Post on 01-Nov-2014
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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
�
�
�
Prof. Dr. N. Ebeling Boundary Layer Theory - 2 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 3 -
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
�����
{
Prof. Dr. N. Ebeling Boundary Layer Theory - 4 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 5 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 6 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 7 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 8 -
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
∆ ∆∆ ∆
Prof. Dr. N. Ebeling Boundary Layer Theory - 9 -
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↑
Prof. Dr. N. Ebeling Boundary Layer Theory - 10 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 11 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 12 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 13 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 14 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 15 -
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�
Prof. Dr. N. Ebeling Boundary Layer Theory - 16 -
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∂ ∂
∂ ∂
Prof. Dr. N. Ebeling Boundary Layer Theory - 17 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 18 -
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
∂ ∂
Prof. Dr. N. Ebeling Boundary Layer Theory - 19 -
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�
Prof. Dr. N. Ebeling Boundary Layer Theory - 20 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 21 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 22 -
Spring :
2
1 arctg x =
x 1 + x
∂
∂
( )( )
y
x
y
x
u = x
∂ ∂Ψ
∂ ∂i
Prof. Dr. N. Ebeling Boundary Layer Theory - 23 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 24 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 25 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 26 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 27 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 28 -
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⇒
Prof. Dr. N. Ebeling Boundary Layer Theory - 29 -
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∫
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 :
(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
Prof. Dr. N. Ebeling Boundary Layer Theory - 31 -
(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
Prof. Dr. N. Ebeling Boundary Layer Theory - 32 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 33 -
( 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
Prof. Dr. N. Ebeling Boundary Layer Theory - 34 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 35 -
( 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
Prof. Dr. N. Ebeling Boundary Layer Theory - 36 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 37 -
5Re = 5 10i
, sudden δ τ↑ ↑
Prof. Dr. N. Ebeling Boundary Layer Theory - 38 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 39 -
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 ∂ ∂
Prof. Dr. N. Ebeling Boundary Layer Theory - 40 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 41 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 42 -
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
Prof. Dr. N. Ebeling Boundary Layer Theory - 43 -
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
∂
∂
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 )
burbling from
point A on
Prof. Dr. N. Ebeling Boundary Layer Theory - 45 -
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 )
(nach : Gersten, K. : Einführung in die Strömungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 111 )
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→
Prof. Dr. N. Ebeling Boundary Layer Theory - 46 -
creeping flow :
d'Alembert : no friction (and no burbling)
(nach : Gersten, K. : Einführung in die Strömungsmechanik, Bertelsm. Univ.Verlag, 1st edition, page 114 )
(nach : Gersten, K. : Einführung in die Strömungsmechanik,Bertelsm. Univ.Verlag, 1st edition, page 112 )
f d =
uSr
i
Prof. Dr. N. Ebeling Boundary Layer Theory - 47 -
Periodic stream due to debonding :
Strouhal - Number :
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 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.