Critical Flow Computation
Transitions Problems
Transition Problem (By changing width)
a. Compute the contraction for width of the channel for
producing critical flow
b. The depth and change in water level produced by
i. A smooth contraction in width to 1.5 m
ii. A smooth contraction in width to 0.75 m
iii. A smooth expansion in width of 3.5 m
Assume Q does not change, neglect head loss and α = 1.0
Transition Problem (Combinations of transitions)
Water flows in a 6 m wide rectangular channel at a depth of 2
m and a velocity of 2 m/s. the channel is contracted to a width
of 3 m. How much the channel bottom is to be simultaneously
raised or lowered for the flow to be possible as specified?
Neglect energy loss and take α = 1.0
Transition Problem
The flow is taking part a section shown in Figure. The step
height is 4.57 cm. The upstream depth 45.7 cm. The water
surface drops by 7.63 cm from its original level on the step.
Determine the discharge.
Uniform Flow
Introduction
 a flow is said to be uniform if its properties remain constant
with respect to distance
 the depth of flow, flow velocity , discharge remain constant
along the channel
 therefore, uniform flow is possible only in prismatic channels
 the slope of the energy line Sf , slope of the water surface Sw
and bottom slope S0 will all be equal to each other
Uniform Flow Formulas
CHEZY EQUATION/ CHEZY FORMULA
Uniform Flow Formulas
CHEZY EQUATION/ CHEZY FORMULA
there is no acceleration in uniform flow. By applying the
momentum equation to a control volume encompassing Sections 1
and 2, distance L apart as
P1 − W sin θ − Ff − P2 = M2 − M1 ………………………………………..(1)
where P1 and P2 are the pressure forces and M1 and M2 are the
momentum fluxes at Sections 1 and 2 respectively W = weight to
fluid in the control volume and Ff = shear force at the boundary
Uniform Flow Formulas
CHEZY EQUATION/ CHEZY FORMULA
Since the fl ow is uniform, P1 = P2 and M1 = M2
Also, W = γ AL and Ff = τ0 PL
where τ0 = average shear stress on the wetted perimeter of length
P and γ = unit weight of water. Replacing sinθ by S0 (= bottom
slope
Eq. (1) can be written as
γ AL S0 = τ0 PL …….. (2) or τ0 = γ A/P S0 = γ R S0 ……….. (3)
where R = A/P is defined as the hydraulic radius
Uniform Flow Formulas
CHEZY EQUATION/ CHEZY FORMULA
Expressing the average shear stress τ0 as τ0 = k ρ V ^2 ,
where k = a coefficient which depends on the nature of the surface
and flow parameters, Eq. (3) can be written as
K ρ V ^2 = γ R S0 leading to V = C (R S)^(1/2) ……………………(4)
where C = (γ /ρ * 1 /k ) ^ ½ = a coefficient which depends on
the nature of the surface and the flow
Uniform Flow Formulas
CHEZY EQUATION/ CHEZY FORMULA
Equation 4 is known as the Chezy formula after the French
engineer Antoine Chezy, who is credited with developing this basic
simple relationship in 1769
The dimensions of C are [L^1/2 T ^(−1)] and it can be made
dimensionless by dividing it by g
The coefficient C is known as the Chezy coefficient
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
Pipe Flow: A surface can be termed hydraulically smooth, rough or in
transition depending on the relative thickness of the roughness
magnitude to the thickness of the laminar sub-layer. The classification is
as follows:
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
For pipe flow, the Darcy–Weisbach equation is
where hf = head loss due to friction in a pipe of diameter D and
length L; f = Darcy–Weisbach friction factor
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
For smooth pipes, f is found to be a function of the Reynolds
number only
For rough turbulent flows, f is a function of the relative
roughness (εs/D) and type of roughness and is independent of
the Reynolds number
In the transition regime, both the Reynolds number and relative
roughness play important roles
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
variation of f in various regimes of flow:
It is usual to show the variation of f
with Re and εs/D by a threeparameter graph known as the
Moody chart
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
Open Channels: For purposes of flow resistance which essentially
takes place in a thin layer adjacent to the wall, an open channel
can be considered to be a conduit cut into two
The hydraulic radius would then be the appropriate length
parameter and prediction of friction factor f can be done by using
eqn 1 to 4
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
Darcy–weisbach equation can then be written for an open channel
flow as
which on rearranging gives
Uniform Flow Formulas
DARCY–WEISBACH FRICTION FACTOR FORMULA
Noting that for uniform flow in an open channel hf / L = slope
of the energy line =Sf = S0 , it may be seen that
If f is involved on both sides of the equations. Simplified
empirical forms of, which are accurate enough for all
practical purposes, are given by Jain as follows: