Aerodynamics(1)-Chap 2 Governing Equation of Fluid Mechanics(v2)_2

8/3/2019 Aerodynamics(1)-Chap 2 Governing Equation of Fluid Mechanics(v2)_2

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Aerodynamics (1) - AERO3260

.

Prof. Chul-Ho Kim

- Conservation Law- ,- Navier-Stokes equations

- Vorticity and Circulation- Stream Function and Streamlines

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 1School of Aerospace, Mechanical & Mechatronic Engineering


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. rs0.8B$/7Yrs (2002)

1.3B$/3Yrs (2005)

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 2


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Lecture Topics

General Information of Fluid Dynamics and

Thermodynamics

Governing Equations of Fluid Dynamics (2wks)

2-D Airfoil Section Theory (2wks)

3-D Finite Wing Theory (2wks)

Viscous Flow and Boundary Layer Theory (1wk)



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The Fundamentals of fluid mechanics for the further

development of Aerodynamics is introduced withphysics and

mathematics of the one-dimensional fluid motion.

- Physical laws governing the change of the properties of air

- Application to subsonic/transonic/supersonic flow regions

2-D approach to have G.E. of the fluid flow phenomenon int s c apter.

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 4Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved


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Contents2.1 Introduction

(Conservation Analysis)

2.2 One-dimensional flow; the basic equation propert es n u ynam cs an t ermo ynam cs

2.3 The measurement of air speed

2.4 Two-dimensional flow (The Continuity equation)

2.6 The momentum equation (moments)

2.5 The Stream function and streamline

2.7 Rotational flow and vorticity. e av er- to es equat ons

2.9 Properties of the Navier-Stokes equations

-


.


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. n ro uc on

nd

momentum

- Solid mechanics; amaF ==

- Fluid mechanics; (mass flow rate)

c

vmmvd

amaF ==== &)(c

- Body forces : weight by gravity, acceleration, electro-magnetic force- Surface forces : pressure, shear force by viscosity



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For the viscous force in the Newtonian fluid;

V

ratestrainStress

==hA

Newtonian fluid a fluid whose stress versus rate of strain curve is

linear and passes through the origin.

(common fluids: water, air, other gases etc)

.

Exceptions : Rarefield gas dynamics. (for aerospace; re-entry vehicle)



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2.1.2 Comparison of steady and unsteady flow

,

1 Ground-fixed coordinate s stem

With the variation of time, the properties

of the air particle changes continuously.i.e.

Properties of air (T) constant

It is called that the flow is unsteady state.

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 8( quasi-steady state)


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(2) Vehicle-fixed coordinate system

With the variation of time, the properties

of the air article at the fixed oint does

not change, that is,

Properties of air (T) = constant

.

Therefore the governing equations of air flow phenomena about a vehicle


w e s mp e ecause e me er va ve erms are e m na e .


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True unsteady flow

From the flow around a bluff body, the wake flow zone occurs at the

rear of the body. At point P, sometimes, it is in the wake region andsomet mes not. n t e ow proper-

ties at P changes rapidly with time. Q

Point Q is well outside of the wake,

and the variation of the properties

is very small and it can be regarded

as steady flow with little error.



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- tream ne : an mag nary ne t at s nstantaneous y tangent to t e

velocity vector of the flow and no flow across the line- Streamtube : a bundle of streamlines or a region bounded by streamlines

- Streakline : the locus of oints of all the fluid articles that have assed

- Pathline : the trajectories that individual fluid particles



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2.2 One-dimensional flow : Basic Equation(Control Volume Analysis)

For the real flow analysis,

- onserva on aw mass, momen um an energy

- Equation of state of perfect gas

If a real flow field can be modeled by simplified assumptions, the handlingcomplexity may be reduced considerably. However, we need the experiences to

u ge w at t e reasona e assumpt ons or t e pro em are.

For exam le air is considered as an incom ressible and inviscid fluid in muchof aerodynamics. However, the viscosity plays important role to transmit the

aerodynamic forces to the body.



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2.2.1 One-dimensional flow : the basic equations of conservation

n ere, t e govern ng equat ons o u ynam cs s er ve rom t e

conservation law.

- Conservation of Mass : ContinuityContinuity EquationEquation

- Conservation of Momentum : MomentumMomentum E uationE uation

- Conservation of Energy : EnergyEnergy EquationEquation

et s t e propert es o u an n s t e propert es per un t mass o

the fluid. (N : mass, momentum, energy)

= dnN



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The volume of fluid is V1 and V2 at time t1and t2 respectively. That is,

N = VA

+ VB

at t1= +

And the property of the fluid at each time is

expresse as;

= N + at t

N2 = NB2 + NC2 at t2

)()(

)()(1122 BACB

NNNN

NNNNN

+=

++=



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v e y t me ;

tNN

tNN

tN ACBB

+

=

)()( 1212

To know the rate of change of the property in the limited time,

NNNNN CBB )()( 1212(1)ttt ttt 000

WheredNN

=

lima e o c ange o u proper y n e ys em

systemtt

12

0

BB

t

NNN

=

)(lim

Rate of chan e of fluid ro ert N in and out CV

VolumeControl

NNNAC =

12)(

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 15tt

tt

00


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From the above concepts introduced,

Conservation Law

a e o ange o e roper y system =

Difference of the ro ert enterin and leavin CS

+ (Accumulated value of the property in CV)CV

That is, the change rate of the fluid property(N) is;

(The rate of change of the property entering and leaving through CS)CS+ (The rate of change of the accumulated property in CV)CV



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(1) The rate of change of fluid property(N) in and out through CSAs shown in the figure, the fluid velocity(V) can be

the infinite control surface area (dA). Thus the mass

flow through the control surface is,

= ddm

ere

That is, the amount of the mass flow through the control surface (dA) is,

n=

tdAVdm n=

Therefore, the fluid property(N) leaving the control volume through dA is;

==


n


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or e ra e o c ange o e proper y,

dAVnN

=

in the limit time,

t

dAVnt CS

nt=

0lim

,

leaving CS is,dAVn

NCS

nt =

0lim

Accumulation rate of fluid property(N) in the CV

(2) The rate of change of the fluid property(N) accumulated in the

CV;)( tdAVn

N

=

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 18 tt volumecontrol


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ere

=

dnN

tn=

in the limit time, tt volumecontrol

N(3)

=

CStn

tt

0m

+= dndAVndN

The rate of chan e of the fluid ro ert N =

CVCS n

system tdt

(The rate of change of the property entering and leaving through CS)CS


CV


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2.2.2 The Continuity, Momentum and Energy equation

on nu y qua onIf N is mass(m) of the fluid in the flow field; n=N/m=1. Thus from the

conservation e uation 4

+=CVCS

n ddAVdM

From the conservation of mass, the rate of change of mass is zero. Then,

system

0

+=

CVCS

n d

t

dAV

It means that the incoming mass flow rate through CS is equal to the sum

of the outgoing mass flow rate through CS and the accumulated mass flow

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 20rate in the CV.


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forsteady flow, the time derivative term is zero, thus

0=CS

ndAV

for steady, incompressible flow, ( =constant )

0=CSndAV

The equation means that the incoming mass flow rate of the fluid through

the CS is equal to the outgoing mass flow rate through the CS.

Ann

VdAV =



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dAVV

n=

Flowrate

k /sec2

m3/secVolume flow rate1

kgf/secWeight flow rate3



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As shown in the figure, the

parabolic equation of velocity

s r u on n a p pe s g ven.

Find the average velocity of the

fluid in the pipe and the volumeflow rate.

FromdAVn

A=

Thus ,

sftddrrRr

V /5.2)/1(5

2

0

22

0 =

=

The volume flow rate; 28547V .==



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For summary, the continuity equation with the averaged velocity ( ) is,V

CVoutin tVV

+=

forsteady flow without storage of mass,

=outin

forsteady flow, incompressible flow,

&

( mass transfer across control surface + mass storage within control volume )



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omen um qua on

From the conservation of momentum the linear momentum and an ular

momentum equations are delivered.

near omen um qua on

If N is momentum(mv) of the fluid in the flow field; n=N/m=v. Thus from the

conservation equation (4), Vm

mn ==

(8)

+=

CVCS n

system

dV

t

dAVVmV

dt

d)(

)()( mVdAVVmVd

CSn

+=


system


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n a - mens ona ow e , e orce s expresse y ew on s aw.

zzyyx mVd

FmVd

FmVd

F )(,)(,)( === from the momentum equation,

(9)x

CSn

system

xx mVt

dAVVmVdt

F )()(

+==

(10)yCS nsystem

yy mVtdAVVmVdtF )()( +==

(11)zCS

nzz mV

t

dAVVmV

dt

dF )()(

+==

where Vn is the vertical component of the velocity to CS and Vx, Vy, Vzare the velocity

components of the main flow. Thus the two components of the velocity are not always


t e same n t e contro vo ume.


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For summary, the momentum equation with the averaged velocity( ) is,V

d ==CV

CSn

system tdt

The is the summation of all force components acting on the CV. They couldbe gravity, electric/magnetic force in the field, surface tension effects, pressure

orces an v scos y.

The terms on the right side of the equation mean,

(momentum transfer across control surface + momentum storage within control



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For 1-D steady flow, the momentum equation is,

CSnii

If the CV has single inlet and outlet on the surface,

(14)iniouti

iii

AVVAVV =

=

11112222

for1-D stead and incom ressible flow

(15)diriinouiii VVVVmF == )()( 12&



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(2) Angular Momentum Equation

For the desi n and erformance anal sis of turbo-machine such as turbine um

compressor etc angular momentum equation is required.

Lets assume that the infinite volume of particle located at the radius(r) from the origin

- .

may be considered as rotor blade of a turbine.

tangential and normal direction of the radius(r). However, the rotating energy of theparticle is the tangential force. Thus, the torque energy

(16)ts

rdFdT =

where can be obtained from the previous linear momentum equation.t

dF



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a s,

+=

CV tCS nttdT

tdAVVF

in differential form,

dAVVdmVt

dFnttt

+

= )(

from (1), (2),

dAVrVdmVrdT +

=t

ns

Total Torque;

+= tnts dmrVdAVrVT )(



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+= CVnCSs dmrVdAVrVT sin)(sin

rewrite with vector notation,

(17)+

=

CVCVs

rmrt

for stead flow;

(18) = CSs AdVVrT ))((


A d i (1) AERO3260


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3) Energy Equation

Energy can be neither created nor destroyed but only converted from oneorm o ano er.

With the 1st Law and some assumptions, we can produce Energy eq. to

analysis energy exchanges between systems and environments.

- Steady : Conditions at any section of the system is independent of time

- Continuity : The weight entering must equal to the weight leaving the


.

Aerodynamics (1) AERO3260


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o se up e energy eq. , we ave en e energy erms a

apply to various situations.

o en a nergy : , ne c nergy : g

Internal Energy : U , Flow Work : W , Heat : q




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from the law of the conservation of Energy, we can set up an EnergyEquation.

)(exitinlet

Energywhere : =

11

1

11

2Z

g

V

ruqW inin +++++

2

2

2

2

22

2 Z

gV

ruqW outout +++++=

for net work and net heat ;

worne inout ==

hnet ==eat




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2

222

21

211

122

ZV

r

PuWZ

V

r

Puq ++++=++++

where

Enthal = Internal Ener + Flow Work S. Enthalpy = S. I. E. + S. F. W.

ruh+=

Thus ZZVVhhWq ++=12

2

1

2

212

)()(1

)(

(19)ZVh ++=21




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'

Many engineering problems have been solved since it was introduced by Daniel

ernou n . as e m a ons o app y or e pro ems g ven e ow.

- Hi h-S eed Flow of Com ressible Fluids c- Fluid Flow with Heat Transfer

- Fluid Flow with large Pressure Change due to friction ( P c )

For the practical flow problems, the theory of fluid dynamics based on theconservation law should be solved to get the detailed information of the flow

field characteristics.

-- Energy Equation

- Momentum Equation


- State equation of Gas



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y ( )

or a sma s ream u e, rom ew on s n aw o o on ;

dLPdWdAdPPdAPF += cos)(

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 38am =



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y ( )

ere

dZdUadLdAgW === cos,,

is Perimeter of Cross-section Area of the stream tube.d

dL

dAdUUdLPddZdAgdAdP =

dA

=

(20)0=+++dA

PddZgdUUdP

The above equation is called Eulers equation for viscous fluid.

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 39Forinviscid flow, the Shear Stress ( ) is zero.



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0=++dP

ZgUUP

0=++ dZdUg

02

2

=++ dZUddPei ..

forincompressible fluid, ( = const. )

(22)02

2

=

++ ZUP

d




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1-dimensional steady, incompressible, inviscid flow ( Bernoulli Eq. )

0

2

=

++ Z

UP

d

,

11 222111

gr

011

12

2

1

2

212=++ )()()( ZZUU

gPP

r




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ZUP

ZUP

++=++ 2222

1

211

UP2

ernou equat ong

==2

( where H : Bernoulli constant or Total Head )

Pressure Head Velocity Head Potential Head::: ZUP

2

g 2




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2.2.2 Comments on the Momentum and Energy Equations

constZ

g

=++

2

(Pressure or Internal energy) + (Kinetic energy) + (Potential energy) = constant

It is not considering the viscosity effect, thus Bernoullis equation can not beapplied along a streamline in a boundary layer.

In aerodynamics, viscosity is important reason in the energy dissipation in flow

field.




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. easuremen o a r spee o -s a c u e

-. .

The pitot tube was invented by the French engineerHenri Pitot in the early 1700s and

was modified to its modern form in the mid 1800s by French scientist Henry Darcy.

Pitot tube Static tube




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22 BBAA VPVP +=+From Bernoullis equation,

airair

BABV

=

=

where the air density,o

oair

RT

P=

It can be used to measure the flow velocity in the wake and boundary layerregions even though the Bernoullis equation has a limitation to inviscid flow.

Stagnation Pressure Coefficient (Cpo)

2

1120 1

=

=

=

V

V

V

PPCp




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2.3.2 The pressure coefficient (Cp)

2

2

11

=V

p

wherePis static pressure at some point where the velocity is q.P

and V

is static pressure and velocity of the undisturbed flow.

2

For incompressible flow;

1

=V

Cp

(2) if q = V then P = P and Cp = 0 ( free stream )

(3) if q < V then P > P and Cp > 0 (positive value)




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. . -

1) Indicated air speed (IAS)

2) Equivalent air speed (EAS)

3) True air speed (TAS) oVV

E == ,

1) IAS : Uncorrected reading by the actual-air-speed indicator. The speed of

an aircraft as shown on its pitot static airspeed indicator calibrated to

reflect standard atmosphere adiabatic compressible flow at sea level

uncorrected for airspeed system errors.

2) EAS : The airspeed at sea level which represents the same dynamic

pressure as that flying at the true airspeed (TAS) at altitude

3) TAS : The actual aircraft speed relative to the air

= = > 1.0), the




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(Example 2-1) Calculating the TAS from IAS

IAS = 950km/h at SL in ISA

TAS = 891km/h (59km/h difference)

P

airair

BA

BV

==

TRair

=




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2.4 Two-dimensional flow (Differential Analysis)

, -

dimensional flow.

1-D flow 2-D flow

- ow

wkvjuiV ++=vjuiV +=uiV=




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2.4.1 Component VelocitiesThe flow velocity at P(x,y) in the field,

In a Cartesian coordinate, a particle moves

vjuiV +=

rom x,y to x+ x, y+ y , t en t e ve oc ty

of the particle is ;

In the horizontal direction ; dtdxu /=the vertical direction ;

the ma nitude of velocit

dtdyv /=

22

the direction of velocity ; )/(tan dydx1=




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In a polar coordinate, a particle moves from P(r,) to Q(r+r, +), then thevelocity of the particle is ;

- the radial velocity ;dtdrq

n

/=

- the tangential velocity ; dtrdqt

/=

- the distance travelled ;

- the ma nitude of velocit

22 )( rss +=

222 +=

- the direction of velocity ;

n

)/(tannt

qq1=




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( Fluid acceleration ) Velocity at Q(x+x, y+y),

x-dir :

-

cce era on a x+ x, y+ y ,

x-dir :

y-dir :

Pressure at Q(x+x, y+y),




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2.4.2 Continuity Equation in Cartesian coordinate

, .

The local velocity components and density

are; U(x, y, t), V(x, y, t), (x, y, t)

Mass flow er unit area UA

(x-dir) in: 12

yxx

uu ))((

out : 12

+ y

x

x

uu

)

)((

Mass flow accumulated in x-dir; (out)-(in) 1

yxx

u

v


- -

yxy



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The total accumulated mass flow in CV;

vu )()(

The rate of change of mass of the fluid through the CS.

yx

(2)

=

=

Qdydx

tVolume

t )()( 1

That is, (1) = (2) vu )()(

Continuit E uation 2-D flow

yxt

0=

+

+

+

+ vu

vu




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Continuity equation for steady and incompressible flow in 2-D flow;

vu=

yx

Continuity equation in 3-D flow;

0=

+

+

+

+

+

+

zw

yv

xu

zw

yv

xu

t

for steady and incompressible flow in 3-D flow (=const)

0=+

+

z

w

y

v

x

u




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2.4.3 Continuity Equation in Polar coordinate

flow rate (mr);

for Tangential direction; the accumulated

mass flow rate (mt);

Thus total accumulated mass flow rate in

CV;




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The rate of change of mass of the fluid through the CS.

=

=

drrdVolume

That is, (3) = (4)

tt

01

=

+

++

tnn

q

rr

q

r

q

t

)(

for steady flow;0

1=

+

+

tnn

qqq )(

for stead and incom ressible flow =const

rrr

01

=

+

+

tnn

q

rr

q

r

q




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2.6 The momentum equation

( the force acting on the fluid within CV) =

(the rate of increase in momentum in CV, ) +

the net rate at which momentum flows out thou h CS

)( vm

vmvm

The fluid force ;

(Body force) : gravity (or weight) of the fluid in the CV

Surface force : ressure force, viscous force on the CS

(a) pressure force : a stress acting perpendicular to the CS




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(b) viscous force : acting on the surface of CS.

(2) top/bottom face :

stress tensor for 2-D flow :

n eng neer ng pro em, t e rect stress xx, yy are neg g e compare w t

the shear stress. Even the shear stress is ignorable in a still fluid.

(the force acting on the fluid within CV, (III)+(IV)+(V) ) =(the rate of increase in momentum in CV, (I) )+(th t t t hi h t fl t th h CS (II) )Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 63

(the net rate at which momentum flows out though CS, (II) )



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(I) t e rate o ncrease n momentum n CV

v =

=

)(

tt

vuv)( =yx

ttyx

t

e ne ra e a w c momen um ows ou oug

for x-direction 1133 VmVm &&

1

+

+

+

= yxvu

uu

vu

u

(continuity eq.)

yxyx

1

+

= xu

vu

u


yx



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for y-direction, 2244 VmVm &&

1

+

= yxv

vv

u

(III) Body force acting on the CV (weight)

= = -

11 = yxgggyxyx

),(

x y ,

(IV) Pressure force acting on the CV

- 1

=

x

P

Net pressure force in x-direction ;

xxx

121

= yxP

PPyy



t t


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scous orce act ng on t e

Viscous force in x-direction ;

1

+

= yxyx

F xyxxx

Viscous force in x-direction ;

1 += yxyxFyyyx

y

Put the terms (I~V) into the concept;

- =

(2.66a)pgu

vu

uu

xyxx

x

+

+

=

+

+




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y-direction ; (I)+(II) = (III)+(IV)+(V)

pvvvyx

.yxyyxt

y =

For 3-D momentum equations for continuum fluid;

puuuuxzxxx

zyxxzyxtx =

. a, , czyxx

gz

wy

vx

ut

yzyyyx

y

+

+

+

=

+

+

+

zyxz

pg

z

ww

y

wv

x

wu

t

wzzzyzx

z

+

+

+

=

+

+

+



2 6 1 E l ti


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2.6.1 Euler equation

Then the equation is called Eulers equation.

or - u er s equat ons or nv sc ow;

puuuu

xzyxtx =

xg

zw

yv

xu

ty

=

+

+

+

z

pg

z

ww

y

wv

x

wu

t

wz

=

+

+

+



I i i bl


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In engineering problem,

- continuity equation

-

can be solved to get the velocity (U, V, W) and pressure (P) in the flow field.

But because the pressure term does not appear on the continuity equation.

,

which is called Laplace Equation.



2 5 The Stream function and Streamlines


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2.5 The Stream function and Streamlines

. . ream unc on

Lets think about a shallow river with constant depth,

1m. If = 2m/s the volume flow rate is assinthrough OA line is ;

/smV 3802140 ===

Even the shape of the rope is changed, the amountof water passing through is the same and is not

affected by the shape of the rope.

ream unc on : e amoun o u quan y pass ng an area per untime. (m3/s) volume flow rate ( )

Q




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Let the flow velocity be q passing over a small length (S) of line;3 smsq

,

op

dsq sin

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 71(Integration of normal velocity component from O to P)



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If this quantity remains constant irrespective of the path of integration, it iscalled the stream function of P with respect to O.

Stream Function : =op

P dsq sin

< Sign Convention for Stream Function >

When looking in the direction of integration, if the flow across from left to right,

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 72the sign is positive.


2 5 2 Streamline


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2.5.2 Streamline

If no flows across the PP P Pmmmmmm ===== .....P2P3, P3 P4. That is, the velocity of the flow must along or tangential to the lines.

4321

con on o s ream ne- No flow across the streamline

- The stream function is constant alon the lineP

- Flow velocity is always tangential to the

stream line

-

Lets assume,

OPnOPOPOPOPOP ===== .....4321



If there is no flow cross the line PP That is the flow must be1OPOP =


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If , there is no flow cross the line PP1. That is the flow must be

along or tangent to PP1. Therefore, PP1, P1P2, P2P3, P3P4, ., Pn-1Pn are lines

1OPOP

. .

A streamline is a line of constant .

The velocity of fluid particles on a streamline can be changed in magnitude

but the direction is always that of the tangent to the line.

2.5.3 Velocity component in terms of a artes an oor nate

The amount of fluid flowing across between

P and is .

i.e. : the change of stream function between


x, y an x x, y y



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,

The total flow across in the line PR is

=

.

yx ,

xvyu == ,


x, y


( ) Po ar Coor nate


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( )

,

rqrqrqrrqrqnntnt

++=++= )(

Thus,

rqrqnt

+=




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The total flow across in the line PRQ is ,

+= rrr ),(

Thus

=

= qq

nt

1,

are velocity components at a point (r, ) in a flow given by stream),( nt qq

From the above, the velocity(q) in any direction is found by differentiating the

stream function ( ) with respect to the direction n normal to velocity (q).

nq =



2.7 Rates of strain, rotational flow and vorticity


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, y

, . ,

(the viscous stress) (the rate of strain)

d

dUF=

(the rate of strain) (the velocity gradient)

2.7.1 Distortion of fluid element in flow field

,

(1) Translation


(2) Dilation/Compression (shape remain invarient but volume changes)


n genera , e rans orma on o a u e emen compr ses e o ow ng opera ons;


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1 Translation

(2) Dilation/Compression (shape remain invarient but volume changes)

(3) Distortion (shape changes with keeping the volume)




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( Transformation of a Fluid Element )



2.7.2 Rate of strain


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The control volume, ABCD at t = t deformed ABCD at t=t ;

The velocity at ( t = ti),

22

y

y

Ux

x

UUUA

=

22

y

y

Vx

x

VVVA

=

22

y

y

Ux

x

UUU

B

+

=

22

y

y

x

x

VVB

+

=

22

y

y

Ux

x

UUUC

+=

22 yxVV

C

+=

xu A

'=

yv A

A

'=


,


The deformed angle (, ); t= t=


82/97

Thus,x y

-

xdt =

ydt =

+

=

+

=+

=uv

tu

tvdd xy 11

yxtyxdtdt 222

+

=y

u

x

v

dt

xy

2

+

=z

u

x

w

dt

xz

2

(2.72a, b, c)

+

=

z

v

y

w

dt

d yz

2

1



2.7.3 Rate of direct strain


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Thus the rates of direct strain are obtain;

x

u

dt

dxx

=

v

dt

dyy

=

z

w

dt

dzz

=

The rate of strain tensor for 2-D flow;

xyxx


yyyx


. . or c y ro a on,


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The vorticity() is an instantaneous rate of rotation of a fluid element.

dt

d )(

=

It is called the vorticity in z-axis.

uvdd

For 3-dimensional flow, the vorticit is a vector iven b ;

yxdtdt

==

),,(),,(u

x

v

x

w

z

u

z

vw

==

Mathematically, the vorticity is given for 3-D flow; (3-D vorticity, )


=


2.7.5 Vorticity (rotation, ) in polar coordinates


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The vorticity() in polar coordinates is ; from equation (2-54), p73

sec)/(1

radianq

rr

q

r

qntt

+=

2.7.6 Rotational and Irrotational

Vorticity is associated with the effects of viscosity.

,

not rotate or distort as they move through the flow field.

For an inviscid flow; - Vorticity is zero ;- Irrotation or undistorted flow (pure translation flow)

0),,( ==


-


2.7.7 Circulation


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Circulation () : The total amount of vorticity passing through any plane

region within a flow field.

= A dAn

If Area(A) in the X-Y plane and , then;kn =k=

-

Circulation is a measure ofthe combined strength of the total number of vortex line

== AAn

Copyright 2009 Prof Chul-Ho KIM Seoul National University of Technology. All r ight Reserved 86(vorticity flux)passing through the area (A).


Circulation is the property of the region A in CV whereas vorticity is a flow

d fi d i


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defined at a point.

- vorticity Point ( ) - circulationArea ( )),,( = = A dA

In 2-D flow, in the absence of viscosity, circulation is conserved;

0=++v

xu

t

Circulation is also calculated by an integration around the perimeter (C);

a l in Stokes theorem

(Circulation is important concept to the theory of lift.)sqc =



(Example 2.2) page89

P th t th i l ti l b l t d b th


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Prove that the circulation can also be evaluated by the.

(sol) uvfrom Eq. (2-81) ==

AA

yx

Where C1 ; C3 ;dxdsituiq === ,, dxdsituiq === ,,

C2 ; C4 ;dydsjtvjq === ,, dydsjtvjq === ,,


c=


. e av er- to es equat ons (p89)


89/97

. .

In solid mechanics b Hookes Law : stress

strain

In fluid mechanics (Newtonian Fluid) : (viscous stress) (rate of strain)For 2-dimensional flow, the above concept can be written in;

(2.87)

It is good for an incompressible fluid. For the compressible fluid, direct

stress generated by dilation should be considered.



Claude Louis Navier (10 Febr ar 1785 in Dijon 21 A g st 1836


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Claude-Louis Navier (10 February 1785 in Dijon 21 August 1836in Paris)

-Established the elastic modulus as a property of materials in 1826

-Navier-Stokes equations, central to fluid mechanics in 1822

George Gabriel Stokes (13 August 18191 February 1903,

Cambridge, England), a mathematician and physicist, who at

-Terminal velocity or Settling velocity (Falling viscometer)

-



For compressible flow, the term given below should be considered for direct

t b dil ti f fl id


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stress by dilation of fluid;

(2.88)

Stokes hypothesis : 3+2=0 or =-2/3

2''

The bulk viscosity( ) is ignorable in most engineering problemsbut important

3

'for the propagation of sound waves in liquids and gases.

, ,

is ignored and only eq.(2-87) is valid.



2.8.2 The derivation of Navier-Stokes equations


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- -, . . .

the momentum equation (2-66a) becomes;yxx

pg

y

uv

x

uu

t

uxyxx

x

+

+

=

+

+

Thus the momentum e uation written in the velocit terms for viscosities of

fluid;

+

+

=

+

+

2

2

2

2 uupg

uv

uu

ux


- -


Thus the momentum equation written in the velocity terms for viscosities of fluid;


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22 vvpvvvy- rec on; -

The above equations (2-92a, b) are 2-D Navier-Stokes equations.

22 yxyyxt

y



Navier-Stokes equations for 3-dimensional incompressible flow ;


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(2.94)0=

+

+

z

w

y

v

x

u

(2.93a)

+

+

+

=

+

+

+

2

2

2

2

2

2 uuupg

uw

uv

uu

ux

2.93b +++= +++222 vvvpgvwvvvuv

zyxyzyxt

222 wwwpwwww.

222 zyxzzyxt z



2.9 Properties of the Navier-Stokes equations (p91)


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3-D incom ressible flow N-S e uation Newtons 2nd Law of Motion

It is impossible to obtain the exact solutions from N-S equations because,

(Limitations of Navier-Stokes Equations)

(1) Non-linearity of the equation

2 Com lex effect of viscosit



Non-dimensional N-S equations with an example given below;


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- - , , , 2 incorporated.

. -

put the equations into eqs. (2.94) and (2.95);

-



(non-dimensional momentum equation) with body force terms omitted;


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(2.100 a, b, c)

The equations have the Reynolds number term that has very serious effect on the

flow pattern of the flow field.

For the simulation of the flow with no flow separation on the body surface, the N-

S e uations without the viscous term can be a licable but in the real case with

separation , the calculation results obtained from N-S equations is not reliable.


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Aerodynamics(1)-Chap 2 Governing Equation of Fluid Mechanics(v2)_2