II. Kinematics of Fluid Motion...FLUID KINEMATICS Lecture 3 Kinematics Part I 1 . Definitions,...
Transcript of II. Kinematics of Fluid Motion...FLUID KINEMATICS Lecture 3 Kinematics Part I 1 . Definitions,...
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