-
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
-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
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
R
V = ∫ u (r) i 2π i dr
0
π i R4  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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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-
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 :
Boundary Layer Theory
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
Prof. Dr. N. Ebeling
Boundary Layer Theory
 ∂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
bi l
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
assumption :
Boundary Layer Theory
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 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Bronstein :
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Definition :
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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

Prof. Dr. N. Ebeling
∂v
ν i u∞
=
i
∂y
2ix
−
ν i u∞
2ix
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
∂ 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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 :
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 34 -
convection :
∆Qc = ( m i cp i ∆T)
 ∂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
friction :
dP =
τ
i dx i dz i
τ
=η i
∂u
i dy
∂y
>0
∂u
∂y
Compare the conservation equations for heat with Navier-Stokes !
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
For gases Pr ≈ 1.
Independent from the condition u and T behave equal.
q = α i ( Tw - T∞ )
 ∂T 
q = -λ i 

∂
y

w
 ∂T 
-λ i 

∂
y

u
α=
Tw - T∞
( ) 
 ∂ TT
∞
-λ i 
 ∂y

α =
Tw
-1
T∞

w
- 36 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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


Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
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 )
Prof. Dr. N. Ebeling
low speed
Boundary Layer Theory
-
- 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 
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 44 -
Result :
(nach : Schlichting, H. , Gersten, K. (2006): Grenzschicht - Theorie, Springer , 10th Ed., page 37 )
(nach : Schlichting, H. , Gersten, K. (2006): Grenzschicht - Theorie, Springer , 10th Ed., page 39 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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 )
Prof. Dr. N. Ebeling
Boundary Layer Theory
Periodic stream due to debonding :
Strouhal - Number :
Sr =
f id
u
- 47 -
Prof. Dr. N. Ebeling
Boundary Layer Theory
- 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.