convection 2

8/3/2019 Convection 2

1/64

Convection 2


2/64

DIFFERENTIAL ANALYSIS OF FLUID FLOW Finite control volume approach is very practical and useful, since itdoes not generally require a detailed knowledge of the pressure and

velocity variations within the control volume Problems could be solved without a detailed knowledge of the flowfield

Unfortunately, there are many situations that arise in which details of

the flow are important and the finite control volume approach will notyield the desired information

How the velocity varies over the cross section of a pipe, how thepressure and shear stress vary along the surface of an airplane wing

In these circumstances we need to develop relationships that apply ata point, or at least in a very small region infinitesimal volume within agiven flow field.This approach - DIFFERENTIAL ANALYSIS


3/64

DIFFERENTIAL ANALYSIS PROVIDES VERY DETAILED KNOWLEDGA FLOW FIELD

Control volume analysisInterior of the CV is

BLACK BOX

Differential analysis

All the details of the flow aresolved at every point withinthe flow domain

Flow out

Flow in

Flow out

Control volume

F

Flow out

Flow out

Flow domain

F

Flow in


4/64

LINEAR MOTION AND DEFORMATION

Generalmotion

++ +

=

Translation Lineardeformation

Rotation Angulardeformation

Element at t0 Element at t0+ t


5/64

TRANSLATION

If all points in the element have the same velocity which isonly true if there are no velocity gradients, then theelement will simply TRANSLATE from one position to

another.

v t

u t

vu

O

O


6/64

LINEAR DEFORMATION

Because of the presence of velocity gradients, the element willgenerally be deformed and rotated as it moves. For example,consider the effect of a single velocity gradient

On a small cube having sides

xu

z and y , x

AO

B C

O

B C

A

uu x x

uu x

x

x

y

u

u

u x t

x

C

A

y

x


7/64

x component of velocity of O and B = u

x component of velocity of A and C =

This difference in the velocity causes a STRETCHING of the volumeelement by a volume

Rate at which the volume V is changing per unit volume due

the gradient

x xu

u

t z y x xu

x

u

xu

t

t xu

0 t Lim

t d V d

V 1


8/64

zw yv xu t d V d V 1

If the velocity gradients are also present zw

& yv

This rate of change of volume per unit volume is called theVOLUMETRIC DILATION RATE

Volume of the fluid may change as the element moves from onelocation to another in the flow field

Incompressible fluid volumetric dilation rate = zero

Change in volume element = zero; fluid density = constant

(The element mass is conserved)


9/64

Variations in the velocity in the direction of velocity causeLINEAR DEFORMATION

zw

& yv

, xu

Linear deformation of the element does not change the shape of the element

Cross derivates cause the element to ROTATE and undergoANGULAR DEFORMATION

x

v ,

y

u

Angular deformation of the element changes the shape of theelement


10/64

ANGULAR MOTION AND DEFORMATION

Consider x-y plane. In a short time interval t line segment OA andOB will rotate through angles and to the new positions OAand OB

AO

B C

uu y

y

vv x

x

x

y

u

v

AO

B C

u y t

y

v x t

x

x

y

B

A


11/64

Angular velocity of OA , OA

t a

Lim0 t

oA

t xv

x

t x xv

Tan

For small angles

xv - positive oA - counterclockwise

t Lim

0 t oB

t yu

y

t y yuTan

xv

t

t xv

Lim0 t

oA

yu

t t y

u

Lim0 t

oB

y

u - positive oB

- clockwise


12/64

Rotation z of the element about the z-axis is defined as the averageof the angular velocities oA and oB of the two mutuallyperpendicular lines OA and OB. Thus, if counterclockwise rotation isconsidered positive, it follows that

yu

xv

21

z

Rotation x of the element about the x -axis

zv

yw

21

x

Rotation yof the element about the y -axis

xw

zu

21

y

k

z j

yi

x


13/64

V 21

V curl 21

Vorticity is defined as the vector that is twice the rotation vector

V 2Fluid element will rotate about the z axis as an undeformed block(ie., oA = - oB ) only when Otherwise, the

rotation will be associated with an angular deformation xv

yu

yu

xv

Rotation around the z axis is zero.

0V Rotation and vorticity are zero;

FLOW FIELD IS IRROTATIONAL


14/64

In addition to rotation associated with derivatives

These derivatives can cause the fluid element to undergo an angulardeformation which results in change of shape

Change in the original right angle formed by the lines OA and

OB is SHEARING STRAIN

= +is positive if the original right angle is decreasing

Rate of Shearing Strain or Rate of Angular Deformation

xv

& yu

xv

yu

t

t yu t x

v

Lim0 t t

Lim0 t


15/64

xv

yu

Rate of angular deformation is related to a corresponding shearingstress which causes the fluid element to change in shape

x

v

y

u

Rate of angular deformation is zero;

Element is simply rotating as anundeformed blockRotation


16/64

Incompressible flow field

Fluid elements may translate, distort,and rotote but do not grow or shrinkin volume

Compressible flow field

Fluid elements may grow or shrink involume as they translate, distort orrotate

Volume = V 2

Volume = V 2= V1

Volume = V 1

Time = t 2

Volume = V 1

Time = t 1

Time = t 1

Time = t 2

(a)

(b)


17/64

CONSERVATION OF MASS OR CONTINUITY EQUATION

Time rate of changeof the mass of thecoincident system

Time rate of change of the mass of thecontents of the

coincident controlvolume

Net rate of flow of mass through thecontrol surface

0 dA n

V V d t cs cv

z y x t

V d t cv

cs cv

sys dA n

bV V bd t t D

DBx1

y1

z1

dx

dy

dz

x

y

z


18/64

x

y

z

j

i

K z yu

z y x xu

z yu

z xv

z y x

y

v z xv

z y x zw

y xw

y xw

cs dA n

V


19/64

z yu y xw z xv z yu z y x t

0 dA n

V V d t cs cv

0 z y x zw

y xw z y x yv

z xv z y x xu

0 z y x zw

z y x yv

z y x xu

z y x t

0 zw

yv

xu

t


20/64

t xu

yv

0 zw

0 zw

zw

yv

yv

xu

xu

t

0 zw

yv

xu

zw

yv

xu

t

0V

. t D

D


21/64

volume control of contents

cs cv F dA n

V V V Vd t

CONSERVATION OF MOMENTUM

RATE OF INCREASEOF

x -MOMENTUM

RATE AT

WHICH x -MOMENTUMENTERS

RATE ATWHICH x -MOMENTUMENTERS

- +SUM OF THEX-COMPFORCESAPPLIED TO

FLUID IN CV

=

z y x tu

V Vd t cv

SURFACE FORCES NORMAL STRESSES

SHEAR STRESSES

PRESSURE

BODY FORCES GRAVITY FORCES

CORIOLIS FORCES

CENTRIFUGAL FORCES


22/64

x

y

z

j

i

K

cs dA n

V V

z yuu z y x xu

z yuu 2

z xvu

z y x yuv

z yvu

y xwu

z y x zuw

y xwu

u


23/64

LHS z y x zuw

z ywu

z y x yuv

z xvu z y x xu

z yuu

y xwu z xvu z yuu z y x t

u

2

z y x LHS

zuw

yuv

xu

tu 2

z y x LHS

zu

w yu

v xu

u tu

yw

yv

xu

tu

z y x LHS

Dt Du


24/64

xx

xy

xz

yy

yz

yx

xy

xz

xx

First subscript denotes the direction of the normal to the

plane on which the stress actsSecond subscript denotes the direction of the stress


25/64

x

y

z

j

i

K

z y xx z y x

x z y xx xx

z x yx

z y x y

z x yx yx

z y P z y x x P

z y P

z y x RHS

f z y x x

P x

zx yx xx

x f z zx

y yx

x xx

x P

Dt Du

CAUCHYS EQN


26/64

V

. 3 2

xu

2 xx xv

yu

xy xw

zu

xz

x f xw

zu

z xv

yu

yV

. 3 2

xu

2 x x

P Dt Du

x 2

2

2

2

2

2 f V

. 3

2

z

w

y

v

x

u

x z

u

y

u

x

u

x

P

Dt

Du

y 2

2

2

2

2

2 f V

. 3 y z

v

y

v

x

v y

P Dt Dv

x f V

. 3 x 2

z

u 2

2 y

u 2

2 x

u 2

x P

Dt Du


27/64

y 2

2

2

2

2

2 f V

. 3 y z

v

y

v

x

v y

P Dt Dv

x 2

2

2

2

2

2 f V

. 3 x z

u

y

u

x

u x

P Dt Du

z 2

2

2

2

2

2

f V

. 3 z z

w

y

w

x

w z

P Dt Dw

VISCOUS COMPRESSIBLE FLUID WITH CONSTANT VISCOSITY

f V

. 3

V

P Dt

V

D 2


28/64

VISCOUS COMPRESSIBLE FLUID WITH CONSTANT VISCOSITY

f V

. 3V

P DtV

D 2

VISCOUS INCOMPRESSIBLE FLUID WITH CONSTANT VISCOSITY

f V

P Dt

V

D 2

INVISCID INCOMPRESSIBLE FLUID WITH CONSTANT VISCOSITY

f P

Dt

V

DEULERS EQN


29/64

g x p

zu

w yu

v xu

u tu

s z

g s p

su

uAlong a stream line ggsin zs

ds s z

g ds s p

ds su

u

gdz dpudu

C gz P 2u

2

C gz 2u p 2

D


30/64

y 2

2

2

2

2

2 f V

. 3 y z

v

y

v

x

v y

P Dt Dv

x 2

2

2

2

2

2

f V

. 3 x z

u

y

u

x

u x

P Dt Du

z 2

2

2

2

2

2 f V

. 3 z z

w

y

w

x

w z

P Dt Dw

0V

. t D

D

Navier French mathematician Stokes English MechanicianFOUR EQUATION AND FOUR UNKNOWNS U,V,W AND P

Mathematically well posed

Nonlinear, second order partial differential equations

Continuity equation

X- momentum

Y- momentum

Z- momentum


31/64

Relation between Stress and Rateof Strain

R l ti b t St d R t f St i


32/64

Relation between Stress and Rate of Strain In elasticity, the relationship between the stress and strain of a solid body within

the elastic limit is governedby Hookes Law.

The generalised Hookes law states that each of the six stress components may beexpressed as a linear function of the six components of strain and vice versa

The validity of this assumption has been verified by experiments for continuous,homogenous and isotropic materials.

In a fluid, the physical law connection the stress and rate of strain can also be madeby the following simpleand reasonable assumptions:

a. The stress components may be expressed as a linear function of the rates of strain components

b. The relations between stress components and rates of strain componentsmust be invariant to a coordinate transformation consisting of either arotation or a mirror reflectionof axes.

c. The stress components must reduce to the hydrostatic pressure p when allthe gradients of velocities are zero

DCBA


33/64

3333

2222

1111

DC B A

DC B A

DC B A

xy yy xx xy

xy yy xx yy

xy yy xx xx

where the As, Bs, C s and Ds are constants to be determined. Theassumption (b) requires that the stress-rate of strain relation remainsunaltered with respect to a new coordinate system.

3333

2222

1111

DC B A

DC B A DC B A

y x y y x x y x

y x y y x x y y

y x y y x x x x

(1)

(2)

Transformation of stress components


34/64

222

2222

2222

cos sin

sin cos

sin cos

xy yy xx

y x

xy yy xx yy xx

y y

xy yy xx yy xx

x x

Transformation of stress components

(3)

x

y

y x' y' x'x'

xx

xy

xy yy

1 X

1Y

x

B

C

o

2 X

y

xy yy

y

x E

D

xy

xx

2Y

xo

Substituting equation (1) into equation (3 )


35/64

Substituting equation (1) into equation (3 )

3333

2222

1111

DC B A

DC B A

DC B A

xy yy xx xy

xy yy xx yy

xy yy xx xx

222

2222

2222

cos sin

sin cos

sin cos

xy yy xx

y x

xy yy xx yy xx

y y

xy yy xx yy xx

x x

(1)

(3)

2

2

223333

2222111122221111

sin DC B A

cos DC B A DC B A DC B A DC B A

xy yy xx

xy yy xx xy yy xx xy yy xx xy yy xx x x

2212

212

2212

212

2212

212

2212

212

321

321

321

321

sin D cos D

cos D

sinC cosC

cosC

sin B cos B

cos B

sin A cos A

cos A

xy

yy xx x x

222222 2121 B B A A


36/64

2212

212

2212

212

2212

212

2212

212

321

321

321

321

sin D cos D

cos D

sinC cosC

cosC

sin B cos cos sin A cos cos

xy

yy xx x x

(4)

We know that

22

222

22

22

2222

cos sin

sin cos

sin cos

xy yy xx y x

xy yy xx yy xx

y y

xy yy xx yy xx

x x

(5)

Substituting eqn (5) in eqn (3)

1111

111

111

222

22

2212

212

2212

212

D cosC sin B

sin A

sinC cos B

cos A

sinC cos B

cos A

xy

yy xx x x

(6)

Comparing eqn (4) and eqn (6)


37/64

Comparing eqn (4) and eqn (6)

2212

212

2212

212

2212

212

2212

212

321

321

321

321

sin D cos D

cos D

sinC cosC

cosC

sin B cos B

cos B

sin A cos A

cos A

xy

yy xx x x

(4)

1111

111

111

222

22

2212

212

2212

212

D cosC sin B

sin A

sinC cos B

cos A

sinC cos B

cos A

xy

yy xx x x

(6)

A B A 21

B A B 21

C C B AC 2331

D D D 21

2221

3

B A A AC

03 D

It should be noted that the corresponding transformation applied to


38/64

xy yy xx xy

xy yy xx yy

xy yy xx xx

B AC

DC A B

DC B A

2

(7)

It should be noted that the corresponding transformation applied to

y x y y &

Now let us consider the new coordinate system ( x 1, y 1) which is related

to the original coordinate system ( x , y ) by y y and x x 11

Thus, the new coordinate system is a mirror reflection of the originalsystem with respect to the y-axis. With reference to the newcoordinate system, the velocity components are

vv and uu 11

Rates of strain and stresses are


39/64

Rates of strain and stresses are

xx x x xu

xu

1

1

11

yy y y yv

yv

1

1

11

xy y x yu

xv

yu

xv

1

1

1

1

11

xy y x

yy y y

xx x x

11

11

11

(8)

(9)

Equations (8) and (9) into equation (7)

DCBA


40/64

1111111

111111

11111111

21

11

y x y y x x y x

y x y y x x y y

y x y y x x x x

B A

C

DC A B

DC B A

(10)

According to assumption (b), the relations between stress components and rates of strain components must be invariant to a coordinate transformation consisting of either a rotation or a mirror reflection of axes, Equation (7) and (10) independent of the coordinate system. Hence, C = 0

xy yy xx xy

xy yy xx yy

xy yy xx xx

B AC

DC A B

DC B A

2

(7)

p DAccording to assumption(c), Eqn. (7) gives

22

B A

The constant (A-B)/2 in the last equation of equation (10) is the proportionality constantconnecting the shearing stress and rate of

shearing strain which is generally denoted bythe dynamic coefficient of viscosity,

The relations between stress and rate of strain in the two dimensional case given in


41/64

xy xy xy

yy yy

xx xx

p yv

xu

B

p yv

xu

B

2

2

2

gequation(7) are reduced to

3

2 B

The relations between stress and rate of strain can be extended to three dimensionalflows. They are

wvu


42/64

xz zx zx zx

zy yz yz yz

yx xy xy xy

zz zz

yy yy

xx xx

p zw

yv

xu

B

p z

w

y

v

x

u B

p zw

yv

xu

B

2

2

2

2

2

2

The sum of the three normal stresses is

For an incompressible fluid, zw

yv

xu

q .q . B p zz yy xx 323

0q .

The sum of the three normal stresses is


43/64

zw

yv

xu

q .q . B p zz yy xx 323

For an incompressible fluid, 0q .

p zz yy xx3

DESIRABLE SITUATIONS OF TURBULENT FLOW


44/64

To transfer the required heat between a solid and an adjacent fluid such as in thecooling coils of an air conditioner or a boiler of a power plant would require anenormously large heat exchanger if the flow were laminar Turbulence is also of importance in the mixing of fluids. Smoke from a stack wouldcontinue for miles as a ribbon of pollutant without rapid dispersion within thesurrounding air if the flow were laminar rather than turbulent Although there is mixing on a molecular scale (laminar flow), it is several orders of magnitude slower and less effective than the mixing on a macroscopic scale(turbulent flow). It is considerably easier to mix cream into a cup of coffee (turbulent

flow) than to thoroughly mix two colorsof a viscous paint (laminar flow)

DESIRABLE SITUATIONS OF LAMINAR FLOW Pressure drop in pipes - power requirements for pumping can be considerablylower if the flow is laminar rather than turbulent

Blood flow through a persons arteries is normally laminar, except in the largestarteries with high blood flowrates Aerodynamic drag on an airplane wing can be considerably smaller with laminarflow past it than with turbulent

DEFINITION OF TURBULENCE:


45/64

Taylor and Von Karman (1937)

Turbulence is an irregular motion which in general makes its

appearance in fluids, gaseous or liquids, when they flow past solidsurfaces or even when neighbouring streams of same fluid past overone another

Turbulent fluid motion is an irregular condition of flow in whichvarious quantities show a random variation with time and spacecoordinates, so that statistically distinct average values can bediscerned


46/64

MEAN MOTION AND FLUCTUATIONS

'uu

Time ( t )

u

t0 t0+T

t u

MEAN MOTION AND FLUCTUATIONS


47/64

uuu ;uuuT t

t

dt t , z , y , xuT 1

u o

o

T t

t

dtuT t

t

dtuT 1

T t

t

dtuT t

t

dtuT 1

T t

t

dtuuT 1

u o

o

o

o

o

o

o

o

o

o

0T uuT T 1

T uT t

t

dtuT 1

u o

o

T t

t

dtT 1

_______ o

o

2u 2u

0u

21


48/64

u

2

u

______

IntensityTurbulence

T o

t

o t

dt 2uT 1

2u

uu 2 2u 2u 2uu 2u

uu 2 2u 2u 2u________

_____________________

0 zeroT u

T t

t

dtuT u

T t

t

dtuuT 1________ o

o

o

o

uu 0________

uu

u .u 2u 2u

_____________


49/64

vuuvvuvuvvuuuv_________________________________________

vuvuuv__________

RULES

f f

g f ____

g f

g . f ____

g . f

s f

s

___ f

ds f ______

ds . f

REYNOLDS EQUATIONS AND REYNOLDS STRESSES


50/64

0 zw

yv

xu

2 z

u 2

2 y

u 2

2 x

u 2

x P

zu

w yu

v xu

u tu

2 z

v 2

2 y

v 2

2 x

v 2

y P

zv

w yv

v xv

u tv

2 z

w 2 2 y

w 2 2 x

w 2 z P

zww

ywv

xwu

tw

ppp;www;vvv;uuu


51/64

p p p ;www ;vvv ;uuu

0

z

w

y

v

x

u

0 z

_____ww

y

_____vv

x

_____uu

x

__u

xu

x

_____uu

0 zw

yv

xu

222P


52/64

2 z

u 2

2 y

u 2

2 x

u 2

x P

zu

w yu

v xu

u tu

2 z

u 2

2 y

u 2

2 x

u 2

x P

z

uw

y

uv

x

2u

tu

tu

t

__u

tu

t

_____uu

FlowsSteady Mean for zero tu

____22

____2

__________2

___2


53/64

xu

u 2 x

2u x

2u x

2u x

2uu x

2u

xuu 2

x

____

2u x

___

2u

y

____vu

y

uv

y

vu

y

___uv

y

___uv

____vu

____vuvu

y

_____________

vvuu y

___

uv

z

____wu

zu

w zw

u z

___uw

_______


54/64

x p

x p

x p p

x p

x

_____ p p

2 z

u 2

2 y

u 2

2 x

u 2

x p

z

____wu

z

uw

z

wu

y

____vu

y

uv

y

vu

x

uu 2

x

____ 2u

uuuwvu


55/64

z

____wu

y

____vu

x

____ 2u

2 z

u 2

2 y

u 2

2 x

u 2

x p

zu

w yu

v xu

u zw

yv

xu

u

z

____wu

y

____vu

x

____ 2u

2 z

u 2

2 y

u 2

2 x

u 2

x p

zu

w yu

v xu

u


56/64

z

____wu

y

____vu

x

____ 2u

u 2 x p

zu

w yu

v xu

u

z

____wv

y

____ 2v

x

____vu

v 2 y p

zw

w yv

v xv

u

z

____ 2w y

____wv

x

____wu

w 2 z p

zw

w yw

v xw

u

____wu

____vu

____ 2u2puuu


57/64

zwu

yvu

xu

u 2 x p

zu

w yu

v xu

u

z

____wv

y

____ 2v

x

____vu

v 2

y p

zw

w yv

v xv

u

z

____ 2w

y

____wv

x

____wu

w 2 z p

zw

w yw

v xw

u

u 2 x p

zu

w yu

v xu

u

v 2 y p

zw

w yv

v xv

u

w 2 z p

zw

w yw

v xw

u

Steady state Navier Stokesequations, if the velocitycomponents and pressurecomponents are replaced bymean components or timeaverages

Resultant surface force per unit area due to the additional terms


58/64

Resultant surface force per unit area due to the additional terms

z

' zz

y

' yz

x

' xz k

z

' yz

y

' yy

x

' xy

j z

' xz

y

' xy

x

' xxi P

___ 2u' xx

___ 2v' yy

___ 2w' zz

___vu' xy

___wu' xz

___wv' yz

' xz

' xy'xx2 puuu


59/64

z yy

x xxu 2

xp

zw

yv

xu

z

' yz

y

' yy

x

' xy

v 2

y p

zw

w yv

v xv

u

z

' zz

y

' yz

x

' xzw 2

z p

zw

w yw

v xw

u

zz yz xz yz yy xy

xz xy xx

___ 2w

___wv

___wu

___

wv

___ 2

v

___

vu

___wu

___vu

___ 2u

=

In turbulent flow, laminar stresses must be increased by additional

streses REYNOLDS STRESSES (1895 - Reynolds)

V l it fil Average velocityy y


60/64

Laminar flow shear stresscaused by random motionof molecules

Turbulent flow as a series of random three dimensional eddies

(1)

(2)

Velocity profileu = u(y)

u 1< u 2

AA

u

u u y

Average velocityprofile

Turbulenteddies

AA

u

y


61/64

_____vu

dy du

turblam

Laminar flow 0v0u0_____

vu

Pipe wall Viscous sublayer


62/64

VISCOUS SUBLAYER :lam >>> turb OVERLAP LAYER: lam ~ turb TURBULENT LAYER:

lam


63/64

VISCOUS SUBLAYER

OVERLAP LAYER

TURBULENT LAYER

5 y yu

30 y 505 . 3 yln 5u

30 y 5 . 5 yln 5 . 2u

u

uu ;wu ;

yu y

25


64/64

30 y 5 . 5 yln 5 . 2u

5 y ; yu

30 y 505 . 3 yln 5u

Pipe centerline

Experimental data

uu*

20

15

10

5

1

010 10 2 10 3 10 4

yu*

Viscous sublayer

convection 2

Documents