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