Influence of Relative Roughness and Reynolds Number on the Roll-Waves Spatial Evolution

Influence of Relative Roughness and Reynolds Number onthe Roll-Waves Spatial Evolution

Cristiana Di Cristo1; Michele Iervolino2; Andrea Vacca3; and Barbara Zanuttigh4

Abstract: The paper investigates the influence of the resistance coefficient variability onto the spatial development of roll-waves. Twomodels, based on time-asymptotic solutions of the linearized St. Venant equations, subject to either impulsive or oscillating perturbation,have been modified by including the dependence of the resistance coefficient on flow conditions, wall roughness, and fluid viscosity.Independently of the perturbation type, it has been shown that the hypothesis of constant resistance coefficient leads to underestimate thedisturbance spatial growth. Theoretical predictions are finally compared with results of a fully nonlinear model and with literatureexperimental data for several combinations of Froude and Reynolds numbers and relative roughness values. The representation ofvariability of the resistance coefficient fundamentally improves the performance of minimum channel length criteria, whereas its neglectmay lead to noncautious channel design.

DOI: 10.1061/�ASCE�HY.1943-7900.0000139

CE Database subject headings: Bed roughness; Friction; Open channel flow; Stability; Water waves; Reynolds number; Flowresistance.

Author keywords: Bed roughness; Friction; Open channel flow; Stability; Water waves.

Introduction

Open channel flows on slopes may generate a succession ofwaves, termed “roll-waves,” the amplitude of which increases asthe current descends, up to the formation of surges replacing theinitial continuous flow. This behavior is usually explained interms of an instability of sheet-flow configuration, pertaining notonly to Newtonian fluids, but also to multiphase, power-law flu-ids, and granular flows �Woods et al. 2000; Engelund and Wan1984; Forterre and Pouliquen 2003; Pascal 2006; Pascal andD’Alessio 2007�, gravity currents �Alavian 1986; Cenedese et al.2004� and even in ocean �Swaters 2003� and lakes �Fer et al.2002�. In the present paper only clearwater flows will beconsidered.

Under controlled laboratory conditions, roll-waves were ex-perimentally investigated by Brock �1969,1970�, Julien and Hart-ley �1986�, and Liu et al. �2005�, who, in analyzing roll-wave

1Assistant Professor, Dipartimento di Meccanica, Strutture, Ambientee Territorio, Università di Cassino, Via Di Biasio 43, 03043 Cassino �FR�,Italy �corresponding author�. E-mail: [email protected]

2Assistant Professor, Dipartimento di Ingegneria Civile, Seconda Uni-versità di Napoli, Via Roma 29, 81031 Aversa �CE�, Italy. E-mail:[email protected]

3Associate Professor, Dipartimento di Ingegneria Civile, SecondaUniversità di Napoli, Via Roma 29, 81031 Aversa �CE�, Italy. E-mail:[email protected]

4Assistant Professor, Dipartimento di Ingegneria delle Strutture, deiTrasporti, delle Acque, del Rilevamento del Territorio, Università di Bo-logna, Viale del Risorgimento 43, 40136 Bologna, Italy. E-mail:[email protected]

Note. This manuscript was submitted on October 1, 2008; approvedon July 10, 2009; published online on July 13, 2009. Discussion periodopen until June 1, 2010; separate discussions must be submitted for indi-vidual papers. This paper is part of the Journal of Hydraulic Engineer-ing, Vol. 136, No. 1, January 1, 2010. ©ASCE, ISSN 0733-9429/2010/1-
24–33/$25.00.
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formation and development, identified three different phases. Inthe first �initial or instability dominant�, roll-waves are character-ized by exponentially increasing amplitude in the streamwise di-rection and present almost constant mean wave period. In thesecond �transition�, waves begin to overtake each other, increas-ing their period. Finally, in the third �overtaking dominant� thespatial density of waves, characterized by steep fronts, decreasesand the mean period linearly increases.

It is commonly accepted that the one-dimensional �1D�shallow-water wave equations, with or without viscous dissipa-tion, can predict the roll-waves spatial growth. Linear, weaklynonlinear, along with fully nonlinear models, have been widelyused. Fully nonlinear models can reproduce the entire roll-wavesdevelopment process, illuminating the role of coalescence on themodification of wave parameters �i.e., period and height� in thefinal phase �Zanuttigh and Lamberti 2002; Balmforth and Mandre2004; Chang et al. 2000; Que and Xu 2006�. Weakly nonlinearanalyses are valid until the end of the transition phase and predictthe wave tendency to combine into longer and higher waves�Kranenburg 1992; Balmforth and Mandre 2004; Yu andKevorkian 1992�. Finally, linearized models are able to describeonly the initial phase and to define, through temporal stabilityanalysis, the limiting values of the relevant dimensionless param-eters above which roll-waves are expected �see for instance Brock1970; Berlamont 1976; Ponce and Simon 1977; Berlamont andVanderstappen 1981; Trowbridge 1987�. The linearized equationsperturbed with a pointwise small disturbance �i.e., linear spatialanalysis� predict the cited exponential growth of the waves in theinitial phase. Both instantaneous �Vedernikov 1946; Stoker 1957;Liggett 1975� or periodically time-varying �Supino 1960; DiCristo and Vacca 2005; Di Cristo et al. 2008� disturbances havebeen considered. Following Stoker �1957�, in the former case theperturbation is investigated in the neighborhood of the character-istic of the uniform flow, finding that it travels downstream withthe celerity of the faster shallow-water wave. In the latter, a ca-
nonical spatial linear analysis in the complex plane is carried out
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and it is found that the perturbation causes the appearance of twowaves propagating downstream, whose celerities strongly dependon the imposed oscillation period. The spatial wave growth ratededuced by linear analyses constitutes the theoretical basis ofsemiempirical criteria for the prediction of roll-waves, widely em-ployed in hydraulic engineering. For both turbulent �Montuori1961,1963, Di Cristo et al. 2008� and laminar �Julien and Hartley1986� flows, the above criteria state that in unstable conditions aminimum channel length is also required for perturbations togrow to an observable threshold.

The above spatial linear analyses assume for the bed shearevaluation a constant resistance coefficient �, despite its well-known dependence on both Reynolds number, Re, and relativeroughness, � /h, where �=characteristic roughness height; and h=flow depth, measured normally to channel bed. Even in the fullyrough regime, in which � can be safely assumed independentfrom Re, a residual dependence of � on � /h persists. The presentwork investigates the influence of � /h and Re on the spatial evo-lution of roll-waves in their initial phase. Spatial linear analyseswith instantaneous and oscillating disturbance are both general-ized evaluating � through a Colebrook-White type resistance for-mula. Theoretical results are finally compared with numericalcomputations with the nonlinear model of Zanuttigh and Lamberti�2002� and with experimental data �Fedorov 1954,1956;Arseniscvili 1955,1956; Montuori 1961,1963�.

Governing Equations

Let us consider an incompressible fluid flowing over a fixed bedof constant slope S, in an infinitely wide channel, without lateralinflow or outflow. Under the hypotheses of negligible viscousdissipation and vertical acceleration, bed shear stress as in uni-form flow, the 1D flow field may be described by the followingequations �St. Venant equations�:

�u

�t+ g cos �

�h

�x+ u

�u

�x+

�b

�h− g sin � = 0 �1�

�h

�t+ u

�h

�x+ h

�u

�x= 0 �2�

where x=streamwise coordinate; t=time, g�the acceleration dueto gravity; �=water density; u=depth-averaged velocity; sin �=S the bed slope; and �b=bed shear stress, expressed by

�b = ��u2

8�3�

The resistance coefficient � is evaluated by the implicitColebrook-White type formula.

1��

= a ln� b

Re��+ d

�

h� �4�

in which Re=uh /�=Reynolds number; with �=kinematic viscos-ity, a, b, and d=dimensionless experimental coefficients, whosevalues, for the case under investigation, are a=−2.03 / ln�10�,b=0.425, d=1 /10.95 �Rouse 1959�.

In the fully rough regime, typical of most hydraulic engineer-
ing problems, the effect of Re is negligible and Eq. �4� reduces to
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1��

= a ln�d�

h� �5�

By denoting with the subscript 0 the reference uniform base state,Eqs. �1�–�3� can be rewritten in the following dimensionless form

� u

� t+ u

� u

� x+

1

F2

� h

� x+

�

sin �

u2

8h=

1

F2 cos ��6�

� h

� t+ u

� h

� x+ h

� u

� x= 0 �7�

in which x=xS /h0, t= tSu0 /h0, u=u /u0, h=h /h0, and F=u0 /�gh0 cos �=Froude number of the base flow.

Spatial Linear Analysis

Impulsive Disturbance

Owing to the hyperbolic character of the linearized St. Venantequations, the asymptotic time behavior of the solution of Eqs. �6�and �7� perturbed by an impulsive pointwise disturbance has beenstudied, with � constant �Liggett 1975�, by introducing the fol-lowing transformation of variables:

� = x, = c+t − x , �8�

with c+=1+1 /F the propagation celerity of the perturbation. Inwhat follows this analysis is generalized to account for the depen-dence of � on relative roughness and Reynolds number. For thispurpose, Eqs. �6� and �7� are rewritten in the new reference frame

�1 +1

F� � u

�+ u� � u

��−

� u

�� +

1

F2� � h

��−

� h

�� +

�

sin �

u2

8h

=1

F2 cos ��9�

�1 +1

F� � h

�+ u� � h

��−

� h

�� + h� � u

��−

� u

�� = 0 �10�

and the unknown functions u, h are expanded in a series in aneighborhood of =0

u = 1 + u1 + u22 + . . . h = 1 + h1 + h22 + . . . �11�

with u1, u2 , . . . , h1, and h2 , . . . =unknown functions of � only.Moreover, the resistance term in Eq. �9� is expanded through thefollowing power series:

�u2

h= ��0 +

��

� hh1 +

��

� uu1 + . . .��1 + 2u1 + . . .��1 − h1 + . . .�

�12�

By substituting Eqs. �11� and �12� in Eqs. �9� and �10�, and ne-glecting the terms of order greater than one, the followingordinary differential equation is obtained

dh1

d�= h1

2 − �h1 �13�

in which

NAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2010 / 25


=3

2�F + 1��14�

� = −�2 + A� − �1 − B�F2F cos ��F + 1�

�15�

The A and B coefficients in Eq. �15� account for the � variabilityand are

A =2ab��0

d Re ��0 − ab��0 + b�16�

B =2a��0d Re �� + b��0�

d Re ��0 − ab��0 + b�17�

in which ��=� /h0 denotes the relative roughness. The solution ofEq. �13� is

h1 =�

�

�+ e−��

�18�

with the integration constant � given by

� =

� � h

��

=0

1 −

�� h

��

=0

�19�

In the initial development phase, for small values of ��h /��=0,Eq. �18� can be approximated �Liggett 1975� as

h1 = �e�� 20�

Thus, the flow depth perturbation grows exponentially in �, pro-vided that � is greater than zero, which from Eq. �15� occurswhenever F exceeds the following critical value

Fc =2 + A

1 − B�21�

The critical value Eq. �21� resembles the limit value deduced bythe temporal stability analyses of linearized St. Venant equationsdiscussed in Berlamont and Vanderstappen �1981� and Trow-bridge �1987�.

In the case of constant resistance coefficient �i.e., A=B=0�,Fc=2, whereas � is

� = −2 − F

2F cos ��F + 1��22�

which generalizes to steep slope channels the findings of Liggett�1975�.

In the fully rough regime �Re→ �, A=0 and B=2a��0

Fc =2

1 − 2a��0

�23�

� = −2 − F�1 − 2a��0�2F cos ��F + 1�

�24�

Because a�0, the comparison of Eq. �22� with Eqs. �15� and �24�implies that the hypothesis of �=�0 induces an underestimationof �, independently on Re and �� values. Fig. 1 presents, for fully
rough regime, the spatial growth rate � versus F predicted in case


of constant �Eq. �22�� and variable resistance coefficient �Eq.�24��, for different �� values. The underestimation is about 50% inthe range of F between 3 and 5 and for relatively high values of�� between 10−4 and 10−2�. Furthermore, results not shownherein indicate that such underestimation increases when finite Revalues are considered.

Periodic Disturbance

The analysis carried out in the previous section considers thespatial evolution of roll-waves in the initial phase as being gen-erated by an instantaneous pointwise disturbance. A different rep-resentation, based on the assumption of a monochromatic time-oscillating pointwise disturbance, was suggested by Di Cristo etal. �2008�, whose theoretical analysis was carried out under thehypotheses of unbounded domain �i.e., infinitely long channel�and constant resistance coefficient. This latter hypothesis is re-laxed in the present section. The analysis is performed superpos-

ing a small perturbation �up , hp� to the base uniform flow andlinearizing the system of Eqs. �6� and �7�. The following lineardifferential problem is obtained:

� up

� t+

� up

� x+

1

F2

� hp

� x+

1

F2 cos ��2 + A�up − �1 − B�hp� = 0

�25�

� hp

� t+

� hp

� x+

� up

� x= 0 �26�

with A and B given by Eqs. �16� and �17�, respectively. The sys-

tem Eqs. �25� and �26� can be reformulated in terms of hp only

�2hp

� t2+ 2

�2hp

� x � t+ �1 −

1

F2� �2hp

� x2

+1

F2 cos ��3 + A − B�

� hp

� x+ �2 + A�

� hp

� t� = 0 �27�

By assuming that the flow depth perturbation is generated by apointwise disturbance, which varies periodically in time with di-

mensionless pulsation �e=2�fe /S=2� / Te, with Te and fe the di-

Fig. 1. Wave spatial growth rate versus Froude number, in fullyrough regime for different relative roughness values ��. Dashed line:constant resistance coefficient. Solid lines: variable resistancecoefficient.

mensionless oscillation period and the frequency respectively, the


time asymptotic behavior of hp is expressed �Bers 1975� by

hp = i �l=1,2

U�x�e−i��et−klx�

�dD/dk�k=kl,�=�e

�28�

in which i=imaginary unit; U�x�=Heaviside operator �Abramow-itz and Stegun 1972�; D�k ,��=dispersion operator associatedwith the differential problem of Eq. �27� with

�dD

dk�

k=kl,�=�e

= 2i F2 cos ��kl − �e� − 2ikl cos �

+ �A + 2� − �B − 1� �29�

The complex spatial branches kl �l=1,2� in Eq. �29� are given by

kl =2F2 cos ��e + �3 + A − B�i � �r

2 cos ��F2 − 1��30�

with

r = 4F2 cos ��e�cos ��e + i�1 − B�� − 9

+ 8i cos ��e − �A − B��A − B + 6� + 4Ai cos ��e �31�

The time-asymptotic solution of Eq. �28� shows that the oscillat-ing disturbance causes the appearance of two waves, which

Fig. 2. �a� Faster and �b� slower wave celerity versus disturbance pulvariable resistance coefficient.

Fig. 3. �a� Faster and slower �b� wave celerity versus disturbance puvariable resistance coefficient.

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propagate with celerities cl=�e /R�kl� �l=1,2� and amplify ordecay in space at a rate �l=−I�kl�, R�kl�, and I�kl� being real andimaginary part of kl, respectively. The analysis of celerities im-plies that, independently of �e, under supercritical flow condi-tions �F�1�, both waves propagate downstream. Moreover,whenever F exceeds the value expressed by Eq. �21�, the faster�slower� wave grows �decays� exponentially while propagating.The dependence of wave celerity cl and spatial growth �or decay�rate �l on the disturbance pulsation �e and on the three dimen-sionless parameters Re, F, and �� is expressed by Eq. �29�. In thefollowing, the subscript 1 denotes the faster �slower� wave. In thecase of constant resistance coefficient, i.e., A=B=0, the results ofDi Cristo et al. �2008� are retrieved.

With reference to fully rough regime, Figs. 2 and 3 depict thedependence of phase speed of both faster and slower waves on�e, for several values of �� and two values of Froude number,namely F=2.5 and F=5. The curves corresponding to the hypoth-esis of constant � are also plotted. Independently of F, �� anddisturbance pulsation, the celerity of faster �slower� wave is al-ways larger �smaller� than the corresponding under the constant�0 assumption.

Figs. 4 and 5 represent the counterpart of Figs. 2 and 3 interms of spatial growth and decay rates. As far as the faster waveis concerned, Figs. 4�a� and 5�a� show that in the investigated

�e �F=2.5�. Dashed line: constant resistance coefficient. Solid lines:

�e �F=5�. Dashed line: constant resistance coefficient. Solid lines:

sation

lsation



range of the dimensionless parameters the hypothesis of constant�0 leads, similarly to the impulsive case presented in the previoussection, to an underestimation of the spatial growth, indepen-dently of the disturbance pulsation. Moreover, the damping of theslower wave is less pronounced when a constant resistance coef-ficient is assumed �see Figs. 4�b� and 5�b��. A similar trend, interms of both celerity and growth �decay� rate, can be observedalso for finite value of Reynolds number �results not shownherein�.

As a final remark, it can be observed that, for F�Fc andindependently on �� and Re values, in the limit for �e→ theasymptotic values of the celerity �c1� and growth rate ��1� of thefaster wave coincide with those predicted by impulsive analysis�i.e., c1→c+, �1→��. Therefore the impulsive analysis resultsinto a disturbance growth rate larger than the oscillating one,independently on the �e value.

Comparison with Fully Nonlinear NumericalSimulations

Numerical Model and Data Analysis

Numerical simulations are carried out with the 1D fully nonlinearmodel by Zanuttigh and Lamberti �2002�, based on the weighted

Fig. 4. �a� Wave growth and �b� decay rate versus disturbance pulsavariable resistance coefficient.

Fig. 5. �a� Wave growth and �b� decay rate versus disturbance pulsvariable resistance coefficient.



average flux �WAF� method. The WAF method �Toro 1992,1997�is a development of Godunov’s and Roe’s schemes and is “shock-capturing” in the sense that discontinuities are correctly treatedwithout the need for a special tracking algorithm. Bottom resis-tance is modeled according to Eq. �4�. The full details of thenumerical integration scheme, along with the geometry used inthe computations, can be found in Zanuttigh and Lamberti �2002�.The set of the 23 simulated flow conditions is listed in Table 1.Simulations were carried out until the occurrence of the overtak-ing dominant phase. Waves are identified from computed flowdepths according to the zero down-crossing method, with a heightthreshold value equal to �1% of the normal flow depth. For eachnumerical test, wave statistics were computed based on the final200 s of a 500-s simulation run, containing about 200 waves inthe initial phase. The roll-waves development in the initial phasewas characterized by period and spatial growth rate, which havebeen computed according to the methodology discussed in Zanut-tigh and Lamberti �2002�. In addition, the celerity of the roll-waves has been estimated from the cross-correlation analysis offlow depths at adjacent channel sections �i.e., from the position ofthe cross-correlation peak in time, being known the distance be-tween the numerical channel sections�. The computed dimension-

less values of period TI, growth rate �NM and celerity cNM of thewave are listed in Table 1.

e �F=2.5�. Dashed line: constant resistance coefficient. Solid lines:

�e �F=5�. Dashed line: constant resistance coefficient. Solid lines:

tion �

ation


Results

This section compares wave growth rate and phase speed pre-dicted by linear analysis, for both impulsive and periodic distur-bance, with the computations performed with the fully nonlinearmodel described in the previous section. In the following, thesenumerical results are assumed as reference. For each test, simi-larly to Di Cristo et al. �2008� the frequency of the disturbance in

the periodic analysis was computed assuming Te= TI.Theoretical predictions of wave celerity and spatial growth

rate are compared with the corresponding values from the fullynonlinear model in Figs. 6 and 7, respectively. Fig. 6 shows thatthe two linear models both approximate with comparable accu-racy the computed values. No matter if pointwise impulsive or

Table 1. Investigated Flow Conditions and Results of Nonlinear Simula

S F ��

0.20 7.88 0.002

0.21 6.80 0.030

0.24 8.42 0.012

0.30 7.17 0.065

0.30 8.19 0.031

0.30 8.70 0.016

0.30 9.21 0.015

0.30 9.73 0.008

0.35 7.23 0.095

0.35 9.30 0.024

0.40 8.36 0.068

0.40 9.40 0.020

0.40 9.40 0.036

0.40 10.45 0.019

0.40 12.53 0.005

0.45 10.58 0.010

0.50 10.21 0.050

0.50 11.28 0.020

0.50 12.89 0.010

0.55 16.30 0.001

0.55 17.62 0.001

0.60 13.42 0.005

0.60 15.65 0.005

Fig. 6. Linear �c+ ,c1� versus nonlinear �cNM� estimates of wave ce-lerity

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oscillating disturbance is considered, both theoretical analysesgive an error percentage between �15% and 10%. However, theimpulsive analysis systematically results in an underestimation ofwave celerity, while the periodic leads to both over and underpre-diction.

Fig. 7 represents the counterpart of Fig. 6 in terms of spatialgrowth rate. Differently from the celerity, a clear influence of theperturbation type is observed onto the spatial growth rate. Theassumption of a pointwise impulsive disturbance leads to a sys-tematic and relevant overestimation of the growth rate, with amean error �on the whole data set� greater than 130%, which isinstead less than 4% when the oscillating disturbance is consid-ered.

TI �NM cNM

15.82 0.0341 1.16

15.32 0.0348 1.16

18.22 0.0283 1.15

20.55 0.0271 1.31

20.99 0.0268 1.17

21.59 0.0293 1.16

22.26 0.0249 1.17

22.21 0.0250 1.22

21.88 0.0301 1.33

24.36 0.0247 1.22

25.25 0.0246 1.29

26.40 0.0242 1.20

27.04 0.0232 1.21

28.41 0.0208 1.21

30.03 0.0182 1.17

30.61 0.0227 1.28

33.15 0.0178 1.22

33.33 0.0193 1.21

34.48 0.0155 1.24

38.73 0.0129 1.30

40.44 0.0131 1.13

42.86 0.0156 1.34

42.55 0.0136 1.32

Fig. 7. Linear �� ,�1� versus nonlinear ��NM� estimates of wavegrowth rate

tions



The present results indicate that, particularly as far as the spa-tial growth rate is concerned, a linear analysis with a pointwiseoscillating disturbance allows a correct prediction of the roll-waves evolution in the initial phase, independently of F, Re, and��.

Comparison with Experimental Data

The spatial growth rates deduced in the previous section havebeen used to estimate the minimum channel length for roll-wavesoccurrence and to compare it with available experimental evi-dence. We considered several flow conditions coming from bothlaboratory and field experiments �Fedorov 1954,1956; Arsenis-cvili 1955,1956; Montuori 1961,1963� used for the validation ofthe semiempirical criterion of Montuori �1961,1963�. For eachtest, the channel slope, S, and length, L, along with the uniformflow conditions �h0 and u0� are provided. To comply with ourhypothesis of wide channel, only the tests with R0 /h0�1.1 havebeen selected from the whole data set of about 70 configurations,obtaining a set with 26 experimental conditions, 18 with roll-waves and 8 without roll-waves �see Table 2�. Table 2 reports alsofor each test the dimensionless parameters F, Re, and �, com-puted from the experimental flow data, and the correspondingrelative roughness �� obtained by Eq. �4�. All dimensionless pa-rameters span at least one order of magnitude. Moreover, thevalues of the A and B coefficients, expressed by Eqs. �16� and�17�, are given in the same table. Their magnitude, compared withthe leading term O �1�, quantifies the effect of the friction coeffi-

Table 2. Experimental Data

N. Test S0

h0 u0 L

�m� �m/s� �m�

No roll-waves NR1 0.552 0.040 10.10 24

NR2 0.552 0.064 11.70 24

NR3 0.552 0.068 12.20 24

NR4 0.552 0.072 12.50 24

NR5 0.602 0.094 14.11 80

NR6 0.602 0.122 10.80 80

NR7 0.044 0.090 1.92 213

NR8 0.707 0.011 6.10 14

Roll-waves R1 0.602 0.030 6.61 80

R2 0.602 0.038 5.21 80

R3 0.602 0.039 8.02 80

R4 0.603 0.040 10.00 149

R5 0.602 0.050 6.32 76

R6 0.148 0.080 6.80 1158

R7 0.148 0.060 5.00 200

R8 0.148 0.120 8.65 1158

R9 0.046 0.120 3.25 670

R10 0.040 0.020 0.90 500

R11 0.133 0.040 2.37 115

R12 0.203 0.040 3.28 97

R13 0.148 0.060 5.15 200

R14 0.148 0.130 8.00 200

R15 0.242 0.080 7.15 60

R16 0.053 0.100 3.30 59

R17 0.603 0.120 17.90 151

R18 0.046 0.220 5.10 670

cient variability onto the spatial growth rate of the roll waves.



Table 2 shows that the influence of Reynolds number is generallymarginal, i.e., AO�10−3� while, the correction due to the rela-tive roughness is far to be negligible �BO�10−1��.

The experimental dimensionless channel length Mexp

=LSg /u02 �i.e., the Montuori number� is compared with the theo-

retical threshold value for roll-waves occurrence, expressed as

Mmin = ln��F2 cos �/� �32�

where �=disturbance spatial growth factor. In the Montuori’s cri-terion, theoretically based on a spatial growth-rate computed con-sidering an impulsive pointwise disturbance under constantfriction factor hypothesis, � is expressed by Eq. �22� with cos �=1 �Liggett 1975; Montuori 1961,1963�. Accounting for the vari-ability of the friction factor, � is given by Eq. �15� or by theimaginary part of Eq. �30�, for impulsive or oscillating perturba-tion, respectively. The dimensionless pulsation of the oscillatingdisturbance in Eq. �30� is evaluated assuming the dimensionlessfrequency fe constant and equal to 0.013 �Di Cristo et al. 2008�.The coefficient � in Eq. �32� is fixed to 10−4, as found by Mon-tuori �1961,1963�.

In Table 3 for all tests, the ratio of the predicted dimensionlesschannel length, with impulsive, Mmin

I , and oscillating, MminO , dis-

turbance, with respect to the experimental value Mexp is given.Both constant �A=B=0� and variable �A and B given by Eqs. �16�and �17�, see Table 2� friction factor cases have been considered.For completeness the Mexp values are also reported. According tothe minimum channel length criterion, roll-waves are expectedwhenever the dimensionless channel length Mexp exceeds the the-

F �� Re � A B

�-� �-� �-� �-� �-� �-�

7.66 0.0017 4.04·105 0.0170 �0.011 �0.229

6.17 0.0037 7.49·105 0.0203 �0.003 �0.251

6.36 0.0034 8.30·105 0.0198 �0.003 �0.248

6.29 0.0035 9.00·105 0.0200 �0.003 �0.249

6.44 0.0055 1.33·106 0.0223 �0.001 �0.263

1.05 0.0666 1.32·106 0.0494 0.000 �0.392

2.04 0.2201 1.73·106 0.0843 0.000 �0.512

2.08 0.0010 6.71·104 0.0164 �0.075 �0.217

3.64 0.0200 1.98·105 0.0324 �0.002 �0.317

9.55 0.1329 1.98·105 0.0661 0.000 �0.453

4.51 0.0134 3.13·105 0.0286 �0.002 �0.298

7.87 0.0028 4.00·105 0.0189 �0.007 �0.242

0.10 0.1032 3.16·105 0.0591 0.000 �0.429

7.72 0.0036 5.44·105 0.0201 �0.004 �0.249

6.55 0.0122 3.00·105 0.0279 �0.002 �0.294

8.02 0.0027 1.04·106 0.0186 �0.003 �0.240

3.00 0.0404 3.90·105 0.0410 �0.001 �0.357

2.03 0.1853 1.80·104 0.0775 �0.002 �0.490

3.80 0.1707 9.48·104 0.0743 �0.001 �0.481

5.29 0.1035 1.31·105 0.0592 �0.001 �0.429

6.75 0.0099 3.09·105 0.0263 �0.003 �0.285

7.12 0.0068 1.04·106 0.0236 �0.001 �0.271

8.19 0.0152 5.72·105 0.0297 �0.001 �0.304

3.33 0.0330 3.30·105 0.0382 �0.001 �0.344

8.47 0.0022 2.15·106 0.0177 �0.002 �0.235

3.47 0.0166 1.12·106 0.0305 0.000 �0.308

1

1

1

1

1

1

2

1

1

1

1

1

oretical Mmin value. So, referring to Table 3, a correct prediction


corresponds to a ratio value smaller �larger� than 1 for cases with�without� roll-waves, respectively.

Considering the experiments in which roll-waves were not de-tected, Table 3 indicates that the assumption of variable frictionfactor in the theoretical developments does not lead to substantialdifferences in the minimum channel length prediction. In contrast,the disturbance shape strongly influences the performance of thecriterion. Indeed, the oscillating disturbance leads to correctlypredict the roll-waves absence in all tests, whereas an impulsiveone allows to comply with only 50% of the cases.

Conversely, whenever the roll-waves occur and independentlyon the shape of the disturbance, the performance of the criterionstrongly improves when � variability is accounted for. In fact, thenumber of cases in which the presence of roll-waves is predictedincreases from 14 to 17 and from 12 to 15 for impulsive andoscillating disturbance, respectively. These results are consistentwith the underestimation of the disturbance spatial growth due tothe constant friction factor hypothesis, for both shapes of the per-turbation �see Figs. 1, 4�a�, and 5�a��. Moreover, such a hypoth-esis could result into a nonconservative channel design, since itmay lead to predict the roll-waves absence even for conditions inwhich they would occur.

Table 4 summarizes the criterion performance, showing theoverall percentage of successful predictions for all consideredanalyses. It follows that accounting for the � variability leads to asignificant improvement of the minimum channel length criterion,independently on the shape of the disturbance. Moreover, accord-ingly with the numerical simulation �see Fig. 7�, Table 4 indicates

Table 3. Comparison between Experimental Data and Theoretical Predic

N. Test Mexp MminI

No roll-waves NR1 1.26 1

NR2 0.94 1

NR3 0.86 1

NR4 0.83 1

NR5 2.36 0

NR6 4.03 0

NR7 24.91 25

NR8 2.53 0

Roll-waves R1 10.76 0

R2 17.31 0

R3 7.31 0

R4 8.80 0

R5 11.17 0

R6 36.37 0

R7 11.61 0

R8 22.47 0

R9 28.63 0

R10 242.22 3

R11 26.70 0

R12 17.95 0

R13 10.95 0

R14 4.54 0

R15 2.79 1

R16 2.81 6

R17 2.78 0

R18 11.62 1

that the best performance of the criterion is achieved considering

JOUR


the dependence of the friction factor on Re and �� in the linear-ized flow model, perturbed with a periodic disturbance. Neverthe-less, since Eq. �15� leads to overestimate the disturbance growthfactor and therefore to underestimate the minimum channellength, even the easier impulsive model, with a variable frictioncoefficient, may be safely used for channel design purposes.

Conclusions

The analysis of linearized Saint Venant equations forms the the-oretical basis of engineering criteria aiming to predict roll-wavesoccurrence in natural and manmade channels. In the presentpaper, two linear models �based on impulsive or time-oscillatingpointwise disturbance of a base uniform flow� have been consid-ered for the description of the initial phase of roll-waves devel-opment, both accounting for variable resistance coefficient.

Independently of the perturbation type, resistance coefficientvariability strongly affects the spatial growth of roll-waves, evenin fully rough regime. In general, the assumption of constant re-

Wrong Predictions Are Italicized

�=const �=��Re ,��

MminO /Mexp Mmin

I /Mexp MminO /Mexp

2.58 0.78 2.27

3.94 1.14 3.40

4.25 1.23 3.66

4.40 1.28 3.83

1.67 0.44 1.43

1.88 0.37 1.49

28.57 1.01 1.25

1.21 0.30 1.07

0.49 0.12 0.41

0.58 0.1 0.44

0.65 0.16 0.55

0.39 0.11 0.35

0.80 0.15 0.63

0.14 0.07 0.11

0.57 0.28 0.44

0.21 0.12 0.17

0.95 0.42 0.48

3.84 0.11 0.13

0.75 0.24 0.44

0.67 0.22 0.46

0.57 0.29 0.45

1.26 0.66 1.00

2.10 0.85 1.68

7.11 3.43 4.02

1.19 0.34 1.04

1.50 0.80 0.90

Table 4. Percent Agreement with Experimental Data

Friction factor

Disturbance

Impulsive Periodic

Constant 73% 77%

Variable 81% 88%

tions;

/Mexp

.10

.64

.77

.84

.66

.64

.4

.46

.18

.18

.25

.16

.26

.10

.41

.15

.86

.56

.49

.38

.41

.91

.22

.40

.48

.39



sistance coefficient underestimates the wave growth rate.The comparison of the theoretical results with fully nonlinear

simulations indicates that linear analysis with a pointwise oscil-lating disturbance more correctly describes both propagation andgrowth of the roll-waves in the initial phase. This conclusion isconfirmed by the comparison with experimental evidence, whichalso shows that the performance of the minimum channel crite-rion, widely used in channel design, is improved when the con-stant friction factor assumption is removed.

Acknowledgments

The writers are grateful to Professor Willi Hager for inspiring thepresented study. The writers thank Professor D. A. Lyn for hisconstructive comments, which improved the quality of the paper.

Notation

The following symbols are used in this paper:A, B � dimensionless coefficients;

a, b, d � dimensionless experimental coefficient in theColebrook-White type formula;

c1,2 � faster �slower� wave celerity obtained by theperiodic disturbance analysis;

cNM � wave celerity predicted by numerical model;c+ � slope of faster characteristic line;F � Froude number;

Fc � critical Froude number;fe � perturbation dimensionless frequency;g � gravity;h � flow depth defined normal to channel bed;i � imaginary unit;

k1,2 � complex spatial branches;L � channel length;

Mexp � experimental dimensionless channel length;Mmin � dimensionless channel length;

MminO , Mmin

I � predicted dimensionless channel length byoscillating �impulsive� method;

r � dummy variable;R � hydraulic radius;

Re � Reynolds number;S � bed slope;

Te � perturbation period;TI � wave period predicted by numerical model;

t � time;U�x� � unit-step function;

u � depth-averaged velocity in the streamwisedirection;

x � streamwise coordinate;�1,2 � growth rate of faster �slower� wave obtained

by the periodic disturbance analysis;�NM � growth rate predicted by numerical model;

� � bed inclination angle;� � wall roughness;

�� relative roughness;� � wave growth rate;� � disturbance spatial grow factor;� � integration constant;� � resistance coefficient;� � threshold value;
� � kinematic viscosity;


� integration constant;� � water density;

�b � bed shear stress;�e � perturbation dimensionless pulsation;

�, � transformed variables;�x � space step of the numerical grid along x;

. � dimensionless variable �superscript�;0 � uniform base state variable �subscript�;p � perturbed variable �subscript�;I � imaginary part; andR � real part.

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Influence of Relative Roughness and Reynolds Number on the Roll-Waves Spatial Evolution

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