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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 ρ = 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) dy

dx

iDu

= dmDt

idF

∂

∂ = 0

u

t

∂

∂i

u = u

x

Du

Dt

if

= dx dy dzdm ρ i i i

x

y

z

i e x u

j e y v general definitions

k e z w

�

�

�


1) General fluid mechanics / Newton Fluids

General definitions

Acceleration :

stationary :

frequently :

volume force: (e.g. g )

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂i i i

u u u u = + u + v + w

t x y z

Du

Dt

1dF↓

2dF↑

x

F f dx dy dz = dx

x

∂ρ

∂i i i i i

dp dy dzdF = i i

x

p f =

xρ

∂

∂i

x 1 2 f dV + dF = dFρ i i


1.1) Euler's law of hydrostatics

x ↓xf

= + u

yτ η

∂

∂i

u =

yτ µ

∂

∂i

inertial force

friction force

+ dyy

∂

∂

ττ i

τ

��

�� ( ) = dy dx dzFR

Fy

dA

τ∂

∂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 dzy

dm∂

ρ∂

∂τ

∂

��

i i i i i

i i i

υ

u u ~ ; =

y y

u

x d yη

τ∞ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ i

u ~

y dyη

τ ∞∂ ∂ ∂ ∂ i

v dRe =

νi with =

ην

ρ


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

ρ ρηη

i ii i

i

AA

FC =

p si

21 u

2p ρ ∞→ i i

( or ) analogouswC ζ

( )R2

m

d dp = or

dx u

2

ζ λρ

i

i

�Froude -number

vFr

g d=

i

Rw

FC =

p si


ascending force

s

Bernoulli :

Pipe :

Gravity influence :

ν

64 =

ReR

ζ

r dp du = = - 2 dx dr

− τ ηi i

22R dp ru(r) = - 1

4 dx R

η

i i


1.4) Laminar flow in a tube extremely high

nearly no initial forces, no influence of dm or ρ !

Hagen-Poisseulle :

Derivation :

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

2

d dp 64 64 = =

dx v d v d v

2

η νρ ρ

i ii

i i ii

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

dx

π π τ π

i i i i i

v + = 0

y

u

x

∂ ∂

∂ ∂

R

0

V = u (r) 2 drπ∫ i i

4 R dpV = -

8 dx

π η

i ii

2

2

V R pu = =

R 8 l

∆

π η

i

i i

( )u 2RRe =

ρ

η

i i

64 = laminar !!

Reξ


2) Conservation equations

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

2.1) Mass balance for ρ = const.

212

p d =

u l

∆ξ

ρi

i 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 xρ

∂ ∂

∂ ∂i i

x

Du udV = dV u = + dF

Dt x

∂ρ ρ

∂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

x

u - p u = + f

x x

∂ ∂ρ ρ

∂ ∂i i i

22 2 2

1 1 1

u = p - g h

2ρ ρi i i

( ) = dy dx dzR

dFy

τ∂

∂i i i

u u p u + v = -

x y x

DuDt

∂ ∂ ∂ρ ∂ ∂ ∂

��

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

yητ ∂

=∂i

2 2

x 2 2

u u p u + v = f - + +

x y x

u u

y xρ ρ η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

i i i i

2 2

y 2 2

p v + u = f - + +

v v v v

y x y y xρ ρ η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

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 = +

yR

uf

xη

∂ ∂ ∂ ∂ i

x Rx

Du p = f - + f

Dt xρ ρ

∂

∂i i

2

2

u u ~

x∞ ∞ρ η

δi i

x ~

u

νδ

∞

i

2

u u ~ ; ~

u

x x yη

δτ∞ ∞∂ ∂

∂ ∂i

2 2

2

u u = =

u

x y yρ η

τ∂ ∂ ∂

∂ ∂ ∂i i i

99 (x)

x = 5

u

νδ

∞

ii


Thickness of a boundary layer, laminar on a plate

inertial force = friction force ( Navier -Stokes )

( ) = f xδ

u→∞

( )i

y = 0

(x) = U - u(x,y) dy U δ∞

∫i i

valuelow

99 is arbitraryδ

99 (x) 5 x =

lRel

δi


Dimensionless :

A non - arbitrary value : displacement thickness

3.2) Friction forces on a plate :

high value

99

1 3

iδ δ≈ i

u( ) =

yw

w

x ητ ∂

∂ i

�

W Ww

2

S(Surface)

F F = c = =

E u b l

2∞

ζρi i

( )l

W W

0

F = b x dxτ∫i i

l 1

3 2W

0

F ~ b u x dx−

∞µ ρ ∫i i i i i


xu

~ with ~ u

w

ηρ

η δδ

τ ∞

∞

i

i

3 u ~ w

x

η ρτ ∞i i

1

3 2W

F ~ b u 2 l∞µ ρi i i i i

3

24 2

b 2 u l ~

b u l4

wc

η ρ

ρ∞

∞

i i i i i

i i i

l ~

Rew

c

1,1328 =

Rew

c


3.3) Boundary layer on an obstacle :

Navier - Stokes :

Far away from the obstacle (stream line) :

( )dU l dpU = - no friction

dx dxρi i

dU dp and are related to Bernoulli

dx dx

= w dsΓ ∫ i�


4) Potential and Stream functions

For describing vortex streams ( and comparable ) :

Circulation :

Potential streams (no friction ) : no rotation

Mass balance ; conservation equation :

11

2

2

v = =

t x

= = -t

1 v u = -

2 x y

u

y

γω

γω

ω

∂ ∂∂ ∂∂ ∂

∂ ∂

∂ ∂ ∂ ∂

v u = 0 ; - = 0

x yω

∂ ∂

∂ ∂

v + = 0

y

u

x

∂ ∂

∂ ∂

u = ; v = - y x

∂Ψ ∂Ψ

∂ ∂

+ - = 0y xx y

∂ ∂Ψ ∂ ∂Ψ ∂ ∂ ∂ ∂

2 2

2 2 + = 0

x y

∂ Ψ ∂ Ψ

∂ ∂

= ; v = y

ux

φ φ∂ ∂

∂ ∂


Stream function (definition ) :

Conservation equation :

No rotation :

Potential function :

Potential streams

1 v u v u = - ; - = 0

2 x y x y

∂ ∂ ∂ ∂ω ∂ ∂ ∂ ∂

i

2 2

2 2

u u + = 0

x y

∂ ∂

∂ ∂

( )p = f u, v

2

2

v u u v - - + = 0

y x y x x x y

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

u u p u + v = - + 0

x y x

∂ ∂ ∂ρ ∂ ∂ ∂ 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

∂ ∂ ∂ Φ ∂ Φ↔

∂ ∂ ∂ ∂ ∂ ∂

2 2

2 2 + = 0

x y

∂ Φ ∂ Φ

∂ ∂


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

∂Φ ∂Φ

∂ ∂

u = ; v = - y x

∂Ψ ∂Ψ

∂ ∂

= w dsΓ ∫��

i�


from definition :

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

Circulation :

Example :

here : Γ = 0 ( all possible ways )

airfoil : high speed

low speed

u + v = 0x y

∂Ψ ∂Ψ∂ ∂

i i

0Γ ≠

( ) = 2 r rΓ π ωi 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,yΦ ( )x,yΨ ( )u x,y ( )v x,y

U x + V y∞ ∞

E ln r

2π

2

Γ ϕπ

1

2

E r ln

2 rπ

2

M x

2 rπ

U y - V x∞ ∞

E

2ϕ

π

ln r2

Γ−π

( )1 2

E -

2ϕ ϕ

π

2

M y

2 r−

π

U∞

2

E x

2 rπ

2

y

2 r

Γ−π

2 2

1 2

E x + h x -

2 r r

π

2 2

4

M y - x

2 rπ

V∞

2

E y

2 rπ

2

x

2 r

Γπ

2 2

1 2

Ey 1 1 -

2 r r

π

4

M 2xy 2 r

−π

l

rw ~ = 0Γ


assumption : ; obviously :

One exception : including the centre :

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

radyield : E = w 2 rπi i

for x = r : E = u 2 xπi i

Eu =

2 x ( or r )π

2 2E = ln x + y

2Φ

πi

2 2 2 2

E 1 1 1u = = 2x

x 2 2x + y x + y

∂Φ∂ π

i i i i

2

E xu =

2 rπi

E E y = = arctg

2 2 x

Ψ ϕ π π

i i

E yu = = arctg

y 2 y x

∂Ψ ∂ ∂ π ∂

i

radspring : V = w 2 r hπ i i i


Spring :

2

1 arctg x =

x 1 + x

∂

∂

( )( )

y

x

y

x

u = x

∂ ∂Ψ

∂ ∂i


Bronstein :

the rest is the same

For application :

airfoil :

( )2

2 2y

x

E 1 1 xu =

2 x x1 + πi i i

2

E xu =

2 rπi

stream = of model streams∑

AF =b l p∆i i

( )

2

A

1F =b l 2u

2∞ρi i i i

2

AF = 2 l b u∞ρi i i i


5) Law of Kutta - Joukowski

simple example : flat plate :

Kutta - Joukowski

= 2 u l∞Γ i i

AF = b u∞Γ ρi i i

: u = Uy ∞→ ∞

2

2

u u + v =

x y

uu

yν

∂ ∂ ∂

∂ ∂ ∂i i i

= 0 : u = 0, v = 0y

u

y

x ν

∞

i


6) Exact calculation of the Boundary layer thickness

Boundary layer on a plate :

For similarity y/δ (x) is important

v + = 0

y

u

x

∂ ∂

∂ ∂

( ) x v x ~

uδ

∞

i

m = =

y m yu

∂Ψ ∂Ψ ∂

∂ ∂ ∂i

( ) = u f mu ∞ ′i

( )1 1 = 2 u f m

2 xxν ∞

∂Ψ

∂i i i i i

= y 2 x

um

ν∞ii i

( ) = 2 x f m 2 x

uu uν

ν∞

∞ ′i i i i ii i


Definition :

( the factor 2 is arbitrary but helpful )

Idea :

Stream function :

= - x

v∂Ψ

∂

( ) = 2 x u f mν ∞Ψ i i i i

( )�

Schlichtingsays

dimensionlessstream function

= 2 x u f m∞

η

Ψ ν��

i i i i

( ) m 2 x u f m

xν ∞

∂ ′+ ∂ i i i i i

3-2

u 1 = y - x

2 2

m

x ν∞∂

∂ i i i

i

3-

122

u u = f - y x x

x 2 x 2 x∞ ∞

ν∂Ψ ∂ ν

ii i i i i

i i i

( ) uv = - = m f - f

x 2 x∞ν∂Ψ ′

∂i

i ii

( )3

-2

1 = u f m y - x

2 2

uu

x ν∞

∞∂ ′′ ∂

i i i i ii

u v = m f - f

2 x y 2 x

u

y y

ν ν∞ ∞ ∂ ∂ ∂′

∂ ∂ ∂

i ii i i

i i

( ) 1 = f m -

2 x

uu m

x∞

∂ ′′ ∂ i i i

i

( ) 2 x u f m∞′νi 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

ν νν ν ν

∞ ∞ ∞ ∞ ∞∂ ′ ′′∂

��

i ii i i i i

i i i i i i i i

u v + = 0

x y

∂ ∂⇒

∂ ∂

( ) = m f - f

2 x

uv

ν ∞ ′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∞

∞

∂ ′′∂ ν

i ii 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 uνν

∞ ∞′−i

i ii i i

( ) 1 = m f m

2 x

vu

y∞

∂ ′′∂

i i ii

( )2

2 = f m

2 x 2 x

u uuu

y ν ν∞ ∞

∞∂ ′′′∂

i i ii i i i

2

2

u u uu + v =

x y y

∂ ∂ ∂ν

∂ ∂ ∂i i i

f + f f = 0

Blasius - Equation

′′′ ′′i

m = 0 f = 0 , f = 0

m : 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 - f

2 x∞ν ′ii i

i

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 - fη β → ∞ η η

wf ′′

( )2

0

= f 1 - f d∞

′ ′β η∫


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∞δ i

99 = y for f = 0.99′δ


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

Attention : f and η 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∞

ν

df u =

dm u∞

2

2

d f

dm

uu = u f y

2 x∞

∞

′ ν

ii i

w

w

uu = u f

y 2 x∞

∞ ∂ ′′ ∂ ν

i ii i

w

uu 0,4696 = u

y x2

∞∞

∂ ∂ ν

i ii

l

w w

0

F = b dxτ∫ i i

l 1

2w

0

uF = 0,332 u b x dx

−∞

∞ην ∫i i i i i i

l 1

2

0

with x dx = 2 l−

∫ i


6.3) Friction :

Plate ( 1 side ) :

w

u = 0,332 u

x∞

∞τ ην

i i ii

ww

Fc

u b l 2

∞

=ρi i i


( see also 3.2 )

7) Thermal boundary layer

Conservation equation for heat :

convection :

22

p 2

T u c u + v = +

x y

T T

y yρ λ η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

i i i i i i

w

1.328c

Re=

( ) T dx dz dy

y

∂− λ ∂ i i i

( )c pQ = m c T∆ ∆ i i

( )p

Tc dy dz u dx

x

∂ ρ ∂ i i i i i i

udP = dx dz dy

y ∂

∂τ i i i i

u =

yητ ∂

∂i


convection :

> 0

conduction :

< 0

friction :

> 0

Compare the conservation equations for heat with Navier-Stokes !

T A

yλ

∂− ∂ i i

2

p 2

T T T c u + v =

x y y

∂ ∂ ∂ρ λ ∂ ∂ ∂ i i i i

2

2

u u u + v =

x y y

∂ ∂ ∂η ν ∂ ∂ ∂ i i i

2

2p

2

2

u u uu + v

cx y y =

TT Tu v

yx y

∂ ∂ ∂ ν ρ∂ ∂ ∂

∂λ ∂ ∂ ∂∂ ∂

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∞α i

w

Tq = -

y

∂λ ∂

i

u

w

T-

y =

T - T∞

∂λ ∂ α

i

( )T T

w

w

- y

= T

- 1T

∞

∞

∂ λ

∂ α

i


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

w

u

u =

y

∞

∂ α λ

∂

i

u = 0,4696

2 x∞α λ

νi i

i il

1

0 2

1b dx

u x = 0,4696 2 b l

∞α λν

∫i i

i i ii i

2

u 4 l = 0,4696

2 l

∞α ληρ

i ii i

i i

1+

2 = 0,664 Rel

λα i i

Nu = 0,664 Rei

wwith u = 0


5Re = 5 10i

, sudden δ τ↑ ↑


There is evidence that for Pr ≠ 1 :

see also : Vauck, W.R.A., Müller, 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

11

32Nu = 0,664 Re Pri i

2

A A AAB 2

c c cu + v = D

x y y

∂ ∂ ∂

∂ ∂ ∂i i i

fx

50 =

cRe

2

v

x

δ

i

( ) ( ) ( )( )

u x,y,t = u x,y + u x,y,t

v x,y,t = v....

′

u = average ; u = 0′

p (x,y,t) = p (x,y) + p (x,y,t)′

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

x x x x

′ ′∂ ∂ ∂ ∂ ′ ′ρ ∂ ∂ ∂ ∂ 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

.....

′ ′∂ ∂ ∂ ∂′ ′ρ

∂ ∂ ∂ ∂

=

i i i i

( )2

2

u u du uu + v = u + - u v

x y dx y y

∂ ∂ ∂ ∂′ ′ρ ρ η ρ ∂ ∂ ∂ ∂

i i i i i i

��

u uu = 0 , u = 0

x x

′∂ ∂′ ′

∂ ∂i i

2 2

2 2

d u u - + +

dx y y

′ ρ ∂ ∂= η ∂ ∂


Average :

2 2

2 2

dp u u - + +

dx y y

′ ∂ ∂= η ∂ ∂

i

( )dp dU - U Bernoulli

dx dx= ρ i i

( )u uu + v u v

x y y

′ ′∂ ∂ ∂′ ′ ′ ′≈

∂ ∂ ∂i i i

l

u =

y

∂τ η

∂i

( )t = - u v

is usually negative

u v

′ ′τ ρ

′ ′

i i

i

t

- u v = + with =

u

y

u

y′ ′∂∂

∂τ ρ∂

ε ε ii i

~ l u

y

u ∂∂′ i

2

tu u

= l y y

∂ ∂ρ

∂ ∂τ i i i


laminar sheer stress :

turbulent sheer stress :

ε : turbulent kinematic viscosity

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

laminar sublayer

v ~ u′ ′

dpFlat plate : = 0

dx

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 2 213

u

u + v + w T =

u∞

′ ′ ′i

2

2

u u dp u u + v = - +

x y dx y

∂ ∂ ∂ρ η ∂ ∂ ∂ i i i i


low speed - high pressure

When friction and pressure increase, debonding occurs.

In the layer :

2

2

dp uIf has a high value, must

dx y

become positive

∂

∂


Result :



burbling from

point A on


Turbulent flow : η + ε · ρ instead of η : burbling occurs later

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


w c = 0→

u D u DRe = =∞ ∞ρ

η ν

i i i

laminar→

laminar, but burbling→

} turbulent

ww 2

2

sphere :

Fc =

u D

2 4∞ π

ρi i i

w c = 0→


creeping flow :

d'Alembert : no friction (and no burbling)


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

f d =

uSr

i


Periodic stream due to debonding :

Strouhal - Number :


11) Bibliography

- Gersten, K. : Einführung in die Strömungsmechanik, 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., Müller, 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.