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040993B.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 I
****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
================================================================
ABSTRACT
A theoretical study is conducted to determine the dominant
length of sand waves in the lower flow regime, which are usually
formed on the cohesionless erodible bed of rivers and channels.
A chaarcteristic dimensionless wave number, that is the dominant
over all others, is studied by using the two dimensional linear
stability analysis, taking account of factors which represent
the effect of suspended sediments, the asymmetric distribution
of bed shear stress, the centrifugal effect of curvilinearity of
bed and water surfaces, and the non-equilibrium of bed load
transport process.
The wave number value has been numerically evaluated for
different values of flow velocity, depth, bed size material,
shear velocity and bed load discharge of interest. Then, the
effect of variables on the dominant wave number is examined and
an equation for the dominant wave number is obtained. Finally,
a practical formula for wave number is given in order to be used
for further study. The theoretical results are compared with
data from laboratory flumes and rivers and show a reasonable
agreement with them.
KEYWORDS: Sand waves, Dunes and ripples, Dominant wave
length, Linear stability analysis, Suspended sediments, Tractive
force, Bed load
1. INTRODUCTION
In recent years it has been made clear that until a
fundamental understanding of the reasons of formation and the
characteristics of bed forms is achieved, precise approach to
problems of bed configurations remains largely on a rather
roughly empirical formula. In general, it might be suggested
that the prediction of the sand wave geometrical shape requires
a full understanding of two main fundamentals: one is the
initiation process of sand waves, the second is the growth
process of the sand waves. If the two processes were precisely
understood, it would be possible to predict the shape of sand
waves. in particular, it might be essential to correlate the
wave length or height with the flow resistance, bed load
transport rate and bed roughness.
Most of previous studies, for the geometrical shape of sand
waves in open channels and alluvial rivers, concentrated on
height of dunes and antidunes. On the contrary, length of bed
forms such as dunes and ripples has a lack of interest. The
biggest part of these studies introduced the wave number and the
Froude number relation as a boundary for bed stability.
Anderson[1] studied the sand waves in equilibrium condition
using a model of small surface waves over a sinusoidal bed which
was given by Milne-Thomson[2]. Through his analysis, the
relation between wave number and the Froude number was expressed
in stability diagram. Kennedy[3], based on the kinematics of
fluid and sediments motion, used the potential flow theory for
two dimentional moving wavy bed to express the wave length
depending on the assumption that the form of bed is taken as a
sinusoidal wave. He assumed that the dominant wave length is
that its growth rate of amplitude is the greatest. His
representation is good for the profile of antidunes but it is
much poorer for ripples and dunes which are generally asymmetric
because of the exsistance of separation zone in downstream of
there crests. He also ignored the dynamics of the particle
entrainment and movement. Yalin[4] treated the problem by using
the dimensional analysis and concluded that length of sand
waves, which formed on a cohesionless movable bed under the
condition of free surface tranquil flow, is either a linear
function of flow depth or a linear function of grain size. In
this analysis he ignored the effect of some factors, such as the
bed load, the suspended sediment and the flow condition by means
of the Froude number, also data from rivers and experimental
flumes showed that his expressions are not generally the case.
Mikhalilova[5] assumed that the length of initiated sand waves
over a flat bed is controlled by the flow depth and velocity,
but the length of fully developed dunes varies within a wide
range, and the bed slope is one of the most characteristic
effective parameters. Tsuchiya and Ishizaki[6] introduced the
interaction between surface waves and sand waves to illustrate
the mechanics of dune formation. Through their derivation, the
relation between dunes wave number and the Froude number is
expressed. Also they experimentally observed that the surface
wave have the half wave length of dune. Hayashi[7] developed
the potential flow analysis and studied the regions of occurence
of sand waves as stability problem. In his study, the relation
between wave number and the Froude number has been introduced in
a simple applicable form as a boundary stability of bed forms.
Nakagawa and Tsujimoto[8] studied the instability of sand bed
particularly in relation to load motion which described by
Eulerian stochastic model. Furthermore, they modified the
potential flow model by considering the effects of flow
convergence and divergence. On the stability diagram the
relation between wave number and the Froude number is derived.
Fredso[9] studied the influence of suspended sediment on dune
length and produced an expression for calculating that length,
but without proving the validity of this expression. Watanabe
and Hirano[10] conducted a theoritical study on sand wave
forming process by using linear stability analysis. The theory
can be applied to determine the functional form of length of
fully developed sand waves in the lower flow regime, though it
is based on assumption of infinitesimal wave.
The purpose of this study is to explain the geometrical
shape of sand waves from a more theoritical point of view
according to the study by Watanabe and Hirano[10].
Characteristic dimensionless wave number equation is derived
from the real part of the dimensionless complex propagation
velocity, which represents the logarithmic growth rate of an
initial disturbance. The functional relation is numerically
solved, and then the effect of various parameters on the wave
number is clarified. As a result, a simple form of wave number
equation is obtained. Finally a comparison between theoritical
results and data collected from rivers and experimental flumes
are performed.
2. LENGTH OF SAND WAVES
2.1 LINEAR STABILITY ANALYSIS
In the theory, according to the study by Watanabe and
Hirano[10], small scale sand waves are treated as two
dimensional bed configurations in the sence that their formation
is independent of the channel width.
First, continuity and momentum equations for one
dimensional flow are used to describe the flow properties as
follws:
Continuity equation for flow
a'h/a't + a'(u(sub m)h/a'x = 0. . . . . . . . . . . . . . . . . . (1)
where t is the time, x is the coordinate in downstream direction
on the undisturbed bed, h is the local water depth, u(sub m) is
the local mean velocity over the cross section in x-direction.
Momentum equation for flow
a'u(sub m) + u(sub m)(a'u(sub m)/a'X) =
g sinT' - g(a'/a'x)(L'h + z)cosT' - t'(sub b)/p'h . . . . . . . . (2)
where g is the acceleration gravity, T' is the slope angle of
undisturbed bed, L' is the Jaeger coefficient, z is the
elevation of bed taken from the average bed surface, p' is the
mass density of water and t'(sub b) is the bed shear stress.
Second, a continuity equation for sediments and an equation
for bed load discharge in non-equilibrium state are used to
describe the sedimentation phenomena, respectively.
Continuity equation for sediments:
(a'z/at) + [1/1 - e'](1/l)(q(sub B(sub e'))
- q(sub B) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . (3)
where c is the sediment porosity, l is the average step length of
a sediment particle, q(sub B(sub e')) and q(sub B) are transport
rates of bed load per unit time and unit width of bed in
equilibrium and non-equilibrium state, respectively.
Equation for bed load discharge in non-equilibrium state:
a'(C(sub B)a(sub *))/a't + a'q(sub B)/a'X =
[1/l][q(sub B(sub e')) - q(sub B)] +
C(sub B)w(sub o){f'(s')-F(s')*
[1- (C(sub B){f'(s') - F(s')}/
C(sub B){f'(s') - F(s')}]. . . . . . . . . . . . . . . . . . . (4)
++++NOTE: SOMETHING IS MISSING, I THINK, IN LAST EXPRESSION AS
NUMERATOR AND DENOMINATOR ARE THE SAME (PW)++++
where C(sub B)is the bed load concentration in the bed layer,
a(sub *0 is the thickness of bed layer, w(sub o) is the fall
velocity of sediment particle and f'(s'), F(s') are functions of
s' = (w(sub o)/O.93u(sub *)), in which u(sub *) is the shear
velocity.
The formation of sand waves is a result of local erosion and
deposition produced by the irregularity of sediment transport in
the flow direction. The flow over wavy beds accelerates and
decelerates, espesially in shallow water depths, and a separation
zone is formed in the downstream face of bed forms. Considering
these factors, the local bed shear stress is given by
t'(sub b)/p' =
[1/f'^2]((1 - DEL(sub o)/3)/(1 - DEL(sub o)))^2*u(sub m)^2*
[(1-DEL)/1-(DEL/3)]^2 . . . . . . . . . . . . . . . . . . . . . . . (5)
where
DEL = DEL(sub o) + A'(a'h/a'X) ,
DEL(sub o) = 6/(f'(sub o) + 2) . . . . . . . . . . . . . . . . . . (6)
where DEL is the velocity defect at the bed, DEL(sub o), f'(sub o)
are values of DEL and f'(sub o) = u(sub m)/u(sub *) for uniform
flow, respectively, and a is the empirical parameter denoting
asymmetric distribution of bed shear stress.
By using the following dimensionless quantities:
z' = z/h(sub o), X = x/h(sub o),
T = t*q(sub o)/{1 - e')(h(sub o)^2} . . . . . . . . . . . . . . . . (7)
a sinusoidal bed is introduced in the form
z' = ze^(g'T + iB'X). . . . . . . . . . . . . . . . . . . . . . . . (8)
where z' is the normalized amplitude of disturbance, g' is the
dimensionless complex propagation velocity, and B' is the
dimensionless wave number defined by (2PI*h(sub o)/L), where L is
the wave length. Subscripts 'o'indicates the undisturbed flow
quantity.
By using an infinitesimal wave theory and assuming a quasi-
uniform flow, g' can be derived from above equations as
g' =
[B'^2*M*(1 - L'B'^2F^2){B'^2(E(sub *)qA' + 2/B'^2) +
iB'(qA' - 2E(sub *)}]/
[{B'^2E^2 + (1 + EW)^2}[-3S(sub o) +
iB'{1 - S(sub o)qA' - (1 + L'(sub 2)B'^2)F^2}]]. . . . . . . . . (9)
in which
F = U/SQRT(gh(sub o),
E = [L'(sub d)d/h(sub o)](1 + A(sub o)f'(sub BO),
E(sub *) = E + (1/B'^2)(1 + EW)W,
W = {[k(sub o)w(sub o)]/
x'(sub o)u(sub *o)}{f'(s') - F(s')} . . . . . . . . . . . . . . . .(10)
M = m/(1 - S'(sub c)/S'(sub o),
q = (4/3)/(1 - DEL(sub o)(1 - DEL(sub o)/3),
k(sub o) = 1/k(1 - u(sub *c)/u(sub *c)) . . . . . . . . . . . . . .(11)
where F is the Froude number, U is the mean flow velocity, P' =
q(sub Bo)/SQRT(sgd^3), s is the specific weight of sediment in
water, X'(sub c) = u(sub *o)^2/sgd is the dimensionless tractive
force, X'(sub o) = u(sub *o)^2/sgd is the dimensionless critical
tractive force, in which u(sub *o) is the critical shear velocity,
S(sub o) is the longitudinal slope for undisturbed bed, a = 5,
According to Tsubaki and Saito [11] L(sub d0 = 100, A(sub *) = 10,
k = 8.5, 0.05 and x' = 3/2. The parameter (E) represents the non-
equilibrium state of bed load transport process. The parameter
(W), which increases rapidly with the decrease of wo/u*, denotes
the effect of suspended sediment. The parameters L'(sub 1), and
L'(sub 2) represent the centrifugal effects of curvilinearity of
bed surface and water surface, respectively. These were
introduced by Iwasa and Kennedy[12] as sub-coefficients in the
equation of pressure correction coefficient L', which known as the
Jaeger coefficient. The value of L'(sub 1) was given in the range
of (0.4 - 0.55), and the value of L'(sub 2) was given in the range
of (0.2 - 0.4).
2.2 LENGTH OF DUNES AND RIPPLES
2.2.1 DOMINANT WAVE LENGTH CONCEPT
Flow-generated bed configurations in the lower flow regime,
such as dunes and ripples, have a characteristic wave length which
depends on the properties of flow, fluid and bed material. From
stability analysis by using perturbation technique, the bed wave
length could be obtained successfully from linear terms of the
expanded equations. Based on Yalin's study[4] the sand wave
length can be measured as soon as the sand wave begins to form and
does not change noticeably with the time. Accordingly, the linear
stability analysis may be used to obtain the length for fully
developed sand waves even the theory is developed for the
initiation of sand waves. The real part of Eq. 9 is represnted by
g'(sub 1) which is derived in the form
g' =
[B'^2*M*(1 - L'B'^2F^2)[{1-(1 + (qA'/f'(sub o)^2)*F^2}
{qA' - 2E(sub *)} - 3(F^2/f'(sub o)^2)(E(sub *)qA' +
(2/B'^2)]] /
{B'^2E^2 + (1 + EW)^2}[{1 - (qA'/f'(sub o)^2 +
L'^2B'^2)F^2}^2 + (9F^4/B'^2F(sub o)^4] . . . . . . . . . . . . . . (12
The parametery,represents the logarithmic growth rate of sand
waves and its sign identificates the stable and unstable condition
of bed. The dominant wave number B'(sub d) corresponds to the
maximum growth rate of bed disturbance. Therefore, B'(sub d) is
obtained by differentiating g'(sub 1) with respect to B', and then
the first derivative is equated with zero.
a'g'(sub 1)/a'B' = 0 . . . . . . . . . . . . . . . . . . . . .(13)
The general solution for B'(sub d) from Eq. 13 is given in
the form
SUM {6, n = 0} A(sub n)B'(sub d)^2n. . . . . . . . . . . . . .(14)
in which, the coefficients from A(sub 0) to A(sub 6) contain the
parameters F, f'(sub o), E, W and q. Also they contain the
constants A', L'(sub 1), and L'(sub 2). According to Tsubaki and
Saito[11], the parameter A' is taken as 5. According to Iwasa and
Kennedy[12], the values of parameters L'(sub 1) and L'(sub 2) may
be taken as constants and equal to 0.5 and 0.4, respectively by
using the formula of Robillard and Kennedy[13].
The solution of this equation gives more than one value for
wave number, B'(sub d). To clarify which value is the dominant,
the relation between wave number B' and the growth rate of
amplitude of sand waves is shown in Fig. 1 by using Eq. 12 for
Froude number F = 0.3, velocity coefficient f'(sub o) = 16,
dimensionless tractive force S'(sub o) = 0.1 and suspended
sediment coefficient w(sub o)/u(sub *) = 0.1. In the figure, when
the sign of g' is negative, the bed is stable and sand waves do
not occur. When it is positive, the bed is unstable and the sand
waves are formed. Also it is quite clear that the wave number B'
= 5.5 is the dominant wave number B'(sub d) which gives the
maximum growth rate of amplitude for sand waves g'(sub m).
From the previous discussion, the value of B'(subd) which
gives the maximum growth rate of amplitude for bed forms g'(sub m)
should achieve the following two conditions:
a'^2/a'B'^2 < 0. . . . . . . . . . . . . . . . . . . . . . . .(15)
and
g'(sub m) is the maximum of the extreme values.
+++++
END Part I
See Next message for completion of Length of Sand Waves,
Comparison with experimental results, Conclusions, List of Figure
Captions, and References.
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END. Msg. B.ENG
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