CEE 262A

HYDRODYNAMICS

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

vSteady 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

)(~tx

0~0

~)( 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

ii

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

ii

ux

F

t

F

dt

dF

-- (3)

ii x

Fu

t

F

Dt

DF

Local rate of change at a point

~x

Advective change past

~x

Fut

F

Dt

DF

~

jF a

i

ji

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

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.

sds

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 33

(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

dtx 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

X1t0

X2

X1t1

(b) Rotation

X2

X1t1

• No change in dimensions of control volume

Relative Motion near a Point

(c) Straining (need for stress):

Linear (Dilatation) – Volumetric Expansion/Contraction

X2

X1t0

X2

X1t1

(d) Angular Straining – No volume change

X2

X1t1

Note: All motion except pure translation involves relative motion of points in a fluid

x2

x1t0

x2

x1t1

~x

~~xx

~x

~u

~~duu

'

~xP

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

ux

u

x

u

x

u

x

u

x

ux

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

ux

u

x

u

x

u

x

ux

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

21

21

( )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

t

v

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

x1

x3

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

t0tu 2

tu 1

1x

1x2x

2x1 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

212

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

22

11 x

x

uu

A

B

11

22 x

x

uu

1u

2u

2x

1xO

ttt 01

X2

X1

t1

O

A

B

txx

u 22

1

txx

u 11

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

11

2

12

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.

)( 22

11 x

x

uu

(iv) Rotation

1x

2x

A

B

11

22 x

x

uu

1u

2u

2x

1xO

ttt 01

t1

O

A B

txx

u 22

1

txx

u 11

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

22

1

22

1

2

1

11

2

11

2

1

2

1:

1:Rotation due to

due to

1. Relative motion near a point is caused by

2. This tensor can be decomposed into a symmetric and an anti-symmetric part.

(a) Symmetric

* : Dilation of a fluid volume

* : Angular straining or shear straining

(b) Anti-symmetric

* : 0

* : Rotation of an element

ji

ji

ji

ji

~u

~u

x

u

j

i

ije

ijr

Summary