Chapter 3

All rights reserved. 2003 A. Jaafar

Chapter 3: The Bernoulli

EquationNewton’s Second Law

F=ma along a streamline

F=ma normal to a streamline

Physical interpretations

Static, Stagnation, Dynamic and Total Pressure

Examples of use of the Bernoulli Equation

The energy line and the hydraulic grade line

Restrictions of use of the Bernoulli Equation


Newton’s 2nd Law

The net force acting on the fluid

particle must equal its mass times its

acceleration

For inviscid fluid, we are assuming

that the fluid motion is governed by

pressure and gravity forces only


Newton’s 2nd Law (cont.)

Streamlines – lines that are tangent to the velocity vectors throughout the flow field

Along the streamline,

Normal to the streamline,

Note : For steady, inviscid, incompressible flow, the pressure variation across streamline is merely hydrostatic (because of gravity alone), even though the fluid is in motion

s

VVas

R

Van

2



The equation of motion along the

streamline direction is

A change in fluid particle speed is

accomplished by the appropriate

combination of pressure and particle

weight along the streamline

s

VV

s

p

sin



(cont.)Rearranging and integrating the

equation for inviscid, incompressible

flow gives

zVp 2

2

1Constant along a streamline

Applicable to steady flows only


Example 1

Some animals have learned to take advantage of the Bernoulli effect.

For example, a typical prairie dog burrow contains two entrances

– a flat front door and a mounded back door. When the wind blows

with velocity Vo across the front door, the average velocity across

the back door is greater than Vo because of the mound. Assume

the air velocity across the back door is 1.07Vo. For a wind velocity

of 6 m/s, what pressure difference, p1-p2, is generated to provide a

fresh air flow within the burrow.


F=ma normal to a

streamlineFor steady, inviscid, incompressible

flow

zdnR

Vp

2

Constant across the streamline


F=ma normal to a

streamline (cont.)When the fluid travels along a curved path, a net force directed towards the center of curvature is required, due to either gravity or pressure or both.

When the streamlines are straight, the centrifugal effect is negligible and the pressure variation across the streamline is due to gravity aloneeven though the fluid is in motion.


Physical Interpretation

An equivalent form of the Bernoulli

Equation

Hzg

Vp

2

2

Constant along a streamline

Elevation head-related to potential energy of the particle

Velocity head-vertical distance needed for the fluid to fall freely (neglecting friction) if it is to reach V from rest

Pressure head-height of the column of fluid that is needed to produce the pressure p

Total head

Pressure head + Elevation head = Piezometric head


Static, stagnation, Dynamic

and Total Pressure

TpzVp 2

2

1Constant along a streamline

Dynamic pressure

Hydrostatic pressure

Total pressure

Static pressureActual thermodynamic pressure



and Total Pressure (cont.)For 2 points at the same height with

V2=02

1122

1Vpp

Figure 1 : Measurement of

static and stagnation

pressures



and Total Pressure (cont.)Then, p2 is called the stagnation

pressure

The pressure at stagnation point, p2,

is greater than the static pressure, p1

There is a stagnation point on any

stationary body that is placed into a

flowing fluid


Example 2Air is drawn into a small open circuit wind tunnel as shown.

Atmospheric pressure is 98.7 kPa (abs) and the temperature is

27C. If viscous effects are negligible, determine the pressure at

the stagnation point on the nose of the airplane. Also determine

the manometer reading, h, for the manometer attached to the

static pressure tap within the test section of the wind tunnel if the

air velocity within the test section is 60 m/s.


Pitot Static Tube

Fluid speed can be calculated if we

know the values of the static and

stagnation pressures in a fluid.

Figure 2 : The Pitot static tube

43

14

2

21

3

2 ppV

ppp

Vpp


Examples of use of the

Bernoulli Equation

Free Jets

Assumptionsz1=h, z2=0

Reservoir is large, V1=0

Reservoir is open to atmosphere, p1=0 gage

Fluid leaves as a free jet, p2=0

Once outside nozzle, the stream continues as a free jet, p5=0

Figure 3 : Vertical flow from a tank



Bernoulli Equation

Free Jets (cont.)

Figure 4 : Vertical flow from a tank

ghh

22v2

Hhg 2v5


Example 3

For the system in the figure, h= 36 ft and the diameter of the side

opening is 2 in. Find the

(a) Jet velocity in units of ft/s

(b) Volume flow rate in units of gallon per min. (gpm)


Example 4

A smooth plastic, 10-m long garden hose with an inside diameter of

15 mm is used to drain a wading pool as shown. If viscous effects

are neglected, what is the flowrate from the pool?


Example 5

Water is siphoned from the tank

as shown. The water

barometer indicates a

reading of 30.2 ft. Determine

the maximum value of h

allowed without cavitation

occurring. Note that the

pressure of the vapor in the

closed end of the barometer

equals the vapor pressure.


Example 6

Water flows from a large tank as shown. Atmospheric pressure is

14.5 psia and the vapor pressure is 2.88 psia. If viscous effects are

neglected, at what height, h, will cavitation begin?



Bernoulli EquationFree Jets (cont.)

– If exit of tank is

not smooth, well

contoured nozzle,

the diameter of the

jet will be less than

the diameter of the

hole – vena

contracta effect

– Contraction coef.,

Cc=Aj/Ahholejet

Figure 3 : Typical flow patterns and

contraction coef. for various round exit

configurations



Bernoulli EquationConfined Flows

In many cases, fluid is confined and its

pressure cannot be prescribed a priori –

need to use the concept of conservation

of mass

Figure 5 : Steady flow into and out of a tank



Bernoulli EquationConfined Flows (cont.)

In such case, mass is conserved, i.e. inflow rate must equal to the outflow rate

In general, following Bernoulli, an increase in velocity (could be due to reduction of flow area) is accompanied by a decrease in pressure

For flows of liquids, this may result in cavitation, a potentially dangerous situation that results when liquid pressure is reduced to vapor pressure and the liquid “boils”.

ible)incompress(ifor 211222111 VAVAVAVA

AVmAVQ ,



Bernoulli EquationFlowrate measurement

Assumptions – steady, inviscid and

incompressible

Figure 6 : Typical devices for measuring

flowrate in pipes



Bernoulli EquationFlowrate measurement (cont.)

Between points (1) and (2)

2

12

212

2211

2

221

2

2

121

1

1

)(2

hence

and

AA

ppAQ

VAVAQ

VpVp



Bernoulli EquationFlowrate measurement (cont.)

The actual measured flowrate, Qactual will

be smaller than this theoretical results

because of the assumptions made in

deriving the Bernoulli Equation

Other flowmeters based on Bernoulli

equation are used to measure flowrates

in open channels such as flumes and

irrigation ditches.


The Energy Line and the

Hydraulic Grade Line

Energy line is a line that represents

the total head available to the fluid

Under the assumptions of the

Bernoulli equation, the energy line is

horizontal

If the fluid velocity changes along the

streamline, the hydraulic grade line

will not be horizontal




(cont.)

Figure 7 : Representation of the

energy line and the hydraulic

grade line

Measures the sum of

the pressure head and

the elevation head.

The sum is called

piezometric head




(cont.)

Figure 8 : Representation of the energy line and the hydraulic grade line for flow from a

tank




(cont.)The distance from the pipe to HGL in

Fig. 7 indicates the pressure within

the pipe.

If the pipe lies below HGL the pressure

within the pipe is positive

If the pipe lies above HGL the pressure

is negative


Example 7

Draw the energy line and the hydraulic grade line for the flow of

Example 6.


Restrictions of use of the

Bernoulli Equation

Assumptions involved in deriving the

Bernoulli equation

Fluid is incompressible – ok with liquids

Flow is steady

Inviscid flow

In the absence of viscous effects, the total energy of

the system remains constant

There are no mechanical devices in the system

between the two points along the streamline to

which the equation is applied

Chapter 3

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