roll waves
  


The dynamic hydraulic diffusivity reexamined

Nicole R. Nuccitelli and Victor M. Ponce

05 May 2014


ABSTRACT

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.


1.  INTRODUCTION

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.


2.  THE HYDRAULIC DIFFUSIVITY

Hayami's hydraulic diffusivity, or kinematic hydraulic diffusivity, is (Hayami, 1951):

              qo   
νk  =   _______
             2 So
(1)

in which qo = unit-width discharge, and So = bottom slope.

Dooge's dynamic hydraulic diffusivity, applicable for Chezy friction in hydraulically wide channels, is (Dooge, 1973: Eq. 33b):

              qo               Fo 2
νd  =   _______  (1 - _____ )
             2 So              4
(2)

in which Fo = normal-flow Froude number.

Dooge et al.'s dynamic hydraulic diffusivity is (Dooge et al., 1982):

              qo   
νd  =   _______  [1 - (β - 1)2 Fo 2]
             2 So
(3)

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.

Ponce's dynamic hydraulic diffusivity is (Ponce, 1991b):

              qo   
νd  =   _______  (1 - Vo 2)
             2 So
(4)

in which Vo is the Vedernikov number, defined as follows:


Vo = (β - 1) Fo
(5)

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).


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 yo = flow depth, Ao = flow area, Qo = flow discharge, Sf = 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 = αAβ (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 To = 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 Sf  with So :

           1                       qo    
X =   _____  { 1  -  ____________   [1 - (β - 1)2 Fo 2] }
           2                  So co Δx
(27)

in which co is the flood wave celerity, co = m uo = β uo.

Replacing Eq. 5 in Eq. 27:

           1                       qo    
X =   _____  [ 1  -  ____________   (1 -  Vo 2) ]
           2                  So co Δx
(28)

The kinematic hydraulic diffusivity νk is given by Eq. 1. The grid diffusivity is νg = cox /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 So, 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 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.


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.


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