Upper Paraguay river at Porto Murtinho, Mato Grosso do Sul, Brazil,
featuring a flood hydrograph lasting one year (the maximum possible), clearly the quintessential kinematic flood wave.

THE PONCESIMONS NUMBER
Victor M. Ponce
Professor Emeritus of Civil and Environmental Engineering
San Diego State University, San Diego,
California
January 29, 2024
ABSTRACT.
The three diffusivities relevant in fluid mechanics and openchannel flow (molecular, hydraulic, and spectral), are
appropriately defined and explained. This article focuses on the PonceSimons number, properly the ratio of
hydraulic and spectral diffusivities, while being affected with the factor 2π.
This dimensionless number
characterizes the spatial scale and associated properties of surface disturbances in unsteady openchannel flow.

1. INTRODUCTION
In hydraulic engineering, viscosity, or its synonym, diffusivity, is a fundamental fluid property.
Diffusivity
is the first moment of the flow velocity.
Therefore, the units of diffusivity are (L/T)L, or its equivalent expression L^{2}/T.
The equality ν = 1 m^{2}/sec
describes the mathematical certainty
that a given disturbance will diffuse at the rate of ν = 1 m^{2}/sec.
In fluid mechanics, diffusivity
relates to the process of diffusion;
in engineering hydrology, to flood wave
attenuation, or dissipation.
In hydraulic mathematical modeling, diffusivity is described by the
secondorder term of a differential
equation (Table 1).
Table 1. Comparison of velocities and diffusivities in openchannel flow. 
Property 
Symbol 
Units 
Process 
Order 
Velocity  u  L/T 
Convection, advection  First 
Diffusivity 
ν 
L^{2}/T 
Diffusion, dissipation  Second 
The fluid properties listed in Table 1 describe the
flow up to
second order. In this article, we focus on the PonceSimons number,
a ratio of diffusivities which characterizes the spatial
scale of the wave phenomena.
Increased understanding of this dimensionless number significantly
enhances the comprehension of
wave phenomena in unsteady openchannel flow.
2. DIFFUSIVITIES IN OPENCHANNEL FLOW
Three diffusivities are recognized in openchannel flow:
Molecular diffusivity,
Hydraulic diffusivity, and
Spectral diffusivity.
In fluid mechanics,
the molecular diffusivity ν_{m}
is commonly referred to as kinematic viscosity ν,
a measure of the fluid's
internal resistance to flow at the molecular level.
In openchannel flow, the hydraulic diffusivity ν_{h} is expressed in terms of the unitwidth discharge
and bottom slope. In unsteady openchannel flow, the spectral diffusivity
ν_{s}
is defined in terms of the wavelength of the sinusoidal perturbation to the steady flow.
These propositions are further explained in Box A.
Box A. Diffusivities in openchannel flow.
Newton's law of viscosity is: τ /ρ =
ν (∂u/∂s), in which
τ = shear stress, ρ = mass density of the fluid,
ν = kinematic viscosity of the fluid (molecular diffusivity)
and (∂u/∂s) = velocity gradient in the direction s perpendicular to
the direction of τ. For our purpose:
τ /ρ =
ν_{m} (∂u/∂s)
The molecular diffusivity
may be expressed as
ν_{m} = u (L_{m} /2),
in which L_{m} = (2ν_{m} /u) is a characteristic
molecular length (Chow, 1959).
The hydraulic diffusivity
is defined as ν_{h} =
u (L_{o} /2), in which L_{o} = (d_{o} /S_{o}) is a characteristic
hydraulic length, defined as the
distance measured along the channel wherein
the flow drops a head (i.e., an elevation) equal to its equilibrium depth
(Hayami, 1951; Ponce and Simons, 1977).
The spectral diffusivity ν_{s}
is defined as ν_{s} =
u (L /2), in which L = characteristic
wavelength of the sinusoidal surface disturbance
(Ponce, 1979).
Note that all three diffusivities
(molecular, hydraulic, and spectral)
are defined in terms
of their respective characteristic lengths: (1) molecular length L_{m}, (2)
hydraulic length L_{o}, and (3) spectral wavelength L. Pointedly,
we observe that the three diffusivities
share a similar algebraic structure: A product of the convective
velocity times onehalf of the respective characteristic length.

3. THE PONCESIMONS NUMBER
The three diffusivities identified in Box A give rise to only two independent dimensionless numbers
(Ponce, 2023b):
The ratio of hydraulic to molecular
diffusivity, clearly a type of Reynolds number; and The ratio of hydraulic to spectral diffusivity,
a type of PonceSimons number.
In their seminal work on shallow wave propagation, Ponce and Simons (1977) defined a dimensionless wavenumber
as follows:
σ_{*} = (2π /L)L_{o}.
It is observed that the PonceSimons number is indeed a surrogate for a ratio of diffusivities, since:
σ_{*} = (2π /L)L_{o} = 2π (ν_{h} /ν_{s}).
The PonceSimons dimensionless wavenumber σ_{*}
classifies the entire realm of unsteady flow disturbances into
four spectral ranges (Fig. 1):
Kinematic (extreme left),
Diffusion (leftofcenter),
Mixed kinematicdynamic (rightofcenter), and
Dynamic (extreme right).
The precise domains of these ranges have been examined by
Ponce (2023a):
Kinematic flow:
σ_{*} < 0.001.
Diffusion flow:
0.001 ≤ σ_{*} < 0.17.
Mixed kinematicdynamic flow:
0.17 ≤ σ_{*} < 1 to 100,
depending on the Froude
number (refer to Fig. 1).
Dynamic flow:
σ_{*} ≥ 10 to 1000, depending on the
Froude number (refer to Fig. 1).
Fig. 1 Dimensionless relative wave celerity
c_{r}_{*} vs
dimensionless wavenumber σ_{*}.

The findings of Ponce and Simons (1977), depicted in Fig. 1, elucidate
the behavior of all wave types
in unsteady openchannel flow. These include
both "long" waves, of a kinematic nature, towards the far left
side of Fig. 1, and "short" waves, of a dynamic nature, towards
the far right, both of which ostensibly feature constant
celerity. Also included are the
diffusion waves, in the leftofcenter range and displaying properties
that are shown
to be quite practical, and the
mixed kinematicdynamic waves in the rightofcenter range.
The latter are, for the most part, impractical
due to their extremely strong dissipative tendencies
(Ponce, 2023a).
4. SUMMARY
The three diffusivities relevant in fluid mechanics and openchannel flow [(1) molecular, (2) hydraulic, and
(3) spectral)], are
appropriately defined and explained. This article focuses on the PonceSimons number, properly the ratio of
hydraulic and spectral diffusivities, while being affected with the factor 2π.
This dimensionless number
characterizes the spatial scale and associated properties of surface disturbances in unsteady openchannel flow.
REFERENCES
Chow, V. T. 1959. Openchannel hydraulics. McGrawHill, Inc, New York, NY.
Hayami, I. 1951.
On the propagation of flood waves. Bulletin, Disaster Prevention Research Institute,
No. 1, December, Extract.
Ponce, V. M. and D. B. Simons. 1977.
Shallow wave propagation in open channel flow.
Journal of Hydraulic Engineering ASCE, 103(12), 14611476.
Ponce, V. M. 1979.
On the classification of open channel flow regimes. Proceedings,
Fourth National Hydrotechnical Conference, Vancouver, British Columbia, Canada.
Ponce, V. M. 2023a.
When is the diffusion wave applicable?
Online article.
Ponce, V. M. 2023b.
Ths states of flow
Online article.
