ARCHIEF
118
INTERNATIONAL ASSOCIATION FOR HYDRAULIC RESEARCH
Delft
AREAL VARIATIONS OF BED-FORM CHARACTERISTICS IN MEANDERING STREAMS
by
Emmett M.
O'Loughlin
Research Engineer
Institute of Hydraulic Research
The University of Iowa
Iowa City, Iowa, U.S.A.
and
David Squarer
Research Associate
Institute of Hydraulic Research
The University of Iowa
Iowa City, Iowa, U.S.A.
Synopsis
The reliability of expressions for resistance coefficients and sediment
discharge in alluvial channels is open to question when they are applied to sinuous
channels because no account is taken of the distorted channel section and the
sec-ondary currents induced by channel curvature.
Bed-form geometry is related in a
fundamental way to resistance and sediment discharge, and it is shown in this paper
that channel curvature results in significant variations in bed-form geometry.
Following a procedure similar to that described by Nordin and Algert,
autocovariance and spectral density functions of a process defined by the bed
ele-vation as a function of the distance along the channel are computed in several
zones in a laboratory model of an alluvial channel having an equilibrium bed
con-figuration.
These data are compared with other data of the same type obtained
with nominally identical flow conditions (same mean depth and mean velocity) and
bed material in a straight flume.
Details of spectrum computations are discussed in terms of the length of
record and the variability of the estimated spectral density.
Stationarity
re-quirements indicate that future analyses should probably be made in the time
do-main, and in such cases the possibility of relatingtime and space spectra needs
close examination.
Sommaire
Les relations exprimant la resistance
el'ecoulement et le debit solide
soot sujettes
acaution quand on les applique au cas de canaux sinueux du fait que
l'on ne tient pas compte de la distorsion de leur section et des courants
secon-daires induits par la courbure du canal.
La geometrie du lit est reliee d'une
maniere fondamentale
ala resistance et au debit solide, et nous montrons dams cat
article qua la courbure du canal change dune maniere appreciable la geometrie du
lit.
Suivant une methode semblable
acells decrite par Nordin et Algert,
l'auto-covariance et l'analyse spectrale d'un ensemble forme par la cote du lit en
fonction de la distance le long du canal sont calculees pour plusieurs regions
d'un modele reduit de canal alluvial dont l'etat du lit est stationnaire.
Les
re-sultats des calculs sont compares avec d'autres du meme genre obtenus, pour le cas
d'un canal rectiligne, dams des conditions d'ecoulement (meme profondeur et meme
vitesse moyenne) et des materiaux du lit pour ainsi dire identiques.
Les calculs de l'analyse spectrale soot examines en detail en fonction de
la longueur de l'echantillon et de la variabilite du spectre approxime.
Les
condi-tions exigees pour qu'un phenomene soit stationnaire indiquent qua des analyses
futures devraient sans doute etre faites dans le temps et en de tels cas la
possi-bilite de relier les spectres temporels at spaciaux necessite un examen detaille.
INSTITUTE OF HYDRAULIC RESEA
UNIVERSITY OF I DWA
Reprint No. 235
Lab. v. ScheepshowakunJe
Introduction
During the last decade, knowledge gained from research on the mechanics of flow over alluvial beds has contributed significantly to the understanding of phenomena such as sediment transportation and boundary resistance. It is notewor-thy that most of the contributions have been based upon, inspired by, or verified with laboratory experiments which have been confined to observations made in straight flumes of one type or another. Those responsible for such research have usually appreciated that the validity of semi-empirical laws derived from straight-channel experiments is open to question when applied to sinuous straight-channels, the reason being that no account is taken of sectional distortion or secondary cur-rents induced by boundary curvature.
On a gross scale, patterns of erosion and deposition occurring in a bend affect and are affected by the flow configuration within the bend; the state of equilibrium within a bend for a given discharge is determined by the mutual inter-action of the bed geometry and the mean-flow variables. Thus, any quantity which depends upon the three-dimensional configuration of the channel, such as a bend-loss coefficient, should ultimately be expressible in terms of the mean-flow
vari-ables. It is, of course, understood that the sediment properties will influence the form of this type of relationship, as well the planimetric shape of the bend; the following discussion presupposes that factors such as these are not permitted
to vary.
In attempting to describe the processes associated with the flow in a bend formed in alluvial material, one is immediately confronted with the problem of selecting which of the variables can be regarded as independent, that is, which set of variables will uniquely determine any of a number of remaining variables which may be of interest. If, for example, the problem is to predict the total sediment load in the comparatively simple case of a sinuous channel confined laterally between fixed boundaries, it must be appreciated that the channel-geometry parameters and the flow characteristics, both of which influence the
sediment load, are themselves influenced by the sediment load. It is not yet possible to formulate explicit relationships between these variables for bend
flow.
The dilemma raised by this situation has relevance not only to flow within a bend itself, but also to the flow through reaches between bends. Even
if the sediment transportation is considered to depend only upon conditions exist-ing in the straight reaches, these same conditions cannot be accepted as beexist-ing independent of the processes which occur in the preceding bends, because the bed level in the straight reaches is determined in part by the extent of bed scour in the preceding bends. Material scoured from bends is deposited in the straight
reaches or crossings; for large amounts of scour in the bends, the depth of de-posited material in the crossings may also be large. The relationship between
the fluid and sediment discharges in the straight reaches must therefore depend to some extent on the state of flow in the bends. In fact, if the channel is composed of a succession of bends joined by short straight reaches, the sediment load will possibly be determined by the bend phenomena alone.
In view of the apparent interdependence of bend- and straight-channel flow, it was deemed advisable that a set of experiments be conducted to provide some basis for clarifying the nature and extent of the influence of channel curva-ture upon conditions existing in the intervening straight reaches.
Bed-Form Geometry
Curvature effects may be directly evaluated by making comparisons of some relevant quantities measured under nominally the same flow conditions in both straight and meandering channels. Rather than comparing gross quantities
1,
characterizing the flow in the two channels, a more directly measured parameter, the local bed-form geometry, was selected for comparison. Justification for this was based upon the relevance of bed-form characteristics to phenomena of primary
interest in alluvial-stream mechanics, such as channel roughness, and sediment transported as either bed or suspended load. Because the usefulness of express-ing a bed friction factor in terms of an equivalent sand roughness has been found to diminish due to the dependence of the Karman "constant" upon the areal concen-tration of roughness elements, statistical methods have been used in the present study to describe the various bed configurations. It has been shown by Simons et al (1) that the characteristics of bed forms in alluvial channels are related to the sediment transported as bed load, and Nordin and Algert (2) have demonstrated
that the statistical parameters describing bed forms correlate well with the mean-flow characteristics. It might also be expected that for a given sediment, con-sistent correlations between these statistical parameters and total sediment load can also be obtained, because the diffusivity and hence the entraining power of flow near a rough boundary is determined by the geometry of the roughness ele-ments, which in this case is best described by the statistical properties of the bed forms.
The bed-form geometries in the straight and meandering channels were computed in terms of the statistical descriptors used by Nordin and Algert (2). Autocovariance and spectral density functions of a process defined by the bed
ele-vation as a function of the distance along the channel were computed in several zones in a laboratory model of a meandering alluvial channel having an equilibrium bed configuration. These data were compared with other data of the same type ob-tained with nominally identical flow conditions (same mean depth and mean veloci-ty) and bed material in a straight flume. The observations which have been made can be grouped to illustrate two fundamental comparisons; in the first instance, that between the bed form geometry in the straight reach separating two bends and in a straight channel; and in the second instance, between the ripple-geometry parameters measured at several locations within the bend. These two comparisons serve to emphasize the dubious applicability of existing predictors for sediment transportation, bed roughness, and related quantities, when they are applied to sinuous channels.
3. Experimental Facilities
Straight-channel measurements were made in a recirculating tilting flume with a test section length of 90 feet and a width of 3 feet. The reinforced-concrete meandering channel (Fig. 1) is 116 feet in overall length, and has a rec-tangular section 7 feet 8 inches wide by 18 inches deep. The slope is adjustable, and two variable-speed axial-flow pumps recirculate the sediment-fluid mixture. Further details of the meandering channel are given in a report by Ben-Chie Yen
(3). The sand in both channels had a median grain size of 0.25 mm., and was placed to a depth of 9 inches.
In each channel, the mean depth is uniquely determined by the volume of water placed in it. Discharges were measured at calibrated constrictions in the return pipelines. The mean depth and velocity of flow in each channel were 0.383 foot and 1.053 feet per second. The duration of the tests in each channel was sufficient to ensure that steady uniform flow was achieved; such a condition was reached in the straight channel in a fairly short time by making slope adjustments
Simons, D.B., Richardson, N.V., and Nordin, C.F., "Bedload Equation for Ripples and Dunes," U.S. Geological Survey Professional Paper 462-H, 1965.
Nordin, C.F., and Algert, J.H., "Spectral Analysis of Sand Waves," J. Hyd. Div., ASCE, Vol. 92, No. H15, September 1966.
Yen, Ben-Chie, "Characteristics of Subcritical Flow in a Meandering Channel," Ph.D. Dissertation, Univ. of Iowa, June 1965.
during a run. However, in the meandering channel, the overall slope could not be conveniently changed, so sufficient time was allowed for the bed to adjust its
slope until no further significant changes in local depth occurred. The flow was classed as uniform insofar as the depth at the exit of the second bend was the same as that at the exit of the first bend. No more precise criterion for uni-formity could be established because the approach conditions to each bend were necessarily dissimilar. In fact, the channel has been designed so that the func-tion of the first bend is to impose realistic entrance condifunc-tions for the second bend, in which all measurements were made.
Details of bed features were measured by lowering a carriage-mounted point gage to the bed at longitudinal intervals of 0.025 foot; the motion of the pointer was transmitted to a ten-turn precision potentiometer by means of a rack-and-pinion arrangement. Voltage measured across two terminals of the potentiome-ter was thus linearly related to the poinpotentiome-ter displacement. Use was made of the Institute's IBM 1801 Data Acquisition and Control System to convert the potenti-ometer voltage to a digital value, and to provide both punched and printed records of bed-form data. The punched output was thereafter fed directly to a larger ca-pacity digital computer, which was programmed to provide a continuous plot of the bed form, together with the results of the autocovariance and spectral density function analyses.
Shown in Fig. 2 are the positions at which bed-form measurements were made in the bend. In locating these traverse lines, an attempt was made to
fine each line to a zone in which the boundary shear stress was approximately
con-stant. Guidelines for selecting such zones were available in results obtained by Chin-lien Yen (4), who used a Preston tube to measure the shear distribution in the same channel as that used in these experiments; the use of a Preston tube re-quires a smooth surface, so Yen covered the sand with a thin mortar cap after the bed features had been smoothed out. Although the boundary roughness was thus
forcibly made uniform, it would be expected that the measured shear-stress distri-bution would at least approximate the distridistri-bution on the undisturbed bed. Of
course, the question remains as to whether shear-stress distribution provides a sound basis for selecting reaches to be analyzed; this important point is related to the stationarity of the process, and is considered in the following section.
4. Analysis of Bed-Form Data
The discussion of the computational procedure given below is presented in some detail. It is the authors' opinion that statistical measures of bed-form geometry will provide consistent and systematic descriptors which will prove to be useful in formulating expressions for quantities such as boundary resistance, to-tal sediment load, diffusion, dispersion, and so on. Until quite recently, it has been impractical to make detailed measurements of bed geometry, but the continuing development and growing acceptance of sonic sounding devices with high resolution capabilities now permit such data to be obtained with ease in either the field or the laboratory.
The statistical analysis should provide descriptors which characterize both dune heights and wavelengths. From a roughness point of view, we are inter-ested in not only an average roughness height, expressible as the standard devia-tion of the bed level measured from its mean elevadevia-tion, but also in the relative contribution of dunes of a given wavelength to the overall standard deviation. The two measures would provide information comparable to measures of absolute roughness height and areal concentration of roughness elements, both of which are required in an expression for the roughness of even a rigid boundary. The auto-covariance and spectral density functions provide just this type of information.
(4)
Yen, Chin-lien, "Bed Configuration and Characteristics of Subcritical Flow in a Meandering Channel," Ph.D. Dissertation, Univ. of Iowa, February1967.
In a spatial domain, the autocovariance of the bed elevation X(X) mea-sured from its mean level is given by
lim 1 fL/2
c(x)
=,
x(
R.) X(9. + x)dt (1)and represents the mean value of the product appearing in the integrand. If the "lag distance" is zero, then C reduces to the variance of the variable X. This expression for C(X) requires that X(R) be stationary in the statistical sense,i.e., the environment which determines X must be unchanging throughout the length L, and have zero mean. (If the latter condition is not fulfilled, an additional term appears in the expression for C(X).)
To meet the requirement that the process be stationary, the continuous records of bed elevation along the sections shown in Fig. 2 were divided into sub-sections; for these subsections, lines representing the mean bed elevation were fitted by computation, using the criterion
ave iX(R)1 = IL X(t)d.R. 0
L 0
It should be recognized that if the stochastic process may be described by a dispersive-wave phenomenon, that is to say waves whose celerities depend upon their wavelengths, the ergodic property is not necessarily valid even though the process might well be stationary. It is beyond the scope of this paper to pursue this point, but it should be pointed out that even if the strict mathematical con-ditions for a stationary process to be ergodic are not fulfilled, one might still expect to find some empirical relationship between averages in the time and space
domains. The latter possibility appears to find justification in the results of Nordin and Algert (2).
An infinite number of dune wavelengths contribute to the variance of X. Wave numbers (cycles per unit length) between f and f + df contribute an amount given by VP(f)df, where P(f) is known as the spectral density function. Taylor (5) has shown that P(f) and C(A) are simply Fourier Transforms of each other, and because both are even functions, the relationship between them may be expressed as a cosine transform
CO
P(f)
= LC(A) cos 27fAdX
(2)Determining the spectrum of dune wave numbers amounts to making the transformation defined by Eq. (2).
In a physical situation, estimates of C(X) and P(f) must usually be made on the basis of a finite number of discrete observations of X. In such a case, the summation equivalent of Eq. (1) is
C(J) 1 E X(I) X(I + J.) (3)
N-m N-M
I.1
where C(J) is the autocovariance at a lag distance of A JAX, AX being the dis-tance between successive points at which X is measured. The maximum lag distance is defined as MAX, and MAX is the total length of the sample.
Application of Eq. (2) to obtain P(f) requires that C(A) be defined for all possible values of X. For a finite length of record, this is clearly
impossi-ble. It has been shown by Bartlett
(6)
that the use of a modified autocovariance function, D(X)C(A), where D(A) is defined as zero for values of 'XI greater thanTaylor, G.I., "The Spectrum of Turbulence," Proc. Roy. Soc.,
A164, 476-490,
1938.
Bartlett, M.S., "Periodogram Analysis and Continuous Spectra," Biometrika,
June
1950.
=
=
MAX, allows this difficulty to be overcome. A judicious choice of D(A) results in a smoothing effect on the estimate of P(f), so that a reduction occurs in the
spectrum-curve irregularities which may arise from unwanted data "noise" or from the limitations imposed by a finite length of record.
After the transformation of the modified autocovariance function is made, the convolution theorem may be used to show that the result obtained for any wave number fk represents an estimate of the smoothed spectral density P'(fk), and that the mean value of the smoothed estimate of the true spectrum, P(f), is
ave fP'(fk)} =
fo(Q(f
fk) Q(f - fk)) P(f)df(4)
The function Q(f) is the Fourier Transform of DM , and may be visualized as a weighting function of total weight unity which gives maximum weight to the wavenumber fk. Many alternatives are available in choosing the transform pair D(A) and Q(f), and the common choices are discussed in detail in Blackman and Tukey (7). The function D(A) adopted in this analysis is the same as that used by Bartlett (6), and by Nordin and Algert (2),
D(X) = 1 - (5)
Writing an expression for the one-sided cosine transform of the modified autoco-variance function in the form of a summation, one obtains
M-1
P'(f )k = 2AA(C_o + 2 E (1 -
k)C(J)
cosNE)
(6)
J.1for Ks positive integer, and JIM = fkAA.
For the purposes of planning an experimental program, an approximation to the variability of the estimate P'(fk) is given by
var{P'(fk)/ M
(ave{p,(fk)})2 N - M/3
It is thus seen that the use of lag distances which are more than a small fraction of the total length of record may lead to erratic variation in the computed
spec-trum. On the other hand, the resolution between successive estimates is given by Af = 1/MA, which is inversely proportional to the lag distance. Therefore, if the total length of record is limited, the spectrum estimate must represent a com-promise between the desired resolution and the tolerable variability. The point of this discussion is illustrated in Fig. 3(a), in which are shown spectra of bed
forms measured in a 35-foot length of the straight tilting flume. Both curves have been derived from the same data, but the smoother curve, computed for a maxi-mum lag distance of 5 feet (M = 200) yields a resolution which is inferior to the
curve for a maximum lag distance of 10 feet (M = 400). Tilting-flume data repre-sented in Fig. 3(a) were limited by the length of uniform flow, i.e., the length in which the process could safely be assumed stationary. Other samples may easily be generated by making several flume runs, in which case the spectrum may be com-puted as the average of the spectra obtained for all runs.
The condition of stationarity poses more difficult problems in the case of the meandering channel, because uniform flow comparable to that in a straight
channel simply does not occur. Short lengths chosen on the basis of approximate uniformity lead to spectral estimates of great variability. However, even though one still has recourse to the possibility of combining many estimates to obtain a stable spectrum, estimates at low wave numbers (long dunes) are hampered by the
(7)
(7)
Blackman, R.B., and Tukey, J.W., "The Measurement of Power Spectra From the Point of View of Communications Engineering," Dover Publications, Inc.,N.Y.,
1958.
+ +
-severe limitation that the lag distance can only be a small fraction of the length of record, which is already short.
Experimental results for four locations in the meandering channel are presented in Figs. 3 and 4 in the form of spectral density and autocorrelation
(ratio of C(A) to C(0)) functions. The autocorrelation curves are marked by prom-inent periodicities, which are manifested also as peaks in the spectral density
curves. The positions and relative magnitudes of the peaks supply a quantitative estimate of the predominant ripple wavelengths at the various locations in the
channel. If the straight-channel spectrum is adopted as the basis for comparison, it is seen that the outside of the bend is characterized by long dunes, and the point bar by short dunes; just as striking is the comparison of standard devia-tions of the bed elevation, which for the deeper part of the channel is approxi-mately twice that on the point bar.
Shown in Fig. 3(a) are spectra computed from data measured in the straight flume and in the straight reach between bends in the meandering channel. The standard deviations of the two sets of data are comparable, but the presence of a pronounced peak in the meandering-channel spectrum at a wavelength of 0.5
foot, in contrast to the fairly featureless spectrum measured in the straight flume, suggests that a significant difference in boundary roughness exists in the two situations.
Conclusions
It is likely that many sets of statistical estimates will be obtained in the course of future research on alluvial-channel flow; comparisons between these sets may be grossly misleading unless it is appreciated that the computed statis-tical descriptors are in fact estimates based upon a finite sample of the
popula-tion. In the first place, inevitable sampling errors must be expected; secondly, the parameters used in the analysis of given data exert a pronounced influence up-on the appearance and variability of spectral estimates. The two effects become interconnected in the analysis, and the researcher must be able to recognize the limitations of his raw data in terms of the variability of the resulting spectrum (and to convey these limitations to others who may wish to use the results).
The material discussed in the preceding sections represents the outcome of experiments in which bed forms were observed in a straight and in a meandering
channel. It is clear that consistent estimates of the statistical parameters can-not be made unless repetitive experiments are carried out. In any case, long dunes will escape analysis in all but the longest channels. It therefore seems inevitable that future experiments be made not in the space, but rather in the time domain, in which the possible length of record is limited only by the tempo-ral variability of the flow. However, boundary roughness is probably more con-veniently related to a spatial domain, while the rate of sediment transportation is likely to be easily expressed in terms of the temporal domain. The future uti-lization of the statistical parameters depends upon whether the two spectra of the nonergodic but stationary process can be shown to be related to each other.
Areal nonuniformities in dune amplitude and wavelength throughout bends must have a strong influence upon the gross behavior of flow between curved
bound-aries. The nature of the interaction between bed and flow is a complex one, and
it is to be hoped that some understanding of the interrelationship may be gained through experimental results of the type presented herein.
Acknowledgments
The writers are grateful to Dr. J.R. Glover, and Messrs C. Farell, R. Chevray, T. Pandit, and S. Mehrotra for their help in various phases of the study. This work was supported in part by the United States Department of the Interior as authorized under the Water Resources Research Act of
1964,
Public Law88-379,
andis being undertaken by D. Squarer as dissertation research under the direction of E.M. O'Loughlin. The authors' names are given in alphabetical order.
5.
Datum
Fig. l, Geometry of Meandering
Channel-CaracteristiqUes geometriques du eanal sinueUx,
,B /17,.=
F =0.3
14, =1,053 fp.s. Contours at z/ h. shown
Fig,. 2. Bed Configuration. and Location of Sections in Bend
Topographie du lit et emplacement des SectionS, longitudtriales den§ un m6andre.
Straight flume
./7=
2,84Xle N = .07M 200
-M , 400 ---Affes, /12 96 80 64 48 32 00Fig. 3(a). Spectral-Density Functions in Straight Flume and Straight Reach of Curved Channel.
Functions spectrales pour an canal rectiligne et pour an bief rectiligne dun canal sinueux.
Simtjahebiend = er.rfcT`fr.
N=2.
M=100 2 3 flayal48/1,211 5Fig. 3(b). Autocorrelation Functions in Straight Flume and Straight Reach of Curved Channel.
Auto-correlations pour un canal rectiligne et pour an bief rectiligne d'un canal sinueux.
1,0 Straight flume AA 0,025 fr. 0.2 06 0. 4 o. 0.2 0.40 A
/.0 0,6 0.4 0.2
-02
05 1,0 /,5 2,0 2,5(f')
05 1.0 /.5 2.0 2.5 3 4 ( feet)Spectral Density and Autocorrelation Functions at Three Locations within Curved Channel.
Fonctions spectrales et auto-correlations trois emplacements dans le meandre. 2 5 fr,awmov 48 SECTION 2 IN BEND
