05 long waves in open channels

8/12/2019 05 Long Waves in Open Channels

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Alternative Hydraulics Paper 5, June 11, 2012

Long waves in open channelstheir nature, equations,

approximations, and numerical simulationJohn D. Fenton

Institute of Hydraulic and Water Resources Engineering

Vienna University of Technology

Karlsplatz 13/222, 1040 Vienna, Austria

http://johndfenton.com/Alternative-Hydraulics.html

[email protected]

Abstract

This is a document that is not ready. It is intended for referees of a paper submitted to Agricultural

Water Managementso that they can judge some of the results. The paper should be ready by the

middle of June.

Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . 2

2. The long wave equations . . . . . . . . . . . . . . . . . . 2

3. Non-dimensionalisation . . . . . . . . . . . . . . . . . . 4

4. Linearisation of the equations . . . . . . . . . . . . . . . . 5

4.1 Mass and momentum conservation equations . . . . . . . . . . 6

4.2 The Telegraphers equation . . . . . . . . . . . . . . . 7

5. Propagation behaviour of waves . . . . . . . . . . . . . . . . 9

5.1 Dimensionless Telegraphers equation . . . . . . . . . . . . 9

5.2 Solutions for waves periodic in time . . . . . . . . . . . . 9

5.3 Approximate solutions in limits of not-so-long and very-long waves . . 10

5.4 Presentation of results . . . . . . . . . . . . . . . . . 12

5.5 Stability offlowroll waves . . . . . . . . . . . . . . . 14

6. The momentum equation relative importance of terms and some

approximations . . . . . . . . . . . . . . . . . . . . . 14

7. The slow-change/slow-flow routing equation . . . . . . . . . . . . 18

8. Numerical solution of the long wave equationsan explicit finite difference

scheme . . . . . . . . . . . . . . . . . . . . . . . . 20

8.1 A Forward-Time-Central-Space scheme . . . . . . . . . . . 20


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Long waves in open channelstheir nature, equations, approximations, and numerical simulation John D. Fenton

8.2 Linear stability . . . . . . . . . . . . . . . . . . . 20

8.3 Approximate stability criteria . . . . . . . . . . . . . . 22

8.4 Test of linear stability . . . . . . . . . . . . . . . . . 22

8.5 Determining stability limits for unsteadyflow problems . . . . . . 23

9. Characteristics . . . . . . . . . . . . . . . . . . . . . 24

References . . . . . . . . . . . . . . . . . . . . . . . 25

10.Introduction . . . . . . . . . . . . . . . . . . . . . . 27

10.1 Approximate stability criteria . . . . . . . . . . . . . . 28

10.2 Test of linear stability . . . . . . . . . . . . . . . . . 29

1. Introduction

This is not ready

2. The long wave equations

Consider the long wave equations in the form obtained by Fenton (2010 b, equations 19):

In terms of cross-sectional area A and discharge Q: For some theoretical studies, this is thesimplest formulation:

A

t +

Q

x i= 0, (1a)

Q

t

+ 2Q

A

Q

x

+gAB

Q2

A2A

x

+ PQ |Q|

A2

gAS= 0, (1b)

where t is time, x is a horizontal streamwise co-ordinate, i is inflow per unit length of stream, g isgravitational acceleration,B is surface width,is the Boussinesq momentum coefficient, is a dimen-sionless resistance coefficient, which is defined below, and where Sis the local downstream bed slope ina mean sense, evaluated around the perimeter. In equation (1b) some relatively unimportant terms have

not been shown: the inflow contribution to momentum has been neglected; there are no converging or

diverging vertical walls (otherwise an extra contribution to the slope term would have to be made);and

where the Boussinesq momentum coefficienthas been assumed to be constant.

In terms of surface elevation and Q: In practice it is more convenient to use surface elevationinstead of area as the dependent variable, where the derivatives are related by

A

t =B

t and

A

x =B

x+ S

, (2)

provided there are no vertical walls that move in time t or converge or diverge in x. Equations (1)become

B

t+

Q

x i= 0, (3a)

Q

t + 2

Q

A

Q

x +

gA Q

2B

A2

x+ P

Q |Q|

A2 Q

2B

A2S= 0. (3b)

The resistance term What to do: In most situations, however, the geometry is poorly known and onlyan approximate Sis used, and similarlyis usually poorly known. Certainly the slope correction S2 inthe expression for is almost always unnecessary.

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The resistance term in both momentum equations is P Q2/A2, in which is the dimensionless resis-tance coefficient

=

8

1 +S2

, (4)

where is the dimensionless Weisbach resistance coefficient.This definition allows forfinite slopes tobe used, however in almost all rivers and canals, Sis so small that =/8to good approximation, andis simply related to ChzysCby = g/C2. We willfind that using the dimensionless makes anumber of our expressions simpler.

In physical terms connects the mean stress on the wetted boundary to the mean velocity in the streamQ/Aby (e.g. eqn (6.11) of White 2003)

=

8

Q |Q|

A2 =

Q |Q|

A2 ,

where is fluid density, and so the manner in which the resistance appears in the momentum equations(1b and 3b), as P Q |Q| /A2, has a simple physical significance as a dimensionless coefficient (in which

the necessary horizontal resolution of force has been included) multiplied by the wetted perimeter andthe square of the mean velocity with sign attached.

Other, more familiar, ways of writing the resistance term are

Manning!

gA Q |Q| /K2: whereKis the conveyance of the stream

K=

rgA3

P, (5)

which is a function of the resistance coefficient and the stream geometry. Although this form loses

some of the physical significance ofP Q |Q| /A2, occasionally it is more convenient to use it as a

shorthand.

gASf: where Sfis called the resistance slope. This terminology has its pitfalls, for example in someworks (e.g.Lyn 1987, Lyn & Altinakar 2002) it has been assumed to be a constant even whereQandAvary.

The introduction ofmakes notation rather simpler further below. If we consider the case of steady uni-

formflow in a prismatic canal of constant slopeS0 and resistance 0, when all derivatives in equations(1) are zero andQ=Q0,A=A0, andP =P0are all constant, equation (1b) becomes

0P0Q20A20

gA0S0= 0, (6)

In general, the resistance coefficient as given by equation (4) foris a function of the relative rough-ness of the channel boundary = ks/(A/P) where ks is equivalent sand grain roughness and thechannel Reynolds numberR = (Q/P)/whereis kinematic viscosity, as given by Yen (1991, eqn30), as explained in Fenton (2010a). For a particular channel shape,Pis a known function ofAso thatequation (6) actually involvesA0andQ0in quite a complicated manner. If the resistance coefficient 0is taken as independent of Reynolds number and so independent ofQ0, the equation can be written as anexplicit expression for mean velocityU0 = Q0/A0, the familiar Chzy-Weisbach form of the uniformflow equation:

U0 =

r g

0

A0P0

S0. (7)

This has an interesting significance if we re-write it to give an expression for the Froude numberF0

of

theflow

F20 = U20

gA0/B0=

S00

B0P0

. (8)

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P0. We could have used B0 for this, but the end result is slightly more concise if we use a separatescale. Now introducing the area scale A0, the dependent variables A and Q are scaled as A=AA0andQ = QU0A0. In addition we write the channel slope and friction factors in terms of reference valueswith a 0 subscript as S=S0S, and = 0. The equations we obtain are

LU0TAt

+ Qx= 0, (13a)

U0gS0T

Qt

+ U20gS0L

x

Q2A

+

A0/B0S0L

AB

Ax

+ PQ2A2

AS= 0, (13b)

where we have used the uniform flow equation (7) to write 0 in terms ofU0, and where it has beenassumed that flow does not reverse so that we can use Q2 in the resistance term.

In the traditional approach (e.g. Woolhiser & Liggett 1967, and subsequent researchers), the length

and time scales of disturbances, and hence their velocity of propagation, are assumed related by the

mean fluid velocity, such that it is assumed that T = L/U0, the equations become, after dividing themomentum equation through byU20 /gLS0:

At

+ Qx

= 0,

Qt

+

AB

1

F20 Q

2

A2

Ax

+ 2QA

Qx

+ LS0

F20 A0/B0

P

QA

2 AS

!= 0.

There are two dimensionless flow parameters: the Froude number appearing as 1/F20 , and the quantityLS0/

F20 A0/B0

, which Liggett (1968) called the Kinematic Flow Number. In fact it can be written as

1/F20 timesLS0/ (A0/B0), the Drop/depth ratio, which is the ratio of the drop in channel bed LS0overa disturbance length scale ofL to the mean depth A0/B0, so the Drop/depth ratio could be thought of asthe second parameter.

Strelkoff & Clemmens (1998, eqn 12) made the additional assumption that one can attach a magnitudetoL, such that in present terms L = (A0/B0) /S0, so that the drop/depth ratio is 1, and the KinematicFlow Number becomes1/F20 . The momentum equation can be written

F20

Qt

Q2

A2

Ax

+ 2QA

Qx

+

AB

Ax

+P

QA

2 AS= 0,

in terms of a single parameterF20 , and which suggests that the leading terms can be ignored forF20 1,

leading to the so-called Zero-inertia model.

The problem with these formulations is that in the first a relationship between L and Thas been assumedand additionally in the second a value ofLhas been assumed in terms of channel geometry. This means

that we cannot be sure what the magnitudes of the derivatives are for example in the dynamic wavelimit, when much experience suggests that waves travel at a velocity related to

pgA0/B0, then L/T/U0,

assumed to be 1 is actually of a magnitudep

gA0/B0/U0 = 1/F0, and the non-dimensionalisation asgiven does not tell us the real relative importance of terms.

In this work, we prefer not to use the above approach. In fact, we believe that such scalings cannot be

done at this stage, as we assert that while the input time scale Tis an independently-specifiable quantityfrom the boundary conditions imposed on the problem, the length scale L is determined by the actualsolutions to the equations. To obtain the relative importance of various terms in the equations without

solving them seems not to be possible. We now proceed to a solution of the linearised equations and

then use that to examine the relative importance of terms in the momentum equation.

4. Linearisation of the equations

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4.1 Mass and momentum conservation equations

To linearise the equations, it is actually more insightful to go back to their dimensional form. We

consider relatively small disturbances about a uniform flow, where there is no inflow i= 0, and we write

A= A0+ A1 and Q=Q0+ Q1,where is a small quantity. AsA0 andQ0 are constant, all derivatives ofA andQ in equations (1) areof order so that the coefficients need only be written to zeroeth order, and the linearisation is simple.The only non-trivial operations are in the remaining resistance and slope terms of equation (1b). We

introduce the functionfor those terms

= PQ2

A2 gAS, (14)

where we have writtenQ |Q|= Q2, as we consider small perturbations about a uniform flow, which isunidirectional. For small perturbations in area and discharge ofA1andQ1the Taylor expansion ofis

=0+ Q1Q

0

+ A1A

0

+ O

2

, (15)

where in the case of steady uniform flow

0 = 0P0Q20A20

gA0S0 = 0.

For thefirst derivative in (15) we obtain from (14)

Q

0= 20

Q0P0A20

1 + 12

Q

0Q00

. (16)

As 0 is in general a function of discharge Q0 (because of its variation with Reynolds number) thisexpression is a somewhat complicated function ofQ0. Below we will see that it, with units of T

1 isactually an important channel parameter, determining the nature of wave behaviour, so it helps us here

to obtain a more physical feel for it. We obtain an approximation by noting that the variation of with

Q is relatively unimportant so that we neglect the derivative /Q and as is independent ofQ in thisapproximation we can use the Chzy-Weisbach resistance equation (7) to eliminateQ0 = U0A0, givingthe expression in terms of the geometrical and roughness properties of the channel

Q

0

2s

gS00A0/P0

, (17)

which would be quite accurate in most applications.

For the other derivative we obtain

A

0

= gS0+0P0Q

20

A30

2 +A0

P0

P

A

0

+A00

A

0

and we can make use of the uniform flow relationship (7) to give

A

0

= 3gS0

1 13

A0P0

P

A

0

+A00

A

0

(18)

In this case, both derivative terms in P/Aand /A in general may have finite contributions, thefirst expressing the effect offinite channel width, and the second because the resistance coefficient has a

strong variation with relative roughness, as shown and quantified in Fenton (2010a, p4).

Now to combine the results into linear equations. The mass conservation equation (1a) is already linear.

For the momentum conservation equation (1b), we take the derivative terms with coefficients from the

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zeroeth order solution and combine with the linearised resistance-slope contribution (15) to give the

equations

A1t

+ Q1

x = 0, (19a)

Q1t

+

C20 2U20

A1x

+ 2U0Q1x

+ Q

0

Q1+ A

0

A1= 0, (19b)

in which the mean fluid velocity U0 = Q0/A0 has been used for simplicity, as well as the speed C0which is given by

C0=q

gA0/B0+

2

U20 , (20)

as obtained by Keulegan & Patterson (1943, eqn 87) for the wave speed of small disturbances when

there is no resistance, and who noted that a similar expression had been obtained by Boussinesq (1877,

p285, eqn 265). This result is a generalisation that puts into context the fact that in most textbooks this

is written with= 1such thatC0 = pgA0/B0, an approximation, and which is usually said to be the

"celerity" or "long wave speed" or "dynamic wave speed". Remarkably, the same expression, but with

no limitations of linearisation, is valid for the speed of characteristics, shown in 9. However, it is not,

in general, the speed at which disturbances travel. Below it will be shown that it is actually the speed

of waves only in the limit of shorter waves, but where they are still long enough that the hydrostatic

approximation holds. That limit is achieved relatively rarely, for example, when waves are due to rapid

gate movements. The dynamic wave speed seems to be a much less-important quantity in hydraulics

than is generally believed.

It is occasionally useful when we consider velocities of propagation to use the coefficient of the sec-

ond term in the form C20 2U20 , as written in equation (19b), while at other times when we con-sider the relative importance of terms it will be useful to use equation (20) to write it in the form

(gA0/B0) 1 F20 .

4.2 The Telegraphers equation

A pair of equations such as (19) has been obtained and analysed by Ponce & Simons (1977) for wide

channels, requiring a number of elementary matrix operations for subsequent manipulations. It is math-

ematically and physically simpler if, instead, the equations are combined into a single equation similar

to equations for wide channels obtained by Deymie (1938, eqn 6), Iwasa (1954, eqn 8), Lighthill &

Whitham (1955, eqn 28) and Ponce (1990, eqn 4). Continuing our analysis of prismatic channels of

arbitrary section we introduce a perturbation upstream volume v (x, t)(see equation 11 above) such thatA1 = v/x and Q1 =v/t, so that it satisfies the mass conservation equation (19a) identically.The momentum equation (19b) becomes the Telegraphers equation:

vt

+ c0 vx

+2v

t2 + 2U0

2vxt

(C20 2U20 ) 2vx2 = 0. (21)

We have introduced the symbols andc0, where

1. The quantity is simply /Q|0as given by equations (16) and (17):

=

Q

0

=2gS0

U0

1 + 12

Q00

Q

0

2

sgS00A0/P0

, (22)

in which we have shown two forms, one in terms ofU0 and the other approximation in terms ofthe fundamental geometrical channel quantities and resistance where it has been assumed that

does not depend onQ. It has dimensions of T1;we see here that it increases with both slope andresistance and decreases with depth.

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2. The termc0is the ratio of the two derivatives

c0= /A|0/Q|0

, (23)

and using equations (16) and (18):

c0 = 32U0

1 13

A0P0

P

A

0

+A00

A

0

1 + 12Q00

Q

0

(24)

= U0, (25)

where for convenience, the symbol has been introduced, which from equation (24) is roughly3/2. In most practical situations we know the resistance coefficient only approximately, andincluding the derivatives/Q and/A might be thought excessively enthusiastic. However,for possible theoretical studies, they will be retained here. The actual derivatives of with respect

to A and Q could be obtained from the formulae (7) of (Fenton 2010a), which give the derivatives of( 8) with respect to relative roughness and Reynolds number. In many situations the Reynoldsnumber of theflow is so large that there is no longer any variation with it, such that /Q= 0. Amore important situation is where bed forms can develop, when would show a stronger variation

with dischargeQ.

The quantity c0is a wave speed, as will be shown below. It has previously been called the "kinematicwave speed", but as we shall show, that is actually a misleading term: it is actually the speed of

propagation of very long waves.

The definition ofc0in equation (23) is a generalisation of the usual Kleitz-Seddon equation

c0=dQ0(A0)

dA0,

whereQ0(A0)is the discharge given by any of the usual resistance formulae. The generalisation isnecessary for the model we have set up here, where the resistance coefficient may be a function

ofQ, such as when it depends on channel Reynolds number.

Nature of wave propagation in limiting cases: At this stage it is already possible to anticipate

results that will be established more fully below. The terms in the Telegraphers equation (21) can be

grouped: the first two terms can be characterised asfirst derivatives, and the last three terms are allsecond derivatives. We now examine the governing equation in two limits:

Very-long waves: For disturbances that have a long period, such that 2/t2 /t, which we will

call "very long waves", the last three terms in equation (21) can be neglected, and the equation becomesv

t + c0

v

x= 0,

with a general solution v = f1(x c0t), where f1(.) is an arbitrary function given by the upstreamconditions. This solution is a wave propagating downstream at speedc0= U0.

Not-so-long waves: In the other limit, for disturbances which are shorter, such that 2/t2 /t,for which we use the term "not-so-long" waves, equation (21) becomes

2v

t2 + 2U0

2v

xt (C20 2U20 )

2v

x2 = 0,

which is a second-order wave equation with solutions

v= f21(x (U0+ C0) t) + f22(x (U0 C0) t)

where f21(.) andf22(.) are arbitrary functions determined by boundary conditions. In this case the

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solutions are waves propagating upstream and downstream at velocities ofU0 C0. It is noteworthythat the water carries disturbances at a theoretical transport velocity ofU0, rather than just U0, its meanvelocity. The waves have a speed ofC0 relative to the transport velocity. If= 1, with no velocityprofile or turbulence, equation (20) givesC0=

pgA0/B0, which in many places is referred to as "the"

long wave speed. What we have shown here is that it is the speed of waves just in the shorter limit and

then only for= 1. Accordingly we will not callC0the long wave speed. In fact, it will be shown thatthere is no such thing as a long wave speed.

We shall now spend some effort on the intermediate general case of long waves intermediate between

the limits of "not-so-long" and "very long".

5. Propagation behaviour of waves

5.1 Dimensionless Telegraphers equation

Now it is simpler to make the Telegraphers equation (21) dimensionless. We introduce dimensionless

varibles t = t andx = x/U0, and simply considerv to have been made dimensionless in somemanner that does not matter. The equation becomes

v

t+

v

x+

2v

t2+ 2

2v

xt (F20 )

2v

x2= 0. (26)

We note that the physical parameters and are constant for a particular channel, which leaves theequation containing a single parameter determining the nature of solutions, the Froude number in the

formF20 . It might be thought awkward to have a quantityF20 which is often large, but it is simpler to

have it appearing in this way, in only one place and where it actually expresses the relative importance

of this term, rather than multiplying the whole equation through by F20 and having that appear as anartefact in every term but one.

It might be thought that there is a large problem with thisin 3 on conventional non-dimensionalisations

we were critical of exactly the scaling forxandtwe have used here. However, we are about to obtainan analytical solution of equation (26) for the wavelength in terms of the period, and with that solution,

in 6 below we calculate the magnitudes of the various terms in the equation for different periods and

Froude numbers, the results being independent of our scaling.

5.2 Solutions for waves periodic in time

We start by obtaining solutions of the form

v= exp (i (x

t

)) , (27)

where i =

1, and where in general and are complex quantities, whose nature determines thebehaviour of the solutions. Substituting this solution into the dimensionless Telegraphers equation (26)

and dividing through by common terms gives the polynomial equation

i2 + 2i+i2

F20

= 0, (28)

where the order of the terms corresponds to those in equation (21). Ponce & Simons (1977) obtained

a similar equation for a wide channel, however different terms had different coefficients, each of which

could take a magnitude of 0 or 1, so that they could be switched in or out according to different approx-

imations. We will not follow that path, preferring to let the equation make its own approximations.

Another departure here is that we are not so interested in the response of waves that are periodic in space,as in rivers and canals usually the lengths of disturbances are not fixed. Instead we will consider waves

that are periodic in time, as any input disturbance can be written as a Fourier series in time. Then from

the governing equation we will determine the behaviour in space. In this case, is real, and equation

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(28) is a quadratic equation forwith solutions

= (i/2 + ) +

q(i/2 + )2 +

F20

(2 + i)

F20 , (29)

where = 1 and where we have neither expanded the products nor have we written it in standardcomplex form, as either gives a much longer expression.

For any frequency of input , this equation (29) gives the complex value of determining the wavepropagation characteristics. If we write it as=r+ii, the solution (27) can be written

v= exp(ix)exp

ir

x

rt

, (30)

so that in (x, t) space, the decay coefficient in x is i, the length of the waves is 2/r, and thepropagation velocity is /r. In terms of the physical independent variables, as t = t andx =x/U0, the solution is

v= exp

iU0

x

exp

i rU0

xU0

rt

, (31)

so that the physical propagation velocity is U0/r, for which we will use the symbol c. To plot theresults for dimensionless propagation velocity, the widely-used dimensionless velocity c/

pgA0/B0 is

suggested, which is related to thercalculated from equation (29) by

cpgA0/B0

= U0

rp

gA0/B0=

F0r

, (32)

and asrcalculated from equation (29) is a function of Froude numberF0already, as well as a functionof frequency , this does not complicate the representation. For plotting the spatial decay rate, it seems tobe too complicating to use anything other than the dimensionlessi, the decay rate in the dimensionlessspace variablex.

5.3 Approximate solutions in limits of not-so-long and very-long waves

Unfortunately, before presenting the results, some more mathematical manipulations are necessary, to

relate the present work to known limiting behaviour and to explain some of the behaviour to be shown

graphically.

Above we have obtained solutions in terms of dimensionless frequency , whereas dimensionless periodseems to have more significance. Consider the time variation in equation (27):t = t, which variesby2while physical time varies by a period T, so that

= 2

T, (33)

andTis the dimensionless period. We will consider two limits forT.

5.3.1 Not-so-long waves,Tsmall

Thefirst limit is where the dimensionless wave period is actually relatively small. The theory we have

been using, based on an approximation that the waves are much longer than the depth, is correctly

called a long wave theory, such that the pressure distribution is hydrostatic, however in the case we are

considering the waves are short enough that Tis small, and we call this the "not-so-long" wave limit.

In this case whereT

is relatively small so that

is large, the solution (29) can be written as an asymp-

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totic series, valid for large. The solution forbecomes

=F0(F0+ C0)

1 F20+i

F0F0C0 +

1 + F20 ( 1)

2

1 F20

C0

+ O

1

, (34)

whereC0 =C0/p

gA0/B0=q

1 +

2 F20 . The solution shows the first two terms of the seriesthe first is of order1, which is real, and the next0 which is purely imaginary. The next neglectedterm isO

1

which is also real. From equation (32) these real terms give the physical velocity of

propagation, which after several steps of simple mathematics, including rationalising surds,

c=U0 C0+O

2

, (35)

where C0 is the dynamic wave speed, given by equation (20). This is the result, of course, widely-believed to be the case for all long waves, where the waves are carried downstream at an advective

velocity (curiously, U0and not just U0), but propagate up- and down-stream relative to this at velocities

U0. The error term in equation (35) which we can write as O (T)2

shows how the result applies

only to not-so-long waves, and not to all long waves. We will see below how limited is the range ofwave periods for which this is a good approximation.

Having established that slightly unexpected result, other results (for example the next term in equation

34) are more complicated, and it is just as insightful here to take the common case = 1, forfluidvelocity constant over the depth, and = 3/2, for a wide channel, to give the result for the leadingimaginary term in equation (34), which as it is independent of and hence period, is a decay rate. It canbe shown to be

i=F0

2

12F01 + F0

+ O

2

, (36)

and so forF0small it is approximatelyi=F0/2, so that it is positive for downstream propagation so

that the wave decays downstream, and decays upstream for propagation in that direction.

5.3.2 Very-long waves,T large

The other limit is where the dimensionless wave period is relatively large, which we will call the "very-

long" wave limit, where T is large, small. The series expansion of the solution (29) in terms of:

= iF20 /2

1 F20( 1)+

+

F20 ( 1)1 F20

+i

2

F20 3

1 F20

2 (2 1)

+O

3

. (37)

Taking the real part, equation (32) gives for the velocity of propagation

c = U0+ O

2 for downstream propagation,= 1

(38a)

c = U0

1 + 2F20 /

1 F20+ O 2 for upstream propagation,= 1 (38b)

thus in the very long wave limit waves that travel downstream do so at a velocity c=U0= c0, where c0is the kinematic wave speed, while those that travel upstream do so at a similar, slightly smaller, speed.

For upstream propagationc has a value of approximately c0, where the value is actually modified bya function ofF20 . There is no apparent advection component from the flow in either direction, unlike"not-so-long" waves. It is interesting that this theory has shown that the condition that waves travel at

the kinematic wave speed is that the waves be very long, whereas it has traditionally been believed that

the requirement is that the Froude number be small.

Considering the decay rate of the waves, for downstream propagation = +1, the imaginary part of

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equation (37) has a zero first part, leaving

+i = 2

23F20

1 F20

2 (2 1)

+ O

4

, (39)

which is of a diffusion-like nature, that the decay rate in space varies like the square of the frequency in

time, so that shorter waves decay faster, much faster, while in the limit of sufficiently long waves there

is no diminution.

For upstream propagation

i = F201 F20

+i , (40)

and we see that the decay rate consists of a frequency-independent term plus the negative of the diffusion

term for downstream waves. These waves decay as xbecomes increasingly negative, as we might expect.

5.4 Presentation of results

To estimate over what range we should obtain and present solutions we consider equation (22) for ,which gives

T 2gS0U0

T (41)

and takeg 10ms2, and a representative value ofU0 = 1 ms1. To estimate the larger magnitudesofT, we consider a slow-rising flood such that the period might beT = 4days on a stream of steepergradientS0= 103, which gives

T 2 4 24 3600 10 103

1 6900.

For the smallest likely value, consider the fast movement of a gate T= 20min(time to open10 min)on an irrigation canal of gentle slopeS0= 104, giving

T 2 20 6010 104

1 = 2.4.

In view of these values we will plot results over a range of dimensionless wave periods 1 6 T6 104.

Figure 1 shows the results for wide channels with = 3/2, and for= 1as a function of dimensionlessperiodT with Froude number as parameter. For the moment we are concerned only with the solidblack lines;the coloured dashed lines are an approximation that will be considered below.

The upper part (a) of the figure shows the dimensionless propagation velocity c/pgA0/B0as computedfrom equations (29) and (32). The set of curves for positive/negativec are for downstream/upstreampropagation. It has been obvious since the work of Lighthill & Whitham (1955) and more recently

Ponce & Simons (1977) that propagation velocity is in general a function of wave length. Here, we have

put it in the context of wave period rather than length; the figure shows that waves of different periods

travel at different velocities. In mathematical terms this means that the physical system of long waves

with resistance actually shows the phenomenon of dispersion. It is clear and noteworthy that shorter

waves travel faster than longer waves, whether going upstream or downstream. There is no such thing

as a unique propagation speed. The results are more complicated than is widely believed.

For not-so-long waves,T < 2, say, at the left of the figure, corresponding to rapid changes such asdue to gate movements, the curves are almost horizontal and as shown by equation (35), c U0 C0is a good approximation to the wave velocity, for both downstream and upstream propagation. For

T >2, say, and certainly forT >10, U0 C0 is no longer a good approximation to the speed ofpropagation, contradicting the widely-held belief in hydraulic engineering. For2 < T < 100or200the propagation velocity continuously depends on wave period. ForT > 100or200, the curves are

12


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-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Velocitycp

gA0/B0

F0 = 0.1

0.20.40.6

0.8

F0 = 0.1

F0 = 0.3

c U0+ C0

c U0 C0

c U0

c eqn (38b)

Downstream propagation

Upstream propagation

(a) Propagation velocity

-0.3

-0.2

-0.1

0.0

0.1

1 10 20 50 100 1000 10000

Decayrate

i

Dimensionless period T

(b) Spatial decay rate

F0 = 0.1

F0 = 0.2

F0 = 0.3

F0 = 0.1

F0 = 0.8

Downstream propagation


i eqn (36), = +1

i eqn (36), = 1

i eqn (39)

i eqn (40)

Floodwaves

Fast gatemovements

Linear theory, eqn (29)

Figure 1. Dependence of propagation properties on wave period and Froude number

again almost horizontal, but this time with a propagation velocity corresponding to equation (??) withc

c0= U0for downstream propagation, more honoured in the hydrology literature, and with a similar

magnitude for upstream propagation. The conventional view is that this is a low-inertia or small Froude

number approximation. Figure 1 shows graphically that it is actually a slow variation approximation for

sufficiently long period waves, as shown by the neglected terms O

2

= O

(T)2

in equation (??)

and apparently valid for any (sub-critical) Froude number as predicted by the theory.

Figure 1(b) shows the spatial decay rate of disturbances. Again it can be seen that there is a dependence

on wave period, in this case rather more marked than for wave speed. In mathematical terms, this

dependence of decay rate on period is known as diffusion. Classically that refers to decay in time of

disturbances in space, here we use the same term, to describe the damping in space of disturbances in

time. The figure shows that in all cases, just as in classical diffusion, shorter disturbances decay more

quickly. For downstream propagation, shown by positive decay rates, for sufficiently long waves the

decay rate goes to zero which means that very long waves suffer little decay. The behaviour is like 2so that if we write equation (39) asi =

2whereis a diffusion-like coefficient, giving the spatial

13


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decay rate in terms of the square of the frequency in time, for downstream propagation

= 1

3

F20 +

(2 1) 2

+ O

2

,

and in the common approximations = 1and for wide channels = 3/2,= 8/F20 2 /27.For upstream propagation the behaviour is very different, and we find that

lim0

i=

F20 2

such that there is a contribution independent of frequency plus a negative diffusion term, the same as the

downstream one in magnitude, but meaning that waves decay as they propagate in the x direction. Themost noteworthy feature of this is the leading term independent of frequency, very much greater than for

downstream-travelling waves, so that the upstream-waves decay much more quickly.

5.5 Stability offlowroll waves

The stability limit offlow in the channel can be determined by writing the solution (29) in standard

complex form and satisfying the condition for stability, thati = = () > 0. After a number of longalgebraic manipulations, the limiting condition leads to the limit in terms of Froude number

F06 1q

2 (2 1)(42)

for stability. ForF0 above this limit the flow becomes unstable and roll waves develop. This result canbe obtained quickly from equation (39) for very long waves, for the decay coefficient+i to be positive.It can also be obtained from the imaginary term of equation (34) for not-so-long waves, after some

manipulations, or, for the case of a wide channel = 3/2and with the traditional= 1,the condition isobtained from equation (42) (or immediately from approximation 36 for not-so-long waves) that F0 6 2for stability, the result obtained by Jeffreys (1925). If we were to use Gauckler-Manning resistance we

wouldfind = 5/3for a wide channel, and using this with the more realistic value of= 1.05givesF0 6 1.75for stability. For = 3/2and= 1.05, the limiting value isF0 6 2.58. It can be seen thatthis stability limit is quite sensitive to the values ofand, unlike most of the results in this work.

6. The momentum equation relative importance of terms and

some approximations

Having established the general nature of wave propagation now we examine the relative importance

of terms in the momentum equation, using the dimensionless Telegraphers equation (26), so that weknow when approximations can be made. Consider the various cells of the second row of table 1 with

individual terms taken from the full momentum equation (1b), where the resistance and slope terms are

best considered as a single term shown bracketted. In the third row are shown the corresponding terms in

the dimensionless Telegraphers equation, where the resistance and slope terms are now separated into a

time and a space derivative. In the fourth row are shown the terms in polynomial (28) after substitution

of the periodic solution. We could substitute solutionsfrom equation (29), but it is more insightful toconsider the approximations of small and large.

Not-so-long waves, Tsmall, large: we have the asymptotic solution (34). Results for the relativeimportance of terms are shown in the fifth row of table 1. The first term corresponding toQ/tin the

original equation is the same for all linear solutions here, as it contains a contribution i2

and nonefrom aterm. Examining the approximations in the subsequent terms along the row it can be seen thatthis and the next two terms in the momentum equation are of order2, whereas the resistance and slopeterms are of order. In this limit of large,this would suggest an approximate equation, in which the

14


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Term number 1 2 3 4 5

Momentum eqn (1b) Q

t 2

Q

A

Q

x

gA

B

Q2

A2

A

x

P

Q2

A2 gAS

Telegraphers eqn (26)

2

vt22

2

vxt (1/F20

)

2

vx2vt

vx

Polynomial (28) i2 i2 i2

F20

Substituting the solution() from equation (29) results shown just for limiting cases from (34) & (37):

Not-so-long waves, large i2 O

2

O

2

O ()

Very-long waves, small

Downstream i2 O

2

i

1 F20

2F2

0

2 + O

3

+ O

2

Upstream i2 2F20

1 F20

+ O 2

i 2F201 F2

0

+ O () i 2F201 F2

0

+ O ()

Table 1. Terms in momentum equationand their relative magnitudes in the limits of not-so-long waves and very

long waves

latter terms could be omitted. However, this dropping of the relatively simple resistance and slope terms,

and retaining the terms including the time derivative and two space derivatives does not give much of a

simplification, and we will not consider this limit further.

Very long waves,T large, small: substituting the solution (37) the values are given in the lasttwo rows of the table, for downstream and upstream propagation respectively, as they are different:

Downstream propagation: in this case thefirst two terms are of magnitude2, which we shall sub-sequently ignore in this limit of small . The third term is also of this order, but in the limit of smallFroude number the term varies like i2/

2F20

, which is not necessarily small relative to the remain-

ing two terms of magnitude in columns four and five. This means that if we retain all of term three aswe arenotmaking a low-inertia (small Froude number) approximation here, plus terms four and five we

would approximate the momentum equation (1b) bygA

B Q

2

A2

A

x+ P

Q2

A2 gAS= 0. (43)

To examine the wave propagation properties of the approximation compared with the full momentum

equation, we consider the corresponding terms in the second row of table 1, excluding terms 1 and 2, to

give the quadratic equation in

i

F20

2 + = 0, (44)

with solution

=i/2 + i

q2/4 i

F20

F20 . (45)

Figure 2 is a repeat of part offigure??, but for slightly longer waves,T> 10and with smaller verticalrange, plus results from the approximation (43) in the form of the linear solution (45) shown by blue

dashed lines. Here, for the moment, we confine our attention to downstream-propagating waves, the

upper part of each of the figures 2(a) and (b). The result is immediately clear that equation (43) is

indeed a slow-change approximation, and not a low-inertia approximation, because agreement of both

propagation velocity and decay rate with the full linear theory is very close for all waves longer than

T 50, and shows very little dependence on Froude number at all.Now we introduce a further approximation, where we neglect the inertial term in the brackets in

15


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-0.5

0.0

0.5

1.0

1.5

Velocitycp

gA0/B0 F0 = 0.1

0.2

0.4

0.6

0.8

F0 = 0.1

F0 = 0.3

c U0

c eqn (38b)Downstream propagation


(a) Propagation velocity

-0.1

-0.1

-0.0

0.0

0.1

10 20 50 100 1000 10000

Decayrate

i

Dimensionless period T

(b) Spatial decay rate

F0 = 0.1F0 = 0.8

F0 = 0.1

F0 = 0.2

F0 = 0.3

Downstream propagationUpstream propagation

Linear theory, eqn (29)Approxmn: eqn (45) based on (43)Approxmn: eqn (48) based on (46)

Figure 2. Dependence of propagation properties on wave period and Froude number

equation (43), of relative magnitude F2, believing that now for the first time we are making a low-inertiaapproximation, to give

gA

B

A

x + P

Q2

A2 gAS= 0. (46)

This is a well-known approximation (*************), however, it is widely believed that the approxi-

mation just consists of ignoring terms of orderF2, as suggested by the traditional scaling of the dimen-sionless equation (16), as also done by the author previously (Fenton & Keller 2001, Appendix B.2).

To examine the wave propagation properties of this approximation we ignore the terms in equation(44) to give the polynomial

i2F20

+ = 0, (47)

with solution, similarly simply obtained from equation (45)

= iF20

/2 +

q2/4 i/F20

. (48)

16


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This result is plotted as red dashed lines on figure 2. Examination of the top halves offigures (a) and

(b) shows the very surprising result that, if anything, agreement with the full linear solution is even

better, especially for velocity of propagation, and results are even better for large Froude number. For

these downstream travelling waves (propagation velocity and decay rate both positive) the accuracy of

the approximation does depend only on the period, such that it is accurate forT & 20and does not

deteriorate with increasing Froude number. In this case, it really does seem to be a "slow-change" ap-proximation. For decay rate, there is some loss of accuracy with increasing Froude number, nevertheless

it is clear that the approximation is still very much a very-long-wave approximation, that forT& 50is an excellent approximation, for all (sub-critical) Froude numbers considered. Even though formally

equation (46) was obtained from the the very-long-wave approximation (43) by dropping a term of order

F2, it shows almost no loss of accuracy with increasing Froude number.

Hence, for downstream-propagating waves the simplified momentum equation (46) in effect is just a

long period orslow-change approximation.

Upstream propagation: A different situation arises when we consider upstream propagating waves,

possibly caused by control changes such as gate operations or tides. . Considering the last row of table 1

it can be seen that thefi

rst, unsteady, termQ/tstill has a relatively small contribution ofi2

in thislong wave limit;the next term involvingQ/xhas a contributionO

F20;and in the last three cells

that each has a contribution of orderO ()and in the third and fifth there are leading order terms in F20that cancel. The result is that the second term, neglected in the above approximations, has a magnitude

relative to the last three terms ofO

F20

, which is as was previously believed to be the order of accuracy

of the approximation (46). In this case the approximate equations (43) and (46) are both aslow change

and aslow-flow approximation.

Considering the results of the linear solutions (45) and (48) for propagation velocity plotted in the

lower half offigure 2(a), it can be seen the results are now dependent on Froude number, even for the

longest waves, and already forF0 = 0.3are not accurate, while for all results there is little differencebetween the two simplified equations (43) and (46). For decay rate, shown in the lower half offigure

2(b), the nominally more accurate equation (43) agrees closely with the full linear solution for all Froudenumber, while the nominally low-inertia approximation (46) really is a low-inertia approximation for

upstream propagating waves. It is accurate forF0 = 0.2 but again forF0 = 0.3 it is no longer. Inany case, it can be seen that the decay rates for upstream travelling waves are rather greater than for

downstream propagation.

Combining the results from downstream and upstream propagation, we can conclude that the use of

the nominally low-inertia approximation (46) is preferable to the in-principal more accurate equation

(43). It seems generally to mimic the effects of time and space derivatives in the full equation that

both approximations have ignored. For downstream propagating waves the approximate equation (46) is

only that the change of boundary conditions (flood inflowetc.) be slow, while for upstream propagating

waves both change should be slow and flow should be slow such that F2 is small. In view of these

two slightly-different results for downstream and upstream propagation we will refer to equation (46) asthe slow-change/slow-flow momentum equation, where the terminology is an abbreviation for the rather

longer term "slow-change and possibly slow-flow momentum equation".

We can estimate typical values of wave period for which the simplified momentum equation should be

accurate. For example, in a channel of slope104 and velocity0.5 ms1, from equation (41)T = 20corresponds, to a physical periodT = 20 U0/2gS0 20 0.5/(2 10104) = 5000s, or the time-to-rise of a flood half of that, roughly40 min. If the slope were103, the corresponding limitation (forthe sameU0) would be about4 min. For any changes slower than those the simplified theory should beaccurate. Of course, that limit can easily be calculated in any application.

As we obtained equation (47) with an approximation neglecting 2 and higher order terms, it is ap-

propriate to write the solution (48) as a power series, nominally in , but equation (48) shows that theexpression is a function of/F20 so that this is the effective expansion parameter. The leading terms

17


18/26


are

=

+ O

3/F40

+i

F20

2 ( 1) +

3F202 + O

4/F60

. (49)

From the real part and equation (32), we have to lowest order, c = U0, as in equation (38a) from

the full linearised solution for downstream propagating waves, and also as an approximation to equa-tion (38b) forF20 small. The notable and surprising result here, although obvious on figure 2, is thatusing the slow-change/slow-flow momentum equation (46), and the algebraic equivalent equation (47)

+i2F20 = 0, gives the expected result to lowest order of = for wave propagationdownstream at physical velocity+U0but also, not at all obvious from the equation, for upstream prop-agation, a velocity of U0 as an approximation, so that very-long waves travel at the same speed inboth directions.

From the imaginary part of equation (49) similar results for the decay coefficient are obtained, which

are smallF20 approximations to equations (39) and (40).

7. The slow-change/slow-fl

ow routing equationWe now examine the use of the slow-change/slow-flow momentum equation (46) and develop a simple

single equation which can be used for simulation purposes for longer waves, say forT& 20.

If the resistance is not a function ofQthen equation (46) can be solved forQ

Q=

rgA3

P

rS 1

B

A

x =K

rS 1

B

A

x, (50)

where the conveyanceKhas been introduced for convenience, from equation (5). From here we willnot necessarily limit ourselves to Chzy-Weisbach friction, so that forKone could use the formulationused above, where

8, or the Chzy form or the Gauckler-Manning formula:

K=

rg

A3

P =C

rA3

P =

1

n

A5/3

P2/3.

To use equation (50) to give a single equation in a single unknown, there are two ways that we can

proceed.

Routing equations in terms ofA(x, t) or (x, t): The first is to substitute Q from equation (50)into the mass conservation equation (1a) to give the equation

A

t

+

xKrS

1

B

A

x!=i. (51)

Routing problems could be solved by solving forA down the channel and if necessary using equation(50) at any stage to calculate the dischargeQ. However the free surface elevation has more directsignificance. Equations (14) and (16) of Fenton (2010b) are

A

t =B

t and

1

B

A

x =

x+ S,

provided the banks of the stream are not both vertical and converging/diverging. Equation (51) becomes

the even simpler equation in terms of surface elevation :

t+

1

B

xKr

x!= i

B. (52)

In many problems, however, the discharge is specified as a boundary condition, which does not naturally

lead to a boundary condition onAor as we have considered here.

18


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The slow-change/slow-flow routing equation in terms of upstream volume V(x, t): A moregeneral formulation is obtained if we use the upstream volume identity (11) forQ: Q = Q(x0, t) +Rxx0

i(x0) dx0 V/tused directly in equation (50) to give the partial differential equation in terms ofV:

Vt

+ K(Vx)s

S 1B(Vx)

2

Vx2

=Q(x0, t) +Z xx0

i(x0) dx0, (53)

where we have shown that both KandB are functions ofVx = V /x, as they are functions offlowarea. In practice, this equation would be solved by finite differences, stepping forward in time.

The author has used such an equation in several studies (, Fenton et al. 1999, Fenton & Keller 2001, Bar-

low et al. 2006) where its simplicity and/or its relative stability for zero or low flow was an advantage.

Here, the inclusion of the inflow at the upstream boundaryQ(x0, t)in the definition ofV /t(equation11) is more consistent because it now includes inflow both at the upper end and distributed inflow in the

same term, such that V really is the volume in the channel between the upstream boundary and the pointunder consideration. This modification has been suggested by Fatemeh Soroush (2011, Personal Com-

munication). Previously the present author called the resulting equation the Volume Routing Equation.Fatemeh Soroush also suggested that that is an ambiguous term in irrigation engineering. In view of that

and the discovery above of the real nature of the approximation to the momentum equation (46) here we

propose the name slow-change/slow-flow routing equationfor equation (53).

Almost any common boundary condition can be used with the equation, as (equation 11) Q can beexpressed as a function ofV/t and the surface elevation ,viacross-sectional area A can be expressedas a function ofV/x, which variables are often connected at control structures:

Upstream boundary: It is usually the upstream boundary condition that drives the whole model,

where a flood or wave enters, via the specification of the time variation ofQ = Q (x0, t)at the bound-ary. However, the boundary condition on V is exceptionally simple as there is no flow upstream ofthe most upstream point, we have simply the boundary condition V(x0, t) = 0. The actual inflow hy-

drograph appears instead in the partial differential equation (53). The surface elevation at the upstreamboundary is obtained as part of the computations, as if we know the variation ofV withx(probably in-terpolated from point values) at any time level, then from equation (11) we have A = V/x, and thesurface height comes from knowledge of the cross-section.

Downstream boundary: If a control exists: if there is a structure downstream that restricts or con-

trols the flow with a flow formula such as Qout = f(out), then as out is related to area Aout =V/x|outfrom the cross-section, and Qoutis related toV/t|out, from equation (11), the dischargeequation for most boundary structures gives an equation forV/t|outin terms ofV /x|out, whichcan be evaluated as part of a numerical solution and the value ofVupdated there, in a similar senseto equation (53) giving a value ofV/tat interior points which would be similarly used to give Vat the next time level.

Open boundary condition: often the computational domain might be truncated without the presence

of a control structure on the stream. One solution is to use a uniform flow boundary condition there,

just as if it were a control. The author prefers a different approach, and this is simply to treat the

boundary as if it were just any other part of the river (which it is!) and to use the slow-change/slow-

flow routing equation to updateVthere, calculating the necessaryx-derivatives from upstream finitedifference formulae. I have found that it works very well in practice, but a senior hydraulic engineer

was horrified when he heard this, believing it to violate the physics of the problem. To me it is a

sensible step. One still has to truncate somewhere. If one has truncated the computational domain,

one has already abandoned any idea of information coming from downstream anyway. So, if all

information is coming from upstream, we just use that and the solution at the exit point from the

equations, using approximations to derivatives from the conditions immediately upstream. To me,

it is more sensible than applying the wrong boundary condition such as a uniform flow boundarycondition at the downstream end.

19


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8. Numerical solution of the long wave equations an explicit

finite difference scheme

A Forward-Time-Centred-Space scheme is probably the simplest possible finite difference scheme for

numerical solution of the equations. Liggett & Cunge (1975) suggested that such a scheme was uncon-ditionally unstable and named it as such. The author believes that the deductions from their analysis are

wrong. The scheme has a quite acceptable stability limitation, and it opens up the possibility of a much

simpler method for computations offloods and flows in open channels than the implicit methods often

used.

8.1 A Forward-Time-Central-Space scheme

I think I should put Manning in here

Consider the forward-timefinite difference approximation to equations (3):

(x, t +) = (x, t) +

iB

1B

Qx

, (54a)

Q (x, t +) = Q (x, t) +

Q2B

A2S PQ |Q|

A2 2Q

A

Q

x

gA Q2B

A2

x

,(54b)

where all quantities on the right are evaluated at (x, t), and we use the central difference expressions forthe derivatives:

x

(x,t)

= (x + , t) (x , t)

2 (55a)

Qx(x,t)

= Q (x + , t) Q (x , t)

2

. (55b)

This combination of equations (54) for stepping forward in time using the central difference expressions

(55) for space derivatives, gives the scheme and its name.

8.2 Linear stability

Now we examine the stability properties of the scheme. We consider, in the same way as in 4, the case

of no inflowi= 0, and small perturbations about a uniformflow such that we write

= h0 S0x + h1 and Q=Q0+ Q1,

In fact, the linearisation would be the same as that done previously so that the linear equations in termsofh1 andQ1 are obtained more simply by substituting A1 = B0h1 into the linearised equations (19)plus from equation (22) andc0from equation (23), and we replace derivatives by the finite differenceapproximations to give the linearised finite difference equations

B0h1(x, t +) h1(x, t)

+

Q1(x + , t) Q1(x , t)2

= 0, (56a)

Q1(x, t +) Q1(x, t)

+

C20 2U20

B0h1(x + , t) h1(x , t)

2

+2U0Q1(x + , t) Q1(x , t)

2 + (Q1 c0B0h1) = 0. (56b)

We look for solutions of the form h1(x, t)Q1(x, t)

=

h10Q10

exp ikx exp t,

20


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where h10 andQ10 are coefficients and i=

1. This has assumed a periodic solution inx,expikx,possibly as part of a Fourier series, as computations are performed on a finite length of channel. The as-

yet unknown behaviour in t is written asexp t, where the nature of determines the solution behaviourin time. Substituting into the equations (56a) and (56b), introducing the notation

K= sin(k)

and M=exp(

) 1

, (57)

and dividing through by common factors gives the matrix equation M iK

i

C20 2U20

K c0 M+ + 2iU0K

B0h10

Q10

=

00

.

For non-trivial solutions to exist, the determinant of the2 2matrix must be zero, giving the quadraticinM:

M2 + M(i2U0K+ ) +iKc0+ K2

C20 2U20

= 0,

and solving forMand using equation (57) we have the solution for the multiplication factor at each time

step:

exp() = 1 /2 iK U0 q

(/2)2 K2C20 iK (c0 U0), (58)

where= 1for downstream/upstream propagation. It is convenient to introduce the modified Froude

numberF00 =U0/C0, where from equation (20),C0 =q

gA0/B0+

2

U20 ; to use the relation-

ship (25)c0= U0;and to expressKusing the dimensionless-frequency-like term

=2KC0

=

2C0

sin(k)

, (59)

so that the solution (58) becomes

exp() = 1

2

1 +iF00 +

q1 2 i2F00( )

. (60)

We write the square root term in standard complex form asq1 2 2iF00( ) = +i,

where = 12

p + 1 2 and = 1

2

p 1 +2,

inwhich =

q(1 2)2 + 42F020 ( )

2,

so that the solution is written

exp() = 1

2

1 +iF0

0 + (+i)

.

The condition for stability of the numerical solutions is that the magnitude of the factorexp() bywhich the solution changes at each time step must be less than or equal to 1, so that |exp()|2 6 1.Multiplying the solution forexp ()by its complex conjugate then gives the condition for stability ofthe FTCS scheme:

6 4 1 +

(1 + )2 + (F00+ )2 , (61)

giving a limiting value of. As and are functions of, the dimensionless frequency parameterdefined in equation (59), and F00( ) this condition (61) means that the stability criterion is a functionof, ,F00 andF

0

0. Asand are constant for practical purposes it is a function of, andF0

0. In

any particular problem we have to examine both values of= 1and all the values of from equation(59) over the range of values ofk = 2m/L, wherem can take any value between1 and N/2forNcomputational points.

21


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8.3 Approximate stability criteria

We obtain an approximation to the stability criterion (61) by writing it as a series in terms of Froude

number. It is necessary to consider two alternatives, depending on whether 1. The results are:

6 4

1 +

1 2 + O

F0

20

for 6 1, (62a)

6 4

2

1 + O

F00

for > 1. (62b)

It can be seen that neglected terms in the first are proportional to the square of the Froude number, and

in the second, the Froude number itself. Usually we do not need a particularly accurate result for the

limiting step size, so these should be adequate for practical purposes. It is interesting that the inverse

time scale has again appeared, and that the criteria have been able to be expressed simply in terms ofit.

The most limiting case for the first condition (69a) is when = +1. For we make the furtherapproximations that = 1 plus the wide-channel approximation such that A0/B0 = A0/P0 = d0,where d0 is the depth offlow. In equation (59) in this caseC0 =gd0 and from equation (22) =2p

gS00/d0, and we use, fork small, sin(k) / k = 2m/L, which in this case, the smallestvalue possible is whenm= 1, giving

min=2C0

sin(k)

= 2

d0/LS00

From equation (8) we have for a wide channel

0 S0F20

which gives

min= 2F0d0/L

S0

and is a minimum, min, for whichm= 1

Formaxwe takesin (k) = 1. This gives

max= d0/

S00=N

d0/LS00

=N F0d0/L

S0

min = 2 h0/LS00

, and (63a)

max = h0/

S00=N

h0/LS00

, (63b)

where we assume thatN >2 6.

In the limiting case of zero slope, S0 0 the scheme is unstable, as max and equation (73)shows that the time step for stability 0.

8.4 Test of linear stability

Now we test the stability theory numerically. To do this we considered a wide model stream of rectan-gular section. Resistance coefficients and slopes were taken from the data given in section ??, generallylimited by the region shown dashed in figure **** ?? ****. Four values of resistance were used:[0.003, 0.01, 0.03, 0.1], and six Froude numbers, [0.05, 0.1, 0.2, 0.4, 0.6, 0.8]. In each case the slope

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0.003

0.01

0.03

0.1

105

104

103

102

1

10

100300

(sec.)

From computationsLinear theory, eqn (61)Approximations, eqns (72 & 73)

Slope

(sec.)

Figure 3. Limiting time step for stability, as determined theoretically and by performing computations

was determined fromS= F2. It was thought not unreasonable to take depth of1 mfor a depth scale,a large width of100 m, a computational lengthL= 10km, andN= 20such that= 500 m.

To test the time step in actual calculations, the initial normal flow was calculated, and then a flood with a

peak at24 hwas introduced, but with a peak increase of only 1%of the normal flow to test the stability.A series of calculations using the FTCS scheme were performed, varying the time step until the scheme

was just stable. Results are shown in figure 5. It can be seen that over the region of most streams as

given in figure??, the computations were stable with time steps between 10 sand180 s. In the limit ofvery rough and steep streams with Froude numbers of up to 0.8, the limiting step went down to4 s. Ofcourse, the implicit Preissmann Box scheme would allow much larger time steps.

The results from the linear theory, equation (61) with a scan of both values and all values ofmbetween1 and N/4 were found to be highly accurate. Also, the approximate theory presented here, equations(72 & 73), gave results good enough for practical purposes. For large Froude numbers the results could

be in error by a factor of 2, but as an estimate only is required, that seems not to be a problem.

8.5 Determining stability limits for unsteady flow problemsIn all the above we have considered only small variations about a uniform flow. Now we proceed to a

method by which we might estimate what is the time limitation for an unsteady flow computation. We

will base this on the assumption that the above theory holds even for the unsteady case.

If we consider the stability criterion (69a) for 6 1 and take the most stringent limitation, . 2.Substituting equation (22) for and using the wide-channel approximation

.s h0

gS00, (64)

and the minimum time step is dictated by the minimum depth. To express this in terms of discharge

we use the wide-channel approximation, from the Chzy-Weisbach expression (7), and introduce q0 =

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U0h0, the discharge per unit width, to give

h0=

q200gS0

1/3(65)

from which we deduce

6

q0

(gS0)20

1/3, (66)

determined by the minimum flow.

Now we consider the other condition (73) for > 1. As we are considering approximations only, weneglect the term in Froude number to give the condition . 4/2. Substituting equation (22) for,using = maxfrom equation (71b) with the wide-channel approximation gives

. 2

sh0S00

g

h0

2. (67)

In this case the most demanding criterion is when h0is a maximum. Using equation (66) to express it interms ofq0the criterion becomes

. 2S0

2

q0. (68)

It is interesting that resistance has disappeared completely.

The procedure we suggest tentatively to determine the time step such that computations are stable is

then:

1. Withq0the minimum flow per unit width expected, use equation (69) to estimate h0then calculatemin from equation (71a). Ifmin 6 1 then calculate the limiting value of from equations (64)

or (66).2. Withq0the maximumflow per unit width expected, use equation (69) to estimate h0then calculate

max using equation (71b). Ifmax 6 1 the previous criterion applies, else ifmax > 1 thencalculate the limiting value of from equations (67) or (68).

3. The minimum of the two values determines the real limiting value of.

As an example, we consider the100 mwide channel used above, but now with conditions more likely ina practical problem: a longer stream, withL= 100 km,N= 100, and aflow that rises from100 m3s1

to a peak of1000 m3s1 after 24 hours and then diminishes again. We computed for96 h. A number ofdifferent cases of resistance and slope were considered. In each case the approximate limit to stability

was found by trial and error, and compared with the results from the procedure suggested immediatelyabove. Results are shown in figure 4. It can be seen that over the range of most natural rivers and

canals, the estimates of the approximate formulae here are quite sharp. For streams on a small slope,

they underestimated the stability limits, but the order of magnitude was correct. Computational times

for each simulation varied between less than a second to several seconds.

That is not a conclusive proof of the applicability of the limits in real streams and floods, of course,

but provides a guide to the estimation procedure that might be followedand to the sorts of time steps

expected: very small in the case of streams on small slopes, but finite for the majority of streams.

While the time steps obtained can sometimes be small, a typical run time on a personal computer was

only second(s). Such small time steps do not seem to be a problem in practice, and are a small price to

pay for being able to use such a simple scheme.

9. Characteristics

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0.003

0.01

0.03

0.1

105104

103102

1

10

100

(sec.)

From computationsApproximations, eqns ()

Slope

(sec.)

Figure 4.

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05 long waves in open channels

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