Applicability of Kinematic and Diffusion Models Victor M. Ponce, Ruh-Ming Li, and Daryl B. Simons Online version 2015 [Original version 1978]

1.  INTRODUCTION

The kinematic and diffusion wave models have found wide application in engineering practice. Both are approximations to the unsteady open channel flow phenomenon described by the complete Saint Venant equations. The diffusion model assumes that the inertia terms in the equation of motion are negligible as compared with the pressure, friction, and gravity terms. The kinematic model assumes that inertia and pressure terms are negligible as compared with the friction and gravity terms.

2.  WAVE PROPAGATION IN OPEN CHANNEL FLOW

Recently, two of the writers (4) have developed an analytical solution for wave propagation in open channel flow, based on a linearized form of the Saint Venant equations as presented by Lighthill and Whitham (3). They took the linearized equations and sought a solution in sinusoidal form which led to a system of homogeneous linear equations.

3.  DEFINITIONS

The following definitions are advanced: u_o = steady uniform flow mean velocity; d_o = steady uniform flow depth; S_o = bed slope; L = wavelength of sinusoidal perturbation to steady equilibrium flow; T = wave period of sinusoidal perturbation to steady equilibrium flow; c = wave celerity; L_o = reference channel length; F_o = steady uniform flow Foude number; σ_*= dimensionless wave number of the unsteady component of the motion; and τ_* = dimensionless wave period of the unsteady component of the motion, such that

L            c  =  ____           T

(1)

do            Lo  =  ____            So

(2)

uo            Fo  =  ____________             ( g do )1/2

(3)

in which g = the acceleration of gravity.

The propagation celerity c can be expressed in dimensionless form by dividing it by u_o. The dimensionless propagation celerity c_* is

c            c*  =  _____             uo

(6)

The logarithmic decrement δ (5) is defined as

δ  = ln (α1) - ln (αo)

(7)

in which α_o and α₁ = the wave amplitudes at the beginning and end of one wave period, respectively.

Two of the writers (4) have shown that for the dynamic model (that based on the complete Saint Venant equations), the dimensionless propagation celerity c and logarithmic decrement δ are functions of F_o and σ_* (see Appendix I). In practice, however, it is desirable to express the space parameter σ_* as a function of the time parameter τ_*. Combining Eqs. 1, 4, 5, and 6

2 π            τ*  =  ________             c* σ*

(8)

Thus, c_* and δ can be expressed as a function of F_o and τ_* by use of Eq. 8. Furthermore, the results of the theory suggest that for the comparison of diffusion and full dynamic models, a more appropriate parameter is τ_*/F_o. Making use of Eqs. 2, 3 and 5, τ_*/F_o is expressed as

τ*              So uo            _____  = T  ________    Fo                 do Fo

(9)

or

τ*                     g        _____  = T So ( _____ ) 1/2       Fo                    do

(10)

4.  KINEMATIC WAVE VERSUS DIFFUSION WAVE

The kinematic model breaks down when the neglect of the pressure term is not justified. Accordingly, it is of interest to compare the kinematic and diffusion models. Both models have a propagation celerity equal to 1.5 times the equilibrium flow velocity. They differ, however, in the attenuation. The logarithmic decrement of the kinematic model is 0, i.e., the kinematic model does not allow for physical attenuation. The attenuation often observed in numerical schemes based on the kinematic model is of an artificial nature (numerical damping due to truncation errors) (1). The logarithmic decrement of the diffusion model is (4)

τ* do          171 × 10            T  ≥  ________ =  _____________ ≅ 66 days            So uo        0.0001 × 3

(15)

If water discharge and channel friction are given, u_o and d_o can be calculated by the use of the appropriate uniform flow formula (Manning or Chezy).

TABLE 1. - Dimensionless period τ* versus attenuation factor eδd.
eδd	τ*
0.99	873
0.95	171
0.90	83

An explanatory note is necessary here. The criteria of Table 1 are based on a comparison of the attenuation (described by the logarithmic decrement δ) of the analytical solutions for the kinematic and diffusion models. In a numerical solution, however, often the truncation errors may mask the nondiffusive character of the analytical solution of the kinematic wave, with the result that the numerical solution of the kinematic wave may resemble the analytical solution of the diffusion wave (1), further complicating the modeling.

5.  DIFFUSION WAVE VERSUS DYNAMIC WAVE

The next step in the analysis is to compare the propagation celerity c_*d and logarithmic decrement δ_d of the diffusion model with those of the full dynamic model. For F_o < 2, the propagation celerity of the diffusion wave, c_*d = 1.5, is a lower bound for the dynamic celerity. Since only the primary dynamic wave (that which travels downstream) is of interest here, the dynamic wave celerity will be referred to as c_*1.

Fig. 4.  Attenuation factor of dynamic model e δ1 versus τ*/Fo.