Integral Relation for a Control Volume (Part 2)

8/11/2019 Integral Relation for a Control Volume (Part 2)

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Lecture 6

Mechanics of Fluids 1: Lecture 6: Integral Relations for CV (Part 2) Department of Mechanical Engineering MEHB223

Integral Relation forIntegral Relation fora Control Volumea Control Volume

(Part 2)(Part 2)


Integral Relation for CV :Integral Relation for CV :Pressure Variation in FlowingPressure Variation in FlowingFluids and Bernoulli EquationFluids and Bernoulli Equation


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Chapter Summary


Bernoulli Equation Applications of Bernoulli Equation Separations and Its Effect on Pressure

Variation

6.1. IntroductionPressure


mpor ance o now ng pressure variations

Force Structure Interaction Hydraulic Turbomachines Cavitation Aerodynamic Lift and Drag

Variation

Medical Blood Pressure Meteorological Cyclonic Storms


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6.2. Basic

Cause

of

Pressure

Variation Pressure in fluid can varies due to weight,

acceleration and viscous resistance


To Accelerate a mass of fluid in certain direction, there must be a net force in the direction

e.g. in pipe

6.2. Basic Cause of Pressure Variation Pressure Variation due to weight and acceleration


( ) ll

a z p =+

Note that if acceleration is zero. The equation is reduced back to hydrostatic equation


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6.2. Basic

Cause

of

Pressure

Variation

Euler E uation o Motion


ll

=

6.3. Bernoulli Equation Bernoulli Equation along a Streamline :


Applying Euler Equation along a pathline :

Tangential component of acceleration is given by : ( ) t a z ps

=+

t

a n

For steady and incompressible flow t V

sV

V a t +

=

( )

=

=+

2

2V ss

V V z p

s


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6.3. Bernoulli

Equation

Bernoulli Equation along a Streamline :


Integrating along a streamline :

02

2

=

++

V

z ps

C V

z p =++2

2

Daniel Bernoulli

This is the BE which states that the sum of piezometrichead and kinetic pressure is constant along a streamline for steady flow of an incompressible fluid

P1, z1, V1 P2, z2, V2 22

22

22

21

11V

z pV

z p ++=++

6.3. Bernoulli Equation Bernoulli Equation for Irrotational Flow :

For irrotational low BE is also a licable normal to the


streamline Thus BE is applicable anywhere in the irrotational flow.

Example 3 :

C V

z p =++2

2

P2, z2, V2 , A2

P1, z1, V1 , A1 22

22

22

21

11V

z pV

z p ++=++


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6.4. Applications

of

BE Stagnation Tube :

A l in BE rom 12 :


Or :

Usin H drostatic E n :

2

21

1 2 p

V p =+

( )12212

p pV =

Hence :

( ) and 21 d g pgd p +== l

( ) lg p pV 22 121 ==

6.4. Applications of BE Pitot Tube :

P1 measure the sta nation ressure


P2 measure the static pressure


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6.4. Applications

of

BE Pitot Tube :

A l in BE rom 12 : 22 V V


V 1 = 0, z1 ~ z2. Solving for V 2 gives :

If manometer is connected to both taps

2211 22gz pgz p ++=++

( )2122

p pV =

and assuming m >> f :

Where h1 & h2 are height of the manometer columns

( )122 2 hhgV f m

=

6.4. Applications of BE Pitot Tube is commonly used to measure speed



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6.4. Applications

of

BE Pitot Tube (Example 4)


ow me er ng ev ce cons s sof a stagnation probe at station2 and a static pressure tap atstation 1. A 2=0.5A 1. Air with adensity of 1.2 kg/m3 flowsthrough the duct. A water

between the stagnation probe

and the pressure tap, and adeflection of 10 cm ismeasured. What is the velocityat station 2.

6.4. Applications of BE Velocity of fluid exiting from a large tank

A l BE AB alon the


streamline

P A = PB = Patm V A = 0

A

B B

B A A

A gzV

pgzV

p ++=++22

22

z AzB = h Hence :

ghV B 2=


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6.4. Applications

of

BE Pressure variation in a tornado


1

2

3

2

2max

2V

p p o =2

max1 V p p o =

Pressure difference between the centre and outside part of the vortex gives rise to secondary flow radially inward

6.4. Applications of BE Pressure variation near curved boundaries


Applying BE along a streamtube (assuming irrotational) :

oo

o gzV

pgzV

p ++=++22

22


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6.4. Applications

of

BE Pressure variation near curved boundaries :


. .

Combining with MCE and defining Pressure Coefficient, C p :

( )222

V V p p oo =

2

2 1

== o p

V p pC

If n is the streamline spacing :

oo

22

2 112

=

==

nn

V V

V

p pC o

oo

o p

6.4. Applications of BE Pressure variation near curved boundaries :


2

2 12

==

V V

V

p pC

oo

o p

2

1

=

nn

C o p


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6.4. Applications

of

BE Pressure variation around a circular cylinder :


22

2 112

=

==

nn

V V

V

p pC o

oo

o p

6.4. Applications of BE Pressure variation around a circular cylinder :


2

2 12

==

oo

o p V

V

V

p pC


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6.4. Applications

of

BE Pressure variation around a circular cylinder.


The pressure coefficientdistribution on a cylinder in across flow is given by

2sin41= pC

Where is the angulardisplacement from the forwardstagnation point. Assume that 2

pressure taps are located at +-30 o as shown and connected toa water manometer. Thecylinder is immersed in air

with a density of 1.2 kg/m 3

and a velocity of 50 m/s in thedirection as shown in the figur What will be the deflection onthe manometer, in cm ?

6.5. Separation and Its Effect on Pressure Variation


, . This is called no slip condition Boundary layer will be developed very close to the body

If the main stream is accelerating (pressure decreasing or negative pressure gradient), boundary layer will remain thin

If the main stream is deccelerating (pressure increasing or adverse pressure gradient), boundary layer will separate

Separation point


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6.5. Separation

and

Its

Effect

on

Pressure Variation

Hence on the front part streamlines are similar to ideal


However downstream, separation occurs

Separation starts to occur at Reynolds Number of 50.

6.5. Separation and Its Effect on Pressure Variation


, Separation occurs at boundary discontinuity


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6.5. Separation

and

Its

Effect

on

Pressure Variation


Pressure remains almost constant within the separation zone This causes force imbalance towards the right (drag)


End ofEnd of Lecture 6Lecture 6

Integral Relation for a Control Volume (Part 2)

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