FLOOD06873

• County: Statewide
• Basin: Statewide
• Title: Hydraulic Engineering volume 1
• Date: 1/1/1994
• Prepared By: American Society of Civil Engineers
• Floodplain - Doc Type: Educational/Technical/Reference Information

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CIlE2Y FOIlMULA VERSUS TIlEORY AND COMMON SENS~.
Yltold Gustaw Slrupczewskl. Professor Or.
ABSTRACT. Using the theoretical concepts of the uniform flow formula
and the Inverse solution of the Chezy equation it is shown that a
channel shape may have considerable Inf'luence on the value of the
Hanning roughness coefficient.
INTRODUCtION. A varlabllity of the Hanning roughness coetflcient n
Is usuallY put down to a bed lining but not to a channel shape. The
aim of the paper Is to present theoretical findings pointing out a
dependence of the Hanning n on a channel shape. The theoretical
concepts of uniform flow is used for that. Then the two paradoxes
are shown which solely arise from the assumption that the n value
reflects the surface roughness but not a channel shape. It comes
from the common sense that for steady state in a channel wi th
uniform lining and without flow obstruction: (1) any increase of
water level is related with increase of flow discharge: (2) an
increase of channel width above any level causes an inc~ease in flow
capacu.y for any other level above it. By solvIng the two Inverse
problems of Chezy formula, I.e. by derivation of channel shape over
an initial level for the given flot.l area-floW discharge and flow
depth-rlow discharge curveS, violation of the both axioms is shown.
THEORETICAL coNCEPTS OF UNIFORM FtOV. For practical purpose, a
turbulent uniform flow formula Is usually expressed as

Q = K 5112 = a A. Rr-tst/2 (l)

where a denotes resistance factor. Q and A flow discharge and flow
area respectively, R - hydraulic radIuS, S - bottom slope and the
term K is known as the conveyance.
SInce both surface roughness andwater surface slope are assumed
here to be constant, it is convenient to express the Chezy equation
as the product of the slope-friction factor (SF) and the geometric
factor (Cl

Q = SF . G
where SF as the constant would be further neglected while
C = Ar Ipr-t
where p_wetted perimeter, r equals 3/2 and 5/~ for Chezy and
friction law respectively.

(Ial

(21
Hanning

1
Institute of Geophysics, 64 Ks. JanuSZa, 01-452 Warsaw, PL

662

.

CHEZY fORMULA V. THEORY

663

Let us see what can be deduced about a relation of t.he SF w1th
a channel shape shap" from the rational b
r I r asis of any monomial
ormu a or uniform flow in an open channel
(al For uniform flow there is a balance between bo
and gravity so that undary resistance

T
.

P = r

131

A S

where ~. is the average boundary sh nd
ear a 7 the specific gravity of
the water.
(b) In the case of rough turbulent now in a uniform chann I
relgular shape, the simplest assumption for lhe velocity (:) o~ a
g 'lien distance from the rough bou d ita a
the flow velocity to the so-called ns~;:r V~lO~i~;P~:~~net:easratio of

or after substitution the

'1,1.= .,J'{ Ip
.
value of T
.

from equation (31

(4)

v.=.,JgRS
where p and g are the fluid density and
acceleration respectively, as a function of the
from the boundary, I.e.
~. = ~[~.r

where Yo Is a representative roughness height.

. In the case of the Chezy formula and the Hanning formula this
unIversal velocity relationship is wrItten as

v y p

-.=a(yl
v .
with p : 0 for Chezy and p = 1/6 for Hanning
~~:_ In the two-dimensional case of a wi'de open channel or the
be ~ymmetric case of a semi-circular channel flowing full th
un ary shear will be the same at all points on the ~tte~
~~~~:~e~l ~~dal\hehalengtt: of the normals from the boundary to the
'lie e same value (y ). Under these conditions
~:e ~elocity distribution formula of equ:;~on (Sa) can be integrated
the gr:~atlo:ns~~; mean velocity (V) in the wide rectangular seelion
h
Iv dA
.

(4al
the gravitational
relative distance

(5)

15a)

v
.

.
a v B h
= - fyPdy =
A y P 0
o
channel, and In

a v. B h1p.ftJ

(6!

=

A

(n+O A yP
o
the semi-circular

section the

W'h~re B -width of the
relationship

h
Iv dA
.

1f a v. h(P+2)

.
. a v
.
.. - fyP.ft dy =
A yP 0
equation (4a) and

depth (hI by

(7)

v
..

.

(n+2) A yP
expressing the

A
Substituting for y.