Page 1
- FLOW O
SHALLOW
 

VER CURVED BEDS
y
A KUMARAN

Page 2

tr. Pathmanaba Iyer
27-189High Street Plaistouv fondon E13041D
Tes: O2O&471 5636
SHALLOW-FLOW OWER CURWED BEDS
BY
N, S SIVAKUMARAN
A Dissertation submitted in partial fulfillment of the requirement
for the Degree of Doctor of Engineering
Examination Committee : Professor Robert B. Banks (Chairman)
Dr. Tawatchai Ting sanchali (Co-chairman) Professor Roger J. Hosking
Dr. Huynh Ngoc Phien
Professor Anat Arbhabhiranna
Nagalingam Subramaniam Sivakumaran
Nationality : Sri Lankan
Previous Degree : B. Eng., University of Ceylon, Peradeniya, Sri Lanka
M. Eng. , Asian Institute of Technology, Bangkok, Thailandi
Scholarship donor: His Majesty the King of Thailand
Asian Institute of Technology
Bangkok Thailand
98.

Page 3
ACKNOWLEDGEMENTS
I wish to express my deep gratitude and sincere appreciation to my adviser Professor Robert B. Banks, President of the Asian Institute of Technology, for his inspiring guidance and encouragement during the research period. I am deeply indebted especially to Dr. Tanatchai Tingganahali - who selected this topic for my research - for his critical review of this dissertation, the many valuable suggestions and his interest and patience' which greatly exceeded the expectation from a co-adviser.
I find no words to thank enough Dr. Roger J. Hosking, Professor of Applied Mathematics, for his deep interest, encouragement, invaluable assistance in reviewing the whole work from both the mathematiqą and physical standpoint, and editing the dissertation.
Sincere appreciation is also due to Dr. Huynh Ngoc Phien and Professor Anat Arbhabhirama for their encouragement. I am grateful to Dr. André Daubert, Chief du Service IMA, Direction des Etudes et Recherches for his kind acceptance to act as the external examiner and to review the dissertation.
The able assistance that I received from the Water Resources Engineering Laboratory technicians is gratefully acknowledged. Thanks are also due to the Water Resources Engineering Division secretaries for their invaluable help in typing the manuscript.
I also wish to express my gratitude to His Majesty the King of Thailand, who through the Asian Institute of Technology, provided a scholarship for the period of my doctoral programme.
I am grateful to my wife, Inpah, a source of constant strength and encouragement throughout the period of my programme, and my parents, grandmother and brother for their sacrifices in order to fulfill ty educational objectives.
Finally, I thank my friends who helped me to successfully complete the experiments, and I am proud to dedicate this work to them. Will they read it

ABSTRACT
General equations are derived for shallow-flow over a two-dimen
sionally curved bed; the Saint-Venant and the recent Dressler
equations are recovered as special cases. The concept of Froude
number is generalized, and the validity of the Dressler equations
discussed. The Dressler equations are solved for steady flow,
including transition profiles. Application of these equations to
a te sted spillway crest reproduces its head-discharge relation
ship and the pressure distribution in remarkable agreement with
the experimental data. Predictions for a spillway toe also com
pare with earlier theory and experiment. Finally, new experiments
were carried out for steady flow over a symmetric and an unsym
metric profile, and the Dressler equations are found to be appli
cable in the range - 2 S. k h g 0.54 for steady frictionless flow
over curved beds. Roil waves in curved bed open channels are
briefly discussed in an appendix.

Page 4
II.
IV
W
TABLE OF саNTENTS
Acknowledgeтeтt8 Abstraat Table af Contents List of Symbols
INTRODUCTION
LITERATURE REVIEW
THEORY OF SHALLOW-FLOW OVER CURVED BEDS
3.l Basic Assumptions and Fundamental Equations 3.2. Geometry 3.3 Shallow-Flow Assumptions 3. 4. General Shallow-Flow Equations 3.5 Special Cases
3.5A Saint-Venant Equations 3.5B Dressler Equations 3.6 Generalized. Froude Number 3.7 Validity of the Dressler Equations
STEADY SOLUTIONS OF THE DRESSLER EQUATIONS
4. Steady Flow 4. 2 Stability 4.3 Transition. Profile 4.4 Flow Over a Spillway Crest 4. 5 "Flow Over a Spillway Toe
EXPERMENT
5.1 Experimental Setup 5.2 Measurements
(PLATES-I, II á III appear between pages 5-4 and 5-5) 5.3 Experimental Data and Theory 5 - 4 Results and Discussion
5.4A Symmetria Profile 5.4B Unaymetric Profile
CONCLUSIONS AND RECOMMENDATIONS
6. l. Conclusions 6.2 Recommendations
Appendiac-A On the Geometry of Curved Bed Profiles Appendiac-B Roll Waves References
- iv -
ii iii

LIST OF SYMBOLS
The page numbers in this list refer to the pages on which the symbols first appear. For those symbols having more than one meaning, an entry is made at each appropriate page number. See, for eacample, the symbol h listed opposite page numbers l-l, 2-l and 2-5.
Subscripts h, O sor o) denote values at the free surface and at the bed respectively.
Bold type a ignifies vector or matria character.
Page Sყmbo1 Meaning
l-l. K bed curvature
h flow depth normal to bed
2-l X 2 horizontal and vertical coordinate
t time
bed level from 2 is 'O
岛 gravitational acceleration h flow depth measured vertically
u, w flow velocity components in x, z-directions
三 defined by
flow rate per unit channel width
E total energy head
(), y-axes streamline and equipotential line hq0 | hy scale factors in (), p-directions * * curvatures of equipotential line and streamline V7 flow speed along streamline K (-K K.)
2-3 P pressure
О fluid density
o vertical acceleration of flow
k unit basis in 2-direction
O null vector in x2-frame
if (ac) function of x

Page 5
Page
2-5
2-6
2-7
3-1
2
Symbol
W
S
Ο
X
Meaning
coordinate normal upwards from bed radius of curvature of streamline hydraulic mean radius
Chézy and Manning roughness coefficients magnitude (nodulus)
arc length measured downstream along curved bed flow depth normal to bed
flow velocity at bed
bed slope
flow velocity components in s, n-directions Kh , ;
Dressler's local Froude number
Քo < Ph when 狩 > 芳。 Dressler equations are elliptic when if > 5
Dressler's local critical Froude number
measure of channel surface roughness
flow velocity
scalar product of two vectors
gravitational potential constant atmospheric pressure
vector product of two vectors
body force smooth two-dimensional Riemannian manifold
generally non-orthogonal Gaussian coordinates on R.
belongs to metric tensor of R“; O, 8 e til, 2} determinant of matrix of as
scalar field
surface vector in R
contravariant and covariant components of F in t-direction; O e il, 2}
e? = -e' s l and zero otherwise
three-dimensional Riemannian manifold defined by (ξ, ξ", η)
- vi -

Page
3-9
3-0
4-l
G
oBr 28
X, Y
Хg” Х.
Y (χ)
Y' (X)
X da
Meaning
metric tensor of R. i., d e (l, 2, 3} normal curvature of R in 5'-direction 1-Kn
determinant of matrix of 9i
l for a cyclic (even) permutation of ijk -li for an anticyclic (odd) permutation of ijk 0 for any two equal indices
space vector in R3
contravariant and covariant components of w in R defined by (E, E, n)
defined by (3.13) and (3.14) associated contravariant metric tensor of R
contravariant and covariant components of flow velocity V (space vector) in -direction
component of V in n-direction
angle n makes with 2
small positive number << l second order symmetric matrix defined by ( 3.24) Kronecker delta (=l. for O-8, =0 for O73)
coordinates on flat bed, along flow and normal to it
l-kn
storage (c. f. figure 3.6)
radius of curvature of bed
celerity
generalized Froude number
base of natural logarithms :
lower and upper bounds of X for applicability of Dressler equations
2g (E-Ç)/(qK) * -2geo80/ (qoko) [ (1-x) lin (1-x) ] T“
dy/dx
horizontal coordinate
a vii -

Page 6
Symbol Meaning
Գd discharge coefficient
Se energy slope
A free parameter
, Y defined by (4., lé)
Ο g'k/g
of X singular point i.e. solution of d e i = 0
defined by (4,17)
)3n quasi-normal discharge c. f. (4.2l؟
)4.22( .4c qçi tical dişcharge c. f؟
hor h c normal and critical depths
in flat bed channel flow
T (X), m (x), e (x,x) defined by (4.25a,b,c)
ζ ι, ζη, ζει di/dx, dg/dx, dig/dx
球a design head of spillway
F g/ (2gh)
n upper nappe coordinate
H operating head of spillway հ, h2 ، initial and central depths c.f. figure 4.9
W, 1 initial velocity
o (Kh, ) " "
S2 l-kh
Քo bed pressure at point of symmetry
Cp pressure coefficient
2c) toe angle
F u/ (gh)
2a/F
li a*-2a* (a-1) F*cos(þ
Ps hydrostatic pressure at point of symmetry Pc centrifugal pressure at point of symmetry Cs Co hydrostatic and centrifugal pressure
coefficients root of (4.36)
ar Wiii -

Page
5-1
Sყmbo1
(X፤ , Zu = &1 +Du )
(xi, 21.)
(x2, 2 ک( 82, K2 H2 h2
O2, 82 მ 1 , K1 C. , 3.
Po
Pc
A (X) A' (x) Ape, Ak f
R
E.
I.
I V = (X , Y) 2
Vo
Vm
S
(5, C) ξ , ξ", ζ , ζ"
Meaning
total discharge (am/s)
average level difference in mercury manometer attached to orifice (cm)
water, depth at flat bed section (cm)
point on x-axis
(X) vertically measured water depth at X (cm) experimental location of free surface
theoretical location of free surface on same bed-normal through (X, Z)
base of bed-normal through (X, Z) bed slope and curvature at (x2, .)
experimental and theoretical water depths normal to bed at (x2, .2)
Ot, 8 at (3*, 2) bed slope and curvature at (X, ) o, 3 at (X, )
bed pressure
centrifugal pressure
defined by (5.15)
dA/dx error in centrifugal pressure and in curvature
friction factor
Reynolds number relative roughness of channel surface
length of model bed profile ac-interval Xi < ac < Xi+1
vertices
Oli, m-l
(0,0)
(LO)
parameter e (o, l}
point on spline curve.
a prime denotes d/ds
- ix -

Page 7
Page
в-4
d
mb
o
T
Meaning
coordinate downstream along flat bed coordinate normal upwards from x
fiow area
flow depth riormal to bed
local flow yelocity in x-direction average of u over H
chánnel ińclination momentum coefficient = (UA) '{u°dA hydraulic mean radius
shear stress magnitude at channel surface
Partial derivatives with respeat to x, t are denoted by subscripts.
dA/d
A/ (d
H
A/dH)
F, if
b و a
ᏙᎩ
F
ثG و
free surface width
hydraulic depth
Froude number and friction factor defined by (B. 4)
c. f. figure B. l.
constants depend on assumed velocity profile characteristic bed roughness height kinematic viscosity of fluid
Reynolds number E 4UH/W
critical Froude number below which no roll waves are formed
wave-length of roll waves
bed slope
l-Kh arc length measured downstream along curved bed
frictional dissipation
Bar designates steady value Partial derivatives with respeat to s, t are denoted by subscripts
(þ, s)
F
M N و مH
و 0 , 6
0 r م0
small perturbations in , h Froude number defined by (3.35)
defined by (B. l6) real constants
complex phase-velocity critical phase-velocity measure of channel roughness
-- Χ --

Ι INTRODUCTION
Open channel hydraulics is one of the oldest disciplines of human civilization. However, it was during the relatively recent Western scientific revolution that the Saint-Venant (187l) equations were established to study various problems in open channel flow-uniform, nonuniform, steady or unsteady. The basic assumption is that vertical acceleration of fluid particles is negligible, or equivalently that the vertical pressure distribution is hydrostatic. Although the basic equations are inviscid, energy loss due to friction at channel boundaries has been incorporated by invoking either the Chézy (l769) or the Manning (la89) formula. The simplicity of the Saint-Venant equations, and their successful application to various problems in hydraulics, has led to their wide acceptance by engineers who by experience tolerate the errors introduced by the basic assumptions.
Using an asymptotic approximation in terms of a "shallowness" parameter, Friedrichs (1948) re-derived in a Cartesian frame the Saint-Venant equations for flow over flat beds. Keller (l948) applied Friedrichs' method to twoand three-dimensional flows over curved bed channels, and obtained higherapproximation equations that are either not very different from Saint-Venant or otherwise difficult to solve. On the other hand potential theory conformal mapping techniques applied to flow over curved beds, such as spillways and sills, theoretically demonstrated the importance of bed curvature. However these solution procedures are usually lengthy, and there is no convenient way of including viscous dissipation.
Recently Dressler (1978) produced new shallow-flow equations with bed curvature by applying an asymptotic approximation in terms of a "shallowness" parameter to the exact formulation of the problem in curvilinear coordinates -one coordinate directed along the bed and the other normal upwards from it. In terms of a local Froude number, he identified regions of flow separation, subcritical flow, and supercritical flow. He suggested the range -0.85 S Kh S 0.5 (K : bed curvature, h : free surface location normal to the bed) for applicability of his equations, subject to future experimental verification.
After a detailed literature review in Chapter II, this thesis emphasises theoretical and experimental work on the Dressler equations. In Chapter III the generalized shallow-flow equations are derived for flow over a twodimensional Riemannian manifold; the Saint-Venant and Dressler equations are recovered as special cases. Further, the concept of Froude number is generalized, and the validity of the Dressler equations discussed. In Chapter IV, steady solutions of the Dressler equations are obtained, and applied to steady flow over an experimentally tested Spillway crest and spillway toe. New experiments to verify the Dressler equations, involving steady flow over a symmetric and an unsymmetric bed profile, are described in Chapter W. Conclusions and recommendations are presented in Chapter VI. The experimental bed geometry of Chapter W is detailed in Appendiac A , and roll waves over curved bed open channels are briefly discussed in Appendiac B.

Page 8
I LITERATURE REVIEW
free Burfge
减 Figure 2. Definition sketch : Saint-Venant equations
For more than a century, almost all open channel flow analyses have been based upon the Saint-Venant (187l) equations (for a rectangular channel; figure 2. l) viz.
continuity, 器 + s 0 , q S uh = flow per unit width. (2.l.)
0E 京” th": total energy head. (2.2)
მu momentum t 3 O E
The classical derivation of these equations (Lamb (1945), Stoker (l948) and Fox (1977)) assumes two-dimensional incompressible irrotational invisaid flow in the constant gravitational field, over a linear bed (either horizontal or with only a small inclination, al/ax), with negligible particle acceleration in the vertical direction of flou).
In fact, the vertical acceleration is negligible (i.e. the vertical pressure distribution is hydrostatic and the vertical velocity component vanishes identically) whenever the streamlines have neither substantial curvature nor divergence. From figure 2.2, the respective intrinsic equations of conti." nuity and irrotationality are
d
(hv) as O or 嵩器· Kyv = 0 ,
Il Gv 5y(афv) α. Ο or ಸ್ಮಶy KV as O
ah 8h where ky - - - op and K E es mm Pè. are respectively the curvatures of
hᏠᏂᏛ 3) hh 0.
the equipotential line and the streamline (figure 2.3) ; equivalently
grad v s vK , k- (-ky K).
2
 
 

Literature Revie , 2-2
equipotenti line
Strograf y-axis
He -axis
که مرمر ^í/r ペ ”r* W í Gertre of crwcture of Centre o curator oSupotania ing g o of tranra at P
figure 2.2 Definition sketch intrinsic equations.
- dx
ه A. ۷۷ سامرار де ށ ر" 器沙/
V \since : <०) From similar triangles () and (2) `v
ձփ Հձնհրծվ)}óփ \, - ..., - y l/ky h'qb *业 hh ೫ * 8, and similarly from G) and G)
)ಕ್ಲಬ್ಮರಿಛಿಠಿ في نهh
ôh - - - .قبل مسيل - س مع ج -1/** ಕ್ಷೌರಿ) h¢hህ al
Figure 2.3 * ond

Page 9
Literature Review 2-3
Hence from Euler's equation of steady motion under gravity
- grad (p+pgz) am pv grad v aus pvos
the vertical acceleration az is
8 5(p+Pgz) = ه = pv'K-k (k. vertical unit basis) or az = 0 *** K = 0 +-+ p+pgz = f(ac). (2.3)
Therefore strictly speaking (2.2) is applicable with great accuracy to gradually varied flow (since the change in depth of flow is so mild that the streamlines have neither appreciable curvature (K-0) nor divergence (k0)) as well as to uniform flow.
Further, the kinematic boundary conditions at both the curved bed and the non-horizontal free surface are violated, since (2.3) defines the instantaneous streamlines to be straight and horizontal.
y - axis
(-axis
()- axis
Case of substantial streamline curvature Coso of streamline divergence
Friedrichs (l948) re-derived the Saint-Venant equations for flow over a flat horizontal bed, using asymptotic approximations in terms of a "shallowness" parameter, and Keller (l948) obtained equations for two- and three-dimensional flows over a curved bed. However, Keller's equations for two-dimensional flow are similar to (2. l) and (2.2), with the slight nonlinear bed effects only accounted for by the term 3/8x. Keller did derive higher-order approximate equations by extending this asymptotic method to better include the effects of bed curvature, but these equations proved difficult to solve and have not been exploited by engineers, (Dressler, l978).
Another approach is to treat flow over a curved bed by potential theory. Watters and Street (l964) considered two-dimensional steady ideal fluid flow over sills in open channels. By means of complex functions and conformal mapping, they cleveloped a theory for flow over a single step, then extended it to flow over a sill made out of finite line segments, and finally gene ralized it to flow over a smooth sill. Their general theory enables the calculation of velocity, pressure and free surface location for an arbitrary local change in the channel bed.
 
 
 

Literature Review , 2 4 -به
Cassidy (1965) studied irrotational flow over circular weirs, and spillways of finite height shaped after Weir-nappe profiles; he suggested a numerical technique to solve the Laplace equation. His work involved the mapping of the problem into the complex-potential plane (rectangular), the sketching of an initial approximate flow net, and the use of numerical methods. Although his theoretical discharge coefficients for irrotational flow were slightly but notably greater than experimentally measured values, the pressure and velocity distributions and free surface coordinates were in remarkable agreement with experiment. Curves for minimum pressure on the spillway surface were also developed in this study. However, the required computer storage and time (6 hours) were high.
Ali (1972) investigated flow over rounded spillways, assuming the velocity distribution ov/on = v/r (consistent with irrotational flow) ; linear variation of streamline curvature (r) between the upper and lower nappes, parabolic or cubic forms of normals to the streamlines; and cubic form for the upper nappe (free surface). This approach does not require any initial approximate flow net. He found that the choice of parabolic normal (in this event the discharge need not be known) greatly simplified the solution, and that the experimental upper nappe can be fitted quite well by general cubic equations. His calculated discharges, pressure and velocities agreed well With experimental results provided that these were measured downstream of the crest. The required computer time was around one minute.
Flow in open channels with smooth curved boundaries was analysed by Moayeri (1973). He derived a pair of integro-differential equations expressing the potential flow over a smooth step in open channels in terms of the approach Froude number and an unknown distribution of elevation as a function of velocity potential on the flow boundaries. Numerical solutions using appropriate quadrature and differentiation formulae were obtained for flows with approach Froude number F = 0.4 and l. 7. Free surface geometry, flow net and pressure distribution were also given for each value of F.
As pointed out by many investigators, (including Watters and Street (1964), and Ali (1972)), the flow of a real liquid with air above its free surface has several complications. For example, in the case of spillways, the development of a turbulent boundary layer downstream of the crest, and the consequent aeration of the flow, are governed by viscosity of the liquid and the roughness of the Solid boundary, amongst other parameters. Unfortunately there is no convenient way of modifying potential theory to include the nonconservative viscous dissipation. On the other hand, friction may be treated as an external force in the Euler's equation; an extra term (as given appropriately by the Chézy or Manning formula) representing lumped frictional effects may be "glued" to Saint-Venant and Keller equations. For example, the Saint-Venant equation (2.2) becomes
: Chézy (l769) friction term, (2.4) C છેu + B , g at 9x
7 Manning (l889) friction term, (2.5)

Page 10
Literature Review 2ー5
where C and n are Chézy and Manning roughness coefficients, respectively; and R is the hydraulic mean radius. However, the lack of attention to bed curvature effects may make these classical equations inaccurate.
Centre of curvature at P
Fro0 surfaco, n = h (s, t) (Constant atmospheric prIIIur, P, o
r(חא - !)
wts, n, t)
Gravity,
h(s,t)
ufs,n,t) q(s,t) Radius of curvature at P is r(s)
curvature at P is 1/r 5 k(s) 8 - exts
impervious bed, n - O
+ n cose (s)
Arbitrary digturn x - exis
Figure 2.4 Definition sketch: Dressler's derivation
Dressler (1978) derived new nonlinear shallow-flow equations with bed curvature used (sin) coordinates, where s is the arc length measured downstream along the curved bed, and n is normal upwards from it figure 2.4. He relaxed the assumptions of linear bed and negligible vertical acceleration of particles and applied the more familier types of boundary layer asymptotics (Prandtl (l905), Friedrichs and Dressler (l96l)) to the exact formulation of the flow problem, by introducing new independent variables (see p. 3-6) that stretch the flow domain to keep it from vanishing as the shallowness parame ter approaches its limit.
Dressler's first order results are (figure 2.4):
oٹ 2 {(l-Kh) ہوت] ۔ hنھیں جیh + ..........l? 5E (l-kh) ناos (l-Kh) K 9s
1 đKI_Kh lin (l-kh)
ਰੰ - l-Kh [2.6( , 0 = ܘܪ)
8h [ ܘܶo l buo kuنانهٔ
1- - - -- it 6( + تبعیت || -یسسسسس
3t 2 o3s (s O8 总寸 θε
2
dK Կo a K. 6 إسي سيسيكتشسجسسيصيب تحت ـ h + a 29 as O 27
s (l-Kh) 3. (2,7)
 
 
 
 
 
 
 
 
 
 
 

Literature Review 2-6
where uo (s,t)
u (s, n , t) = - (2.8)
(ln (l-Kn))?"o – 1 dk{ Kn ln (Il-Kn) | w (synt) 辟 k2ds (l-kn) + -Kn to (2.9)
and པས་
р (s , n, t) = pg (h-n) со8 Ө + 蒜唱国 (aa)?] (2.0) 萨&曼源 2"o t(1-«հ) 2 (l-ka) 2 t
Also, he derived the mass conservation relation
h Gh Øq as o (l-Kh) - 5급 - Οι α ξ u din K ln (Il-Kh) , (2.ll)
O
from (2.6) and (2.8); we have shown (see Section 3.5B) that this could be realized independently from the principle of mass conservation itself. Moreover, his momentum equation (2.7) failed to exhibit compactness in terms of any physically understandable quantity.
Dressler showed that his equations are hyperbolic', and give the Saint-Venant equations as the bed curvature approaches zero. With the definition of a local Froude number (j=u/ghcose figure 2.5) he identified regions of flow separation; and supercritical and subcritical flows.
Where s u
A cos to
elipte
, ?- f tisporato { bod prassurus Pg P 2
as eras P g 2= t s uportritika 2- .
4. -
in - i.
Figure 2.5 Regions of flow separation; and
supercritical and subcritical flow
(after Drassler, 1978 )
Suberti
ဂန္ဒီ 안
l l iAAS S SSLSZS SS0LSLLL iSiSLSLZ SLSLS0S S S 0S SS LLS0S S S SYS S LSZS0SLS S0
r-e
' The type of a partial differential equation illas aon invariant undero frame
transformations.

Page 11
Literature Review 2-7
Further, by assuming the drag at the channel surface as -u (s,n, t), he suggested a generalization of the Chézy formula for vide channels; and mentioned that the term
Xu ы. 굵은 (2. l.2)
where A : measure of roughness,
h : "hydraulic radius" (wide channels),
can be added to the right hand side of his momentum equation (2.7) when needed. He also stated that, for steady flow, the new equations define a
generalization of the Bresse (1860) profile equation, when a Chézy resistance term (modified for curvature) is added.
Dressler did not present any experimental verification of his theory, but tentatively suggested the range
- 0.85 S Kh S 0.5 (2.3)
subject to experimental check.

III THEORY OF SHALLOW-FLOW OWER CURWED BEDS
The fundamental flow equations in vector form stated below are first represented in general coordinates suitable for flow over a curved bed (a twodimensional Riemannian manifold). General shallow-flow equations are then derived, and the Saint-Venant and the Dressler equations are recovered as special cases. The theoretical validity of the Dressler equations is discu88ed in terma of a generaliaed Froude numbero.
3.l. Basic Assumptions and Fundamental Equations
For an incompressible irrotational inviscid flow under constant gravity (g) with a stable free surface of negligible surface tension, over an impervous stationary bed, the fundamental governing equations are:
div v = 0 (continuity) (3.l.)
+ grad (gE) = 0 (momentum) (3.2)
сиrlv = 0 (irrotationality) (3.3)
subject to boundary conditions of kinematic type:
Gh
a W E. 3.4 5 * V, grad h w (stable free surface) (3.4)
wo = O (impervious stationary bed), (3.5)
and of dynamic type:
P * (negligible surface tension), (3.6)
where the subscripts h, O denote values at the free surface and at the bed respectively (this notation is used throughout this chapter) and
- time
- flow velocity 2 - total energy head, gE E S2 + R + - gravitational potential p
pressure
- fluid density - normal coordinate from the bed (c. f. figure 3.l.) - flow depth normal to the bed - velocity component in the n-direction - constant atmospheric pressure
(3.7)
y
In deriving (3.2), we make use of (3.3) and the identity

Page 12
$3.2 Geometry 3-2
v • gradv = grad (v*,/2) - v x curliv
in the Euler equation of motion:
òv l
* vgradv ہے P حس it grad p
with the body force F given by the gravitational field ograd SR.
3.2 Geqretry
The choice of coordinates is commonly determined by reference to the boundary, and the geometry associated with a curved boundary (the bed) is strictly speaking not Euclidean but Riemannian. Thus the (s , n) - coordinates adopted by Dressler (1978) lead to direct involvement of bed curvature (K), characteristic of the differential geometry of the space.
Regarding the bed as a smooth two-dimensional Riemannian manifold R, we may introduce Gaussian coordinates , , , (generally non-orthogonal) with covariant symmetric metric surface tensor *og (see figure 3.1). The operations
2 = 8 axis (opposite te gravity)
Figure 3.l. Geometry of the bed ds - (να, δξ + νας δξ*αοεφ) + (να δξ*ε έηφ) , 率 aృ్యర్డోరిడో, where a, r a va, , a,, ,a'o8 qb.
vac,
Greek indices are assigned to the surface and Italic indices are assigned to the space; thus O., 8 e (1,2} and i,j,k e }1 3 و2 و{.
 

$3.2 Geometry 3-3
grad, div and curl in R are (Kyrala, 1967) :
grago - o,
l C. άει F - (Vа в“), (3.8)
l of é2
cur F a ya * 8لا و
where () is a scalar field, 08م and Fox are the respective contravariant and
•ಣ್ಣಯ್ಬant components of a surface vector F , a denotes the determinant laoBl e
the usual completely antisymmetric double index symbol (of Levi-Civita), and 9/05'' is denoted by an a following a comma.
With coordinate in normally upwards from the bed so that n=0 is R and n=h (****, t) is the free surface of flow the space" is a threedimensional Riemannian manifold R' with covariant symmetric metric space ten So
2
Ja, o 2.
2. to, , , , ;, 0, (3.9)
O O 1.
Οι centre of mornia curvature O,ኖሪ A
of r in ξ director W
M W.
w O- cortre of ornal curvur oa
- dregfloa
Figure 3.2 Geometry of the space G8ܝܟ ܨ PA/PeAo - J, and PB/PoBe - J. , ds”= gạ8ôt“ổt”+ (ồn)”,
0 ||||||||||||||||12 ص || 0 |J where tgas) -:
22 ق م] [J2 0
A 8turface vector at any point P on the surface is any vector on the tangent plane to the surface at P.
* Rhenever K> 0 or k > 0, the condition for unique representation of points
** the flou domain by the normal coordinate n i 8 n < min. Kī, K2 }.

Page 13
63.2 Gaomatry 3-4
where
)3.0( ,n) = 1 - Kon,یخ, پیج) پJo
and Ko (Ęo,Ę*) deņotes the normal aurvature of Ro in the to-direction (c.f. figure 3.2). In R* , we recall the well known formulae (Kyralia, 1967)
92:4ငှါ ဖုံ = 4,း
l dë v - f (ʻa v ( هi (3. ll)
... — li elijk aur у - e "kij
where v" and vi are the respective contravariant and covariant components of a space vector v , g denotes the determinant gii, and e the usual completely antisymmetric triple index symbol (of Levi-Civita). Further, from (3.9)
石- و تات a O (3.12)
Moreover, if we define the covariant symmetric tensor
2 -Jo J. J. Q
212 {go8} = (3.3)
2 J2a22 هته الدولة
oß - C٪ ۱ the associated contravariant tensor (defined by g oßY ಛಿ। ) is
프 22 - - 12 - ل
2 J o Ꮰ,Ꮰ, O )3.14( = {8گلاهى}
طبایی لوك ـ طـ Ꮰ,Ꮰ , C J: C
and at the bed (n=O)
3{O8 = ,0x - ܒܒ 908o " %8 and g = " , (3.15)
where of is the associated contravariant metric surface tensor of R?.
Hence the fundamental equations may be re-written
;f༧, ༧, ༦༧ g*} ܚܕܪ {JJ va w} = 0 (3.l.)

S3.2 Geometry 3-5
Gv 3E
O at + g Şट्ट५ = O (3.2O)
2ow GE 孟+8孟一9 (3.2n) it
8v Ow
- *R òn Bहुप्। O (3.3c)
8v θν )3.3n 0 = - - بخش۔ 25" ago (3 . Зn)
Gh c ôh 3 * v- a (3.4) t
w sa O (3,5)
p - II (3.6)
where ? = gz se g (Ç+nceo89) , and
oß 2 9. vove (3.7)* E E + neoa 6 + i +
፭ + nG 2g
Here 6 denotes the angle in makes with z (c. f. figure 3.3) ; (v, v', w) and (v , vz ,ҹ) are respectively the contravariant and covariant components of the flow velocity v (a space vector).
gorginta M
o)
Figure 3.3
Flow over a curved bed

Page 14
$3.4 General Shallow-Flo.) Equations 3-6 3.3 Shallow-Flow Assumptions
For shallow-flow it is assumed that the normal length scale (n-direction) relative to the characteristic length (5'-direction) is small. One can infer from continuity (3.1)* that w << vo, so that in terms of a "tag" e << 1 the terms in the fundamental equations are ordered as tabulated below.
h E v°, v, W 0ج
l l є : є 1 1 є
a àe ' ?go є 1 , l є (3.16)
er el l
The following alternative approach to the shallow-flow approximation has been widely used by hydraulic engineers in the study of open channel flow (Wehausen and Laitone, 1960). Here it is essential to nondimensionalize the variables so that the vertical (n) and the horizontal (s) distances are stretched by different amounts. Let L be a scale for horizontal measurement and H one for the vertical measurement. Defining a 'shall owneas parameter' of E (H/L) * and introducing new i nondimensional variables iš , ñ . . . by the equations
s = syo v is v/vg Ě = Evo fi = n W = w vo/g h = h È = tygo f = p/pg
we can arrive at the same conclusion as (3.16) regarding the relative order of magnitude of terms in the fundamental equations. This ordering is of course the initial step of the asymptotic derivation given by Dressler (1978) for plane shallow-flow over a curved bed (c. f. figure 2.4).
3.4 General Shallow-Flow Equations
Using (3. 26 ) , neglecting o(e") terms with respect to ο (ει) for i-j & 2, the fundamental equations reduce to the general shallow-flow equations:
Oh . . Wx )3.7( 0 ܡܫ h 孟+ άές Gv(وت وت)
მvo d (3.8)
t 十 grag (gE)=0 :
)3.19( ۷ . (و) است.
cur: Vo

$3.4 General Shal Lou)-Flou) Equatton8 3s 7
where Ρ
E (Ęo,Ę*, t) = t + heose + 诺 + gas, 0 (3.20)
)t( (3.2l,*ټ, *É*,n, t) = vo (E, *٤) vc
14 سلم - = 2 ميج 1 w (8 克° Gv (3.22) p(t,5°,n,t) — р ww
-- h = (h- aß – „0ßl 00'80
P3 (h-n) cose + kg -g 2g (3.23)
and
Ω 622 J O o 1器* or 구 )3.24( . an { = Oناهيتېت" } = {G = {G*P
... O 2
- in 321. fill s ته an
O o J2
O
For shallow-flow, (3.2n) * and (3.30) * imply that the total energy head E and the covariant space velocity components v, v2 are uniform across the flow depth - i.e. E = E (E”,5*,t) = E, and vo - voo (****,t).
From (3.l.) , (3.5) * and (3.2l) we get (3.22); and since E is independent of in from (3.2l) we get (3.23). In deriving (3.17) we integrate (3.l) * over the flow depth using (3.2l) to get
h 8 {ತತ್ಥva * ER | data, , «а givedn
O
h 0 òh
Fill s 3లోav్క - }Jو تva vه{ ga
Ο
and then invoke the kinematic boundary conditions at the bed and free surface (viz. (3. 5) * and (3.4) *) . Equations (3. l8) and (3. l9) follow immediately from (3.20) * and (3.3n) * respectively, on using (3.2l).
We now show that both Saint-Venant and Dressler equations are special cases of these general shallow-flow equations.

Page 15
.83.5 Special Ca8e8 3-8
3.5 Special Cases
3.5A Saint-Venant Equationa
Saint-Venant equations describe plane shallow-flow over a flat bed (c. f. fig
ure 3.4). In this Euclidean limit K K = 0 and, for Cartesian coordinates Х У Оn Ro, რo8 “ °a8 thus the Jacobian J 1, the metric tensor მoგ * .8
and 08ی = n&08.
Figure 3.4 Definition sketch: Saint-Venant equations
hrkts šta Στο
Since W (u, OW), the shallow-flow equations are
+ thu) в O, (c.f. (2.l)
მu 9E
,0 = كرة 8 -4 و where 2 (c.f. (2.2)
9 E(x,t) = t + hoose + + ,
Pg 2g
u (xn,t) in vo (x,t) •
- , ou °莎
w (х , n , t)
and
Ph سه (p(x, n , t
Og (h-n) aos 6 : hydro8 tatila.
3.5B Drea8 ιer Eαιμαίιoγιε
Dressler equations describe plane shallow-flow over a curved bed (c. f. figure 3.5). Orienting the axes as shown we have k = k and K2 = 0, and with arc lengths s, r measured orthogonally on R? gே " 'B' The Jacobitian J = ll-Kn,
 
 

53.5 Special Савев 3-9
Figure 3.5 Definition sketch: Dressler equations
Norizontal datum
Fo
ჟ* 0 oß lт Ј O the metric tensor {g aa} *R , and G' - 曾
O li O " -
Noting that the physical, the contravariant and the covariant components of the velocity W are related by
v = (u,0,w) = (Jv“,0,w) - (v1/J,0,w),
and at the bed
as w Ο R Vo حنا
the shallow-flow equations are
-- Pin O, c.k. (2.1l) (3.25)
3uo QE Зs“ + 8 5 - о, where (c.f. (2.7) ) (3.26)
E (s,t) hoose ت 2-تe
- - pg "h 2g '
ս (s, t) u (sin, t) = 그의 그 (3.27)
من ة 1 w (sin, t) a J ln 미
Ln J18uo - 1 dk.ln J. . Kn
JK 牌 - * (3.28)
and
p (sin, t) Ph
- "2 - ,סבן 2רץ - - (hーn)のo66 + J. 1动 (3.29)

Page 16
$3.6 Generaliaed Froude Number
rate of incregge of tro urtaco storage a لdه :{
አ مه ( - ) يلج - = مه به " . » bod, mae o レイ O 2K"h
The rat of increase of stora ا وهل
h Inflow rate, - *In 4,
Figure 3.6 Mass conservation: + = 0
NOTES: l Equation (3.25) defines maa a conservation for unit channel width
c. f. figure 3.6.
2 Equation (3.26) is a more compact form of Dressler's equation (2.7), and displays the relation with energy.
momentum
3 Equation (3.27) may be re-written
(R-n) pu = Rpuo where R = k is radius of curvature -thus angular momentum about the local centre of curvaturve ita aona tant,
3.6 Generalized Froude Number
The characteristice of Dressler equations (2.6) and (2.7) are defined by
)3.30( , , ]능an h ी مت / it
where X E Kh. Hence, for aritical flow defined by ds/dt = 0
Equation (2, 6 ) and (2... ?) can be written as
s + T = f, where r = (h uo), S = (: αγιά Τ == ( . 3r 9r dt in dr., the real eigenvalue8 given by 룹 տ0,
Noting that ds + 5
i.e. as a t wbc correspond to the characteristia direations.
 
 

S3.7 Validity of the Dressler Equations 3rall
2 c ln(l-X) Kulلt (1-X) 2 (1-X) K ဖြုံး၀os၆ 十
u (2-X) * ln(1-X)
chaos 6 x1 in (1-x) ,
Dressler (1978) defined the left member as the local Froude number 3 and identified the right member as the local critical Froude number f.
OA (3.3)
Recalling that u = u/(1-x) , (3.31) can be re-written as
)332( ، *a = ܕܲu݂
= / (1-X) Ит (1-X) where ○ コ XIl--lin (l-X) ghcos 8 (3.33)
is identifiable as the celerity (i.e. the speed of small disturbances at the free surface) in curved bed flow, and clearly as K - 0 (flat bed) the known result c = vehicose is recovered.
Thus at the critical flow, any small disturbance at the free surface travels with the same fluid particles. One may therefore preserve the definition of Froude number, originally introduced for flow over flat beds (K = 0), as the ratio of free surface speed to celerity
E.
| un{/e , (3.34)
which is l for critical flow irrespective of bed curvature; we recall that the flow is subcritical/supercritical according as F is l. From (3.31) note
that
F = У ју је . (335)
3.7 Validity of the Dressler Equations
The singularity in the celerity (c.f. (3.33) ) occurring where l+lin (l-X)=0 or X = l-e 0.632l defines an absolute upper bound x = 0.6321 for validity of the Dressler equations. Dressler (1978) suggested X = 0.5 and lower bound X = -0.85 (c.f. (2.13)). Within (X. · X) ang small ästurbance at the free strface sprade faster over a concave bed (KPO), and slower over a conveac bed (K<0), tham over a flat beả (K=0) - c. f. figure 3.7.
The shallow-flow approximation of the fundamental equations (3.l) to (3.7) due to Dressler (l978) is first Order, and in theory could be extended to higher order to extend the range of validity. Experimental verification of
* Eliminating geo86 in (3.30) u8ing (3.33), 19e get
ds — — “h + — a — VA . ( - r2 ) 1 (1-v
dt l-X 土 l-X li + (l - F“) lin (l-X) , which in the flat bed limit (k -> 0) reduces to the well known result associated with the Saint-Venant equations-ds/dt = u it vghcose .

Page 17
$3.7 Validity of the Drea8 ler Equations 3-2
Figure 3.7 "Relative celerity" c/yghcosé versus dimensionless curvature X. (The curve is imaginary for X > 0. 632l, and has minimum 0.8776 at X is -5.009.l.)
s
س
so - d.s d o.5 d.s32
جسس X
the Dressler equations for 8 teady flow is considered in subsequent chapters.
For a teady flow we note that the kinematio boundary condition (3.4)
ah h ah
wh expressed as
| - 노 의모 նih Jds implies
|dh | << | Jds (336)
under the shallow-flow assumption w << u, (3.16). It is clear that (3.36) is more readily satisfied for contea beds (K < 0) as h increases - c. f. figure 3.8.
concave bed
W rive W
Figure 3.8 Geometrical interpretation of dh<<|J,ds
 
 
 

IW STEADY SOLUTIONS OF
THE DRESSLER EQUATIONS
Steady flow solutions of the Dressler equations are considered in this Chapter. Sμbαγίίίααί, αγίίίσαί αγιά 8ιμpeγαγίίίσαι 8οίμίίοη8 βασιεί αγιά τηe Iίοααίίοη of the critical flow is identified. Flow stability a discussed, and the transition profile derived. Finally, application of the theory to flow over a teated 8pilluay ta described.
4.l. Steady Flow
For steady flow, (3.25) and (3.26) with Ph E 0 reduce to
- لاo q = -k ln (Il-kh) s constant (4.l.)
u E E + hade 6 + हूँ (l-kh) r constant, (4.2)
so that eliminating u gives the equation for the free surface (defined by h)
2.2 E a ζ + hαοεθ + [ (Il-kh) ln (Il-kh) ) * (4.3)
o
γ (χ) = α + βχ , (4.4)
where
a = 2g (E-g)/(qk)
β Ξ -2geo8θ/ (ακ)
X = Kh and Y (x) = [ (1-x) lin (l-x) ] To .
(45)
The graphical solution of (4.4) is sketched in figure 4.l, and is interpreted as follows.
ጎ((2‹)
Figure 4.l Solution of Y (X) = (+8x
4-l

Page 18
$4.l Steady Flow 4-2
Theorem (1) IF & 1 --- Y " (x) & 8. (4., 6) To prove this we recall (3.31) and (3.35); using (4.l.) to eliminate حاليا
(-kg/ln (1-x)' s - (1-X) in (1-X)
j i Ĵe --- ghcos 9 s x Il-ln (1-x))
2geo8è. v > - 2 [l+ ln (1-X) }
X ਟੋ - -
Comparing with (4.5), we get :. . .
戸 愛芳。++-8x & -Y' (X) X .
d
Therefore, when
x > 0 : s j ---8 & -Y' (x) x < 0 : J & J -- 8 & Y" (x) Hence in figure 4.l. we identify the roots
} -- | 8 | è |Y' (x) | QED.
, χ* - 8ιbαγίίίσαι fία)
X. - αγίίίσαι fίοω
s
X supercritical flow Note that
2 - fotalo - Ĵ - Yo (x) - 2 — li+bra (l-X) F = F. 6 ß ( (1-X) ln (1-x)] 3 (4.7)
Differentiating (4.3) with respect to the horizontal coordinate we have
2
dE - d - gké dx + hсо8Ө + 2g γ (χ)
aς , ακ dk. qʻkʻ — vi dih - (1-x)設+ 3. (Y(x)+Y'(x)x/2); + “ 2faර86 Y (χ) σοsθ : = ο, (4.8)
since d6/dx = (d8/ds) / (dx/ds) = K/cos 0, tar 9 = d/dx and X E Kh. Using (4.4), (4.5) and (4.7) this reduces to ,
- .t1 - IF 2 dh + کاd 2 که به یق ۱ - 1) = dE dx (l Х0 2 o-- (lit. F /2)BX埼 - (- Ff) cos 0 dx О. (4.9)
The above equation relates d/dx, dk/dx and dh/dx, and implies :
Theorem (2) In ideal shallow-flow over bed profile q = q (ac), the cerritical flow normally occurs at a point defined by
= ՑՏ ک3 + ٫۱36(l-X); + 2 (a+38x/2): 0. (4.10)
where X is given by (4.4).

54 - 2 8ίαbίίίίμ 4-3
critical depth
Figure 4.2 Symetrical convex Crest
NOTE: For symmetrical profiles d'Mdx = dk/dx is 0 at the point of symmetry,
where (4.1.0) is trivially satisfied.
At the critical 8ection, from (4.7) 8 Y' (X) or
Hence, the di 8 charge coefficient (eacaeluding any viascoua effect)
- (v3y (v) 1... / I (X) (l-X. Ca 2gco8 瓦环 (X ү (χ) ) 2X3 (1 + ln (Il-X) ] (4. llll)
For example, for k - 0: ca - l/y2 as o. 707.
4.2 Stability
dh - li diX - X diko Since dx R K dx K2 dx " from (4.9)
de - ti- a, ... (1 - ro saolao dX ... air. 2 /2 Y Av- it - r? dk. it - (-x); (l- F -sie at (l: F/2) 8x-(l F')0088కి
V 39. - ани а. - 289289 = d = 30 d = 93 or, recalling that ds είγιθ, 2k3: ٦ 8 and *百”
E - (1-v) ai - R2 αΧ. ακ dK ds (l-X) sin6 + (l- F“) cos 6 ՃՅ + 2g (2a+38x) is (4.2)
In a steady flow with viscous dissipation the momentum equation is
= - S. . (4.13)
where the energy slope S may be given by a generalized Chézy formula. Hence, from (4.12) and (4.13), we get the differential equation for the free surface
2.2 .dk ܬܵ3ܸ - aܗ?a ܙv- 11 - dХ - Sε - (1-χ) 8ιηθ 2g (2a+3.8x) is
d9 (l-F?)cose When there is no dissipation (SEO), the solution of (4.14) is of course (4.3).
(4.14)
To study the stability of flow, let us take the simplest case of frictionless flow over a constant curvature bed, when (4.i.4) reduces to

Page 19
54.2 Stability 4-4.
dX - - - (l-X) 8è20 de T :) deos 6 * (4.15)
Let us introduce an independent variable A (parameter) such that
E = Φ(θ.χ) = - (1-χ) είηθ, 驚 ، 2-به R l+ ln (Il-X) (4., l6) 3式 = Y (0,x) (4- F“) cos8 = cos6 + o ( (1-X) ln (1-X) ) *
where O = q^k/g. The autonomous system (4.16) has a 8ίηgμίαr pοίηί Ρ(θοΞ0,χο) defined by (0,x) = (6.X) = 0.
Expanding and as Taylor series about P:
Φ (θ, χ) = Φ(0,χο) + 關。 }پنج)x-xo( + . . . = - (1-X) 6 + . . . Y (0,x) = Y(0,x) + 關。 -- (4,- + = om (1-xo) (x-xo)
where : 25? Ill
)翡。 = 3 > 0 (4.l7 = (ܘm* (1-X
The integral curves of (4..6) define the free surface, and their nature in the vicinity of the singular point P can be determined from the approximate
linear systema:
, 6 (مx = - (1-x d6 2 (4.18) Å = om (LX) (XX) :
thus s
- - x - d8 (1-X) dà = e cm2 (x-x) d
Թd6 + σm (χ-Χολάχ st 9,
whence
e° + Omo (X-X) o = Constant, (4.19)
The integral curves are thus ellipses when O > 0 (i.e. k > 0) and hyperbolae when O ( 0 (i.e. k < 0).
NOTES : 1, Concave Bed
When k > 0, the singular point of the approximate linear system (4.8) is a centre (also called portea); hence, is either a centre or a focus for the original autonomous system (4.16). But dy/de = d/ has an integral (given by Y(X) se c + 3X), therefore the singular point of the original autonomous system is a centre and weak ly stabZe (Plaat, 1971).

S4.3 Transition Profila 4-5
2 Convex Bed
When k < 0, the singular point is a saddle and always unstable.
3 At the singular point P, (0,x)=0; hence, from (4.6) F=l - i.e. the
flow is critical at P and X X, or X
4.3 Transition Profile
In (4.8), writing derivatives with respect to s rather than x, and noting that d/ds is sine and dE/ds = -se, we get the backwater curve for flow over curved beds :
2
= -(1-x) tan0 t (4.20)
where the quasi-normal discharge q, and the cerritical diascharge qe are given by 圣 Se z(1-x) e.png dk " (4.2l)
↑ * É[2Y(X)+ጎ'(X)X]፰
a。手 亲器 (4.22)
In the flat bed limit (Ki > 0)
Se/g' + l/ (cono), (с : Chézy coefficient) 3 h/2 ـ جـ (3y" (X مK
so that
)4.23( , his tra6ةC- جـ في
و OS6فيgh ج- q
where he and he denote respectively the normal and the critical depths for flat bed channel flow.
The transition profile is the locus of the transition points defined by 역n = 역e (Escoffier, 1958); that is
- (1-X) sine : -2gcose Še . -K dK KY" (X) '
፵፪2Y(X)+ኘ (X)Xl
S T(x) = n(x) + e (x,x): (4.24)
where, on using Y(X) = ( (1-x) ln (l-X))*, d8/ds as K, and dx/ds = cos 6 and k = (df/dx') cose, we have
- Ω=X)Y (X) -- = (l-X) littlr (1-X) ) (X) = X+lin (l-X) (4.25a)

Page 20
64.3 Trang ition Profile 4一é
(a) centre (K > 0) " (b) saddle (k < 0)
Figure 4.3 Flow profiles around a singular point f-flow profiles f-flow profiles passing through P and asymptotic to q as 'n'
= eot6 d< - է" [1+ (%')*1 - = * - an - - (նոյ:
et (x, X) =E 22 y(x) +y' (x)x) (4.25c)
a prime (') in m(x) denotes d/dx. Figure 4.3 depicts flow profiles around a singular point (c.f. section 4.2).
m (x) 3, (4.25b)
and
At a point where dg/dx dk/ds as 0 the transition profile is not defined however, in a frictionless flow the transition profile passes through the critical point (Theorem (2)).
The transition profile for a friationless flow over the bed (x) is given by
( (4.24) with S = 0)
T(x) as m(x). (4.26)
From figure 4.4, note that (4.26) can be solved when k > 0 only if n < 0, and when k < 0 only if m < -4.908.
a .342 O O632
Figure 4.4 Sketch of the 4998- - - - - - - -ح>>یہ سسلم سمیٹتے۔
solution for T (X) = m (x)
r
ኾ(፮C
 

$4.4 Flouy Over a SpilluJay Crest 4-7
4.4 Flow Over a Spillway Crest
We apply the shallow-flow equations with bed curvature to steady flow over a spillway crest. In dimensionless form (4.3) becomes
E - ζ - (αρ8θ KHd 2 鼠 牌k e (4.27)
where H denotes a reference head ("design head")
2
X = Kh, F = (4.28)
gh
The dimensionless pressure head at the bed (n=0)
KH 2 음 = || - 품 || - F-보-', (4.29) p8H Hd H ln (1-X)
where p denotes the pressure at the bed.
chow (1959) has summarised model tests by the U.S. Army Engineers Waterways Experiment Station of so-called WES shapes for high overflow spillways. We consider the case of vertical upstream face without piers described in Section 14. 6 of that book.
The spillway crest is given nondimensionally by
85 S = - 翡 (4.30)
2 Ha
and experimental coordinates of the upper nappe profile are given for dimenSionless operating heads (excluding the velocity head) H/Ha = 0.50, 1.00 and l. 33. We consider the nappe coordinate domain o. 2 < x/H < 1.8, for which the ranges of X are summarised in the table :
卒n 2n H/ 8 - 1 كانت ثالث- 0.2 = -شش
d H 크
0.50 -0 - 320 -O. Oles 1.00 -O. 745 -O. O59 l. 33 -l - 029 -O.ll.0
It is notable that x< -o.85 near the crest for H/Ha = l. 33 somewhat Outside the tentative range of validity suggested by Dressler (1978) for experimental check of the equations.
Setting E = H, we calculated the values of the parameter F (related to the Froude number) in (4.27) to fit the experimental upper nappe profiles at the nine tabular points x/H =0.2 (0.2) 1.8. The pointwise deviation of F

Page 21
S4.4 Flo, Over a Spillvay Creet 4-8
Polinta H/H rov.
() o, so 0.02ss
Α.Ο. O .00 0.2622 A i.SS 0.6847
O s
2.0
O 巽 i " . . 吕器 “n /ዞ4 -ሙ
o 薰一一z。 雀日 به ه
s A A a .2س خشک
C
O → 4. O οι 0.2 0.3 o 2.0
H/H- Figure 4.5 Pointwise percentage Figure 4.6 Discharge versus
deviation of F operating head
from its average value is not more than 4, as shown in figure 4.5. The inferred dimensionless flows per unit width (vF) for operating heads н/на = 0.50, l. OO and l.33 are 0.163, 0.5l2 and 0.8l. 5; weirs of simpler shapes are often used for flow measurements (see for example Ackers et al., l978). Further, a logarithmic plot of averaged vf against operating head H/H shown in figure 4.6 yields the formula
H SS 7 = 0 5 默 (4.3l)
g Hi * {Ha
H 5
物 72ন্তু দ্বিসূত্র 0.5!, (4.31)
Introducing a local Froude number as defined by Dressler (l978) viz. 2 oلا... ۔ 疗三 ghaos 8 (4.32)
with E H, (4.2) reads
j н = , + + 2 (1-x) }hoeae
whence
KH as 2 (l- 唱 - , l- “d- - (4.33) 升 2 (l-x) Ha
7 Equation (4.27) defines the relation between (E/Hd, F, X). To test thia relation earperimentally one must at least know either (E/Hd, X) or (F,X). For instance, given the energy (E/Ha) and the spillway and nappe profilea (X for different x), (4.27) definea F uniquely; energy loss due to the build up of a turbulent boundary layer as we go down from the creat accounts for any slight error trend in estimating F (c.f. figure 4.5). However, given either the energy or the flow (q), using (4.31) and (4.27) one can aolve numerically for the unknown nappe.
 

$4.5 Flou) Over a SpilluJay Toe 4-9
p i | f il soprate f
Pg P. i Figure 4.7 Curves for flow separation
| and critical flow
4 : - - 2 (1-x) 容 χ (2-χ)
ے
7. - - 2
X (lt-in (l-X)
oلسبب بیبیلا --سسی
The local Froude number as a function of X (shown in figure 4.7) corresponds to supercritical flow with or without separation.
Using the averaged values of F found from the upper nappe profiles, we computed theoretical pressure profiles from (4.29) for H/H - 0.50, l.00 and 1.33, to compare with the experimental profiles in the range 0 < x/Hd < l.2 reproduced in Chow (l959) figure l4-l3: see figure 4.8. These experimental pressure profiles at the bed are affected by separation at larger heads and build up of local turbulence, particularly behind curvature discontinuities that should be avoided (Rouse and Reid, l935) but are clearly indicated in Chow's figure. Slight modification of the results to allow for the influence of the neglected approach velocity head might be expected. We also fitted the experimental pressure profiles shown in Chow (l959) as best we could to (4.29), to obtain new parameter values YF is 0.160, 0.507 and 0.797 for the respective dimensionless operating heads H/H = 0.50, l. 00 and l. 33: we ob
tained
65 ਸੂ - 0.50
d d
4.5 Flow Over a Spillway Toe
Assuming negligible potential energy, various authors have given analytic solutions for Steady ideal flow over a spillway toe. Douma (1954) and Balloffet (96l) used a "free-vortex" approximation, and Henderson and Tierney (1963) used a hodograph transformation to study irrotational flow for large curvature. A detailed discussion of their assumptions may be found in Henderson (l966) and also in Dobson (1967), who computed solutions by finite
difference methods.

Page 22
54.5 Flot. Over a Spillway Toе 4-0
t .
r
/0},
ア
x/h.
H/ 1.33
量 K片 - ... t -#ti. osxo. as) (;)
d
2
Figure 4.8 ---- tested spillway data (c.f. Chow, l959) - theory
Pressure profiles along the spillway crest
 
 
 

$4.5 Flow Over a Spillway Toe 4-ll
contre of cut voture سمس
V A . χνίδι
w A. V A. V A h \
V
w
V
V Figure 4.9 Flow at a a V. Na A. M
spillway toe
g gravity P pressor
The irrotational nature of the shallow-flow equations implicit in (3.27) leads to an identical solution if the potential energy is neglected. It follows from the Bernoulli equation (c.f. (3.7) or (4.2)) that the particle speed at the free surface is constant (u, say); hence from (4.l) and (3.29) we have the dimensionless curvature
a“ = Kh = M lnMo (4.34)
and the pressure coefficient
Ρ
三
O s -O2 Cp 千五 l-S2, (4.35)
2 2 pu:
where 2 at l-kh, p is the bed pressure at the point of symmetry, and h, h; are the initial and central depths respectively (c. f. figure 4.9). This solution is identical with that of "free-vortex" theory and is valid when Kh S 1/6 or Kh2 S O. l'85 according to Henderscon and Tierney (1963).
Of course the shallow-flow equations also readily permit a solution including the potential energy. In the symmetric case shown in figure 4.9 for example, it follows from (4.l) and (4.2) that
(õ õ)? XS + 1, (4.36) where À = 23 Flo, E a-2a (a-l) F.' coach, C. El (Kh)“ as before and F = ui/ (gh).
The bed pressure at the point of symmetry (Po) now consists of a hydrostatic component
Pe = pgh2

Page 23
$4.5 Flow Over a Spillway Toe 4-2
so
}-2
(اقت 6 Figure 4.10 Root بهمA
tinfinite F ) of equation (4.36)
and a centrifugal component
l p = i puĝ [ (1-kh,) "*-l]
or correspondingly
P 2 )4.37( (1-1) كي - وقع = ca
; pui *
- Pe 62 ・2 ce = - Il-62*) (a linn)?. (4.38)
2 pui
When the potential energy is neglected (F. oo) the hydrostatic part vanishes, so that Cp = ce. The root e* ś Â < 1 of (4.36) that corresponds to the low potential energy limit (infinite Froude number) described above is shown in figure 4.10. Corresponding centrifugal pressure profiles for various Froude numbers are shown in figure 4.ll, and we note that the bed pressure at the point of symmetry is increased when the potential energy is included. Thickening of the flow layer associated with increasing centrifugal pressure is illustrated in figure 4.12. We note that the solution validity is as before and that we have continued to neglect surface disturbances that may occur at high Froude number.
o Since x ł l-eo (eo. 6321), ve have N = 1-x # eo
 
 
 
 
 

$4.5 Flou) Over a SpilluJay Toe 4-3
C. r
N is
运 F -in ܚܠ SY
4.
d iod, and Infinity
s -
4. 2
1/(Kh) –-
Figure 4.ll Maximum centrifugal pressure versus toe curvature, for 2} = 45°
O experiment by Henderson and Tierney (1963)
. o, .
حبس اh/h
Figure 4.12 Flow layer thickening with d increasing centrifugal pressure, for 2b F 4.5

Page 24
V EXPERMENT
The eacperimental setup to test the Dressler equations for steady flow is illustrated and the measurements made are out Zined. The proce asing of eacperimental data to a form readily comparable with theoretical values is first described, before reaults for a symmetric and an unsymmetria profile are discussed. ޅ; v,
5.l Experimental Setup
The experimental setup is shown in figure 5.l. The experiments were carried out in a 915 cm x 75 cm x 44.5 cm flume made of a steel frame with glass windows on both vertical sides. The bed was elevated by lo cm using .5 cm thick plywood, to house the plastic tubes connecting the piezometer tappings along the centre line of the curved bed model (c. f. figure 5.2) and the piezometers. The flume width was vertically partitioned along the entire channel length into two compartments, again using l. 5 cm plywood. The larger compartment was 30 cm wide and served as the test channel for steady flow over curved bed models. The test section was at a distance of 366 difi froLa the inlet box. The bed pressure-piezometers were set up within the smaller compartment. The inflow to the inlet box through a 15.24 cm (6 in) dia, gast iron pipe was controlled by a gate valve.
The Dressler equations were examined for steady flow over two curved bed models, one a symmetric profile shaped after the normal distribution and the other an unsymmetric profile fashioned by B-Splines (c.f. Appendiac-A). In each case the model was fabricated as described in figure 5. 2.
5.2 Measurements
A Discharge q
A 7. 26 cm (3 in) dia. orifice, placed well before the control valve in the l5.24 cm (6 in) dia. inflow pipe, was employed to rate the inflow. Because of rapid oscillation of the mercury column in the U-tube manometer attached to both sides of the orifice, about 25 readings of the simultaneous mercury levels in both legs of the manometer were recorded at about losec. intervals to estimate the average mercury level difference H (am). The accuracy of the manometer scale was 0.1 cm. The orifice equation (at 27°C)
Q = 30 x 157.03 yh am/s. (5.)
was used to give the steady unit width discharge
q = 2/30. am/s.am. (5.1) it
5-l

5ー2
đṁas TequəurȚIəđxq I og ə Infiț¢I
Note
•v•wu, Isøys ựļļa saopuła ssos, L-L-:)----→
町 wuɔ os --— ! *るシV peq peaunɔ*|-- - - - - -}!» 義』シ る eるこga*シ シd系的守
ww -- Not L'OES
●ApA ●pg シゅ aるシ
uxo ou
əuunų真心的如
•Astya •ų 194ļuỌo aoụų
xeq se!!!no

Page 25
pƏq pəAȚnɔ əųą go UOȚąɔn ŋsuwoo əųą buȚMoqs uoȚąɔəssȚII zog ərn6ț¢
 

5 - 2 Medaluroeneritas 5-4
B Energy Head, E
A portable trolley carrying a point gauge of accuracy O. Ol Cin was placed on the two rails fixed along the flume top. At a flat bed section of the channel (e.g. the section 350 cm from the inlet box), the water depth D (cm) was measured using this point gauge to give the energy head
2
(5.2) E D
C Free Surface
The point gauge mentioned above was used to measure the water depth at every 5 cm horizontal interval along the centre line of the curved bed model.
D Bed Pressure, po/pg
Along the centre line of the curved bed model, 0.32 cm (1/8 in) dia. copper piezometer tappings were fixed at 5 cm horizontal intervals (c. f. figure 5.2) ; these were connected by long plastic tubes (0.63 cm = 1/4 in internal dia . ) to vertical water-piezometers (0. 63 cm s li/4 in external dia. glass tubes) of reading accuracy 0.1 cm.
Reading durin teady flow
Reading just after the flow dry bod
hydrostatic
-- Arbitri Det
Figure 5.3 Reading the bed pressure
As depicted in figure 5.3, the recorded piezometric level difference between the steady flow and the dry bed condition (i.e. just after slowly draining all water from the flume) gave the bed pressure head p/pg - c.f. Plate
E Fou Pattern
Potassium permanganate solution was injected from an overhead container through a 0.15 cm dia. nozzle at different points in the flow field, to trace
the flow pattern - c. f. Plate:S-III & III.

Page 26
PTE
3.
Es.
LSSDL L L LSLSLSLSLL LSSLSLSSLSLSSLSLSSLSL
First --- runn
 
 

FII
{ uz Boyɛ = 3 ptio uzooo/, væ ! ' 5III -Đ) otsiasoraaeum,*ųā sānā Haets spesił 5 (
针) No= 歴|-s 현----飓|-|× 灵
---- 圈 醋
劑』
藏
*T)
|-
3đYHOEBĘ

Page 27
{ uso costro = a pun uo'oy, wo so9TTI *b)|- =IȚĮGIẢ ĐỊTĘŁmaesīEETTI BỊ ĐẨÐ MÆTHỊ Ảge=35
|- - - - - :-)| ()
i No.
 

553 Erpérreri tal DItz (td Té2 5
5.3 Experimental Data and Theory
Although the theory des Finos the Free surfaer by the esbrdinate n normal ta LC LLLCLLS GCCCLLMLLLLLL LLLLLLLLH LLLLL LLLLaL LLLLLLCC LHHLLLLLLL CCLC LCCL LHC LLCO LLaL KCCLLLSL LLCLLL LLLLLLLLLL S SS S S LLLLLLL M CLLMMMHLLSYLLC SS LLLLLLLCLLL CCTLLCCL LLL LLLLLL
LTLL LMCCL MLMMLL CHLCL LLMLL LLH LLC LLLLLL tkLLLLLL LLLLLLLCC CCCCLC LL LLL LLLLMCLLLL CLLLHLH LLL LC CMMC S SLLeLeL S LLLLLLLLS LLLLL LCLLLLLLLS OCL LC LLeLeMLLL tC TLLLLLLL LLC LMMT LL LLLT LML L LLL0LCLCCCLLLHHL L LLaH CCCCCLLMMLCHLLLLH GCLLLLLCLLLLL S SCCLCLLLLLCCL LCLLLLLSS TCL CLaeLeL LL figure 5.4, at point X, the vertical depth D, on the int ga Liga gives the CCCCLLMLGLGLLLLL LLLLCC GGGLLOLC LOLOLOLLLLLC S SES SKSS S S SSEK S LLLLL LCL GGLe Me eLLLLL LLLLLL LLLHLLLLLLLS S S LLLCLC LLLLCLLLLC LLLCCLL LLLLH * 후, ) on this sang bod-normul thLEO Lighl (X, E) as follows.
- - -- Tħar riiigl dr Lurf Lily
prrhl trh kurld h
Figure 5.
Given the bed profile () the solution of
- x, (x + 1 } (5.
is the base ( x, ... = g(x)) of this bed-normal, and at this point We have
ಶಿಷ್ಠಿ = {" {೫}
K L" |x, Joe's | 5, 도)
the experimental flow depth
н; - Уix-х) “+(2-)*,
Td
= H్న (5.)
*experi
S GCCkk kTkL LLkLL LLLL CeLeLlTGLTT LLTTtLLLLSS TTT TkLeLe MMLL kuT kukcctkLkykT
ικαλ με Β.

Page 28
S5.4 Results and Discussion 5-6
The theoretical X is the solution of (4.4); that is
Y(X) = 0 + 32X , (5, 8)
where from (4.5), a = 2 g (E-2)/(qk) and 8 = -2.goog02/(qk). The theoreetical flow depth is
h2 theory /K2 (59)
hencę the theoretical location of the free surface is given by
X;= * - hαθιηθα (5.10)
h z ceo882 + 1 ܧ
As previously described, the bed pressure was measured by a piezometer. Since uo = -qk/lin (l-X), from (3.29) the theoretical bed pressure at X is given by
99. 22 s
- x + x(2-x)v(x), (5.11)
Hydrostatic Centrifugal part part
where X is the solution of
Y(x) = a + 8X (5. l.2)
in which O = 2g (E-1) / (qk), B = -2geo80/(qk); tane and k are respectively the bed slope and curvature at X (c. f. figure 5.4), calculated by similar formulae as (5.4) and (5.5).
Finally, the experimental and theoretical Froude numbers were calculated
from (4.7); i.e.
- /2 l-it-lrn (l-X) )5.3( , )E.I (1-x) in (1-x ܒܚ ܲ8ܐ
(Newton-Raphson iteration was used to solve (5.3), (5.8) and (5.12).
5. 4 Results and Discussion
5.4A Symmetria Profile
Figures 5.5 to 5.9 show the experimental and theoretical free surface and bed pressure for various q and E. (Although not shown, the theoretical bed pressure profiles are also symmetre about x = 0.) Agreement is excellent, although for larger q the theoretical free surface is slightly below the experimental points in the subcritical region; the inadequacy of the shallowflow approximation where the flow is deep probably accounts for this. The critical flow occurs exactly at the crest, as predicted by Theorem (2) (see

$5.4 Resultas and Discussion 5-7
also figure 5.10). Near the crest in the subcritical region, the experimental points lie below the theoretical free surface of subcritical flow as the flow accelerates into a transition region from sub to supercritical flow. For low q, no solution of (4.4) exists near the transition point-C. f. the discontinuity at the crest in figures 5.8 and 5.9. Change of velocity gradir ent may be large in the transition region, so that the basic assumptions of the Dressler equations are questionable (viz. irrotational inviscid, flow) .
TABLE
E *min. *max.
g 3 C O
m/8.am am experi. theory x (am) experi.. theory x (cm)
19.7 34.8 -0.380 -0.385 -5. O. 427 0.47 -55 lOl4.4 34. O -0.360 -0.370 -5 O. 413 O. 407 ー55 770.3 31.7 -0.304 -O. 3 lO 0.38 5سl. O. 377 -55 56. O 29. 6 -0.253 -O.256 -10 O. 355 O352 -50 359.9 27 2 م || --O - 2Ol -0.20 - 10 0.328 0.326 -50
Table-l summarises extreme X values and their locations, for the flows (various q and E) for which the symmetric profile was tested; and all values fall within the range suggested by Dressler (c.f. (2. l3)).
Figures 5.5 to 5.9 also show that the total bed pressure is accurately predicted, at least when centrifugal pressure is small. From (5.ll), the theoretical centrifugal pressure is
2 Oq s = az A (X), (5.l-4)
Λ (χ) Ξ Τα αίθάν Σ (5.15)
[ (l-X) tn (Il-X) ]? "
Considering the logarithmic partial differentation of (5.l4) with respect to K, we get
9
where
>h'Aل(8)بلe = A Pe Λ(χ) iPo/Pc - A(X)x. 2(x-3X3) in (1-x)+2xxil (5.l6)
AK/K Λ (χ) (2-X) (l-X) lin (Il-X) Ꮫ
where APG is the error in centrifugal pressure due to an error AK in bed Curvature.
Figure 5.ll depicts the variation with X of the ratio between error in the centrifugal pressure and in the bed curvature. Both small local turbulence and curvature error introduced by the flat end piezometer tappings may be the cause of the systematic error pattern in the measured bed pressure profiles in the supercritical region-c. f. figures 5.5 to 5.9.

Page 29
55.4 Rевиlta and Dü8сиваіот 5-8 س
SO 50 40 so 2 O I 2C - so - 40 - so -60
(a) Free surface
(b) Bed pressure
Ceritrifugal ܓ ܔ" ”ص صے سے
ーチ= so so 40 so nao d O ro -30 -40 -3d -so
«om- x cm) ܢܠ
ܓܠ
a.
Figure 5.5 Steady flow over the symmetric profile for q = lll.9.7 am/s.am and E = 34.8 cm
subcritical --- critical theory
- - supercritical
experiment
 
 

55.4 Rевиltв атd Ditваив8іот
o 49 29 O -O -2 s 40 است -Bo
q-e ow
(a) Free surface
s
(b) Bed pressure
ཕ༤ 86 حس سطع سے
ܘ ܢ <
contrifuga صے ہے- ~5 R* ۔ ---- ح- ص" سے سے سے s سمتی
ܓ ܔ"
-- ل- i -- Ya--───────────
o so osao KO -0 عد on 4 -so
a tem
Figure 5.6 Steady flow over the symmetric profile for q = 10l4.4 cm/s.am and E = 34.0 an
subcritical - - - critical theory
- - - www. supercritical
experiment

Page 30
85. 4 Regμίίε αγιά Dί8σμ88ίοη
5-10
30 (b) Bed pressure
28
20
15 ܘ
O
--A-ro---- O ്~" ri مگس صص O ~کہ صومہ ح
Confrifugal ۔ ۔ ۔ہ ح "ست سی سے ܢ ܓܐ*" "مصيمي
Na ■ܚ so şo 40 g ~ 10 2って m30 wad so -60
e- x (em) T~ ~ ~ ~ -ہے
Steady flow over the symmetric profile
theory
Figure 5.7
for q = 770.3 cm/s.am and E = 31.7 am
subcritical
--- critical
a - - - - supercritical J.
experiment :
 
 

55.4 Кевиltв ата Diвоиват от
(a) Free surface
(b) Bed pressure
so
2.
2O
༄
毅 Togo ܘ
حيح يجده كـنه ده الاسم س - ག༠- ལ་ عنہ بے سہی يح عكس مدن ح"ثة
གམ། ཁ་ て一」 དེ། ཟ།---- 丛 義@ so 4. so so ܚܘ 33 خد سب مخمدQ38
se- en - - -
ors
Figure 5.8 Steady flow over the symmetric profile for q = 561.0 cm/s. em and E = 29.6 cm
--- critical
«8 6 «6 4 supercritical
subcritical
8 experiment
theory

Page 31
š5.4 Rезиlt8 and Dівсизвton 5-2
(a) Free surface
(b) Bed pressure
A.逻
Tete
Contrifugal
r
Figure 5.9 Steady flow over the symmetric profile for q = 359.9 cm/s. em and E = 27.2 cm
subcritical
--- critical theory
super critical
g experiment
 

85.4 Кевиlta and Diвсизвъот 5-3
SUPERCRTCA
5. - 브
c x+in (1-x)
SUBCRITICAL
Experi. point c
| 9.7 O4.4 O 7ro.3 O 58 . Ο A 359.9
1. t OOOss l - l -O.3 -O2 -O. Ο o, O.2 O.3 Ο,4 Os
Figure 5.10 Theoretical and experimental if versus X for the symmetric profile

Page 32
S5. 4 Re 8utt8 and Dias au88 ítor 5-4
ΔP /P. AK Mk
2,337
0. r
l o.O. ー2,5 22 عح . O - 1.5 • 1.0) - O. O 0.632
Figure 5.ll Variation of error ratio
P/Pc Ak/k versus X (equation (5.16))
 

$5.4 Resultas and Disau88ion 5-5
5.4B Unsymmetric Profile
In the symmetric profile, the experimental X values fell within the range suggested by Dressler (c.f. (2.3)) . To test the validity of this range 3】。 unsymmetric bed profile, skewed upstream, was designed using a B-splined shape. Figures 5.12 to 5.15 show the experimental and theoretical free Sur face and bed pressure for different q and E. (Again, the theoretical bed pressure profiles are not continued through the crest.) The transition Zone in which the flow changes from sub to supercritical is more extensive in this case, and apparently the critical flow does not occur at the crest but somewhat downstream as expected (c.f. Theorem (2), Chapter IV). The following table gives the extreme X values and their locations. These extremes are outside Dressler's suggested range (2.13), especially for negative X (c.f. also section 3.7).
TABLE-2
E *min. Âmax.
3. C *AYS e
/s. cm experi. theory x (cm) experi. theory x (cm)
lills, 5 44。7 一2。808 -3 O20 40 0.543 0.543 5 905.3 42.9 -2.236 -2.445 45 O526 O527 5 745.8 4[ 6 | 5 0.523 0.52 45 2.260- 070 . 2-سسه 375. O 37.8 -1.608 ーl。692 40 0.502 O506 5
Bed pressure cannot be predicted for certain x (e.g. 346 cm give B. Let us call this phenomenon "normal-crossing" - since it corresponds to bed normals crossing each other. If normal-crossing occurs within the flow, then the bed normals between the respective crossing normals appear redundant for free surface prediction but necessary for bed pressure. If the domain of no solution (X 2 0.632l) does not exist within normal-crossing, then these bed normals give a third free surface prediction Dressler's non-zero Jacobian at the free surface, i.e.
소스 = 1-Y > o
l-X F
seems insufficient for uniqueness of the predicted free surface.
There is no continuous prediction of either free surface or bed pressure across the transition point (c. f. figures 5. l.2 to 5. lis). Apart from curvature error introduced by the piezometer tappings, other model fabrication curvature errors probably accounts for larger systematic errors in the bed

Page 33
55.4 Rевиltв атd Di8си88іот 5-6
pressure. (Matched B-splines give a class-2 curve - i.e. continuous together with its first two derivatives -therefore K is continuous but not dk/dx everywhere; the discontinuity in dk/dx causes the kinks in the theoretical bed pressure. )
The theoretical X E Kh values are plotted against the experimental X values in figures 5.l6 and 5.17, for the symmetric and the unsymetric profiles respectively. There is remarkable agreement for
-2 S Kh S 0.54 , (5.l7)
beyond Dressler's recommended range of validity for his equations.

85.4 Кевиlta and Diacиза от 5
r 50
محصے
40
|
20 p.
s mam
”” °---o--سo۔۔۔۔۔۔۔۔
lso
(a) Free surface 43
-40
(b) Bed pressure
fo
so سسم^بر () WO W or'n 8 کمرح \\
/ N ---0-g O N
.8 *ـــــــــــه --
N O イつTN
O ~പr o ο
í 鲇 t O ad 30 2. O iod ed O 7d SO so 4d SC 20 O
e- x (cm)
Figure 5.12 Steady flow over the unsymmetric profile
for q = lll6.5 cm/s.am and E = 44.7 an domains where in a 0.632.1/k
- subcritical
--- critical theory
----- supercritical
C. D intervals where bed pressure cannot be predicted
O experiment

Page 34
55.4 Rевиltв атd Diасиввtoт
(a) Free surface
- که سیاسی
40
C s
(b) Bed pressure
o O
s
g n 20 a
Mor- ---- ---- می ----حمام است ه
N O ১৯২ سکا۔ سم۔۔۔۔۔ ہوس<<سم°
w ح= حے o *"ܘܵܐܣܓܢܔ ༠༽
l l
s 40 O 20 IIO 30 80 d ?ס so is 40 O O
army (cm)
Figure 5.13 Steady flow over the unsymmetric profile
for q = 905.3 cm/s. cm and E = 42.9 am
domains where n 2 0.632.1/k
subcritical critical -------- supercritical C, D
C experiment
theory
intervals where bed pressure cannot be predicted
 
 
 
 

55.4 Rевиlts and Düасив8іот . . 5-19
一、一式 4
(a) Free surface -- r 43
i II
H レイ* a o: o :
(b) Bed pressure -зо" o 鼠
궁 |- 25-S o o
o
i
is
e -較》 s
ح- --سے صحیحصے - S ܓܠܠ o °سی سے سی۔ ܓܥܖܐ,ܘ
ov - ~ بیم؟ s so \
且 i l l
l so 30 援Q d loo o o so so 4. so 20 o o
s (sw)
Figure 5.14 Steady flow over the unsymmetric profile for q = 745.8 am/s. am and E = 4l.6 an
domains where n 2 0.632/K - subcritical --- critical theory ----- supercritical C, D intervals where bed pressure cannot be predicted
experiment

Page 35
$5.4 Resultas and Disasau88ion
5-20
lso 40 O 20 IIO O0 90 o 7ס so 4s 24 O c
(a) Free surface D 4s
40
s པ། s
ܒ -૩૦ ર્ક
·
S* (b) Bed pressure o -25 co
20
。 - lO
s
o O ;\ حorہ صص .0 പ്-- ”6 سسo-چہ۔-o۔۔۔۔ O ۔ ۔ مصر لسـسسسـتل مسلم ”ک*2سبم?***................ |ნo ”9’’ ’’ V* O 2c 0. O סד so 4Ο 30 20 Ο
‘ H- x { ema)
Figure 5.15 Steady flow over the unsymmetric profile
for q = 375.0 cm/s. cm and E = 37.8 am
domains where n 29.632/k
- subcritical
- - - critical theory
----- supercritical
interval where bed pressure cannot be predicted O experiment
 

85.4 Rевиltв атd Dїваив8іот
.4
0.3ح
a ܘ
· R 8
O.
o,
l i -0.4 ses aro, O. O. 0.2 9. 0,4
*experi. o
so...!
Por q {cm/s. ora)
is. 7 so,2 04 4 O 77Ο. 3 O 56, O d 359. )
-0.
-4
Figure 5.16 Theoretical versus experimental x for the
symmetric profile. The error (X
experi.
*theory has
mean O.OOl3 and standard deviation 0.0023

Page 36
š5.4 Rевиltв атd Dівси88іот 5-22
l- -حلـ ف l ܥܝܢܝ -l -i i -l i -l a S.O. a 2.0 aus 1.0
0
O r.0- O
*ܘ
Point a cm3a.cm)
2.0 6 as o 90s. 3 O 745, O sts, o
- O.3۔ـــــــــ
Figure 5.l7 Theoretical versus experimental X for the unsymmetric profile. The error "experi. s-a *theory has
mean 0.0265 and standard deviation 0.0599
 

WI CONCLUSIONS AND
RECOMMENDATIONS
6.l Conclusions
From comparison with tested spillway and new experimental data, we reach the following major conclusions.
(a)
(b)
The Dressler equations are easy to use to accurately predict the free surface and bed pressure (hydrostatic + centrifugal) for steady flow over curved beds, when friational effects are negligible.
The equations are valid for conveac beds of larger aurvature than is the case for concave beds the eacperimental range was
-2 S Kh S 0.54
with remarkable agreement between the theory and eacperiment (c.f. Dressler's range: -o.85 s kh so. 5).
Other conclusions are :
(a)
(b)
(c)
6.2
(a)
(b)
the location of the critical flow can be accurately predicted;
in supercritical flow, any error in the bed curvature affects the bed pressure considerably;
near the singular point (i.e. transition point), the basic assumptions of the Dressler equations are questionable.
Recommendations
The present study emphasises steady flow, and solution of the Dressler equations for unsteady flow is recommended for future research. Dynamic pressure variation on spillway crests during the spill of a short duration flood wave, difficult to assess experimentally, is an important engineering problem for example.
Experimental formulae are available for the "friction factor" ff ( R, e) of a flat bed, where R and E denote the Reynolds number and relative roughness of the channel surface respectively. Dimensional analysis suggests that for curved beds frf (R, E, Kh), and its evaluation is important in order to generalize the existing Moody diagram to extend the Chéay formula to curved beds. The discussion on roll waves given in Appendiac-B could then be extended.
6-l

Page 37
APPENDIX-A On the Geometry of
Curved Bed Profiles
Al Symetric Profile
The symmetric profile of length l?0 am is described by the normal distributton: -
. . 2 = 20emp (A.)
A.2 Unsymmetric Profile
The unsymmetric profile of length L is l50 am is designed using B-aplines
(De Boor, l978) as follows.
Let II be c-intervals: x < ac S i+1 , and V 三 (X , Ұ) be vertices, where t = 0, 1, • • • ,m-1 . Vo = (0,0) and V 2 (L.,0). In terms of parameter s e {0, l} in It , a point (E(s) , (s)) on the spline curve is given by
r (s) = (so s՞ s l) {-1 3 -3 i] (va-1 (A.2)
3 -6 3 0 | jv -3 0 3 0| |Vi. 1. 4 l 0 J|Vi+2
where n - E when V - X, and n + , when W - Y. Artificial vertices
2Vo — V1 . , } Vm+1 = 2Vm - Vm-l
s
V.-
(A. 3)
are defined so that the curve has zero curvature at the end points Wo and V. The entire curve from Vo to V is of class-2-i.e. the curve is continuou8 together with its first to derivatives (so that slope 6 and curvature k, but not necessarily its derivatives, are continuous).
At a given abscissa ac, to find the slope tar8 E d/d5 and the curvature k = (d/dz) cose, we splve
(s) - له - O (A. 4)
for s (using Newton-Raphson iteration) to find E', ', " and " (where prime denotes differentiation with respect to s) so that
de - s/ご r
a度下 /8
2. (A. 5) 器- c/(50°-IE/(Eり"器
A-l

For the unsymmetric profile we take m = ll and:
i Χα Yi
O О . 0.
l 9.22568 -0.109890 2 3. Oll056 3.846.53 3 40.1404.5 12.318963 4. 45.67562O 20.524.09 5 48.524623 26.7004.79 6 53. 408628 32.55536 7 7.438746 26, 29.4730 8 88.980463 ll 732882 9 lO7. 890.720 2.787639 O 4l. 578l44 0.0O8453 ll l50. O.

Page 38
APPENDIX-B Roll Waves
B.l. Roll Waves in Flat Bed Open Channels
It has been observed that in steep channels steady, gradually varied, turbulent flow theory fails beyond a certain critical Froude number as free surface instability results in roll waves of various wave-length, amplitude and phase-velocity (Cornish l934. Rouse l938). Dressler (l949) showed theoretically that roll waves cannot occur if the flow resistance is too large, although they cannot be formed without it, He considered Saint-Venant equations (c. f. p. 2-l) with resistance given by the Chézy formula for wide channels (c. f. p. 2-4), and obtained a simple necessary condition for the initiation of roll waves viz. 4g/c < tar8 (g-gravity, c-constant Chézy coefficient, and 8-channel inclination), or equivalently the Froude number F > 2.
Iwasa (l954) described one-dimensional flow in a prismoidal channel by con tinuity and momentum equations (c. f. figure B. l) ;
A + (UA) - 0, (B.l)
て
U + 8UU + gao86 H + (1-8)崇 A = gaine - R • (B2)
y- asia
gyOrage voiocity U,t)
roe surface
O ht,y S T---
is a
7ی
locai velocity profilo u(x,y)
x - Gille
N
flow Groo A(x,t)
Figure B.
where
A - flow area x - coordinate along flat bed g - gravity y - coordinate normal to bed H - flow depth Ti ~, shear stress at channel surface R - hydraulic radius p - fluid density t - time 6 - channel inclination u r local velocity 2 U - average velocity 8 E üエ s udA - momentum coefficient
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

and partial derivatives with respect to x and t are denoted by suffices.
He established a necessary condition for roll waves:
- AdRaff - slo - R (R-1) * F > { 器器 - (3 1) 8 (8. 1)} (B。3)
where the Froude number and friction factor are given by
F is "/ */制。 ! f = 8τ/ρυ". (B. 4)
Since f is assumed to vary (i.e. Chézy coefficient not constant, unlike Dressler's derivation) , (B. 3) gives a family of curves relating the critical
Froude number, Reynolds number and the friction factor (c.f. Rouse . (1965)).
Berlamont (l976) studied the roll wave phenomenon taking into account vertical velocity and acceleration. Assuming that
(a) the velocity profiles are "similar" in all cross-sections:
(b) the flow is gradually varied; (c) the bed shear stress = f(U,н)u”/8:
(d) the product of lower order derivatives of U and H are small
compared with the higher order derivatives;
he integrated the continuity and momentum equations to yield:
Hit + (UH) = 0, (B5)
UJ Ut + BUUx + gco8Ө нx + (1-6) it.
2. 2 HՍ՞ | d: b U * -늄- - Hxtt † Hxxt † باره es{ = g8ίηθ - f ότι , (B. 6)
where constants a, b and c depend upon the shape assumed for the velocity profiles, and the Darcy-Weisbach friction factor f is calculated from the White-Colebrook (Thijsse) formula:
ks/R 器 (B.7)
l - Winnaan
s -2,03 სog, ე. ே + RVf
Here the Reynolds number IR = 4UH/w (v-kinematic viscosity), and ks is characteristic bed roughness height.
Berlamont (l976) derived a fourth-order linearised equation for small amplitude disturbances of uniform flow. He observed a lower critical Froude number Fc for roll wave formation as a function of IR, ks/R, velocity distribu
i.e. u(x,y,t) = 4(洛) U (x,t) . This is eacacet for Zaminar flow (parabolic
velocity profile) and holds approacimately for turbulent flow.

Page 39
Bణా8
tion and wave-length (c. f. figure B. 2) and also explained the existence of an upper critical Froude number, beyond which no roll waves are formed.
Chory
o
2
O
Figure B.2 Lower critical Froude number for 3=l, Xso (after Berliamont, 1976)
1毒怒
پتا
g
26------------one.------
.
s
B.2 Roll Waves in Curved Bed Open Channels
B. 2A General Equation
for Lower Critical Froude Number
Shallow-flow in a curved bed open channel is defined by the pressler equations (c. f. p. 3-9)
მh 2aS. at * 家 O, (B. 8)
buo E.
- * 8 - , (B.9)
where
S2 E -kh
u (q, h, s) = -qk/ilns (B.O)
2.2 E (q,h,s) = + hdroas 9 -- 3-(n linn) o
and F(qh, s) denotes frictional dissipation at the rough channel surface.
In this Section the stability of steady flow is considered.
If as constant and h(s) denote the steady flow, the perturbed quantitlea are
q(s,t) = q (1+d)
; (B.11) h (s ,t) = h (l+ቫ)
where b (s,t), n(s,t) are small (i.e. d), n << l). Introducing (B. ll) into
(B. 8) and (B.9), and retaining only linear terms in b, n we get
п. --it, , (B. l.2)
 

-(器9。+(器)n (B. l3)
From (B. lO)
(帶@-品。(器)--鄂品
oE - in ۵E که F2 (B.14) (器 g) لا (器 հg) gh (Il- F“) cos6
where E l-kh, ūh = 氏/@, and F is given by (3.35). Partial derivatives with respect to s and t are denoted by subscripts.
Differentiating (B.13) with respect to t and eliminating ni using (B.l2), we get a second order linear equation in (b:
ಕ್ಯಲ್ಲಿ + 2ಙ್ * 14 + Mo* No * 0 (B. l5)
where - il- F°) geo86
M = (u). - @器 (B. les)
N = List (h 器)
Equation (B. 15) is general, and from its solution one may determine the lower critical. Froude number based on a stability criterion.
в.2в Special Case:
"Plane Wave" Approacimation
Let us suppose that his is small, to consider a "plane wave" approximation
(b (s,t) = a eap (to (s-ot) ] (B.17)
in which CJ se Op+zoz is a complex constant: a, O, O, and at are real constants and i = Y.-l. (B. l6) now simplifies to (hs F0, Kso0)
L = - 품d- Ελgσοsθ
8Ꭼ" a - (B. l.8) мs - (а : N = 4 gкві _i f, 포 S2 gKasin 6 + h (h 0h

Page 40
B-5
since de/ds = K, and 'F' goose is a function of Kh only.
Setting (B.17) in (B.15), taking real and imaginary parts we get (c. 7 O)
uಂ (ರಕ್ಷಿ - ೦) + 2ua೮ + MC - La = 0, (B. 9).
20 .N = 0, (B - مړdi + MoهOt - 2uمو2ueOO.
on omitting the bars for the steady flow quantities.
The stability depends on the sign of o i : if o > o, the amplitude will grow with time and the flow will be unatable. Considering marginal stability (Oi = 0) , from (B.19) and (B.20) we get
2. 2 )21 .B( ,0 = نc + L و موc - 2uO, مoOfتا )22 .c = N/M, {B ,مO where a, denotes the oritiaat phase-velocity; eliminating or, we have 》23。B《 0 = * برN* + 2u NM + Lہu
c h
to be solved for the critical Froude number F below which no roll waves are formed.
Setting (B. l8) in (B. 23), and using uĝ/ (gheo8 €) = f=荒.F*, g/uo s - k2ns, and S2 is l-X (c.f. Section 3.6) we get
F = l+Zara (l-X) (B.24)
2 [x-1) پx تلا ܘܡ 1 ln (Il-X) + D - X
where
sin 6 + h 3RFAQh B25 ஆர்க்க (B. 25)
D
For the flat bed limit (k -- 0), (B.24) reduces to
Fe = l/ll+Dd. (B. 26)
If F = Au/h" = Aq/h", then aF/0h = -(m+2)F/h, 0F/0q = 2F/q, and hence (B. 26) gives
F is 2/m (B. 27)
This recovers the familiar results F = 2 and l. 5, respectively, for Chéay
(m=l) and Manning (ms4/3) resistance terms (c. f. figure B. 2).

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克法女女女***典

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