1.  INTRODUCTION

Roll waves can occur in an unstable free-surface flow regime, i.e., when the Vedernikov number exceeds 1. The flow condition can be either laminar or turbulent. In the laminar regime, an example of roll waves is the pulsating flow often observed in steep urban catchments following intense rainfall. In the turbulent flow regime, roll waves may require a specific type of boundary condition for their inception.

The Vedernikov number V > 1 (Vedernikov 1945, 1946) is the necessary condition for roll-wave formation. The existence of this criterion was confirmed by Craya (1952) and lwasa (1954). Using the theory of linear stability, Ponce and Simons (1977) confirmed the Vedernikov criterion for turbulent Chezy friction in hydraulically wide channels. In addition, Ponce and Simons identified the scale of the disturbances under which roll waves are most likely to develop. Brock (1967) experimentally studied roll-wave propagation using a laboratory flume. Brock measured crest depths and wave periods under a wide range of flow conditions for turbulent Chezy friction in hydraulically wide channels.

The objective of this note is to use Brock's (1967) laboratory data to test the validity of Ponce and Simons' (1977) theory. Experimental wavenumbers, calculated from wave periods measured by Brock, are compared with wavenumbers predicted by the theory. An approximate match of wavenumbers confirms the validity of the theory. In a practical setting, this knowledge can be used to promote or inhibit the development of roll waves.

2.  VEDERNIKOV NUMBER

The Vedernikov number is defined as (Chow 1959)

V = xγ F

(1)

in which x = exponent of hydraulic radius R in the mean velocity functional relation u = f(R ); γ = cross-sectional shape factor; and F = Froude number. The parameter x is defined as:

1 + b x = _______         2 - b

(2)

in which b = exponent of Reynolds number R in the frictional power law f = α(R)^-b, in which f = Darcy-Weisbach friction factor. Parameter b varies in the range 0-1, with b = 0 for turbulent Chezy friction; and b = 1 applicable to laminar flow. The shape factor γ is defined as:

dP y = 1 - R ______                  dA

(3)

in which P = wetted perimeter; and A = cross-sectional flow area. The shape factor γ varies in the range 0-1, with γ = 0 applicable to a channel of constant hydraulic radius R, an inherently stable channel (Liggett 1975; Ponce 1991), and γ = 1 for the case of a hypothetical channel of constant wetted perimeter P, a hydraulically wide channel.

3.  THEORY OF ROLL WAVE FORMATION

The linear stability analysis of Ponce and Simons (1971) expresses the propagation characteristics of shallow water waves in terms of: (1) the Froude number F of the steady uniform flow; and (2) a dimensionless wavenumber characterizing the unsteady component of the motion. The Froude number is defined as F = u₀ / (gd₀)^1/2; in which u₀ = steady uniform mean flow velocity; d₀ = steady uniform flow depth; and g = gravitational acceleration. The dimensionless wavenumber is defined as σ = (2π/L)L₀; in which L = wavelength of the disturbance; and L₀ = reference channel length, defined as the channel length in which the steady uniform flow drops a head equal to its depth.

The propagation characteristics of shallow water waves are: (1) the dimensionless relative wave celerity c_r; and (2) the logarithmic decrement δ. The dimensionless relative wave celerity c_r = (c - u₀) / u₀, in which c = wave celerity. The logarithmic decrement δ is s a measure of the tendency of the wave disturbance to attenuate or amplify during propagation. For δ < 0, waves attenuate; conversely, for δ > 0, waves amplify. A value of δ = 0 indicates neutral stability, i.e. neither wave attenuation nor amplification. The theory states that roll waves are most likely to form in the unstable flow regime (V > 1; δ > 0), within a narrow band of dimensionless wavenumbers, at or near the peak of the functional relation between logarithmic decrement and dimensionless wavenumber [δ = f(σ)].

The theoretical expressions for dimensionless wave celerity and logarithmic decrement (Ponce and Simons 1977) are included here for completeness.

The dimensionless relative wave celerity is:

C + A cr = ( ________ )1/2                2

(3)

TABLE 1.- Roll-wave data sets from Brock (1967) study.

in which

1        1 A = ____ - _______         F2     σ 2 F4

(4)

and

1 C = ( A2 + _______ )1/2                   σ 2 F4

(5)

The logarithmic decrement of the downstream-propagating shallow wave is:

1          C - A                      ______ - (_______)1/2                     σ F2          2 δ = - 2π _________________________                            |1 + cr |

(7)

4.  RESULTS

The Brock (1967) data were assembled as shown in Table 1, which shows the following: (1) normal depth (i.e., steady uniform flow) d₀; (2) bottom slope S₀; (3) wave celerity c; and (4) wave period T for 28 test cases included in the Brock study. The normal depth and bottom slope were used to calculate the reference channel length L₀ = d₀ / S₀. The wave celerity and wave period were used to calculate the wavelength L = cT. The dimensionless wavenumber is defined as follows (Ponce and Simons 1977): σ = (2π / L)L₀. Assuming the roll wave celerity to be approximately equal to the inertial wave celerity c = u₀ + (gd₀)^1/2, the Froude number was estimated as follows: F = c / (gd₀)^1/2 - 1 (Brock 1967). This is a reasonable approximation for roll waves.

The solid curves of Figs. 1 and 2 show logarithmic decrement δ versus dimensionless wavenumber σ, (7), for the range of Froude numbers suited to this study. A total of 18 σ - F data points (3 < F < 5) obtained from the Brock data are shown in Fig. 1, Ten (10) additional data points (5 < F < 8) are shown in Fig. 2. The agreement between theory and experiments is very good, considering the five order-of-magnitude range of possible dimensionless wavenumbers. This agreement is evidence that roll waves are likely to form within a narrow band of dimensionless wavenumbers, at or near the peak of the functional relation δ = f (σ), i.e, for values of the logarithmic decrement δ > 0.2. The marked tendency for a decrease in dimensionless wavenumber with an increase in Froude number is in agreement with Mayer's (1957) findings.

Fig. 1 Logarithmic decrement δ as function of dimensionless wavenumber σ for Froude numbers in 3-5 range (Ponce and Simons 1977).

Fig. 2 Logarithmic decrement δ as function of dimensionless wavenumber σ for Froude numbers in 5-8 range (Ponce and Simons 1977).

5.  SUMMARY

Using Brock's (1967) laboratory flume data, an experimental verification of a theory of roll-wave formation (Ponce and Simons 1977) was presented. Results show that dimensionless wavenumbers corresponding to the roll waves documented by Brock's study plot near the peak of the theoretical functional relation of logarithmic decrement δ versus dimensionless wavenumber σ. The analysis confirms that roll waves are most likely to form for large positive values of the logarithmic decrement (δ > 0.2), located within a narrow center band of dimensionless wavenumbers.

APPENDIX |. REFERENCES

Brock, R. R. 1967. "Development of roll waves in open channels." Rep. No. KH. R-16, W. M. Keck Laboratory of Hydraulics and Water Resources, California institute of Technology, Pasadena. Calif.

Chow, V. T. 1959. Open-channel hydraulics, McGraw-Hill, New York, N.Y.

Craya, A. 1952. "The criterion of the possibility of roll wave formation." Gravity Waves, National Bureau of Standards Circular No. 521, National Bureau of Standards, Washington, D.C., 141-151.

lwasa, Y. 1954. "The criterion for the instability of steady uniform flows in open channels." Memoirs of the Faculty of Engineering, Kyoto University, Kyolo,Japan, 16(6),264-275.

Liggett, J. A. 1975. "Stability -Chapter 6," Unsteady flow in open channels, Vol. 1, K. Mahmood and V. Yevjevich, eds., Water Resources Publications, Fort Collins, Colo., 259-281.

Mayer, P. G. W. 1957. "A study of roll waves and slug flows in inclined open channels," PhD dissertation, Cornell University, Ithaca, N.Y.

Ponce, V. M., and D. B. Simons. 1977. "Shallow wave propagation in open channel flow." J. Hydr. Div., ASCE, 103(12), 1461-1476.

Ponce, V. M. 1991. "New perspective on the Vedernikov number," Water Resour. Res., 27(7), 1777-1779.

Vedernikov, V. V. 1945. "Conditions at the front of a translation wave disturbing a steady motion of a real fluid," U.S.S.R. Academy of Sciences Comptes Rendus (Doklady), 48(4), 239-242.

Vedernikov, V. V. 1946. "Characteristics features of a liquid flow in an open channel." U.S.S.R. Academy of Sciences Comptes Rendus (Doklady), 52(3), 207-210.