A type of dynamic mechanism of river hydraulic geometry

SCIENCE CHINA Technological Sciences

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*Corresponding author (email: [email protected])

• Article • April 2014 Vol.57 No.4: 847–855

doi: 10.1007/s11431-014-5492-6

A type of dynamic mechanism of river hydraulic geometry

BAI YuChuan1*, JI ZiQing1 & ZHANG MingJin1,2

1 State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China; 2 Tianjin Research Institute for Water Transport Engineering, Key Laboratory of Engineering Sediment

of Ministry of Transport, Tianjin 300456, China

Received September 6, 2013; accepted December 20, 2013

Large-scale structure of river flow is the main driving force for bed erosion-deposition and bank deformation. The structure shapes and retains a corresponding hydraulic geometry form. Therefore, the most stable flow structure is the probable natural river plane formation. Natural coordinate transformation and perturbation methods were adapted to deform the governing equations of sine-generated river basic flow and disturbance flow independently. The stability and retention of perturbation waves were analyzed in our model to explain why meandering rivers followed a certain type of flow path. Computation results showed that all types of perturbation waves in meandering rivers were most stable when the meandering wave number was about 0.39–0.41. We believe that this type of stable flow structure shaped a certain meandering river. The statistical average length-width ratios of Yalin, Habib and da Silva and Leopold and Wolman somewhat confirmed our most stable river mean-dering wave number. In some ways, meandering rivers always tend to diminish internal turbulence intensity.

meandering river, stability, coherent disturbance, river optimization

Citation: Bai Y C, Ji Z Q, Zhang M J. A type of dynamic mechanism of river hydraulic geometry. Sci China Tech Sci, 2014, 57: 847855, doi: 10.1007/s11431-014-5492-6

There is rarely a perfectly straight river channel in nature. Once disturbed, point bars will occur alternately at both bank sides and will develop a certain river hydraulic geom-etry, for non-uniform sediment transport on a lateral bed surface. The flow path tends to meander at lateral plane level and form either a meandering river with just alternate bars or a braided river with several bars in a multi-seriate bar section (Figure 1).

Yalin et al. and Julien [1–3] discerned a statistical hy-draulic geometry relationship between bar length and flow width. It was shown that the average length-width ratio of alternate bars along the flow path was about 6, i.e., L=6Ba . Habib and Da [4] studied the hydraulic geometry relation-ship of multi-seriate bar rivers. It was shown that average data at a cross section resulted in the same number, as

Figure 1 (Color online) Flow path and sand bars. (a) Alternate bars of Huajialing segment of Ganjiang River; (b) multi-seriate bars of Shenbeitan segment of Yellow River.

L=6Ba/n, if any stream reach among neighbor bars was con-sidered a single meandering flow. Leopold and Wolman [5] focused on the meandering flow path itself rather than sand bars. Their research uncovered the length-width relationship

(a) (b)

848 Bai Y C, et al. Sci China Tech Sci April (2014) Vol.57 No.4

of meandering rivers as L= aBm, for m≈1, a = 7−11. Above all, it seems than river streams always follow a flow path with a certain average wavelength.

Yalin and Da [2] regarded the occurrence of large-scale flow construction as a reason for far-ranging morphological bed forms and plan deformations [1]. However, he did not provide strict theory support. According to modern turbu-lence theory, coherent flow structure almost always comes from disturbance near boundaries. River hydraulic geometry is related to the interaction of coherent structure and bound-ary formation. Therefore, we believe that the most probable plane form is that which effectively stabilizes the internal coherent structure. Based on this concept, we address how river meandering alters the development of internal coher-ent structure.

1 Non-dimensional governing equations

A natural orthogonal coordinate system (s, n) was used to follow river meander. The flow path centerline y0=y0(x0) was taken as the s coordinate. A sine-generated curve was adopted as the centerline, since Langbein and Leopold [6] regarded it as the most probable flow path. This curve can be expressed as (s) = 0sin(2πs/M), where is the angle between the centerline and x coordinate axis, and 0 is the amplitude of . M is curve length of one wave number, shown in Figure 2. In addition, L is the down-valley length, B is channel width, and Ba is meander width.

Taking half-channel width b, zero-order maximum of basic longitudinal velocity Um, and b/Um as reference scales of length, velocity, and time, respectively:

2( , ) ( , ); ( , ) ( , ); / ; , s n m s n m ms n b s n u u U u u t tb U p U p

(1)

here, superscript symbol ~ signifies non-dimensional oppo-site physical quantities, us and un are respectively velocity in longitudinal and transverse directions, t is time, and p is pressure. Curvature of centerline is measured by its maximum 02 / m M , expressed as

( ) expi( ). m m f fs s t

yx

b nS

θθ0

(x0, y0)

M

L

BBa

Figure 2 Sketch map of sine-generated river segment.

The sine-generated curve has three important parameters: meandering amplitude 0, wave number f, and radian fre-quency ωf. Non-dimensional quantities are f = 2πb/M and ωf = f Cf /Um, where Cf is river meandering speed. For nat-ural rivers, Cf is nearly zero, so ωf ≈ 0.

Non-dimensional parameter ψ is defined as maximal curvature, ψ=0f. For a slightly meandering river segment, ψ is sufficiently small to be used as the perturbation param-eter.

Lamé coefficients may be expressed as 1 sh n ,

and 1nh .

Non-dimensional governing equations are as follows:

1 1 0

s n

s s n

u uh h u

s n, (2)

1 1

1 12

2 2 2

3 2

2

,

s s ss s n s s n

ss s

n ss s s

ss s n

u u uh u u h u u

t s nf b p

h Re usU

u uh h u

Re s n

uh n h u

Re s s

(3)

1 1 2

2

2 1 2 2

3 2

1

2

,

n n ns s n s s

nn

s ns s s n

ns s s

u u uh u u h u

t s nf b p

un ReU

u uh h h u

Re s n

uh n h u

Re s s

(4)

where fi (i=s,n) are gravity components along the s and n directions, fs =gJcos, fn = gJsin, J is hydraulic slope, g is gravity acceleration, and Laplace operator

2 2

2 2 2 2

1 1

s nh s h n.

For common wide-shallow rivers, the velocity gradient in the vertical direction is much stronger than in the other two directions. As suggested by Engelund and Hansen [7] and Elder [8], eddy viscosity was taken as νt = au*H, where fric-

tion velocity * u gHJ , H is water depth, and a=0.077.

The Reynolds number is defined as the ratio of coherent structure inertial force and bed friction resistance, Re= Umb/νt.

2 Perturbation analysis

According to turbulence theory [9–11], solution of the gov-erning equations is decomposed into two parts, F=Fψ +FT,

Bai Y C, et al. Sci China Tech Sci April (2014) Vol.57 No.4 849

where F is the unknown solution, Fψ is the basic solution, and FT is the disturbance solution, as follows:

T , s nu u pF T

; s nu u pF T T sT nT Tu u pF .

For a slightly bending river segment, curvature ψ is small enough to be used as a perturbation parameter. The basic solution Fψ is expanded into a first-order component by the perturbation method, as follows:

20 1 exp i( ) ( ), f fs t oF F F (5)

where, the first part of the right side is a zero-order pertur-bation solution of basic flow in the real domain, denoted Fψ0, and the second part is a first-order perturbation solution of basic flow in the complex domain, denoted Fψ1.

Compared with a straight channel, the boundary of a si-ne-generated river is a waveform with variable curvature. The meandering boundary is the driving force of fluctuating first-order basic flow perturbation component Fψ1. The dis-turbance solution FT is the fluctuating quantity of disturbed flow. Disturbance solution FT depends on the constant component Fψ0 and fluctuating component Fψ1. Therefore, the stability and evolution of the internal disturbance be-longs to the category of a Floquet-Lyapunov problem.

There are various orders of harmonic waves in a mean-dering river segment. However, high orders of harmonic waves have much less than first-order fluctuation harmonic quantity for small disturbances. Thus, the disturbance solu-tion FT contains −1, 0 and 1 orders of harmonic fluctuating component FTm, expressed as follows:

1

1

2

exp i( ) i ( )

( ).

T T Tm T T f fm

T

s t m s t

o

F F

(6)

2.1 Zero-order basic flow

The zero-order equations of basic flow are written as fol-lows:

0 0

0 0 0 020 0 0

0 0 0 00 0 0

0,

1,

1,

s n

s s ns n s

n n ss n n

u u

s nu u u p

u u Fr J ut s n s Re

u u u pu u u

t n n n Re

boundary condition: 0 0 0 s nu u , 0 0 0 s nu u .

Here, the Laplace operator is 2 2

2 2

s n, Froude num-

ber Fr2 = Um2/gH, width-depth ratio β = b/H, and Reynolds

number Re = Umb/νt.

The zero-order solution corresponds to a straight channel. Therefore, unsolved variables are constant along the river, i.e., ∂Fψ0/∂t = 0 and ∂Fψ0/∂s ̃ = 0. With boundary conditions, the zero-order perturbed velocity is solved as

1 1 0,

sT nT

s s nT

u uh h u

s n (8)

where Um is the reference scale of velocity, specifically ex-pressed as Um=0.51β2u*, and Re depends only on width- depth ratio β, as Re=0.52β3.

2.2 First-order basic flow

The zero-order equations of basic flow are written as

0 1

1 1 0 1 0 010 1

1 1 12 1 10 0 0

0,

,

i ,

s n

s s n ss n

n ns n n f s

u u

s nu u u p p u

u u n Ret s n s s n

u u uu u Re u Re u

t s n

boundary condition: 11, 0. sn u

We denote the first-order and second-order ordinary dif-ferential operators as D and D2, i.e., d/d D n and

2 2 2d /d . D n The first-order equations can be simplified as a single equation:

2 2 2 2 2 2 20 0 1

3 2 20 0 0

ˆ{( ) i [( )( )] }

i ,

f f s f f f s n

f s f s f s

D Re u D D u u

u Reu i D u

this is a fourth-order linear ordinary differential equation in the complex domain. Using a finite difference method, we can solve eq. (10). It is obvious that the first-order solution is dependent on zero-order quantity Fψ0.

2.3 Disturbance flow

The disturbance equations are as follows:

1 1 0,

sT nT

s s nT

u uh h u

s n (11)

01 10 0

1 111 1

1 2 11 1

1

2

ssT sTs s nT s s nT

s ssT sTs s sT n nT

Ts s nT n sT s sT

uu uh u u h u u

t s n

u uu uh u u u u

s s n n

ph u u u u h u

s Re

hRe

2 1 2 2

3 2 ,

nT sTs s s sT

sTs s nT

u uh h u

s n

uh n h u

Re s s

(9)

(10)

(12)

(7)


01 10 0

1 111 1

2 11

2

2

12

2

snT nTs s nT s s sT

n nnT nTs s sT n nT

Ts s sT nT

sTs

uu uh u u h u u

t s n

u uu uh u u u u

s s n n

ph u u u

n Re

uh

Re1 2 2

3 2 ,

nTs s nT

nTs s sT

uh h u

s n

uh n h u

Re s s

boundary condition: 1 n , 0 sT nTu u .

Eqs. (11)–(13) are governing equations of coherent structure. These are a second-order homogeneous linear partial differential equation set. Clearly, the disturbance solution is dependent on zero-order quantity Fψ0 and first-order quantity Fψ1, as well as Lamé coefficient hs. There are three kinds of fluctuation variables: first-order variable Fψ1, disturbance variable components FTm and La-mé coefficient hs. The inertia term of the governing equa-tion reflects dispersion interaction of all three kinds of fluc-tuation variables, and the viscosity term reflects dispersion interaction of the latter two.

The Lamé coefficient represents boundary fluctuation, written with Taylor expansion as follows:

2 2 2 311 c.c ( ),

2 sh n n o (14)

here, c.c means complex conjugate. Taylor expansion of the Lamé coefficient retains three components, as the disturb-ance solution FT contained three harmonic components.

Taking eqs. (6) and (14) into eqs. (11)–(13), we obtain the disturbance amplitude equation set, as follows:

1 1 1 1

21 2 1 2

1ˆ ˆ ˆ î2

1 ˆ ˆ 0,4

T f sTm nTm s sTm n nTm

s sTm n nTm

m u Du u u

u u

220

0

2 1 2 1 2 1

22 2 2 2 2 2

î

ˆ î

1 ˆ ˆ ˆ21 ˆ ˆ ˆ 0,4

T f T f T f s sTm

s nTm T f Tm

s sTm sd sTm n nTm

s sTm sd sTm n nTm

D m Re m m u u

Re Du u Re m p

u Du u

u Du u

220

0

3 1 3 1 3 1

23 2 3 2 3 2

î

ˆ ˆ

1 ˆ ˆ ˆ21 ˆ ˆ ˆ 0,4

T f T f T f s nTm

s nTm Tm

s sTm n nTm nd nTm

s sTm n nTm nd nTm

D m Re m m u u

Re Du u Re Dp

u u Du

u u Du

where m is the m-order harmonic variable, m=0, ±1. T , ωT are respectively wave number and radian frequency of the disturbance wave, and f , ωf are respectively wave number and radian frequency of river meandering. Other coefficients are explained in an appendix.

Eqs. (15)–(17) compose a homogeneous linear equation

set with 9 unknown variables: 0 0 0ˆ ˆ ˆ, , , sT nT Tu u p 1 1

ˆ ˆ, , sT nTu u

1ˆ

Tp , 1 1 1ˆ ˆ ˆ, , sT nT Tu u p . All these variables change at the

cross section. The equation set determines the stability and growth rate of large-scale coherent structure, written simply as

2 0, D DAX B X C X (18)

where X is a vector composed of unknown quantities, DX and D2X are respectively first-order and second-order dif-ferential quantities of vector X at the cross section; A, B and C are 9×9 matrices whose elements are represented by (T , ωT), (0, f , ωf), Re, usψ0 and (usψ1, unψ1), as explained in the appendix. Since the disturbance amplitude equation is ho-mogeneous, vector X is the fundamental set of solutions, written as

TT 1 T0 T 1 T 1 0 T 1 T 1 T0 T 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ[ ] , s s s n nT nu u u u u u p p pX

The fundamental set X solves via the difference method. The stability of disturbance wave is solved with Muller’s method and temporal mode, i.e., wave number T is real and eigenvalue ωT is complex (ωT =ωTr+iωTi). The imaginary part of ωTi reflects growth rate of the disturbance wave. When ωTi > 0, that wave is unstable and grows with time. When ωTi < 0, the wave is stable and damped, and ωTi = 0 indicates the critical condition.

Since matrices A, B and C are dependent on T , ωT, 0, f , ωf and Re, eigenvalue ωT can be expressed as ωT =ωT (0, f , ωf, T , Re). Thus, the increasing or decreasing character of disturbance waves in a bending channel are affected by river meandering parameters (0, f and ωf).

3 Stability and growth of coherent disturbance

When ψ = 0, the coefficient matrix of eq. (18) is singular, so we took the case of 0=f =10−6 as the straight channel con-dition to ensure correctness of our computation results. Those results were compared with Reynolds and Potter's computations and Nishioka and Ichikawa's experimental data [12,13], as shown in Figure 3. The method used in this paper served this purpose well.

3.1 Basic flow

The velocity vector of basic flow ũψ is composed of two parts: zero-order velocity vector ũψ0 and first-order velocity vector ũψ1, i.e., ũψ=ũψ0 +ψũψ1 (Figure 4). The zero-order

(13)

(15)

(16)

(17)


Figure 3 Verification in case of straight channel.

Figure 4 Velocity field of basic flow. (a) Zero-order velocity field:ũψ0; (b) first-order velocity field: ũψ1; (c) velocity field of basic flow: ũψ0 +ψũψ1.

velocity was symmetric, and the first-order velocity nearly asymmetric.

The zero-order solution and first-order solution of basic flow composes a linear model of basic flow. We define the tensor multiplication of first-order velocity vector (ũsψ1, ũnψ1) as a kind of inertia force, denoted by T and expressed as

2 T1 1 , T u u

here, ũψ1 is the first-order velocity vector and the superscript T means vector transposition. The form of tensor T is one part of the inertia force, just like Reynolds stress or radia-tion stress, and is called the additional inertia force tensor. It is the most important part of the second-order basic flow equation with scale of the meandering length, and is a two-dimensional tensor with three elements, Tss, Tsn and Tnn. Take Tsn as an example:

2i *1 1 1 1

1 1ˆ ˆ ˆ ˆe . .4 4

f fs t

sn s n s nu u c c u u c cT ,

where the superscript symbol * means conjugation. The additional inertia force tensor T is nonlinear and composed of three types of harmonic components: 2f , −2f and 0f. The former two components transmit energy from larg-er-scale to smaller-scale vortices. The last component is constant along the river and changes the zero-order velocity distribution at the cross section as nonlinear interaction, just like Reynolds stress. Figure 5 shows the distribution of ve-locity and additional inertia force component Tsn. Tsn has four couples of extremum regions along the flow path. These regions were slightly later than kπ/2 phase positions near the channel banks.

3.2 Growth rate of disturbance wave

For a certain flow magnitude, growth rate of the disturbance wave was affected by meandering parameters (0, f and ωf). Figure 6 shows the growth rate of disturbance wave ωTi

varying with meandering wave number f , with parameters set to 0=0.1; ωf =0; T =1; Re=6000−12000. Figure 7 illus-trates the growth rate spectrum (ωTi >0) varying with dis-turbance wave number T.

For a certain type of disturbance wave with any kind of flow magnitude, the wave growth rate was neither a mono-tonically increasing function of meandering wave number,

Figure 5 Distribution of velocity and additional inertia force component Tsn. (a) f =0.2; (b) f =0.4; (c) f =0.6.

Figure 6 Growth rate of disturbance wave varying with meandering wave number.

(a) (b) (c) (a)

(b)

(c)

−0.6 −0.2 0.2 0.6


Figure 7 Growth rate varying with disturbance wave number.

nor a monotonically decreasing one (Figure 6). With in-crease of meandering wave number above zero, the growth rate increased slowly but upon reaching about 0.18, the rate began to decrease quickly until the meandering wave num-ber reached about 0.39–0.41, whereupon the growth rate minimized and rebounded slowly. However, for a certain flow magnitude, the relationship between growth rate of all-scale disturbance waves and meandering wave number was similar to that shown in Figure 6.

Hence, there was a narrow range of meandering wave number in which all scales of disturbance waves were most stable, at any flow magnitude. This range was 0.39–0.41.

3.3 Neutral curve

When growth rate of the disturbance wave became zero (ωTi

= 0), the coherent structure was in a critical condition. The relationship between Reynolds number and disturbance wave number is given by a neutral curve. Obviously, the neutral curve is also affected by (0, f and ωf), as shown in Figures 8 and 9(a) and (b). The two figures reveal how meandering wave number and amplitude affected the neu-tral curve. Parameters for Figure 8 were 0=0.05 and ωf =0.

Figure 8 Neutral curve for several meandering wave numbers.

The neutral curve variation with meandering wave num-ber shown in Figure 8 corresponds to growth rate variation in Figure 6. There was also the most stable meandering wave number, denoted as fm=0.39 in this run.

Figure 9 shows that the trend of the neutral curve with meandering amplitude was different at meandering wave numbers f =0.18 and f =0.3. This was because growth rate of the disturbance wave had different trends in different meandering wave number ranges, as shown in Figure 6.

3.4 Critical Reynolds number

The minimum Reynolds number on the neutral curve is the critical Reynolds number, denoted as Recr. This number is related to river meandering parameters (0, f and ωf), i.e., Recr = Recr(0, f , ωf). Figure 10 shows the critical Reynolds number varying with meandering wave number for several meandering amplitudes. For a slightly bending assumption, the curvature (ψ=0f ) should be sufficiently small. We took the cases of 0 =0–0.1 and f =0–0.5.

Figure 10 indicates two extremum values with meander-ing wave number variation. One was a minimum around f

=0.18, corresponding to the most unstable flow condition, and the other was a maximum around f=0.39–0.41, corre-sponding to the most stable flow condition. This is in accord

Figure 9 Neutral curve for several meandering wave amplitudes. (a) f

=0.18; (b) f =0.30.


Figure 10 Reynolds number variation with meandering parameters.

with Figure 6. The most stable meandering wave number fm varied

slightly with meandering amplitude. In the straight channel, meandering amplitude was zero (0= 0) and the most stable meandering wave number fm=0.41, corresponding to L= M= 7.6B. With increase of meandering amplitude, the most stable meandering wave number attained a constant value, fm=0.39, corresponding to M= 8.1B.

In some ways, meandering rivers always tend to diminish internal turbulence intensity. In a river channel with mean-dering wave number fm, all disturbance waves were most stable at any flow magnitude. This means that this kind of river meandering stabilized the inner coherent structure more than any other hydraulic geometry. Since the inner coherent structure was the major driving force of river evo-lution, we infer that this narrow meandering wave number is the trend of river evolution in nature.

For the sine-generated river, the ratio of down-valley length L and meander width Ba is dependent on meandering wave number f and amplitude 0, as

2

00

2

00

cos( cos )d.

2 sin( cos )daf

L

B b

Taking the meandering wave number fm and amplitude 0=0–0.2, we calculated the ratio L/Ba = 5.3–6.4. The aver-age ratio was about 6, agreeing well with earlier statistical data from experiments and field observation [1–4].

4 Conclusions

There is a kind of large-scale flow structure that drives a river to meander or “wiggle”. The meandering boundary affects the flow current and disturbance wave inside the meandering river. Some disturbances grow into a large-

scale flow structure to shape a certain hydraulic geometry. In some ways, meandering rivers always tend to diminish internal turbulence intensity. Therefore, we investigated how river meandering affects its inner disturbance and ex-plained the character of river hydraulic geometry formation. Some conclusions are as follows.

1) Stability and evolution of the disturbance wave in a sine-generated meandering river were affected by the me-andering amplitude, wave number and frequency. The me-andering boundary tended to determine disturbance waves inside it.

2) The growth of disturbance waves was influenced by meandering wave number. These waves were most unstable at f =0.18, and most stable at f=0.39–0.41.

3) The growth of disturbance waves was also affected by meandering amplitude. For a slowly varying river channel, the greater the meandering amplitude increase, the more disturbance waves were unstable. On the contrary, in a rap-idly varying river channel, the greater the meandering am-plitude increase, the more disturbance waves were stable.

4) For slight sine-generated river meandering, we ob-tained the length-width ratio L/Ba = 5.3–6.4. The average ratio was about 6, agreeing well with statistical data from prior experiments and field observation.

5) We believe that the river meandering that made the inner disturbance most stable is the common hydraulic ge-ometry form in nature. The statistical average length-width ratios of Yalin, Habib and Da Silva, Leopold and Wolman confirm our assumption. Our model explained why rivers in nature follow certain statistical laws, such as L=6Ba of Yalin and L=aBm of Leopold and Wolman.

Since our theory is a simple linear model, we only reached preliminary conclusions about hydraulic geometry relationships of a slightly bending river. More research is needed regarding nonlinear theory of hydraulic geometry.

Finally, river hydraulic geometry is also indirectly af-fected by other factors, such as bed composition, upstream application of water and sediment, and geologic features. All these influences cause river landscape variation. How to incorporate these factors is a further research direction.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 51279124, 50979066, 51009105) and the Natural Sci-ence Foundation of Tianjin (Grant No. 12JCQNJC05600).

1 Yalin M S. River Mechanics. UK: Oxford, Pergamon Press, 1992 2 Yalin M S, Da S. Fluvial Processes. IAHR Monograph, IAHR, Delft

NL, 2001 3 Julien P. River Mechanics. UK: Cambridge University Press, 2002 4 Habib A, Da S. Regions of bars, meandering and braiding in da Silva

and Yalin’s plan. J Hydraul Res, 2011, 49: 718–727 5 Leopold L B, Wolman M G. River meanders. Geol Soc Am Bull,

1960, 71: 769–794 6 Langbein W B, Leopold L B. River meanders-theory of minimum

variance. US Geol Surv Prof Paper, 1966. H1–H14 7 Engelund F, Hansen E. A Monograph on Sediment Transport in Al-

luvial Streams. Copenhagen: Teknisk Forlag, 1967


8 Elder J W. The dispersion of marked fluid in turbulent shear flow. J Fluid Mech, 1959, 5: 544–560

9 Herbert T. secondary of boundary layers. Annu Rev Fluid Mech, 1988, 20: 487–526

10 Zhang Z S, Lilley G M. A theoretical model of coherent structure in a plate turbulent boundary layer. In: Turbulent Shear Flow III. New York: Springer-Verlag, 1981. 72

11 Zhang Z S. Turbulence. Beijing: National Defence Industry Press, 2002

12 Reynolds W C, Potter M C. Finite-amplitude instability of parallel shear flows. J Fluid Mech, 1967, 27: 465–492

13 Nishioka M, Ichikawa Y. An experimental investigation of the stabil-ity of plane Poiseuille flow. J Fluid Mech, 1975, 72: 731–751

Appendix

Coefficients in eqs. (15)–(17) are expressed as follows:

i( )

1 1 e . i ,

f fs t

s T fn c c m

i( )

1 1 e . ,

f fs t

n c c

2i( )21 2 e . 2 i ,

f fs t

s T fn c c m

2i( )

1 2 e . 2 ,

f fs t

n n c c

i( )

2 1 0

i( )

1

i( )

1

2 i( ) i( )

i( )

e . i

ˆ e . i

ˆ i e .

2 e . e .

i e . i ,

f f

f f

f f

f f f f

f f

s t

s T f s

s t

s T f

s t

f s

s t s t

T f

s t

f T f

Ren c c m u

Re u c c m

Re u c c

m n c c c c

n c c m

i( ) i( )

2 1 1ˆ e . e . ,

f f f fs t s t

sd nRe u c c c c

i( )

2 1 0

i( )

1

i( )

i( )

e .

ˆ e .

2 e . i

i e . ,

f f

f f

f f

f f

s t

n s

s t

s

s t

T f

s t

f

Re c c u

Re Du c c

c c m

c c

2i( )22 2 0

i( ) i( )

1

i( ) i( )

1

i( ) i( )

1

2 2i(2

e . 2 i

ˆ e . e . i

ˆ e . i e .

ˆ e . e .

3 e

f f

f f f f

f f f f

f f f f

f

s t

s T f s

s t s t

s T f

s t s t

f s

s t s t

n

s

T f

Ren c c m u

Ren c c u c c m

Ren c c u c c

Re c c u c c

m n

)

2i( )

2i( )2

. 2

e . 2

3i e . i ,

f

f f

f f

t

s t

s t

f T f

c c

c c

n c c m

2i( )

2 2 e . 2 ,

f fs t

sd n c c

2i( )

2 2 0

i( ) i( )

1

2i( )

2i( )

e . 2

ê . e .

2i e .

4 e . 2 i ,

f f

f f f f

f f

f f

s t

n s

s t s t

s

s t

f

s t

T f

Ren c c u

Re c c u c c

n c c

n c c m

i( )

3 1 0

-i( )

1

e . 2

î e . ,

f f

f f

s t

s s

s t

f n

Re c c u

Re u c c

i( )

3 1 0

i( )

1

i( )

1

2 i( )

i( )

e . i '

î e .

ˆ e .

2 e .

i e . ,

f f

f f

f f

f f

f f

s t

n T f s

s t

T f s

s t

n

s t

T f

s t

f T f

Ren c c m u

Re m u c c

Re Du c c

n m c c

m n c c

i( ) i( )

3 1 1ˆ e . e . ,

f f f fs t s t

nd nRe u c c c c

2i( )

3 2 0

i( ) i( )

1

i( ) i( )

1

2i( )

2i( )

e . 2 2

î e . e .

ˆ e . 2 e .

4 e . 2 i

2i e . ,

f f

f f f f

f f f f

f f

f f

s t

s s

s t s t

f n

s t s t

s

s t

T f

s t

f

Ren c c u

Ren c c u c c

Re c c u c c

n c c m

n c c

2i( )23 2 0

i( ) i( )

1

2 2i( )2

2i( )

2i( )2

e . 2 i

ˆ e . e . i

3 e . 2

e . 2

3i e . i ,

f f

f f f f

f f

f f

f f

s t

n T f s

s t s t

s T f

s t

T f

s t

s t

f T f

Ren c c m u

n c c u c c m

m n c c

c c

n c c m

2i( )

3 2 e . 2 . f fs t

nd n c c

The 9×9 matrices A, B and C involved in eq. (18) are expressed as follows.

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46 47 48 49

51 52 53 54 55 56 57 58 59

61 62 63 64 65 66 67 68 69

71 72 73 74 75 76 77

81 82 83 84 85 86 88

91 92 93 94 95 96 99

A

A A A A A A

A A A A A A

A A A A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A A

A A A A A A A

A A A A A A A

,


14

25

36

41 42 43

51 52 53

61 62 63

74 75 76 77

84 85 86 78

94 95 96 79

B

B

B

B B B

B B B B

B B B

B B B B

B B B B

B B B B

, 41

52

63

74

85

96

C

C C

C

C

C

C

.

A type of dynamic mechanism of river hydraulic geometry

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