Source: http://milestones.sdsu.edu/the_dynamic_hydraulic_diffusivity_reexamined.html
Timestamp: 2019-04-23 08:20:31+00:00

Document:
The concept of hydraulic diffusivity and its extensions to the dynamic regime are examined herein. Hayami (1951) originated the concept of hydraulic diffusivity in connection with the propagation of flood waves. Dooge (1973) extended Hayami's hydraulic diffusivity to the realm of dynamic waves, specifically for the case of Chezy friction in hydraulically wide channels. Dooge's formulation amounts to a dynamic hydraulic diffusivity. Subsequently, Dooge et al. (1982) expressed the dynamic hydraulic diffusivity in terms of the exponent β of the discharge-area rating Q = αAβ. Lastly, Ponce (1991a, 1991b) expressed the dynamic hydraulic diffusivity in terms of the Vedernikov number, further clarifying the mechanics of flood wave propagation.
The concept of hydraulic diffusivity is due to Hayami (1951), who combined the equations of continuity and momentum of unsteady open-channel flow to derive a convection-diffusion equation, i.e., a partial differential equation containing a term of convection (first order) and a term of diffusion (second order). The modeling of flood wave propagation in terms of the convection-diffusion equation has been referred to as Hayami's diffusion analogy (Ponce, 1989).
The coefficient of the second order term of the convection-diffusion equation is the channel hydraulic diffusivity, or Hayami's hydraulic diffusivity. Since he neglected inertia in his formulation, his hydraulic diffusivity is properly a kinematic hydraulic diffusivity.
Dooge (1973) extended Hayami's hydraulic diffusivity to the realm of dynamic waves, specifically for the case of Chezy friction in hydraulically wide channels. Dooge's formulation amounts to a dynamic hydraulic diffusivity. Subsequently, Dooge et al. (1982) expressed the dynamic hydraulic diffusivity in terms of the exponent β of the discharge-area rating Q = αAβ. Later, Ponce (1991a, 1991b) expressed the dynamic hydraulic diffusivity in terms of the Vedernikov number, further clarifying the mechanics of flood wave propagation. These propositions are now explained in detail.
in which qo = unit-width discharge, and So = bottom slope.
in which Fo = normal-flow Froude number.
The variable β in Eq. 3 is the exponent of the rating Q = αAβ. For Chezy friction in hydraulically wide channels, β = 3/2, and Eq. 3 reduces to Eq. 2.
As the Froude number Fo → 0, the Vedernikov number Vo → 0 (Eq. 5), and the dynamic hydraulic diffusivity (Eq. 4) reduces to the kinematic hydraulic diffusivity (Eq. 1), a finite value which is independent of the Froude or Vedernikov numbers. On the other hand, when the Vedernikov number Vo → 1, the dynamic hydraulic diffusivity → 0. The condition Vo = 1 is the threshold of neutral stability, where roll waves tend to develop (Fig. 1).
Fig. 1 Roll waves in a steep irrigation canal, Cabana-Mañazo, Puno, Peru.
Under normal open-channel flow conditions, the Froude number Fo > 0; therefore, the Vedernikov number Vo > 0. Thus, the dynamic hydraulic diffusivity νd (Eq. 4) is always smaller than the kinematic hydraulic diffusivity νk (Eq. 1). In practice, the use of νk in lieu of νd will always exaggerate the amount of wave diffusion. In the limit, for Vo = 1, no diffusion is possible, and kinematic and dynamic waves travel at the same speed and are not subject to dissipation, thus leading to the development of roll waves such as those shown in Fig. 1 (Ponce and Simons, 1977).
in which yo = flow depth, Ao = flow area, Qo = flow discharge, Sf = friction slope, g = gravitational acceleration, and L = reach length Δx.
This variable (m) is the same as the exponent β of the discharge-area rating Q = αAβ (Ponce, 1989: Eq. 9-13).
in which co is the flood wave celerity, co = m uo = β uo.
Equation 29 is the same as Eq. 4. It is confirmed that the expression within parenthesis in the dynamic hydraulic diffusivity is a function not of the Froude number, as presented by Dooge (1973), but of the Vedernikov number, as presented by Ponce (1991b).
The script ONLINE DYNAMIC HYDRAULIC DIFFUSIVITY calculates the kinematic and dynamic hydraulic diffusivities, given a set of hydraulic variables consisting of (mean) velocity u, flow depth y, bottom slope So, and exponent β of the rating.
The Dooge et al. equation for dynamic hydraulic diffusivity, Eq. 3, is shown to be a function of the Froude number Fo and the exponent β of the discharge-area rating Q = α Aβ. The Ponce formulation for dynamic hydraulic diffusivity, Eq. 4, is shown to be a function of the Vedernikov number Vo. In view of the relation between Froude and Vedernikov numbers, Eq. 5, the formulations of Dooge et al. (1982) and Ponce (1991) are equivalent.
The Vedernikov-number dependent formulation for hydraulic diffusivity is recommended for increased modeling accuracy in the following applications: (1) channel routing using the Muskingum-Cunge method (Ponce and Yevjevich, 1978); and (2) overland flow routing using the diffusion wave model of catchment dynamics (Ponce, 1986). An online calculator is provided to round up the experience.
Dooge, J. C. I. (1973). Linear theory of hydrologic systems, Technical Bulletin No. 1468, Agricultural Research Service, U.S. Department of Agriculture, Washington, D.C.
Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski. (1982). Hydrodynamic derivation of storage parameters of the Muskingum model, Journal of Hydrology, Vol. 54, 371-387.
Hayami, S. (1951). On the propagation of flood waves. Bulletin No. 1, Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan, December.
Ponce, V. M., and D. B. Simons. (1977). Shallow wave propagation in open channel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, pages 1461-1476, December.
Ponce, V. M., and V. Yevjevich. (1978). Muskingum-Cunge method with variable parameters. Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, pages 1663-1667, December.
Ponce, V. M. (1986). Diffusion wave modeling of catchment dynamics. Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 8, pages 716-727, August.
Ponce, V. M. (1989). Engineering hydrology: Principles and practices, Prentice Hall, Englewood Cliffs, New Jersey.
Ponce, V. M. (1991a). The kinematic wave controversy. Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 4, pages 511-525, April.
Ponce, V. M. (1991b). New perspective on the Vedernikov number. Water Resources Research, Vol. 27, No. 7, pages 1777-1779, July.

References: V. 
 V. 
 V. 
 V. 
 V. 
 V. 
 V.