UNSTEADY VISCOUS FLOW
upgDt
uD
2
2
21yu
xp
tu
Viscous effects confined to within some finite area near the boundary → boundary layer
In unsteady viscous flows at low Re the boundary layer thickness δ grows with time; but in periodic flows, it remains constant
If the pressure gradient is zero, Navier-Stokes equation (in x) reduces to:
2
2
yu
tu
2
2
yu
tu
Heat Equation– parabolic partial differential equation - linear
Requires one initial condition and two boundary conditions
U
y
Uuy 0@
0@ uy 00,, tyxu
Total of three conditions
Impulsively started plate –
Stokes first problem
2
2
yu
tu
Heat Equation– parabolic partial differential equation Can be solved by “Separation of Variables”
Suppose we have a solution: tTyYtyu ,
Substituting in the diff eq: tTyYy
tTyYt 2
2
May also be written as: tTyYtTyY
Moving variables to same side:
yY
yYtTtT
The two sides have to be equal for any choice of x and t ,
kyYyY
tTtT
The minus sign in front of k is for convenience
This equation contains a pair of ordinary differential equations:
kyYyY
tTtT
00
TkTkYY
0
02
2
TktT
kYyY
Uuy 0@
0@ uy
00,, tyxu tTyYtyu ,
0YU
Y0 00 T
0 Tk
tT
tkTT
tkTT AtkT ln tkAeT
02
2
kYyY ykCykBY sincos
ty
Ueu 4
2
ykCykBY sincos
LyneAu
Lnk
tL
n
nn
sin22
tTyYtyu ,
increasing time
2
2
yu
tu
Uuy 0@
0@ uy 00,, tyxu
New independent variable:t
y
2
η is used to transform heat equation:
dd
tdd
tt 2
dd
tdd
yy 21
2
2
2
2
41
dd
ty
Substituting into heat equation: 2
2
42
dud
tddut
Alternative solution to“Separation of Variables” – “Similarity Solution”
from: 2yu
tu
022
2
d
dud
ud
022
2
d
dud
ud
Uu 0@
0@ u
asu 0
To transform second order into first order: d
duf
2 BC turn into 1
02 fddf
With solution:2Aef
Integrating to obtain u:
BdeAu
0
2
Or in terms of the error function:
deerf0
22 erfUu 1
df
df 2
erf
2e
For η > 2 the error function is nearly 1, so that u → 0
erfUu 1 For η > 2 the error function is nearly 1, so that u → 0
Then, viscous effects are confined to the region η < 2
This is the boundary layer δ
ty
2
t
2
2
t 4
δ grows as the squared root of time
increasing time
erfUu 1
ty
2
2
2
yu
tu
UNSTEADY VISCOUS FLOW
Oscillating Plate
tUuy cos0@
boundeduy @
Ucos(ωt)
y
Look for a solution of the form: tieyYtyYu Recos
tite ti sincos Euler’s formula
Fourier’s transform in the time domain: tieyYu Re 0YU
0Y
B.C. in Y
2
2
yu
tu
Substitution into:
titi YeiYet
2
2
yYYi
2
2
2
2
yYe
yu ti
02
2
YiyY
21 ii
yiByiAY
21exp
21exp
00 BY UAUY 0
yiUY
2
1exp
yiUY
2
1exp
ytUeYeu
yti cosRe
Most of the motion is confined to region within:
2
Ucos(ωt)
y
UUeUe
yy
37.0
@1
UUeUeUe
yy
06.0
/4@24
ytUeu
ycos
2
2
yu
tu
UNSTEADY VISCOUS FLOW
Oscillating Plate
tUuy cos0@
0@ uWy
Look for a solution of the form: tieyYtyYu Recos
tite ti sincos Euler’s formula
Ucos(ωt)
y
W
Fourier’s transform in the time domain: tieyYu Re UY 0
0WY
B.C. in Y
2
2
yu
tu
Substitution into:
titi YeiYet
2
2
yYYi
2
2
2
2
yYe
yu ti
02
2
YiyY
21 ii
yiByiAY
21exp
21exp
UBA
WiBWiA
21exp
21exp0
2
UBA
WiWi
BeAe11
0 sinh ee
sinh
UeB
Wi 1
sinh1 eUA yy BeAeY
Wi
yWiUY)1(sinh
))(1(sinh
WiWiW )1()1(sinh@
WyU
WyWUY 1
tWyUu cos1
Wi
eWiW)1(
)1(sinh@
yiUY )1(exp
Wi
yWiUY)1(sinh
))(1(sinh
sinh ee
tWyUu cos1