This article contains a concise overview of flow instability (roll waves) in clear water. Attention is focused on the aspects that may be of direct interest to students in the field of open channel flow and engineers working with steep channels. Basic equations (Saint Venant equations) are initially introduced and some less known aspects related to their derivation are highlighted. The growth or decay of small disturbances "in time" is shown according to both Vedernikov and Liggett approaches. The subsequent analysis "in space" introduced by Montuori is then presented. The mathematical analysis of fully developed roll waves is accounted for according to Dressler's piecewise approach. Finally, two brief references are made to results concerning an "enhanced" form of the Saint Venant equations and associated experimental data.

1.  INTRODUCTION

The basic classification of free surface flow as subcritical (tranquil) or supercritical (shooting) is well known to engineers since their undergraduate course in Hydraulics. The distinction between subcritical and supercritical flow is based on the Froude number. When F_r < 1 the flow is subcritical, when F_r > 1 the flow is supercritical.

Lesser known is the existence of a second classification which is based on the characteristic of the flow to be stable or unstable. When the flow becomes unstable, pulsating waves, known as roll waves, develop. These conditions are generally related to steep artificial channels. Roll waves have been documented since 1910 (Cornish ***) and have been studied since then. For a concise history of roll waves see Hager (2002).

When completely developed, roll waves consist of two parts: a "head" or "front" formed by a bore of tumbling water which may entrain air, and a "tail" of smoothly flowing fluid.

The basic distinction between stable and unstable flow is based on a nondimensional group known as the Vedernikov number.

dP      m Ve =  ( 1  _  R  _____ ) ____  Fr                          dA       n

(1)

2.  THE BASIC EQUATIONS

Keulegan (1942) has an interesting discussion on the differences arising from the application, to one dimensional steady open channel flows, of the energy method (with the Coriolis coefficient α):

∂h            V    ∂V                  d              τo                    τo                          V 2 ____ + α  ___  ____  =   So _ ____ ( N1 ____ ) _ ( N2 ________ ) = So _ λe  ______ ∂s            g     ∂s                  ds           g ρ               g ρ R                        g R

(2)

and of the momentum method (with the Boussinesque coefficient β)

∂h            V    ∂V                 τo                    V 2 ____ + β ___  ____  =   So _ ____ = So _ λ _______ ∂s            g    ∂s                  g ρ                 g R

(3)

Previous equations make clear that λ_e is an energy loss coefficient while λ is a resistance coefficient.

Keulegan & Patterson (1943) made a quite lucid application of the principles of conservation of mass and momentum to the case of unsteady flows. Their usual well established expression for the continuity equation is:

∂ (VA)        ∂A _______ + _____  =   0    ∂s           ∂t

(4)

In the dynamic equation, they derived an extra term, depending on the velocity profile:

∂h           V   ∂V          1    ∂V                     V  ∂V ____ + β  ___  ____  +  ___  ____ + ( 1 _ α ) ___ ____  =  ( So _ Sf )  ∂s           g    ∂s          g     ∂t                     A   ∂t

(5)

which is usually neglected.

3.  LINEAR STABILITY ANALYSIS OF THE BASIC FORM OF THE SAINT VENANT EQUATIONS

The stability analysis of the "basic" Saint Venant equations may be pursued using both the partial differential form:

∂ (VA)        ∂A _______ + _____  =   0    ∂s           ∂t

(4)

∂h         V   ∂V          1    ∂V ____ +  ___  ____  +  ___ ____  =  ( So _ Sf )  ∂s          g    ∂s         g    ∂t

(6)

or the characteristic form which for a wide channel reads:

ds ____  = V ± c  dt

(7)

d ____ ( V - w ) =  g ( So _ Sf )  dt

(8)

While results are similar, each approach seems to be useful to convey certain information in a more easy and understandable fashion than the other.

Common to both method is the definition of the relative Lagrange celerity w:

g A w = ( _____ ) 1/2             B

(9)

3.1  Vedernikov analysis in time

Vedernikov's (1945) and (1946) analysis considered a cyclindrical channel of any shape. He studied the basic form of the Saint Venant equation in which he took the velocity V(A,t) and the abscissa s(A,t) as unkwowns, and the cross-sectional area A and the time t as independent variables. With the new variables, the change in time of the inverse of the free surface slope becomes:

∂     ∂                        ∂ω ___ ___  s ( A, t ) = B ____  ∂t   ∂h                        ∂A

(10)

He then obtained the expression for the change in time of the inverse of the free surface slope:

∂ω        d     ∂s       3      g                  g   ∂s      i               2 m Mo Vo ____ = ____  ____ = ___  ______ No + ___  ____  ____ [ n  _ _____________ ]  ∂A       dt     ∂A      2    AoBo             2   ∂A     Vo                  Ao                                                                                        ( g ____ )1/2                                                                                               Bo

(11)

where:

dP Mo = 1 _ R  _____                     dA

(12)

and:

1    Ao    dB No = 1 _  ___  ____  _____                   3    Bo    dA

(13)

N_o is a specific term of Vedernikov's analysis which arises due to the generality of the cross section he considered. For the case of a wide channel, N_o = 2/3 because the term:

Ao    dB ____  _____  =  1  Bo    dA

(14)

N_o is a function of the cross section, and it is always positive.

In order for the descending wave to grow, it is necessary and sufficient that the term in parentesis be positive, which is ensured for V_e > 1.

2.2  Liggett analysis in time

Liggett (1975) performed a stability analysis of the characteristic form of the Saint Venant equations. He found that instability may develop on both characteristics according to the following condition:

- 1 < Ve < 1

(15)

V_e is basically a function of channel shape and frictional characteristics. In case the flow is turbulent, V_e also depends on the type of resistance formula.

Liggett (1975) analyzed the effect that different shapes have of the limiting Froude number. For a rectangular section, the limiting Froude number increases linearly according to the following equation:

n        B + 2h Fr  ≤   ___  +  _________             m            B

(16)

which for Manning formula becames:

3            h Fr  ≤  ___ +  3  ___            2            B

(17)

while for the Chezy formula it becames:

h Fr  ≤  2 + 4  ___                      B

(18)

For triangular section:

n Fr  ≤  2  ___               m

(19)

For flow over a plane (wide hydraulic channel) the limiting Froude number is constant:

n Fr ≤ ___         m

(20)

For the case of a wide hydraulic channel a more thoroughly analysis was performed by Ponce and Simons (1977).

For circular shapes, expressing the depth h as a function of the opening angle ϑ it is obtained:

n        2ϑ (1 _ cos(2ϑ)) Fr  ≤  ___  _____________________            m     sin(2ϑ) _ 2ϑ cos(2ϑ)

(21)

Circular (closed) shapes present an interesting feature, while for open shape sections the only possible instability is related to the downstream characteristic, in case of circular shape while the section is approaching saturation there is the possibility to develop instability along the backward characteristics.

Liggett (1975) also showed that instability may arise as a consequence of a rapid rise of the water level. The Liggett (1975) number for the stability of a rapidly increasing level is:

g C Von n                 _____________ ( 1 - Ve ) dh              Rom co Vo ____  <  _________________________ dto                   g         B'                 3  _____  _  ____                      co2        Bo

(22)

This aspect has been confirmed by Sjoberg (1982).

3.3  Montuori analysis in space

The primary goal of stability analysis is to investigate which are the conditions leading to the decay or growth of small disturbances. in this sense, time is the most relevant parameter. It follows that evolution in time means also evolution in space. When the attention is focused toward experiments, aimed to verify theoretical results, or toward application to real engineering cases, the attention shifts from time to space.

Montuori (1965), having assumed a very simple relationship linking time to space, performed his analysis integrating the stability equation over space. He identified an expression for the length (time) needed to develop a roll wave that depended on an undefined parameter. He then defined a stability curve which was a function of a parameter that he developed using empirical data from Russian experiments. Montuori's graph allows, once geometrical and hydraulic parameters concerning the channel are determined, to establish which is the minimum length required to develop roll waves.

4.  GALILEAN TRANSFORMATION OF THE BACKWATER EQUATION

Developed roll waves may be seen as a train of similar waves all moving at the same celerity. Each wave may be considered as composed by two pieces: a continuous profile, which increases from upstream to downstream, and a bore (or moving hydraulic jump). The bore connects the highest depth of an upstream profile with the lowest depth of the downstream profile.

When a Galilean transformation (refer to a reference frame moving at the same speed of the wave) is applied, the system appears steady. The wave celerity is the relative velocity used to perform the Galilean transformation. The idea behind this approach is well exemplified by Thomas (1937) analogy of a conveyor belt.

This approach has been used by Thomas (1937), Craya (1952), and Dreessler (1947). Thomas (1937) and Craya (1952) inserted the "relative velocity" directly into the steady backwater equation (Belanger equation), while Dressler (1947) started from the Saint Venant equations. Both procedures produce the same equation:

(Uy _ q)  |Uy _ q|               So  _  _________________ dy                             C2 y3 ___  =  _________________________ ds                             q2                       1 _  ________                                g y3

(23)

Profiles representing solution to Eq. 23 have been extensively studied by Thomas (1937). A first result is that there is no periodic solution. This conclusion has been confirmed by subsequent studies. All procedures related to the steady backwater equation may be applied to the one derived in the moving reference frame.

The continuous profile of the roll waves may be seen as composed by a supercritical flow upstream, becoming subcritical downstream. Thomas found that the only possibility to have a continuous profile was when the slope was critical.

Dressler (1949) picked up where Thomas had stopped. From the mathematical point of view. the presence of the hydraulic jump adds one extra unknown to the problem, therefore, an extra equation must be taken in. Dressler (1949) identifies the needed equation in the one describing the hydraulic jump.. Dressler's result was the standard hydraulic jump equation written in terms of relative velocity. In other words, Dressler (1949) "built" a roll wave matching two consecutive profiles identified by Thomas with a shock wave condition. Alternate depths at the shock wave have to match the lowest and highest depth of the continuous profile.

Studying the system composed by the continuous profile equation and the shock wave equation, Dressler found that the system had no solution (no roll waves would develop) if: (1) the flow was frictionless; and (2) the friction was larger than a given value.

Dressler (1953) also observed that the form of the resistance law plays an essential role. When the resistance term depends only on velocity, and dependency on depth is not included, the flow is always stable. The same result is obtained if resistance depends only on depth (Coulomb type friction), and the dependency on velocity is not included.

5.  LINEAR STABILITY ANALYSIS OF ENHANCED FORM OF THE SAINT VENANT EQUATIONS

Needham and Merkin (1984) studied the case of a hydraulically wide channel, adding an "enhancement" term to the basic momentum equation. The "enhancement" term takes the form of a turbulent-viscous dissipation term.

∂h         V  ∂V         1    ∂V                                 ∂2V ____ +  ___ ____  +  ___ ____  =   ( So - Sf ) = ν  ______  ∂s         g    ∂s         g    ∂t                                  ∂s2

(24)

The "enhancement" term is an empirical term intended to represent the effect of energy dissipation by shearing normal to the flow. The "enhancement" term does not affect the general criterion for flow stability, but it allows the presence of periodic solutions. The inclusion of the extra term makes the limiting Froude number for turbulent flow also dependent on the Reynolds number.

6.  EXPERIMENTAL DATA

The roll wave experiments by Koloseus and Davidian (1966) and Brock (1967) are reported here.

6.1  Koloseus and Davidian experiments

Koloseus and Davidian (1966) investigated the length required to develop a roll wave. Experiments were performed on surfaces covered with artificial roughness. For each given discharge, they increased the slope of the flume and determined which was the minimum distance where the roll wave appeared to develop. Koloseus and Davidian (1966) experiments were concentrated around a narrow range of Vedernikov numbers close to one. They may not be readily compared with the ones used by Montuori, which referred to real hydraulic structures with a very large range of Vedernikov numbers.

Results of experiments by Koloseus and Davidian.s (1966) show the following features:

1. Roll waves increase the overall channel resistance, decreasing the carrying capacity;

2. The length required for the development of roll waves is a function of: (a) flow depth, (b) channel shape and roughness, and (c) level of initial instability;

3. The length required for the development of roll waves increases with increasing discharge;

4. The length required for the development of roll waves decreases with increasing roughness; and

5. For a given discharge, the length required for the development of roll waves increases as the slope decreases.