10.1 GOVERNING EQUATIONS
Three conservation principles are applicable in openchannel flow:
Steady gradually varied flow combines the statements of steady conservation of mass and conservation of energy (Chapter 7). Unsteady gradually varied flow combines the statements of unsteady conservation of mass and conservation of momentum (Table 101). Thus, unsteady gradually varied flow differs from steady gradually varied flow in its description of the temporal variation of the flow variables (discharge, stage, flow depth, mean velocity, and so on). In practice, steady gradually varied flow is simply referred to as "gradually varied flow" (GVF), while unsteady gradually varied flow is commonly referred to as "unsteady flow" (UF).
Figure 101 depicts the forces acting on a control volume. One body force and two surface forces are shown. The body force is the component of the gravitational force resolved along the direction of motion (W sin θ). The surface forces are: (1) the force due to the pressure gradient (due to the difference in flow depths), ΔP = P_{2}  P_{1}, and (2) the force developed along the bottom boundary due to friction (F_{f} ). When these three forces are in equilibrium along the direction of motion, the flow is steady from the force standpoint. When the three forces are NOT in equilibrium along the direction of motion, the flow is unsteady and a fourth force arises (the inertia force) to produce a balance.
Governing equations The derivation of the governing equations of unsteady flow (Fig. 102), also referred to as unsteady gradually varied flow or the shallowwater equations (Liggett, 1975) considers the statements of mass and momentum conservation in a control volume (Fig. 103).
The statement of conservation of mass is:
For a unitwidth channel (Liggett, 1975):
Simplifying, and neglecting secondorder terms as Δx → 0:
Equation 103 is the unsteady conservation of mass equation, commonly referred to as the equation of continuity. However, its complete name is: the differential equation of water continuity. For steady flow: ∂h/∂t = 0, and Eq. 103 reduces to: q = uh = constant. In general, for a channel of flow area A, the steady water continuity equation is: Q = uA = constant. The statement of conservation of momentum is:
The momentum, per unit of channel width, is: ρ u (uh). Therefore, the net rate of momentum entering the control volume F_{Δ} (force per unit of width) is:
The forces acting on the control volume, resolved along the direction of motion, are: (1) gravitational force, (2) pressuregradient force, and (3) frictional force (Fig. 101). The gravitational force, per unit of width, is:
The pressuregradient force, per unit of width, developed along the sides of the control volume, is:
The frictional force, per unit of width, developed along the channel bottom, resembles the gravitational force, but it is opposite in sign:
in which S_{f} = friction slope. [As an exception, for channels of adverse bottom slope, for which S_{o} < 0, the gravitational and frictional forces are of the same sign].
The size of the control volume, per unit of width, is:
The conservation of momentum is:
Replacing Eqs. 104 to 108 into Eq. 109:
Simplifying Eq. 1010:
Equation 1011 is in conservation form. For certain applications, it must remain in this form. However, it is often expressed in reduced form, by operating on the derivatives:
The second and third terms of Eq. 1012 have implicit in them the continuity equation (Eq. 103). Thus, Eq. 1012 reduces to:
By dividing by g, Eq. 1013 is expressed in slope units:
Equation 1014 is referred to as the equation of motion.
It is expressed in terms of slopes as follows:
in which S_{a} = local acceleration slope, S_{c} = convective acceleration slope, S_{p} = pressuregradient slope, S_{f} = friction slope, and S_{o} = bottom slope.
Equation 1015 indicates that the momentum balance is essentially a balance of slopes. In the general case, when all forces are present, all slopes are acting and the solution is the most general. In certain cases, however, one or more slopes may be reduced to zero, or assumed to be negligible (compared to the remaining slopes). This simplification gives rise to several types of waves, described in Table 102.
10.2 LINEAR SOLUTION
Equations 102 and 1014 are the governing equations of continuity and motion, also referred to as the SaintVenant equations (SaintVenant, 1871). They are repeated here for convenience, as Eqs. 1016 and 1017, respectively.
These equations constitute a set of two nonlinear (actually, quasilinear) partial differential equations, which when appropriately combined, result in a secondorder partial differential equation of the hyperbolic type, featuring two solutions. To this date, there is no closedform analytical solution of the set of Eqs. 1016 and 1017. An approximate solution may be obtained by linearizing the equation set and using the tools of linear stability analysis (Ponce and Simons, 1977). The friction slope S_{f} is directly related to the bottom shear stress τ by the expression (similar to Eq. 616):
In the usual manner of stability calculations, Eqs. 1016 and 1017 must satisfy the unperturbed flow, for which u = u_{o} , h = h_{o} , and τ = τ_{o} . They must also satisfy the perturbed flow, for which u = u_{o} + u' , h = h_{o} + h' , and τ = τ_{o} + τ' . The superscript ' represents a small perturbation to the steady uniform flow. Thus, all quadratic terms in the fluctuating components may be negledted due to an orderofmagnitude reasoning. Substitution of the perturbed variables in Eqs. 1016, 1017, and 1018, yields, after linearization (Lighthill and Whitham, 1955):
in which:
The boundary shear stress τ can be related to the mean velocity u as follows (Eq. 53):
in which the friction factor f is (Eq. 512):
In view of Eq. 1022, Eq. 1020 is converted to:
The solution for a small perturbation in the depth of flow is postulated in the following exponential form (Ponce and Simons, 1977):
in which the subscript _{*} indicates dimensionless variables, and i = (1)^{1/2}. The quantity σ_{*} = dimensionless wavenumber, β_{*} = dimensionless complex propagation factor, and x_{*} and t_{*} are dimensionless space and time coordinates, such that:
and β_{I}_{*} is an amplitude propagation factor. The quantity L_{o} is the length of channel in which the uniform flow drops a head equal to its depth:
The depth disturbance is associated with a velocity disturbance of the form:
The substitution of Eqs. 1025 and 1031 into Eqs. 1019 and 1024 yields the set:
in which
Equations 1032 and 1033 constitute a homogeneous system of linear equations in the unknowns u_{*} and h_{*}. For the solution to be nontrivial, the determinant of the coefficient matrix must vanish. Therefore, the following relation holds:
Equation 1035 is the characteristic equation governing the propagation of small amplitude
The solution of Eq. 1036 is (Ponce and Simons, 1977):
in which
The equations for dimensionless celerity and attenuation for the primary and secondary waves are:
in which
The dimensionless relative wave celerity is:
Figure 104 shows a plot of the dimensionless relative wave celerity c_{r}_{*} versus the dimensionless wavenumber σ_{*}. Figure 105 shows a plot of the primary wave logarithmic decrement δ_{1} versus the dimensionless wavenumber σ_{*}, applicable for Froude numbers F < 2. Figure 106 shows a plot of the primary wave logarithmic increment +δ_{1} versus the dimensionless wavenumber σ_{*}, applicable for Froude numbers F > 2. Based on these figures, the characteristics of shallow waves are described in the box below.
10.3 KINEMATIC WAVES
A kinematic wave is an idealization (of gradually varied unsteady openchannel flow) that neglects both acceleration terms (local and convective) and the pressuregradient term (Table 102). By neglecting these terms, the equation of motion (Eq. 1014) is reduced to a statement of steady uniform flow:
The unsteadiness of the phenomenon, however, is preserved through the timevarying term in the continuity equation (Eq. 103). The combination of Eqs. 103 and 1049 gives rise to the kinematic wave equation. Since q = uh, Eq. 103 may be expressed in terms of the unitwidth discharge:
In terms of discharge Q, the continuity equation is:
A statement of uniform flow (Eq. 1049) may be properly represented by the dischargearea rating:
in which α and β are coefficient and exponent, respectively. The coefficient α varies as a function of type of friction, crosssectional shape, and bottom slope. The exponent β varies as a function of type of friction and crosssectional shape. Assuming for the sake of simplicity that α and β are independent of A, Eq. 1052 yields:
in which V = Q / A = mean velocity. The kinematic wave equation is obtained by combining Eqs. 1051 and 1055 to yield:
In terms of unitwidth discharge q :
Convective celerity Equation 1056 (or Eq. 1057) is a firstorder partial differential equation. It describes the convection of the quantity Q (or q) with the convective velocity or celerity c_{k}, where c_{k} is:
Given Eq. 1055, the convective velocity may also be expressed as:
Given that dA = T dy (Eq. 311), where T = channel top width, and y = stage, the convective velocity may also be expressed as follows:
Equation 1060 was originally derived by Kleitz (1877) and was later discovered from actual field observations by Seddon (1900). It often referred to as the KleitzSeddon Law, or simply Seddon's Law. Equation 1058 is used when β is known with certainty, Eq. 1059 in theoretical formulations, and Eq. 1060 in practical applications. Since Eq. 1056 is a firstorder partial differential equation, it does not allow for wave diffusion (wave attenuation, or wave dissipation). Diffusion can only be obtained through the agency of a secondorder term. Under the assumption of linearity (constant convective celerity), a kinematic wave will convect its discharge with no wave diffusion; that is, the discharge will retain its shape and remain constant in space and time upon propagation.
Kinematic shock When the linearity assumption is relaxed, the kinematic wave may change its shape by becoming either (a) steeper [Fig. 107 (a)], or (b) flatter [Fig. 107 (b)]. Whether a wave will steepen or flatten out will depend largely on the channel crosssectional shape. Two asymptotic limits are recognized: (1) waves propagating in hydraulically wide channels, while (2) waves propagating in inherently stable channels (Chapter 1). In hydraulically wide channels, the waves will steepen, while in inherently stable channels, they will flatten out (Ponce and Windingland, 1985).
When allowed to proceed unchecked, the steepening will eventually result in a kinematic wave becoming a kinematic shock. Thus, a kinematic shock is an unsteady openchannel flow feature intrinsically related to the kinematic wave: A wave must be kinematic before it can develop into a kinematic shock (Lighthill and Whitham, 1955). Kibler and Woolhiser (1970) sought to clarify the occurrence of kinematic shock phenomena by stating:
Thus, the kinematic shock is real but rare in the physical world, where spatial irregularities manifest themselves as diffusion, with the net effect of arresting shock development. On the other hand, the computational world is likely to be much more regular, thereby inhibiting diffusion and promoting "numerical" shock development. Kinematic wave celerity The relative kinematic wave celerity, ie., the kinematic wave celerity taken relative to the flow velocity, is:
Furthermore, the dimensionless relative kinematic wave celerity is:
According to Eq. 111, the relative dimensionless kinematic wave celerity is:
Thus, for V = 1, i.e., for neutrally stable flow, the Froude number is:
Table 103 shows the variation of: (a) the exponent β, (b) the dimensionless relative kinematic wave celerity c_{drk}, and (c) the neutralstability Froude number, with selected types of friction and crosssectional shape.
The following conclusions can be drawn from Table 103:
Note that the value of β may be less than 1 for cases other than those shown in Table 103; for example, when
the cross section does not grow monotonically with stage, as in circular culvert flow.
Also, note that since the Froude number has an upper limit (corresponding to a realistically achievable lower limit on the bottom friction),
the value F_{ns} = ∞ is of limited practical value. If the maximum attainable Froude number is conservatively assumed to be
In summary, kinematic waves have the following properties:
Kinematic wave rating Kinematic waves are based on a singlevalued dischargearea rating, Eq. 1052. Thus, a kinematic wave rating is singlevalued, exhibiting a onetoone correspondence between (a) discharge, and (b) flow area, depth, or stage. A kinematic wave rating is calculated by using a uniform flow formula such as Manning or Chezy, for a range of (a) flow depths, in artificial channels, or (b) flow stages, in natural channels. Applicability of kinematic waves A kinematic wave is a simplified type of wave, wherein three terms in the equation of motion (Table 102) have been either neglected or assumed to be too small to be of any practical significance. Thus, the kinematic wave does not apply to the general case. Its use is recommended for cases where the flow unsteadiness is relatively small. In practice, a kinematic wave will apply provided the following dimensionless inequality is satisfied (Ponce, 1989; Ponce, 2014):
in which t_{r} = timeofrise of the hydrograph, S_{o} = bottom slope, V_{o} = average flow velocity, and d_{o} = average flow depth. 10.4 DIFFUSION WAVES
A diffusion wave is an idealization that neglects both acceleration terms in the equation of motion (Table 102). By neglecting these terms, Eq. 1014 is reduced to the following statement:
The unsteadiness of the phenomenon, however, is preserved through the timevarying term in the continuity equation (Eq. 103). The combination of Eqs. 103 and 1068 gives rise to the diffusion wave equation. The kinematic wave equation was derived by using a statement of steady uniform flow in lieu of the equation of motion (Section 10.4). In deriving the diffusion wave equation, a statement of steady nonuniform flow (friction slope is equal to watersurface slope) is used instead (Fig. 108). In this case, the dischargearea rating, using the Manning formula in SI Units (Eq. 517), is:
in which the term within parentheses (...) is the watersurface slope S_{w}.
The difference between kinematic and diffusion waves lies in the pressuregradient term (dh/dx). When this term is included in the formulation, the resulting equation is of second order and, therefore, it is able to simulate diffusion. Lighthill and Whitham (1955) referred to this situation as the "diffusion of kinematic waves," i.e., a type of kinematic wave, still with no inertia in its formulation, that is nevertheless able to diffuse. To derive the diffusion wave equation, Eq. 1051 is repeated here in a slightly different form:
Equation 1069 is expressed in a more convenient form (Cunge, 1969):
in which m is the reciprocal of the square of the channel conveyance K (Eq. 534), repeated here for convenience:
With dA = T dh, in which T = top width, Eq. 1071 changes to:
Equations 1070 and 1072 constitute a set of two partial differential equations describing diffusion waves. These equations can be combined into one equation with Q as dependent variable. However, it is first necessary to linearize the equations around reference flow values. For simplicity, a constant top width is assumed (i.e., a wide channel assumption). The linearization of Eqs. 1070 and 1072 is accomplished by small perturbation theory (Cunge, 1969). The variables Q, A, and m can be expressed in terms of the sum of a reference value (with subscript o) and a small perturbation to the reference value (with superscript ' ): Q = Q_{o} + Q' ; A = A_{o} + A' ; m = m_{o} + m'. Substituting these into Eqs. 1070 and 1072, neglecting squared perturbations and subtracting the reference flow, leads to:
and
Differentiating Eq. 1073 with respect to x and Eq. 1074 with respect to t gives:
Using the chain rule and Eq. 1073 yields:
Combining Eq. 1076 with Eq. 1077:
Combining Eqs. 1075 and 1078 and rearranging terms, yields:
By definition: mQ ^{2} = S_{f} (Eq. 1070). Therefore:
and also
Substituting Eqs. 1080 and 1081 into Eq. 1079, using the chain rule, and dropping the superscripts for simplicity, the following equation is obtained:
The left side of Eq. 1082 is recognized as the kinematic wave equation, with ∂Q/∂A as the kinematic wave celerity.
The right side is a secondorder (partial differential) term that accounts for the physical diffusion effect.
The coefficient of the secondorder term has the units of diffusivity
The hydraulic diffusivity, a characteristic of the flow and channel, is defined as follows:
in which q_{o} = Q_{o} /T is the reference discharge per unit of channel width. From Eq. 1083, it is concluded that the hydraulic diffusivity is small for steep bottom slopes (e.g., those of mountain streams), and large for mild bottom slopes (e.g., those of large rivers near their mouths). Equation 1082 describes the movement of flood waves in a better way than Eq. 1050. While it falls short from describing the full inertial effects, it does physically account for wave attenuation. Equation 1082 is a secondorder parabolic partial differential equation. It can be solved analytically, leading to the diffusion analogy solution for flood waves (Hayami, 1951), or numerically with the aid of a numerical scheme for parabolic equations. An alternative approach is to match the hydraulic diffusivity (Eq. 1083) with the numerical diffusion coefficient of the Muskingum flood routing method (Section 10.5). This approach is the basis of the MuskingumCunge method (Section 10.6). Diffusion wave rating Diffusion waves are not based on a singlevalued dischargearea rating. Thus, a diffusion wave rating is not singlevalued, exhibiting a loop. In general, however, the loop is relatively small and may be neglected on practical grounds. A kinematic wave rating may be used as an approximation in diffusion wave routing. Diffusion wave celerity According to Eq. 1082, the diffusion wave celerity should be the same as the kinematic wave celerity (Ponce and Simons, 1977). However, diffusion waves attenuate; therefore, the actual dischargearea rating is not exactly singlevalued. In practice, the difusion wave celerity equals the kinematic wave celerity only as an approximation. Applicability of diffusion waves A diffusion wave is a simplified type of wave, wherein two terms in the equation of motion (Table 102) have been either neglected or assumed to be too small to be of any practical significance. Thus, while the diffusion wave applies for a wider range of cases than the kinematic wave, it is still not suited to the general case. Its use is recommended for cases where the flow unsteadiness is small to medium size (where the wave remains within 30% of its original strength, within one period of propagation). A diffusion wave will apply provided the following dimensionless inequality is satisfied (Ponce, 1989; Ponce, 2014):
in which t_{r} = timeofrise of the hydrograph, S_{o} = bottom slope, g = gravitational acceleration, and d_{o} = average flow depth. Diffusion waves apply to problems of flood wave propagation (see Hayami's diffusion analogy of flood waves in the box below). While kinematic waves apply to flood waves that do not diffuse, diffusion waves apply to flood waves that attenuate appreciably. Where the diffusion wave fails to account for the wave propagation, only the mixed kinematicdynamic wave (read "dynamic wave", Table 102) is able to solve the problem correctly. In practice, however, diffusion waves apply to a wide range of flood propagation problems.
Dynamic hydraulic diffusivity The hydraulic diffusivity (Eq. 1083) is a fundamental property of diffusion waves. It states that the coefficient of diffusion is directly proportional to the unitwidth discharge and inversely proportional to the channel slope. This conclusion ia applicable to diffusion waves, which are governed by the convectiondiffusion equation represented by Eq. 1082.
Using concepts of linear theory, Dooge (1973) has developed a convectiondiffusion equation using the complete equation of motion. Dooge's approach extends
the concept of diffusion wave to the realm of dynamic waves (Table 102). When all terms are included in the
formulation, the hydraulic diffusivity is essentially a dynamic hydraulic diffusivity, expressed, for hydraulically wide channels, as follows:
Ponce (1991) has expressed the dynamic hydraulic diffusivity in terms of the Vedernikov number, as follows:
Equation 1086 is altogether better than Eq. 1083. They are equivalent only if V = 0, i.e., for very small Froudenumber flows. For V = 1, using Eq. 1086, the hydraulic diffusivity vanishes, while this is not the case for Eq. 1083, for which the hydraulic diffusivity remains finite. 10.5 MUSKINGUM METHOD
The Muskingum method of flood routing was developed in connection with the design of flood protection schemes in the Muskingum River Basin, Ohio (Fig. 1010) (McCarthy, 1938). It is the most widely used method of flood routing, with numerous applications in the United States and throughout the world.
The method is based on the differential equation of storage (Fig. 1011):
in which I = inflow, O = outflow, and S = storage.
In an ideal channel, storage is a function of inflow and outflow. This is in constrast with an ideal reservoir, in which storage is solely a function of outflow. In the Muskingum method, storage is a linear function of inflow and outflow:
in which S = storage volume; I = inflow; O = outflow; K = a time constant or storage coefficient; and To derive the Muskingum routing equation, Eq. 1087 is discretized on the xt plane (Fig. 1012), to yield:
Equation 1088 is expressed at time levels 1 and 2:
Substituting Eqs. 1090 and 1091 into Eq. 1089 and solving for O_{2} yields:
in which C_{0}, C_{1} and C_{2} are routing coefficients defined in terms of Δt, K, and X as follows:
Since C_{0} + C_{1} + C_{2} = 1, the routing coefficients may be interpreted as weighting coefficients. Given an inflow hydrograph, an initial flow condition, a chosen time interval Δt, and routing parameters X and K, the routing coefficients can be calculated with Eq. 1093, and the outflow hydrograph with Eq. 1092. The routing parameters K and K are related to flow and channel characteristics, K being interpreted as the travel time of the flood wave from upstream end to downstream end of the channel reach. Therefore, K accounts for the translation portion of the routing (Fig. 1013). The parameter X accounts for the storage portion of the routing. For a given flood event, there is a value of X for which the storage in the calculated outflow hydrograph matches that of the measured outflow hydrograph. The effect of storage is to reduce the peak flow and spread the hydrograph in time (Fig. 1013). Therefore, it is often used interchangeably with the terms diffusion and peak attenuation.
The routing parameter K is a function of channel reach length and flood wave speed; conversely, the parameter X is a function of the flow and channel characteristics that cause runoff diffusion. In the Muskingum method, X is interpreted as a weighting factor and restricted in the range 0 ≤ X ≤ 0.5. Values of X greater than 0.5 produce hydrograph amplification (i.e., negative diffusion), which does not correspond with reality (under the Froude numbers applicable to flood flows). With K = Δt and X = 0.5, flow conditions are such that the outflow hydrograph retains the same shape as the inflow hydrograph, but it is translated downstream a time equal to K. For X = 0, Muskingum routing reduces to linear reservoir routing. In the Muskingum method, the parameters K and X are determined by calibration using streamflow records. Simultaneous inflowoutflow discharge measurements for a given channel reach are coupled with a trialanderror procedure, leading to the determination of K and X (see Example 101). The procedure is timeconsuming and lacks predictive capability. Values of K and X determined in this way are valid only for the given reach and flood event used in the calibration. Extrapolation to other reaches or to other flood events (of different magnitude) within the same reach is usually unwarranted. When sufficient data are available, a calibration can be performed for several flood events, each of different magnitude, to cover a wide range of flood levels. In this way, the variation of K and X as a function of flood level can be ascertained. In practice, K is more sensitive to flood level than X. A sketch of the variation of K with stage and discharge is shown in Fig. 1014.
Example 101 has illustrated the predictive stage of the Muskingum method, in which the routing parameters are known in advance of the routing. If the parameters are not known, it is first necessary to perform a calibration. The trialanderror procedure to calibrate the routing parameters is illustrated by Example 102.
The estimation of routing parameters is crucial to the application of the Muskingum method. The parameters are not constant, tending to vary with flow rate. If the routing parameters can be related to flow and channel characteristics, the need for trialanderror calibration would be eliminated. Parameter K could be related to reach length and flood wave velocity, whereas X could be related to the diffusivity characteristics of flow and channel. These propositions are the basis of the MuskingumCunge method. 10.6 MUSKINGUMCUNGE METHOD
The Muskingum method requires a calibration in order to identify the routing parameters (Example 102). The procedure is based on measured input and output hydrographs, as shown in Cols. 2 and 3 of Table 105. Therefore, a gaging station is an absolute necessity for the Muskingum method to be properly used. This limits the applicability of the method to reaches which have a streamgaging station. Unlike the Muskingum method, the MuskingumCunge method requires hydraulic rather than hydrologic data in order to calculate the routing parameters. The hydraulic data consists of geomorphic data such as channel slope and crosssectional characteristics. Thus, the MuskingumCunge method does not explicitly require a streamgaging station, being applicable to any channel reach as long as the geomorphic data is available. The Muskingum method is derived by combining the differential equation of storage, i.e., the continuity equation expressed in total differential form (Eq. 1087) with a linear inflowoutflowstorage relation (Eq. 1088). This leads to the Muskingum routing equation (Eq. 1092) with appropriate routing coefficients (Eq. 1093). The MuskingumCunge method is derived by discretizing the kinematic wave equation (Eq. 1055) in a linear mode, in a manner similar to that of the Muskingum method, which leads to the same routing equation. The coefficients, however, are defined based on measurable channel characteristics. The similarities appear to end there. The Muskingum method is lumped across the channel reach, based on storage, and able to describe flood wave diffusion (Example 101). The MuskingumCunge method is distributed, with data specified at cross sections, and based on a discretization of the kinematic wave equation, which ostensibly does not diffuse. Yet actual calculations using the MuskingumCunge method shows that it is able to describe wave diffusion, in a manner similar to the Muskingum method. Cunge (1969) traced the diffusion of the discretized analog of the kinematic wave equation to the numerical diffusion of the scheme itself (see box). Thus, he was able to explain the paradox. The Muskingum and MuskingumCunge methods have the same theoretical basis. The routing parameters of the Muskingum method are hydrologic, based on storage, and determined by calibration using streamflow data. The routing parameters of the MuskingumCunge method are hydraulic, distributed (at a cross section), and based exclusively on geomorphic data.
MuskingumCunge routing equation To derive the MuskingumCunge routing equation, the kinematic wave equation (Eq. 1055) is discretized on the xt plane (Fig. 1017) in a way that parallels the Muskingum method, centering the spatial derivative and offcentering the temporal derivative by means of a weighting factor X:
in which c = βV is the kinematic wave celerity.
Solving Eq. 1094 for the unknown discharge leads to the following routing equation:
The routing coefficients are:
By defining the travel time as
it is seen that the sets of Eq. 1093 and Eq. 1096 are the same.
Equation 1097 confirms that K is in fact the flood wave travel time, i.e., the time it takes a given discharge to travel the reach length Δx with the kinematic wave celerity c.
The Courant number is defined as the ratio of physical celerity (c) to the numerical, or grid, celerity
Equation 1095 is a numerical analog of Eq. 1055 and, therefore, subject to numerical diffusion and dispersion. Numerical diffusion is the secondorder error; numerical dispersion is the thirdorder error. The following conditions hold:
These relations are summarized in Table 106.
In practice, the numerical diffusion can be used to simulate the physical diffusion of the actual flood wave. By expanding the discrete function Q (jΔx, n Δt) in Taylor series about grid point (jΔx, n Δt), the numerical diffusion coefficient of the Muskingum scheme is derived (Cunge, 1969) (Appendix B):
in which ν_{n} is the numerical diffusion coefficient of the Muskingum scheme. This equation reveals the following:
A predictive equation for X can be obtained by matching the hydraulic diffusivity ν_{h} (Eq. 1083) with the numerical diffusion coefficient of the Muskingum scheme ν_{n} (Eq. 1099). This leads to the following expression for X:
With X calculated by Eq. 10100, the Muskingum method is referred to as MuskingumCunge method. Thus, the routing parameter X can be calculated as a function of the following numerical and physical properties:
It should be noted that Eq. 10100 was derived by matching physical and numerical diffusion (a secondorder processes), and does not account for dispersion (a thirdorder process). Therefore, in order to simulate wave diffusion properly with the MuskingumCunge method, it is necessary to optimize numerical diffusion (with Eq. 10100) while at the same time minimizing numerical dispersion by keeping the value of C ≅ 1.
An improved version of the MuskingumCunge method is due to Ponce and Yevjevich (1978). The grid diffusivity is defined as the numerical diffusivity for the case of X = 0. From Eq. 1099, the grid diffusivity is:
The cell Reynolds number D is defined as the ratio of hydraulic diffusivity (Eq. 1083) to grid diffusivity (Eq. 10101). This leads to:
Therefore:
Equations 10101 and 10102 imply that for very small values of Δx, D may be greater than 1, leading to negative values of X. In fact, for the characteristic reach length
the cell Reynolds number is D = 1, and X = 0. Therefore, in the MuskingumCunge method, reach lengths shorter than the characteristic reach length result in negative values of X. This should be contrasted with the classical Muskingum method (Section 10.4), in which X is restricted in the range 0.0 ≤ X ≤ 0.5. In the classical Muskingum, X is interpreted as a weighting factor. As shown by Eqs. 10101 and 10102, nonnegative values of X are associated with long reaches, typical of the manual computation used in the development and early application of the Muskingum method. In the MuskingumCunge method, however, X is interpreted in a momentmatching sense or diffusionmatching factor. Therefore, negative values of X are entirely possible. This feature allows the use of shorter reaches than would otherwise be possible if X were restricted to nonnegative values. Routing coefficients The substitution of Eqs. 1098 and 10100 into Eq. 1096 leads to routing coefficients expressed in terms of Courant and cell Reynolds numbers:
The calculation of routing parameters C and D can be performed in several ways. The wave celerity can be calculated with either Eq. 1057 or Eq. 1059. With Eq. 1057, c = βV; with Eq. 1059, c = (1/T) dQ/dy. Theoretically, these two equations are the same. For practical applications, if a stagedischarge rating and crosssectional geometry are available (i.e., stagedischargetop width tables), Eq. 1059 is preferred over Eq. 1057 because it accounts directly for crosssectional shape. In the absence of a stagedischarge rating and crosssectional data, Eq. 1057 can be used to estimate flood wave celerity. With the aid of Eqs. 1098 and 10102, the routing parameters may be based on flow characteristics. The calculations can proceed in a linear or nonlinear mode. In the linear mode, the routing parameters are based on reference flow values and kept constant throughout the computation in time. The choice of reference flow has a bearing on the calculated results, although the overall effect is likely to be small (Ponce and Yevjevich, 1978). For practical applications, either an average or peak flow value can be used as reference flow. The peak flow value has the advantage that it can be readily ascertained, although a better approximation may be obtained by using an average value. The linear mode of computation is referred to as the constantparameter MuskingumCunge method to distinguish it from the variableparameter MuskingumCunge method, in which the routing parameters are allowed to vary with the flow. The constant parameter method resembles the Muskingum method, with the difference that the routing parameters are based on measurable flow and channel characteristics instead of on historical streamflow data.
Resolution requirements When using the MuskingumCunge method, care should be taken to ensure that the values of Δx and Δt are sufficiently small to approximate closely the actual shape of the hydrograph. For smoothly rising hydrographs, a minimum value of t_{p} /Δt = 5 is recommended. This requirement usually results in the hydrograph time base being resolved into at least 15 to 25 discrete points, considered adequate for Muskingum routing. Unlike temporal resolution, there is no definite criteria for spatial resolution. A criterion borne out by experience is based on the fact that Courant and cell Reynolds numbers are inversely related to reach length Δx. Therefore, to keep Δx sufficiently small, Courant and cell Reynolds numbers should be kept sufficiently large. This leads to the practical criterion (Ponce and Theurer, 1982):
which can be written as follows: 1 + C + D ≥ 0. This confirms the necessity of avoiding negative values of C_{0} in MuskingumCunge routing (see Eq. 10105a). Experience has shown that negative values of either C_{1} or C_{2} do not adversely affect the method's overall accuracy. Notwithstanding Eq. 10106, the MuskingumCunge method works best when the numerical dispersion is minimized, that is, when C ≅ 1. Values of C substantially less than 1 are likely to cause the notorious dips, or negative outflows, in portions of the calculated hydrograph. This computational anomaly is attributed to excessive numerical dispersion and should be avoided. Nonlinear MuskingumCunge method The kinematic wave equation, Eq. 1055, is nonlinear because the kinematic wave celerity varies with discharge. The nonlinearity is mild, among other things because the wave celerity variation is usually restricted within a narrow range. However, in certain cases it may be necessary to account for this nonlinearity. This can be done in two ways: (1) during the discretization, by allowing the wave celerity to vary, resulting in a nonlinear numerical scheme to be solved by iterative means; and (2) after the discretization, by varying the routing parameters, as in the variableparameter MuskingumCunge method (Ponce and Yevjevich, 1978). The latter approach is particularly useful if the overall nonlinear effect is small, which is often the case. The variable parameter MuskingumCunge method represents a small yet sometimes perceptible improvement over the constant parameter method. The differences are likely to be more marked for very long reaches and/or wide variations in flow levels. Flood hydrographs calculated with variable parameters show a certain amount of distortion, either wave steepening in the case of flows contained inbank or wave attenuation (flattening) in the case of typical overbank flows. This is a physical manifestation of the nonlinear effect, i.e., different flow levels traveling with different celerities. On the other hand, flood hydrographs calculated using constant parameters do not show wave distortion. Assessment of MuskingumCunge method The MuskingumCunge method is a physically based alternative to the Muskingum method. Unlike the Muskingum method where the parameters are calibrated using streamflow data, in the MuskingumCunge method the parameters are calculated based on flow and channel characteristics. This makes possible channel routing without the need for timeconsuming and cumbersome parameter calibration. More importantly, it makes possible extensive channel routing in ungaged streams with a reasonable expectation of accuracy. With the variableparameter feature, nonlinear properties of flood waves (which could otherwise only be obtained by more elaborate numerical procedures) can be described within the context of the Muskingum formulation. Like the Muskingum method, the MuskingumCunge method is limited to diffusion waves. Furthermore, the MuskingumCunge method is based on a singlevalued rating and does not take into account strong flow nonuniformity or unsteady flows exhibiting substantial loops in dischargestage rating (i.e., dynamic waves). Thus, the MuskingumCunge method is suited for channel routing in natural streams without significant backwater effects and for unsteady flows that classify under the diffusion wave criterion (Eq. 1066). An important difference between the Muskingum and MuskingumCunge methods should be noted. The Muskingum method is based on the storage concept (Eq. 1087) and, therefore, it is lumped, with the parameters K and X being reach averages. The MuskingumCunge method, however, is distributed in nature, with the parameters C and D being based on values evaluated at channel cross sections. Therefore, for the MuskingumCunge method to improve on the Muskingum method, it is necessary that the routing parameters evaluated at channel cross sections be representative of the channel reach under consideration (Fig. 1018). Historically, the Muskingum method has been calibrated using streamflow data. On the contrary, the MuskingumCunge method relies on physical characteristics such as rating curves, crosssectional data and channel slope. The different data requirements reflect the different theoretical bases of the methods, i.e., lumped storage concept in the Muskingum method, and distributed kinematic/diffusion wave theory in the MuskingumCunge method.
10.7 DYNAMIC WAVES
In unsteady openchannel flow, the term dynamic wave is used to refer to two different types of waves:
To avoid confusion, the first type of wave [3] is referred here as true dynamic wave. The second type [5] is referred to as mixed kinematicdynamic wave, for short, mixed dynamic wave.
Conceptually, true dynamic waves are the exact opposite of kinematic waves. While kinematic waves lie to the left side of the wavenumber spectrum, true dynamic waves lie to the right (Fig. 103). Thus, their dimensionless wavenumber is long, that is, the wavelength L is short relative to the reference channel length L_{o} (Eq. 1030). While the dimensionless relative celerity of a kinematic wave is constant and equal to 0.5, that of the true dynamic wave is equal to the reciprocal of the Froude number (Fig. 103):
The relative celerity of a true dynamic wave is:
The celerity of a true dynamic wave is:
Therefore, a true dynamic wave has two components, with celerities:
Kinematic and true dynamic waves share a distinct property: They do not attenuate. This is due to the constancy of the dimensionless relative wave celerity within the applicable range of dimensionless wavenumbers (Fig. 103). In practice, true dynamic waves apply to the "short" waves that may be present in laboratory flumes and small canals. They do not apply for flood waves, which lie on the left side of the dimensionless wavenumber spectrum.
Mixed kinematicdynamic waves, for short mixed dynamic waves, lie toward the middle of the wavenumber spectrum (Fig. 103).
Conceptually, they are the most complete type of wave, because they
consider all the terms
of the equation of motion (Table 102). However, for Vedernikov numbers V < 1,
(corresponding to
Froude numbers F < 2 under Chezy friction in hydraulically wide channels),
the mixed dynamic waves are subject to strong attenuation. The attenuation is strongest at the point of inflexion
of the dimensionless relative celerity versus dimensionless wavenumber function Lighthill and Whitham (1955) described the impermanence of [mixed] dynamic waves in the following terms:
They followed up with this statement (op. cit., page 291):
Thus, in general, mixed dynamic waves do not apply to flood flows in natural streams. Once generated, mixed dynamic waves tend to dissipate very quickly, with their mass going to join the predominant underlying kinematic or diffusion wave. The exception may be a flood wave generated by a dam breach, which is typically so sudden that it may effectively be a mixed dynamic wave. These waves attenuate very quickly, confirming the correctness of the theory. For example, consider the failure of Teton Dam, in Idaho, on June 5, 1976 (Fig. 1019). The flood wave released at the damsite attenuated to a small fraction of its initial strength (less than 3%) within a relatively short distance downstream. Many other examples of actual dam breaches have confirmed that dambreach flood waves tend to dissipate rather quickly.
In a dynamic wave solution, the equations of continuity and motion are solved by a numerical procedure, either (a) the method of finite differences, (b) the method of characteristics, or (c) the finite element method. In the method of finite differences, the partial differential equations are discretized following a chosen numerical scheme. The method of characteristics is based on the conversion of the set of partial differential equations into a related set of ordinary differential equations, and the solution along a characteristic grid, i.e. a grid that follows characteristic directions. The method of finite elements solves a set of integral equations over a chosen grid of finite elements. In the past four decades, the method of finite differences has come to be regarded as the most expedient way of obtaining a mixed dynamic wave solution for practical applications. Among several numerical schemes that have been used in connection with the mixed dynamic wave, the Preissmann scheme is perhaps the most popular (Liggett and Cunge, 1975). This is a fourpoint scheme, centered in the temporal derivatives and slightly offcentered in the spatial derivatives, by use of a weighting factor θ. The offcentering in the spatial derivatives introduces a small amount of numerical diffusion necessary to control the numerical stability of the nonlinear scheme. This produces a workable yet sufficiently accurate scheme.
The independent variables used in mixed dynamic wave routing are usually discharge Q and
In practice, a mixed dynamic wave solution represents an orderofmagnitude increase in complexity and associated data requirements when compared to either kinematic or diffusion wave solutions. Its use is recommended in situations where neither kinematic nor diffusion wave solutions are likely to adequately represent the physical phenomena. In particular, mixed dynamic wave solutions are applicable to dambreach flood waves, flow over very flat slopes, flow into large reservoirs, strong backwater conditions and flow reversals. In general, the mixed dynamic wave is recommended for cases warranting a precise determination of the unsteady variation of river stages. The current version (Version 4.1) of the U.S. Army Corps of Engineers' HECRAS model (U.S. Army Corps of Engineers, 2010) contains a dynamic wave module suited for practical applications. Mixed dynamic rating curve Mixed dynamic wave solutions are often referred to as hydraulic river routing. As such, they have the capability to calculate unsteady discharges and stages when presented with the appropriate geometric channel data and initial and boundary conditions. Their importance in unsteady flow is examined here by comparing them to kinematic and diffusion waves. Kinematic waves calculate unsteady discharges; the corresponding stages are subsequently obtained from the appropriate rating curves. Equilibrium (steady, uniform) rating curves are used for this purpose. Diffusion waves may or may not use equilibrium rating curves to calculate stages. Some methods, e.g., MuskingumCunge, use equilibrium ratings, but more elaborate diffusion wave solutions may not. Mixed dynamic waves rely on the physics of the phenomena as built into the governing equations to generate their own unsteady flow rating. A looped rating curve is produced at every cross section, as shown in Fig. 1021. For any given stage, the discharge is higher in the rising limb of the hydrograph and lower in the receding limb. This loop is due to hydrodynamic reasons and should not be confused with other loops, which may be due to erosion, sedimentation, or changes in bed configuration.
The width of the loop is a measure of the flow unsteadiness, with wider loops corresponding to highly unsteady flow. If the loop is narrow, it implies that the flow is mildly unsteady, perhaps a diffusion wave. If the loop is practically nonexistent, the flow can be approximated as kinematic flow. In fact, the basic assumption of kinematic flow is that momentum can be simulated as steady uniform flow, i.e., that the rating curve is singlevalued. The preceding observations lead to the conclusion that the importance of mixed dynamic waves is directly related to the flow unsteadiness and the associated loop in the rating curve. For highly unsteady flows such as dambreach flood waves, it may well be the only way to properly account for the looped rating. For other less unsteady flows, kinematic and diffusion waves are a viable alternative, provided their applicability (Eqs. 1067 and 1084, respectively) can be clearly demonstrated. Downstream boundary condition Modeling of a mixed dynamic wave presents an interesting paradox: In order to solve the problem correctly, a dynamic downstream boundary condition (usually a rating curve) must be specified. However, a dynamic downstream boundary condition is not known a priori. Abbott (1976) put it in the right perspective when he stated:
A way out of this difficulty is to artificially extend the channel several subreaches downstream, and to specify a kinematic rating at the newly defined downstream boundary, while giving the loop a chance to develop at the upstream cross section of interest (Fig. 1022). Ponce and Lugo (2001) have used sensitivity analysis to show that the artificial extension of the channel by an amount equal to the channel length (i.e., doubling the channel length) may be sufficient to produce an accurate looped rating at the cross section of interest.
QUESTIONS
PROBLEMS
REFERENCES
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