HYPERBOLIC MODEL FOR AGGRADING CHANNELS

By Mohammad Akram Gill,1 Member, ASCE

ABSTRACT: The problem of bed aggradation due to overloading in a river is formulated using a one-dimensional hyperbolic partial differential equation. The equation and the associated initial and boundary con­ditions are Laplace transformed. General solutions are developed for the bed aggradation and the distribution of bed load transport in space and time. Computed results are compared with those given previously in approximate solutions given in the literature. The formulated solution predicts initially a shock front which has been observed to form in some laboratory experiments. The shock front is predicted to progressively decay which is also in agreement with observation.

INTRODUCTION

When a stream of water flows over a sand bed transporting a constant rate of bed load for a sufficiently long time, it attains a steady uniform condition of flow in due time. The bed remains stable and is free of aggradation and degradation. If now the rate of sediment supply is suddenly increased locally at a given section without altering any other flow parameter, the initial state of equilibrium is disturbed and the stream starts steepening its slope by depositing quantities of sediment over the channel bed. The bed is in a state of aggradation. In due time, the stream may attain another condition of equilibrium.

The problem of bed aggradation in river channels due to excessive sediment supply has previously been discussed in several papers that used a simplified parabolic partial-differential equation to describe the process of aggradation (Gill 1980, 1983a; Jain 1981; Soni et al. 1980). A numerical solution of the nonlinear parabolic equation has recently been published by Zhang and Kahawita (1987). A perturbation solution of the same equation had been developed and published recently by the writer (Gill 1987).

Recently, Ribberink and Sande (1984, 1985) discussed the aggradation problem using a hyperbolic partial differential equation. The proposed hyperbolic model is presumably better than the parabolic (diffusion) model in that some of the acceleration terms that are neglected in the parabolic model are preserved in it. In both the models, the controlling equations are essentially nonlinear, but the solutions are obtained for linear equations. The parabolic solution is characterized by a feather-edge front of the aggrading bed wave, which is typical of all the diffusion processes. The hyperbolic model, on the other hand, predicts a more realistic, abrupt step-like front for the wave. The height of the abrupt front, however, is a decaying function of the distance, so that at a sufficiently large distance from the origin, the step-like front degenerates into a feather edge. This

'Assoc. Civ. Engr., Detroit Water and Sewerage Dept., 735 Randolph Street, Detroit, MI 48226.

Note. Discussion open until December 1, 1988. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on June 30, 1987. This paper is part of the Journal of Engineering Mechanics, Vol. 114, No. 7, July, 1988. ©ASCE, ISSN 0733-9399/88/0007-1245/$ 1.00 + $.15 per page. Paper No. 22633.

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also is the region in which predictions of aggradation from the hyperbolic and parabolic models are close to each other. Practical usefulness of both the models depends on the accuracy with which the numerical values of the linearized parameters can be determined.

Ribberink and Sande could not obtain the general solution of the partial differential equation that they used. They presented exact solutions for some specified values of distance x and approximate solutions for small and large values of time. A general solution of the proposed hyperbolic equation is presented herein; the exact solutions at specified locations given by Ribberink and Sande follow naturally as special cases of the general solution.

HYPERBOLIC MODEL

Write the steady, one-dimensional St. Venant equation as follows:

d(Zo + Zb) a " - ' — 3

h ;/ dx dx

zb) , du dy guJ

dx (1)

in which it = local velocity; z0 = elevation of the initial bed above a horizontal datum, Fig. 1; zh = height of the aggraded bed with respect to the initial bed; y = local depth measured from the aggraded bed; g = gravitational acceleration; C = Chezy's coefficient of hydraulic friction assumed to be constant; q = unit discharge; and x = coordinate direction and also the distance measured from the origin. The flow depth measured from the initial bed is denoted by h so that

h = v + zb (2)

The continuity of water flow is described by q = uy = u(h - zh), or

du dy dx dx

(3)

neglecting the term involving the time rate of change. The continuity of bed-load transport is described by the Exner equation:

dzb dG dt dx

(4)

X=0

t " SEDIMENT SUPPLY

FIG. 1. Schematic of Aggrading Channel

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in which G = volumetric rate of bed-load transport; and t = time. Finally

G = aith (5)

after Ribberink and Sande (1984, 1985) and Soni et al. (1980). In Eq. 5, a and b are constants. Carrying out the linearization of the preceding equations in the same way as was done by Ribberink and Sande, the following equations are derived:

( . -P>£ + F»£ + 3 S < « . - 0 - . : <«

and

dZb dzh d£

i r + C o a F - c » a r 0 . •••(7)

in which c0 = (it{)/y0) dGldu; and F2 = iil/gyQ. The zero subscripted parameters refer to mean steady values; and £ = the deviation of the local depth y from y0 . Eliminating t, and dt,/dx terms from Eqs. 6 and 7 gives

dzh k0 d2zh . d2zh

— T ^ ; _ *o T T = 0 (8)

dt C'/,o dxdt dX in which cW) = c(/(l — F2); lcQ = (ftG(/350); and S0 = initial bed slope. Eq. 8 is the hyperbolic partial differential equation that describes the process of bed aggradation due to an excessive supply of sediment. For subcritical flows in which F —> 1, the term cM —* °°, with the result that the term (/<(/c'W)) • (d

2zh/dxdt) -> 0. Eq. 8 in this case reduces to the parabolic model, which has been discussed in previous publications. The situation F —> 1 is obtained for steep slopes, in which case the term u(duldx) and g, dy/dx in Eq. 1 are exceedingly small quantities compared with the 3(z0 + zh)/dx term (Gill 1983a).

Eq. 8 is to be solved for appropriate initial and boundary conditions for a hyperbolic model.

NONDIMENSIONALIZATION OF CONTROLLING EQUATIONS

Use the following substitutions for nondimensionalizing Eqs. 4 and 8:

z* = -p p 7 (9a) O x - Go •

(•hi) t h = 2^- (9c)

/Co

G* = 7 F" (9cf> u« — Go

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In Eq. 9d, G0 = initial steady rate of sediment supply; and G« = final steady rate of sediment supply at x = 0. Using these substitutions in Eqs. 4 and 8, the following equations are derived:

dz* dG* 2 ^ T + ^ = 0 (10)

o?# OX*

3z* d2Z* 1 d2Z* d/* dx*dt* 2 dx%

INITIAL AND BOUNDARY CONDITIONS

Initially zb = 0, so that the bed gradient equals 50 everywhere. Also, at / = 0, the rate of sediment supply at x = 0 is constant and denoted by G0 . The sediment supply is then instantaneously increased to another constant value denoted by GK . Mathematically,

z*(x*, 0) = 0 (12)

_G* Gx — G0

G*(0, ?*) = G»* = „ ^ (13)

Another condition must be satisfied by any realistic solution; the solution should remain bounded as x* -» oo. .

SOLUTION

Eq. 11 is expressed as follows

dz* 1 d2z* PZ*-pdx-*~2M = 0 . . . . . ( 1 4 )

in terms of Laplace transform variables, z* is the transform of z* ; and/? is the transform parameter. It should be noted that Eq. 12 has been used in writing Eq. 14. The solution of Eq. 14 that remains bounded as x* —»oo is written as follows:

z* = A exp [-x*(p + \/p2 + 2p)] (15)

in which A is constant. In order to evaluate A, Eq. 10 is transformed to

dG* dx

' * *

= - 2pz* = - 2pA exp [-**(/? + \/p2 + 2p)] (16)

and is solved to give

2pA _ G* = + . / 2 , , e xP [~ X*(P + VP

+ 2PK + D (17) p + \fpA + 2p v

in which D is another constant of integration. At x* = 0, G* = G^ = GJ(GX - G0), so that

G»* 2pA

P P + \/p2 + 2p

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z

and as A-* -* °°, G* = G0* = GJ(GK - G„), so that

~ Go* D = (19)

p

From Eqs. 18 and 19

2pA _ GK* — G0H; 1 p + V V + 2 p ~ P = P (20)

since (G„* — Go*) = 1. Using Eq. 20 in Eq. 15 gives

(p + \/p2 + 2p\ „ = ( \ l j e x p [ - A * ( p + V ? + 2p)] (21)

Eq. 17 is reduced to

_ exp [ - x*(p + \fp2 + 2p)] G0H! G* — 1 (22)

P P

INVERSION

Bed-Load Transport Write Eq. 22 as follows:

exp'(-A-*p) exp[-A-H !A/(P+ l ) 2 - 1] Go* G* = * + — ' (23)

P P

In order to invert Eq. 23, consider a standard inversion result from Churchill (1958)

T" '[exp ( - k\Jp2 - or) - exp (-/</>)] = 0, 0 < r* < k (24a)

r " '[exp ( - kVP73^) - exp ( - kp)-]

'= Jfi-k1 tfax/t*-^* l*>k <24/>)

from which

r - '[exp (- k V/'2 " «2)] = W* - k) + m / i l o - i / l i - a , /» ><r (25)

y l 1 v The symbol T"1 = inverse of the expression within the square braces; 8( ) = the impulse function; and /,( ) = the modified Bessel function of the first kind and first order. Another standard result is used with Eq. 25 for the solution:

if(p + P)] = exp , ( - pr*)F(f*) (26)

in which'/O?) = the transform of F(?*). Using the results of Eqs. 25 and 26 in Eq. 23 leads to

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] )« / / . (27)

„_i/exp (-.v»p)exp [-x*\/(p + I)-- 1]] T ' < * : > = exp ( - -v*)

+ I * 'x* Hexp [ - ( /» - .y < , ) ] / |K / ( f« - .v < , ) 2 - .Y J , v > A / ( ' * - - V * ) - - . v i V

It should be noted that the first term on the right side of Eq. 27 is obtained from Jexp[-(f* - JC*)]8(/* - 2**) A* . In order to evaluate the integral in Eq. 27, use 6 = (/* - **) and write

M,=X* f ex^J~e),/iK/e^7|)Je. (28)

The integral in Eq. 28 has been previously discussed by Ford (1958), and the solution is written here after Yudell (1962).

6 — A'sis 6 + .V* M, = - exp ( - jr.) + exp ( - e)/„( W F 7 ? 1 ) + 2 1 - A—j-, —f^j i .... (29)

in which I0( ) = the modified Bessel function of the first kind and zero order. The complete solution is written as

G*-G0* = exp(-e)/o(A/e^Tvl) + 2 | l - y ( ^ y ^ , ^ ^ J |. >* > 2.Y* (30)

The function 7[(9 - x*)/2, (6 + x*)/2] is a tabulated function (Brinkley et al. 1952), and some of its properties are given in Yudell (1962). It is defined for the purposes of this analysis as follows:

x

/(<(.,, 4>2) = 1 - exp [ - (d>, + «>2)] ^ *"*/*(«) (3D

in which £ = 2V"4>14>2 a n d 4* = V<J>2/<|>|. Since Eq. 30 is valid for ?* > 2,v* , the result can be written in a slightly more elegant manner as follows:

(32) G* - G„* = HU* - 2xM exp (- 8) /„(We2-rJ + 2 1 - • / ( - y - . —5—I [

in which #(/* - 2x*) = the unit step function, which is equal to one for t* > 2JC# , and is otherwise zero.

Bed Aggradation Write Eq. 21 as follows:

1 1

2p 2 \Jpl + 2/7 p\fP2 + 2Pj

• exp (-/?**) exp ( - A-*\/pz + 2/7) (33)

Each of the first three terms on the right-hand side of Eq. 33 is defined as follows: factor 1 = 1/2 /?; factor 2 = 1/(2Vp2 + 2/7); and factor 3 = 1 /

+ 2/7). Factor 1 is half of the first term in Eq. 23 on the right side. Its

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inverse has already been obtained in Eq. 30. The inverse of factor 2 is obtained as follows:

exp ( - px*) exp[-x*-\/p(p + 2)]\

2 ' VPTPTZ) I = o, 0 < ?* < 2x* (34a)

, - J e x p (-px*) exp[~x*-\/p(p + 2)]

\/p(p + 2)

= - exp \ - (?* - x*) J7( W-y/ ( * • * $ , r ^ /x^ (34/7)

Factor 3 is (2/p) times factor 2; therefore

= I * exp [ - (f* - jr*)]/0(-v/(f* - x*)z - jrj|) oY*

Write Eq. 35 in terms of 8 as follows:

M2 = T exp (- 8)/„(W82 - x2) </8

(35)

(36)

whose solution is available in Yudell (1962). The complete solution for z* is given by

z* = H(f* - 2x*)-j (1 + 8 - x*) exp ( - 8 ) / , ( W e 2 - x2*)

+ W e 2 - x | exp (- 8 ) / , ( A / 8 2 - x2*)

8 - x* 8 + x!N + (1 - 2x*) 1 - 7 (37)

The special cases considered by Ribberink and Sande are easily obtained from the general solution in Eq. 37.

NUMERICAL RESULTS

Values of z* have been computed using Eq. 37 and are plotted against x* at different values of t* in Fig. 2. The shock front is quite pronounced at small values of t* ; for larger values of r* , the front decays progressively.

For small values of time, Ribberink and Sande obtained the following result:

z* = 2 exp ( - x*) - exp I - - t 1

-V* ^ 'J ^ ' =* (38)

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I APROXIMATE SOLUTION-] | ^ ' 3 | ' ° ^f** *'°

FIG. 2. Plot of z* versus v* for Various Values of /* ; Comparison of Exact and Approximate Solutions at Small and Large Values of Time

Eq. 38 is compared with the exact solution for values of r# up to 2.0. The height of the shock front is correctly predicted by Eq. 38; some discrep­ancy is noted in the overall profile. This discrepancy diminishes for decreasing values of t* .

v-l \

1.0 \ \

2.0 \

3.0 \

4.0 ^ 3.0 ^ N ,

8.0

FIG. 3. Plot of (G* - Gm) versus v* at Different Values of /*

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For values of /* —» <», Ribberink and Sande recommended

z* = V ^ ^ e X p [ - ^ ^ }

+ ^ - ^ e r f c [ V 2 ( ; ; _ ^ ] , * * < , * . . . . . . . . (39)

Predictions from Eq. 39 were compared with those from Eq. 36 for values of t* & 4.0. The discrepancy between the approximate and the exact solutions progressively diminishes for increasing values of t* . The ap­proximate results, Eqs. 38 and 39, should not be used in the range of 1 < /* < 4, as already discussed by Ribberink and Sande, because the discrepancy in this time range is appreciable.

Profiles of bed load transport with respect to .v* and tt. are plotted in Fig. 3. These profiles are similar to those of z*{x* , f*) in Fig. 2. The preceding solutions are valid for 0 =s / < T, in which T = the time taken by the aggrading front to reach the downstream end of a channel. The solutions are inapplicable for t > T. A complete discussion of this point is available in Gill (1983a, 1983b).

CONCLUSION

The main results of the paper are the general solutions of the hyperbolic partial differential equations for z* and G* . The solution for z* , Eq. 36, yields the solutions previously given by Ribberink and Sande as special cases. Qualitatively, the general solutions are capable of predicting the physical conditions more accurately than any other solution presently available. Initially, the aggradation front is large and steep, as has been observed in some laboratory experiments (Kerssens and van Urk 1986) degenerating gradually into a feather-edge type of front. These conditions are predicted by the theoretical solution.

Since the same linear differential equation is applicable to the process of bed degradation with identical initial and boundary conditions, the same solutions, Eqs. 32 and 37, can be adapted for degradation [see Gill (1983a, 1983b)].

ACKNOWLEDGMENTS

J. S. Ribberink, of the Civil Engineering Department at Delft University of Technology sent a copy of Report No. 84-1 to the author on his request; his valuable cooperation is appreciated and acknowledged. Diagrams included in this paper were prepared by George J. Smith III, DWSD, whose assistance is appreciated. J. N. Murphy, Research Director, U.S. Department of the Interior, Bureau of Mines, Pittsburgh, Pa., furnished a copy of the table of/( , ) on the writer's request.

APPENDIX I. REFERENCES

Brinkley, Jr., S. R., Edwards, H. E., and Smith, Jr., R. W. (1952). "Table of the temperature distribution function for heat exchange between a fluid and a porous

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solid." Mathematical tables and other aids to computation, National Res. Council, Washington, D.C., 6, 40.

Churchill, R. V. (1958). Operational Mathematics. 2nd Ed., McGraw-Hill Book Co., Inc., (Kogakusha Company, Ltd.), Tokyo, Japan, 323 and 329.

Ford, F. A. J. (1958). "On certain indefinite integrals involving Bessel functions." / . Math.Phys., 37, 157-161.

Gill, M. A. (1980). Discussion of "Aggradation in streams due to overloading," by J. P. Soni, R. J. Garde, and K. G. Ranga Raju. J. Hydr. Div., ASCE, 106(HY11), 1955-58.

Gill, M. A. (1983a). "Diffusion model for aggrading channels." J. Hydr. Res., IAHR, 21(5), 355-367.

Gill, M. A. (1983b). "Diffusion model for degrading channels." J. Hydr. Res., IAHR, 23(5), 369-378.

Gill, M. A. (1987). "Nonlinear solution of aggradation and degradation in chan­nels." J. Hydr. Res., IAHR, 25(5), 537-547.

Jain, S. C. (1981). "River bed aggradation due to overloading." J. Hydr. Div., ASCE, 107(HY1), 120-124.

Kerssens, P. J. M., and van Urk, A. (1986). "Experimental studies on sedimen­tation due to water withdrawal." J. Hydr. Engrg., ASCE, 112(7), 641-656.

Ribberink, J. S., and van der Sande, J. T. M. (1984). "Aggradation in rivers due to overloading." Communications on Hydraulics, Report No. 84-1, Department of Civil Engineering, Delft University of Technology, Delft, Netherlands.

Ribberink, J. S., and van der Sande, J. T. M. (1985). "Aggradation in rivers due to overloading—analytical approaches." J. Hydr. Res., International Association for Hydraulic Research, 23(3), 273-284.

Soni, J. P., Garde, R. J., and Ranga Raju, K. G. (1980). "Aggradation in streams due to overloading." J. Hydr. Div., ASCE, 106(HY1), 117-132.

Yudell, L. L. (1962). Integrals of Bessel functions. McGraw-Hill Book Co., Inc., New York, N.Y., 271-282.

Zhang, H., and Kahawita, R. (1987). "Nonlinear model for aggradation in alluvial channels." J. Hydr. Engrg., 113(3), 353-369.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A a b C

Cb0

c0

D erf (6)

erfc (0) F G

G0 Goo

G* Go* Goo*

9 H{ )

h

= = = = = = = = = = = = = = = = = = =

constant; constant; constant; Chezy's coefficient; c«/(l - F2); [(dGldu) • (ujy0)]; constant; (2/V^)/oexp (-e2)</e; 1 - erf(G); Froude number; local bed-load transport; initial bed-load transport; increased bed-load transport; G/(Goo - G0); Go Gos - G0); GooAGoo - G0); gravitational acceleration; unit step function; flow depth referred to initial bed;

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/ ( , ) k

*o L

M, M2

P q

So t

u it

«o X

X*

y Jo Zb

z<> •£* a P £ 9 T

* , -e1

«!»

= = = = = = = = = = = = = = = = = = = = = = = = = = = =

function of 0 and x% , as in Eq. 31; constant; bGJlS0; length of channel; notation for integral in Eq. 28; notation for integral in Eq. 36; parameter of Laplace transform; unit discharge; initial bed slope; time; 2cj,o/k0; local velocity; average velocity; coordinate direction, distance from origin; cox/ko ; local depth referred to aggraded bed; mean steady depth; elevation of aggraded bed above initial bed; bed elevation above horizontal datum; zhc0/{G - G0); constant; constant; 2{$M)m; u# — -*W; time at which aggradation front arrives at downstream end; typical variable; another typical variable; and (4>2/4>i)'/2-

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