A modification of Chezy's formula of hydraulics

CHEF

A Modification of Chezy's Formula of Hydriullcs

Br

R. GRAN.OLSSON

Lab. v.

Teisce

'HogccoJ

KONGELiGE NORSKE VIDENSKABERS SELSAS

FORHANDLINER Bind 26 1953 Nr 15

DET KONGELIGE NORSKE VIDENSKABERS SELSKABS FORHANDLINGER Bind 26 1953 Nr 15

532.51

A Modification of Chezy's Formula of Hydraulics

B

H. GBANOLSSON.

(Innsendt til GeneraIsekretren l4de september 1953)

1. The pressure drop in a tube maybe expressed in the following

man-ner [1]

A.. j

where 4p Is the pressure drop, the resistance coefficient, e the density of the fluid, v the mean value of velocity, 1 the length and d the diameter of

the tube. .

For another cross-section we instead of d introduce the hydraulic mean depth R5 ,-defined as the ratio of the cross-section A and the <<wetted

peri-meter>> P ,. i.e. Rh = A/P. The pressure drop may then be written

(i) 'P=-?i- QV2 j-A I

'h

The pressure drop may be expressed as (Fig. 1)

where i is the slope of 1 e v e I. From Eq (1) and (2) we now obtain

which is the Chezy's formula.

On the other hand the pressure drop must counterbalance the shearing stresses on the boundary surface. Hence we have

(p1p2)A=rlP

For the shear stress we set

₌

62 - D. K. N. V. S. FORHANDLINGER BD 26 1953, NR 15

(3a)

v=T'giR

Comparing Eqs (3) and (3a) we have

(5)

We now introduce the Reynolds' number as the ratio between the forces of inertia and the friction forces i.e.

(6)

ru,12 Tii, V2*

where again u, = QV1,? is introduced [2]

By comparison of Eqs (5) and (6) we have

=

V=

and from this by means of Eq (3)

(3b) v=R112VgiR

By taking the acceleration of gravity g under the square root, one has as a coefficient the square root of Reynolds' number.

2. The velocity with which a slight levation is propagated in a canal with shallow water is (Ii = depth of water).

(7)

For a broad river course we have approximately Rh = h and therefore from Eq (3b)

(3c) V = R '121/g I h

The velocity v1 from Eq (7) forms the boundary between <<tranquil flow>> and <<shooting>>. Comparison of Eqs (7) and (3e) gives as condition for the two shapes of flow

(8)

Ri1

where the upper sign corresponds to the <<shooting flow>>, and the lower

sign to the <<tranquil flow>>.

In a coordinate system with a Rand iaxis the hyperbola R i = 1 will

divide the R-i-plane in two regions, each of which corresponds to the dif-ferent types of flow, the region above the hyperbola representing the region of <<shooting flow>>, whereas the region between the hyperbola and the axis corresponds to the <<tranquil flow>>. (Fig. 2).

The vertical velocity component of the velocity from Eq (3c) is ap-proximately

Fig. 2.

A simple graphical construction in Fig. 3 shows the connection between the three different velocities. The circle in Fig. 3

has a radius equal

(1 + i) (gh)l/2/2. Eq (8) enables Us to

determine the actual value of Reynolds' number, which we need for calculation

of the mean velocity according to Eq (3c). Assume we have a water course

as a canal wher,e we may put the hydraulic mean depth equal to the real depth Ii. The water volume flowing past in unit of time may at the begin-fling be so large that the water flow is <<tranquilflow>>. By decreasingwater volume the water depth h also decreases and for a certain value of hwe have the transition from <<tranquil>> to <<shooting flow>>. As the slope i is known

and for state of transition we have, according to Eq (8) Ri= 1

Reynolds' number R = j-1 Thus we obtain Reynolds' number by a very simple experiment.

The product Ri has for this problem a similar meaning as the Mach

number M in the case of subsonic and supersonic flow, where M -- 1

cor-responds to the transition from one state of flow to another.

PRANDTL: _{Essentials of Fluid Dynamics, Blackie and Son Ltd., London and Glasgow}

1952 p. 161.

R. GRAN OLssoN: Some Remarks on Reynolds' Number, Kgl. Norske Videnskabers Sciskabs Forhandlinger, Vol. 26 (1952) No. 3.

I'rvkI lode okiober 1958

I konnnifsjon boa F. Emas Bokhande

Aketrykkeriet I Trondhjem

R. GRAN OLSSON: A Modification of Chezy's Formula 63

= R112 I Vgih

We therefore have

vv17=Ri(gih)

and in

<tranquil>> to <<shooting flow>> because

of Ri=1

(lOa)

vv0=gih.

But according to Eq (3a) with R = h we also have

(3d)

v=gih

and therefore from Eqs (lOa) and (3d) U: U,, = Vu