Energy and Hydraulic
Grade Lines
Prof. Balázs M. Fekete
Office: Steinman Room #188
Hours: Monday – Thursday 12:30pm – 2:00pm
E-Mail: [email protected]
Website: http://tulip.ccny.cuny.edu/CE35000
Username: CE35000 Password: fluidMechanics
2014-10-22
CE 35000 Fluid Mechanics
1
Rectangular sharp-crested weir
1
2
H
dh
h
Crest
L
H
dQ=L dh √ 2 g h
2014-10-20
Q= √ 2 g L ∫0 √ h dh
2
3/2
Q= √ 2 g L H
3
CE 35000 Fluid Mechanics
2
Triangular (V-notch) weir
1
b
2
dh
θ
H
h
θ
dA=2 h tan dh
2
θ
dQ=2 h tan √ 2 g h dh
2
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H
Q= √ 2 g tan θ ∫0 h √ h dh
2
2
5/2
θ
Q= √ 2 g tan H
5
2
CE 35000 Fluid Mechanics
3
Static, Stagnation, Dynamic
and Total Pressure
Stagnation pressure
1
p 2= p1 + ρU 21
2
Total pressure
1
p+ ρ U 2 + γ z= pT
2
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Stagnation Point Flow – Video
On any body in a flowing fluid
there is a stagnation point. Some
of the fluid flows "over" and some
"under" the body. The dividing line
(the stagnation streamline)
terminates at the stagnation point
on the body.
As indicated by the dye filaments
in the water flowing past a
streamlined object, the velocity
decreases as the fluid
approaches the stagnation point.
The pressure at the stagnation
point (the stagnation pressure) is
that pressure obtained when a
flowing fluid is decelerated to zero
speed by a frictionless process.
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CE 35000 Fluid Mechanics
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Pilot-static Probe
p 4= p1= p
1
p3− p 4 = ρU 2
2
p3− p 4
U= 2 ρ
√
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Airspeed Indicator – Video
The stagnation pressure at the tip
of a Pitot tube is greater than the
static pressure by an amount
equal to one-half times the fluid
density times the speed squared.
Thus, the speed can be
determined by measuring the
pressure difference.
The Pitot tube on an aircraft is
connected to the air speed
indicator, a pressure transducer
calibrated to read in knots or
miles per hour rather than psi.
Because the air density varies
with temperature, pressure, and
altitude, the speed displayed on
the air speed indicator (termed
the indicated air speed) equals
the actual speed at which the
airplane is flying through the air
only for standard density
conditions.
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Pressure Distribution along
Pilot-static Tube
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Energy and Hydraulic Grade Lines
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Hydraulic and Energy Grade Lines
Under the assumptions of
the Bernoulli equation, the
energy line is horizontal.
2
p U
γ + 2 g + z=Total Head
2
U
=Velocity Head
2g
p
γ + z= piezometric head
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Energy and Hydraulic Grade Lines
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The Use of Energy and
Hydraulic Grade Lines
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Siphon from Tank
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Compressibility Effect
ρU
Δ p= p stag − p static=
2
2
p U2
+
+ z=constant
ρg 2g
2
dp U
∫ ρ + 2 + gz=constant
Isothermal flow
dp U 2
RT∫ +
+ gz=constant
p
2
p=ρ R T
ρ=
p
RT
2
2
U
U
RT
RT
ln p1 + 1 + z 1=
ln p 2 + 2 + z 2
g
2g
g
2g
( )
2
U 12
p
U
RT
+ z 1+
ln 1 = 2 + z 2
2g
g
p2
2g
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CE 35000 Fluid Mechanics
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Isentropic Flow
2
dp U
∫ ρ + 2 + gz=constant
k is the ratio of specific heat at constant pressure (cp)
And specific heat and constant volume (cv).
1/ k
p
ρ= 1/ k
C
p
=C
k
ρ
C
1/ k
∫p
−1/ k
p2
C
1/ k
∫
p= p 1
−1/ k
p
dp=C
2
U
dp +
+ gz=constant
2
1/ k
[
k
p2
k −1
k −1
k
− p1
k −1
k
2
]
(
p2 p1
k
=
−ρ
ρ
2
1
k −1
)
2
k p1 U 1
k p2 U 2
+
+ z 1=
+
+ z2
ρ
ρ
1
2
k −1
2g
k −1
2g
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Stagnation Flow in Compressible Fluid
V1
Ma 1=
C1
C 1=√ kRT
p1=ρ1 R T
Ma 1=
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V1
√ kRT
[(
[ ]
p2 − p1
k −1
= 1+
Ma 12
p1
2
p2 − p1
V 12
=
p1
2 RT
[
p2− p1 k Ma 12
=
p1
2
)
k
k −1
]
Compressible
Incompressible
]
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Pressure Ratio as a Function of
Mach Number
Maximum 2% difference up to Ma = 0.3
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Compressible Flow – Mach Number
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Unsteady Flow Effect
Steady Flow
U =U ( s )
Unsteady Flow
U =U ( s , t )
ρ
∂U
∂s
∂U
∂U
as=
+U
∂t
∂s
a s =U
∂U
1
ds+dp+ ρ d ( U 2 ) + γ dz=0
∂t
2
Along a streamline
s2
1
∂U
1
p1 + U 12 + γ z=ρ ∫
ds+ p 2 + ρ U 22 + γ z
2
2
s=s ∂ t
1
Valid for incompressible inviscid flow
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Unsteady Flow – YouTube
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Oscillation in U-tube – Video
Many flows are true steady flows.
For others which are not precisely
steady, their unsteadiness may be
of secondary importance and they
can be treated as quasi-steady
flows. However, for other flows, the
unsteadiness is a critical factor in
their behavior.
The oscillation of a liquid in a Utube is an important flow for which
unsteady effects are of critical
importance. When the liquid in the
tube is released from a state with
one end higher than the other, it
oscillates back-and-forth in a way
that is very similar to the oscillation
of a simple pendulum. Because of
viscous effects in the oscillating
fluid within the tube, the
oscillations finally die out, just as a
pendulum will stop swinging
because of the aerodynamic drag
on it.
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Bernoulli Equation Across Streamlines
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Bernoulli Equation Across Streamlines 2
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Flow over a Cavity
In some situations it is valid to apply
the Bernoulli equation across
streamlines. However, for flows that
involve rotational effects, such use of
the Bernoulli equation produces
incorrect results.
The swirling flow in the rectangular
cavity shown is produced by the
interaction of the cavity fluid with the
horizontal flow across the cavity's
open top. The viscous shear stress
produced by the large velocity
gradient at the interface of these two
flow segments drags the cavity fluid
along and produces the circular
motion within the cavity. In addition
to this overall "rotation", there is
considerable rotation of each minute
fluid particle at the interface.
Because of this effect, it is not valid
to use the Bernoulli equation across
the streamlines from the main flow
into the cavity flow.
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Other Restrictions to the Bernoulli Equation
Another restriction on the Bernoulli equation is that the flow is inviscid. The total
energy of the system remains constant.
If viscous effects are important the system is nonconservative “dissipative” and
energy losses occur.
The final basic restriction on use of the Bernoulli equation is that there are no
mechanical devices “pumps or turbines” in the system between the two points
along the streamline for which the equation is applied.
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Hydraulic and Energy Grade Lines
Under the assumptions of
the Bernoulli equation, the
energy line is horizontal.
p V2
γ + 2 g + z=Total Head
V2
=Velocity Head
2g
p
γ + z= piezometric head
Munson; et al. 2012
2014-03-25
CE 35000 Fluid Mechanics
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Open Channel
U2
2g
( U +dU / dl )
2g
dY dZ
+
dl dl
2
Q
H
Y
∂Q
dl
∂l
Q+
Y+
dl
dY
dl
H+
dH
dl
dZ / dl
Z
Z+
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CE 35000 Fluid Mechanics
dZ
dl
27
Critical Flow Conditions
Kinetic
3
Static
Q2
E kinetic =
2g A2
2.5
Specific
E static=Y
Depth [m]
2
Yc
1.5
U 2c
2g
Q2
E specific=
+Y
2g A2
1
Critical State
0.5
0
0
0.5
1
1.5
2
2.5
3
Energy Head [m]
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Hydraulic Jump 1
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Hydraulic Jump 2
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Hydraulic Jump in the Kitchen
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Hydraulic Jump – Video
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