Ponce, V. M. 1979. "On the classification of open channel flow regimes," Proceedings, Fourth National Hydrotechnical Conference, Vancouver, B.C., Canada. |

A coherent treatment of open channel flow regimes is presented. Three representative velocities and three representative diffusivities are identified. From these, at most two independent velocity ratios (the Froude and Vedernikov numbers) and two independent diffusivity ratios (the Reynolds and Ponce-Simons numbers) can be formulated. These ratios establish the criteria for the classification of open channel flow regimes into: (1) subcritical or supercritical, (2) stable or unstable, (3) laminar or turbulent, and (4) kinematic or inertial.
Since the publication of the paper by Robertson and Rouse in 1941 (10), open channel flow has been classified into the following four regimes: laminar-subcritical, turbulent-subcritical, laminar-supercritical, and turbulent-supercritical. The Reynolds number Jeffreys (4) and Vedernikov (11, 12) have laid the foundations for the classification of open-channel flow according to the stability of the free surface. They classified the flow as stable or unstable depending on whether surface disturbances (waves) tend to attenuate or amplify in time. The Vedernikov number (9) is defined as:
in which
In Eq. 2,
in which k = a coefficient. According to the Vedernikov criterion, the flow will be stable for V < 1, and unstable for V > 1. At V = 1, the flow is said to be neutrally stable, or for short, "neutral".
Ponce and Simons (7) have recently made a study of wave propagation in open channel flow. Their study leads to another classification of open channel flow, depending on which forces are dominant in the wave movement. According to Ponce and Simons, free-surface shallow water waves can be classified as: (1) kinematic, if the inertia terms can be neglected; (2) inertial, if the bottom friction and gravity (bed slope) terms can be neglected; and (3) dynamic, if none of these terms can be neglected without incurring a significant loss of information. A coherent treatment of the foregoing classification criteria is presented herein. It is shown that while the Froude and Vedernikov criteria are ratios of velocities (or celerities), the Reynolds and Ponce-Simons criteria are ratios of diffusivities (or viscosities). Three velocities and three diffusivities are identified, from which at most two independent velocity ratios (the Froude and Vedernikov numbers) and two independent diffusivity ratios (the modified Reynolds and Ponce-Simons numbers) can be formulated.
There are three representative velocities in open channel flow:
(1) the average velocity of the particles, (2) the velocity of inertial waves, and (3) the velocity of kinematic waves. The velocity of the particles varies within the flow depth; therefore, in a one-dimensional formulation the depth-averaged velocity The velocity of inertial waves (inertial wave celerity) is that of a wave governed exclusively by Inertia and pressure forces. It is referred to variously as the Lagrangian celerity [after Lagrange (5) who first derived it], dynamic wave celerity (6) and small gravity-wave celerity (1). However, in the context of open-channel flow, the terms inertial wave and inertial wave celerity are preferred. The relative inertial wave celerity (the wave velocity relative to the mean flow velocity) is:
in which The velocity of kinematic waves (kinematic wave celerity) is that of a wave governed by bottom friction and gravity, to the exclusion of inertia. Such a wave can be of the diffusive-kinematic type if the pressure gradient is taken into account, or nondiffusive-kinematic if it is neglected. The kinematic wave celerity is also referred to as the Kleitz-Seddon celerity (1). It is expressed as follows :
in which
In Eq. 6,
There are three representative diffusivities in open-channel flow: (1) the molecular diffusivity, (2) the channel diffusivity and (3) the spectral diffusivity. The molecular diffusivity
in which τ = shear stress, ρ = mass density of liquid, and ∂
in which
is a characteristic molecular length.
The channel diffusivity
in which
is a characteristic channel length, or the channel length necessary for the flow to drop an elevation equal to its normal depth. The channel diffusivity
The spectral diffusivity
in which
The Froude criterion is characterized by the Froude number, defined as the ratio of the depth-averaged velocity
The flow is classified as subcritical for
The Reynolds criterion is characterized by the Reynolds number commonly defined as follows:
Small values of
For the purposes of this paper, a modified Reynolds number
such that
The Vedernikov criterion is characterized by the Vedernikov number defined by Eq. 1. Craya (2) has shown that the Vedernikov number is really the ratio of the relative kinematic wave celerity to the relative inertial wave celerity:
According to the Vedernikov criterion, the flow is classified as stable for
According to Ponce and Simons (7), free-surface shallow water waves can be classified as follows: (1) kinematic, if the inertia terms can be neglected; (2) inertial, if the bottom friction and gravity terms can be neglected; and (3). dynamic, if none of these terms can be neglected without incurring a significant loss of information. They identified a dimensionless wave number σ
In general, small values of σ
in which
For the purpose of this paper, a dimensionless number
such that
The three velocities (average velocity c) and the three diffusivities (molecular diffusivity _{rk}D, channel diffusivity _{m}D_{o}, and spectral diffusivity D) give rise to at most two independent velocity ratios and two independent diffusivity ratios, as follows:
Froude number:
Vedernikov number:
Reynolds number (modified):
Ponce-Simons number:
The velocity ratios provide an exact delineation of the regime limits, i.e.,
A coherent treatment of open-channel flow regimes is presented. Three representative velocities and three representative diffusivities are identified. From these, at most two independent velocity ratios (thy Froude and Vedernikov numbers) and two independent diffusivity ratios (the modified Reynolds and the Ponce-Simons numbers) can be formulated. These ratios establish the criteria for a classification of open channel flow regimes into: (1) subcritical or supercritical (Froude number), (2) stable or unstable (Vedernikov number), (3) laminar or turbulent (Reynolds number), and (4) kinematic or inertial (Ponce-Simons number). While for the celerity ratios there is a precise regime limit (
Chow, V. T., *Open Channel Hydraulics*, McGraw-Hill Book Company, Inc., New York, N.Y., 1959.Craya, A., "The Criterion for the Possibility of Roll Wave Formation," in *Gravity Waves*, National Bureau of Standards Circular 521, 1952, pp. 141-151.Dooge, J. C. I., "Linear Theory of Hydrologic Systems," *Technical Bulletin No 1468,*Agricultural Research Service, Oct. 1973.Jeffreys, H., "The Flow of Water in an Inclined Channel of Rectangular Section," *Philosophical Magazine and Science Journal,*Vol. 49, Series 6, May 1925, pp. 793-807.Lagrange, I. L.,"Mémoire sur la Théorie du Mouvement des Fluides," *Bulletin de la Classe des Sciences Academie Royal de Belgique,*1783, pp. 151-198.Lighthill, M. J., and Whitham, G. B., "On Kinematic Waves, I. Flood Movement in Long Rivers," *Proceedings, Royal Sooiety of London,*London, England, Series A., Vol. 229, 1955, pp. 281-316.Ponce, V. M., and Simons, D. B., "Shallow Wave Propagation in Open Channel Flow," *Journal of the Hydraulics Division,*ASCE, Vol. 103, HY12, Proc. Paper 13392, Dec. 1977, pp. 1461-1476.Ponce, V. M., Li, R. M., and Simons, D. B., "Applicability of Kinematic and Diffusion Models," *Journal of the Hydraulics Division,*ASCE, Vol. 104, 053, Proc. Paper 13635, Mar. 1978, pp. 353-360.Powell, R. W., "Vedernikov's Criterion for Ultra-Rapid Flow," *Transactions, American Geophysical Union,*Vol. 29, No. 6, Dec. 1948, pp. 882-886.Robertson, J. M., and Rouse, H., "On the Four Regimes of Open Channel Flow." *Civil Engineering,*Vol. 2, No. 3, Mar. 1941, pp. 169-171.Vedernikov, V. V., "Conditions at the Front of a Translation Wave Disturbing a Steady Motion of a Real Fluid," *C. R. (Doklady) U.S.S.R. Academy of Sciences,*Vol. 48, No. 4, 1945, pp. 239-242.Vedernikov, V. V.. "Characteristic Features of a Liquid Flow in an Open Channel," *C. R. (Doklady) U.S.S.R. Academy of Sciences,*Vol. 52, 1946, pp. 207-210.
The following symbols are used in this paper:
ρ = mass density of liquid; σ = wave number, (σ = 2π/L);
σ τ = shear stress, Eq. 8;
τ |

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