Post on 11-May-2017
transcript
Chapter 3
Low Reynolds number flows ( Lubrication theory, stoke’s flow, Oseen's flow)
3-1
21' ' : motion pressureV V p V p
Limiting cases for the Navier-Stokes equation
(a) Low Re creeping flow slow motion, larege viscosity
Consider steady, incompressible, Constant flow
Navier-Stokes eq.
0
1
0 1
Make non-dimensconal
Assume single scale velocity =
length =
dimensionle
ss quantities
pressure =
, , ,
V
l
p
V p xV p x y
V p l
2
2 2
, z
Note: ,
y z
l l
l l
3-2
2
0 1
0
substitute
V l p lV V p V
v
inertiaCharacteristic Reynolds number = Re
viscous
2
cm1 mm V=1.4 sec
Re 1 cm in air =0.14
sec
d
Limiting cases for the Navier-Stokes equation
3-3
At small Re, to a first approximation, neglect the inertia
terms in the equation.
2 linear P.D.E. like pa1
rallel flo1 N-S eq.
2 continuit
0 '
y eq.
w
0
p V
V
4 eqations 4 unknows ,v, ,u w p
Limiting cases for the Navier-Stokes equation
3-4
2
2
0
1 '
'
=
contin
= -
uity
p V
p V
V V
0
2 p' is a potentia' 0 l function p
Limiting cases for the Navier-Stokes equation
3-5
2a vorticity eq. 0
0V
2 eqations
2 unknow u,v
(recover p from N-S eq.)
recover p from N-S eq.
4
b vorticity eq. in terms of , stream function
0
4th order linear P.D.E. eq.
known as the bi-harmonic eq.
Limiting cases for the Navier-Stokes equation
2 2 D flow
3-6
x
Φ
r
y
z
R
r
The oldest known solution for a creeping motion
, ,
Stokes flow 1851 (Sphere)
U
3-7
2
2
2
V
vorticit
spherical coordinate
solve using vorticity eq. in terms of
y
sin1 10
sin
1
sin
r r
r
r
r
V V V
rV VV
r r
r V VV
r r r
Vr
1
sinV
r r
Stokes flow 1851 (Sphere)
0 (symmetry)
3-8
2
2 2
2
2 2
2
2 2
2 2
wher
sin 1
sin sin
1or E
sin
sin 1 E
vorticity
= sin
0
E1 sin
e
=-sin
eq.
r r r
r
r r
r r
E1
sin
4 0
0
Stokes flow 1851 (Sphere)
3-9
2
vorticity eq. in terms of
E 0
r 1 , 0 0
Boundary Condit
2 , 0 0
3 , uniform x-directed flow
4
ions
?
r R V
r R Vr
r
linear 4th order P.D.E
Stokes flow 1851 (Sphere)
3-10
θ
θ U
rVV
2
2 2
1sin
sin
1cos
sin
Integrate
= sin2
& compare
r
V Ur r
V Ur
Ur
---------------- B.C. (3)
Stokes flow 1851 (Sphere)
3-11
2
2
22
2 2
2 22 2
2 2 2 2
22
2 2
Assume =
since the B.C. at r holds for all
let g sin
= sin
2sin
2 2sin 0
Must hold for all
2 0
f r g
f r
E fr r
E fr r r r
df
dr r
Stokes flow 1851 (Sphere)
3-12
4 2
4 2 2 3 4
2 4
2 2
4 8 8 0
1, 1, 2, 4
expanding
solution
substitute
Boundary condition r = s in2
n
d f d f dff
dr r dr r dr r
f r
n
Af r Br Cr Dr
r
Ur
2 2
D=0 ; C=2
sin2
U
A UBr r
r
Stokes flow 1851 (Sphere)
3-13
2
3
3
3
1r=R, 0
sin
= cos 2 cos
1r=R, 0 0
sin
sin cos
=4solving
3
4
1 3 11
2 2 2
rVr
A BU
R R
Vr r
A BU
R R
U RA
B U R
R RU
r r
3
2 2sinr
B.C. (1)
B.C. (2)
Stokes flow 1851 (Sphere)
3-14
D
2 2
d
d
2 skin friction
3 D=6
1 pressure
3
6D
1frontal area dynamic head
2
24
Re
d=2R
Drag :
Drag c
oefficie
Re
nt :
va
D
RU
RUC
R U
C
U d
lid Re 1
Drag and Drag coefficient
3-15
dA
pdA
r dAU
drag D
lift L
Drag and Drag coefficient
3-16
3/13/2014
a circular disk perpendicular t
special
o the s
cases
treamdisk
U
R = radius
D 16 RU
b circular disk parallel to the stream
32
D = 3
RU
Ellipsoid in a parallel stream
R = radius
U
disk
3-18
inertia terms
' ' '' ' '
v ' v ' v ' ' ' ' + ' v ' ' j + U ' ' ' k
U u U u U uV V u u v w i
x y z
w w wU u w u v w
x y z x y z
Oseen’s approximation
'
v v ' '
let
'
u U u
V U i V
w w
U u',v',w'
'
As
' ', '
sume
wher ,e 'V
U V u v wx
3-19
2
D
0 '
Hence '
' 1momentum eq
Note:Approximation must fail near the surface, sin
. U ' '
continuity eq. ' 0
24Drag coefficient
R
e
e
c
Cd
u u U
u U
Vp V
x
V
dd
4.5
where Re hold for Re up t 5 oU d
Oseen’s approximation
3-20
NCKU Department of Mechanical Engineering