The recurrence of these pulsating waves in channelized rivers of the La Paz region has been documented by Molina et al. (1995).

In this article, we review the theory of the inherently stable channel, clarify its physical and mathematical basis, and develop an online calculator. It is hoped that this contribution will encourage the hydraulic engineering profession to try the inherently stable channel to avoid the risk of roll waves in steep urban channels.

2.  THEORETICAL BACKGROUND

The criterion for the instability of open-channel flow is due to Vedernikov (1945, 1946). Powell (1948) gave this criterion the name of Vedernikov number. Subsequently, Craya (1952) clarified the concept, enhancing its theoretical basis. The Vedernikov criterion states that the water surface of a steep open channel may become unstable, with the possibility of developing roll waves, when the relative kinematic wave celerity, i.e., the Seddon celerity (Seddon, 1900), equals or exceeds the relative dynamic wave celerity, or Lagrange celerity (Lagrange, 1788). In their comprehensive study of shallow wave propagation in open-channel flow, Ponce and Simons (1977) confirmed the validity of the Vedernikov criterion, which for the case of Chezy friction is equivalent to Froude number F ≥ 2.

Roll waves are a train of waves occurring in steep channels of rigid boundary, lined with either masonry or concrete. Figure 1 shows an early photograph of a roll wave event in the Grünnbach Conduit of the Swiss Alps (Cornish, 1907).

Cornish Fig. 1  Roll waves in a masonry-lined channel in the Swiss Alps, c. 1907.

Roll waves are a common occurrence in steep, lined channels. The following video shows a train of roll waves in a steep urban channel.

Herein we review the theory of free-surface instability in open-channel flow. We start with the kinematic wave celerity (Ponce, 2014a):

ck  =  β u

(1)

in which u = mean flow velocity, and β = exponent of the discharge-flow area rating  (Q = α A ^β ). Thus, the relative kinematic wave celerity, i.e., the celerity relative to the flow velocity, is:

ck  =  (β - 1) u

(2)

The two components of the dynamic wave celerity are (Ponce, 2014b):

cd  =  u  ±  (g D )1/2

(3)

in which D is the hydraulic depth:

A D =  _____           T

(4)

in which A = flow area, and T = top width.

The relative dynamic wave celerity is:

cd  =  (g D )1/2

(5)

The Vedernikov number is the ratio of relative kinematic wave celerity to relative dynamic wave celerity (Ponce, 1991):

(β - 1) u V  =  ____________             (g D )1/2

(6)

For V > 1, the flow becomes unstable, opening up the possibility for the development of roll waves. Ponce and Maisner (1993) have shown that the condition V > 1 (equivalent to Froude number F > 2 under Chezy friction) is necessary but not sufficient; i.e., that it may not always lead to roll waves (see also Montuori, 1965, p. 26). Ponce and Maisner (op. cit.) showed that wave disturbances would amplify within a relatively narrow range of dimensionless wavenumbers, near the peak of the positive log decrement function (Fig. 2). Therefore, we reckon that the scale of the disturbance plays a major role in determining whether roll waves will form.

Fig. 2  Primary wave logarithmic increment for Froude numbers F > 2 (Chezy friction).

Discussion. In their linear stability analysis of unsteady open-channel flow, Ponce and Simons (1977) confirmed the findings of Lighthill and Whitham (1955) that the flow disturbances would amplify when the kinematic wave celerity exceeds the dynamic wave celerity (Fig. 2). Furthermore, Ponce (1992) compared the transport of mass and energy across the dimensionless wave spectrum, concluding that at V > 1 kinematic waves would overcome mixed kinematic-dynamic waves, and that, in turn, the latter would overcome dynamic waves. Since kinematic waves transport mass, and dynamic waves transport energy, it is readily seen that in the overcoming of energy waves by mass waves must lie the true nature of roll waves (Ponce, 2014c).

The Froude number is defined as follows (Chow, 1959):

u F  =  __________           (g D )1/2

(7)

Combining Eqs. 6 and 7:

V β - 1  =  _____                 F

(8)

Equation 8 underscores the significance of the exponent [of the rating] β in open-channel flow. The quantity (β - 1) is the ratio of Vedernikov and Froude numbers. At neutral stability, i.e., in the absence of wave attenuation or amplification, the Vedernikov number V = 1, and the Froude number becomes F = F_ns. Thus:

1 β - 1  =  _______                 Fns

(9)

Therefore:

1 Fns  =  ________                β - 1

(10)

In the turbulent flow regime, under Manning friction, the feasible range is: 1 ≤ β ≤ 5/3. This gives rise to three asymptotic cross-sectional shapes (Ponce and Porras, 1995b; Ponce, 2014):

1. Hydraulically wide, with β = 5/3, for which the wetted perimeter P is a constant (Fig. 3), and F_ns = 1.5;

2. Triangular, with β = 4/3, for which the top width T is proportional to the flow depth d (Fig. 4), and F_ns = 3; and

3. Inherently stable, with β = 1, for which the hydraulic radius R in the upper subsection [the overflow subsection] is constant and equal to that of the lower subsection (Fig. 5), and F_ns = ∞.

Nuccitelli Fig. 3  Mississippi River at Mud Island, Memphis, Tennessee.

Fig. 4  The triangular channel cross-section.

Fig. 5  An inherently stable channel cross-section (Ponce and Porras, 1995c).

Note that while, under Manning friction, the hydraulically wide channel becomes unstable at F = 1.5, and the triangular channel at F = 3, the inherently stable channel becomes unstable as F ⇒ ∞.

The existence of a lower limit for boundary friction imposes a practical upper limit to the Froude number. To estimate this upper limit, we invoke the dimensionless Chezy formula (Ponce, 2014d):

So  =  f F 2

(11)

in which S_o = channel slope, and f = dimensionless friction factor, equal to 1/8 of the Darcy-Weisbach friction factor f_D. Assume a very steep slope (say, 45°, i.e., 100%), for which S_o  =  1; and the lowest possible value of friction factor, f = 0.001875, corresponding to a Darcy-Weisbach f_D = 0.015. This assumption leads to an estimate of the maximum value of Froude number than can be achieved in practice: F_max = 23 (Chow, 1959).

It is concluded that the inherently stable will never achieve the asymptotic Froude number F_ns = ∞ which characterizes it. Therefore, constructing an inherently stable channel, for which β = 1, provides an unrealistically high factor of safety against roll waves. This suggests the possibility of designing a practically stable cross-section, still one where the risk of roll waves would be so small as to be of no concern.

Assume a realistic maximum value of Froude number F_max = 25. Using Eq. 8, at neutral stability, V = 1, and, therefore, β = 1.04. It is concluded that a conditionally stable channel may be designed to correspond to β = 1.04, and that there appears to be no need to impose the asymptotic value of β = 1 in order to achieve hydrodynamic stability. Thus, a channel designed with β = 1.04 is poised to provide complete freedom from roll waves.

3.  THE INHERENTLY STABLE CHANNEL

To derive the equation of the inherently stable channel, the general relation between wetted perimeter P and flow area A is postulated in the following exponential form (Ponce and Porras, 1995d):

P  =  Κ A δ

(12)

from which:

A      dP              dP δ  =  ____  _____  =  R  _____                         P      dA              dA

(13)

For the hydraulically wide channel, the wetted perimeter P is a constant (Section 2). Therefore, from Eq. 12: δ = 0, and the wetted perimeter remains:

P  =  Κo

[Hydraulically wide]

(14)

and:

dP                  ____  =  0  dA

(15)

For the triangular channel, both the hydraulic radius R and the wetted perimeter P vary with the flow area, and the exponent δ has the central value δ = 0.5. From Eq. 12, the relation between hydraulic radius R, wetted perimeter P, and flow area A is:

P  =  Κ A 0.5                [Triangular]

(16)

and:

dP                 P ____  =  0.5  _____  dA                 A

(17)

For the inherently stable channel, the hydraulic radius R is a constant (Section 2). Therefore, from Eq. 12: δ = 1, and the wetted perimeter remains:
                       

P  =  Κo A                 [Inherently stable]

(18)

and:

dP          P        ____  =  _____  =  Κo  dA          A

(19)

For the case of Manning friction, δ = 0 corresponds to β = 5/3, and δ = 1 corresponds to β = 1 (Section 2). Therefore, the following linear relation holds:

5          2 β  =  ____  -  ____ δ            3          3

(20)

Multiplying Eq. 20 by 3/2 and solving for δ:

5          3 δ  =  ____  -  ____ β            2          2

(21)

Given a chosen value of neutral-stability Froude number F_ns, Eq. 9 can be used to calculate β, and Eq. 21 used to calculate δ. Furthermore, assume that the optimum stable channel design is that associated with F_ns = 25, for which β_o = 1.04, and that the factor of safety for this case is postulated to be F.S._i = 1. For a given choice i of F_ns,i, a factor of safety is defined as follows:

βo F.S.i  =  ____               βi

(22)

Table 1 shows selected values of parameters of the stable channel, for suitable values of neutrally stable Froude numbers in the range 10 ≤ F_ns ≤ ∞. We conclude that a choice of F_ns = 20 has a lesser chance of developing roll waves than a choice of F_ns = 10. Furthermore, a choice of F_ns ≥ 25 is practically bound to assure freedom from roll waves.

4.  STABLE CHANNEL DESIGN

Liggett (1975) derived the differential equation of the inherently stable channel, for which δ = 1. Ponce and Porras (1995d) extended this equation to the conditionally stable channel, for which δ < 1. Table 1 shows the values of β and δ corresponding to selected values of F_ns. The conditionally stable channel is stable, i.e., V ≤ 1, provided the Froude number is restricted to F ≤ F_ns, while the hydraulic radius varies mildly with the flow depth.

Due to symmetry, a half-cross-section analysis is appropriate. Hereafter, the asterisk (* ) is used as a subscript to refer to half values of the hydraulic variables, e.g., T_*  = half top width.

The differential wetted perimeter of the stable cross section is:

dP*  =  [ (dh) 2 + (dT* ) 2 ] 1/2

(23)

in which dh = differential flow depth. Dividing by dh, and since  dA_* = T_* dh  (Ponce, 2014):

dP*                       dT* T*  ______  =  [ 1  +  ( ______ ) 2 ] 1/2         dA*                       dh

(24)

Substituting Eq. 13 into Eq. 24:

δ T*                        dT*  ______  =  [ 1  +  ( ______ ) 2 ] 1/2     R                          dh

(25)

The inherently stable channel is that for which δ = 1; therefore, the hydraulics radius is constant and equal to R_o, i.e., Κ_o in Eq. 18. Operating in Eq. 25, the differential equation of the inherently stable channel is obtained:

dT*               T*  ______  =  [ ( _____ ) 2   -  1 ] 1/2     dh                Ro

(26)

subject to T_* > R_o .

Equation 26 may be solved using the following indefinite integral (Spiegel et al., 2013):

dx                ∫  ______________  =  ln [ x + (x 2 - a 2) 1/2 ]      (x 2 - a 2) 1/2

(27)

in which x = T_* , and a = R_o .

The design of a stable channel requires that the hydraulic radius be specified at the start. To achieve this requirement, the cross section must be comprised of two subsections:

1. A lower subsection, of rectangular, trapezoidal, or triangular shape; and

2. An upper subsection, of stable shape (see, for instance, Fig. 5).

The lower subsection (of half bottom width B_* , depth h_o, and size slope z : 1 (H:V) defines the hydraulic radius R_o:

0.5 (2 B*  +   zho) ho  Ro  =  ________________________                B*  +  ho (1 + z 2)1/2

(28)

In addition to defining R_o, the lower subsection serves the purpose of conveying the low flows. In practice, high-velocity flows may occur at relatively small flow depths. This fact should be taken into account in the design of the lower subsection.

Equation 26 constitutes a family of inherently stable channel cross sections, with parameter R_o. A particular solution, where T_*o  is the half top width corresponding to the depth h_o , is:

T*                T*                                   ______  +  [ ( _____ ) 2   -  1 ] 1/2                                      Ro               Ro h  =  ho  + Ro  ln  { _________________________________ }                                     T*o               T*o                                   ______  +  [ ( _____ ) 2   -  1 ] 1/2                                      Ro               Ro

(29)

For the special case of T_*o = R_o , Eq. 29 reduces to the Liggett (1975) solution for the inherently stable channel:

T*                 T*                                       h  =  ho  + Ro  ln  { ______  +  [ ( ______ ) 2   -  1 ] 1/2 }                                   Ro                Ro

(30)

Since boundary friction has a lower limit and cannot realistically decrease to zero, it follows that there is an upper limit to the Froude number that can be achieved in practice. In other words, the inherently stable channel will become unstable as the Froude number F ⇒ ∞, yet the latter cannot be reached in any practical setting. Therefore, there appears to be no need to design a stable channel section with δ = 1.

Alternatively, a channel section with a value δ < 1 may be designed to remain stable, provided the neutrally stable Froude number associated with this value of δ is not exceeded. In practice, given that the maximum Froude number is not likely to exceed F = 25, a conditionally stable channel section, for which δ = 0.94, may be envisaged. Values of δ corresponding to selected values of F_ns are shown in Table 1.

The extension of Eq. 26 to the conditionally stable channel, for which δ < 1, is:

dT*              δT*  _____  =  [ ( ______ ) 2   -  1 ] 1/2     dh                R

(31)

subject to δT_* > R .

Unlike Eq. 26, Eq. 31 cannot be integrated analytically. The shape of the upper subsection T_* = f (δ, R_o, R ) may be obtained by numerical integration, given a value of δ, corresponding to a choice of F_ns, and the hydraulic radius R_o corresponding to the depth  h_o of the lower subsection.

The numerical integration proceeds by selecting the shape of the lower subsection (B_*, h_o, and z ) and the total channel depth h_t , to comprise lower and upper subsections. In the lower subsection, the flow depth varies in the range 0 ≤ h ≤ h_o; in the upper subsection, it varies in the range h_o < h ≤ h_t .

INPUT DATA Half bottom width of the lower subsection B* Depth ho of the lower subsection Side slope of the lower subsection z Relative depth of the upper subsection hu' = (ht - ho)/ho Channel slope So Manning's n Neutrally stable Froude number Fns

The computational algorithm is based on the following recursive procedure.

RECURSIVE PROCEDURE With Fns , calculate β using Eq. 8. With β, calculate δ using Eq. 20. Set counter i = 0 Calculate the half top width T*o  [at ho]: T*o = B* + zho Calculate the half wetted perimeter P*o: P*o = B* + ho (1 + z 2) 1/2 Calculate the half flow area A*o: A*o = 0.5 (2 B* + zho) ho Calculate the hydraulic radius Ro: Ro = A*o  / P*o Calculate the mean velocity Vo: Vo = (1/n)  Ro 2/3  So1/2 Calculate the hydraulic depth Do: Do = A*o  / T*o Calculate the Froude number Fo: Fo = Vo  / (g Do )1/2 Calculate the half discharge Q*o: Q*o = Vo A*o Set Δh = 0.0001 m ⇐ ⇐ ⇐ ⇐ ⇐ ⇐ ⇓ Increment the counter i by 1 Calculate the flow depth hi : hi = hi - 1 + Δh Using Eq. 29, calculate the increment ΔT*i : ΔT*i = Δh [ (δ T*i - 1 / R*i - 1 ) 2 - 1 ] 1/2 Calculate the increment ΔP*i : ΔP*i = [ (Δh)2 + (ΔT*i )2 ] 1/2 Calculate the increment ΔA*i : ΔA*i = 0.5 ( 2T*i - 1 + ΔT*i ) Δh Calculate the value of T*i  [at hi ]: T*i = T*i - 1 + ΔT*i Calculate the value of P*i : P*i = P*i - 1 + ΔP*i Calculate the value of A*i : A*i = A*i -1 + ΔA*i Calculate the value of Ri : Ri = A*i  / P*i Calculate the mean velocity Vi : Vi = (1/n)  Ri 2/3  So1/2 Calculate the hydraulic depth Di  : Di = A*i  / T*i Calculate the Froude number Fi : Fi = Vi  / (g Di )1/2 Calculate the discharge Q*i  corresponding to the flow depth hi : Q*i = (1/n) A*i  R*i 2/3  So1/2 Return to Step 13, and proceed recursively, until hi ≥ ht . ⇑ ⇐ ⇐ ⇐ ⇐ ⇐ ⇐

5.  ONLINE CALCULATOR

The calculator ONLINE INHERENTLY STABLE solves the recursive algorithm explained in the previous section. Input data to the calculator consists of:

• Geometric and hydraulic data:  half bottom width B_* , depth h_o , side slope z , relative depth h_u' , channel slope S_o, and Manning's n; and

• Neutrally stable Froude number F_ns.

To solve for the inherently stable channel, specify a very high value of neutrally stable Froude number, say F_ns = 10,000. To solve for the conditionally stable channel, specify a realistically high value of neutrally stable Froude number, say F_ns = 25 (Section 2).

6.  ANALYSIS

Table 2 shows a summary of typical results of calculations of the stable channel. The following data set was used for the example shown in Table 2:

EXAMPLE INPUT DATA Half bottom width B* = 2.5 m Depth ho = 1 m Side slope z = 0 Relative depth hu' = 1 Channel slope So = 0.012 Manning's n = 0.015 Neutrally stable Froude number:   Ten (10) values, varying in the range:  3 ≤ Fns ≤ 10,000

For increased accuracy, the depth interval is set at Δh = 0.0001 m. In this example, the results of the recursive computation are printed once every 1000 increments, i.e., the depth interval for output is Δh_out = 0.1 m. As expected, Table 2 shows that the hydraulic radius for F_ns = 10,000 remains practically constant and equal to R = 0.714 m throughout the indicated range of flow depths in the upper subsection (1 ≤ h ≤ 2).

The examination of Table 2, supplemented with results obtained using ONLINE INHERENTLY STABLE lead to the following conclusions:

1. For h > h_o, the smaller the choice of F_ns, the smaller the resulting stable top width.

2. For h > h_o, the larger the choice of R_o, the smaller the resulting stable top width.

3. Given F_ns, the larger the choice of R_o, the narrower the resulting stable channel section.

4. Given R_o, the smaller the choice of F_ns, the narrower the resulting stable channel section.

These results confirm and extend the findings of Ponce and Porras (1995e).

Fig. 6  Stable channel cross sections as a function of neutrally stable Froude number F_ns and hydraulic radius R_o:
(a) h_o = 0.5 m, R_o = 0.417 m; (b) h_o = 0.75 m, R_o = 0.577 m; and (c) h_o = 1.0 m, R_o = 0.714 m
(redrawn from Ponce and Porras, 1995e).

A comparison of the inherently stable or conditionally stable channels with a rectangular channel of the same bottom width, channel slope and boundary roughness, shows that for both channels the discharge and Froude number are likely to vary gradually with stage, with the latter either increasing or decreasing, depending on the flow conditions. Theory, however, predicts that while the inherently stable channel will always remain stable, and the conditionally stable channel will most likely remain stable throughout a practical range of Froude numbers, the rectangular channel may be poised to eventually develop roll or pulsating waves.

7.  DESIGN EXAMPLE

Design a stable channel for the following data:  baseflow Q_b = 10 m³/s, peak flow Q_p = 100 m³/s, half bottom width B_* = 2.5 m, size slope z = 0, channel slope S_o = 0.012, and Manning's n = 0.015. Calculate the depth h_o and relative depth h_u'.

Solution. Trial runs with lower subsection depth h_o = 0.8 m, and ratio of upper-to-lower subsection depths h_u' = 2.0, show the following results:

• For the inherently stable channel, at F_ns = 10,000:  Lower subsection discharge Q_b = 10.46 m³/s, and total discharge, corresponding to a total channel depth h_t = h_o (h_u' + 1) = 2.4 m, is Q_p = 112.112 m³/s. The hydraulic radius remains R_o = 0.606 m through the range 0.8 ≤ h_t ≤ 2.4, and the half top width T_* at h_t = 2.4 m is: T_* = 34.468 m.

• For the conditionally stable channel, at F_ns = 25:  Q_b = 10.46 m³/s, and Q_p = 102.902 m³/s. At h_t = 2.4 m, the hydraulic radius is R = 0.692 m and the half top width is: T_* = 25.169 m.

In this example, the design half top width of the conditionally stable channel is equal to 73% of that of the inherently stable channel, i.e., a reduction of 27% is indicated.

8.  CONCLUSIONS

The inherently stable channel and its conditionally stable alternative are reviewed, elucidated, and calculated online. The asymptotic neutrally stable Froude number for the inherently stable channel is F_ns ⇒ ∞. Theoretically, such a channel will become neutrally stable when the Froude number reaches infinity. Since the latter is a physical impossibility, this requirement effectively guarantees that the inherently stable channel will always remain well below the threshold of instability, regardless of flow discharge, thus completely eliminating the possibility of roll wave formation.

The inherently stable channel will never achieve the value of Froude number F_ns = ∞ which characterizes it. Therefore, constructing an inherently stable channel, i.e., that for which the exponents β = 1 and δ = 1, provides an unrealistically high factor of safety against roll waves. This suggests the possibility of designing instead a conditionally stable cross-sectional shape, for a suitably high but realistic Froude number such as F_ns = 25, corresponding to β = 1.04 and δ = 0.94, for which the risk of roll waves would be so small as to be of no practical concern.

The results of this study lead to the following conclusions:

• For h > h_o, the smaller the choice of Froude number F_ns, the smaller the resulting stable top width.

• For h > h_o, the larger the choice of hydraulic radius R_o, the smaller the resulting stable top width.

• Given F_ns, the larger the choice of R_o, the narrower the resulting stable channel section.

• Given R_o, the smaller the choice of F_ns, the narrower the resulting stable channel section.

A design example shows that the conditionally stable channel with F_ns = 25 is about 27% narrower than the inherently stable channel.

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