1.  INTRODUCTION

When dealing with free-surface flows, the modeler often finds himself resorting to various simplifications of the full equations, in the interest of practicality and/or mathematical tractability. It is, therefore, not surprising that despite the perceived trend toward computational sophistication, simplified free-surface flow models continue to hold the attention of researchers and practitioners alike. In practice, modeling proves to be both an art and a science, a duality which requires a judicious compromise between achievable accuracy and resource utilization. It is this duality which is at the root of the diversity in free-surface flow modeling approaches.

The diffusion wave model is well established in the literature [Hayami, 1951; Lighthill and Whitham, 1955; Cunge, 1969; Dooge, 1973]. However, its capability to simulate inertial behavior has not been fully elucidated to date. As shown herein, under certain simplifying assumptions a generalized diffusion wave equation can be derived from the complete equations of continuity and motion (i.e., the shallow-water equations [Liggett, 1975]). Such a generalized diffusion wave model differs from Hayami's diffusion analogy [Hayami, 1951] in that the former includes inertia while the latter does not. This qualifies the generalized model as a diffusion wave model with inertial effects [Ponce, 1986]. Applications of the generalized diffusion wave theory are envisioned in both analytical and numerical modeling, ranging from the linear theory of hydrologic systems [Dooge, 1973] to strategies for two-dimensional depth-averaged free-surface flow computational modeling [Ponce and Yabusaki, 1981].

2.  THEORETICAL DEVELOPMENT

The theoretical background of the diffusion wave approach to unsteady free-surface flow can be traced back to Deymie [1938], Hayami [1951], Lighthill and Whitham [1955], and Dooge [1973]. Following Lighthill and Whitham [1955] the linearized analog of the equation of continuity a unit-width analysis is:

ht + u0hx + h0ux = 0

(1)

in which the subscripts x and t denote differentiation with respect to space and time, respectively, and the subscript denotes a reference flow value, either flow depth h or mean velocity u. The linearizedanalog of the equation of motion [Lighthill and Whitham, 1955], assuming Chézy friction, is:

2u      h ut + u0ux + ghx + gS0 ( ____ - ____) = 0                                        u0      h0

(2)

in which g = gravitational acceleration and S₀ = bottom slope.

The role of the inertial terms in describing runoff diffusion can be analyzed by a procedure similar to that used by Ponce and Simons [1977]. Accordingly, (2) is recast as:

2u      h aut + bu0ux + ghx + gS0 ( ____ - ____) = 0                                             u0     h0

(3)

in which the tracking variables a and b can take values of either 0 or 1, depending on whether the inertia terms are excluded from or included in the formulation.

3.  DIFFUSION WAVE EQUATION WITH INERTIA

Hyperbolic System

The diffusion wave equation with inertia is derived by differentiating Eq. 1 with respect to x, Eq. 1 with respect to t, and Eq. 3 with respect to x. Combining the resulting equations into a second-other equation in terms of flow depth leads to:

a + b                             a ht + 1.5 u0hx  =  (1 - b F 2 )νhxx  -  ( ______ ) F 2 νhxt  -  ( ______)νhtt                                                             u0                              gh0

(4)

In which the double subscripts represent double differentiation. In Eq. 4, ν = noninertial hydraulic diffusivity based on the reference flow [Hayami, 1951]:

u0h0 ν = ______         2S0

(5)

and F = reference flow Froude number:

u0 F = _________        (gh0)1/2

(6)

It can be readily shown that Eq. 4 is parabolic for the special case of a = b = 0 and hyperbolic otherwise, confirming that inertial systems are hyperbolic in nature. However. Ponce and Simons [1977] have shown that shallow-water flows are weakly hyperbolic, with the secondary wave (i.e., that associated with the C- characteristic [Abbott, 1979]) featuring a high degree of dissipation through a wide range of dimensionless wavenumbers. Such a mechanical system is readily amenable to conversion to a parabolic analog, which ostensibly can propagate in only one characteristic direction (C+).

Parabolic Analog to Hyperbolic System

To derive the generalized diffusion wave equation, a parabolic analog to the hyperbolic system of Eq. 4 is derived. To this end, Eq. 4 is differentiated with respect to space and, likewise, with respect to time, with the neglect of third-order terms justified by the usual license of differential calculus. The resulting second-order equations are substituted into the right side of Eq. 4 to yield:

ht + 1.5 u0hx = [(1 + b F 2) + 1.5 (a + b) F 2 - 2.25 a F 2 ] νhxx

(7)

Equation (7) describes convection and diffusion and is of parabolic type. The left side of Eq. 7 describes the convection of the flow depth with a wave celerity c = 1.5u₀ , applicable to a unit-width channel governed by Chezy friction. The right side describes the diffusion of the flow depth with a Froude-number-dependent hydraulic diffusivity. This qualifies Eq. 7 as a generalized diffusion wave equation which includes inertial effects (in its description of runoff diffusion). Accordingly, the diffusivity of Eq. 7 may properly be referred to as inertial hydraulic diffusivity.

By excluding or including the inertia terms, Eq. 7 specializes into the following models: (1) noninertial, (2) convective inertial, (3) local inertial, and (4) full inertial. The noninertial model [Hayami, 1951; Cunge, 1969] is obtained by setting a = b = 0 in Eq. 7:

ht + 1.5 u0hx = νhxx

(8)

in which ν is the noninertial hydraulic diffusivity, i.e., Eq. 5. The convective inertial model [Ponce and Simons, 1977] is obtained by setting a = 0 and b = 1 in Eq. 7:

ht + 1.5 u0hx = [1 + 0.5 F 2]νhxx

(9)

in which [1 + 0.5 F ²]ν is the hydraulic diffusivity including convective inertia only. The local inertial model [Ponce and Yabusaki, 1981] is obtained by setting a = 1 and b = 0 in Eq. 7:

ht + 1.5 u0hx = [1 - 0.75 F 2]νhxx

(10)

in which [1 - 0.75 F ²]ν is the hydraulic diffusivity including local inertia only. The full inertial model [Dooge, 1973] is obtained by setting a = 1 and b = 1 in Eq. 7:

ht + 1.5 u0hx = [1 - 0.25 F 2]νhxx

(11)

in which [1 - 0.25 F ²]ν is the full inertial hydraulic diffusivity.

A generalized diffusion wave equation encompassing all four models can be expressed as follows:

ht + 1.5 u0hx = [1 - α F 2]νhxx

(12)

in which α = 0 for the noninertial model, α = - 0.5 for the convective inertial model, α = 0.75 for the local inertial model, and α = 0.25 for the full inertial model.

The hydraulic diffusity of the full inertial model is Froude-number dependent, decreasing with an increase in Froude number in the range 0-2. Assuming a wide channel and Chezy friction, its neutral Froude number (i.e., the Froude number for which the hydraulic diffusivity vanishes) is F = 2, which simulates that of the complete equations [Lighthill and Whitham, 1955; Ponce and Simons, 1977].

The hydraulic diffusivity of the noninertial model is independent of the Froude number, which limits the noninertial model to low Froude number flows. Therefore, for high Froude number flows, the noninertial model tends to overpredict the hydraulic diffusivity. The hydraulic diffusivity of the convective inertial model is Froude number dependent, increasing with an increase in Froude number. The hydraulic diffusivity of the local inertial model is also Froude number dependent, decreasing with an increase in Froude number in the range 0 - (4/3)^0.5. Its neutral Froude number is F = (4/3)^0.5 = 1.15.

It is seen that the convective inertial (α = - 0.5) and local inertial (α = 0.75) models have an opposite behavior with regard to hydraulic diffusivity. When combined, the resulting full inertial model has α = 0.25, which is closer to the noninertial model (α = 0) than either convective or local models. Therefore, it is concluded that the noninertial model better approximates the hydraulic diffusivity of the full inertial model. For example, for a channel with u₀ = 1 m s^-1, h₀ = 1 m, and bottom slope S₀ = 0.0005, the Froude number is F = 0.32 and the noninertial hydraulic diffusivity (Eq. 5) is 1000 m² s^-1. The convective inertial diffusivity (Eq. 9) is 1050 m² s¹. The local inertial diffusivity (Eq. 10) is 925 m² s^-1. The full inertial diffusivity (Eq. 11) is 975 m² s^-1.

4.  SUMMARY AND CONCLUSIONS

A generalized diffusion wave equation is derived on the basis of the linear analogs of the complete equations of continuity and motion of free-surface flow. Specializations of this equation lead to the following four types of diffusion wave models, depending on whether the inertia terms (local and convective) are excluded from or included in the formulation: (1) full inertial, (2) local inertial, (3) convective inertial, and (4) noninertial.

Analysis of these diffusion wave models reveals substantial differences in their behavior, particularly with regard to the Froude number dependence of the hydraulic diffusivity. The full inertial and local inertial models have neutral Froude numbers, while the convective and noninertial models do not. In addition, the neutral Froude number of the full inertial model under Chezy friction simulates that of the complete equations (F = 2). For low Froude number flows, the noninertial model is shown to be a good approximation to the full inertial model. The noninertial model is a better approximation to the full inertial model than either local or convective models.

APPENDIX I.  REFERENCES

Abbott, M. B. 1979. Computational Hydraulics, 324 pp., Pitman. London.

Cunge. J. A. 1969. On the subject of a flood propagation computation method (Muskingum method), J. Hydraul. Res., 7(2), 205-230..

Deymie, P. 1938. Propagation d'une intumescence allongée (probleme aval), Appl. Mech. Proc. Int. Congr., 5th, 537-544.

Dooge, J. C. I. 1973. Linear theory of hydrologic systems, U.S. Dep. Agric. Tech. Bull., 1469, 327 pp..

Hayami, S. 1951. On the propagation of flood waves, Bull Disaster Prev. Res. Inst., Kyoto Univ., 1-16..

Liggett, J. A. 1975. Basic equations of unsteady flow, in Unsteady Flow in Open Channels, vol. I, edited by K. Mahmood and Yevjevich. pp. 29.02. Water Resources Publications, Fort Collins, Colo.

Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves. I. Flood movement in long rivers, Proc. Soc. London, Ser. A, 229, 281-316.

Ponce. V. M. 1986. Diffusion wave modeling of catchment dynamics. J. Hydraul. Eng., 112(8),716-727.

Ponce, V. M., and D. B. Simons. 1977. Shallow wave propagation in open channel flow, J. Hydraul. Div., Am. Soc. Civ. Eng., 103(HY11). 1461-1476.

Ponce, V. M., and S. Yabusaki. 1981. Modeling circulation in depth-averaged flow, J. Hydraul. Div., Am. Soc. Civ. Eng., 107(HY11). 1501-1518.