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Transcript of Bernoulli's Eqn
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Bernoulli equation
Daniel Bernoulli (1710-1782)
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Bernoulli: preamble
Want to discuss the properties of a moving fluid.
Will do this initially under the simplest possible
conditions, leading to Bernoulli’s equation. The
following restrictions apply.
• Flow is inviscid, there are no viscous drag forces• Heat conduction is not possible for an inviscid
flow
• The fluid is incompressible .
• The flow is steady (velocity pattern constant).
• The paths traveled by small sections of the fluidare well defined.
• Will be implicitly using the Euler equations of
motion (discussed later)
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Coordinates and streamlines
• Each piece of fluid has velocity v .
• Steady flow, nothing changes with time at givenlocation. All particles passing through (1) end
up at (2) with velocity v
• The trajectories followed by the particles are
called streamlines.
• Describe motion is terms of distanced traveledalong streamline, s .
• Velocity given by
|v|
= dsdt
. Normal to velocity
is n̂ .
• Stream-line can bend, R is radius of curvature.
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Coordinates and streamlines
Using body fixed coordinates. If the particles change
speed along stream-line, or if stream-line bends, then
accelerations must be present.
The tangential acceleration
as = dvdt
= ∂v∂s
dsdt
= v ∂v∂s
The normal acceleration
an = v2
R
The radius of curvature R changes along the
streamline.
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Streamline coordinates
It is convenient to use a coordinate system defined in
terms of the flow streamlines. The coordinate along
the streamline is s and the coordinate normal to the
streamline is n . The unit vectors for the streamlinecoordinates are ŝ and n̂ .
The direction of ŝ will be chosen to be in the same
direction as the velocity. So v = vŝ .
s
n^
s^
V
s = 0
s = s1
s = s2
n = n2
n = n1
n = 0
Streamlines
y
x
The flow plane is covered with an orthogonal curvednet of coordinate lines and v = v(s, n)ŝ and
ŝ = ŝ(s, n) for steady flow.
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Forces on streamlines
Any particle travelling along the streamline will be
subjected to a number of forces.
The relevant Forces for Bernoulli’s equation are
gravity and pressure.
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Streamlines F = ma
Will resolve forces in directions parallel ŝ and
perpendicular n̂ to particles motions. y is out of
page, z is down, x is horizontal.
δF s = δm as = δm v ∂v∂s
= ρδV v∂v
∂s
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Streamlines F = ma
Resolve forces
δW s = δW sin θ = γδV sin θ
δW s would be zero for horizontal motion.
The pressure changes with height. Let p be pressure
in middle of fluid slab. Let p + δps be pressure in
front of slab and p− δps be pressure behind slab.From Taylors series
δps = ∂p
∂s
δs
2
The net pressure force
δF ps = ( p−
δps)δnδy−
( p + δps)δnδy
= −2δpsδnδy = −∂p∂s
δsδnδy = −∂p∂s
δV
Net force
δF s = δW s + δF ps
=
−γ sin θ − ∂p
∂s
δV
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Bernoulli equation
Equate two expressions for δF s
δF s =−γ sin θ − ∂p∂s δV = ρδV v ∂v∂s⇒−γ sin θ − ∂p
∂s
= ρ v
∂v
∂s
The change in in fluid particle speed along a
streamline is accomplished by a combination of pressure and gravity forces.
• Now use sin θ = dzds
• And v dv
ds = 1
2
dv2
ds
• And dp = ∂p∂s
ds + ∂p∂n
dn
• Along streamline dn = 0
−γ
dz
ds − dp
ds =
1
2ρ
dv2
ds
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Bernoulli, compact expression
Now making the assumption that density is constant,
Bernoulli’s equation is obtained
γ dz
ds +
dp
ds +
1
2ρ
dv2
ds = 0
d
ds
γz + p +
1
2ρv2
= 0
γz + p + 12
ρv2 = Constant
The constant density assumption (incompressible
flow) is good for liquids (sometimes gases at low
speed). Bernoulli’s equation presented in 1738
monograph Hydrodynamics by Daniel Bernoulli.
If one has compressible fluid
dp
ρ +
1
2v2 + gz = Constant
and knowledge of how ρ varies with p .
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Forces normal to streamline
The acceleration normal to the streamline is an = v2
R
where R is the local radius of curvature of the
streamline.
δF n = (δm)v
2
R = ρ δV v
2
R
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Forces normal to streamline
A change in stream direction occurs from pressure
and/or and gravity forces. Resolve forces
δW n = δW cos θ = γδV cos θ
δW n would be zero for vertical motion.
The pressure changes with height. Let p be pressure
in middle of fluid slab, p + δpn is pressure at top of slab and p− δpn be pressure at bottom of slab.From Taylors series
δpn = ∂p
∂n
δn
2
The net pressure force, δF pn
δF pn = ( p− δpn)δs δy − ( p + δpn)δs δy= −2δpnδn δy = −∂p
∂nδn δs δy = − ∂p
∂nδV
Need to combine pressure and weight forces to get
net Force
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Forces normal to streamline
Combine weight and pressure forces
δF n = δW n + δF pn
=
−γ cos θ − ∂p
∂n
δV =
ρδV v2
R
Pressure and weight forces imbalance produces the
curvature. For gas flows it is common to use
∂p
∂n = −ρv
2
R
The pressures increases
with distance away fromthe center of curvature
( ∂p∂n
is negative since
ρv2/R is positive).
For straight parallel streamlines (in gases), ∂p∂n = 0 .
No pressure change across streamlines
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Forces normal to streamline
Will consider fluid parameters normal to stream line
γ cos θ +
∂p
∂n+
ρv2
R = 0
• ∂p∂n
= dpdn
since s is constant.
• cos θ = dzdn
and so for incompressible flows
dp
dn + γ
dz
dn +
ρv2
R = 0
dp
dn + γ
dz
dn +
ρv2
R = 0
ddn
( p + γz) + ρv2
R = 0
p + γz + ρ
v2
Rdn = Constant
For a compressible substance, the best reduction is dp
ρ +
v2
R + gz = Constant
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Interpretation for incompressible flows
Along the streamline
γz + p + 12 ρv2 = Constant
Across the streamline
p + γz + ρ
v2
Rdn = Constant
The units of Bernoulli’s equations are J m−3 . This
is not surprising since both equations arose from an
integration of the equation of motion for the force
along the s and n directions.
The Bernoulli equation along the stream-line is a
statement of the work energy theorem. As the
particle moves, the pressure and gravitational forces
can do work, resulting in a change in the kineticenergy.
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Dynamic and static pressures
p + 12
ρv2 + ρgz = constant
Static pressure is the pressure as measured moving
with the fluid. (e.g. static with fluid). This is the p
term in Bernoulli’s equation. Imagine moving along
the fluid with a pressure gauge.
Some times the ρgz term in Bernoulli’s equation is
called the hydrostatic pressure. (e.g. it is the change
in pressure due to change in elevation.)
Dynamic pressure is a pressure that occurs when
kinetic energy of the flowing fluid is converted into
pressure rise. This is the pressure associated with
the 1
2ρv2
term in Bernoulli’s equation.
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Dynamic and static pressures
The static pressure at 1 can be estimated by the
height of the column.
p1 = γh3−1 + p3
= γh3−1 + γh4−3
= γh
(1) (2)
(3)
(4)
h3-1
h h4-3
ρ
Open
H
V
V 1 = V V 2 = 0
The dynamic pressure at 2 is estimated by
p2 = p1 + 1
2ρv21
The additional pressure due to the dynamic pressure
will cause the fluid to rise a height of H > h .
The point (2) is called a stagnation point.
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The stagnation point
Stagnation point
(a)
Stagnation streamline
Stagnation point
(b)
When fluid flows around any stationary body, some
of the streamlines pass over and some pass under the
object. But there is always a stagnation point where
the stagnation streamline terminates. The
stagnation pressure is
pstagnation = p + 1
2ρv2
v is velocity at some point on stream-line away from
obstruction.
The total pressure, pT
pT = p + 12 ρv2 + γz
is sum of static, dynamic and hydrostatic pressures.
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The pitot tube
Knowledge of static and stagnation pressures makes
it possible to determine the fluid velocity. Geometry
arranged so that elevation differences have little
impact. The free stream pressure is p .
Stagnation pressure
p2 = p3 = p +
1
2 ρv2
V
p
(1)
(2)
(4)
(3)
Static pressure is just p1 = p ≈ p4 . Combiningequations
p = p4 = p3 − 12
ρv2
Rearranging leads to
v =
p3 − p4ρ
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The pitot tube: complications
The main question is de-sign of pitot tubes is where
to place the orifice to mea-
sure the static pressure.
V
American Blower company
National Physical laboratory (England)
American Society of Heating & Ventilating Engineers
The static pressure doesvary along the length of
the tube. More com-
plicated analysis than
Bernoulli required here.
Also make mouth of tube smooth.
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The free jet
The free jet result was first obtained in 1643 by
Evangelista Torricelli.
p1 + 1
2ρv21 + γz1 = p2 +
1
2ρv22 + γz2
γz1 = 1
2ρv22
• p1 = patm = 0 ; gauge pressure• v1 ≈ 0 ; large surface, so v1
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The free jet
γz1 = 1
2ρv22
v2 =
2γh
ρ =
2gh
Outside nozzle, stream continues to fall and at (5)
v2 = = 2g(h + H )Result v = √ 2gh is speed of freely falling bodystarting from rest.
For the fluid, all the potential energy is converted to
kinetic energy when jet leaves tank. (assume no
viscous forces).
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Free jet, fine details
Horizontal nozzle, veloc-
ity at center line v2 is
slightly smaller than v3
and slightly larger than
v1 .
For d h , OK to usev2 as average velocity.
Streamlines cannot follow sharp corner exactly.
Would take an infinite pressure gradient to achieve
zero radius of curvature
(i.e. R = 0 ). Uniform velocity only occurs at a-a line.
Vena Contracta effect.
Jet diameter, d j is
slightly smaller than
hole diameter dh .
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Free jet, Vena contracta effect
The contraction coefficient, C d = A j/Ah is the ratio
of the jet area A j , and hole area Ah .
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Flow rates
How much water flows down a channel or through a
pipe?
The volume flow rate,
Q1 is defined as the vol-
ume of fluid that flowspast an imaginary (or
real) interface.
• Volume of fluid leaving δV = v1δtA1• Rate of volume change δV
δt = v1A1
• The volume flow rate Q = v1A1• Mass of fluid leaving δm = ρv1δtA1
• Rate fluid leaving ṁ = dmdt = ρv1A1
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Equation of continuity
For a steady state situation, the mass of fluid going
into the tank must be the same as the mass of fluid
leaving the tank.
Mass of water in = Mass of water out
ρ1A1v1 = ρ2A2v2
This is the continuity equation and forincompressible flow
A1v1 = A2v2 or Q1 = Q2
The equation of continuity and the Bernoulli’s
equation are used into conjunction to analyze many
flow situations.
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Flow rate: Example 1
Given the water velocity
at (2) is 8.0 m/s and
the pipe diameter is 0.10
m , what are the volume
and mass flow rates?
Q = vA = vπ(d/2)2
= 8.0× π0.0502 = 0.06283 m3/s
The mass flow is just Q× ρ sodm
dt = 1000× 0.06283 = 62.83 kg/s
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Flow rate: Example 2
A stream of water of wa-
ter d = 0.10 m flows
steadily from a tank of diameter D = 1.0 m
as shown in the figure.
What flow-rate is needed
from the inlet to maintain
a constant water volumein the header tank depth?
The depth of water at the
outlet is 2.0 m .
Can regard outlet as a free jet (note water level at
(1) is not going down).
v2 =
2gh =√
2× 9.8× 2.0 = 6.26 m/s⇒ Q2 = A2v2 = π(0.050)26.26 = 0.0492 m3/s
= Q1
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Flow rate measurement
One way to measure flow-rate is to place a
constriction in a pipe. The resulting change in
velocity (continuity equation), leads to a pressure
difference. The absolute fluid velocity can bedetermined from pressure difference between (1) and
(2) .
The Orifice, Nozzle and Venturi meters analysis here
ignores viscous, compressibility and other real-world
effects.
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Flow rate measurement: 2
Want to determine flow rate, need v2
v1A1 = v2A2
⇒ v1 = A2A1
v2
p2 + 1
2ρv22 = p1 +
1
2ρv21
⇒ 12
ρv22 − 1
2ρv21 = p2 − p1
⇒ 12
ρv22 − 1
2ρ
A22
A21
v22 = p2 − p1
⇒ v2
2 =
2( p2−
p1)
ρ
1− A22A21
So the flow rate is
Q = A2 2( p2 − p1)
ρ
1− A2
2
A21
The pressure differences give the flow rate. Real
world flows are 1% to 40% smaller.
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Bernoulli and Cavitation
The temperature at which water boils depends on
pressure.
T (oC) pvap (kPa)
10 1.23
20 2.34
30 4.24
40 7.34
Q
p
(Absolute
pressure)
(1) (2) (3)
Small Q
Moderate Q
Large Q Incipient cavitation
pv
0 x
The process of cavitation involves
• Fluid velocity increases• Pressure reduction• If p < pvap , water boils
• Bubbles collapse when reach high pressure partof fluid
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Bernoulli and Cavitation
Pressure transients exceeding 100 MPa can be
produced. These transients can produce structural
damage to surfaces.
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Sluice gate flow rate
The height of water in the channel can be used to
determine the flow rate of water out the reservoir.
Q = z2b
2g(z1 − z2)1− z22
z21
≈ z2b
2gz1
(b) is the width of the reservoir.
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Venturi meter problem
Determine flow
rate as a function
of the diameter of the tube.
0.2 m
Q
0.1 m D
Use venturi meter equation
Q = A2 2( p2 − p1)
ρ
1− A22A21
• A2 = πD2/4 m2
• p2− p1 = −γ × 0.20 = −9800× 0.20 = −1960 Pa• ρ = 1000 kgm−3
• 1−A22
/A21
= 1−D2/0.102 = 1− 100D2 m2
Q = πD2
4
−39201000(1− 100D2) = πD
2
0.245
(100D2 − 1)
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Sharp crested weir
• Between (1) and (2) pressure and gravitational
forces cause fluid to accelerate from v1 → v2 .• p1 = γh and p2 ≈ 0 forces cause fluid to
accelerate from v1 → v2 .
• Assume flow is like free jet. Average velocity
across weir is C 1√ 2gH , C 1 = constant.
• Flow rate is
Q = (Hb)× C 2gH = C 1b 2gH 3
The parameter C 1 is determined empirically.
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The energy line and hydraulic grade line
Consider Bernoulli equation divided by γ = ρg
p
γ +
1
2
v2
g + z = H = Constant on streamline
The dimensions of the equation are in length. There
is the pressure head, velocity head, and elevation
head. The sum, H is called the total head.
The energy line gives the total head available to a
fluid. It can be measured by measuring thestagnation pressure with a pitot tube.
The Hydraulic grade line is the line produced from
the pressure and elevation heads. It is measured with
a static pitot tube.
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The energy line and hydraulic grade line
• The energy line will be horizontal along thestream line as long as Bernoulli assumptions are
valid.
• The hydraulic grade line will not be horizontal if the fluid velocity changes along the stream line.
• If forces are present (this does occur in pipe
flows), then there will be a loss in energy and
the energy line will not be constant.
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Example: EL and HGL
A scale drawing can be used to depict the pressure in
the tank/pipe system.
• The energy line is horizontal
• The elevation head at (2) is converted intoincreased pressure head p2/γ and velocity headv22
/(2g) . The HGL decreases.
• At (3) , pressure is atmospheric. So the HGL tothe level of the pipe and the elevation head has
been converted entirely into a velocity head
v23
/(2g) .
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EL and HGL
The EL and HGL can depict whether there ispositive pressure p > patm or negative pressure
p < patm .
• The water velocity will be constant in curved
pipe (equation of continuity).
• The pressure head will increase or decrease asthe elevation head changes. Useful to know for
leaking pipes.
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Limitations on Bernoulli equation
A number of problems can invalidate the use of the
Bernoulli equation, these are compressibility effects ,
rotational effects , unsteady effects .
Compressibility effects
When can compressibility effects impact on gas
flows? Consider stagnation point
• Stagnation pressure is greater than staticpressure by 1
2ρv2 (dynamic pressure), provided
ρ constant.
• ρ will not changes too much as long as dynamic
pressure is not too large when compared to
static pressure.
• So flows at low v will be incompressible
• But dynamic pressure increases as v2
, socompressibility effects most likely at high speed.
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Compressibility effects
The isothermal model for an ideal gas, p = ρR∗T
C =
dp
ρ +
1
2v21 + gz1
C = RT
dp p
+ 12
v21 + gz1
C = RT ln p1 + 1
2v21 + gz1
This can be used to get
v21
2g + z1 +
RT
g ln( p1/p2) =
v22
2g + z2
Now write as p1/p2 = 1 + ( p1 − p2)/p2 = 1 + ∆ p/p2and use ln(1 + x) = x for x
1 .
v21
2g + z1 +
RT
g ln(1 + ∆ p/p2) =
v22
2g + z2
v21
2g + z1 +
RT
g (∆ p/p2) ≈ v
22
2g + z2
This can be reduced to the standard Bernoulliequation. Bernoulli recovered as long as pressure
differences are not large.
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Compressibility effects: Isentropic flow
This is the situation that applies when there is no
heat transfer or friction during the flow (reasonable
for many gases). This gas law is p = ρkD where k
depends on specific heat capacities. Introduce the
Mach number, Ma = v/c (ratio of the flow speed to
the speed of sound). Consideration of the pressure
ratio between free stream and stagnation points
leads to p2 − p1
p1=
kMa212
incompressible
p2 − p1 p1
=
1 +
k − 1Ma21
+ k
k−1
− 1
compressible
The compressible and
incompressible expres-
sions agree to 2% for
Ma
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Unsteady effects
Implicit in the discussion was an assumption that
the fluid flows along steady state streamlines, so
v = v(s) is a function of position along the stream
and does not contain any explicit time dependence.
If v = v(s, t) then then it would be necessary to
include this when integrating along the streamline.
p1 + 1
2ρv21 + γz1 = p2 +
1
2ρv22 + γz2 + ρ
t2t1
∂p
∂sds
The additional term does complicate matters and
can only be easily handled under restricted
circumstances. There are quasi-steady flows where
some time dependence exists, but Bernoulli’s
equations could be applied as if the flow were steady
(e.g. the draining of a tank).
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Rotational effects
Model of wake behind insect
Y. D. Afanasyev, Memorial University of
Newfoundland
Bernoulli equation describes motion of fluid particles
along streamline. If particles spin about the
streamline then Bernoulli is no longer valid.
Need to characterize irrotational and rotational flows.