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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