Boundary Layer Theory

-

Prof. Dr. Norbert Ebeling

Boundary Layer TheoryLecture notes

Prof. Dr. N. Ebeling Boundary Layer Theory - 1 -

Contents : 1) General fluid mechanics / Newton fluids 1.1) Euler's law of hydrostatics 1.2) Friction 1.3) Dimensionless numbers 1.4) Laminar flow in a tube 2) Conservation equations 2.1) Mass balance for = const. 2.2) Euler's and Bernoulli's equations 2.3) Navier-Stokes equations 3) Boundary layers 3.1) Boundary layers on a flat plate 3.2) Friction forces on a plate 3.3) Boundary layer on an obstacle 4) Potential and stream functions 5) Law of Kutta-Joukowski 6) Exact calculation of the Boundary layer thickness 6.1) Conservation of mass (continuity equation) 6.2) Navier-Stokes and Blasius equations 6.3) Friction 7) Thermal Boundary layer

8) Mass Transfer Boundary layer equation 9) Turbulent Boundary layer 10) Burbling 11) Bibliography 12) Acknowledgment

(dz) dydx

iDu

= dmDti

dF

= 0ut

i

u = u

x

DuDt

if

= dx dy dzdm i i i

x

y

z

i e x uj e y v general definitionsk e z w


1) General fluid mechanics / Newton Fluids

General definitions

Acceleration :

stationary :

frequently :

volume force: (e.g. g )

i i iu u u u

= + u + v + w t x y z

DuDt

1dF

2dF

x

F f dx dy dz = dx

x

i i i i i

dp dy dzdF = i i

x

p f =

x

i

x 1 2 f dV + dF = dF i i


1.1) Euler's law of hydrostatics

x xf

= + u

y

i

u =

y

i

inertial forcefriction force

+ dyy

i

( ) = dy dx dzFRF ydA

i i i

{


1.2) Friction

Moving fluid : ( Couette - flow )

Newton fluid

Schlichting :

1.3) Dimensionless numbers :

Reynolds number

Re ~

u dx dy dz u

xRe ~ dx dy dz

y

dm

upcurlybracketleftupcurlybracketmidupcurlybracketright

i i i i i

i i i

u u ~ ; =

y yu

x d y i

u ~

y dy i

v dRe =

i with =


or any comparable speed v else

laminar flow : high friction forces,low inertial forces

avoided by friction

deciding

2

V v

v ddRe = = v

d

i ii i

i

AA

FC = p si

21 u

2p

i i

( or ) analogouswC

( )R2m

d dp = or

dx u

2

ii

Froude -number

vFr g d

=

i

Rw

FC = p si


ascending force

s

Bernoulli :

Pipe :

Gravity influence :

64 =

ReR

r dp du = = -

2 dx dr i i

22R dp ru(r) = - 1

4 dx R

i i


1.4) Laminar flow in a tube

extremely high

nearly no initial forces, no influence of dm or !

Hagen-Poisseulle :

Derivation :

Integration with u (r = R) = 0 leads to :

2

d dp 64 64 = =

dx v d v d v

2

i ii

i i ii

2 2dpp r - p + dx r - 2 r dx = 0dx

pi pi pi

i i i i i

v + = 0

yu

x

R

0

V = u (r) 2 drpi i i4

R dpV = - 8 dx

pi

i ii

2

2V R p

u = = R 8 l

pi

ii i

( )u 2RRe =

i i

64 = laminar !!

Re


2) Conservation equations Important conservation equations for describing continuous flow ( cartesian coordinates ) :

2.1) Mass balance for = const.

212

p d =

u l

ii i

1 y zu i i 2 = u y z i i 2 + v x z i i( )1 2 2u - u y = + v x i i

u p u = -

x x

i i

x

Du udV = dV u = + dFDt x

i i i i i

u v + = 0

x y


2.2) Euler's and Bernoulli's equations Eulers equation ( one direction, pipe ):

Integration : W = F l leads to Bernoulli's equation

xu - p u = + f

x x

i i i

2 2 2 2

1 1 1

u = p - g h

2 i i i

( ) = dy dx dzRdF y

i i i

u u p u + v = -

x y x

DuDt


i i i

v


Mechanical energy balance : Bernoulli incl. hydrostatics

Euler (2 directions ):

v leads to a higher value of u

2.3) Navier - Stokes - equation Bernoulli and Euler neglect friction

u

y =

i

2 2

x 2 2

u u p u + v = f - + +

x y xu u

y x

i i i i

2 2

y 2 2p

v + u = f - + + v v v vy x y y x

i i i i


Navier - Stokes - Equations ( Can be simplified in a boundary layer (later))

3) Introduction to Boundary layers

3.1) Boundary layers on a flat plate

No influence of the viscosity but directly on the wall

Boundary layer phenomena :

( Schlichting )

2 2

2 2

u = +

yRuf

x i

x RxDu p

= f - + fDt x

i i

22u u

~

x

i i

x ~

u

i

2

u u ~ ; ~ u

x x y

i

2 2

2u

u = = u

x y y

i i i

99 (x) x

= 5 u

ii


Thickness of a boundary layer, laminar on a plate

inertial force = friction force ( Navier -Stokes )

( ) = f xu

( )iy = 0

(x) = U - u(x,y) dy U

i i

valuelow

99 is arbitrary

99 (x) 5 x =

lRel

i


Dimensionless :

A non - arbitrary value : displacement thickness

3.2) Friction forces on a plate :

high value

991

3i

i

u( ) = yw

w

x i

W Ww

2

S(Surface)

F F = c = =

E u b l

2

i i

( )l

W W0

F = b x dxi i

l 1 3 2

W0

F ~ b u x dx

i i i i i


xu

~ with ~ uw

i

i

3 u

~ wx

i i

13 2

WF ~ b u 2 l i i i i i3

24 2

b 2 u l ~

b u l4

wc

i i i i i

i i i

l ~

Rewc

1,1328 =

Rewc


3.3) Boundary layer on an obstacle : Navier - Stokes :

Far away from the obstacle (stream line) :

( )dU l dpU = - no frictiondx dxi idU dp

and are related to Bernoullidx dx

= w ds i


4) Potential and Stream functions For describing vortex streams ( and comparable ) :

Circulation :

Potential streams (no friction ) : no rotation

Mass balance ; conservation equation :

11

22

v = =

t x

= = -

t

1 v u = -

2 x y

u

y

v u = 0 ; - = 0

x y

v + = 0

yu

x

u = ; v = - y x

+ - = 0y xx y

2 2

2 2 + = 0x y

= ; v = y

ux


Stream function (definition ) :

Conservation equation :

No rotation :

Potential function :

Potential streams

1 v u v u = - ; - = 0

2 x y x y

i

2 2

2 2u u

+ = 0x y

( )p = f u, v

2

2v u u v

- - + = 0y x y x x x y

u u p u + v = - + 0

x y x i i i


Streams without any rotation :

also conservation equation :

Insert in Navier - Stokes :

leads to Bernoulli for v = 0 - no friction ! - no rotation - no friction

v u - = 0

x y

u v + = 0

x y

2 2v u - = 0 - = 0

x y x y x y

2 2

2 2 + = 0x y


Model frequently used : On the obstacle : boundary layer in the vicinity , but outside the layer : no friction

potential function :

No rotation :

Conservation equations :

Stream function :

Conservation equations o.k.

u = ; v = x y

u = ; v = - y x

= w ds

i


from definition :

Stream line : ( no v : ) -> = constant

Circulation :

Example :

here : = 0 ( all possible ways )

airfoil : high speed

low speed

u + v = 0x y

i i

0

( ) = 2 r r pi i i iPotential- and flowfunctions as well as velocitys for some elementary potential flows

flow streamline

translational flow

source flow

( productiveness E )

potential vortex stream

( circulation I' )

source-drain flow

( productiveness E, distance h )

dipole flow

( dipole moment M )

( )x,y ( )x,y ( )u x,y ( )v x,y

U x + V y

E ln r

2pi

2 pi

1

2

E r ln

2 rpi

2M x

2 rpi

U y - V x

E

2

pi

ln r2

pi

( )1 2E - 2 pi

2M y

2 r

pi

U

2E x

2 rpi

2y

2 r

pi

2 21 2

E x + h x -

2 r r

pi

2 2

4M y - x

2 rpi

V

2E y

2 rpi

2x

2 rpi

2 21 2

Ey 1 1 -

2 r r pi

4M 2xy

2 r

pi

lr

w ~ = 0


assumption : ; obviously :

One exception : including the centre :

(see also: Gersten, K. : Einfhrung in die Strmungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 130 )

radyield : E = w 2 rpii i

for x = r : E = u 2 xpii i

Eu =

2 x ( or r )pi

2 2E = ln x + y

2

pii

2 2 2 2

E 1 1 1u = = 2x

x 2 2x + y x + y pi

i i i i

2E x

u = 2 rpii

E E y = = arctg

2 2 x pi pi

i i

E yu = = arctg

y 2 y x

pi i

radspring : V = w 2 r hpi i i i


Spring :

21

arctg x = x 1 + x

( )

( )yx

yx

u = x

i


Bronstein :

the rest is the same

For application :

airfoil :

( )2

2 2yx

E 1 1 xu =

2 x x1 + pii i i

2E x

u = 2 rpii

stream = of model streams

AF =b l pi i

( ) 2A 1F =b l 2u2 i i i i

2AF = 2 l b ui i i i


5) Law of Kutta - Joukowski simple example : flat plate :

Kutta - Joukowski

= 2 u l

i i

AF = b u i i i

: u = Uy

2

2u u

+ v = x y

uu

y

i i i

= 0 : u = 0, v = 0y

u

yx

i


6) Exact calculation of the Boundary layer thickness

Boundary layer on a plate :

For similarity y/ (x) is important

v + = 0

yu

x

( ) x v x ~ u

i

m = =

y m yu

i

( ) = u f mu i

( )1 1 = 2 u f m2 xx

i i i i i

= y 2 x

um

ii i

( ) = 2 x f m 2 xu

u u

i i i i i

i i


Definition :

( the factor 2 is arbitrary but helpful )

Idea : Stream function :

= -

xv

( ) = 2 x u f m i i i i

( )

Schlichtingsays

dimensionlessstream function

= 2 x u f m


i i i i

( ) m 2 x u f m x

+

i i i i i

3-

2u 1 = y - x

2 2m

x

i i ii

3-

122

u u = f - y x x

x 2 x 2 x

ii i i i i

i i i

( ) uv = - = m f - fx 2 x

i

i ii

( ) 3-21 = u f m y - x2 2uu

x

i i i i i

i

u v = m f - f

2 x y 2 xu

y y

i i

i i ii i

( ) 1 = f m - 2 xu

u mx

i i i

i

( ) 2 x u f m i i i i i


6.1) Conservation of mass (continuity equation)

u u u u uv = f + y f

2 x 2 x 2 x 2 x 2 x

m

y


i ii i i i i

i i i i i i i i

u v + = 0

x y

( ) = m f - f2 xu

v

ii i

i

( ) = f mu u i( ) 1 = f m m - 2 x

uu

x

i i i

i

( ) uu = u f m y 2 x

i i i i


Conti - equation

6.2) Navier-Stokes and Blasius equations Navier-Stokes for the boundary layer on a flat plate :

f 2 x 2 x

u u

ii i

i i i

( ) 1 = m f m 2 xv

uy

i i i

i

( )2 2 = f m 2 x 2 xu uu

uy

i i i

i i i i

2

2u u u

u + v = x y y

i i i

f + f f = 0Blasius - Equation

i

m = 0 f = 0 , f = 0m : f = 1

( )m = 0 i.e. y = 0 u = 0 and f = 0( )m i.e. y u = u and f = 1

( ) uv = m f - f2 x

i

i ii

for y = 0 v and f have to be 0


with :

Inserting and differentation leads directly to :

side conditions :

( )u = u f m i

characteristic parameters for the boundary layer on a longitudinal flown plate

0,4696

1,2168

0,4696

0,7385

( )1 = lim - f wf

( )20

= f 1 - f d


There is a function f(m), but there is no equation. description of f(m) : Thickness of the boundary layer :

(see also : Schlichting, H. , Gersten, K. (2006): Grenzschicht - Theorie, Springer , 10th Ed., page 158 )

(nach : Schlichting, H. , Gersten, K. (2006): Grenzschicht - Theorie, Springer , 10th Ed., page 159 )

2

99 = y for u = 0.99 u i99 = y for f = 0.99


(see also : Incropera, F.P.; DeWitt, D.P.: Fundamentals of Heat and Mass Transfer, Wiley, 4th Ed., page 352 )

Attention : f and deviate from Schlichting in factor !

Flat plate laminar boundary layer functions

0,0 0,0000 0,000 0,470

0,4 0,0191 0,133 0,468

0,8 0,0750 0,265 0,462

1,2 0,1683 0,394 0,448

1,6 0,2970 0,517 0,420

2,0 0,4596 0,630 0,378

2,4 0,6520 0,729 0,322

2,8 0,8704 0,812 0,260

3,2 1,1095 0,876 0,197

3,6 1,3647 0,923 0,139

4,0 1,6306 0,956 0,091

4,4 1,9035 0,976 0,055

4,8 2,1814 0,988 0,031

5,2 2,4621 0,994 0,016

5,6 2,7436 0,997 0,007

6,0 3,0264 0,999 0,003

6,4 3,3086 1,000 0,001

6,8 3,5914 1,000 0,000

f um = y x

df u =

dm u

2

2d fdm

uu = u f y

2 x

ii i

w

w

uu = u f

y 2 x

i ii i

w

uu 0,4696 = u

y x2

i ii

l

w w

0

F = b dx i i

l 12

w

0

uF = 0,332 u b x dx

i i i i i il 1

2

0

with x dx = 2 l

i


6.3) Friction :

Plate ( 1 side ) :

w

u = 0,332 u

x

i i i

i

ww

Fc

u b l 2

= i i i


( see also 3.2 )

7) Thermal boundary layer

Conservation equation for heat :

convection :

22

p 2T u

c u + v = + x yT T

y y i i i i i i

w

1.328c

Re=

( ) T dx dz dyy

i i i

( )c pQ = m c T i i( )p Tc dy dz u dxx

i i i i i i

udP = dx dz dyy

i i i i

u =

y

i


convection :

> 0

conduction :

< 0

friction :

> 0

Compare the conservation equations for heat with Navier-Stokes !

T A

y i i

2p 2 T T T

c u + v = x y y

i i i i

2

2u u u

+ v = x y y

i i i

2

2p

2

2

u u uu + v

cx y y =

TT Tu v

yx y

i ii i

i

i i i

2

2

2

2

u u uu + v

x y y =

TT Tu v

yx y

Pr

i i

i

i i i


( heat from friction neglected )

Navier-Stokes adapted to a boundary layer (see also 6) )

( )wq = T - T i

w

Tq = - y

i

u

w

T-

y =

T - T

i

( )T Tw

w

-

y = T

- 1T

i


For gases Pr 1. Independent from the condition u and T behave equal.

wu

u =

y

i

u = 0,4696

2 x

i i

i il

10 2

1b dxu x

= 0,4696 2 b l

i ii i i

i i

2

u 4 l = 0,4696

2 l

i ii i

i i

1+

2 = 0,664 Re

l

i i

Nu = 0,664 Rei

wwith u = 0


5Re = 5 10i

, sudden


There is evidence that for Pr 1 :

see also : Vauck, W.R.A., Mller, H.A.: "Grundoperationen chemischer Verfahrenstechnik" , Wiley, 11th Edition (2001)

8) Mass transfer boundary layer equation

9) Turbulent Boundary layer Plate : turbulent from on

virtual friction

turbulent layer : 2 layers

viscous sublayer

1132Nu = 0,664 Re Pri i

2A A A

AB 2 c c c

u + v = D x y y

i i i

fx

50 =

cRe 2

v

x

i

( ) ( ) ( )( )

u x,y,t = u x,y + u x,y,tv x,y,t = v....

u = average ; u = 0p (x,y,t) = p (x,y) + p (x,y,t)

u u u u u + u + u + u + v.....

x x x x

i i i i i


Viscous sublayer :

Turbulent boundary layers

Conservation equations :

Navier - Stokes ( for boundary layers ) :

u u v v u v + + + = 0 ; + = 0

x x y y x y

u u u uu + u + v + v

x x y y.....

=

i i i i

( )2

2u u du u

u + v = u + - u v x y dx y y

i i i i i i

u uu = 0 , u = 0

x x

i i

2 2

2 2d u u

- + + dx y y

=


Average :

2 2

2 2dp u u

- + + dx y y

=

i

( )dp dU - U Bernoullidx dx= i i

( )u uu + v u vx y y

i i i

lu

=

y

i

( )t = - u v is usually negative

u v

i i

i

t- u v

= + with = u

y

u y

i

i i

~ l u yu i

2t

u u = l

y y

i i i


laminar sheer stress :

turbulent sheer stress :

: turbulent kinematic viscosity

l = length of mixing way l = f ( distance to the wall )

laminar sublayer

v ~ u

dpFlat plate : = 0dx

du dpU = - dx dxi


Degree of turbulence :

10) Burbling

Stream line along a body different from a flat plate outside the boundary layer ( no friction : )

( see Bernoulli and Navier-Stokes )

( )2 2 213u

u + v + w T =

u

i

22u u dp u

u + v = - + x y dx y

i i i i


low speed - high pressure

When friction and pressure increase, debonding occurs.

In the layer :

2

2dp uIf has a high value, mustdx y

become positive


Result :



burbling from point A on


Turbulent flow : + instead of : burbling occurs later

(nach : Gersten, K. : Einfhrung in die Strmungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 110 )


w c = 0

u D u DRe = =

i i i

laminar

laminar, but burbling

} turbulent

ww 2

2

sphere :F

c = u

D2 4

pii i i

w c = 0


creeping flow :

d'Alembert : no friction (and no burbling)


(nach : Gersten, K. : Einfhrung in die Strmungsmechanik,Bertelsm. Univ.Verlag, 1st edition, page 112 )

f d =

uSr i


Periodic stream due to debonding :

Strouhal - Number :


11) Bibliography

- Gersten, K. : Einfhrung in die Strmungsmechanik, Shaker; 1st edition (2003), ISBN-13: 978-3832210397 - Schlichting, H., Gersten, K. : Grenzschicht - Theorie, Springer Verlag, 10th edition (2006), ISBN-13: 978-3540230045 - Incropera, F.P., DeWitt, D.P.: Fundamentals of Heat and Mass Transfer, Wiley, 5th edition (2001) , ISBN-10: 9755030654 - Vauck, W.R.A., Mller, H.A.: "Grundoperationen chemischer Verfahrens- technik" , Wiley, 11th Edition (2000), ISBN -10: 3527309640 - Bronstein, I.N., Semendjajew, K.A., Musiol, G., Muehlig, H. : Taschenbuch der Mathematik, Deutsch, 7th edition (2008) , ISBN-13: 978-3817120079

12) Acknowledgment I would like to thank my student assistant Matthias Kemper for his contribution to this work.

Boundary Layer Theory

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