mech 221 review

8/12/2019 MECH 221 Review

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MECH 221 FLUID MECHANICS(Fall 06/07)

REVIEW

http://www.galleryoffluidmechanics.com/misc/splashb.htm



MECH 221 – Review

2

What Have You Learnt? 1. Fluid Statics

2. Fluids in Motions

3. Kinematics of Fluid Motion

4. Integral and Differential Forms of Equations of Motion

5. Dimensional Analysis

6. Inviscid Flows

7. Boundary Layer Flows

8. Flows in Pipes

9. Open Channel Flows

On coming week lectures




MECH 221 – Review

3

Fluid Statics

It is to calculate the fluid pressure when thefluid is no moving

Shear stress is due to relative motion of fluid,so no shear stress and only normal stress(Pressure) acting on the fluid

The fluid pressure is only due to body force,Gravitational Force




MECH 221 – Review

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Fluid Statics

Fluid pressure will increase when the positionof the fluid become deeper, we have followingequation:

g dz

dp

z

y

x

g

0




MECH 221 – Review

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Fluid Statics

Total force acting on the surface become:

hdA g A p pdA F atm




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Fluid In Motion (Inviscid Flow)

2 sets equations for solving fluid motion problems

Conservation of Mass

Conservation of Momentum

dV pd d dV )( t S )t ( V S )t ( V

g s s

vvv

0d dV t S )t ( V

s v




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By invoking the continuity equation, themomentum equation becomes Euler’sequation of motion

Bernoulli equation is a special form of theEuler’s equation along a streamline

constantz

2

g 2

v p

Along streamline incompressible flow




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A conical plug is used to regulate the air flowfrom the pipe. The air leaves the edge of thecone with a uniform thickness of 0.02m. If

viscous effects are negligible and the flowrateis 0.05m3 /s, determine the pressure within thepipe.




MECH 221 – Review

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Procedure:

Choose the reference point

From the Bernoulli equation

P, V, Z all are unknowns

For same horizontal level, Z1=Z2

Flowrate conservation

Q=AV




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From the Bernoulli equation,

)(2

2

2

zzlevel,lhorizontiasameat the

Since,

z

2

z

2

2

1

2

221

2

22

2

11

21

2

2

221

2

11

vv p p

g v

g p

g v

g p

g

v

g

p

g

v

g

p




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From flowrate conservation,

smv

smv

mrt A

m D

A

mr mt m D sm

Q

v Av AQ

894.190251.0/5.0

034.120415.0/5.0

Therefore,

0251.0)02.0)(2.0(22

0415.04

23.0

4

2.0,02.0,23.0,5.0Given

2

1

2

2

222

1

3

2211




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21

22

1

22

3

21

2

1

2

221

565.148

)034.12894.19(2

184.10

0 p point,reference becomes pSet

184.1C,25atm,air@1standardFor

894.19,034.12

)(2

m N p

p

mkg

smv

smv

vv p p

Sub. into the Bernoulli equation,





MECH 221 – Review

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Fluid In Motion (Viscous Flow)

In the mentioned fluid motion is inviscidflows, only pressure forces act on the fluidsince the viscous forces (stress) wereneglected

With the viscous stress, the total stress onthe fluid is the sum of pressure stress ( )

and viscous stress ( ) given by:τ

pσ

τ pσ σ




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Dimensional Analysis

The objective of dimensional analysis is to obtainthe key non-dimensional parameters that governthe physical phenomena of flows

After the dimensional analysis or normalization ofthe complicated Navier-Stokes equations (steadyflow), the non-dimensional parameters areidentified

The equations are reduced to simple equationand solvable analytically under certain conditions





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By using proper scales, the variables, velocity(u), pressure (p) and length (L) arenormalized to obtain the non-dimensionalvariables, which are order one

*

U

gL

UL p

U

P *

g2

2

2 ivvv

***

L P p p U //vv

directionnalgravitatioinrunit vecto*

g i





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For simplicity consider the case where thegravitational force has no consequence to thedynamic of the flow, the Navier-Stokes

equations becomes

UL

Re ,

Re

1 2*

2

**vvv

*

p

U

P

,

Re

1 2* **vvv

*

p

When Re >> 1

scalepressureas2U P





MECH 221 – Review

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Inviscid Flow Vs. Boundary Layer Flow

where is the viscous diffusion length in anadvection time interval of .

Here, measures the time required forfluid travel a distance L.

2

22

v

L

U / L

LUL Re

forceviscous

forceinertia

U / L

U / Laτ

aU / L τ





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When , inertia force is much greater thanviscous force, i.e., the viscous diffusion distance ismuch less than the length L.

Viscous force is unimportant in the flow region of

, but can become very important in the region of

near the solid boundary.

This flow region near the solid boundary is called anboundary layer as first illustrated by Prandtl.

1 Re

)( O ) L( O





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Flow in the region outside the boundarylayer where viscous force is negligible isinviscid. The inviscid flow is also called thepotential flow.

U

Boundary layer flowPotential flow





MECH 221 – Review

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Inviscid Flow

Inviscid flow implies that the viscous effect isnegligible. The governing equations areContinuity equation and Euler equation.

We introduce a potential function, which isautomatically satisfy the continuity equation

v

02

2

2

2

2

22

z y x





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Inviscid Flow

The continuity equation becomes Laplaceequation. The flow is described by Laplaceequation is called potential flow

For 2D potential flows, a stream function (x,y) can also be defined together with (x,y)

x y y x

and

C i





MECH 221 – Review

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Inviscid Flow

If 1 and 2 are two potential flows, the sum =( 1+ 2 ) also constitutes a potential flow

We can combine certain basic solutions toobtain more complicated solution

+ =




MECH 221 R i




MECH 221 – Review

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Boundary Layer Flow

The thin layer adjacent to a solid boundary iscalled the boundary layer and the flow insidethe layer is called the boundary layer flow

Inside the thin layer the velocity of the fluidincreases from zero at the wall (no slip) to thefull value of corresponding potential flow.

MECH 221 R i





MECH 221 – Review

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Boundary Layer Flow

There exists a leading edge for all externalflows. The boundary layer flow developing fromleading edge is laminar

MECH 221 Review





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Boundary Layer Flow

When we normalize the governing equations withRe underneath the viscous term and resolve thevariables of y and v inside the boundary flow, thenon-dimensional normalized variables are selected:

V

vv

U

uu

y y

L

x x

L

,,,

V be the scale of v in the boundary layer

L is viscous diffusion layer near the wall (boundary layer)

MECH 221 Review





MECH 221 – Review

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Boundary Layer Flow

These results in the boundary layer equations thatin dimensional form are given by:

0

y x

vu

2

2

y

u

x

p

y

uv

x

uu

y

p

0

Continuity:

X-momentum:

Y-momentum:

MECH 221 Review





MECH 221 – Review

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Boundary Layer Flow

A boundary layer flow is similar and its velocityprofile as normalized by U depends only on thenormalized distance from the wall:

i.e.,

y xU y

x

2/1

g U

u

(*)

MECH 221 – Review





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Boundary Layer Flow

By introduce a stream function

The boundary layer equation in term of thesimilarity variables becomes:

)('

f U y

u

f xvU 21

f f f ' ' asandat 100

02 ' ' ' ' ' ff f

(**)

MECH 221 – Review





MECH 221 Review

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After we solve this ordinary equation, we obtain asolution of

We first find the value of by Equ. (*) based on

coordinate of x and y, then find out the value ofby checking the solution table in the reference. Finallythe u at x and y is calculated by Equ. (**)

Therefore, we obtain following results:

Boundary Layer Flow

5

U

vx

)(' f

)(' f

x

w

U

Re

332.0 2

x

f C Re

664.0

MECH 221 – Review





MECH 221 Review

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Boundary Layer Flow

Laminar boundary layer flow can becomeunstable and evolve to turbulent boundarylayer flow at down stream. This process iscalled transition

MECH 221 – Review





MECH 221 Review

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Boundary Layer Flow

Under typical flow conditions, transition usuallyoccurs at a Reynolds number of 5 x 105

Velocity profile of turbulent boundary layer flows is

unsteady

A good approximation to the mean velocity profilefor turbulent boundary layer is the empirical 1/7

power-law profile given by

71

y

U

u

MECH 221 – Review





MECH 221 Review

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Boundary Layer Flow

51Re

37.0

x x

512

Re

0577.0

2/ x

w f

U C

41

200225.0

U

vU w

For turbulent boundary layer, empirically wehave




MECH 221 – Review




MECH 221 Review

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If i is the unit vector in the body motion direction,

then magnitude of drag F D becomes:

For two-dimensional flows, we can denotes j as

the unit vector normal to the flow direction, F L is

the magnitude of lift and is determined by:

Boundary Layer Flow

)( ss

..itni dAdA p F w

sb

D F

jtn j )( ss

..

dAdA p F w

sb

L F

Pressure Drag

Friction Drag

MECH 221 – Review





MECH 221 Review

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The drag coefficient defined as

For uniform flow passing a flat plate and nopressure gradient is zero and no flow separation, :

Boundary Layer Flow

22

/ AU

F C D

D

Re

072.0

51

L

DC

v

ULC

L L

D ReRe

328.1 whereLaminar Friction Drag

Turbulent Friction Drag

MECH 221 – Review





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Boundary Layer Flow

The pressure drag is usually associated withflow separation which provide the pressuredifference between the front and rear faces ofthe body

For low velocity flows passing a sphere ofdiameter D, the drag coefficient then is expressedas:

D

D D

AU

F C

Re

24

2/2

directionflowtheinspheretheof areaprojectedtheis where 42 / πD A


mech 221 review

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