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Chap. 6

Differential Analysis

of Fluid Flow

Although the control volume analysis gave us accurate

solution of flow physics, it deals with only global information

of a certain volume (control volume). Therefore, it can not

provide the detail information of flow at certain point.

To arrive this goal of gaining the detail flow physics, the

differential analysis should be adopted and this is truly the

governing equation of fluid dynamics written as partial

differential equation (hard to solve by hand CFD

essentially!!!).

6.1 Fluid Element Kinematics

Eulerian derivative:

V

t

Dt

D

zw

yv

xu

tDt

D

termConvective change of rate Convective

termunsteady Local derivate time Local

V s,coordinate

spatial torespectivewith

derivative imesVelocity t

t location, fixed aat

change of Rate

fluid a ofproperty of

change of rate Total

Linear deformation-Stretching

-Rate of the change of volume per unit volume

x

uzyx

x

u

zyx

1

t

zyxzy txx

ux

limzyx

1

dt

Vd

V

1

0t

rate

This rate of change of the volume per unit volume is called

the volumetric dilatation rate, or simply dilatation.

The derivatives, , and cause a linear

deformation (stretching) of the element in the sense that the

shape of the element does not change.

V

z

w

y

v

x

u

dt

Vd

V

1

In general,

x/u y/v z/w

Angular Motion and Deformation

Vorticity(渦度):

If the flow is , then the flow is called “irrotational”.

V

0V

6.2 Conservation of Mass

Conservation of mass: 0mmmt

inoutcv

zyxt

zyxt

mt

cv

z x

2

y

y

vvz y

2

x

x

uumout

z x

2

y

y

vvz y

2

x

x

uumin

Continuity equation:

For incompressible flow,

0z yx

y

vz yx

x

uz yx

t

0

z

w

y

v

x

u

t

0Vt

0z

w

y

v

x

u

Dilatation

0V

or

Ex. 6.2

6.2.2 Cylindrical Polar Coordinates

Cylinderical coordinates

Spherical coordinates

0

z

vv

r

1

r

vr

r

1

t

zr

0Vt

0

v

sinr

1sinv

sinr

1

r

vr

r

1

t

r

2

2

6.2.3 The Stream Function

introduce the stream function, , which satisfy the continuity

equation automatically as

;

substituting these relationships into incompressible continuity

equation,

By introducing the stream function, we can simplify the

problem from two unknowns, u and v, to only one unknown, .

Another merit of using the stream function is that lines along

which is constant are streamlines.

yu

xv

0xyyxxyyxy

v

x

u 22

The change in the value of is given by

Along a line of constant ,

i.e.,

which is the defining equation for a streamline.

Thus, if we know the function (x,y) we can plot lines of

constant to provide the family of streamlines.

udyvdxdyy

dxx

d

udyvdx0d

u

v

dx

dy

Ex. 6.3

6.3 Conservation of Linear Momentum

Newton’s second law of x-direction

gravity,xstress ,xxx FFFa m

gravity,xstress ,xxx FFFa m

Newton’s second law of x-direction

D-2in

y

uv

x

uu

t

u zyx

Dt

Du zyxa m x

zy x yx

z x 2

y

yz x

2

y

y

z y 2

x

xz y

2

x

xF

yxxx

yx

yx

yx

yx

xxxx

xxxxstress,x

xgravity,x g z y x F

Thus, the Newton’s second law in x-direction is

Similarly, in y-direction

x

yxxx

Dt

Du

gyx

1

y

uv

x

uu

t

u

y

yyxy

Dt

Dv

gyx

1

y

vv

x

vu

t

v

The general equation of motion in 3-D is

xzxyxxx

Dt

Du

gyyx

1

z

uw

y

uv

x

uu

t

u

zzzyzxz

Dt

Dw

gyyx

1

z

ww

y

wv

x

wu

t

w

y

zyyyxy

Dt

Dv

gyyx

1

z

vw

y

vv

x

vu

t

v

x-dir:

y-dir:

z-dir:

g1

Dt

VD

6.8.1 Stress-Deformation Relationships

For incompressible Newtonian fluids, it is known that the

stresses are linearly proportional to the rates of deformation

as:

x

u2pxx

y

v2pyy

z

w2pzz

x

v

y

uyxxy

y

w

z

vzyyz

z

u

x

wxzzx

; ;

zzyyxx

3

1p

6.8.2 The Navier-Stokes Equations (Incompressible)

y2

2

2

2

2

2

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

x2

2

2

2

2

2

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

z2

2

2

2

2

2

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

x-dir:

y-dir:

z-dir:

x

2 g ux

p1

Dt

Du

y

2 g vy

p1

Dt

Dv

z

2 g wz

p1

Dt

Dw

; ;

g Vp1

VVt

V 2

(for incompressible flows)

These equations are called Navier-Stokes equations (1822),

named in honor of the French mathematician L. M. H. Navier (1758-

1836) and the English mechanician Sir G. G. Stokes (1819-1903).

Unfortunately, because of the complexity of the Navier-Stokes

equations ( i.e., nonlinear, second order partial differential

equations), they are not amenable to exact solutions except in a few

instances.

CFD (Computational Fluid Dynamics) is essential to solve the Navier-

Stokes equations.

g Vp1

VVt

V 2

Sir George Gabriel Stokes, 1st Baronet FRS

(1819–1903, England)

Claude-Louis Navier(1785–1836, France)

Compressible Navier-Stokes Eq.

For compressible flows,

xzxyxxx gyyx

1

z

uw

y

uv

x

uu

t

u

zzzyzxz gyyx

1

z

ww

y

wv

x

wu

t

w

y

zyyyxyg

yyx

1

z

vw

y

vv

x

vu

t

v

x-dir:

y-dir:

z-dir:

g1

Dt

VD

Continuity eq:

0z

w

y

v

x

u

t

Momentum eq.

Stress relationships:

x

u2pxx

y

v2pyy

z

w2pzz

x

v

y

uyxxy

y

w

z

vzyyz

z

u

x

wxzzx

Energy Equation for Compressible Viscous Flow

Energy Equation:

qt

QVp

Dt

De

agency external

fromin-heat

of Rate

q

t

Q

Dt

Dp

Dt

Dh

or,

Where e=internal energy, Tce v

,/peh Tch ph=enthalpy,

Tkq

: Heat transfer by conduction

: Dissipation function

5 unknowns (u, v, w, p, and ) and 5 equations

6.4 Inviscid Flow (Euler equation)

yg y

p1

z

vw

y

vv

x

vu

t

v

xg x

p1

z

uw

y

uv

x

uu

t

u

zg z

p1

z

ww

y

wv

x

wu

t

w

x-dir:

y-dir:

z-dir:

xg x

p1

Dt

Du

yg y

p1

Dt

Dv

zg z

p1

Dt

Dw

; ;

g p1

VVt

V

Euler’s equation

Swiss mathematician

Leonhard Euler (1707-1783)

Leonhard Euler

6.4.3 Irrotational Flow

An irrotational flow field is defined as the flow

for which , or =0. 0V

0z

v

y

wx

0x

w

z

uy

0y

u

x

vz

That is,

6.4.4 The Bernoulli Equation for Irrotational Flow

For incompressible inviscid irrotational, ( ) flow fields,

the constant of Bernoulli equation is same throughout the

entire flow field and the Bernoulli equation can be applied

between any two points.

(valid between any two points)

- steady incompressible inviscid irrotational flow

0V

22

2

21

1

2

1 gzp

2

Vgz

p

2

V

6.9 Some Simple Solutions for Viscous, Incompressible Fluids

y2

2

2

2

2

2

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

z2

2

2

2

2

2

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

x2

2

2

2

2

2

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

x-dir:

y-dir:

z-dir:

0

z

w

y

v

x

u

t

Continuity :

Incompressible Navier-Stokes Equation

Convective term (nonlinear) Viscous dissipation term (linear)

Mom

entu

m E

quatio

n

in vector form,

- Continuity equation :

- Momentum equation :

0Vt

g Vp1

VVt

V 2

Convective term

(nonlinear)

Viscous term

(linear)

These equations of continuity and Navier-Stokes equations

consist of 4 equations having 4 unknowns (u, v, w, and p).

The unique solution exists mathematically.

CFD(Computational Fluid Dynamics) is essential to solve

the Navier-Stokes and Euler equations.

6.9.1 Steady Laminar Flow Between Fixed Parallel Plates

In this internal flow, v = w = 0.

Continuity eq:

x-dir.

momentum :

“Poiseuille Flow”

0

z

w

y

v

x

u

t

0)(w0

0)(v0

steady0

const)( 0x

u

0

x

dir-zin variationno

0

2

2

2

2

2

2

00

steady0

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

y-dir. momentum:

z-dir. momentum:

g

y

0)v( 0

2

2

2

2

2

2

0) v( 0

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

0

z

0)w( 0

2

2

2

2

2

2

0)w( 0

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

That is,

Continuity:

x-momentum:

y-momentum:

z-momentum:

0x

u

2

2

0)xu/(0

2

2

continuityfrom

0

y

u

x

u

x

p1

x

uu

g-y

p1 0

z

p1 0

x

p1

yd

ud2

2

)x(Cgyp

no variation of u in x-direction, u=u(y only)

no variation of p in z-direction, p=p(x,y)

only) x(fx

p

ode in u

The integration of x-component of momentum gives

Note that is treated as constant because it is not a

function of y.

Two boundary conditions are needed to obtain C1 and C2.

That is, u=0 at

1Cy x

p1

dy

du

dy

du

21

2 CyCy x

p

2

1u

x/p

h y

0C1 2

2 h x

p

2

1C

used for

;

Therefore,

Drag:

Volume Flowrate:

22 hy x

p

2

1u

: Parabolic velocity distribution

y x

p

dy

du

: Linear shear stress distribution

hx

phymax

00ymin ;

hL x

p 2dx h

x

p 2dx2D

L

0

L

0hy

(per unit width in z-dir.)

x

p

3

h2dy hy

x

p

2

1dy uQ

3h

h

22h

h

(per unit width in z-dir.)

( is –’ve value for flowing to +’ve x-direction.)x/p

The volume flowrate is proportional to the pressure gradient

and inversely proportional to the viscosity.

Vmean:

i.e.,

By the way,

Hence,

Here, it was assumed that the present flow is laminar. If the

flow is turbulent , then the above analysis is

not valid.

x

p

3

h2h2VQ

3

mean

x

p

3

hV

2

mean

x

p

2

huu

2

0ymax

meanmax V 5.1u

1400

)h2(VRe mean

6.9.2 Couette Flow

Consider the flow developed by fixing one plate and letting

the other plate move with a constant velocity, U. The flow is

initially at rest.

U

b

x

y

In this internal flow, v = w = 0.

Continuity eq:

x-dir.

momentum :

y-dir.momentum:

z-dir.momentum:

g

y

0)v( 0

2

2

2

2

2

2

0) v( 0

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

0

z

w

y

v

x

u

t

0)(w0

0)(v0

steady0

const)( 0x

u

0

x

dir-zin variationno

0

2

2

2

2

continuityfrom

0x

u0

2

2

0

00

continuityfrom

0steady

0

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

0

z

0)w( 0

2

2

2

2

2

2

0)w( 0

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

Resulting equations are the same as those of Poiseuille flow, except p/x=0.

That is,

Continuity:

x-momentum:

y-momentum:

z-momentum:

0x

u

g-y

p1 0

z

p1 0

0 yd

ud2

2

)x(Cgyp

no variation of u in x-direction,

u=u(y only)

no variation of p in z-direction,

p=p(x,y)

ode in u

The integration of x-component of momentum gives

Two boundary conditions are needed to obtain C1 and C2.

That is, u=0 at y=0 and u=U at y=b

Therefore, ; (constant shear stress throughout the flow)

21 CyCu

b

UC1 0C2 ;

b

yUu

b

U

U

b

x

y

U

The flow in an unloaded journal bearing might be

approximated by this Couette flow if the gap is very small

(i.e., ).iio rrr

irU

io

i

rr

r

;io rrb

U

b

x

y

ri =ri

Couette flow with imposition of streamwise pressure gradient, ( ).

Continuity eq:

x-dir.

momentum :

y-dir.

momentum:

z-dir.

momentum:

g

y

0)v( 0

2

2

2

2

2

2

0) v( 0

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

0

z

w

y

v

x

u

t

0)(w0

0)(v0

steady0

const)( 0x

u

0

x

dir-zin variationno

0

2

2

2

2

continuityfrom

0x

u0

2

2

0

00

continuityfrom

0steady

0

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

0

z

0)w( 0

2

2

2

2

2

2

0)w( 0

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

Couette + Poiseuille Flow

0x/p

That is,

Continuity:

x-momentum:

y-momentum:

z-momentum:

0x

u

g-y

p1 0

z

p1 0

)x(Cgyp

no variation of u in x-direction,

u=u(y only)

no variation of p in z-direction,

p=p(x,y)

ode in u

x

p1

yd

ud2

2

Resulting equations are the same as those of Poiseuille flow.

Only difference is boundary condition.

The integration of x-component of momentum gives

Two boundary conditions are needed to obtain C1 and C2.

That is, u=0 at y=0 and u=U at y=b

Therefore,

1Cy x

p1

dy

du

dy

du

21

2 CyCy x

p

2

1u

0C1

used for

;

U

bx

y

bCb x

p

2

1U 1

2

b

yUbyy

x

p

2

1u 2

0]x/p[

Poiseuille

Couette

Couette+Poiseuille

U

b

x

y

U

U

For positive pressure gradient

For negative pressure gradient

Same and Difference

Flow

Pattern

Resulting

Equation

Boundary Condition

Remark

at y=0 at y=b

Poiseuille 0 0

Couette 0 U

Poiseuille

+Couette0 U

x

p1

yd

ud2

2

x

p1

yd

ud2

2

0 yd

ud2

2

U

bx

y

U

U

2002 WORLD CUP 의 Velocity Vector

2002 WORLD CUP 의 Pressure Contour Line

Take a break

Ex. 6.9

0

z

w

y

v

x

u

t

x2

2

2

2

2

2

g z

u

y

u

x

u

x

p1

z

uw

y

uv

x

uu

t

u

y2

2

2

2

2

2

g z

v

y

v

x

v

y

p1

z

vw

y

vv

x

vu

t

v

z2

2

2

2

2

2

g z

w

y

w

x

w

z

p1

z

ww

y

wv

x

wu

t

w

6.9.3 Steady, Laminar Flow in Circular Tubes

The incompressible Navier-Stokes equations in cylindrical

coordinate system was written as

“Hagen-Poiseuille Flow”

z

vv

r

vv

r

v

r

vv

t

v rz

2

rrr

r

r2

r

2

22

r

2

22

rr g z

vv

r

2v

r

1

r

v

r

vr

rr

1

r

p1

Continuity :

r-momentum :

0Vz

Vr

1rV

rr

1

tzr

z

vv

r

vvv

r

v

r

vv

t

vz

r

r

g

z

vv

r

2v

r

1

r

v

r

vr

rr

1p

r

11

2

2

r

22

2

22

z

vv

v

r

v

r

vv

t

v zz

zzr

z

z2

z

2

2

z

2

2

z g z

vv

r

1

r

vr

rr

1

z

p1

-momentum :

z-momentum :

Since vr = v = 0,

0z

v z

sing

r

p10

cosg

p

r

110

r

vr

rr

1

z

p10 z

Continuity :

r-momentum :

-momentum :

z-momentum :

g

y

x

r

The integration of r- and -direction gives

or

This equation indicates that the pressure is hydrostatically

distributed at any cross-section and is not a function of r or .

The z-momentum equation is

and integration gives

)z(Csinrgp

)z(Cgyp

const.

z

z

pr

r

vr

r

1

2

z Cz

p

2

r

r

vr

r

C

z

p

2

r

r

v 1z

Hence,

Since vz should be finite at r=0, or should be zero at

r=0 from the symmetry condition of velocity profile, C1 must

be zero ( ).

And from vz = 0 at r=R,

Thus,

: Parabolic distribution

21

2

z CrlnCz

p

4

rv

r/vz

)0ln(

2

2 R z

p

4

1C

22

z Rrz

p

4

1v

To obtain a relationship between Q and ,

Mean velocity:

Maximum velocity:

z/p

z

p

8

Rdr r2 Rr

z

p

4

1dr r2vQ

4R

0

22R

0z

z

p

8

R

R

QV

2

2mean

z

p

4

Rvv

2

0rzmax

meanmax V2v

Wall shear stress:

Skin friction coefficient: defined as

Then, the substitution of wall gives

where ReD is called Reynolds number, defined as

mean

RrRr

zwall V

R

4

z

p

2

R

z

p

2

r

r

v

2

mean

wall

f

V2

1C

D

Re

meanmean

mean2

mean

fRe

16

V D 16

V R

18V

R

4

V2

1

1C

D

DVRe mean

D

The flow remains laminar for ReD<2100.

Drag:

z

pLRLV8dz R2 V

R

4 dz R2 D 2

mean

L

0mean

L

0wall

Turbulent

In this flow situation the N-S equations are reduced to the

same equations of the previous section of Hagen-Poiseuille

flow.

Since vr = v = 0,

0z

v z

sing

r

p10

cosg

p

r

110

r

vr

rr

1

z

p10 z

Continuity :

r-momentum :

-momentum :

z-momentum :

6.9.4 Steady, Axial, Laminar Flow in an Annulus

As mentioned before, the integration of r- and -momentum

equation yield that the pressure is hydrostatically distributed

at any cross-section and is not a function of r or .

The integration of z-momentum equation gives

Boundary condition: no-slip condition at r=Ri and r=Ro

Hence,

z/p

21

2

z CrlnCz

p

4

rv

(same as Hagen-Poiseuille)

0vz

o

i

o

2

o

2

i2

o

2

zR

rln

R

Rln

RRRr

z

p

4

1v

Volume Flowrate:

The annular flow is assumed laminar if ReDh <2100, where

ReDh means the Reynolds number based on the hydraulic

diameter, Dh.

i

o

22

i

2

o2

i

4

o

R

Rz

R

Rln

RRRR

z

p

8dr r2vQ

o

i

hmean

D

DVRe

h

io

io

2

i

2

oh RR2

RR2

RR4

perimeter wetted

area sectional-cross4D

Ex. 6.10

p

8

R

z

p

8

RQ

44

i

o

22

i

2

o2

i

4

o

R

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