1.  INTRODUCCIÓN

El desarrollo de ondas de rollo u ondas pulsantes en el flujo de un canal abierto sigue atrayendo el interés tanto de investigadores como de ingenieros profesionales. En 1907, Cornish mostró, por primera vez, una fotografía de este fascinante fenómeno en un artículo publicado en el Journal of the Royal Geographical Society (Fig. 1) (Cornish, 1907). Más tarde, en el Capítulo 8 de su reconocido texto, Ven Te Chow se refirió al fenómeno como la "inestabilidad del flujo uniforme", dando a entender que bajo determinadas condiciones, el flujo podría volverse inestable y romper en un tren de ondas (Chow, 1959). En la Fig. 2 se muestra una fotografía contemporánea del fenómeno de las ondas de rollo, u ondas pulsantes.

Cornish

Fig. 1  Forografia temprana de un tern de ondas de rollo en los Alpes suizos (1907).

Una onda de rollo es un fenómeno inusual y a menudo fascinante, a ser admirado por aquéllos que tienen la suerte de observarlo. Sin embargo, en ciertas circunstancias, puede resultar inquietante e inclusive peligroso. Por lo tanto, es imperativo que los ingenieros hidráulicos comprendan los principios que rigen la formación y propagación de las ondas de rollo, de modo que el diseño pueda tender a minimizar los posibles impactos negativos.

En este artículo enfocamos el estudio de caso de La Paz, Bolivia, en el cual se ha documentado la recurrencia de ondas de rollo con una frecuencia preocupante en algunos ríos canalizados. Localmente, estas ondas de rollo, a menudo de gran tamaño, se denominan ondas pulsantes. La Paz, una ciudad de aproximadamente 800.000 habitantes y sede del Estado Plurinacional de Bolivia, presenta un entorno geomorfológico peculiar: Está construida casi en su totalidad dentro de un inmenso barranco, con varios arroyos de pendiente muy pronunciada que drenan su lado oriental hacia el centro de la ciudad (Fig. 3).

En los últimos 30 años, el desarrollo urbano ha dado como resultado la canalización de varios de los principales arroyos tributarios con canales revestidos de mampostería, destinados a transportar las aguas de avenida del modo hidráulico más eficiente. Sin embargo, tal como se construyeron, los canales de drenaje han modificado la sección transversal de los arroyos de modo que los eventos de olas de rollo ahora son recurrentes (2019), donde no existían antes de la canalización. Con la amenaza del calentamiento global en ciernes, es probable que la intensidad y frecuencia de estos eventos aumenten con el tiempo, lo que requiere un renovado y más decidido enfoque para su mejor gestión.

A modo de ilustración, aqui mostramos dos videos tomados en La Paz en los últimos años. El primer video muestra un evento de onda de rollo en el río Achumani, un afluente del río La Paz (Fig. 4). Este evento ocurrió en el año 2014. El video permite estimar el período de la onda en 19 segundos.

Courtesy of J. Molina

[Haga click encima de la figura para ver el video]

Fig. 4  El río canalizado Achumani, La Paz, Bolivia (2014).

En el segundo video se observa una onda de rollo en el río Huayñajahuira, otro afluente del río La Paz, ocurrido el 24 de febrero de 2016. En este caso, se observa que el flujo pulsante es lo suficientemente grande como para saltar por encima de los límites del canal, constituyendo un peligro para la seguridad pública (Fig. 5). En la actualidad (2019), este tipo de eventos siguen repitiéndose en estos dos ríos canalizados, lo cual clama por una solución al problema.

Cortesia de J. Molina

[Haga click encima de la figura para ver el video]

Fig. 5  El río canalizado Huayñajahuira, La Paz, Bolivia (2016).

En este artículo se revisan los antecedentes históricos y se examina la teoría, experiencia y posibles soluciones para controlar, atenuar y/o manejar las ondas de rollo de manera que la sociedad tenga la seguridad de que en el futuro no causen daños personales y/o materiales.

2.  PERSPECTIVA HISTÓRICA

En 1945, V. V. Vedernikov presentó, en idioma ruso, un análisis matemático exhaustivo de las ondas de rollo (Vedernikov, 1945; 1946). Casi al mismo tiempo, A. Craya publicó un artículo sobre la inestabilidad del flujo en la revista francesa La Houille Blanche (Craya, 1945). Sin embargo, el trabajo definitivo de Craya sobre el tema de la inestabilidad del flujo se publicó sólo siete años después (Craya, 1952).

En un artículo publicado en Transactions of the American Geophysical Union, Ralph W. Powell bautizó el criterio de Vedernikov como el número de Vedernikov (Powell, 1948). Más adelante, Ven Te Chow confirmó la práctica en su reconocido libro de texto (Chow, 1959).

Craya aclaró el criterio de Vedernikov interpretándolo como el umbral en el que la celeridad de la onda cinemática es igual a la de la onda dinámica. Ya en 1900, Seddon, trabajando en el Bajo Río Misisipi, había derivado la expresión para la celeridad de la onda cinemática, es decir, una onda gobernada únicamente por la fricción y la gravedad (Seddon, 1900). Más tarde, Lighthill y Whitham (1955) profundizaron en los fundamentos teóricos y matemáticos de la onda cinemática, destacando su aplicación a la propagación de ondas de inundación.

A diferencia de una onda cinemática, la onda dinámica de la mecánica de fluidos clásica, la cual se propaga con la celeridad de Lagrange, es una onda gobernada únicamente por la inercia y el gradiente de presiones (Lagrange, 1788). Se observa que la celeridad relativa de la onda dinámica de Lagrange es el denominador del conocido número de Froude (Ponce, 2014: Capítulo 1).

Ponce (1991) ha presentado un tratamiento teórico unificado de los números de Froude y Vedernikov, demostrando que son esencialmente independientes entre sí. El análisis siguió el trabajo seminal previo de Ponce y Simons (1977), que sentó las bases para el tratamiento completo del flujo inestable en canal abierto a lo largo del espectro adimensional de números de onda. Ponce confirmó el hallazgo anterior de Craya con respecto al umbral de inestabilidad del flujo, es decir, cuando las ondas de masa de Seddon (es decir, las ondas cinemáticas) superan las ondas de energía de Lagrange (las ondas dinámicas). Este umbral se produce cuando la celeridad de la onda cinemática es igual o superior a la celeridad de la onda dinámica, es decir, cuando el número de Vedernikov V ≥ 1. Por lo tanto, la naturaleza de las ondas de rollo está intrínsecamente ligada al concepto del número de Vedernikov.

3.  EL NÚMERO DE VEDERNIKOV

Ponce (1991) y más recientemente, Ponce (2014) han identificado tres velocidades de interés en el flujo en canal abierto:

1. La velocidad u del flujo permanente, calculable mediante una fórmula como la de Manning o Chezy;

2. La celeridad relativa v de la onda cinemática, definida como v = c_k - u, en la cual c_k es la celeridad de la onda cinemática; y

3. La celeridad relativa w de la onda dinámica, definida como w = c_d - u, en la cual c_d es la celeridad de la onda dinámica.

Se definen las siguientes variables:

Definición de variables Caudal:  Q Área de flujo:  A Velocidad media:  u = Q / A Perímetro mojado:  P Ancho superior:  T Radio hidráulico:   R = A /P Profundidad hidráulica:   D = A /T Curva de gasto caudal-area:  Q = α A β Aceleración gravitacional:  g Celeridad de la onda cinemática (Ponce, 2014: Capítulo 1):  ck = β u Celeridad relativa de la onda cinemática:  v = ck - u  =  (β - 1) u Celeridad relativa adimensional de la onda cinemática:  v / u = β - 1 Celeridad de la onda dinámica (Ponce, 2014: Capítulo 1):  cd = u  ±  (gD)1/2 Celeridad relativa de la onda dinámica:  w = cd - u  =  (gD)1/2 Celeridad relativa adimensional de la onda dinámica:  w / u = (gD)1/2 / u  =  1 / F Número de Froude:  F = u / w  =  u / (gD)1/2 Número de Vedernikov:  V = v / w  =  (β - 1) u / (gD)1/2

Por lo tanto, solamente hay tres velocidades características en el presente análisis: u, v, y w. El número de Froude es la relación entre la primera y la tercera:  F  =  u / w; el número de Vedernikov es la relación entre la segunda y la tercera:  V  =  v / w.

Se observa claramente que los números de Froude y Vedernikov son independentes; las tres velocidades dan lugar a sólo dos números adimensionales independientes, los números de Froude y Vedernikov. La tercera relación es la celeridad relativa adimensional de la onda cinemática v / u  =  β - 1  =  V / F. Por lo tanto, el exponente de la curva de gasto β comprende tanto V como F. Se concluye lo siguiente:

β  =  1 + (V / F )

(1)

Así, la curva de gasto caudal-area puede expresarse en términos de los números de Froude y Vedernikov como sigue:

Q = α A 1 + (V / F )

(2)

En resumen, el número de Froude es la relación entre la velocidad del flujo y la celeridad relativa de la onda dinámica. Para F < 1, prevalecen condiciones de flujo subcrítico y las perturbaciones superficiales pueden trasladarse aguas arriba, lo que hace posible el control aguas abajo. Para F > 1, prevalecen condiciones de flujo supercrítico y las perturbaciones superficiales no pueden trasladarse aguas arriba; por lo tanto, el flujo podría controlarse sólo desde aguas arriba. El criterio de Froude es estrictamente aplicable al flujo uniforme permanente, aunque en la práctica su uso se ha extendido a otras condiciones de flujo.

El número de Vedernikov es la relación entre la celeridad relativa de la onda cinemática y la celeridad relativa de la onda dinámica. Por lo tanto, el criterio de Vedernikov es estrictamente aplicable al flujo no permanente. Para V < 1, las ondas dinámicas viajan más rápido que las ondas cinemáticas y, en consecuencia, el flujo es estable, es decir, libre de ondas de rollo. Por el contrario, para V > 1, las ondas cinemáticas viajan más rápido que las ondas dinámicas y, en consecuencia, el flujo es inestable, es decir, sujeto al desarrollo de ondas de rollo. Sin embargo, el que las ondas de rollo ocurran o no dependerá de la condición de contorno (véase la Sección 5). Por lo tanto, se considera que el criterio de Vedernikov es necesario, pero no suficiente, para la aparición de ondas de rollo.

Los roles de la masa y la energía son fundamentales para el análisis de las ondas de rollo. Está bien establecido que, mientras que las ondas cinemáticas transportan masa, las ondas dinámicas transportan energía (Lighthill y Whitham, 1955). Por lo tanto, se observa que la aparición de ondas de rollo ocurre cuando el transporte no permanente de masa supera al transporte no permanente de energía. En este sentido, las ondas de rollo son una manifestación física de la preponderancia del transporte de masa sobre el transporte de energía en el flujo no permanente en canales abiertos.

4.  EFECTO DE LA SECCIÓN TRANSVERSAL

The Vedernikov number is:

V  =  (β - 1) u / (gD)1/2

(3)

While hydraulic engineers are very familiar with the Froude number, a similar statement does not follow for the Vedernikov number, which continues to be largely ignored, or set aside, in hydraulic engineering practice (Ponce, 2002). To focus on the more familiar Froude number, the Vedernikov number may be expressed as follows:

V = (β - 1) F

(4)

Equation 4 shows that the Vedernikov number and the related flow instability are determined by the product of (β - 1) times the Froude number F. In practice, the Froude number does not vary through a wide range; therefore, (β - 1) is the controlling factor in flow instability.

According to theory, neutral stability, i.e., the threshold of flow instability, corresponds to V = 1. From Eq. 4, the Froude number for neutral stability is:

Fns  =  1 / (β - 1)

(5)

To unravel the physical meaning of flow instability, it is necessary to examine the true nature of β, the exponent of the discharge-area (Q - A) rating. Essentially, β is the ratio of kinematic wave celerity c_k to mean flow velocity u. In other words, β is the factor by which to multiply the mean flow velocity to obtain the kinematic wave celerity. For β ≡ 1, the kinematic wave celerity is equal to the mean flow velocity, averting flow instability.

The value of β is a function of type of boundary friction and cross-sectional shape. In practice, boundary friction may be either:

1. Laminar;

2. Mixed laminar-turbulent; or

3. Turbulent.

Turbulent flow may be expressed in terms of either the Manning or Chezy equations.

The cross-sectional shape may vary widely, and with it, the value of β. There are three asymptotic cross-sectional shapes, in terms of the applicable values of β (Ponce, 2014: Chapter 1):

1. Hydraulically wide, with constant wetted perimeter P;

2. Triangular, for which the top width T is proportional to the maximum flow depth d;

3. Inherently stable, with constant hydraulic radius R (Ponce and Porras, 1995).

Table 1 shows the values of β corresponding to selected combinations of boundary friction and cross-sectional shape. Following Eq. 5, the values of neutrally stable Froude number F_ns are shown in the last column. The practical range of β values shown is:  1 ≤ β ≤ 3, with the higher value (β = 3) corresponding to (hydraulically wide) laminar flow (i.e., surface flow on a plane) and the lower value (β = 1) to the inherently stable cross-sectional shape. We credit Liggett (1975) with pioneering the theory of the stable channel.

5.  PROPAGACIÓN DE ONDAS EN CANALES

The necessary and sufficient condition for roll wave formation may be analyzed using the theory of shallow wave propagation of Ponce and Simons (1977). These authors applied the method of linear stability to the set of equations governing unsteady open-channel flow, commonly referred to as the St. Venant equations. The analysis led to celerity and attenuation functions of various types of shallow-water waves, including kinematic, diffusion, mixed kinematic-diffusion, and dynamic (Ponce, 2014: Chapter 10). The findings are summarized in Figs. 6 to 8.

Figure 6 shows values of dimensionless relative wave celerity c_r* (ordinates) across a wide range of dimensionless wavenumbers σ_* (abscissas). This figure shows that c_r* ⇒ 0.5 as σ_* ⇒ 0 in the kinematic range (to the left); conversely, c_r* ⇒ (1/ F) as σ_* ⇒ ∞ in the dynamic range (to the right). As expected, for midrange values of dimensionless wavenumbers, c_r* is shown to vary strongly with wavenumber, the variation being increasingly marked as the Froude number decreases. Admirably, Fig. 6 confirms the validity of the Seddon formula toward the left of the graph and of the Lagrange formula toward the right.

Figure 6 also shows that for F = 2, c_r* is constant and equal to 0.5, which corresponds to the value of neutrally stable Froude number (V = 1) for a hydraulically wide channel with Chezy friction (Table 1). Therefore, for F = 2, all wave scales travel with the same celerity, confirming the validity of the Vedernikov theory.

Wave attenuation or amplification is produced when the value of c_r* varies across the dimensionless wavenumber spectrum. It can be shown that neither Seddon waves (kinematic) nor Lagrange waves (dynamic) are subject to (any amoiunt of) attenuation or amplification. Figure 6 describes wave attenuation for F < 2 (i.e., for a positive c_r* gradient) and wave amplification for F > 2 (for a negative c_r* gradient), again confirming the validity of the Vedernikov theory.

Figure 7 shows values of logarithmic decrement δ (in the ordinates) across a wide range of dimensionless wavenumbers σ_* (abscissas), for Froude numbers in the range 0.01 ≤ F ≤ 1.5. [The value F = 2 has zero attenuation; therefore, it could not be plotted on a log scale]. In direct correspondence with Fig. 6, in Fig. 7 wave attenuation is shown to be more marked as the Froude number decreases to F = 0.01. Predictably, the peak attenuation is seen to match the point of inflection in the c_r* vs. σ_* curves of Fig. 6. It is confirmed that wave attenuation decreases toward either extreme of the wavenumber spectrum, with -δ ⇒ 0 for both σ_* ≤ 0.001 (kinematic waves) and σ_* ≥ 1000 (dynamic waves).

Figure 8 shows values of logarithmic increment +δ (ordinates) (i.e., wave amplification) across a wide range of dimensionless wavenumbers σ_* (abscissas), for Froude numbers in the range 3 ≤ F ≤ 10. In direct correspondence with Fig. 6, wave amplification is shown to be somewhat more marked as the Froude number increases from F = 3 to F = 10. Predictably, the peak attenuation is seen to match the point of inflection in the c_r* vs. σ_* curves of Fig. 6. It is confirmed that wave amplification decreases toward either extreme of the wavenumber spectrum, with +δ ⇒ 0 for both σ_* ≤ 0.001 (kinematic waves) and σ_* ≥ 1000 (dynamic waves).

The foregoing analysis has confirmed that wave amplification will occur for Froude numbers F > 2, which corresponds with Vedernikov number V > 1 for the case of Chezy friction in a hydraulically wide channel. As shown in Table 1, for the case of Manning friction, the condition V > 1 corresponds to F > 1.5. In practice, flow instability, corresponding to a positive value of δ, may start to occur with Froude numbers as low as 1.5 (under Manning friction)

Practical experience suggests that the Vedernikov criterion (V ≥ 1) is necessary but not sufficient for the occurrence of roll waves. Roll waves do not occur all the time in steep channels where the instability criterion is met. In fact, roll waves are seen to be an unusual occurrence.

In summary, the Ponce and Simons (1977) theory of shallow wave propagation confirms Vedernikov's theory. Most significantly, for Froude numbers in the unstable regime, the value of the logarithmic increment δ (refer to Fig. 8) is not constant, peaking at the point of inflection of the dimensionless relative celerity vs. dimensionless wavenumber function (refer to Fig. 6). This fact clearly establishes a preferential dimensionless wavenumber range for the amplification of open-channel flow disturbances into roll waves.

6.  VERIFICACIÓN EXPERIMENTAL

Figure 8 shows the range of dimensionless wavenumbers where the wave amplification is likely to be stronger. In fact, there is a value of σ_* for which the logarithmic increment +δ is a maximum. Figure 8 shows that for F = 4 (the red curve), +δ_peak = 0.5 corresponds to σ_*@peak = 0.22.

The wave amplification is assured by the nature of the variation of the dimensionless relative wave celerity for F > 2 in Fig. 6. For F = 4, peak wave amplification occurs at the point of inflection of the red curve of Fig. 6, for which σ_*@inflection = 0.22. Note that this value is equal to the σ_*@peak = 0.22 of Fig. 8. Therefore, the existence of a preferential dimensionless wavenumber range for roll wave propagation is confirmed.

To verify the theory, Ponce and Maisner (1993) used the classical Brock laboratory flume data, developed at the California Institute of Technology. Brock (1967) measured crest depths and wave periods of roll waves under a wide range of flow conditions. Ponce and Maisner compared experimental dimensionless wavenumbers, calculated from wave periods measured by Brock, with dimensionless wavenumbers predicted by the theory. Note that the correct theory would show that the laboratory data plots at or near the peak of the curves shown in Fig. 8.

The results of the comparison are shown in Fig. 9. It is observed that Brock's data agrees reasonably well with the theory. All measured data is shown to plot near the peaks of the theoretical logarithmic increment vs. dimensionless wavenumber curves. Therefore, the roll-wave component of the theory of shallow wave propagation has been experimentally verified.

7.  DIFUSIÓN VS. AMPLIFICACIÓN DE LAS ONDAS

Roll waves, occurring under unstable flow, are one of the manifestations of unsteady flow in open channels (V > 1). The other are kinematic waves and dynamic waves occurring under stable flow (V < 1). Under unsteady flow, waves are created in an open channel by action of the boundary conditions and other flow irregularities. Once in the channel, the various scales (sizes) of waves will either: (a) attenuate, (b) hold their stage, or (c) amplify. Theory confirms that waves will: (a) attenuate for V < 1; (b) hold their stage for V = 1; and amplify for V > 1 (Section 3).

The Vedernikov criterion V = 1 is an absolute criterion and, therefore, independent of the type of friction or cross-sectional shape. It may be alternatively expressed as F_ns, i.e., the Froude number corresponding to V = 1 (Section 4). Table 1 shows that F_ns varies with the type of friction and cross-sectional shape. For instance, F_ns = 2 for Chezy friction in a hydraulically wide channel, F_ns = 4 for Chezy friction in a triangular channel, and F_ns = ∞ for an inherently stable channel, regardless of the type of friction. It is observed that F_ns increases with the decrease of β as the cross-sectional shape departs from hydraulically wide; in the limit, for β ⇒ 1, F_ns ⇒ ∞.

Whether a wave diffuses or amplifies will depend on: (1) the effect of boundary friction, and (2) the effect of cross-sectional shape. For V < 1, all waves will attenuate. The strength of the attenuation will depend on the Froude number (of the steady uniform flow) and the dimensionless wavenumber of the perturbation, with lower Froude numbers undergoing markedly greater attenuation (Fig. 7). Wave attenuation peaks near the midrange (and right of midrange) of dimensionless wavenumbers.

For V > 1, all waves will amplify. The strength of the amplification will depend on the Froude number and the dimensionless wavenumber of the perturbation, with higher Froude numbers undergoing somewhat greater amplification (Fig. 8). Wave amplification peaks near the midrange (and left of midrange) of dimensionless wavenumbers.

The strength of the wave diffusion or amplification is related to the rate of variation of dimensionless relative wave celerity with dimensionless wavenumber (Fig. 6). In the stable regime (V < 1), this rate of variation increases markedly as the Froude number decreases, increasing the rate of attenuation (Fig. 7). Conversely, in the unstable regime (V > 1), the rate of variation increases mildly as the Froude number increases, mildly increasing the rate of amplification (Fig. 8).

The exponent of the rating β conditions and produces attenuation or amplification at any scale in the wavenumber spectrum. Values of β ⇒ 5/3 for Manning friction (and β ⇒ 3/2 for Chezy) lead to wave amplification. This is because wave transport being nonlinear, the wave peaks are able to travel faster than the mean flow, with the wave faces poised to steepen and eventually break. Conversely, values of β ⇒ 1 (i.e., approaching 1) lead to wave attenuation, i.e., wave diffusion, because the wave peaks are traveling at nearly the same speed as that of the mean flow and, therefore, are poised not to steepen.

In conclusion, managing β (at the design stage) is a sure way to control flow amplification and thus, avert flow instability. A hydraulically wide channel features β ⇒ 5/3 for Manning friction (and β ⇒ 3/2 for Chezy); therefore, it is preeminently not conducive to wave diffusion. Moreover, a rectangular channel cross-section is that closest to hydraulically wide, with typical values of β in the range 3/2 to 5/3. Therefore, a rectangular cross section is also not conducive to wave diffusion. Values of β ≤ 4/3 for Manning (or β ≤ 5/2 for Chezy) are more conducive to wave diffusion since they raise the value of F_ns above 3 or 4 (Table 1), discouraging flow instability.

8.  CÁLCULO de β

It has been shown that β, the exponent of the discharge-flow area rating, a characteristic of the shape of the cross section, determines to a large extent whether the flow may become unstable or not. A value of β = 5/3, corresponding to a hydraulically wide channel (Manning), more readily leads to flow instability. As β decreases to 4/3, i.e., for a triangular channel (Manning), a larger Froude number is required to produce flow instability, in effect discouraging the latter. As β ⇒ 1, flow instability is no longer possible. Therefore, it is imperative that steep channels (under supercritical flow) be designed by paying close attention to β.

The online script onlinechannel15b calculates β for a prismatic channel of trapezoidal, rectangular, or triangular cross section (Fig. 10). The calculator requires the following input:

To demonstrate the online calculator, we consider two examples: (1) a rectangular cross section, and (2) a trapezoidal cross section; to pass an arbitrary discharge of Q = 50 m³/s. In Example No. 1, assume b = 5.8 m, y = 1.066 m, z₁ = 0, z₂ = 0, n = 0.025, and S = 0.057. The output from onlinechannel15b is shown in Fig. 11.

[Click on top of figure to display]

Fig. 11  Output from onlinechannel15b for Example No. 1.

The results are: Q = 50 m³/s; v = 8.088 m/s; F = 2.501; β = 1.607; F_ns = 1.646; and V = 1.519. Since V > 1, it is concluded that this flow condition may lead to flow instability. Note that for this rectangular cross section, β = 1.607 is near the top of the feasible range (1 ≤ β ≤ 5/3).

In Example No. 2, assume b = 1.2 m, y = 2.391 m, z₁ = 0.5, z₂ = 0.5, n = 0.025, and S = 0.057. The output from onlinechannel15b is shown in Fig. 12.

[Click on top of figure to display]

Fig. 12  Output from onlinechannel15b for Example No. 2.

The results are: Q = 50.03 m³/s; v = 8.735 m/s; F = 2.208; β = 1.4; F_ns = 2.497, and V = 0.884. Since V < 1, it is concluded that this flow condition will not lead to flow instability. In this case, the cross section is trapezoidal, and β = 1.4 is close to a central value within the feasible range (1 ≤ β ≤ 5/3).

We note that the choice of cross-sectional shape and channel dimensions with the objective of reducing β must be done judiciously. For example, a wide trapezoidal channel may not necessarily lead to a reduction in the value of β, because the increase in top width T will result in a decrease in hydraulic depth D, leading to an increase in the Froude and Vedernikov numbers.

9.  LA NATURALEZA DE β

The value of β is smaller not only for triangular channels, but also for narrow channels. Table 3 show calculations of β for six different test channels (See box below) to pass a reference discharge Q = 50 m³/s. To emulate the La Paz case study, we have assumed for these test channels a Manning's n = 0.025 and channel slope S = 0.057 (See Section 11).

Test channels: A rectangular channel. A trapezoidal channel. A triangular channel with the same side slopes (z1 = 1 and z2 = 1). A triangular channel with different side slopes (z1 = 0.5 and z2 = 1). A narrow and deep rectangular channel. A very narrow rectangular channel to pass Q /10 (the diffuser concept, see Section 10).

The following conclusiones are drawn from Table 3:

Conclusions: β is quite large (β = 1.607) for the rectangular channel (i = 1). β decreases somewhat (to β = 1.4) for the trapezoidal channel (i = 2). β decreases to 1.333 for the triangular channel with equal side slopes (i = 3). β decreases to 1.333 for the triangular channel with different side slopes (i = 4). β decreases substantially (to β = 1.356) for the narrow and deep channel (i = 5). β decreases substantially (to β = 1.231) for the very narrow channel (for Q /10) (i = 6).

For the given test cases, Table 3 shows that as β decreases, V also decreases. In particular, for test cases 2 to 6, the Vedernikov number remains V < 1, assuring flow stability and the consequent absence of roll waves. Moreover, for the very narrow section (i = 6), β reduces further to β = 1.269, with a corresponding V = 0.181, the latter seen to be a definitely low value. We conclude that β is of paramount importance in the hydraulic design of open channels to avoid flow instability and associated roll waves.

10.  CONTROL MECÁNICO DE LAS ONDAS DE ROLLO

The Vedernikov criterion is seen to be a necessary condition, but it may be not sufficient for the occurrence of roll waves (See Section 5). Since there is a preferential scale of dimensionless wavenumbers for roll-wave propagation (Fig. 8), whether a roll-wave event will occur may also depend on the upstream boundary condition, i.e., on the (dimensionless) scale of the incoming wave. Therefore, there is a randomness or stochasticity about the occurrence of a roll-wave event, which precludes it from happening too frequently or, otherwise, at regularly timed intervals.

In addition to the effective assessment of the value of the Vedernikov number, roll-wave events may also be controlled by mechanically attenuating (i.e., diffusing) the preferential wavenumbers if, by chance, they happen to be present. This can be accomplished by placing a momentum diffuser in the flow, at or near the upstream boundary. A momentum diffuser separates the flow into a series of narrow channels (Fig. 13). The additional boundary friction, coupled with the small aspect ratio (B /y) of the diffuser channel elements, cause the value of β to drop (Fig. 14). This fact is confirmed by the data of Table 3, line 6, for which β = 1.231 and V = 0.157.

In practice, a properly designed momentum diffuser will most likely require physical and/or mathematical modeling to adequately determine its hydraulic properties and size its geometric features.

11.  ESTUDIO DE CASO:  TRIBUTARIOS DEL RÍO LA PAZ, BOLIVIA

The findings of this article are complemented with a case study of two tributaries of the La Paz river, which flow across the city of Nuestra Señora de La Paz, commonly referred to simply as La Paz, in the department of the same name, Bolivia. The department of La Paz is shown in wine color, immediately east of Peru on the map to the right (Fig. 15). Note the approximate location of the watershed as the small blue dot inside the department (click here ⇒ Fig. 15 to display the map).

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The watershed of the La Paz river comprises 496.4 km² and features six (6) tributaries: (1) Choqueyapu, (2) Orkhojahuira, (3) Irpavi, (4) Achumani, (5) Jillusaya, and (6) Huayñajahuira (Fig. 15). In the rainy season 2018-19, comprising the period from October 2018 to April 2019, several roll-wave events have been documented in the Achumani and Huayñajahuira rivers. The details are shown in Table 4.

The channels are built using stone-masonry walls and unlined channel beds; see Figs. 16 and 17. Most channel reaches have instream vertical drops for slope reduction and grade control (Table 4). An approximate value of wave celerity c = 10.5 m/s has been obtained from available videos of roll-wave events. Therefore, assuming a value of β ≈ 1.5, the mean velocity of the flow (now referred to as v) may be approximated as follows:  v = c / β = 7 m/s.

Pending additional studies, the Manning friction coefficient has been estimated at n = 0.030 for the Achumani river (Fig. 16) and n = 0.025 for the Huayñajahuira river (Fig. 17) (Chow, 1959).

Table 5 shows calculations of β and V  for seven (7) selected channel reaches (stations) with relatively steep local slopes (S_local ≥ 0.034; Table 4). In order to compare events of similar frequency, the flow depths were set to pass the same discharge, chosen as Q₂₅ = 106.1 m³/s for the Achumani river and Q₂₅ = 30.97 m³/s for the Huayñajahuira river (Plan Maestro de Drenaje, 2007). The calculated mean flow velocities shown in Table 5 are confirmed to be around v = 7 m/s, a clear indication that channel friction has been properly estimated.

It is seen that all the channel reaches under consideration have Vedernikov numbers V > 1, confirming the hydraulic conditions under which roll waves are likely to occur. In addition, it is shown that all reaches have β > 1.6, i.e., clearly near the top of the feasible range (1 ≤ β ≤ 1.67).

The existence of channel drops adds a bit of complexity to the analysis, since the flow could no longer be considered to be strictly one-dimensional. Therefore, the present analysis is only an approximation, subject to detailed field verification when appropriate.

12.  CONCLUSIONES

The theoretical foundations and relevant experience with open-channel flow instability are examined with the objective of controlling roll waves. The latter seem to recur with relative frequency in channelized rivers where the Vedernikov number has increased, due to the channelization, above the threshold V = 1 (click on Fig. 18 at the end of this article to watch a video of a pulsating wave event). Channelization with rectangular cross sections typically increases the value of β, the exponent of the discharge-flow area rating, triggering an increase in the Vedernikov number above the threshold of flow instability (V = 1).

It is shown that the choice of β (at the design stage) is of paramount importance in the control of roll waves. Several examples show conclusively the existence of a definite relation between β and V, namely, that as the cross-sectional shape departs from rectangular towards triangular, β decreases accordingly, lowering the value of V. A sufficient decrease in β will cause V  to drop below the threshold of instability.

Shallow wave propagation theory reveals the existence of a preferential scale for the amplification of surface perturbations in the unstable regime (V ≥ 1) (Fig. 8). Therefore, the Vedernikov criterion may not be sufficient to produce roll waves in all cases. The latter may depend on conditions at the upstream boundary, which are likely to be of a stochastic nature. The provision of a momentum diffuser is introduced; the diffuser is designed to attenuate roll waves at the point where they may originate, i.e., at or near the entrance to the channelized stream. Further analysis and testing is recommended to carry the diffuser concept to development stage.

The methodology developed herein is applied to two channelized rivers in La Paz, Bolivia, where large roll waves (pulsating waves) have been shown to recur with increasing frequency. The analysis confirms that where roll waves have occurred in these channelized rivers, they have done so under β > 1.6 and V > 1. The results underscore the promise of β analysis in the hydraulic design of channelized rivers with the objective of controlling roll waves.

REFERENCES

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Powell, R. W. 1948. Vedernikov's criterion for ultra-rapid flow. Transactions, American Geophysical Union, Vol. 29, No. 6, 882-886.

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Vedernikov, V. V. 1945. Conditions at the front of a translation wave disturbing a steady motion of a real fluid. Doklady Akademii Nauk USSR, 48(4), 239-242.

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Fig. 18  Roll-wave event recorded in the Huayñajahuira river, on April 30, 2019.

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