Fig. 1  Roll waves in a steep irrigation canal, Cabana-Mañazo, Puno, Peru.

Under normal open-channel flow conditions, the Froude number F_o > 0; therefore, the Vedernikov number V_o > 0. Thus, the dynamic hydraulic diffusivity ν_d (Eq. 4) is always smaller than the kinematic hydraulic diffusivity ν_k (Eq. 1). In practice, the use of ν_k in lieu of ν_d will always exaggerate the amount of wave diffusion. In the limit, for V_o = 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).

3.  THE DYNAMIC HYDRAULIC DIFFUSIVITY

In deriving the hydrodynamic Muskingum parameters including inertia, Dooge et al. (1982) expressed X in terms of hydraulic variables as follows (op. cit., Eq. 37b):

1         M         R    X =   ___  +  ____  -  ____            2         N         P

(6)

in which:

g yo    M =   ______ (1 - Fo2)               L

(7)

∂Sf    N = g Ao  _____                   ∂A

(8)

∂Sf    P = g Ao  _____                   ∂Q

(9)

2 Qo          1    (∂Sf / ∂A)    R =   ______  +   ___  ___________            Ao L          L    (∂Sf / ∂Q)

(10)

in which y_o = flow depth, A_o = flow area, Q_o = flow discharge, S_f = friction slope, g = gravitational acceleration, and L = reach length Δx.

The ratio M /N is:

M                  1            yo    ______  =   __________ ______  (1 - Fo2)    N              ∂Sf /∂A      A L

(11)

The ratio R /P is:

R                   1             2 Qo               ∂Sf /∂A              1 ______  =   __________ ___________  +  ____________   _________    P              ∂Sf /∂Q      g Ao2 L           (∂Sf /∂Q)2        g Ao L

(12)

Dooge et al. (1982) defined the variable m:

∂Sf / ∂A       Ao    m = -  ___________  _____              ∂Sf / ∂Q      Qo

(13)

This variable (m) is the same as the exponent β of the discharge-area rating Q = αA^β (Ponce, 1989: Eq. 9-13).

Therefore:

∂Sf / ∂A       Ao    ∂Sf / ∂Q = -  ___________  _____                            m            Qo

(14)

With Eq. 14, the ratio R /P reduces to:

R                    m             2 Qo2                m2              Qo2 ______  =   - __________ ___________  +  ___________   _________    P                ∂Sf /∂A       g Ao3 L             ∂Sf /∂A        g Ao3 L

(15)

R                     1              Qo2                ______  =   - __________ ___________  (2 m - m 2)    P                ∂Sf /∂A       g Ao3 L

(16)

From the definition of the Froude number, with T_o = top width:

Qo2 To    Fo2  =   _________                g Ao3

(17)

R                     1            Fo2                ______  =   - __________ ________  (2 m - m 2)    P                ∂Sf /∂A       To L

(18)

R                     1           yo Fo2                ______  =   - __________ __________  (2 m - m 2)    P                ∂Sf /∂A        Ao L

(19)

Replacing Eqs. 11 and 19 into Eq. 6:

1                    1            yo     X =   _____  +   __________    ______  [1 - (m - 1)2 Fo 2]            2             ∂Sf / ∂A       Ao L

(20)

From Eq. 14:

Qo    ∂Sf / ∂A = -  m (∂Sf / ∂Q) _______                                            Ao

(21)

Since:

Qo2     Sf  =   _______               K2

(22)

in which K = conveyance, then:

2 Sf     ∂Sf / ∂Q  =   _______                        Qo

(23)

Replacing Eq. 23 into Eq. 21:

Sf    ∂Sf / ∂A = -  2 m  _______                                Ao

(24)

Replacing Eq. 24 into Eq. 20:

1                  1            yo     X =   _____  -   __________    _____  [1 - (m - 1)2 Fo 2]            2             2 m Sf         L

(25)

Or:

1                      yo     X =   _____  { 1  -  __________   [1 - (m - 1)2 Fo 2] }            2                 m Sf L

(26)

Replacing L with Δx, m with β, and S_f  with S_o :

1                       qo     X =   _____  { 1  -  ____________   [1 - (β - 1)2 Fo 2] }            2                  So co Δx

(27)

in which c_o is the flood wave celerity, c_o = m u_o = β u_o.

Replacing Eq. 5 in Eq. 27:

1                       qo     X =   _____  [ 1  -  ____________   (1 -  Vo 2) ]            2                  So co Δx

(28)

The kinematic hydraulic diffusivity ν_k is given by Eq. 1. The grid diffusivity is ν_g = c_o (Δx /2) (Ponce, 1989). Thus, from Eq. 28, it follows that the dynamic hydraulic diffusivity is:

qo    νd  =   _______  (1 - Vo 2)              2 So

(29)

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 S_o, and exponent β of the rating.

4.  SUMMARY

The Dooge et al. equation for dynamic hydraulic diffusivity, Eq. 3, is shown to be a function of the Froude number F_o and the exponent β of the discharge-area rating Q = α A^β. The Ponce formulation for dynamic hydraulic diffusivity, Eq. 4, is shown to be a function of the Vedernikov number V_o. 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.

REFERENCES

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.