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SCIENCE CHINA

Technological Sciences

RESEARCH PAPER March 2012 Vol.55 No.3: 753 771

doi: 10.1007/s11431-011-4720-6

The nonlinear theory for sediment ripple dynamic process of

straight river

XU HaiJue1,2 & BAI YuChuan1,2*

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China;

Institute for Sedimentation on River and Coastal Engineering, Tianjin University, Tianjin 300072, China

Received December 29, 2010; accepted June 29, 2011; published online January 20, 2012

There are various sand ripples in the natural world. The viewpoint of Yalin is that local disturbances result in laminar instabil-

ity and in sand-ripple formation, namely, local disturbance the instability of the laminar flow the formation of sand ripples.

Based on this viewpoint, a theoretical model of the resonant triad interaction and its nonlinear interaction with the sediment is

established. The purpose of this model is to explain the formation and evolution of the sand-ripple and allow for analysis of the

instability of open-channel flow caused by it and sand-ripple hydro-dynamic process. This model will not only pave a road to

explore the mechanism of interaction between bed-form and turbulence, but also provide a good base for the study of aeolian

sand-ripple formation.

straight river, bed-forms, symmetric sand-ripples, perturbation method, nonlinearity

Xu H J, Bai Y C. The nonlinear theory for sediment ripple dynamic process of straight river. Sci China Tech Sci, 2012, 55: 753 771,

Citation:

doi: 10.1007/s11431-011-4720-6

1 Introduction and by the sands granular size [8]. The flow characteristics

can affect the bed-form, while the bed-form, in turn, can

also affect the flow characteristics [9, 10]. On the one hand,

In sandy river-beds, for example, bed-forms will develop

the complex transport of bed-load and suspended load cre-

when the shear stress exerted by the flow on the surface of

ate various ripple profiles [11]. On the other hand, the

the bed exceeds the value critical to initiating sediment

transport of sediment is also controlled by its flow charac-

transport. Different bed-forms, such as ripples, dunes and

teristics. Thus, a feedback mechanism is formed between

large sand-waves, as shown in Figure 1 [1], are created by

the bed-form development, sediment transport and hydro-

different flow intensities and by different sizes of sediment.

dynamic process [12].

Of these various bed-forms, the ripple is a relatively simple

As the ripple height is very small relative to the overall

type, which is of low height. All types of ripples including

depth of water, its effect on the flow process is minor. In

two-dimensional and three-dimensional ripples [2, 3] can be

general, the vertical profile of an individual ripple is not

seen in many natural areas, such as in a river [4], on a beach

symmetrical the surface facing the direction of flow is

[5] and in a desert [6].

long and smooth, while the surface facing opposite to the

Under certain conditions, the sediment composition and

direction of flow is short and steep. The ratio of one to the

flow characteristics will affect the ripple profile [7]. Hence

other is approximately between 1:2 and 1:4. The sand-ripple

the ripple formation can be determined by flow intensity

height is between 0.5 and 2 cm, while the wave-length is

between 1 and 15 cm [13]. Besides, the natural bed-forms

including sand ripples also exhibit cross-section variations

*Corresponding author (email: ******@***.***.**)

Science China Press and Springer-Verlag Berlin Heidelberg 2012 tech.scichina.com www.springerlink.com

754 Xu H J, et al. Sci China Tech Sci March (2012) Vol.55 No.3

Figure 1 Different periods of sand wave developing. (a) Plane; (b) ripple;

rapid beach or pond.

Figure 2 Order of appearance for sand wave along with flow growth. (a)

Ripple; (b) dune.

and appear to be three-dimensional configuration, such as

dunes with sinuous crests [14], alternate sandbars [15] and

so on.

In the study of Aeolian ripples, Anderson [16] investi-

gated the formation and evolution of aeolian ripples by us-

ing a cellular automaton model of a sandbed. And the phys-

ical process responsible for both the spatial sorting and the

stratigraphic signature of the ripples was identified. Zheng

et al. [17] and Bo el al. [18] simulated the time evolution of

the aeolian sand ripple by using the discrete particle tracing

method. And three more factors were considered in their

papers, including the different sizes of the sand particles,

the particle-bed collision and the ejection or the rebound of

the particles in the sand bed and the saltation and the creep

of the particles. Prigozhin [19], Valance, Rioual [20] and Figure 3 Smoke lines shown in the turbulence [28].

Csah k et al. [21] successively established the continuous

model of ripple formation to study the initial instability of theoretic model of the interaction between the coherent

flat sand surface and the nonlinear dynamics of wind ripples. wave and the sediment particles on the sandy river bed and

All the researches can give the reference to the study of the

the model of formation of the interfacial wave. And aiming

interaction between the flow and the sediment.

at the developing of the bedforms and the feedback of the

The majority of recent investigations into the dynamic

hydraulic process, Bai and Xu [30, 31] studied the instabil-

sediment ripple process have focused on laboratory work.

ity characteristics of the laminar flow on the regular and the

These include detailed assessments of flow-fields above a

irregular sandy rippled bed, respectively. Besides, Bai et al.

particular type of bed-form, either on dunes or ripples. Sig-

[32] realized that it was not enough to merely consider the

nificant progress has been made in the mechanism of to-

river bed and the flow structure, so they discussed the

pographic forcing in terms of the flow characteristics, sepa-

nonlinear dynamic characteristics of the bed load under the

rated-flow dynamics, internal boundary layer development

action of the flow as well. And the catastrophe line that dis-

and associated turbulence structure [22 24]. Based on the

tinguishes the different kinds of bedforms and the explana-

knowledge on the fluid mechanics and the experimental

tion to the data given by Chatou Library is got.

study, Yalin [25] pointed out that the evolution of sand-

This article is based on previous works and establishes

ripples is caused by local disturbance from unstable laminar

the the nonlinear theory of sonant triad interaction with

flow. But at that time, Yalin did not establish a theoretical

bed sediment, using perturbation method (Stuart [33],

model to develop this concept further. Bai [26], Bai et al.

Zhou, Fujimura [34] and Wu [35]) to analyze the dynamic

[27] compared the sand-ripple phenomena with turbulent

process of sand ripple, establishes a theoretical model to

smoke patterns and discovered that there appears to be a

explain the asymmetric ripple formation and development.

similar mechanism underlying their development (see Fig-

The nonlinear theoretical model of the disturbance wave

ures 2 and 3 [28]) and established a linear model of the

instability in a resonant three-wave model will be intro-

formation of the sand ripples under the coherent disturbance

in the open channel. Later, Bai and Luo [29] established a duced in the second part; the third and fourth parts will be

755

Xu H J, et al. Sci China Tech Sci March (2012) Vol.55 No.3

the discussion and analysis of the theoretical results, and 2.1.2 Bed deformation equation

some important conclusions. qbx qbz f

(5),

x z t

2 Theory model

where qb is the bedload transport rate, is the void ratio

of the sediment and f is the depth of the sand ripples.

2.1 Basic equations

2.1.3 Bedload transport formula

2.1.1 Flow equation

qb mU n, (6)

Aiming at the open channel with sloping bed, we select the

coordinate system shown in Figure 4. L, Um, L/Um and

where U is the surface velocity of the bed layer, m and n are

0U m are adopted to non-dimensionalize the length (x, y),

coefficients. According to the results given by Kennedy [36]

the velocity (u, v), the time t and the pressure p. Then the and Taizo [37], n=4 and m is a experimental coefficient.

non-dimensionalized N-S equations under the coordinate

system are obtained: 2.2 Hydrodynamics instability analyses

u u uv uw

2.2.1 Perturbation analysis

t x y z Divide the velocity and the pressure into two parts, which

p 1 2 u 2 u 2 u sin are the averaged quantities and the disturbance quantities.

(1),

x R x 2 y 2 z 2 Fr 2 u u u, v v v,

w w w, p p p,

(7)

v uv v 2 vw

t x y z

where u, v, w and p are the averaged quantities, and u,

v, w and p are disturbance quantities, respectively.

p 1 v v v

2 2 2

(2),

y R x 2 y 2 z 2 The fluctuation in the velocity results in the variation in

the bedload transport rate and the bed deformation, i.e.,

w uw vw w2

qb qb qb, f f f .

(8)

t x y z

p 1 2 w 2 w 2 w cos 2.2.2 Solution of constant uniform open channel flow, (3)

z R x 2 y 2 z 2 Fr 2 For the laminar flow in the open channel, when the flow is

uniform, the non-dimensionalized equations can be simpli-

u v w fied as

0, (4)

x y z sin

1 2

2 u 0, (9)

R y Fr 2

Um L Um

R Fr

where is Reynolds number, is

gL p cos

0. (10)

Froude number. Select depth of water as the characteristic y Fr 2

length and select surface velocity of the open channel

By integrating the above formula and taking Reynolds

U m gh 2 sin 2 as the characteristic velocity.

number, Froude number and the maximum velocity of the

laminar flow of the open channel into consideration, the

expression of the velocity distribution is obtained.

u 2 y y2. (11)

Substituting the above results into the averaged bedload

transport rate formulae and the bed deformation equation,

we have

qb const, (12)

and f is not related to the time.

2.2.3 Disturbance equation

Substituting the divided velocity into the original equation

Figure 4 The coordinate system.

756 Xu H J, et al. Sci China Tech Sci March (2012) Vol.55 No.3

D 2 b2 2 iR b u b v0b y

and subtracting the equation that the averaged quantities

satisfy, we can obtain the disturbance equations:

dp y

Flow equation: R 0b 0, (21)

u v w

0, (13)

D 2 b2 2 iR b u b w0b y

x y z

u u

u du p u i R p0b y 0,

(22)

Lu v u v w,

(14)

t dy x x y z

dv0b

i b u0b w0b 0, (23)

v v

v p v

Lv u v w,

(15)

t y x y z

where D d/dy. After simplification, the three dimen-

w w

w p w

Lw u v w

(16), sional Orr Sommerfeld equation becomes

t z x y z

2 iR ( b u b )

where L u is an operator. 2 2

x R b

Bed deformation equation:

D 2 b2 2 b D 2 u ( y ) 0. (24)

qbx qbz f

(17)

x z t 2.3.2 The amplitude equation for the sediment transport

rate and the bed deformation

Bedload transport rate equation:

Substitute eq. (19) into eq. (18); then

qb 4mU 3u .

(18)

qb 0b 4mU 3 u0b . (25)

Considering eqs. (17) and (19), we have

2.3 Solving process for linear analysis

b u0b v0b

After the higher-order disturbance terms are omitted, the (1 )i b f 0b . (26)

iqb 0b

equation can be analyzed linearly. Construct the solution of u0b 2 v0b 2 w0b 2

the equation in the following form:

From eqs. (25) and (26), we have

u0 a y u0b y

p 4m U 3 u0b b u0b v0b

p y i a x a t p y

f 0b (27)

A 0a B 0b

qb 0 a y qb 0b y (1 ) b

qb u0b 2 v0b 2 w0b 2

f f y

f 0b y

0a Therefore, under the coherent disturbance, the linear ex-

e c.c,

i b x b z b t

pressions of the sand ripples can be written as

(19)

4m U 3 u0 a b u0 a

where the quantities taking 0a as their subscripts are the

f x, z, t Ae a a

i x t

two-dimensional quantities, the quantities taking 0b as their (1 ) a u0 a w0 a

2 2

subscripts are the three-dimensional quantities, and c.c

4mU 3 u0b b u0b v0b

represents the conjugated complex function.

(1 ) b u0b 2 v0b 2 w0b 2

2.3.1 The equation for amplitude configuration

B ei b x b z b t ei b x b z b t c.c. (28)

Substitute eq. (19) into eqs. (13) (16), and omit the nonlin-

ear terms on the rightside. Take the three-dimensional wave

e b b b as an example to explain.

i x z t

2.4 Solving process for nonlinear analysis

If the basic flow of the open channel is a parallel flow,

In order to study the nonlinear interactions, the amplitude

i.e., u (u ( y ), 0, 0), then the amplitude equations of the

evolution and the bed deformation between the disturbance

disturbance quantities are

waves, nonlinear analysis is taken and the spatial model is

D 2 b2 2 iR b u b u0b y adopted as follows:

a ar i ai, b br i bi,

Rv0b y i b R p0b y 0, (20) a Ae ai x, b Be bi x, a ar x a t,

757

Xu H J, et al. Sci China Tech Sci March (2012) Vol.55 No.3

b1 br x b z b t, b 2 br x b z b t. (29)

2 2

vi i wi i, (36)

Then eq. (19) can be written as i 0 i 0

u0 a y

qbix qbiz i f i i

p 0 a y i (37)

x z i 0 t

q a q

e a

b0a y

i 0

when i=0, the above equations become the linear distur-

f f y

0 0a bance condition. The latest modified method of the weakly

u0b y

non-linear theory of the flow stability [35 37] is adopted in

p y i b 1 this paper.

b 0 b i b 2 c.c. (30)

qb 0b y e e Analysis by using the spatial model: For the linear sec-

tion, i.e. 0 term, each disturbance velocity and pressure are

f y

0b still assumed to be U 0 ( y, a, ), then U0, p0 imply x through

a, and U0, p0 are the implicit function of x; for the nonlin-

2.4.1 Perturbation expansion

ear section, i.e. n, n 1 terms, Un, pn are the function of

The disturbance velocity, pressure, the bedload transport

(y, x, ) and is also the implicit function of x. The two

rate and the bed deformation can be expanded into

sections are still the exponential function of t. Then, the

u ui i, v vi i, w wi i,

expressions of / x, / x 2 are different in the linear term

i 0 i 0 i 0

and in the non-linear terms.

p pi i, qb qbi i, f f i i .

For the linear term, n, n=0

(31)

i 0 i 0 i 0

ai a Ai (a, b) i A,

Substituting eq. (31) into disturbance eqs. (13) (16) leads

to the equations that each order of velocity satisfies: i 1

u v w ai b Bi (a, b) i B,

xi yi zi i 0, (32) dx i 1

i 0

d a

ar Ci (a, b) i C,

u dU p

ti Lui vi dy xi i dx i 1

i 0

d b

u i

u u

br Di (a, b) i D,

uk i k vk i k wk i k, (33)

x y z

i 0 k 0 i 1

vi pi i

A B C D (38),

Lvi a b

x a b

i 0

v v

2 2 2 2 2

uk i k vk i k wk i k i, (34) A2 2 B 2 2 C 2 2 D 2 2

x y z a b

x a b

i 0 k 0

2 2 2

w p

2 AB 2 AC 2 AD

t i Lwi zi i a a a b

a b

i 0

A A B B

wi k i

wi k wi k

A B A B

uk,

vk wk (35) a b a a b b

x y z

i 0 k 0

C C D D (39)

A B A B .

1 2 2

2 b a a b b

where L u ( y ) 2 2 2 is an operator.

x R x y z

For the nonlinear terms, n, n 1,

Substituting eq. (31) into the disturbance equations yields

the equations that each bed deformation and sediment d a

ar Ci (a, b) i C,

transport rate satisfy:

dx i 1

qbi i m u ui i d b

br Di a, b i D,

i 0 i 0

dx i 0

758 Xu H J, et al. Sci China Tech Sci March (2012) Vol.55 No.3

d a d b

0 T0, n Tn, n 0,

C D (46),

x x d x a dx b dx x a b x

1 2 1 2

2 2 C D

L0 u 0

S0 2 2

R x R y z

x a x b R

x x

2 2

Ln u n S n, n 1.

2C 2D (47)

x a x b

Then the equations that each order of disturbance quanti-

2 2 2

2CD C 2 2 D2 2 . (40) ties satisfies can be obtained and shown in the following

a b a b

part.

Substitute eqs. (38) (40) into eqs. (32) (35), and introduce

2.4.2 Equations that each order of disturbance quantities

the following operators:

satisfies

n An Bn Cn Dn (41),

2.4.2.1 First-order term ( 0)

a b

a) The coherent part of the flow perturbation:

2 2

n n

S n Ai An i Bi Bn i 2

u0 du

a 2 i 0 b Lu0 v0 0 p0 0, (48)

i 0

t dy

2 2

Ci Cn i Di Dn i

a b 2

v0 p

i 0 i 0

Lv0 0 0, (49)

2 2

n n

t y

2 Ai Bn i 2 Ai Cn i

a a

a b

i 0 i 0

w0 p

Lw0 0 0, (50)

2 2

n n

2 Ai Dn i 2 Bi Cn i t z

a b b a

i 0 i 0

v0 w0

2 2

n n

0 u0 0.

2 Bi Dn i 2 Ci Dn i (51)

y z

b b a b

i 0 i 0

An i B

n n

b) The corresponding sediment transport rate and bed

Ai Ai n i

deformation:

a a i 0 a b

i 0

Cn i Dn i

n n

qb 0 4mu 3u0,

Ai (52)

a a i 0 a b

i 0

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