FLUID KINEMATICS

Lecture 3

Kinematics Part I

1

Definitions, conventions & concepts

( , , , )V V x y z t

Dimensionality Steady or Unsteady

• Given above there are two frames of reference for describing

this motion

Lagrangian

“moving reference frame”

Eulerian

“stationary reference frame”

• Focus on behavior of group of

particles at a particular point

•

Pathline

• Focus on behavior of particular

particles as they move with the

flow

• Motion of fluid is typically described by velocity V

v

v

v Steady flow

y

x

Streamlines

• Individual particles must travel on paths whose tangent is

always in direction of the fluid velocity at the point.

In steady flows, (Lagrangian) path lines are the same as (Eulerian)

streamlines.

Lagrangian vs. Eulerian frames of reference

X2

X1

t0 t1

~x

0~x particle path

* Following individual particle as it moves along path…

At t = t0 position vector is located at

Any flow variable can be expressed as ),(~

txF

following particle position which can be expressed as ),(0~~

txx

Lagrangian

)(~

tx0~

0~

)( xtx

* Concentrating on what happens at spatial point

Any flow variable can be expressed as ),(0~

txF

Local time-rate of change:

Local spatial gradient:

This only describes local change at point in Eulerian description! 0~

x

Material derivative “translates” Lagrangian

concept to Eulerian language.

0~x

Eulerian

t

F

ix

F

Material Derivative (Substantial or Particle) ),,,( tzyxFConsider ; ),,(

~zyxx

• As particle moves distance d in time dt ~x

i

i

dxx

Fdt

t

FdF

-- (1)

• If increments are associated with following a specific particle

whose velocity components are such that

dtudx ii -- (2)

Substitute (2) (1) and dt

i

i

ux

F

t

F

dt

dF

-- (3)

i

ix

Fu

t

F

Dt

DF

Local rate of change

at a point ~x

Advective change

past ~x

Fut

F

Dt

DF

~

jF a

i

j

i

jj

x

au

t

a

Dt

Da

Vector Notation:

ESN: e.g. if

Along ‘Streamlines’:

s

Fq

t

F

Dt

DF

n

s

Magnitude of ~u

Streamlines

• A Streamline is a curve that is

everywhere tangent to the

instantaneous local velocity

vector.

• Consider an arc length

• must be parallel to the local

velocity vector

• Geometric arguments results

in the equation for a streamline

dr dxi dyj dzk

dr

V ui vj wk

dr dx dy dz

V u v w

Pathlines

, ,particle particle particlex t y t z t

• A Pathline is the actual path traveled by an individual fluid particle over some time period.

• Same as the fluid particle's material position vector

• Particle location at time t:

• Particle Image Velocimetry (PIV) is a modern experimental technique to measure velocity field over a plane in the flow field.

start

t

start

t

x x Vdt

Streaklines

• A Streakline is the

locus of fluid particles

that have passed

sequentially through a

prescribed point in the

flow.

• Easy to generate in

experiments: dye in a

water flow, or smoke

in an airflow.

Pathlines, Streaklines & Streamlines

t0

t1

nozzle

nozzle

a b c d e

a b c d

Pathlines: Line joining positions of particle “a” at successive times

Streaklines: Line joining all particles (a, b, c, d, e) at a particular

instant of time

Sreamlines: Trajectories that at an instant of time are tangent to the

direction of flow at each and every point in the flowfield

Streamtubes

• No flow can pass through a streamline because velocity

is always tangent to the line.

• Concept of streamlines being “solid” surfaces forming

“tubes” of flow and isolating “tubes of flow” from one

another.

s

ds

c

No flux

Calculation of streamlines and pathlines

Streamline

),,(~

wvuU

),,(~

dzdydxds

By definition: (i) 0~

dsU

3

~

2

~

1

~

3

~

2

~

1

~

000)(

)()(

aaaaudyvdx

awdxudzavdzwdy

udyvdxwdxudzvdzwdy ;;

w

dz

v

dy

u

dx

dt

dzw

dt

dyv

dt

dxu ;;

1

0

1

0

1

0

1

0

1

0

1

0

;;z

z

t

t

y

y

t

t

x

x

t

t u

dzdt

v

dydt

u

dxdt

Pathline

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x

2

1 1 3 3 u x u x

Example 1: Stagnation point flow

x3

Stagnation-point velocity field:

1 1

3 3

u a t x

u a t x

(a) Calculate streamlines

33 3 3 31

1 3 1 1 1 1

3 33 1

1 1

3 3 31 1 1

1 3 1 3 1 3

1 3

1 1 3

3

(an ode in and )

or...

ln ln ln

streamlines are hyperbolae

a t xdx dx u xdx

u u dx u a t x x

dx xx x

dx x

dx dx dxdx dx dx

u u ax ax ax ax

a x a x a C

Cx x x C

x

Cleverly chosen

integration constant

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

x1

x 3

Stagnation pt flow with a=1

(b) Calculate pathlines

1

1

1 11 1

1

( )

1 11 1

1 1(0) 0

33

3 3

1

which can be integrated

( )ln or ( ) (0)exp

(0)

By a similar argument using , we find that

( ) (0)exp

But, despite all,

x t t

x

dx dxu ax adt

dt x

dx x tadt at x t x at

x x

dxu

dt

x t x at

x x3 1 3(0) (0) '

pathlines are (also) hyperbolae

x x C

0 2 4 6 8 10

-2

-1

0

1Velocity vectors

0 2 4 6 8 10

-2

-1

0

1Streamlines

0 2 4 6 8 10

-2

-1

0

1Pathline

0 2 4 6 8 10

-2

-1

0

1Streakline

Example 2: a (more complicated) velocity field: in a surface gravity wave:

Stream/streak/path lines are completely different.

(1) Basic Motions

(a) Translation

X2

X1 t0

X2

X1 t1

(b) Rotation

X2

X1 t1

• No change in dimensions

of control volume

Relative Motion near a Point

(c) Straining (need for stress):

Linear (Dilatation) – Volumetric Expansion/Contraction

X2

X1 t0

X2

X1 t1

(d) Angular Straining – No volume change

X2

X1 t1

Note: All motion except pure translation involves

relative motion of points in a fluid

x2

x1 t0

x2

x1 t1

~x

~~xx

~x

~u

~~duu

'

~x

P

O

P’

O’

Consider two such points in a flow, O with velocity

And P with velocity moving to O’ and P’

respectively in time t

),( 0~~

txu

),}({ 0~~~~

txxudu

General motion of two points:

tTherefore, after time

ttxuxtxOtuxttxuxxx ),()||(||),()( 0~~~

2

~~~0

~~~~

'

~

tx

uxtuxxxseparationinChange

j

ij

~~~

'

~

tuxxx ~~~

'

~

Relative motion of two points depends on the velocity

gradient, , a 2nd-order tensor. j

i

x

u

to first order

ttxux ),( 0~~~

O’:

P’:

txOuxtxuxx

ttxxuxx

)}||(||),({)(

),()(

2

~~~0

~~~~

0~~~~~

Taylor series expansion of ),( 0~~~

txxu

-- (A)

O() means “order of” =

“proportional to”

(2) Decomposition of Motion

“…Any tensor can be represented as the sum of a symmetric

part and an anti-symmetric part…”

i

j

j

i

i

j

j

i

j

i

x

u

x

u

x

u

x

u

x

u

2

1

2

1

ij ije r

={ rate of strain tensor} + {rate of rotation tensor}

3

3

3

2

2

3

3

1

1

3

2

3

3

2

2

2

2

1

1

2

1

3

3

1

1

2

2

1

1

1

2

2

2

2

1

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

eij

Note:

(i) Symmetry about diagonal

(ii) 6 unique terms

Linear & angular straining

0

0

0

2

1

3

2

2

3

3

1

1

3

2

3

3

2

2

1

1

2

1

3

3

1

1

2

2

1

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

rij

Note:

(i) Anti-symmetry about

diagonal

(ii) 3 unique terms (r12, r13, r23)

Rotation

Terms in 1 1ij 2 2

~( ) ijk k ijk kr u

3 2

3 1

2 1

01

02

0

ijr

1 112 123 3 32 2

1 1 132 321 1 1 12 2 2

. .

( )

e g r

r

Let’s check this assertion about rij

2

mijk k ijk klm

l

mjik klm

l

mim jl il jm

l

ji

j i

ij

u

x

u

x

u

x

uu

x x

r

The recipe:

(a) m = i and l =j

(b) l = i and m = j

gives

Interpretation

~

( )

1

2

1

2

1( )

2

i j ij ij

ij j ijk k j

ij j ikj k j

i ij j i

u x e r

e x x

e x x

u e x x

tx

uxxx

j

ij

~

'

~

'

~ ~

ii j

j

x x uu x

t x

Relative velocity due

to deformation of

fluid element

Relative velocity due

to rotation of element

at rate 1/2

x1

x3

tv

2 tv a

Consider solid body rotation about x2

axis with angular velocity

= 2 x {Local rotation rate of fluid elements)

General result:

Simple examples:

3 1

312

3 1

,0,

2

0

u a t x a t x

uua

x x

e

x1

x3

u1(x3) u3(x1)

Consider the flow

1 3,0,

0

0 0

0 0 0

0 0

u a t x a t x

r

a

e

a

x

1

x

3

t0

t1

t2

What happens to

the box?

It is flattened and stretched

(3) Types of motion and deformation .

(i) Pure Translation

X2

X1

t1

t0 tu 2

tu 1

1x

1x

2x

2x

1 0 t t t

(ii) Linear Deformation - Dilatation

X2

X1

t1

1x

'

1x

’

2x

2x t0

a

b

1 0

22

2

11

1

t t t

ua x t

x

ub x t

x

In 2D - Original area at t0

- New area at t1

21 xx

))(( 21

'

2

'

1 bxaxxx

)]())(1[(

...))((

2

2

2

1

121

2

12

2

221

1

121

'

2

'

1

tOtx

u

x

uxx

tOxtxx

uxtx

x

uxxxx

Area Strain = and Strain Rate = 21

21

'

2

'

1

xx

xxxx

t

StrainArea

t

A

Adt

dA

A t

0

00

1lim

1

and )(

1

2

2

1

1

0

tOx

u

x

u

t

A

A

2

2

1

1

0

1

x

u

x

u

dt

dA

A

In 3D

i

i

x

uu

x

u

x

u

x

u

dt

dV

V

~3

3

2

2

1

1

0

1

* Diagonal terms of eij are responsible for dilatation

In incompressible flow, ( is the velocity) 0 U U

Thus (for incompressible flows),

(a) in 2D, areas are preserved

(b) in 3D, volumes are preserved

(iii) Shear Strain – Angular Deformation

1x

2x

2

2

11 x

x

uu

A

B

1

1

22 x

x

uu

1u

2u

2x

1xO

ttt 01

X2

X1

t1

O

A

B

txx

u 2

2

1

txx

u 1

1

2

d

d

t0

Shear Strain Rate Rate of decrease of the angle formed by 2

mutually perpendicular lines on an element

1

2

2

1

1

1

2

1

2

2

1

2

)(1

)(11

x

u

x

u

txx

u

xtx

x

u

xtt

dd

dd ,Iff small

Average Strain Rate jiex

u

x

uij

1

2

2

1

2

1

The off diagonal terms of eij are responsible for angular

strain.

)( 2

2

11 x

x

uu

(iv) Rotation

1x

2x

A

B

1

1

22 x

x

uu

1u

2u

2x

1xO

ttt 01

t1

O

A B

txx

u 2

2

1

txx

u 1

1

2

d

d

t0

jirx

u

x

u

t

ddij

2

1

1

2

2

1

2

1

Average Rotation Rate (due to superposition of 2 motions)

txx

u

xt

x

ud

x

u

txx

u

xt

x

ud

x

u

2

2

1

22

1

2

1

1

1

2

11

2

1

2

1:

1:Rotation due to

due to

Stream Function & Velocity

Potential Stream lines/ Stream Function (Y)

Concept

Relevant Formulas

Examples

Rotation, vorticity

Velocity Potential(f)

Concept

Relevant Formulas

Examples

Relationship between stream function and velocity potential

Complex velocity potential

Stream Function

Assume

xVy

Instead of two functions, Vx and Vy, we need to solve

for only one function Stream Function

Order of differential eqn increased by one

Then 2

xy

VV dy dy

x x y

2

dyy x x

Stream Function

What does Stream Function mean?

Equation for streamlines in 2D are given by

= constant

Streamlines may exist in 3D also, but stream function

does not

Why? (When we work with velocity potential, we may get a

perspective)

In 3D, streamlines follow the equation

x y z

dx dy dz

V V V

Rotation

Definition of rotation

Time=t

x

y

ROTATION

2z

d

dt

x

y

y x xV

y xV

x yV

x y yV

Assume Vy|x < Vy|x+x

and Vx|y > Vx|y+y

Rotation

To Calculate Rotation

1tany

x

y1

x

1 y yx x xy V t V t

arctan

y yx x xV V t

x

arctan

x xy y yV V t

y

Similarly

Rotation To Calculate Rotation

ROTATION

2z

d

dt

0

1lim

2

t t t

t t

For very small time and very small element, x, y

and t are close to zero

0 0

arctan arctan

1 1lim lim

2 2

y y x xy y yx x x

t t

V V t V V t

x y

t t

0x 0y

0x 0y

0 0

arctan

lim lim

y y y yx x x x x x

t t

V V t V V t

x x

t t

0x 0y

0x 0y

Rotation To Calculate Rotation

For very small , i.e. ~ 0

cos 1 tan

arctan

arctan

y y y yx x x x x xV V t V V t

x x

sin

Rotation To Calculate Rotation

0

limy y yx x x

x

V V V

x x

1

2

y xz

V V

x y

Simplifies to

0 0

arctan arctan

1 1lim lim

2 2

y y x xy y yx x x

zt t

V V t V V t

x y

t t

0x

0y

0x

0y

1

2z V

To write rotation in terms of stream functions

2 2

2 2

1 1

2 2

y xz

V V

x y x y

xVy

yVx

21

2

2 2 0z

That is

For irrotational flow (z=0)

2 0

Rotation in terms of Stream Function

This equation is “similar” to continuity equation

Vx and Vy are related

Can we find a common function to relate both Vx

and Vy ?

For irrotational flow (z=0)

1

02

z V

0V

0y x

V V

x y

Rotation and Potential

Velocity Potential

In 3D, similarly it can be shown that

Assume

xVx

f

y x

V V

x y

0y x

V V

x y

2

y x

f

2

x y

f

yVy

f

zVz

f

Then

f is the velocity potential

Velocity Potential vs Stream

Function

Stream Function () Velocity Potential (f)only 2D flow all flows

viscous or non-viscous flows

Irrotational (i.e. Inviscid or

zero viscosity) flow

Incompressible flow (steady or

unsteady)

Incompressible flow (steady

or unsteady state)

compressible flow (steady

state only)

compressible flow (steady or

unsteady state)

Exists

for

In 2D inviscid flow (incompressible flow OR steady

state compressible flow), both functions exist

What is the relationship between them?

Stream Function- Physical

meaning Statement: In 2D (viscous or inviscid) flow

(incompressible flow OR steady state compressible

flow), = constant represents the streamline. Proof

d dx dyx y

0

y xV dx V dy

If = constant, then d0

y

x

Vdy

dx V

If = constant, then

Vx

Vy

Reynolds—Transport Theorem

(RTT)

There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields).

Reynolds—Transport Theorem

(RTT) • Material derivative (differential analysis):

• General RTT, nonfixed CV (integral analysis):

• In Chaps 5 and 6, we will apply RTT to conservation of mass, energy, linear momentum, and angular momentum.

Db b

V bDt t

sys

CV CS

dBb dV bV ndA

dt t

Mass Momentum Energy Angular

momentum

B, Extensive properties m E

b, Intensive properties 1 e

mV

V

H

r V

Reynolds—Transport Theorem

(RTT)

• Interpretation of the RTT:

– Time rate of change of the property B of the

system is equal to (Term 1) + (Term 2)

– Term 1: the time rate of change of B of the

control volume

– Term 2: the net flux of B out of the control

volume by mass crossing the control surface

sys

CV CS

dBb dV bV ndA

dt t

RTT Special Cases

For moving and/or deforming control volumes,

• Where the absolute velocity V in the second

term is replaced by the relative velocity

Vr = V -VCS

• Vr is the fluid velocity expressed relative to a

coordinate system moving with the control

volume.

sys

rCV CS

dBb dV bV ndA

dt t