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041093B.ENG
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Country: Japan
From: Memoirs of the Faculty of Engineering, Kyushu
University, Vol. 53, No. 1, March 1993
ON THE DOMINANT LENGTH OF SAND WAVES IN THE LOWER FLOW REGIME
(Received December 7, 1992)
by
Kunitoshi WATANABE
Associate Professor, Department of Civil Engineering, Faculty of
Science and Engineering, Saga
University
Muneo HIRANO
Professor, Department of Civil Engineering Hydraulics and Soil
Mechanics
Hossam NAGY
Graduate Student, Department of Civil Engineering Hydraulics and
Soil Mechanics
In Two Parts
Part II
****SYMBOL TABLE FOR FORMULAE****
MATHEMATICAL
a' . . . . . . . . . . . . . . . . . . . . . . . . .partial differentiation
e. . . . . . . . . . . . . . . . . . . . . . . . .base of natural logrithms
DEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .del
PI . . . . . . . . . . . . . . . . . . . . . .transendental number 3.14etc.
SQRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .square root
sub. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .subscript
SUM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . summation sign
*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . multiplication
^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .to the power of
GREEK LETTERS
A' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .capital ALPHA
B' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . capital BETA
g' . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case gamma
e' . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case epsilon
z' . . . . . . . . . . . . . . . . . . . . . . . . . . . . .lower case zeta
T' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .captial THETA
L' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . captial LAMBDA
x' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .lower case xi
p' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case rho
s' . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case sigma
t' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case tau
F' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .capital PHI
f' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lower case phi
S' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .capital PSI
================================================================
See previous message for Abstract, Introduction, Length of Sand
Waves (first part)
2.2.2 CHARACTERISTICS OF DOMINANT WAVE NUMBER
To examine the main factors which have effect on wave
number, Eq.14 could be expressed as the following function
B'(sub d) = f(sub 1)(F, p'(sub o), E, W, q) . . . . . . . . . . (16)
By using the relations S(sub o) = F^2/f'^2 and S'(sub o) =
h(sub o)S(sub o)/sd, E is rewritten as a function of S'(sub o)
E = [L'(sub d)/s]*[1/f'(sub o)^2]*
[1 + A(sub *)F'(sub Bo)/S'(sub o)]*F^2. . . . . . . . . . . . . . (17)
Based on the model of Hirano[14] for suspended sediments,
the parameter W is a function of wo/usub *), as already
mentioned. The parameter q is a function of f'(sub o) from Eqs.
6 and 10.
By using the above relations, Eq. 16 reduces to more
simple and fundamental one:
B'(sub d) = f2(F, S'(sub o), wo/u(sub *)) . . . . . . . . . . . . (18)
Numerical solution for Eq.14 is derived in order to
investigate the four previous variables effects on wave number
values. Figure 2 shows the relation between the Froude number
F and the wave number B'(sub d) for S'(sub o) value equal to
0.1, f'(sub o) equal to a reasonable moderate value 14 and
wo/u(sub *) equal to 0.1, 0.3, 0.8 and infinity, respectively.
The figure shows a descending in B'(sub d) with F, and the curve
changes from linear to curvilinear form. It is also noticed that
B'(sub d) increases with the decrease of wo/u(sub *) for F > 0.6,
and the curves for different values of wo/u(sub *) tend to merge
into the same single line with the decrease of the Froude number
F. In Fig. 3, the relation between B'(sub d) and S'(sub o) with a
parameter wolu* is shown for F = 0.3 and f'(sub o) = 14 by using
Eq. 14. The figure shows that neither S'(sub o) nor wo/u(sub *)
has any effect on the wave number value. The effect of the
velocity coefficient f'(sub o) on the wave number B'(sub d) is
investigated. Figure 4 shows a relation between B'(sub d) and
the velocity coefficient f'(sub o) with a parameter wo/u(sub *)
for F = 0.3 and S'(sub o) = 0.1 by using Eq. 14. It is noticed
that wo/u(sub *) does not affect B'(sub d) value, while F'(sub o)
has a little effect on that value, especially for f'(sub o) < 10.
As been concluded from Figs. 2, 3 and 4,the suspended
sediment parameter wo/u(sub *) may be eliminated in Eq. 18 for
low values of Froude number. Also from Fig. 3, the effect of
parameter S'(sub o) may be eliminated in Eq. 18. Since the value
of B'(sub d) is approximately constant for the change of f'(sub
o) value as shown in Fig. 4, the parameter f'(sub o) may be
eliminated in Eq. 18 without fear of accuracy. Consequently, the
only dominant variable which has effect on the wave length
formation is the Froude number F. From above figures, Eq. 18 may
be simplified by neglecting the less effective parameters as
B'(sub d) = f(sub 3)(F) . . . . . . . . . . . . . . . . . . . . . (19)
2.2.3 PRACTICAL FORMULA FOR WAVE LENGTH
For the sake of treating with sand wave steepness, there is
more particular interest of getting a practical formula of sand
wave length in the lower flow regime is demanded. From Fig. 1,
it is noticed that the dominant value of B' is quite near the
marginal value B'(sub c), which gives the wave length before sand
waves are washed out, or which separates stable and unstable beds
(g'(sub 1) = 0). On the other hand, from the study by Watanabe
and Hirano[10], the three boundary conditions for bed stability
are expressed as follows
1 - L'(sub 2)B'^2F^2 = 0. . . . . . . . . . . . . . . . . . . . . (20)
1 - (1 + L'(sub 2)B'^2)F^2 = 0. . . . . . . . . . . . . . . . . (21)
qA' -2E(sub *) = 0. . . . . . . . . . . . . . . . . . . . . . . . (22)
Figure 5 shows the stability diagram represented by B'
and the Froude number F. The solid lines represent the three
boundary curves which separate stable and unstable conditions, in
other words, sand waves and flat bed form. The dotted line
represents Eq. 14 with all parameters. The boundary curve from
Eq. 21 shows good agreament with Eq. 14 curve, especially for F <
0.6. From this conclusion one can put B'(sub d) = B'(sub c) in
the lower flow regime. By using Eq. 21, the practical equation
for dominant wave number equation may be expressed in the form
B'(sub d) = 1.589SQRT((1/F^2)-1) . . . . . . . . . . . . . . . . .23)
The practical equation of sand waves length in the lower
flow regime is expressed in the form
4F/SQRT(1 - F^2). . . . . . . . . . . . . . . . . . . . . . . . . (24)
Most of previous studies of wave number and wave length have
been made from distinctly different approaches. Based on the
theory of potential flow, using the complex stream function,
Anderson[1] introduced the relation between the Froude number and
wave number as follows
F^2 = [sin(h2B')]/[B"^2(tan(hB')sin(h2B'-2)]. . . . . . . . . . . (25)
Ten years later, Kennedy[3] modified Anderson's theory and
proposed equation which relates the wave number of the dominant
wave length to the parameter F,
F^2 = [2 +B'tan(hB')]/[B'^2 + 2B'tan(hB')] . . . . . . . . . . . (26)
Tsuchiya and Ishizaki[6] assumed that the surface wave has a
small amplitude, which is half the wave length of the dune, then
the following expression was introduced:
F = SQRT[(2/B')tan(h2B')] - SQRT[(1/B')tan(hB')]. . . . . . . . . (27)
Hayashi[7] modified the study of Kennedy[3] and introduced
the region of occurrence of sand waves in F - B' plane by using
the following equation:
F(sub 1 and 2)^2 = [1/4B'tan(hB')]*
{c +2 +/- SQRT[(c + 2)^2 - 8ctan(h^2b')]} . . . . . . . . . . . . (28)
in which c is a dimensionless parameter and taken equal to 2. From
Eq. 28, the Froude number has two values F(sub 1) and F(sub 2) for
one value of R and there relation could be represented by two
curves Hayashi I and Hayashi 2, as shown in Figs. 6 and 7.
Nakagawa and Tsujimoto[8] introduced the limiting curve for
F< 1 by using the following equation
F^2 = tan(hB')/B' . . . . . . . . . . . . . . . . . . . . . . . . .(29)
From all previous expressions, it is needless to say that the
only dominant variable which has effect on wave number value is F.
3. COMPARISON WITH EXPERIMENTAL RESULTS
The theory is examined by using the data from rivers and
experimental channels, presented by Guy, Simons and
Richardson[15], Shinohara and Tsubak[16] and others[17 to 30]. In
Figs. 6 and 7 the data for dunes and ripples are plotted with the
curves given bN, Eqs. 23, 25, 26, 27, 28, and 29, respectively.
In Fig. 6, the data for dunes show scattering in a wide
range, and neither the presented curve nor reference curves,
except one by Eq. 27, give a close agreement with scattered data.
A modification is conducted in Eq. 23 to obtain the fitting curve
for B'(sub d) with the plotted data as
B'(sub d) = 0.63SQRT[(1/F^2)-1)]. . . . . . . . . . . . . . . . . .(30)
Equation 30 is completely compatible with Eq. 27 by Tsuchiya
and Ishizak[6] curve for F > 0.20 as shown in Fig. 6.
In Fig. 7, despite considerable scattering of the plotted
data for ripples, the curve given by Eq. 23 shows the trend of
data and lies the average of scattering more than the other
comparable curves. A modification is conducted in Eq. 23 to
obtain the fitting curve for B'(sub d) with the plotted data as
B'(sub d) = 1.4SQRT[(1/F^2) - 1)] . . . . . . . . . . . . . . . . .(31)
4. CONCLUSIONS
Using linear stability analysis, bed forms in the lower flow
regime was examined analytically for determining the dominant wave
length. The result was compared with the extensive bed forms data
collected from rivers and experimental flumes. From this study
the following conclusions are obtained.
1. The dominant wave number value lies in the unstable zone
for bed and it is very close to the marginal stability value.
It indicates that the linear stability analysis is applicable
approach to the problem.
2. The suspended sediment factor w(sub o)/u(sub *) has a weak
effect on the dominant wave length. For the low Froude
number F, one could neglect this factor in the wave number
equation. On the contrary, for Froude number F > 0.6,
suspended load should be considered.
3. For different values of Froude number, the dimensionless
tractive force S'(sub o) has no effect on wave number value.
4. The velocity coefficient Oodoes not strongly affect the
wave number value. The wave number B'(sub d) slightly
increases with the increase of velocity coefficient f'(sub o)
effect and for more simplification f'(sub o) may be
neglected.
5. The apparant fact in this study is that the Froude number F
is the major factor which affects the wave length.
6. Using the previous results, a practical equations for wave
number B'(sub d) for both dunes and ripples are obtained.
7. To demonstrate the applicability of the theoritical
approach, the practical equations had been compared with the
previous studies and data from rivers and flumes in (F - B')
plane for dunes and ripples, respectively. The data show a
good agreement with the proposed equations.
8. Some further principle or parameter, as yet unknown, must
be responsible for distinguishing between ripples and dunes
wave numbers.
FIGURE CAPTIONS
Fig. 1 Relation between growth rate f' and B'
Fig. 2 Relation between F and B'(sub d) with parameter
w(sub o)/u(sub *)
Fig. 3 Relation between f'(sub o) and B'Sub d) with parameter
w(sub o)/u(sub *)
Fig. 4 Relation between f'(sub o) and B'(sub d) with parameter
w(sub o)/u(sub *)
Fig. 5 Stability diagram for lower regime
Fig. 6 Comparison with various theoretical results and the
experimental data for dunes
Fig. 7 Comparison with various theoretical results and the
experimental data for ripples
REFERENCES
1) Anderson, A. G. (1953): The Chracteristics of Sediment Waves
Formed by Flow in Open Channels, Proc. 3rd Midwestern Conf.
in Fluid Mech., Minneapolis, pp. 379-395.
2) Milne-Thomson, L. M. (1960): Theoretical Hydrodynamics. New
York: MacMillan.
3) Kennedy, J. F. (1963): The Mechanics of Dunes and Antidunes
in Erodible Bed Channels, J. Fluid Mech., Vol. 16, Part 4,
pp. 521-544.
4) Yalin, M. S. (1964): Geometrical Properties of Sand Waves,
Journal of the Hydraulics Division, ASCE, Vol. 90, No. HY. 5,
pp. 105-119.
5) Jopling, A. V. (1965): Discussion of "Geometrical Properties
of Sand Waves", by Yalin, M., Proc. ASCE, Vol. 91, No. HY3,
pp. 348-374.
6) Tsuchiya, A. and Ishizaki, K. (1967): The Mechanics of Dune
Formation in Erodible-Bed Channels, Proc. 12th Congress of
the IAHR, Vol. 1, pp. A59.1-A59.8.
7) Hayashi, T. (1970): Formation of Dunes and Antidunes in Open
Channels, Proc. of the ASCE, Vol. 96, No. HY2, pp. 357-366.
8) Nakagawa, H. and Tsujimoto, T. (1980): Sand Bed Instability
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ASCE, Vol. 106, No. HY12, pp. 2029-2050.
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12) lwasa, Y. and J. F. Kennedy (1968): Free Surface Shear Flow
over a Wavy Bed, Proc. ASCE, Vol. 94, No. HY2, pp. 431-454.
13) Robillard, L., and Kennedy, J. F. (1967): Some Experimental
Observations on Free Surface Shear Flow over a Wavy Boundary,
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Load, Annuals, Faculty of Engineering, Kyushu University,
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15) Guy, H. P., Simons, D. B. and Richardson, E. V. (1966):
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Sand Waves Formed upon the Beds of the Open Channels and
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7, No. 25, pp. 15-45.
17) Vanoni, V. A. and Li-San Hwang (1967): Relation Between Bed
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144.
18) Mantz, P. A. (1983): Semi-empirical Correlations for Fine and
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Approach and Analysis, Proc.
ASCE, Vol. 99, No. HY11, pp. 2041-2059.
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Small-scale Current Ripples:
An Experimental Study Using Medium Sand, J. of Sedimentology
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21) Carl F. Nordin, JR. (1971): Statistical Properties of Dune
Profile, U. S. Geological Survay Professional Paper 562-F, pp.
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22) Martinec, J., Ing., Csc, (1967): The Effect of Sand Ripples on
the Increase of River Bed Roughness, Proc. 12th Congress of the
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23) Znamenskaya, N. S. (1963): Experimental Study of the Dune
Movement of Sediment, Transactions of the State Hydrologic
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24) Michael Gee, D. (1975): Bed Form Response to Nonsteady Flows,
Proc. ASCE, Vol. 101, No. HY3, pp. 437-449.
25) Laursen, E. M. (1958): The Total Sediment Load Streams, Proc.
ASCE, No. HY1, pp. 1530(l)1530(36).
26) Williams, G. P. (1967): Flume Experiments on the Transport of
a Coarse Sand, U. S. Geological Survay Professional Paper 562-
B, pp. 1-31.
27) Hubbell, D. W. and Sayre, W. W. (1964): Sand Transport Studies
with Radio-active Tracers.
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29) Tanaka, Y. (1969y: Study on Sand Waves-A Consideration on the
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END. Msg. B.ENG
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