Acta Mechanica 51, 31--48 (1984)

ACTA MECHANICA
9 by Springer-Verlag 1984

Approximate Treatment of Plane Cl~annel Flow

By

W. H. Hager, Lausanne, and K. Hutter, Zurich, Switzerland

With 6 l~igures

(Received J u l y 26, 1983)

Summary

The governing equation for steady two-dimensional channel flow are deduced from the
equations of inviscid fluid dynamics. Streamline inclination and curvature effects are
approximately accounted for. The integration procedure aims at the elimination of the
spatially transverse coordinate; through it one arrives at a second order non-linear ordinary
differential equation for the free surface profile having the form of an (extended) Bernoulli-
equation and providing formulae for the pressure and velocity distributions. A second
derivation, applicable to channels with horizontal Thalweg and arbitrary prismatic cross
section, leads to a similar equation for the free surface profile. Illustrations concern solitary
waves in rectangular and trapezoidal cross sections. Comparison of theoretical predictions
of wave geometry with experiments indicate advantages of odr formulation over previous
more extensive ones.

Notation

The following symbols may appear in the main text in combination with the sub-
scripts " o " and " u " , denoting upper and lower boundaries, respectively.

a horizontal projection of radius of the flow section

b bottom width of the trapezoidal channel
d differential
] = Q2/gb~ho s non-dimensionalflow. Froude-number in rectangular channel
g gravity constant
h pressure head
k streamline curvature
m cotangent of side slope of trapezoidal channel
m* mass of a fluid particle
n coordinate, normal to streamlines
p pressure
q specific discharge (per unit width)
r quotient of boundary curvatures r ~ R o / R u ~ k u / k o
 32 W. It. ttager and K. ttutter-

8 streamline coordinate
t water depth
horizontal velocity component
V vertical velocity component
x horizontal coordinate
Y non-dimensional pressure head
z bottom height function
z* vertical coordinate
17 cross sectional area
H specific energy
5r length of plane flow section measured along n
Q discharge
R radius of curvature
S specific momentum
V algebraic velocity (along streamlines)
X non-dimensional horizontal coordinate
Z non-dimensional bottom profile
slope factor
non-dimensional geometrical factor for trapezoidal cross section/~ = mho/b
specific weight
c~ bottom amplitude
non-dimensional pressure head
non-dimensional horizontal coordinate
non-dimensional vertical coordinate
non-dimensional normal coordinate
@ density
non-dimensional water depth
0 inclination angle of streamlines relative to horizontal direction
auxiliary variable
A~Vayn dynamical pressure head

Introduction

'Common theoretical treatments of channel flows make use of the predominance

of the flows along the channel axis. Variations of typical physical variables are
primarily in this direction. The usual procedure is, therefore, to smear over the
transverse directions and to obtain one-dimensional equations involving as
independent spatial variable the coordinate along the channel axis. This is
achieved b y averaging the spatially two or three-dimensional equations in the
transverse direction(s). Various degrees of sophistication are possible, but it is
probably fair to say t h a t Boussinesq [2], [3] was the first to incorporate two-
dimensional features into channel flow hydraulics. Many other researchers in this
century have worked along his lines, among these mention m i g h t be made of
Serre [13], Matthew [9], Puzanov [12] and Wehausen and Laitone [15], the latter
providing a review of the literature.
I n the light of Boussinesq's model curvature and slope effects of the stream
lines are accounted for in his hydraulic equations. From a practieioners view point
this is also the aim of this study. I n this respect our study is indeed reminiscent
 A p p r o x i m a t e T r e a t m e n t of Plane Channel Flow 33

of the work of Dressler [4] who adopted two-dimensional curvilinear coordinates

and carried out a detailed asymptotic analysis, retaining terms to first order in the
"shallowness parameter". But, while Dressler aimed at a model in which bed
curvature had a dominant significance (see also Sivakumaran et al. [14]), a reduc-
tion in spatial dimensions of the governing equations was not sought (even though
:this is the result in certain limiting cases). Our principal idea is the reduction of
the number of independent spatial variables paired with a simultaneous account of
curvature and slope effects. Approximations are thus different, as we must in
general delimit attention to small slopes and curvatures.
We start from the Euler equations of motion and demonstrate two methods
from which higher order hydraulic equations can be deduced for channel flow in
which transverse variation of the field variables is approximately accounted for.
In the /irst method the steady Euler equations are formulated in or~hogonal
curvilinear coordinates along and perpendicular to the streamlines. P l a n e / l o w
is assumed and the Euler equations are integrated in the direction perpendicular
to the streamlines. Their curvature and slope angle with respect to the horizontal
direction are approximately accounted for by assuming a linear variation between
the bounding streamlines forming the free and the bottom surfaces. In this
fashion velocity- and pressure profile formulas can be deduced, which go beyond
the usual hydraulic relationships. Im particular, the velocity is non-uniformly
distributed and the pressure has a non-hydrostatic part, and both contributions
are expressible in terms of the pressure head function, the bottom profile and
derivatives of these. The differential equation for pressure head follows from a
Bernoulli-type equation. The novelty of this approach lies mainly in its application
to arbitrary bottom geometry and in a careful distinction of pressure head and
flow depth. The discrimination of these quantities seems to have been glossed over
in previous analyses. The application of our model equations to the solitary wave
and the comparison with known experiments will clarify the need for the distinc-
tion.
In the second method the bottom profile is assumed to be nearly horizontal but
channels may have arbitrary smooth cross sectional areas. This implies that of the
transverse directions flow variations in the vertical are more pronounced than in
the horizontal direction. The flow is, therefore, again nearly two-dimensional.
Euler-, continuity- and zero-vorticity equations in two-dimensional Cartesian
coordinates allow approximate deduction of the differential equation for pressure
head on the basis that the horizontal velocity component is uniform. The governing
equation for pressure head is again of Bernoulli type, and, for rectangular cross
sections, can be brought into coincidence with the corresponding equation obtained
with the first approach. The model equations are applied to the solitary wave
problem in trapezoidal channels. Physical considerations delimit solutions to a
certain range which indicates that solitary waves on trapezoidal channels are
likely to become unstable when the crest height reaches approximately twice the
undisturbed flow depth.

3 Acta Mech. 51/1--2

34 W.H. Hager and K. Hurter:

Governing Equations and Flow Geometry

Consider steady, plane potential flow with free surface. Its Euler equations,
formulated in a curvilinear, orthogonal coordinate system with axes along the
streamlines (s) and perpendicular to them (n) read (see e.g. 135s [1])

aV 1 ap az*
v sT= ~o a8 g sT' (1)

V~ 1 ap 8z*
_R q an an (2)

in which V is the algebraic velocity, o density, p pressure a n d / t the radius of

curvatt~re of the streamlines; moreover, q denotes the gravity constant and z* the
vertical position of the material point in question, measured from an arbitrary,
but fixed reference level. With y ~- ~og, Eq. (1) can be integrated to yield

H =- z* + p + V2 ;
- - (3)
7 2g

.H, a constant of integration, is called energy head. At the free surface, where the
relative pressure vanishes, Eq. (3) implies H = z* 't- V2/2g, or, when z(x) denotes
the elevation of the channel bottom, H ----z ~ h + V2/2g, where h is the pressure
head. Eliminating p from (2) and (3), finally gives

~V V
- ~v, (4)
On R

in which R is the radius of curvature of the streamline. Equations (3) and (4) will
henceforth replace the original Euler equations.
In ensuing developments three quantities, namely the flow depth t (which is
the vertical distance from the bottom to the surface), the length of the orthogonal
trajectory of the streamlines 37, and the pressure head h (which equals the vertical
projection of the normal N), will be important, see Fig. 1. As is evident, the flow
depth t only nearly equals h, but for small slopes 0 of the streamline with respect
to the horizontal the error is of order 02, a restriction which will be imposed in
ensuing developments (p. 35 below). The two bounding streamlines, denoted by
"o" and " u " , respectively, are connected by the orthogonal trajectory of length N,
which is assumed to form an arc of a circle with radius R*. Simple geometric
relations allow the deduction of the equations, see Fig. 1

t = a(tan 0u -- tan 0o), (~)

h ----R*(sin 0~ -- sin 0o), (6)

 Approximate Treatment of Plane Channel Flow 35

st O st

#l R~

Fig. 1. Definition of flow geometry. At a position x the pressure head h differs from the
flow depth, which nearly equals t, if 0 remains small

where 0~ and 00 are the inclination angles of t h e b o t t o m and top streamlines,

respectively. Further, with ~ =/~*(0= - - 0o), and since a = / ~ * cos ~ , Eq. (5)
and (6) t a k e the alternative forms

h sin O~ - - sin Oo O~ + Oo + 02
1 (7)
N O ~ - Oo -- 6

t cos 0~(tan 0u - - t a n 0o) 2(0fie + 002) -- 0~

1 + , (8)
N 0~ - - 0o 6

in which the a p p r o x i m a t i o n s on the far right are carried to second order in 0u and
0o. To the same order of a c c u r a c y

O= = z', Oo = z' + h', (9)

in which primes denote differentiation with respect to x; to second order Eqs. (7)
and (8) a n d the ratio hit m a y thus also be written as

h 3z '~ + 3z'h' + h '2

-- = 1 -- (10)
N 6 '

t 3z '2 + 6z'h' + 2h '2

= 1 -~ (11)
N 6 '

h 3z '2 + 3z'h' + h '~

_ 1 - (12)
t 2 "

The third of these m a y be used to deduce the (measurable) d e p t h t from the

pressure head function h and the b o t t o m profile z(x), once h is known, whereas
the first, h / N , will be needed in the d y n a m i c a l equations (3) and (4).

3*
36 W.H. Hager and K. ttutter:

Approximate Integration Of the Euler Equations

in Transverse Direction

Equations (3) and (4) canflot be further integrated unless additional assump-
tions are invoked. These concern the geometry of the streamlines and may be
explained with the aid of Fig. 2. What is needed are representations of the in-
clination angle 0 and the curvature k : I/R of an arbitrary streamline situated

R~

Fig. 2. Explaining the linear approximations (13) in a flow section

between the bounding top and bottom streamlines in terms of the transverse
coordinate. Since both, 0o, 0~ and ko, ku are assumed small, a linear variation will
be sufficiently accurate so that

0(~) = O~ + (Oo -- 0~) v,

(13)
k(~) = k~ + (& - k.)

are appropriate, with v = n/N, measured along the orthogonal trajectory and
having values between 0 (bottom) and I (top streamline). Relations (13) were first
introduced by Matthew [8]; they may be regarded as truncated (Cauchy) series
expansions which incorporate the boundary conditions at v = 0 and v = 1. As
shown b y Hager [5] sharp crested weits exhibit a fair correlation with the ana-
lytical relations (13). These, incidentally, constitute the essential step by which the
mathematically 2/) problem can be reduced to 1D form, which is possible through
a shape function expansion of 0 and k in terms of the transverse coordinate v.
Substituting the second of relations (13) into Eq. (4) and using ~/0n = N - ~/0v,
straightforward integration leads to

ln(V/Vo) : ~ r(v-- 1 ) + ( l - - r ) (14)

with Vo = V (v : 1) and r = Ro/R~ : ku/ko. For small values of N[Ro and N/Ru,
 Approximate Treatment of Plane Channel Flow 37

this reduces to

V/Vo = 1 +--~-- r O, -- 1) ~- (1 r) (15)

In the following we shall focus attention on this case.

Integrating the velocity along the transverse coordinate ~ gives for discharge
per unit width

q=lgfv(~)dr=VoN l (1 N(26Ro+r!) , (16)

0

or with (10), when only first order terms are considered

3z '2 + 3z'h' + h '2 h(2 -t- r)~

q~ Voh 1-]- 6 6Ro ]" (17)

Using the definition of r, ]~q. (10) and

k~ ~_ z", ko ~ z" § h" (18)

the surface speed V0 may be eliminated from Eqs. (15) and (17), to yield

V(v)----T -- 6 + v-- + - - . (19)

In practice it is more convenient to regard V as a function of the vertical co-

ordinate # ~-- (z* -- z)/h. Now, since d(z* -- z) ~- cos 0 dn, and consequently
d[(z* -- z)/N] -~ cos O(v*) d~* one has

z* z iv 0(()
/~ = " ~ - h cos (0(~*)) & * ~ -~- 1 d~*,
O 0

which, in view of (9) and (10) may also be written as

"z'h' h ts ]
/~=v 1-~T(1--v)~-~-(1--v 2) --~v[1-]-O(z'h',h'S)],

or when inverted and to orders quadratic in z' and h'

z'h' tt ~ ]
v__~# 1 - - 7 ( 1 - - tt) -- T ( 1 - - t~2) ----it+ O(z'h',h'2).

Equation (19) permits calculation of the transverse velocity distribution, once the
function h(x) is determined.
The horizontal and vertical velocity components, u and v, are given by

u -~ V cos O, v ---- V sin O, 0 = z' ~- tth ' -+ O(z'~ z'h',h'2). (20)

38 W.H. Hager and K. Hurter:

With the aid of (19) and the usual second order expansions this yields

u [ z'h' h '2
= ( l - - z '2--2 (1+2#) g (1+3# 2)
(21)
hz ' " hh " )
+ ~ (2~--J)+-~(3~ ~-1) ,

V
Zt
Tl ~h' (22)
q/h
and for a flat bottom with z' ----z" ~ 0

u h'2 hh" v
r = 1 ~ - (1 + 3~ 2) + T (3~2 - 1), ~/q'--r = ~h'. (23)

There remains the derivation of the differential equation for the energy head H.
This can be deduced from the Bernoulli equation (3) evaluated at the free surface,
for which z* ~ h + z, see Fig. 1, and p = 0. Thus

Vo 2
H-~z+h+ (24)
2g '

or with the aid of (17)

2hh" -- h '2
H=z+h+2--~- 1+ + h z ' ~ - h'z' -- z'2). (25)

This expression was first presented by Matthew [9]. It constitutes the essential
differential equation from the sohltion of which h(x) may be determined. For
given energy head H and prescribed boundary conditions at two positions solutions
may be determined by analytical or numerical integration techniques. Provided
that H = eonst., the solitary wave problem treated below is one for which ana-
lytical solutions can be found.
With the solution of (25), the velocity distribution follows from (19)Or from
(21) and (22) and the pressure distribution can be deduced from (3) when writing
it down for z* --~ z + #h, viz.
V2
P --~ H -- z - - # h - - ~ (26)
y 2g '

or in view of (19) and (25)

q~
P -- (1 --/~) + (2hz"(1 -- #) + hh"(1 -- #2)). (27)

Notice t h a t when z" = h" 0 the pressure distribution is hydrostatic. The

second term on the right-hand side accounts for dynamic effects; it depends on the
 Approximate Treatment of Plane Channel Flow 39

local :Froude-number q2/gh3, the curvatures of the bottom and the top surfaces
and on the position ~u. At the bottom, where/x -----0, it obtains
q2
Apdyn/~h -- 2g h3 (2hz" %- hh"). (28)

A Relation Between Specific Momentum S and Energy Head H

In the curvilinear coordinate system of Fig. 1 cross sections are defined by the
orthogonal trajectories of the streamlines. Referred to such a cross section the
specific momentum in the horizontal direction is defined b y
1 1

S -~ h
f
0
P (cos 0 dr) %-
Y f
0
__u dq(v) ,
g
(29)

where cos 0 dv -~ dtt and dq(v) = V N dr. Because the horizontal velocity com-
ponent u is expressible in terms of the algebraic velocity, u = V cos 0, one may
write Eq. (29) also as
1 1

S=h
f P--d~+;V ; - - d ~ .
0
j
0
g
(30)

Integration is performed with the aid of Eqs. (19) and (27), the result being

h~ q~ ( h h " - - h '~ hz" z'h'%- --z '~1 (31)

Differentiation of both, S and H (see Eq. (25)) with respect to x now gives

S ' = h h ' - - - ~ -q2h'

~ 2 ( 1%- 2hh" h '23- h2h'"/h' %- hz" -- h'z' --z'~
X

hez ''" hz'h" 2hz' z"] (32)

- 2h-------=+ ~ + --V-]'

H' = h' q%'ha

g ~(1 -l- 2hh" ~ h '~3- h2h'"/h' %- hz" -- h'z' -- z '~
(33)
h2z''' hz'h" hz' z"
2h-- + h----z--+ - V - ] '

two relations, which differ only in the last two terms in paranthesis. For practical
purposes these terms are negligibly small. To the order of such an approximation
we thus have
S" -~ hH', (34)
40 W.H. Hager and K. Hurter:

relation first deduced b y Serre [13] for a channel with horizontal bottom,
z' : z" : 0. Our derivation shows that it holds approximately for slowly varying
bottom profile. We notice further that in this case H ' -~ 0 whenever S' -~ 0 and
vice versa. Equations (25) and (31) have identical solutions in this case.
~or given S or H Eqs. (25) and (31) are second order differential equations
for h, in the derivation of which the streamline curvature and slope angle has been
taken into account. Omitting their influence the well known hydraulic equations

q2 h2 q2
H = h + 2gh2 , S ~ + gh (35)

emerge. In general, both H and S are functions of position; if h should be deter-

mined, either from (25), (31) or else (35) their functional relations must be pre-
scribed. To lowest order it is customary to assume that H is conserved, Whence
constant, and determined in a reference cross section. In the approximation (34)
this is tantamount to assuming conservation of the specific head. However, the
more general relations (25) and (31) do not conform with this. Constant S- and
constant H-postulates will yield different profiles for h, even though deviations
are small. This should be borne in mind.

Illustrations

In this section one simple application of the above model will be given. No
attempt is made to fully discuss the mathematical solutions of the equations and
to compare these with experimental results. For these subjects the reader is
referred to Naheer [11], Puzanov [12], and Lee et al. [8] and work under progress
by the principal author.
The simplest non-trivial problem for which Eq. (25) can be solved is the
solitary wave for pseudo-uniform flow conditions, i.e. H ' ~ 0. With h(x --->4-oo)
~- h0 and h'(h -~ he) ---- O, Eq. (25) becomes

h o - k 2gh~ ----h~- 2 - ~ lq- 3 ' (36)

or after introducing the scalings

q2
X=x/ho, y =h/ho, /= gh~ ' (37)

/
1~- ~ : y - f -
/(
~y~ 1~-
2yy"--Y'2)
3 '
(38)

with y = 1, y' = 0 at X = i ~ , in which primes now denote derivatives with

respect to X. The solution must be symmetric in X, since (38) is invariant under
 Approximate Treatment of Plane Channel Flow 41

the reflection X -+ --X. I t reads

~ = 1 -- Tanh 2~ = s e c h 2 $,
(39}
_ ]/3(/--1) X, ~7 Y 1
v 4 =]--1"

The function (39), plotted in Fig. 3 as the light solid line, is universally valid
for all Froude numbers / _>_ 1 and was first given by Boussinesq [2]. If h were
the surface profile, observed in experiments, all experimental points should
cluster around this curve. However, in our improved formulation t marks.the
surface profile while h is pressure head. It is thus more appropriatd to plot

T--1 t
~:(~e,]) _ . ~ , T ~ 7-, (40}
1 ]'----
"
ao

versus ~, in which T can be expressed in terms of h with the aid of Eq. (12).
This yields approximately T ~- y[1 + (dy/gX)2]2]. For any g i v e n / , a r-curve
is obtained, corresponding to the free surface of a solitary wave. In Fig. 3, we
have also plotted z(~) for ] ---- 1.6 (heavy solid line) and compared the result
with an experiment reproted by Naheer [10] (crosses). Deviations from the Bous-
sinesq curve are significant. Naheer compared his experiments with the theo-
retical predictions of McCowan [10] and Laitone [7]. Using perturbation techniques
these authors deduced equations which correspond to (25) ; since their formulation
is in terms of Cartesian rather than curvilinear coordinates their systematic
perturbation scheme is very complicated if sufficient agreement between experi-
ments and theoretical predictions is sought. Our reconciliation of the problem
demonstrates that formulations can be relatively simple, if the transformation
h-->t is accounted for. Furthermore, Naheer [11] concluded that among the
models of Boussinesq, McCowan and Laitone the only single approximate theory
that gives a fair representation of the surface profile, wave speed and fluid
particle velocity in horizontal and vertical directions is that of Boussinesq.
Our model, evidently is an improvement of the latter.

1
0.8 [-- HagerlHutter
0.6
// - Exp. Naheer
0.4 ./7 t,, ~(~)
(12
0 J I I
-2 -1 0 1 2
Fig. 3. Solitary wave solution compared with experiment. The light solid line is Eq. (39).
The heavy curve represents the correction according to Eq. (40)
42 W.H. Hager and K. Hu~ter:

0
-2 -1 0 1 2 3
:Fig. 4. Dynamic pressure Ap/yho and horizontal velocity profiles u/(qho) for a solitary wave
with / = 2

To estimate the non-uniformity in the horizontal velocity component and

the deviation of the pressure distribution from hydrostatic conditions we have
plotted in Fig. 4, valid for / ~ 2, at the positions X = (0, 1, 2, 3) the vertical
distributions of dimensionless dynamic pressure 3p[?ho and the dimensionless
horizontal velocity component uho/q. It is evident that at the wave crest the
pressure is under-hydrostatic, and horizontal velocity grows with depth. At
the points of inflection the pressure is strictly hydrostatic (see Eq. (27)) and the
velocity distribution nearly uniform. As X--> i c e , h' = h " - + 0 , implying
again hydrostatic pressure and uniform velocity distributions.

Flow in the Almost H o r i z o n t a l Channel of Arbitrary Profile

The previous analysis holds for flat, weakly curved plane free surface flow
and is approximate when cross sections are nearly rectangular. Results were
derived b y using Bernoulli's equation and integrating it along trajectories which
cross streamlines orthogonally.
When cross sections are far from rectangular, this procedure is no longer
appropriate. We shall illustrate an alternative approach by restricting ourselves
to prismatic channels with horizontal Thalweg. Direct integration of the Euler
equations is now more convenient. We again suppose nearly two-dimensional
flow with a velocity.field which varies negligibly in the direction of the channel
width. The partial differential equations

u ~u v ~u 1 @
---- -+- -i - - - § ---0, (41)
9 ~x g ~z* y ~x
u ~v V ~v 1 @
§ -- 1, (42)
g ~x 9 ~Z* y ~z*
~u ~v
~X § ~z* = o, (43)

c~u ~v
= 0 (44)
 Approximate Treatment of Plane Channel Flow 43

then represent horizontal and vertical m o m e n t u m balances, mass balance and

the condition of vanishing velocity and suffice, in principle, to determine u, v
and p as functions of x and z*, which are the Cartesian coordinates in a vertical
plane.
A direct integration of the above set of equations m u s t be iaborious; it can
be simplified b y supposing a dominant flow in the x-direction with lIvll~ u
and b y ignoring a dependence on transverse coordinates of the u-velocity u = u(x).
Hence, u -= Q/F, where Q is discharge and F the wetted cross sectional area.
Clearly, as only prismatic sections are considered, F = F[h(x)]; Eq. (43) thus
implies

~u Q (Fhh')- ~v (45)
~x /w ~z*

with Fh =-- 0F/~h. Integrating Eq. (45) with respect to z* yields

v(z*) = Q Fhh'(z* - - z), (46)

-Fr

where z* = z marks the bottom surface and m a y v a r y in the direction of the

channel width. With/~hh = ~ F [ ~ h ~ and z = eonst, we deduce from (46)

~v Q
= (--2Fh2h '2 -[- FFhhh '2 @ FFhh") (z* -- z), (47)
~x Ea

an equation which in conjunction with (45) can be used in (42) to evaluate the
pressure. F o r channels which are wide and shallow (z* - - z) hardly varies within
the cross section. With the appropriate choice of the origin of the coordinate
system (at the lowest point) we m a y thus approximately set z = 0 and obtain
from (42) and with # = z*/h

P (tt) = h(1 - - re) -- ~ (Fh~h '~ -- FFhhh '~ -- FFhh") h ~. (48)

Alternatively, from a combination of (42) and (44), one m a y deduce that

z* + p / y + (u ~ + v2)/2g is independent of z* with value H, common to all
streamlines. Hence,
h

It = ~* + P-- + - - ~z*, (49)

7 2g
0

or after integration

H =h ~ 2-~ 1+ ~ 2Fh~h '2 § 2Fhh''

44 W.H. Hager and K. Hurter:

In the special case of a prismatic, rectangular profile (F ----bh, Fh = b, Fhh = O)

Eq. (50) reduces to

H = h +
zg-f~Q2(
-=--~-~. 1+
2hh,, h,~
,
(51)

which agrees with the reduced Eq. (25) for z = z ' ~ - z " = 0. Two different
approaches have thus led to the same result. In the first, (leading to Eq. (25))
the equations of motion, formulated in the streamline direction and perpendicular
to it, were integrated by invoking the additional assumption of linear variation
of the inclination and the curvature of the streamlines as given in (13). By
contrast, the second approach made use of a direct integration procedure of the
Euler equation in Cartesian coordinates by neglecting the vertical variation
of the longitudinal velocity component. The general formulae, Eqs. (25) and (50),
are valid under slightly different conditions; one incorporates longitudinal
variation of the bottom profile z(x) and is restricted to plane flow, the other assumes
an almost horizontal Tha!weg of nearly prism_atic chan_nels with arbitrary cross
sections F(h). The equations are composed of the classical terms, known in
traditional hydraulic formulations, but incorporate small higher order corrections,
which vanish for a horizontal flow with parallel streamlines.

Illustration: Solitary Waves in Trapezoidal Channels

To further illustrate the foregoing analysis we consider a prismatic, trapezoidal

channel with flat bottom and cross sectional area

F = bh + mh 2, (52)

where b is the width of the base and m the cotangent of the slope angle of the
sides. In a prismatic channel both quantities are constant, thus

1rh = b + 2mh, .Fk~ = 2m.

For pseudo-uniform flow H is a constant and the differential Eq. (50) becomes

Q' ( 2h2(b + 2mh) h'' b'h2h ' ' )

H =h+~ 1+ 3F 3F 2 (53)

which, for a solitary wave, must be subject to the boundary conditions

Q~
H=h0+ 2gF~' h'=0, as x-++cr (54)

With the scalings

X = x/ho, y = h/ho, fi = mho/b~ / = Q~/gb~ho3 (55)

 Approximate Treatment of Plane Channel Flow 45

the differential Eq. (53) transforms to

f t 1+
9y(1 + 2fy)v" • :/ . (56)
1 + 2(1 + fl)~ ---- y + 2(y + fy~)2 3(1 + fly) 3(1 + fly)S/

A first integral is

3 (1+2fly) ~ (y-- 1 ) + (2fl-- 1/2)(y~-- 1 ) + 5f~-3 4fl

+ f~(~)(y'-1) _ 2fl~ ) 1
T(YS- 1) + ( l + f l ) ~
(
9 (y-- 1)+2f(y ~-l)+~(ya-
5fl~ f8
1) + - ~ - ( y a - -
)(57)
1)

+ 1 --y y 2f in y]

and satisfies the condition y' : 0 for y = 1, appropriate for X --> ~:c~. A further
integration can only be carries out numerically. However, important inferences
can already be drawn from equation (57). The maximum value y ~-- Ymax > 1
follows from (57) by setting y' = 0. The zeros of the transcendental equation that
obtains when the term in brackets is set to zero, are plotted in Fig. 5 as functions
of ] and are parameterized for various significant values of f, where f = 0 cor-
responds to the rectangular cross section. Positive values of fl correspond to
trapezoids of which the width at the water line is larger than that at the base.
As is physically plausible, for fl < ( > ) 0 maximum amplitudes are enhanced
(diminished) relative to those of the rectangular cross section.
Solutions of (57) are only physically meaningful as long as hmax ~ H. Ac-
cording to (50) this corresponds to

1 + -~ 2Fhhh'2 -]- 2F~h" --

or, at the wave crest where h' = 0

h2
1 + . - ~ . 2Fhh" ~ O. (59)

This inequality can only be violeted at wave crests where h" < 0. For trapezoidal
cross-sections Eq. (59) with equality sign reads

1 + ~ yy" (1.iyk-~-~-
+ 2fy
/ =0. (60)
46 W . H . Hager and K. Hurter:

Using (57) y " m a y be calculated. L e n g t h y , b u t straightforward manipulations

yield

2y'" ,~ y(1 § fly)

3 -- (1 4- 2fly) a + (1 + 2fly) ~

[---~ (1 -}- y(4fl - - 1) ~- y~(5fi 2 - - 4fl) + fi2ya(2fl - - 5) -- 2f13y 4) (61)

l+4fly+5f12y2~ 2/~3ya 1 y 2/~y.]

§
(1 -}- fl)2

where ~5 is the t e r m in brackets in Eq. (57). The m a x i m u m curvature Y~:~x for

solitary waves arises at the wave crests when y = Y~ax for which y' = ~5 = 0.
Combination of (60)and (61) for y = Yln~x', Y' = 0 yields the separation of two
d o m a i n s ; below the curve 9 in J~Sg.5 H ~ hm~.~, above H < hm~x which is
unphysical. Actual observations indicate t h a t for H----hma ~ wave crests are
definitely breaking, see P u z a n o v [12] ; however it is also k n o w n t h a t for rectangles
such breaking occurs for Ym~ ~ 1.78. One must, therefore, interpret the separating
curve as the location where wave breaking has definitely occurred.

YmGx

---0"1 o.41
l

2 3 4 5 f
Fig. 5. Maximum amplitude of a solitary wave in a channel with trapezoidal cross section
plotted as a function of / and ft. Plotted is also the line . . . . . . . Wave crests with Ymax
above this line are definitely breaking

I t remains to discuss the solitary wave profile; this profile can be obtained
from (57) b y a further integration, l~fore convenient is, however, to directly
integrate the original second order differential Eq. (56) with the b o u n d a r y con-
ditions y(y' 0) Xmax and X ( y -=- Ym~) = 0. An e x a m p l e is shown in :Fig. 6
for / --~ 3 and fl = 0.4 in which X is restricted to positive values. I t shows dimen-
sionless pressure head y (top) a n d d y n a m i c pressure at the b o t t o m . This can be
c o m p u t e d from (48), or

Ap Q~ h2
-- gF4 (Fh2h '2 - - FFhhh '2 - - F F h h " ) - ~ , (62)
 Approximate Treatment of Plane Channel Flow 47

1.77

0 Ap

•
-1
0 2 3 4
Fig. 6. Pressure head and dynamic bottom pressure of a solitary wave in a trapezoidal
channel with ] = 3 and fl = 0.4

which for trapezoidal cross sections reduces to

yho -- 2y~(1 -t- fly)4 ((1 ~- 2fly ~- 2fl~y ~) y'~ -- (1 ~- 3fly -~ 2fi2y 2) yy"). (63)

As evident from Fig. 6 bottom pressure is under-hydrostatic below the crests

and over-hydrostatic below the troughs.

Conclusions

A set of hydraulic equations for the description of steady, nearly plane channel
flow was presented. The formulation accounts for the slope and curvature of
the streamlines and allows determination of non-uniformity of velocity and non-
hydrostaticity of pressure distribution. The two types of model equations apply
either to plane flow appropriate for rectangular channels with v a r y i n g bottom
profile or to prismatic channels of arbitrary cross section and with nearly hori-
zontal Thalweg. In this article our attention was focussed on a systematic deri-
vation of the governing equations and a clear distinction between pYessure
head and flow depth. I t was pointed out that these quantitie s are often confused
in the literature. Applying the formulation to the solitary wave in rectangular
and trapezoidal channels indicated that theory and experiment can be brought
into sufficient coincidence by this relatively simple procedure: for both cases
the free surface profiles can realistically be predicted. For trapezoidal channels
it was further shown that supercritical flow is unrealistically predicted when
the crest amplitude of the solitary wave is approximately twice the water depth
of steady conditions.
The equations presented in this article are basic to many channel flow prob-
lems in which streamline inclination and curvature are likely to play a significant
role, i.e., weir and sluice flow, back water profiles and so on. This has already
been demonstrated for plane free overfalls, see Itager [6]. Further applications
on undular bores are in preparation.
48 W . H . Hager and K. Hurter: Approximate Treatment of Plane Channel Flow

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Dr. W. H. Hager Doz. Dr. K. Hutter

Ghaire de Gonstructions Hydrauliques, COIl Laboratory o/ Hydraulics,
Gdnie Givil, GC Hydrology and Glaciology
Eco!e Polytechniquc Fdd~rale ETH
EPFL Glorias~rasse 37/39
GH-1015 Lausanne CH. 8092 Zurich
~qwitzerland ~witzerlanct