Advanced Huid dynamics
College 1
The Huid mechanica approach
a) Non Dimensionaal Numbers
-
(i) Physik Principles should be Independent of arbitrair Chosen dimensies
y
&
quantities.fm: h : :)
( II) Ome can be describe the behaviour of a physical system in terms of
parameters with no dimensies =
nondimensional parameter ( t )
(III ) Why are nondimensional parameters important in AFD ?
because they can quantify the
magnitude of computing physical
relativa
mechanisms (inertie viscosity
, , gravity surfacetenten) ,
b) Dimensionaal
Analysis : The
buckingham I there
consider a
Physik System : we can describe Lexpress its behaviour in terms of :
number of
"
Physik parameters
''
•
a m
( µ f. l ,
,
u .
V. 9 .
. . .
)
these parameters include independent units
" "
•
m n
(M ,
S
, kg ,
K . . . . )
The behaviour of System can be described in terms of
''
m -
n
"
Independent
non -
dimensionaal Numbers IT , Th . .
_
Tm - n
Corolla : we ( alt Ip f (IT )
r D= m N Ir , Tip
- =
- . . - i
, , .
,
in particular If p =/
,
The I ,
=
corset .
Example :
maart
GL f?
l f 1-
Physik parameters :
-4 l f
9
m -
: M
, , ,
m.sk t Ig E '
2- Independent units
3-
Dynamics can be written in terms of a- 3 :
kg ,
M ,
S
M -
n = 4-3=1 non -
dimensionaal group
{
[m) :X = 0 D= -
x = É 8=0
I, =
ga .
f? mr fs .
[ s) : -
2x -
5=0 x =
-
E
( m.54.ms kg
' '
>
.
5 kg) : 8=0 S =
I
I, =
ff
from Corolla
r that
: I
,
-
_
cont .
→
FNV (f = # VE )
ijij
, Advanced Huid Dynamics
College 2
Sealing Analysis
• Determine the interdepandance of between physical parameters b
readning
based on
Physik intuition (
grounded in the
knowledge of
lans of
Physics relevant to the problem)
Etam
Warned : f
|
acceleratie
9h (
µ m mg
-
in ↳
°
Find a
scoding for the accelerator
↳• for er l
scoding
~
position
↳• f. f
scoring for
vetocity
-10 ~
↳*
sacting for acceleratie in f- ff
'
l f rent
µ g
-
- -
f n
#
Example
III. f ne viscose
I
→
Find dimensionaal for
a non -
scoring Law
drug
i) Dimensionaal
Analysis
1- Relevant physical parameters : U
,
A
,
µ P , ,
D
t t tegen
'
M = 5 ms m
2-
Independent units
in =3
Buckingham
dimensionale
3- I theoren : P =
5 -
3=2 non -
tof ÷ En ~
.
~ ND
,
=
↳ ←
Draag
COEFFICIENT
ijijij
, Tk t Re m'n
'
Reynolds
'
5 mi
in #
←
= =
m .
^ number
ft )
at tpg)
t ,
=
=
Co =
#Re)
Ii)
scoding Analysis
ASSUME sphere large vetocity U:
is Maring at
•
↳
assume effects of Velo
City
are
negligible
D Ps of ✓ via
t
N -
→ =
↳ ~
constant
/
↳ 42
Assume very smalt vetocity dominante
a
viscosity
•
D. ~
visgous
stress -
arr µ -
u -
a
µ I
M
- -
Troista
Cont
Re
from find that Red Co
soaling Analysis 1 ~
we :
,
constant
Red '
,
Co ~ Ee
(D= f (Re)
Co Ret
} for Large
=
a = 0 Re
× = -
I for smalt Re
109 ( Co )
=
109 (Ret )
109109 Plot t
log ↳ =
x
109 Re
÷
X X
1094 y
=
¥ "
IJ
, Advanced Huid Dynamics
College 3
Notaties and other mathematica reminders
.fi/: :
notaties fuII notatie index notaties
Nath notatie in Class (Kander)
Scot
Ranko
p µ PT P µ PT P µ PT PAPT
-
Vector
§)
U
→
Rank 1 U -4 U Ui ii. 2,3
Lto simpele
e-
÷:÷ ÷
dinge n '
Ranko div Ü
Ittf 3G I. k
JA
ta p
Ranko ⑦ 4)
die
Idiots IIY
t
dia
product À XE à tb-flb.br) a- b- aibj
nam
÷! :÷ :
