Velocity Profile in Open channel
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Velocity Profile in Open Channel Flow In open channel flow, the velocity is not constant with depth. It increases from zero at the invert of the channel to a maximum value close to the water surface. Depth Velocity v1 v2 v3 The velocity difference results from the resistance to flow at the bottom and sides of the channel. The shearing stress, τ, at any point in a turbulent flow moving over a solid surface has been given by Prandtl as τ = ργ 2 dv dy 2 where ρ is the density of water and γ is a characteristic mixing length. In the region near the solid surface, Prandtl made two assumptions: 1. The mixing length is proportional to y (γ=0.4y) 2. The shear stress is constant (τ 0 ) Applying these assumptions to the above equation dv = 2.5 τ 0 ρ dy y
Transcript of Velocity Profile in Open channel
Velocity Profile in Open Channel Flow
In open channel flow, the velocity is not constant with depth. It increases from zeroat the invert of the channel to a maximum value close to the water surface.
Dep
th
Velocity
v1
v2
v3
The velocity difference results from the resistance to flow at the bottom and sides ofthe channel. The shearing stress, τ, at any point in a turbulent flow moving over asolid surface has been given by Prandtl as
τ = ργ2 dvdy
2
where ρ is the density of water and γ is a characteristic mixing length.
In the region near the solid surface, Prandtl made two assumptions:
1. The mixing length is proportional to y (γ=0.4y)
2. The shear stress is constant (τ0)
Applying these assumptions to the above equation
dv = 2.5 τ0ρ
dyy
(1)v = 2.5 τ0ρ ln
yy0
where y0 is the height up to which the velocity is zero. τ0 can be evaluated byconsidering the forces acting on a flow section.
dx
h
h-dh v
datum
Net hydrostatic force : -ρgdhShear force : -τ0Pdx
F = ma a = vdvdx
− ρgAdh− τ0Pdx= ρAdx.vdvdx
⇒ τ0 = −ρgRdhdx
+ vg
dvdx
= −ρgR d
dxh + v2
2g
τ0ρ = −gRi = v∗
2
where i is the slope of the energy line.
The quantity expressed by v* has the units of velocity and is known as the frictionvelocity or the shear velocity.
In turbulent flow, where κ is the Nikuradse sand roughness.y0 ≈ κ33
Substituting for τ0 and κ in equation 1 yields:
v = v∗2.5 ln33y
κ
The average velocity is given by
v =∫0
d
vdy
d= 2.5v∗
d ∫0
d
ln33y
κ dy
v = 2.5V∗ln 12.14dκ
v = 2.5 ln 12.14d
κ g R
12 i
12
This theoretical equation can be compared with empirical equations that have beendeveloped to express the mean velocity in an open channel:
Chezy Equation
v = CR12 i
12
⇒ C = 2.5 ln 12.14dκ g
Errors can arise in applying the Chezy equation over a wide range of depth since theconstant is actually a function of depth
Manning Equation
v = 1.49n R
23 i
12 =
1.49R
16
n
R12 i
12
⇒ 1.49R16
n ≈ 1.49d16
n = 2.5 ln 12.14dκ g
The changes in the coefficient with changes in depth is reflected in this equation. Itseems to be more suitable if n can be determined with some accuracy.
Uniform FlowUniform flow is flow in which the flow depth does not vary. In other words, thewater surface profile is parallel to the channel bed. Although this scenario rarelyoccurs, it is a reasonable approximation for the flow in open channels that aredevoid of controls. The governing equations will now be derived.
H = E + z
dHdl
= dEdl
+ dzdl
= dddl
• dEdd
+ dzdl
−i = dddl
1 − V2
gD − s
⇒ dddl
= s − i1 − V2
gD
= s − i1 − Fr2
If the flow is uniform then dddl
= 0
Therefore, in uniform flow, s is equal to i. In other words, the slope of the watersurface profile is equal to the slope of the energy line, which is also equal to theslope of the channel bed. The velocity and the depth at uniform flow can beobtained by replacing i in the Manning Equation with s.
vn = 1.49n Rn
23 s
12
Q = 1.49n AnRn
23 s
12
⇒ PnRn
53 =
Qn
1.49s12
where the subscript (n) indicates that flow is occurring at normal (uniform)conditions. Both the wetted perimeter and the hydraulic radius are functions of thedepth that depend on channel geometry.
