Boundary Layer Theory

-

Prof. Dr. Norbert Ebeling

Boundary Layer Theory Lecture notes

Prof. Dr. N. Ebeling

Boundary Layer Theory

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

-1-



1) General fluid mechanics / Newton Fluids General definitions

ex x u   j ey y v  general definitions  k ez z w  i

dy (dz)

dm = ρ i dx i dy i dz

dx

Acceleration : dFi = dm i

Du Dt

Du ∂u ∂u ∂u ∂u = +u i +v i +w i Dt ∂t ∂x ∂y ∂z ∂u =0 ∂t

stationary :

frequently :

Du ∂u =u i Dt ∂x

volume force:

fi

(e.g.

g)

-2-



1.1) Euler's law of hydrostatics

↓ dF1

x↓

fx ↑ dF2

ρ i fx i dV + dF1 = dF2

ρ i fx

∂F i dx i dy i dz = i dx ∂x

dF = dp i dy i dz ∂p ρ i fx = ∂x

-3-



-4-

1.2) Friction

τ = +η i

Moving fluid :

∂u ∂y

( Couette - flow )

Newton fluid

Schlichting :

∂u τ =µ i ∂y

1.3) Dimensionless numbers : Reynolds number

Re ~

inertial force friction force

{ τ

τ + ∂τ

∂y

i dy

∂τ FFR = i dy i ( dx i dz ) ∂y

dA



dm ∂u ρ i dx i dy i dz i u i ∂x Re ~ ∂τ i dx i dy i dz ∂y

∂u u∞ ∂τ ∂  ∂u  ~ ; = η i  ∂x ∂y ∂y  ∂y  d ∂τ ∂  u∞  ~ η i   ∂y ∂y  d 

or any comparable speed v else

V d = ρ ivid v η η i 2 d

ρivi Re =

v i d with ν = η Re = ρ ν laminar flow : high friction forces, low inertial forces avoided by friction

υ

deciding

-5-



ascending force

FA pis

CA =

s

Bernoulli :

Pipe :

1 i ρ i u2∞ 2 Cw ( or ζ ) analogous p →

ζR =

d ρ i u2m 2

Gravity influence :

i

dp dx

Fr Froude number

Cw =

( or λ )

=

v gid

FR pis

-6-



1.4) Laminar flow in a tube

ν

extremely high

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

Hagen-Poisseulle :

ζR

64 = Re

dp 64 i η 64 i ν i = = ρ 2 dx v i ρ i d v i d iv 2 d

Derivation :

dp   p i π i r2 -  p + dx  i π r2 - τ i 2π r i dx = 0 dx   r dp du − i =τ = -η i 2 dx dr Integration with u (r = R) = 0 leads to : 2  R2 dp   r  u(r) = i i  - 1 4η dx   R  

-7-



R

= u (r) i 2π i dr V ∫ 0 4 π i R  dp 

= V i  8iη  dx 

V R2 ∆p u= = i π i R2 8iη l u i ρ i ( 2R ) Re = η ∆p d ξ= 1 i 2 i ρ i u l 2 64 ξ= laminar !! Re

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

2.1) Mass balance for ρ = const.

∂u ∂v + =0 ∂x ∂y

-8-



u1 i ∆y i ∆ z = u 2 i ∆y i ∆z + v 2 i ∆x i ∆z

(u1 - u2 )

i ∆ y = + v2 i ∆ x

∆u ∆v + =0 ∆x ∆y 2.2) Euler's and Bernoulli's equations Eulers equation ( one direction, pipe ):

Du ∂u dV i ρ i = ρ i dV i u i = + dFx ∂x Dt ρ iui

∂u ∂p =∂x ∂x

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

-9-



Mechanical energy balance : Bernoulli incl. hydrostatics

∂u -∂p ρiui = + ρ i fx ∂x ∂x u2 ρi 2

2 1

=p

2

-ρ i gi h

1

2 1

Euler (2 directions ):

↑v

Du Dt

 ∂u ∂u  ∂p ρ i u i +vi = ∂x ∂y  ∂x  v leads to a higher value of u

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

τ

∂u =η i ∂y

∂τ dFR = idy i ( dx i dz ) ∂y

- 10 -



- 11 -

 ∂u2 ∂ 2u  fR = η i  2 + 2  ∂ y ∂ x   Navier - Stokes - Equations ( Can be simplified in a boundary layer (later))

ρi

Du ∂p = ρ i fx + f Rx Dt ∂x

 ∂2u ∂2u   ∂u ∂u  ∂p ρ u i + v i  = ρ ifx + ηi  2+ 2 ∂y  ∂x ∂x   ∂x  ∂y

 ∂ 2v ∂ 2v   ∂v ∂v  ∂p ρ v i + u i  = ρ ify + η i  2+ 2 ∂x  ∂y ∂x   ∂y  ∂y

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 )



Thickness of a boundary layer, laminar on a plate

u∞

→

δ = f (x )

u∞ ∂u u∞ ∂τ ~ ; ~ηi 2 x δ ∂x ∂y

∂u ∂ 2u 2 ∂τ ρ iui =ηi = 2 ∂x ∂y ∂y inertial force = friction force ( Navier -Stokes )

u2∞ u∞ ρ i ~ηi 2 δ x ν ix δ ~ u∞

δ 99 (x) = 5 i

ν ix u∞

- 12 -

Prof. Dr. N. Ebeling Dimensionless :


δ99 (x)

=

l

5 Re

i

- 13 -

x l

δ 99 is arbitrary

A non - arbitrary value : displacement thickness

U i δ i (x) =

∞

∫ (U - u(x,y) )

idy

y=0

δi

≈

1 i δ 99 3

3.2) Friction forces on a plate :

high value

low value



 ∂u  τ w (x ) = η i    ∂y w

τ

w

~η i

τ ζ = cw =

w

u∞

δ

~

η ix ρ

with δ ~

u∞

η i ρ i u3∞ x

FW FW = ρ E i u∞2 i l b 2 S

(Surface)

l

FW = b i

∫ τ ( x)

i dx

W

0

l

FW ~ b i µ i ρ i u∞3 i ∫ x

−

1 2

i dx

0

FW ~ b i µ i ρ i u3∞ i 2 i l cw ~

1 2

b i 2 i η i ρ i u3∞ i l b i u4∞ i

ρ2 4

i l2

- 14 -



1,1328 cw = Re

cw ~

l Re

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

Far away from the obstacle (stream line) :

dU l dp Ui =- i ( no friction ) dx ρ dx

dU dp and are related to Bernoulli dx dx

- 15 -



- 16 -

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

∂γ1 ∂v ω1 = = ∂t ∂x ∂γ 2 ∂u =ω2 = ∂t ∂y

ω=

Circulation :

Γ=

1  ∂v ∂u    2  ∂x ∂y 

∫ w i ds

Potential streams (no friction ) : no rotation

∂v ∂u ω =0; =0 ∂x ∂y Mass balance ; conservation equation :

∂u ∂v + =0 ∂x ∂y



Stream function (definition ) :

u=

∂Ψ ∂Ψ ; v=∂y ∂x

Conservation equation :

∂  ∂Ψ  ∂  ∂Ψ  + =0     ∂x  ∂y  ∂y  ∂x  No rotation :

∂2Ψ ∂2Ψ + =0 2 2 ∂x ∂y Potential function :

∂φ ∂φ u= ;v= ∂x ∂y Potential streams

1  ∂v ∂u  ∂v ∂u ω= i  ; =0  2  ∂x ∂y  ∂x ∂y

- 17 -



- 18 -

Streams without any rotation :

∂v ∂u =0 ∂x ∂y also conservation equation :

∂u ∂v + =0 ∂x ∂y

∂  ∂v  ∂ 2u ∂  ∂u  ∂  ∂v  - 2 + =0       ∂y  ∂x  ∂y ∂x  ∂x  ∂x  ∂y  ∂ 2u ∂ 2u + =0 2 2 ∂x ∂y Insert in Navier - Stokes :

 ∂u ∂u  ∂p ρ i u i +vi = +0  ∂x ∂y  ∂x 

p = f ( u, v ) leads to Bernoulli for v = 0 - no friction ! no rotation - no friction



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

potential function :

∂Φ ∂Φ u= ;v= ∂x ∂y

No rotation :

∂v ∂u ∂ 2Φ ∂ 2Φ =0 ↔ =0 ∂x ∂y ∂x∂y ∂x∂y Conservation equations : 2 2

∂Φ ∂Φ + =0 2 2 ∂x ∂y

Stream function :

u=

∂Ψ ∂Ψ ;v=∂y ∂x

Conservation equations o.k.

- 19 -



from definition :

∂Ψ ∂Ψ ui +vi =0 ∂x ∂y Stream line :

Circulation :

( no v : )

-> ψ = constant

Γ = ∫ w i ds

Example : here : Γ = 0 ( all possible ways )

airfoil : high speed low speed

Γ ≠ 0

- 20 -


assumption :


w~

l ; obviously : r

One exception : including the centre :

Γ = 2π i r i

- 21 -

Γ=0

( ω i r)

Potential- and flowfunctions as well as velocitys for some elementary potential flows Ψ ( x, y )

u ( x,y )

v ( x, y)

flow

Φ ( x, y)

translational flow

U∞ x + V∞ y

U∞ y - V∞ x

U∞

V∞

source flow

E ln r 2π

E ϕ 2π

E x 2π r 2

E y 2π r 2

streamline

( productiveness E )

potential vortex stream

Γ ϕ 2π

−

E r ln 1 2π r2

E 2π

Γ ln r 2π

−

Γ x 2π r2

Γ y 2π r 2

( circulation I' )

source-drain flow

( ϕ1 - ϕ2 )

E x+ h x  - 2  2π  r12 r2 

Ey  1 1 - 2  2 2π  r1 r2 

( productiveness E, distance h )

dipole flow

M x 2π r2

−

M y 2π r 2

M y2 - x 2 2π r4

−

M 2xy 2π r 4

( dipole moment M )

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



- 22 -

spring : V = wrad i 2π i r i h yield : E = wrad i 2π i r

for x = r : E = u i 2π i x E u= 2πx ( or r )

E Φ= i ln x2 + y2 2π ∂Φ E 1 1 u= = i i i 2 2 ∂x 2π 2 x +y Spring :

1 2

2

x +y

E x u= i 2 2π r E E  y Ψ= iϕ= i arctg   2π 2π  x

∂Ψ E ∂   y  u= = i arctg    ∂y 2π ∂y   x 

i 2x


Bronstein :


- 23 -

∂ 1 arctg x = ∂x 1 + x2

∂

() y x

∂Ψ u= i ∂x ∂ yx E 1 u= i 2π 1 + y x

()

()

2

1 i x

E x u= i 2 2π r the rest is the same

For application :

stream =∑ of model streams airfoil :

x2 i 2 x



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

FA =b i l i ∆p 1 2 i ρ i( 2u∞ ) FA =b i l i 2

Γ = 2 i u∞ i l

FA = 2 i ρ i l i b i u∞ FA = Γ i ρ i b i u∞ Kutta - Joukowski

2

- 24 -



6) Exact calculation of the Boundary layer thickness Boundary layer on a plate :

∂u ∂u ∂2u ui +v i =νi 2 ∂x ∂y ∂y ∂u ∂v + =0 ∂x ∂y y = 0 : u = 0, v = 0

y → ∞ : u = U∞

xiv δ ( x) ~ u∞ For similarity y/δ (x) is important

y x iν u∞

- 25 -


Definition :


u∞ m=y i 2iν ix

( the factor 2 is arbitrary but helpful )

Idea : Stream function : dimensionless stream function

Ψ =

2 i ν i x i u∞

i f ( m)

Schlichting says η

u=

∂Ψ ∂Ψ ∂m = i ∂y ∂m ∂y

u = 2 i ν i x i u∞ i f ′ ( m ) i

u∞ 2 iν ix

u = u ∞ i f ′ (m ) v =-

∂Ψ ∂x

Ψ = 2 i ν i x i u ∞ i f (m ) ∂Ψ 1 1 = i i 2 i ν i u ∞ i f (m ) 2 ∂x x

- 26 -



- 27 -

 ∂m  ′ + 2 i ν i x i u ∞ i f (m ) i    ∂x  3 ∂m u∞  1 =y i i -  i x 2 ∂x 2 i ν  2

 ∂Ψ ν i u∞ = i f - yi  ∂x 2ix 

 u∞  i 2iνix

-

xix

3 2

i

1 2

i 2 i ν i x i u∞ i f ′ ( m )

∂Ψ ν i u∞ v == i ( m i f′ - f ) ∂x 2ix

6.1) Conservation of mass (continuity equation) ∂u = u ∞ i f ′′ (m ) i y i ∂x

3 u∞  1 i -  i x 2 2 i ν  2

∂u 1   = u∞ i f ′′ (m ) i m i   ∂x 2 i x  

∂v ∂ = ∂y ∂y

 ν i u∞  ∂  ν i u∞  i m i f ′ if     2 i x  ∂y  2 i x 


∂v ν i u∞ = i ∂y 2ix

−

ν i u∞ 2ix


- 28 -

m u∞ ν i u∞ u∞ u∞ i f ′+ iy i f ′′ i 2 iν ix 2 ix 2 i ν ix 2 i ν ix

i f′ i

u∞ 2 iν ix

∂v 1 ′′ = u∞ i m i f (m ) i ∂y 2ix

∂u ∂v ⇒ + =0 ∂x ∂y

Conti - equation

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

u = u∞ i f ′ (m ) ∂u 1   = u∞ i f ′′ (m ) i m i  ∂x  2 i x ν i u∞ v = i(m i f ′ - f ) 2 ix

∂u = u∞ i f ′′ ( m) i ∂y

u∞ 2iνix



∂ 2u =u∞ i 2 ∂y

u∞ i f ′′′ ( m ) i 2 iν ix

with :

- 29 -

u∞ 2 i ν ix

∂u ∂u ∂ 2u ui +vi =νi ∂x ∂y ∂y 2 Inserting and differentation leads directly to :

f′′′ + f i f′′ = 0 Blasius - Equation side conditions :

m=0 f = 0 , f′ = 0 m → ∞ : f′ = 1

u = u∞ i f ′ ( m)

m = 0 ( i.e. y = 0)

u = 0 and f′ = 0

m → ∞ ( i.e. y → ∞ )

u = u∞ and f′ = 1

ν i u∞ v= i (m i f ′ - f ) 2ix ⇒ for y = 0 v and f have to be 0



- 30 -

There is a function f(m), but there is no equation. description of f(m) : Thickness of the boundary layer : characteristic parameters for the boundary layer on a longitudinal flown plate

′′ fw

0,4696

β1 = limη → ∞  η - f ( η)  

1,2168

∞

β2 =

∫ f ′ ( 1 - f′ )

dη

0,4696

0

0,7385 (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 )



- 31 -

Flat plate laminar boundary layer functions

f

df u = dm u∞

d2 f dm2

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

m=y

u∞ νx

(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  2 !

δ99 = y for u = 0.99 i u∞

δ99 = y for f′ = 0.99



- 32 -

6.3) Friction :

  u∞ u = u∞ i f ′  y   2iν ix  ∂u  u∞  ∂y  = u∞ i 2 i ν i x i fw′′  w

 ∂u  0,4696 i u∞ i =  ∂y  2  w τ w = η i 0,332 i u ∞ i

u∞ ν ix u∞ ν ix

Plate ( 1 side ) :

l

Fw =

∫τ

w

i b i dx

0 l

Fw = η i 0,332 i u ∞ i l

with

−

∫x 0

1 2

1 − u∞ i b i ∫ x 2 i dx ν 0

i dx = 2 l



- 33 -

Fw

cw =

ρ i u∞ i b i l 2 1.328 = Re

cw

( see also 3.2 )

7) Thermal boundary layer

Conservation equation for heat : 2

 ∂T ∂T  ∂ 2T  ∂u  ρ i c p i u i +v i = λ i + η i    2 ∂x ∂y  ∂y   ∂y 

convection :



- 34 -

convection :

i cp i ∆T) ∆Q c = ( m  ∂T  cp i ρ i ( dy i dz ) i u i  i dx   ∂x 

>0

conduction :

 ∂T  − λ i A i   ∂y    − λ i 

( dx

i dz )

∂T  i dy  ∂y 

0

∂u ∂y

Compare the conservation equations for heat with Navier-Stokes !



- 35 -

 ∂T ∂T  ∂ 2T ρ i cp  u i +v i =λ i  2 ∂ x ∂ y ∂ y   ( heat from friction neglected ) Navier-Stokes adapted to a boundary layer (see also 6) )

 ∂u ∂u  ∂ 2u  η i ∂x + v i ∂y  = ν i ∂y2  

 u i   u i   u   u 

∂u +vi ∂x ∂T ivi ∂x

∂u  ∂ 2u ν i ρ i cp ∂y  ∂y2 = i 2 ∂ T λ ∂T  ∂y2 ∂y 

∂u i +vi ∂x ∂T i ivi ∂x

∂u  ∂y  = ∂T  ∂y 

Pr

∂ 2u ∂y 2 i 2 ∂ T ∂y 2



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

q = α i ( Tw - T∞ )  ∂ T 

= -λ i  q  ∂ y  w  ∂T  -λ i   ∂ y  u α= Tw - T∞

( ) 

 ∂ TT ∞ -λ i   ∂y  α = Tw -1 T∞

 w

- 36 -



- 37 -

  u   ∂  u∞     α =λ i with uw = 0  ∂y     w

α=λ i

u∞ i 0,4696 2iν ix l

bi α = 0,4696 i λ i

u∞ i 2iν

α = 0,4696 i λ i

∫

1

0

1 2

i dx

x bil

u∞ i 4 i l η 2 i i l2 ρ

1 + λ α = 0,664 i i Re 2 l

Nu = 0,664 i Re



- 38 -

There is evidence that for Pr ≠ 1 :

1 2

Nu = 0,664 i Re i Pr

1 3

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

8) Mass transfer boundary layer equation

∂c A ∂c A ∂ 2c A ui +vi = DAB i ∂x ∂y ∂y2

9) Turbulent Boundary layer Plate : turbulent from Re = 5 i 10 5 on

sudden δ ↑ ,

τ↑

virtual friction turbulent layer : 2 layers viscous sublayer



- 39 -

Viscous sublayer :

δv x

50

=

Re x i

cf 2

Turbulent boundary layers

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) Conservation equations :

∂u ∂u′ ∂v ∂v ′ ∂u ∂v + + + =0; + =0 ∂x ∂x ∂y ∂y ∂x ∂y Navier - Stokes ( for boundary layers ) :

∂u ∂u ∂u′ ∂u′   ρ i u i + u′ i +ui + u′ i + v.....  ∂x ∂x ∂x ∂x  



- 40 -

 ∂ 2u dp ∂ 2u′  = +η i  2 + 2  dx ∂ y ∂ y   dp dU = -ρ i Ui ( Bernoulli ) dx dx Average :

∂u ∂u′ u′ i = 0 , u′ i =0 ∂x ∂x  ∂u ∂ u′ ∂u ∂ u′  ρ u i + u′ i +vi + v′ i  ∂x ∂x ∂y ∂y   = .....  ∂ 2u dρ ∂ 2u′  = -

+ η  2 + 2  dx ∂ y ∂ y  

∂u′ ∂u′ ∂ u′ i + v′ i ≈ ( u′ i v′) ∂x ∂y ∂y  ∂u ∂u  du ∂ 2u ∂ ρ u +vi = ρ i u i + η i ( u′ i v′) i ρ  2 ∂y  dx ∂y ∂y  ∂x



τl = η i

laminar sheer stress :

turbulent sheer stress :

τt

∂u ∂y

(

= - u′ i v′

)iρ

u′ i v′ is usually negative

∂ u τt = + ε i ρ i ∂y

with ε

- u′ i v′ = ∂u ∂y

ε : turbulent kinematic viscosity

u′ ~ l i ∂u ∂y

v′ ~ u′ ∂u ∂u τt = ρ i l i i ∂y ∂y 2

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

- 41 -



Degree of turbulence :

Tu =

1 3

(

i u′2 + v′2 + w′2

- 42 -

)

u∞

10) Burbling

dp Flat plate : =0 dx Stream line along a body different from a flat plate outside the boundary layer ( no friction : )

du dp Ui =dx dx ( see Bernoulli and Navier-Stokes )


low speed


-

- 43 -

high pressure

When friction and pressure increase, debonding occurs. In the layer :

 ∂u ∂u  dp ρ i u i +vi +ηi  =∂x ∂y  dx  dp ∂ 2u If has a high value, must 2 dx ∂y become positive

 ∂ 2u   2  ∂y 



- 44 -

Result :





- 45 -

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

burbling from point A on

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

(nach : Gersten, K. : Einführung in die Strömungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 111 )



- 46 -

creeping flow :

d'Alembert : no friction (and no burbling)

→ cw = 0 → cw = 0 (nach : Gersten, K. : Einführung in die Strömungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 114 )

Re =

ρ i u∞ i D u∞ i D = η ν sphere : cw =

Fw π u iρi i D2 2 4 2 ∞

→ laminar

→ laminar, but burbling

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



Periodic stream due to debonding : Strouhal - Number :

Sr =

f id u

- 47 -



- 48 -

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 Verfahrenstechnik" , 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.