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Monroe L. Weber- Shirk School of Civil and Environmental Engineering Elementary Fluid Dynamics: The Bernoulli Equation CEE 331 October 26, 2022

03 Bernoulli

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Page 1: 03 Bernoulli

Monroe L. Weber-Shirk

School of Civil and Environmental Engineering

Elementary Fluid Dynamics:The Bernoulli Equation

Elementary Fluid Dynamics:The Bernoulli Equation

CEE 331

April 21, 2023

Page 2: 03 Bernoulli

BernoulliAlong a Streamline

ji

z

y

x

ks

n

ˆp g a kSeparate acceleration due to gravity. Coordinate system may be in any orientation!k is vertical, s is in direction of flow, n is normal.

s

zp

ss

d

da g

Component of g in s direction

Note: No shear forces! Therefore flow must be frictionless.

Steady state (no change in p wrt time)

Page 3: 03 Bernoulli

BernoulliAlong a Streamline

s

dV Va

dt s

s

p dza

s ds

p pdp ds dn

s n

0 (n is constant along streamline, dn=0)

dp dV dzV

ds ds ds

Write acceleration as derivative wrt s

Chain rule

ds

dt

dp ds p s dV ds V s and

Can we eliminate the partial derivative?

VVs

Page 4: 03 Bernoulli

Integrate F=ma Along a Streamline

0dp VdV dz

0dp

VdV g dz

21

2 p

dpV gz C

2'

12 pp V z C

If density is constant…

But density is a function of ________.pressure

Eliminate ds

Now let’s integrate…

dp dV dzV

ds ds ds

Page 5: 03 Bernoulli

Assumptions needed for Bernoulli Equation

Eliminate the constant in the Bernoulli equation? _______________________________________

Bernoulli equation does not include ___________________________ ___________________________

Bernoulli Equation

Apply at two points along a streamline.

Mechanical energy to thermal energyHeat transfer, Shaft Work

FrictionlessSteadyConstant density (incompressible)Along a streamline

Page 6: 03 Bernoulli

Bernoulli Equation

The Bernoulli Equation is a statement of the conservation of ____________________Mechanical Energy p.e. k.e.

212 p

pgz V C

2

"2 p

p Vz C

g

Pressure headp

z

pz

2

2

V

g

Elevation head

Velocity head

Piezometric head

2

2

p Vz

g

Total head

Energy Grade Line

Hydraulic Grade Line

Page 7: 03 Bernoulli

Bernoulli Equation: Simple Case (V = 0)

Reservoir (V = 0)Put one point on the surface,

one point anywhere else

z

Elevation datum

21 2

pz z

g- =

Pressure datum 1

2

Same as we found using statics

2

"2 p

p Vz C

g

1 21 2

p pz z

We didn’t cross any streamlines

so this analysis is okay!

Page 8: 03 Bernoulli

Mechanical Energy Conserved

Hydraulic and Energy Grade Lines (neglecting losses for now)

The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)?

p

z

2

2

V

g

Elevation datum

z

Pressure datum? __________________Atmospheric pressure

Teams

z

2

2

V

g

2

"2 p

p Vz C

g

Mechanical energy

Page 9: 03 Bernoulli

Bernoulli Equation: Simple Case (p = 0 or constant)

What is an example of a fluid experiencing a change in elevation, but remaining at a constant pressure? ________

2 21 1 2 2

1 22 2

p V p Vz z

g g

( ) 22 1 2 12V g z z V= - +

2 21 2

1 22 2

V Vz z

g g

Free jet

Page 10: 03 Bernoulli

Bernoulli Equation Application:Stagnation Tube

What happens when the water starts flowing in the channel?

Does the orientation of the tube matter? _______

How high does the water rise in the stagnation tube?

How do we choose the points on the streamline?

2

p"C2

p Vz

g

Stagnation point

Yes!

Page 11: 03 Bernoulli

2

p"C2

p Vz

g

2a1a

2b

1b

Bernoulli Equation Application:Stagnation Tube

1a-2a_______________

1b-2a_______________

1a-2b_______________

_____________

Same streamline

Crosses || streamlines

Doesn’t cross streamlines

2 21 1 2 2

1 22 2

p V p Vz z

g g

z

21

22

Vz

g 21 2V gz

V = f(p)

V = f(z2)

V = f(p)

In all cases we don’t know p1

Page 12: 03 Bernoulli

Stagnation Tube

Great for measuring __________________How could you measure Q?Could you use a stagnation tube in a

pipeline?What problem might you encounter?How could you modify the stagnation tube to

solve the problem?

EGL (defined for a point)Q V dA

Page 13: 03 Bernoulli

Pitot Tubes

Used to measure air speed on airplanesCan connect a differential pressure

transducer to directly measure V2/2gCan be used to measure the flow of water

in pipelines Point measurement!

Page 14: 03 Bernoulli

Pitot Tube

VV1 =

12

Connect two ports to differential pressure transducer. Make sure Pitot tube is completely filled with the fluid that is being measured.Solve for velocity as function of pressure difference

z1 = z2 1 2

2V p p

Static pressure tapStagnation pressure tap

0

2 21 1 2 2

1 22 2

p V p Vz z

g g

Page 15: 03 Bernoulli

Relaxed Assumptions for Bernoulli Equation

Frictionless (velocity not influenced by viscosity)

Steady

Constant density (incompressible)

Along a streamline

Small energy loss (accelerating flow, short distances)

Or gradually varying

Small changes in density

Don’t cross streamlines

Page 16: 03 Bernoulli

Bernoulli Normal to the Streamlines

ks

n

ˆp g a k Separate acceleration due to gravity. Coordinate system may be in any orientation!n

zp

nn

d

da g

Component of g in n direction

Page 17: 03 Bernoulli

Bernoulli Normal to the Streamlines

2

n

Va

R

n

p dza g

n dn

p pdp ds dn

s n

0 (s is constant normal to streamline)

2dp V dzg

dn R dn

R is local radius of curvature

dp dn p n

n is toward the center of the radius of curvature

Page 18: 03 Bernoulli

Integrate F=ma Normal to the Streamlines

2

n

dp Vdn gdz C

R

(If density is constant)

2dp V dzg

dn R dn

2

n

p Vdn gz C

R

2

"n

Vp dn gz C

R

Multiply by dn

Integrate

Page 19: 03 Bernoulli

n

Pressure Change Across Streamlines

1( )V r C r

If you cross streamlines that are straight and parallel, then ___________ and the pressure is ____________.

p gz Cr+ =hydrostatic

2

"n

Vp dn gz C

R

21 "np C rdr gz C

221

"2 n

Cp r gz C

dn dr

r

As r decreases p ______________decreases

2

'

1n

p Vdn z C

g R

Page 20: 03 Bernoulli

End of pipeline?End of pipeline?

What must be happening when a horizontal pipe discharges to the atmosphere?

2

"n

Vp dn gz C

R

Try applying statics…

Streamlines must be curved!

(assume straight streamlines)

Page 21: 03 Bernoulli

Nozzle Flow Rate: Find Q

D1=30 cm

D2=10 cm

Q

90 cm

1

2 3

1 3, 0p p

2 3, 0z z

1z h

z

Coordinate system

Pressure datum____________

Crossing streamlines

Along streamlineh

h=105 cm

gage pressure

Page 22: 03 Bernoulli

Solution to Nozzle Flow

2

p"C2

p Vz

g

2

'

1n

p Vdn z C

g R

2ph

D1=30 cm

D2=10 cm

Q

h=105 cm

1

2

3

z

2 2 3 3Q V A V A 2 22

4QV

d

1 21 2

p pz z

Now along the streamline

223 32 2

2 32 2

p Vp Vz z

g g

h

2232

2 2

VVh

g g

Two unknowns…_______________Mass conservation

Page 23: 03 Bernoulli

Solution to Nozzle Flow (continued)

2 2

2 4 2 42 3

8 8Q Qhg d g d

22 4 4

3 2

8 1 1h Q

g d d

2

4 43 2

1 18

hgQ

d d

D1=30 cm

D2=10 cm

Q

h=105 cm

1

2

3

z

Page 24: 03 Bernoulli

Incorrect technique…

2

p"C2

p Vz

g

2

'

1n

p Vdn z C

g R

23

2

Vh

g p" 'C nC

223 31

1 3

12

p Vp Vdn z z

g R g

D1=30 cm

D2=10 cm

Q

h=105 cm

1

2

3

z

The constants of integration are not equal!

Page 25: 03 Bernoulli

Bernoulli Equation Applications

Stagnation tubePitot tubeFree JetsOrificeVenturiSluice gateSharp-crested weir

Applicable to contracting streamlines

(accelerating

flow).

Page 26: 03 Bernoulli

Ping Pong Ball

Why does the ping pong ball try to return to the center of the jet?What forces are acting on the ball when it is not centered on the jet?

How does the ball choose the distance above the source of the jet?

n r

Teams

2

p"C2

p Vz

g

2

'

1n

p Vdn z C

g R

Page 27: 03 Bernoulli

Summary

By integrating F=ma along a streamline we found…That energy can be converted between pressure,

elevation, and velocityThat we can understand many simple flows by

applying the Bernoulli equationHowever, the Bernoulli equation can not be

applied to flows where viscosity is large, where mechanical energy is converted into thermal energy, or where there is shaft work.

mechanical

Page 28: 03 Bernoulli

Jet Problem

How could you choose your elevation datum to help simplify the problem?

How can you pick 2 locations where you know enough of the parameters to solve for the velocity?

You have one equation (so one unknown!)

Page 29: 03 Bernoulli

Jet Solution

The 2 cm diameter jet is 5 m lower than the surface of the reservoir. What is the flow rate (Q)?

z

p

z

2

2

V

g

Elevation datum

Are the 2 points on the same streamline?

2 5 mz =-

2 21 1 2 2

1 22 2

p V p Vz z

g g

2 22V g z

2 22 2

2 224 4

d dQ V g z

Page 30: 03 Bernoulli

What is the radius of curvature at the end of the pipe?

2

n

p Vdn gz C

R

2

n

p Vdn gz C

R

dn dz

2

n

p Vz gz C

R

2

n

p Vz g C

R

2Vg

R

D1=30 cm

D2=10 cm

Q

h=105 cm

1

2

3

z

2

1

2VR

g

R1 2 0p p

Uniform velocityAssume R>>D2

Page 31: 03 Bernoulli

Example: Venturi

Page 32: 03 Bernoulli

Example: Venturi

How would you find the flow (Q) given the pressure drop between point 1 and 2 and the diameters of the two sections? You may assume the head loss is negligible. Draw the EGL and the HGL over the contracting section of the Venturi.

1 2

h

How many unknowns?What equations will you use?

Page 33: 03 Bernoulli

Example Venturi

2 21 1 2 2

1 21 22 2

p V p Vz z

g g

gV

gVpp

22

21

2221

4

1

22

221 12 d

d

g

Vpp

412

212

1

)(2

dd

ppgV

412

212

1

)(2

dd

ppgACQ v

VAQ

2211 AVAV

44

22

2

21

1

dV

dV

222

211 dVdV

21

22

21

d

dVV