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

 On the classification of open channel flow regimes Victor M. Ponce Online version 2016[Original version 1979]

ABSTRACT

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.

1.  INTRODUCTION

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 R = ud /ν (u = mean velocity, d = flow depth, and ν = kinematic viscosity) is used to characterize laminar or turbulent flow, a small R indicating laminar flow and a large R indicating turbulent flow. In general, values of 1000 < R < 3000 correspond to a transitional range in which the flow is neither laminar nor turbulent; however, the precise limits of the transitional range are not clearly defined. The Froude number F = u / (gd)1/2 (g = acceleration of gravity) is used to characterize subcritical or supercritical flow, the flow being referred to as subcritical for F < 1, and supercritical for F > 1. For F = 1, the flow is said to be at critical state.

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:

 m V = ___ MF         n (1)

in which M is a cross-sectional shape number defined as follows:

 dP M = 1 - R ____                  dA (2)

In Eq. 2, R = hydraulic radius, P = wetted perimeter, and A = cross-sectional flow area. In Eq. 1, m and n are the exponents of R and u in the friction relationship.

 un Sf  =  k ______               Rm (3)

in which Sf = friction slope, and 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.

2.  VELOCITIES IN OPEN CHANNEL FLOW

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 u is taken as the representative 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:

 cri = (gdo)1/2 (4)

in which do is the normal flow depth.

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 :

 ck = βu (5)

in which β is the exponent of A in the steady-state discharge-area relation

 Q = αAβ (6)

In Eq. 6, Q = discharge and α = a coefficient. The relative kinematic wave celerity is:

 crk = (β - 1)u (7)

3.  DIFFUSIVITIES IN OPEN-CHANNEL FLOW

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 Dm is commonly referred to as kinematic viscosity in the relation

 τ           ∂u ____ = ν ____   ρ           ∂s (8)

in which τ = shear stress, ρ = mass density of liquid, and ∂u/∂s = velocity gradient in a direction perpendicular to τ. The molecular diffusivity can also be expressed as follows:

 uLm Dm = ν = _______                      2 (9)

in which

 2ν Lm = _____             u (10)

is a characteristic molecular length.

The channel diffusivity Do is defined as follows:

 uLo Do = _____             2 (11)

in which

 do Lo = _____            So (12)

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 Do is usually referred to as "coefficient of diffusivity" (6), or in a slightly modified way, as "hydraulic diffusivity" (3).

The spectral diffusivity D is defined as follows:

 uL D = _____           2 (13)

in which L is the spectral wavelength of a sinusoidal surface wave. From Eqs. 9, 11 and 13, the similarities among the three diffusivities are apparent. All are products of the average velocity u times one-half of a certain length. In the case of the molecular diffusivity, it is the characteristic molecular length, defined by Eq. 10. In the case of the channel diffusivity, it is the characteristic channel length defined by Eq. 12. For the spectral diffusivity, it is the spectral wavelength.

4.  THE FROUDE CRITERION:  SUBCRITICAL, CRITICAL OR SUPERCRITICAL FLOW

The Froude criterion is characterized by the Froude number, defined as the ratio of the depth-averaged velocity u to the relative inertial wave celerity cri.

 u              u F = _____ = _________          cri        (gd) 1/2 (14)

The flow is classified as subcritical for F < 1 and as supercritical for F > 1. For F = 1, the flow is said to be at critical state. From a physical point of view, the critical state refers to the stationarity of secondary inertial waves. In practice, this means that in subcritical flow, surface disturbances have two directions of propagation (upstream and downstream), while in supercritical flow they have only one (downstream, since the secondary inertial waves cannot travel upstream).

5.  THE REYNOLDS CRITERION:  LAMINAR, TRANSITIONAL OR TURBULENT FLOW FLOW

The Reynolds criterion is characterized by the Reynolds number commonly defined as follows:

 ud R = _____            ν (15)

Small values of R are used to describe laminar flow, while large values correspond to turbulent flow. For a range of intermediate values of R the flow is neither laminar nor turbulent, and it is referred to as transitional.

For the purposes of this paper, a modified Reynolds number R* is defined as follows:

 Do        Lo R* = _____ = _____          Dm        Lm (16)

such that

 R R* = _____           2So (17)

6.  THE VEDERNIKOV CRITERION:  STABLE, NEUTRAL OR UNSTABLE FLOW

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:

 crk V = _____          cri (18)

According to the Vedernikov criterion, the flow is classified as stable for V < 1 and as unstable for V > 1. For V = 1, the flow is said to be neutrally stable, or for short, neutral. From a physical point of view, the neutrally stable state is that in which the wave celerity is the same throughout the wavelength spectrum (the kinematic, dynamic and inertial wave celerities are all equal!).

7.  THE PONCE-SIMONS CRITERION:  KINEMATIC, DYNAMIC OR INERTIAL FLOW

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* to characterize these regimes, σ* being defined as follows:

 2π σ* = (_____) Lo             L (19)

In general, small values of σ* describe kinematic flow, while large values correspond to inertial flow. Intermediate values of σ* characterize dynamic flow, the σ*-range for dynamic flow being Froude number-dependent. To partially offset this dependency with the aim of analyzing the applicability of the kinematic models, Ponce et al. (8) introduced a dimensionless wave period normalized with respect to the Froude number τ*/Fo, defined as follows:

 τ*                   g _____ = TSo (_____)1/2   Fo                 do (20)

in which T = wave period, and Fo = Fronde number corresponding to the normal flow depth do.

For the purpose of this paper, a dimensionless number P is defined as follows:

 Do         Lo P = ______ = _____           D           L (21)

such that

 σ* P = ____         2π (22)

8.  OPEN-CHANNEL FLOW REGIMES

The three velocities (average velocity u, relative inertial wave celerity cri, and relative kinematic wave celerity crk) and the three diffusivities (molecular diffusivity Dm, channel diffusivity Do, and spectral diffusivity D) give rise to at most two independent velocity ratios and two independent diffusivity ratios, as follows:

Froude number:

 u F = _____          cri (14)

Vedernikov number:

 crk V = _____          cri (18)

Reynolds number (modified):

 Do        R* = _____           Dm (16)

Ponce-Simons number:

 Do P = _____          D (21)

The velocity ratios provide an exact delineation of the regime limits, i.e., F = 1: critical flow is the limit between the subcritical (F < 1) and supercritical (F > 1) regimes; V = 1: neutral flow is the limit between the stable (V < 1). and unstable regimes (V > 1). The diffusivity ratios do not provide an exact delineation of the regime limits. Therefore, it is necessary to define an intermediate range. For the Reynolds criterion, this intermediate range is referred to as transitional flow; for the Ponce-Simons criterion, the intermediate range corresponds to dynamic flow. Table 1 provides a ready reference to the four criteria fox the classification of open channel flow regimes.

 TABLE 1.-  OPEN-CHANNEL FLOW REGIMES

9.  SUMMARY AND CONCLUSIONS

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 (F =1: critical flow; V = 1: neutral flow), for the diffusivity ratios there is no such clear-cut specification. Rather, the term transitional flow regime is used to describe a condition that is neither laminar nor turbulent, while the term dynamic flow describes a condition intermediate between the kinematic and inertial flow regimes.

APPENDIX | - REFERENCES

1. Chow, V. T., Open Channel Hydraulics, McGraw-Hill Book Company, Inc., New York, N.Y., 1959.

2. Craya, A., "The Criterion for the Possibility of Roll Wave Formation," in Gravity Waves, National Bureau of Standards Circular 521, 1952, pp. 141-151.

3. Dooge, J. C. I., "Linear Theory of Hydrologic Systems," Technical Bulletin No 1468, Agricultural Research Service, Oct. 1973.

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. Powell, R. W., "Vedernikov's Criterion for Ultra-Rapid Flow," Transactions, American Geophysical Union, Vol. 29, No. 6, Dec. 1948, pp. 882-886.

10. 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.

11. 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.

12. 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.

APPENDIX || - NOTATION

The following symbols are used in this paper:

A = cross-sectional flow area;

ck = kinematic wave celerity, Eq. 5;

cri = relative inertial wave celerity, Eq. 4;

crk = relative kinematic wave celerity, Eq. 7;

D = spectral diffusivity, Eq. 13;

Dm = molecular diffusivity, Eq. 9;

Do = channel diffusivity, Eq. 11;

d = flow depth;

do = normal flow depth;

F = Froude number;

Fo = Froude number for normal flow;

g = acceleration of gravity;

k = coefficient in Eq. 3;

L = spectral wavelength;

Lm = characteristic molecular length, Eq. 10;

Lo = characteristic channel length, Eq. 12;

M = cross-sectional shape number, Eq. 2;

m = exponent of R in Eq. 3;

n = exponent of u in Eq. 3;

P = wetted perimeter;

P = Ponce-Simons number, Eqs. 21 and 22;

Q = discharge;

R = Reynolds number;

R* = modified Reynolds number, Eqs. 16 and 17;

Sf = friction slope;

So = channel bed slope;

T = wave period;

u = depth-averaged flow velocity;

V = Vedernikov number;

α = coefficient in Eq. 6;

β = exponent of A in Eq. 6;

ν = kinematic viscosity;

ρ = mass density of liquid;

σ = wave number, (σ = 2π/L);

σ* = Ponce-Simons dimensionless wave number, Eq. 19;

τ = shear stress, Eq. 8;

τ* = dimensionless wave period, in Eq. 20.;

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