1. RATIONALE

1.01
A kinematic wave simplifies the momentum equation to a statement of steady uniform flow.

1.02
A diffusion wave simplifies the momentum equation to a statement of steady nonuniform flow.

1.03
A third type of open channel flow wave, the dynamic wave, considers the complete momentum equation, including its inertial components.

1.04
As such, the dynamic wave contains more physical information that either kinematic or diffusion waves.

1.05
However, dynamic wave solutions are more complex.

1.06
In a dynamic wave equation, the equations of mass and momentum conservation are solved by a numerical procedure, typically using finite differences.

1.07
The partial differential equations governing the motion are discretized following a chosen numerical scheme.

1.08
Among the various schemes that have been used for this purpose, the Preissmann scheme is perhaps the most popular.

1.09
This is a four-point scheme, centered in the temporal derivatives, and slightly off-centered in the spatial derivatives.

1.10
The off-centering of the spatial derivatives produces a small amount of numerical diffusion, which is necessary to control the numerical stability of the nonlinear scheme.

1.11
This produces a workable yet sufficiently accurate scheme.

2. METHODOLOGY

2.01
The stream channel reach is divided into several subreaches for computational purposes.

2.02
The application of the Preissmann scheme to the governing equations for the various subreaches results in a matrix solution requiring a double-sweep algorithm.

2.03
A double-sweep algorithm is a type of Gauss elimination technique which accounts only for the nonzero entries of the coefficient matrix, which are located within a narrow band surrounding the main diagonal.

2.04
The upstream boundary condition is a discharge hydrograph.

2.05
The downstream boundary condition is a stage hydrograph.

2.06
The solution of the set of hyperbolic equations marches in time until a specified number of time intervals is completed.

2.07
In practice, a dynamic wave solution represents an order-of-magnitude increase in complexity and associated data requirements, when compared to either kinematic or diffusion wave solutions.

2.08
Its use is recommended in situations where neither kinematic or diffusion waves are likely to adequately represent the physical phenomena.

2.09
In particular, dynamic wave solutions are applicable to flow over very mild slopes, flow into large reservoirs, strong backwater conditions, and flow reversals.

2.10
In general, the dynamic wave solution is recommended for cases warranting a precise calculation of the unsteady variation of river stages.

3. RELEVANCE TO ENGINEERING HYDROLOGY

3.01
Dynamic wave solutions are often referred to as hydraulic river routing.

3.02
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.

3.04
Kinematic wave solutions calculate unsteady discharges; the corresponding stages are subsequently calculated from the appropriate rating curve.

3.05
An equilibrium, that is, a steady uniform, rating curve is used for this purpose.

3.06
Diffusion wave may or may not use equilibrium ratings to calculate stages.

3.07
Dynamic waves rely on the physics of the phenomena, as built into the governing equations, to generate their own unsteady rating curves.

3.08
A looped rating curve is produced at every cross section, as shown here.

3.09
For any given stage, the discharge is higher in the rising limb of the hydrograph, and lower in the receding limb.

3.10
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.

3.11
The width of the loop is a measure of the flow unsteadiness, with wider loops corresponding to highly unsteady flows, that is, dynamic wave flow.

3.12
If the loop is narrow, it implies that the flow is mildly unsteady, perhaps a diffusion wave.

3.13
If the loop is practically nonexistent, the flow can be approximated as a kinematic wave.

3.14
In fact, the basic assumption of kinematic flow is that momentum can be simulated as steady flow, that is, that the rating curve is single valued, with no loop.

3.15
The preceding observations lead to the conclusion that the relevance of dynamic waves to engineering hydrology is directly related to the extent of flow unsteadiness and the size of the associated loop in the rating curve.

3.16
For highly unsteady flows such as dam-breach flood waves, it may well be the only way to account for the looped rating.

3.17
For other less unsteady flows, kinematic and diffusion waves remain a viable alternative, provided their applicability can be clearly demonstrated.

3.18
The applicability of kinematic and diffusion waves is a function of the product of hydrograph time-to-peak and bottom slope.

3.19
A source of complexity in dynamic wave solutions is the need to specify a downstream boundary condition, typically a discharge-stage rating.

3.20
This discharge-stage rating is usually specified as a single-valued rating, that is, a kinematic rating.

3.21
This contradicts the solution at the downstream boundary, because the dynamic wave solution generates its own looped ratings at the interior points.

3.22
A finely balanced model will usually go unstable in this situation, but models with heavy numerical diffusion may survive, only to give erroneous results.

3.23

Narrator: Victor M. Ponce

Music: Fernando Oñate

Editor: Flor Pérez

Copyright © 2011

Visualab Productions

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