Open channel hydraulics for engineers. Chapter 7 unsteady flow
This chapter introduces issues concerning unsteady flow, i.e. flow situations in which hydraulic conditions change with time. Many flow phenomena of great importance to the engineer are unsteady in character, and cannot be reduced to steady flow by changing the
viewpoint of the observer. The equations of motion are formulated and the method of characteristics is introduced as main part of this chapter. The concept of positive and negative waves and formation of surges are described. Finally, some solutions to unsteady flow equations are introduced in their mathematical concepts.......
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter UNSTEADY FLOW
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7.1. Introduction
7.2. The equations of motion
7.3. Solutions to the unsteady-flow equations
7.4. Positive and negative waves; Surge formation
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Summary
This chapter introduces issues concerning unsteady flow, i.e. flow situations in which
hydraulic conditions change with time. Many flow phenomena of great importance to the
engineer are unsteady in character, and cannot be reduced to steady flow by changing the
viewpoint of the observer. The equations of motion are formulated and the method of
characteristics is introduced as main part of this chapter. The concept of positive and
negative waves and formation of surges are described. Finally, some solutions to unsteady
flow equations are introduced in their mathematical concepts.
Key words
Unsteady flow; method of characteristics; positive and negative waves; surge; numerical
solution.
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7.1. INTRODUCTION
In unsteady flow in an open channel, velocities and depths change with time at any
fixed spatial position. Open channel flow in a natural channel almost always is unsteady,
although it often is analyzed in a quasi-steady state, e.g. for channel design or floodplain
mapping. Unsteady flow in open channels by nature is non-uniform as well as unsteady
because of the free surface. Mathematically, this means that the two dependent flow
variables (e.g. velocity and depth or discharge and depth) are functions of both distance
along the channel and time for one-dimensional applications. Problem formulation requires
two partial differential equations representing the continuity and the momentum principle
in the two unknown dependent variables. The full differential forms of the two governing
equations are called the Saint-Venant equations or the dynamic wave equations. Only
through rather severe simplifications of the governing equations analytical solutions are
available for unsteady flow. This situation has led to the extensive development of
appropriate numerical techniques for the solution of the governing equations.
A complete theory of unsteady flow is therefore required, and will be developed in this
chapter. The equations of motion are not soluble in the most general case, but we shall see
that explicit solutions are possible in certain cases which are physically very simple but are
real enough to be of engineering importance. For the less simple cases, approximations and
numerical methods can be developed which yield solutions of satisfactory accuracy.
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7.2. THE EQUATIONS OF MOTION
7.2.1. Derivation of Saint-Venant equations
Although the governing equations of conservation of mass and momentum can be
derived in a number of ways, we apply a control volume of small but finite length, x, that
is reduced to zero length in the limit to obtain the final differential equation. The
derivations make use of the following assumptions (Yevjevich, 1975; Chaudhry, 1993):
1. The shallow-water approximations apply, so that vertical accelerations are
negligible, resulting in a vertical pressure distribution that is hydrostatic; and the
depth, h, is small compared to the wavelength, so that the wave celerity c = (gh)½.
2. The channel bottom slope is small, so that cos2 in the hydrostatic pressure force
formulation is approximately unity, and sin tan = io, the channel bed slope,
where is the angle of the channel bed relative to the horizontal.
3. The channel bed is stable, so that the bed elevations do not change with time.
4. The flow can be presented as one-dimensional with a) a horizontal water surface
across any cross section such that transverse velocities are negligible, and b) an
average boundary shear stress that can be applied to the whole cross-section.
5. The frictional bed resistance is the same in unsteady flow as in steady flow, so that
the Manning or Chezy equations can be used to evaluate the mean boundary shear
stress.
Additional simplifying assumptions made subsequently may be true in only certain
instances. The momentum flux correction factor, , for example, will not be assumed to be
unity at first, because it can be significant in river overbank flows.
7.2.2. The equations of motion
We proceed to obtain equations describing unsteady open channel flow. The terms
used are defined in the usual way, and are illustrated in Fig.7.1.
V2
2g h
B
b
h h+h A
H
x P
z
Datum
Fig.7.1. Definition sketch for the equations of motion
Consider the channel section shown in Fig. 7.1; assuming that the slopes are small and the
pressure distribution hydrostatic, the pressure difference along any horizontal line drawn
longitudinally through the element has a magnitude of gh, where h is defined as the
amount by which the water surface rises from the upstream to the downstream face of the
element. The total horizontal hydrostatic thrust on the element, taken positive in the
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downstream direction, is therefore equal to - gbh, if h/h is small. The summation of
this force over the whole cross-section clearly gives the result - gAh, where A is the
cross-sectional area.
The acting shear force is equal to oPx, where P is the wetted perimeter of the section and
o the mean longitudinal shear stress acting over this perimeter. The two forces are not
quite parallel, but it is consistent with our assumption of small slopes to regard the two
forces as parallel. The net force in the direction of flow is therefore equal to:
- gAh - oPx (7-1)
We now consider the state of uniform flow, in which the channel slope and the cross
section, as well as the flow depth and the mean velocity, remain constant as we move
downstream. In this state there is no acceleration, and the net force on any element is zero.
Hence from Eq. (7-1):
ôo = ñgRio (7-2)
where R = A/P is termed the hydraulic mean radius and io is the bed slope. io = -dz/dx (in
the limit), which is equal to the water surface slope – dh/dx (in the limit) in the case of
uniform flow. Note that we define these slopes so as to get positive numbers when the
surface concerned is dropping in the downstream direction.
Consider now the more general case in which the flow is non-uniform; the velocity may
therefore be changing in the downstream direction. The force given by Eq. (7-1) is no
longer zero, since the flow is accelerating. We consider steady flow, in which the only
acceleration is convective, and equal to:
V
V
x
The force given by Eq. (7-1) applies to a mass Ax; therefore the equation of motion
becomes:
V
-ñgAh - ô o PÄx = ñAV x
x
dh V dV d V2
i.e. in the limit ôo = - ñgR - ñgR h +
dx g dx dx 2g
ô o = ñgRi f (7-3)
where if = - dH/dx, the slope of the total energy-head line, and may be termed the “energy-
head slope” or “friction slope”. We see therefore that for any state of steady flow the shear
stress o can be written as:
ô o =ñgRi (7-4)
dH
We know that, when the flow is steady, the gradient, , of the total energy-head line is
dx
V2
equal in magnitude and opposite in sign to the “friction slope” if 2 . Indeed this
CR
statement was taken as the definition of if; however, in the present context we have to
recognize the two independent definitions:
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ôo V2
if 2 (7-5)
ñgR C R
H V2
and z h (7-6)
x x 2g
introducing partial derivative operators, because the quantities involved may now vary
with time as well as with x.
To allow for variation with time, we need only to reproduce, with appropriate extensions,
the argument leading up to Eq. (7-3). The acceleration term VdV/dx in that argument must
now be replaced by the more general expression:
V V
ax V
dV
(7-7)
dt x t
V V
where ax is the fluid acceleration in the x direction of flow; is the local and V is
t t
the convective acceleration, respectively. The equation of motion therefore becomes:
V V
Ah o Px Ax V (7-8)
x t
(z h) V V 1 V
i.e. in the limit o R (7-9)
x g x g t
H 1 V
so that o R (7-10)
x g t
from Eq. (7-6). Substituting from Eq. (7-5), we now have:
H 1 V V 2
0 (7-11)
x g t C 2 R
and this equation may be rewritten
ie + ia + if = 0 (7-12)
naming the three terms of Eq. (7-11) the energy-head slope, the acceleration slope and the
friction slope, respectively.
A more radical restatement of Eq. (7-11) may be made by using Eq. (7-6), and recalling
that the bed slope io is equal to -z/x. We have, from Eq. (7-6):
H z h V V
x x x g x
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H h V V
io
x x g x
H 1 V
if (7-13)
x g t
from Eq. (7-11). Hence, Eq. (7-11) can be written:
h V V 1 V V 2
if i o (7-14)
x g x g t C 2 R
steady uniform flow
steady non-uniform flow
unsteady non-uniform flow
this equation being applicable as indicated. This arrangement shows clearly how non-
uniformity and unsteadiness introduce extra terms into the dynamic equation.
Like the steady-flow equations of which they are an extension, Eqs. (7-11) and (7-14) are
true only when the pressure distribution is hydrostatic, i.e., when the vertical components
of acceleration are negligible.
The equation of continuity for unsteady flow can be derived by considering a cross section
of the channel with a very short length x, as shown in Fig.7.2.
1 2
Q1 Q2
h +z
x
datum
Fig. 7.2. Definition sketch for the equation of continuity
In Section 1.1.3, Chapter 1, the equation of continuity is written in the form:
Q1 = Q2 = constant
But in this case, the discharges at the two ends are not necessarily the same, but will differ
by the amount:
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Q
Q 2 Q1 x
x
and this term gives the rate at which the volume within the region considered is decreasing.
The partial derivative is necessary, because Q may be changing with time as well as the
distance x along the channel.
Now if h + z is the height of the water surface above the datum plane, then the volume of
water between sections 1 and 2 is increasing at the rate:
h z
B x (Note that 0)
t t
where B is the water-surface width. The two terms derived must therefore be equal in
magnitude but opposite in sign, i.e.
Q h
B 0 (7-15)
x t
When the channel is rectangular in section, the substitution Q = Bq leads to:
q h
0 (7-16)
x t
Q (AV)
An alternative form of Eq. (7-15) may be written by expanding the term ,
x x
leading to:
V A h
A V B 0 (7-17)
x x t
the three terms of which are known as the prism-storage, wedge-storage, and rate-of-rise
term, respectively. The significance of this terminology will become apparent in the
treatment of flood routing problems.
7. 3. SOLUTIONS TO THE UNSTEADY-FLOW EQUATIONS
7.3.1. Characteristic differential equations
The treatment of the method of characteristics dates back from the nineteenth
century. A practical recent account is due to Stoker (1957). It has been further developed
by many other authors, most notably by Lai (1965), McLaughlin et al. (1966), Amein
(1967), Liggett (1967, 1968), Evangelisti (1969) and Strelkoff (1970). Following Courant
and Friedriechs (1954) and Lai (1965), one converts the two partial differential equations
of Saint-Venant into a set of four ordinary differential equations, which are called the
“characteristic differential equations”.
The unsteady flow equations of conservation of momentum, energy and mass were first
developed by Saint-Venant (1871). Keulegan (1942), Liggett (1967, 1975), Ktrelkoff
(1969) and Yen (1973), among others, made them -under several forms- suitable for the
solution of particular problems. General expressions for the continuity equation and the
momentum equation are introduced by Sergio Montes (1997) as:
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A Q
Continuity: ql (7-18)
t x
V V h
V g g io i f l (V v l )
q
Momentum: (7-19)
t x x A
where q l is the lateral discharge Ql per unit length. The lateral inflow Ql may be due to
seepage through the bed of the channel, precipitation over the free surface or, in the case of
a side-overflow spillway, due to the weir discharge into the channel. Similarly, vl is the
lateral inflow velocity involved in the lateral inflow Ql. In Eq. (7-19), the term
V V h
V g g i o if is another form of Eq. (7-14).
t x x
To do this, for the case of a prismatic channel, the continuity equation (7-18) is multiplied
by an unspecified coefficient and added to the momentum equation (7-19), having set the
lateral inflow velocity vl = 0:
V V h h A V h
V g g i o i f l V V l
q q
(7-20)
t x x A t B x x B
h A V h q l
Applying Eq. (7-20): V , where B is the channel width at the
t B x x B
free surface.
Eq. (7-20) results in:
V A V h g h q
V V l V g i o if l
q
t x A
(7-21)
B x t B
In order to give some physical meaning to this algebraic manipulation, it may be remarked
that the terms involving the differentials of V and h on the left hand side of Eq. (7-21) have
a form closely resembling a perfect differential with respect to time:
dV V dx V dh h dx h
and
dt t dt x dt t dt x
The terms within the square brackets on the left hand side of Eq. (7-21) could be reduced
dx
to perfect differentials, if the ratio could be identified with the quantities
dt
A g
V B and V+ . That is: V V
dx A g
dt B
This double identity allows one to give a more definite expression to , as this parameter
must now satisfy the condition:
2 g = g
B B
(7-22)
A A
gA
One recognizes that the term corresponds to the velocity of translation of an
B
infinitesimal wave on still water, c, so that Eq. (7-21) can be written as:
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V V B h h
V c c V c g io if l V c
q
(7-23)
t x A t x A
To reduce the number of dependent variables the depth h may be replaced by the “stage
variable” as a measure of water level in the channel (Escoffier, 1962), defined by:
dA c dh g dh
A h h
c B B
(7-24)
o
A o
A o
A
Following from its definition, has the dimension of a velocity. For any given cross-
section and c are only functions of the depth h and, according to Sergio Montes (1998),
their relation /c equals 2 for a rectangular section and /c = 4 for a triangular section,
with values comprised between these limits for trapezoidal and parabolic sections.
It is possible to replace the derivatives of the depth by similar expressions in terms of :
h A
. (7-25a)
t gB t
h A
and . (7-25b)
x gB x
The momentum equation is transformed into:
V V
V c V c g io if V c
ql
(7-26)
t x t x A
It is useful now to conceive of a fictitious observer moving in the x,t-plane with the
velocity V c, which is the absolute velocity in the infinitesimal wave. The slope of the
trajectory of the observer dx/dt equals the speed V c. As V and are functions of x and t,
the substantial derivatives (i.e. derivatives following the fluid motion) are:
DV V V dx D dx
and (7-27)
Dt t x dt Dt t x dt
It follows from Eqs. (7-26) and (7-27) that the observer will perceive the rates of change of
V and as:
V g(i o i f ) l V c
D q
(7-28)
Dt A
which together with the differential equation for the path of the observer:
Vc
dx
(7-29)
dt
represent the “characteristic system of equations” that replaces the partial differential
equations of the unsteady-flow motion.
The paths described by the observer and defined through Eq. (7-29) are called the
“characteristic directions” or simply the “characteristics” of the system. There is a forward
characteristic C1 defined by:
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Vc
dx
dt
and a backward characteristic C2 defined by:
Vc
dx
dt
Both are shown in Fig. 7.3.
backward characteristic forward characteristic
t
P
Vc
dx
Vc
dx
dt dt
dt
dx
x1, 0 V1, 1 x2, 0 V2, 2 x
P1 P2
Fig. 7.3. Forward and backward characteristics
The integration of the system of equations is now conducted along a very special path:
along the forward and backward characteristics as defined above.
7.3.2. Initial condition
A solution of the original system of Eqs. (7-18) and (7-19) requires that at some
initial time, say t = 0, the values of the (new set of) dependent variables V, are specified
for all values of x. The same conditions are needed for the solution of the characteristic
equations. Suppose then that x, V and (or V, h and c) are known at the points P1 and P2,
both at t = 0 , and that forward and backward characteristics C1 and C2 are drawn from
these points. One may seek the values of V and at the point of intersection, P, of these
characteristic paths (Fig. 7.3).
To accomplish this, one can integrate Eqs. (7-28) and (7-29), the “characteristic
equations”, along the characteristic trajectories, and obtain:
along the forward characteristic:
Vp p V1 1 g i o i f l V c dt
tp
q
(7-30)
0
A
x p x1 V c dt
tp
(7-31)
0
along the backward characteristic:
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Vp p V2 2 g io i f l V c dt
tp
q
(7-32)
0
A
x p x 2 V c dt
tp
(7-33)
0
The set of Eqs. (7-30) to (7-33) is a set of four equations in four unknowns: Vp, p, xp, tp.
Accepting for the moment that a solution is possible, it is seen that the values of the
dependent variables are determined at the points of intersection of the characteristics
emanating from the points with previously determined values of Vp and p. It is
furthermore possible to progressively extend the solution from the initially known
conditions at t = 0, through a network of characteristics as shown in Fig. 7.4, until the
region of interest in the x,t-plane is covered by an appropriate number of points. It is clear
that the intersections of the characteristic lines do not occur at regular intervals of x or t, at
the nodes of a uniform grid in the x,t-plane. Fig. 7.5. shows such a uniform grid
superimposed on the network of characteristics. The conditions at grid point M, for
example, may be deducted by bi-variate interpolation from the known values at points P, Q
and S.
S1
t
R1 R2
Q1 Q2 Q3
x
P1 P2 P3 P4
Fig. 7.4. Network of characteristics
t
x
M
Q2
t
Q1
S1 S3
S2
x
P1 P2 P3
Fig. 7.5. Characteristics do not generally intersect at grid-points M
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7.3.3. The simple-wave problem
A simple wave is defined to be a wave for which io = if = 0, with an initial
condition of constant depth and velocity and with the water extending to infinity in at least
one direction. Though neglecting gravity and friction forces may not be very realistic, the
simple-wave assumption is useful for illustrating the solution of an unsteady flow problem
in the characteristic plane.
We have specified that the undisturbed flow is also uniform: this property defines the
“simple-wave” problem. It may be thought that the conditions set out above are so special
as to be of limited practical interest, but this is not so. There are many practical situations
in which the flow changes so quickly that the acceleration terms in Eq. (7-14) are large
compared with io and if, which may therefore be neglected – to a good first approximation
anyway. An example is the release of water from a lock into a navigation canal; in this case
the initial surge (which is of most interest to the engineer) can be accurately treated by
neglecting slope and resistance, whose effect becomes appreciable only after the wave has
traveled some distance. Moreover, quite apart from immediate applications of the simple-
wave problem, its study will, as we shall see, disclose principles whose interest and
usefulness extend far beyond the simple wave problem alone.
We now consider Eq. (7-26) in detail, for a simple case, assuming that the lateral inflow,
ql, is neglected. Since io = if = 0, Eq. (7-26) can be written as:
V V
t V c x t V c x 0 (7-26a)
For a rectangular cross-section we have seen that 2 , so that:
c
c c
2 and 2
t t x x
Substitution in Eq. (7-26a) yields:
V V c c
t V c x 2 t V c x 0 (7-26b)
from which it follows:
(V 2c) (V 2c)
V c 0 Dt V 2c
D
t x
and
(V 2c) (V 2c)
V c 0 Dt V 2c
D
t x
D
being the total-derivative operator, representing the rate of change from the viewpoint
Dt
of an observer moving with the velocity V+c and V-c, respectively. The total derivatives in
these above equations are zero; this means that to observers moving with velocity (V c),
the quantities (V 2c) appear to remain constant. The paths of these observers can be
traced on the x,t-plane, as in Fig. 7.6, giving rise to two families of lines, called
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characteristics. Along each member of the first group, which we shall designate the C1-
family, the inverse slope of the line is (V + c), and (V + 2c) is a constant; similarly, along
each member of the second group (the C2-family) the inverse slope is (V – c), and (V – 2c)
is a constant.
C1 family
t
B
A
C2
G
E
D
C2
F
1
Vo + co
zone of quiet
0
x
Fig. 7.6. Characteristic curves on the x,t-plane
The two families of curves are therefore contours of (V + 2c) and (V - 2c). We shall see
that in the simple-wave problem the members of the C1-family are also contours of (V + c)
and are therefore straight lines, as in Fig. 7.6. To establish this result it is first necessary to
prove the following introductory theorem:
THEOREM: If io = if = 0, and if any one curve of the C1- or C2-family of characteristics
is a straight line, then so are all other members of the same family.
To prove the theorem we consider the two C1-lines AB, DE, in Fig. 7.6. DE is a straight
line, and we are to prove that AB (which may be any other member of the family) is also
straight. Both (V + c) and (V + 2c) are constant along DE, so that their difference c must
be constant, and hence V also. It follows that cD = cE, VD = VE; also we can write, from the
C2-characteristics AD and BE:
VA – 2cA = VD – 2cD
VB – 2cB = VE – 2cE (7-34)
whence VA – 2cA = VB – 2cB (7-35)
And since AB is a C1-characteristic, we have
VA + 2cA = VB + 2cB (7-36)
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Eqs. (7-35) and (7-36) can be satisfied only if VA = VB, cA = cB, i.e. if AB is a straight line.
The theorem is therefore proved for the C1-family of characteristics; a similar proof can
readily be obtained for the C2-family.
We now formulate the simple-wave problem in detail. We consider a channel in which the
flow is initially uniform, i.e.. V and c are constant and equal to Vo and co, respectively. A
disturbance is now introduced at the origin of x, at the left-hand end of the channel; this
disturbance takes the form of a prescribed variation of V (and/or c) with the time t. We
postpone for the present the question of whether V(t) and c(t) may be prescribed
independently, or whether the one will be dependent on the other. Also, we assume that the
disturbance propagates into the undisturbed region with a velocity co relative to the
undisturbed fluid, i.e. with a velocity (Vo + co). This implies that the disturbance sets up a
wave front small enough to have a velocity co, i.e. we assume for the present that an abrupt
wave front of finite height will not form.
It follows that we can draw a straight line OF, of constant inverse slope (Vo + co), dividing
the undisturbed flow, or “zone of quiet”, from the disturbed region above OF. This line
will also be a C1-characteristic – the first of that family – and since it is straight, so are all
the other members of the family, as in Fig. 7.6. However, the C2-characteristics are not
straight lines. If we could now calculate the values of V and c appropriate to every C1-
characteristic, we could obtain V and c at every point on the x,t-plane, and we should have
the complete solution to the problem.
This calculation is, in fact, easily carried out, given the prescribed values of V = V(t)
and/or c = c(t) along t-axis. Consider any point G on this axis, of ordinate t; the C1-
characteristic issuing from this point will have an inverse slope equal to:
V(t) c(t)
dx
(7-37)
dt
We can examine the interdependence of V(t) and c(t) by drawing a C2-characteristic
(shown dotted) from G to OF; whatever the form of this line may be, it indicates the result:
V(t) – 2c(t) = Vo – 2co (7-38)
and this equation tells us that V(t) and c(t) are not independent; only one or the other need
be prescribed (indeed only one of them can be prescribed), as a description of the
disturbance at the origin of x. From Eqs. (7-37) and (7-38) it follows that the inverse slope
of the C1-characteristic issuing from G can be expressed in either of the two forms:
V(t) Vo c o
dx 3 1
(7-39a)
dt 2 2
3c(t) Vo 2c o
dx
(7-39b)
dt
and from these equations V and c can readily be obtained at any point in the x,t-plane.
The argument leading up to Eq. (7-39) is undoubtedly circuitous, but the end result, in the
form of Eq. (7-39), presents an extremely simple treatment of problems which are
physically quite real, and indeed of some practical importance. Consider, for example,
uniform flow in a river discharging into a large lake or estuary. Initially the water level in
the estuary is the same as the water level at the river mouth; then under tidal action the
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estuary level begins to fall. It is clearly of some interest to ask how long it will be before
the water level falls by a certain amount at some specified distance upstream from the
mouth. Eq. (7-39) can give the solution.
7.3.4. Numerical solution of the characteristic differential equations
Although it is possible to apply the simple-wave solution to a number of cases, a
more comprehensive utilization of the method of characteristics demands that the slope,
friction and lateral flow terms of the complete characteristics equations, Eqs. (7-28) and (7-
29), be retained.
There are three currently used methods which are based on the characteristics equations:
i). The first involves the solution of the characteristics equations by finite
differences, and was advocated by Stoker in his book “Water waves” in 1957.
ii). The second is a modification of the general method of integrating the
characteristics equations along the characteristics by restricting the range of
integration to a specific time-interval t, and a fixed space-interval x. The
method is attributed to Hartree (1958).
iii). The third involves the construction of a general network of characteristics in
which the values of the dependent variables, say V and h, are calculated at non-
regular points with time and space intervals which may vary at different locations
in the x,t-plane.
Conceptual schemes for methods (i) and (ii) are shown in Fig. 7.7, Fig. 7.8 and Fig. 7.9.
Due to the limitation of time and knowledge requirements, it is impossible to go further
into details for each method. Students may have a chance to meet this subject again when
climbing up to the Master’s training programme in Water Resources Engineering.
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t
t + t P
(j+1)t
t
L M R
t jt
x
x
(i –1)x ix (i +1)x
Fig. 7.7. Finite-difference grid for numerical solution by Stoker’s method
forward characteristic backward characteristic
t
t
Q t + t
P
C_ C+ C_
N S t
L M R
x
x
x=0 x-x x x+x
Fig. 7.8. Determination of conditions at points P and Q from those at the previous time step
t
t t + t
P Q
Q
C+ C_ C+
N t
L S M R L S M
x
x
x-x x x+x x=L
Fig. 7.9. Method of characteristics with specified time intervals
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7.4. POSITIVE AND NEGATIVE WAVES; SURGE FORMATION
In the simple-wave problem discussed in Section 7.3.3 the disturbance introduced
at one end of the channel may be positive, i.e. such as to increase the depth, or negative,
reducing the depth. An important difference between the two consequent types of wave
becomes apparent on considering Eq. (7-39b). If the disturbance is negative, c(t) deceases
as t increases, so the inverse slope dx/dt given by Eq. (7-39b) diminishes as we move up to
the t-axis. It follows that the slopes of the C1-characteristics increase as t increases, i.e. that
the characteristics diverge outwards from the t-axis, as shown in Fig. 7.10. From this it
follows that negative waves are dispersive, i.e. that sections having a given difference in
depth move further apart as the wave moves outwards from its point of origin.
t
xs
A
t
envelope of
V+c intersections
F
O x (b)
(a)
Fig. 7.10. The convergence of characteristics and steepening
of the wave front in a positive wave
When the disturbance is positive, on the other hand, the C1-characteristics converge, as in
Fig. 7.10a, and must eventually meet. Such an intersection implies that the depth has two
different values in the same place at the same time – an obvious anomaly. What in fact
happens is equally obvious: the wave becomes steeper and steeper, as in Fig. 7.10b, until it
forms an abrupt steep-fronted wave – the surge, or “bore”. While the intersections of
neighbouring characteristics will form an envelope as in Fig. 7.10a, the surge will actually
form at the “first” point A of the envelope – i.e. the point having the least value of t.
The front of the surge will not necessarily be broken and turbulent; it may, like the
hydraulic jump, consist of a train of smooth unbroken waves if the depth ratio is small
enough. It follows that a surge would certainly not break at the first point of formation, as
at A in Fig. 7.10a, for the depth ratio there approaches unity. Breaking would only occur
after subsequent development of the surge beyond the point A; tracing this development
would be a matter of some difficulty. However, in all the subsequent argument the term
surge will be applied to any abrupt change in depth, as indicated by the point A, whether it
is undular or broken.
The intersection of any neighbouring pair of characteristics can be located by an
elementary geometrical argument. With the terms defined as in Fig. 7.10a, the following
results are obtained:
t sin t sin 2
x s sin
(7-40)
xs
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and (tan ) cos 2 (7-41)
t tan 2
xs
(tan )
whence (7-42)
V c
2
xs
d V c dt
(7-43)
where (V+c) equals the inverse slope of the characteristic, as given by Eq. (7-39).
Substituting Eq. (7-35b) into Eq. (7-43), we obtain:
3c(t) Vo 2c o = (V + c)
dx
dt
V c 3c(t) Vo 2c o
2 2
xs
d V c dt d 3c(t) Vo 2c o dt
3c(t) Vo 2co
2
xs (7-44)
3dc(t) dt
From this equation an envelope of intersections can be traced, given a specified
disturbance c = c(t) along the t-axis (Henderson, 1966). If, as in the case of the release of
water from a lock, the disturbance is specified as a variation in q = q(t), corresponding
values of c and dc/dt can readily be obtained (Henderson, 1966) by the use of Eq. (7-38)
and inserted in Eq. (7-44).
Once a surge does develop, there is of course an energy loss across the surge, and
characteristics cannot be projected from one side of the surge to the other. But it can be
shown (Henderson, 1966) that the flow on each side of the surge can be described by a
separate system of characteristics.
We may also note here that in the case of the negative simple wave, Eq. (7-38) can be
written in the more general form:
V – 2c = Vo – 2co, i.e. a constant (7-45a)
or V 2 gh Vo 2c o (7-45b)
applicable to the whole of the x,t-plane, because a C2-characteristic can be drawn from any
point on the plane to the line OF (Fig. 7.6) bounding the zone of quiet. In many cases it is
more convenient to solve problems by the direct use of Eq. (7-45) and the concept of wave
motion discussed in Section 7.3, rather than by use of the x,t-plane.
It need hardly be emphasized that, if a surge forms, Eq. (7-45) could be applied only by
using different constants on opposite sides of the surge. The form of the negative-wave
profile can readily be deduced by recalling the significance of Eq. (7-39b). This equation
gives the speed at which a section of constant depth h moves; since for a given h, this
speed is constant, we may replace dx/dt by x/(t - t1) where t1 is the value of t at x = 0 (e.g.
as at the point G in Fig. 7.6). Substituting for c, we can then rewrite Eq. (7-39b) as:
3 gh(t1 ) Vo 2 gh o
x
(7-46)
t t1
where h = h(t1) prescribes the initiating disturbance along the t-axis. Eq. (7-46) defines the
wave profile completely in space and time.
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Chapter 7: UNSTEADY FLOW 146