1. INTRODUCTION Over the years, several investigators have attempted to clarify the phenomenon of propagation of shallow waves in open channel flow. Perhaps the most illuminating of these studies is that reported in the classical paper by Lighthill and Whitham, which analyzed in detail the concept of kinematic wave and contrasted it to the dynamic wave (5). The Lagrange formula for the celerity of shallow gravity waves has also been a subject of considerable attention in the literature (3). Despite the progress made in the understanding of the physical phenomenon, a coherent theory that accounts for celerity as well as attenuation characteristics has yet to be formulated. In this respect, the theory of linear stability can be used as an effective tool not only to furnish a first approximation analysis but also to provide valuable insight into the underlying physics of the problem (6). The analysis presented herein endeavors to apply the theory of linear stability to the set of equations governing the motion in open channel flow. The equations are the so-called Saint Venant equations, which are analytical expressions of the principle of mass and momentum conservation (1). The conclusions relate to the magnitude of celerity and attenuation of various types of shallow water waves, expressed as a function of the Froude number of the steady uniform flow and a dimensionless wave number of the unsteady component of the motion. 2. GOVERNING EQUATIONS The governing equations for the one-dimensional unsteady flow in broad channels of rectangular cross section, expressed per unit of channel width, are the equation of continuity (4):
and the equation of motion:
in which u = velocity averaged in a vertical section, d = depth of flow; g = acceleration of gravity; S_{f} = friction slope; and S_{o} = bed slope. The friction slope S_{f} is directly related to the bottom shear stress τ by the expression (2):
in which γ is the unit weight of water.
In the usual manner of stability calculations, Eqs. 1 and 2 must satisfy the unperturbed flow for which u = u_{o}; d = d_{o}; and τ = τ_{o}, as well as the perturbed flow for which u = u_{o} + u' ; d = d_{o} + d' ; and Substitution of the perturbed variables in Eqs. 1, 2, and 3 yields, after linearization (5):
in which
The boundary shear stress τ can be related to the mean velocity by
in which
where C_{f} is the Chezy coefficient; and ρ = the density of water. Considering Eq. 7, Eq. 5 can be rewritten as
For most practical applications, the Manning equation is preferred over the Chezy equation. The latter, however, has the advantage of nondimensionality, and it is for this reason that it is used here. Furthermore, the constant C_{f} is consistent with the small perturbation assumption. 3. WAVE MODELS The propagation of shallow water waves is controlled by the balance of the various forces included in the equation of motion. In Eq. 2, the first term represents the local inertia term, the second term represents the convective inertia term, the third term represents the pressure differential term, and the fourth accounts for the friction and bed slopes. Various wave models can be construed, depending on which of these four terms is assumed negligible when compared with the remaining terms. The following chart provides a ready reference to the various wave models that are recognized. Term I II III IV
Wave model and terms used to describe it are: (I) Kinematic wave IV; (2) diffusion wave III + IV; (3) steady dynamic wave II + III + IV; (4) dynamic wave I + II + III + IV; and (5) gravity wave I + II + III.
4. SMALL PERTURBATION ANALYSIS In order to provide for a convenient way to take explicit account of the various wave models, Eq. 9 is recast as
in which l, a, p, and k are integers that can take values of 0 and 1 only, depending on which terms of Eq. 9 are used to describe the motion. The solution for a small perturbation in the depth of flow is postulated in the following exponential form:
in which d' is a small perturbation to d_{o}; d_{*} is a dimensionless depth amplitude function; σ_{*} is a dimensionless wave number β_{*} is a dimensionless complex propagation factor, and x_{*} and t_{*} are dimensionless space and time coordinates such that
in which L = the wavelength of the disturbance; and T = the period. The value L_{o} = the horizontal length in which the steady uniform flow drops a head equal to its depth, and it is defined as L_{o} = d_{o}/S_{o}. The dimensionless celerity of the disturbance is given by
The wave attenuation follows an exponential law in which the amplitude at a given time t = the initial amplitude at time t_{o} multiplied by (e^{ β*I t* }), in which t_{* } = (t - t_{o}) u_{o} / L_{o}. When comparing wave amplitudes after one propagation period, the value of t_{* } is t_{* } = T u_{o} / L_{o}, or likewise, t_{* } = 2 π / |β_{*}_{R}|. The logarithmic decrement δ is defined as δ = β_{* } T u _{o} / L_{o}, or δ = 2 π β_{*}_{I} / |β_{*}_{R}| (10). The value of δ is a measure of the rate at which the unsteady component of the motion changes upon propagation. For δ positive, amplification sets in; for δ negative, the motion attenuates and dies away. The depth disturbance is associated with a velocity disturbance of the form
in which u_{*} is a dimensionless velocity amplitude function. The substitution of Eqs. 11 and 18 into Eqs. 4 and 10 yields, respectively:
in which
Equations 19 and 20 constitute a homogeneous system of linear equations in the unknowns u_{*} and d_{*}. For the solution to be nontrivial, the determinant of the coefficient matrix must vanish. Accordingly,
Equation 22 is the characteristic equation governing the propagation of small amplitude water waves. In the treatment that follows, successive simplifications will be made to conform to the various types of wave models considered in the previous section.
5. KINEMATIC WAVE MODEL In the kinematic wave model, the inertia and pressure terms are negligible as compared to the friction-bed slope term. Accordingly, in Eq. 22, l = a = p = 0; and k = 1, resulting in
Since all the imaginary terms have dropped out in Eq. 23, β_{*}_{I} = 0; and β_{*}_{R} = β_{*}. The logarithmic decrement δ_{k} , is then
and the dimensionless celerity of the kinematic wave is
Equations 23-25 warrant the following conclusions regarding kinematic wave propagation: (1) Since Eq. 23 is of the first order, kinematic waves propagate only in the downstream direction; (2) the celerity of a kinematic wave is independent of F_{o} and σ_{*} and equal to 1.5 times the mean flow velocity; and (3) the attenuation of a kinematic wave is zero, i.e., a kinematic wave propagates downstream without dissipation.
6. DIFFUSION WAVE MODEL In the diffusion wave model, the inertia terms are considered negligible, but the pressure term is taken into account in the calculations. Accordingly, in Eq. 22, l = a = 0; p = k = 1, resulting in
and
and the celerity of the diffusion wave is
The logarithmic decrement of the diffusion wave is
The following conclusions are derived for diffusion waves: (1) Because Eq. 26 is of first order in β_{*}, diffusion waves propagate downstream only, and their celerity is independent of F_{o} and σ_{*} and equal to 1.5 the mean flow velocity; and (2) diffusion waves attenuate as they propagate downstream, and the rate of attenuation is controlled by the dimensionless wave number σ_{*}. The larger the dimensionless wave number, the larger the attenuation.
7. STEADY DYNAMIC WAVE MODEL In the steady dynamic wave model, the convective inertia term is brought into the problem, and the local inertia term is neglected. Accordingly, in Eq. 22, l = 0; and a = p = k = 1, resulting in
and
or
From Eq. 32, the celerity of the steady dynamic model is
The logarithmic decrement of the steady dynamic model is
The following conclusions are derived regarding the steady dynamic model: (1) The propagation of the steady dynamic wave is in one direction, since Eq. 30 is of first order in β_{*}; and (2) the celerity and attenuation characteristics are functions of the Froude number of the steady uniform flow F_{o} and the dimensionless wave number σ_{*}.
8. DYNAMIC WAVE MODEL In the dynamic wave model, all terms in the equation of motion are considered. Thus, in Eq. 22, l = a = p = k = 1. It follows that
Equation 35 is of the second order, resulting in two roots. Physically speaking, dynamic waves propagate along two characteristic paths, which can face either: (1) one upstream and another downstream; or (2) both downstream. In the case of propagation in different directions, it is expedient to define as primary wave the wave that travels downstream, and identify its celerity and logarithmic decrement by c_{*}_{1} and δ_{1}, respectively. The secondary wave is defined as the wave traveling upstream with celerity c_{*}_{2} and logarithmic decrement δ_{2}. In the case of both waves traveling downstream, primary wave is the faster wave, and secondary wave is the slower wave. The solution of Eq. 35 is
in which
An expression of the same type as Eq. 37 is referred to in the literature as kinematic flow number (9). The complex square root argument can be expressed in polar form as
in which
and the root of the complex argument is
and by use the half-angle relations
in which use has been made of the fact that because ζ ≥ 0, θ/2 falls in the first quadrant. Since
it follows that
or
From Eq. 47, the following two roots obtained:
Now call
and
and the celerity and attenuation functions are given by: For the primary wave:
For the secondary wave:
9. GRAVITY WAVE MODEL In the gravity wave model, the friction and bed slope terms are excluded from the momentum equation. It follows that, in Eq. 22, l = a = p = 1; and k = 0, resulting in
Since Eq. 56 is of the second order in β_{*} and contains no imaginary terms, gravity waves have two characteristic directions and are not subject to attenuation. Solving for β_{*}_{R} in Eq. 56:
and the celerity of a gravity wave is given as follows:
or in the more usual way of expressing it (Lagrange formula):
Gravity waves propagate along two characteristic directions. In subcritical flow, one direction is downstream. with celerity c_{1} = u_{o} + (g d_{o})^{1/2}, and another is upstream, with celerity c_{2} = u_{o} - (g d_{o})^{1/2}. In critical flow, c_{1} = 2u_{o}; and c_{2} = 0. In supercritical flow, both waves travel downstream, with celerities c_{1} = u_{o} + (g d_{o})^{1/2}; and c_{2} = u_{o} - (g d_{o})^{1/2}. A summary of the propagation characteristics of the various waves described is given in Table 1. The celerity shown in Table 1 is the relative celerity c_{*}_{ r} , in which
10. ANALYSIS OF DYNAMIC MODEL Equations 52-55 enable the calculation of the propagation characteristics of the dynamic wave model, as a function of the Froude number F_{o} of the steady uniform flow and the dimensionless wave number σ_{*} of the sinusoidal perturbation superimposed on the steady uniform flow. Figure 1 shows the calculated values of the dimensionless relative celerity c_{*}_{r} versus the dimensionless wave number σ_{*} for Froude numbers between 0.01-10. The following conclusions are derived from this figure:
A corollary resulting from Fig. 1 is that for F_{o} = 2, c_{*}_{r} = 0.5 for all values of σ_{*}. Thus, at F_{o} = 2, all waves, kinematic, dynamic, and gravity, have celerities equal to the kinematic value. The conclusions of Fig. 1 regarding the limiting value of c_{*}_{r} , in the kinematic and gravity bands can be obtained analytically based on limit theory. It can be shown that as σ_{*} → ∞ , c_{*}_{r} → 1/F_{o} , and as
The calculation of the logarithmic decrements
δ_{1} and δ_{2} as a function of σ_{*} and F_{o}, by means of
Table 2 summarizes the conclusions of the foregoing paragraph.
Primary Wave Attenuation Analysis, F_{o} < 2. Figure 2 depicts the variation of the logarithmic decrement δ_{1} for F_{o} < 2, as a function of σ_{*}. The following conclusions are drawn: (1) The logarithmic decrement δ_{1} is maximum (in absolute value) at a value of σ_{*} corresponding to the point of inflexion of the c_{*}_{r} versus σ_{*} curve (Fig. 1). The value of σ_{*} for which δ_{1} is a maximum (in absolute value) decreases with increasing F_{o}; and (2) the logarithmic decrement δ_{1} is minimized (in absolute value) at both ends of the σ_{*} spectrum. As σ_{*} → 0, δ_{1} → 0, corresponding to the kinematic wave case; as σ_{*} → ∞, δ_{1} → 0, corresponding to the gravity wave case. A general conclusion derived from Fig. 2 relates to the fact that for F_{o} < 2, primary waves will be subject to very strong attenuation in the dynamic band, and to very weak attenuation in the gravity and kinematic bands.
Primary Wave Amplification Analysis, F_{o} > 2. Figure 3 depicts the variation of the logarithmic decrement δ_{1}, for F_{o} > 2, as a function of σ_{*}. The following conclusions are drawn: (1) The logarithmic decrement/increment δ_{1} has a maximum positive value at a value of σ_{*} corresponding to the point of inflexion of the c_{*}_{r} versus σ_{*} curve (Fig. 1). The value of σ_{*} for which δ_{1} is a maximum decreases with increasing F_{o}; and (2) the logarithmic decrement δ_{1} is minimized at both ends of the σ_{*} spectrum. As σ_{*} → 0, δ_{1} → 0 corresponding to the kinematic wave case; as σ_{*} → ∞, δ_{1} → 0 corresponding to the gravity wave case. A general conclusion derived from Fig. 3 relates to the fact that for F_{o} > 2, primary waves will be subject to very strong amplification in the dynamic band, and to very weak amplification in the gravity and kinematic bands.
Secondary Wave Attenuation Analysis. Figures 4 (a) and 4 (b) depict the variation of the logarithmic decrement δ_{2} for 0.01 ≤ F_{o} 10 as a function of σ_{*}. The following conclusions are drawn: (1) In subcritical flow, F_{o} < 1, the attenuation of the secondary wave is very strong, and the strength decreases as σ_{*} increases in the gravity band (large σ_{*}); (2) at critical flow, F_{o} = 1, the attenuation of the secondary wave is very strong, with a minimum around the center of the dynamic band (intermediate σ_{*}); and (3) in supercritical flow, F_{o} > 1 the attenuation of the secondary wave has the pattern shown. For F_{o} ≥ 2, δ_{2} decreases as σ_{*} increases.
11. DYNAMIC WAVE PROPAGATION The propagation of a dynamic wave has been shown to be a function of two dimensionless parameters, the wave number σ_{*} and the Froude number F_{o}. The wave number σ_{*} can be interpreted as a ratio of two lengths L_{o} and L, in which L is the wavelength and L_{o} is the horizontal length in which the steady uniform flow drops a head equal to its depth. The square of the Froude number F_{o} is the ratio of twice the velocity head to the flow depth, where the velocity head h_{u}_{o} is given by
For F_{o} < 2, the celerity of a dynamic wave that propagates downstream is larger than the kinematic wave celerity, and smaller than the gravity wave celerity. In fact, the kinematic wave celerity constitutes a lower bound to which the dynamic value tends as the wave number σ_{*} decreases. Conversely, the gravity wave celerity is an upper bound to the dynamic wave celerity as the wave number σ_{*} increases. To place this conclusion in the proper perspective it is perhaps of interest to quote here from Stoker's work [(8), p. 486], regarding the celerities of small disturbances and progressive waves:
In reviewing Stoker's work, it is apparent that his reference to a small disturbance is to a wave with a large σ_{*} (gravity wave) and to progressive wave to that with a small σ_{*} (kinematic wave). He proceeds to further elaborate on the limitations of methods of water routing based solely on the kinematic approach, for the cases in which the dynamic effects cannot be disregarded. For F_{o} ≥ 2, the celerity of a dynamic wave that propagates downstream is smaller than the kinematic wave celerity and larger than the gravity wave celerity. In fact, the kinematic wave celerity is an upper bound to the dynamic value, whereas the gravity wave celerity is a lower bound to the dynamic value. A significant conclusion regarding dynamic wave propagation can be obtained from the summary presented in Table 2. For primary waves, F_{o} = 2 is the threshold dividing the attenuation (F_{o} < 2) and amplification (F_{o} > 2) tendencies. For secondary waves, however, F_{o} = 1 is the threshold dividing the propagation upstream (F_{o} < 1) or downstream (F_{o} > 1) for gravity waves. Thus, F_{o} = 2 is verified to be as important a threshold value as F_{o} = 1 in describing the dynamics of open channel flow phenomena.
12. SUMMARY AND CONCLUSIONS The propagation characteristics of various types of shallow water waves in open channel flow are calculated on the basis of linear stability theory. The celerity and attenuation functions of kinematic, diffusion, steady dynamic, dynamic, and gravity waves are derived. For the most general case, i.e., the dynamic wave model, the propagation characteristics are expressed as a function of the steady uniform flow Froude number and the dimensionless wave number of the unsteady component of the motion. For the dynamic model, the wave number spectrum is divided into three bands: (1) A gravity band corresponding to large wave number, in which the wave celerity is the gravity wave celerity; (2) a kinematic band corresponding to a small wave number in which the wave celerity is the kinematic wave celerity; and (3) a dynamic band corresponding to midspectrum values of the wave number, in which the wave celerity falls between the gravity and kinematic celerity values. Primary dynamic waves propagate downstream, and they attenuate for F_{o} < 2 and amplify for F_{o} > 2. At the F_{o} = 2 threshold, primary dynamic waves neither attenuate nor amplify. For F_{o} ≤ 1, secondary dynamic waves either propagate upstream or downstream, depending on the wave number. At F_{o} = 1, secondary dynamic waves remain stationary or propagate downstream, depending on the wave number. For F_{o} > 1, secondary waves propagate downstream. Secondary waves attenuate throughout the entire wave number spectrum.
13. APPLICATIONS The analysis of the foregoing sections provides an appropriate framework for the systematic study of shallow water waves in open channel flow. Numerous applications are envisioned, among which some of the most important are:
APPENDIX I. REFERENCES
APPENDIX II. NOTATION The following symbols are used in this paper: A = parameter, defined by Eq. 39; a = integer; B = parameter, defined by Eq. 40; C = parameter, defined by Eq. 41; C_{f} = Chezy coefficient; D = parameter, defined by Eq. 50; d = depth of flow; E = parameter, defined by Eq. 51; F_{o} = Froude number of steady uniform flow; f = friction factor, defined by Eq. 7; g = acceleration of gravity; h_{u} = velocity head; k = integer; L = wavelength; L_{o} = length in which the steady uniform flow drops a head equal to its depth; l = integer; p = integer; S_{f} = friction slope; S_{o} = bed slope; T = wave period; u = mean velocity; β_{*}_{I} = amplitude propagation factor; β_{*}_{R} = dimensionless frequency; γ = unit weight of water; δ = logarithmic decrement; ζ = a type of kinematic flow number, as defined by Eq. 37; θ = parameter, defined by Eq. 43; σ_{*} = dimensionless wave number; τ = bottom shear stress; and ∞ = infinity. Subscripts d = diffusive wave; g = gravity wave; I = imaginary; k = kinematic wave; o = steady uniform flow; R = real; r = relative to the main flow; s = steady dynamic wave; 1 = primary dynamic wave; and 2 = secondary dynamic wave. Superscripts ' = perturbed variable; and _{*} = dimensionless function. |
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