bottom slope S.
The methodology consists of the following steps:
Choose an appropriate value of
design discharge Q;
Choose appropriate values of side slopes z_{1}
and z_{2};
Determine a testing set of values of
bottom width b;
Using the online calculator, for each value of bottom width b,
calculate, by trial and error, the flow depth y corresponding to the chosen
discharge Q; and
Note the output of the online calculator,
consisting of the following: (a) [confirming the value of] discharge Q;
(b) flow velocity v; (c) Froude number F; (d) exponent of the rating
β; (e) neutrally stable Froude number F_{ns}; and
(f) Vedernikov number V.
Keeping in mind considerations of flow stability (V < 1) or instability (V ≥ 1),
the results are analyzed to choose the optimal design crosssectional shape
compatible with prevailing site and cost constraints.
Fig. 5 Sample calculation for z = 0.25 and b = 5 m,
shown in Table 2, Col. 2, using
onlinechannel15b.

4.2 Testing program
The testing program is designed to determine
the hydraulic conditions in a series of alternative channel crosssections
for which the calculated Vedernikov number varies in the range V ≷ 1.
Several values of side slope z
are specified, ranging from high (z = 0.25; trapezoidal) to low (z = 0; rectangular),
and varying the bottom width b within a suitable range (5 ≥ b ≥ 0).
Experience indicates that the chosen range
of side slopes (0.25 ≥ z ≥ 0) is likely to provide a desired range of Vedernikov numbers
V for appropriate channel flow stability/instability analysis.
The following six (6) side slopes are considered in this study:
z = 0.25;
z = 0.20;
z = 0.15;
z = 0.10;
z = 0.05; and
z = 0.0.
Tables 2 to 7 show the results of the calculation
using onlinechannel15b.
Generally, when reducing the bottom width b
in the chosen range 5 ≥ b ≥ 1, the smaller the value of
side slope z, the faster the Vedernikov number V
decreases to values less than 1.
Indeed. Table 7 shows that the lowest value of V (V = 0.05)
is obtained for the case of z = 0 (rectangular channel) and b = 1, i.e.,
the narrowest value of b within the chosen test range (5 ≥ b ≥ 1).
A detailed analysis follows.
Table 2. Results for Series A (z = 0.25).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.754 
2.078 
2.581 
3.408 
4.769 
Output flow variables 
P 
8.615 
8.283 
8.320 
9.025 
10.83 
T 
5.877 
5.039 
4.290 
3.704 
3.384 
A 
9.539 
9.391 
9.408 
9.719 
10.45 
R 
1.107 
1.133 
1.130 
1.076 
0.965 
D 
1.623 
1.863 
2.192 
2.624 
3.089 
Results 
v 
10.48 
10.65 
10.63 
10.29 
9.569 
F 
2.62 
2.49 
2.29 
2.02 
1.73 
β 
1.56 
1.53 
1.48 
1.40 
1.32 
F_{ns} 
1.76 
1.87 
2.07 
2.45 
3.12 
V * 
1.48 
1.32 
1.10 
0.82 
0.55 
* Two stable
values of V were found, shown in bold. 
Table 3. Results for Series B (z = 0.20).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width
b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.783 
2.128 
2.680 
3.625 
5.260 
Output flow variables 
P 
8.636 
8.340 
8.466 
9.393 
11.72 
T 
5.713 
4.851 
4.072 
3.450 
3.104 
A 
9.55 
9.417 
9.476 
9.878 
10.79 
R 
1.105 
1.129 
1.119 
1.051 
0.92 
D 
1.671 
1.941 
2.327 
2.863 
3.477 
Results 
v 
10.47 
10.62 
10.56 
10.13 
9.27 
F 
2.58 
2.43 
2.21 
1.91 
1.58 
β 
1.56 
1.53 
1.47 
1.39 
1.30 
F_{ns} 
1.77 
1.88 
2.09 
2.50 
3.26 
V * 
1.46 
1.29 
1.05 
0.76 
0.48 
* Two stable values of V was found,
shown in bold. 
Table 4. Results for Series C: (z = 0.15).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.814 
2.184 
2.795 
3.900 
5.950 
Output flow variables 
P 
8.668 
8.416 
8.652 
9.887 
13.03 
T 
5.544 
4.655 
3.838 
3.170 
2.785 
A 
9.563 
9.451 
9.556 
10.08 
11.26 
R 
1.103 
1.122 
1.104 
1.019 
0.863 
D 
1.724 
2.030 
2.489 
3.180 
4.043 
Results 
v 
10.46 
10.58 
10.46 
9.925 
8.888 
F 
2.54 
2.37 
2.11 
1.77 
1.41 
β 
1.56 
1.52 
1.47 
1.38 
1.28 
F_{ns} 
1.77 
1.89 
2.11 
2.58 
3.47 
V * 
1.43 
1.25 
0.99 
0.68 
0.40 
* Three
stable values of V were found. 
Table 5. Results for Series D: (z = 0.10).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.849 
2.248 
2.935 
4.269 
7.019 
Output flow variables 
P 
8.716 
8.518 
8.889 
10.58 
15.10 
T 
5.369 
4.449 
3.587 
2.853 
2.403 
A 
9.586 
9.497 
9.666 
10.36 
11.94 
R 
1.099 
1.114 
1.086 
0.979 
0.790 
D 
1.785 
2.134 
2.694 
3.630 
4.969 
Results 
v 
10.43 
10.53 
10.35 
9.661 
8.377 
F 
2.49 
2.30 
2.01 
1.61 
1.20 
β 
1.56 
1.52 
1.46 
1.37 
1.26 
F_{ns} 
1.77 
1.90 
2.15 
2.68 
3.82 
V * 
1.40 
1.20 
0.93 
0.60 
0.31 
* Three
stable values of V were found. 
Table 6. Results for Series E: (z = 0.05).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.887 
2.322 
3.108 
4.799 
9.037 
Output flow variables 
P 
8.778 
8.649 
9.223 
11.60 
19.09 
T 
5.188 
4.232 
3.310 
2.479 
1.903 
A 
9.613 
9.557 
9.806 
10.74 
13.12 
R 
1.095 
1.104 
1.063 
0.925 
0.687 
D 
1.852 
2.258 
2.962 
4.334 
6.892 
Results 
v 
10.40 
10.47 
10.20 
9.307 
7.628 
F 
2.44 
2.22 
1.89 
1.42 
0.92 
β 
1.56 
1.52 
1.45 
1.35 
1.22 
F_{ns} 
1.78 
1.91 
2.19 
2.84 
4.52 
V * 
1.36 
1.15 
0.86 
0.50 
0.20 
* Three
stable values of V were found. 
Table 7. Results for Series F: (z = 0).

Q = 100 m^{3}/s 
n = 0.025 
S = 0.06 
Variable 
Bottom width b (m) 
5 
4 
3 
2 
1 
Input 
y 
1.929 
2.407 
3.330 
5.690 
16.54 
Output flow variables 
P 
8.858 
8.814 
9.660 
13.38 
34.08 
T 
5.000 
4.000 
3.000 
2.000 
1.000 
A 
9.645 
9.628 
9.990 
11.38 
16.54 
R 
1.088 
1.092 
1.034 
0.850 
0.485 
D 
1.929 
2.407 
3.330 
5.690 
16.54 
Results 
v 
10.37 
10.39 
10.01 
8.795 
6.050 
F 
2.38 
2.13 
1.75 
1.17 
0.47 
β 
1.55 
1.51 
1.44 
1.31 
1.11 
F_{ns} 
1.79 
1.93 
2.24 
3.13 
8.36 
V * 
1.33 
1.10 
0.78 
0.37 
0.05 
* Three
stable values of V were found. 
5. ANALYSIS
The results of Tables 2 to 7 are analyzed to determine the
crosssectional shape, which in this paper is varied
from trapezoidal (z = 0.25; Table 2)
to rectangular (z = 0; Table 7),
under which the Vedernikov number decreases from the
unstable range, V > 1,
to the stable range, V ≤ 1. At the outset, it is recognized
that the Froude and Vedernikov numbers (Eqs. 1 and 2, respectively) vary inversely with hydraulic depth D.
Thus, the larger the value of D, the smaller the values of both Froude and Vedernikov numbers,
eventually leading to the condition of stable flow, i.e., V ≤ 1. We pose that herewith is the solution of the stability/instability dichotomy:
The larger the hydraulic depth, the more stable is the flow likely to be.
To further explain the findings,
the variation, with hydraulic depth D,
of the rating exponent β, Froude number F,
and Vedernikov number V is shown in Figs. 6 to 8, respectively.
Figure 6 shows that the decrease in
β is gradual for the trapezoidal shapes
(0.25 ≥ z ≥ 0.05),
and sharp (to β = 1.11) for the (asymptotic)
rectangular shape (z = 0).
Figure 7 shows that the decrease in
F is gradual for the trapezoidal shapes
(0.25 ≥ z ≥ 0.05),
and sharp (to F = 0.47) for the (asymptotic)
rectangular shape (z = 0).
Figure 8 shows that the decrease in
V is gradual for the trapezoidal shapes
(0.25 ≥ z ≥ 0.05),
and sharp (to V = 0.05) for the (asymptotic)
rectangular shape (z = 0).
It is concluded that the fastest way to decrease the
Vedernikov number below 1 and, thus, assure hydrodynamic stability,
is to choose a bottom width b, in conjunction with a
side slope z, that will assure that V < 1.
In practice, a suitable value of V < 1 may be used
as a design objective. The results of Tables 2 to 6
indicate that, for the example presented herein,
a V = 0.93 is obtained for
b = 3 m and z = 0.10. Furthermore,
a somewhat lower and, therefore, somewhat
more stable
V = 0.86 is obtained for
b = 3 m and z = 0.05.
The analysis presented here purposely
considers only the question of hydrodynamic
stability. In an actual design situation, issues
such as cost, rightofway, and constructability may also
have a role in determining the optimal crosssectional shape.
Fig. 6 Rating exponent β vs. hydraulic depth D.

Fig. 7 Froude number F vs.
hydraulic depth D.

Fig. 8 Vedernikov number V vs.
hydraulic depth D..

6. CONCLUSIONS
A study of the effect of crosssectional shape on freesurface channel
hydrodynamic
instability is accomplished. At the outset, the rating exponent β, Froude number
F, and Vedernikov number V are identified
as the controlling variables. The rating exponent characterizes the
dischargeflow area rating Q = α A^{ β}.
The Froude number characterizes the flow regime as either: (a) subcritical, (b)
critical, or
(c) supercritical.
The Vedernikov number describes a flow type that is either: (a) stable, (b) neutral,
or (c) unstable.
A steep, lined channel is specified for the analysis.
The selected design discharge is Q = 100 m^{3}/s, with bottom slope
S = 0.06 and Manning's n = 0.025, closely resembling the flow conditions of the
Huayñajahuira river, in La Paz, Bolivia, where roll waves have been shown
to recur with worrisome regularity. The testing program considers
the variation of the bottom width b in the range 5 ≥ b ≥ 1, at 1m
intervals (five channel widths),
and the side slope z in the range 0.25 ≥ z ≥ 0, at 0.05 intervals
(six side slopes).
The online calculator onlinechannel15b
is used to calculate the relevant hydraulic variables, culminating in the
values of rating exponent β, Froude number F, and
Vedernikov number V for each of thirty (5 × 6 = 30) cases.
The results show conclusively that as the channel width b is reduced from 5 to 1 m,
and the side slope z reduced from 0.25 to 0,
that β, F,
and V are reduced, first gradually, and then sharply as z →
0, with the asympotic value z = 0 corresponding to a rectangular channel.
For a given design application,
these findings may be used to determine optimal geometric crosssectional values b and
z in order to assure that V < 1 and, therefore, avoid flow hydrodynamic instability and the associated roll waves.
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