Boundary Layer
Advanced Fluid
M ech anics
2. F ri c t i onal D rag F orc es i n
Boundary Layer
8
Chapter 2
FRICTIONAL DRAG FORCES IN
BOUNDARY LAYER
2.1.
FORCES DUE TO A FLOW ALONG A SURFACE
Let us consider the boundary layer developing over a flat plate
Θ-
the angle of inclination of velocity vector with respect to
the wall
Our task is to determine the frictional drag force F acting on a
plate. The shear force F can only be found by applying the
momentum theorem:
shear (drag) force = change of momentum
→
F =
or
∫ ρU Udy − ∫ ρU Udy
→
outlet
→
int let
(1)
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Fx =
2. F ri c t i onal D rag F orc es i n
Boundary Layer
∫ ρU
cos Θdy −
2
outlet
Fy =
∫ ρU
2
outlet
∫ ρU
2
int let
sin Θdy −
Let us define the control volume
cos Θdy
(1a)
sin Θdy
(1b)
∫ ρU
2
int let
A - a leading edge
QP - a streamline
AP, ZQ - normal to a plate (x axis)
− F = ∫ ρU cos Θdy − ∫ ρU2∞ dy
Q
2
Z
Since U∞ is constant all over |AP|
9
P
(2)
A
2
2
U
dy
=
U
∞ AP
∫ ∞
P
A
Continuity equation gives
∫ U∞ dy = U∞ AP
P
A
so
= ∫ Udy = ∫ Udy ⇒
Q
δ
Z
0
U2∞
F = U∞ ∫ ρUdy − ∫ ρU2 cos Θdy
δ
δ
0
0
AP = U∞ ∫ Udy
δ
0
(2a)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
Frictional drag force for low-viscosity fluids
F = ρ ∫ (U∞ − U) Udy
δ
0
(3)
as Θ → 0 ⇒ cosΘ → 1
2.2. FRICTIONAL DRAG FORCES IN LAMINAR
BOUDARY LAYER
Prandtl (1904) first deduced the relation between velocity gradient
and shear stress
∂U
τ =µ
( 2.1)
∂y
where µ is the coefficient of dynamic viscosity
an engineer wants to know the value of shear stress τ0 on the
surface ⇒ the knowledge of the frictional drag forces opposing
the flow or movement of travelling object
measurement of shear stress is very difficult or even
impossible
measurement of velocity cannot be performed with sufficient
accuracy due to very small thickness of laminar boundary
layer
stress distribution: maximal value τ0 on the surface and zero
at the outer edge of the boundary layer
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1 1
2. F ri c t i onal D rag F orc es i n
Boundary Layer
The solution of the problem
1. Assumption of the shear stress distribution across the boundary
layer - possible stress distributions
2. Calculations of the velocity profile enabling evaluation of drag
forces
3. Comparison with experiment (experimental verification)
Step 1
linear dependence between stress and coordinate
y
τ = τ 0 1 − 
 δ
( 2.2)
taking into consideration Prandtl’s formula (2.1) we have
µ
dU
 y
= τ 0 1 − 
dy
 δ
( 2.3)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
Step 2
integrating each side we have
τ0 
y2 
U =  y −  + C
2δ 
µ
( 2.4)
boundary conditions:
U = 0 for y = 0 ⇒ C = 0
U = U ∞ for y = δ
⇒ U∞ =
τ0 δ
⋅
µ 2
(2.5)
( 2. 6)
shear stress on the surface
τ 0 = 2µ
U∞
δ
( 2. 7 )
velocity distribution across the laminar boundary layer
U∞ −U =
U∞
δ
2
(
)
δ
−
y
2
(2.8)
Conclusion:
if the stress varies linearly then velocity is described with
parabolic equation
Frictional drag force per unit width (the change of momentum
of the fluid passing through the boundary layer)
F = ρ ∫ U (U ∞ − U ) dy
δ
0
( 2.9)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
substituting for U from equation (2.8) it is found that
F=
2
ρU ∞2 δ
15
( 2.10)
F - total force over all the surface from the leading edge to
the point where boundary layer thickness is δ
From the other side force F may be calculated by integration of
the elementary forces τ0⋅dx from x = 0 to x = x0
0
2
2
F = ρU ∞δ = ∫τ 0 dx
15
0
x
( 2.11)
substitutingτ0 = 2µU∞ /δ the above equation may be solved by
separation of the variables δ and x giving
ρU ∞δ 2
x0 =
30 µ
( 2.12)
µ x0
ρ U∞
( 2.13)
or
δ 2 = 30
or
U x 
= 30  ∞ 0 
x0
 ν 
δ
−
1
2
= 30 (Re )−0.5
( 2.14)
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2. F ri c t i onal D rag F orc es i n
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Step 3
The obtained result should be verified experimentally:
the measurement of δ cannot be performed with sufficient
accuracy
dU/dy is to small for y = δ
boundary layer thickness δ is to small for penetration, e.g.:
♦ air at 320km/h – maximum thickness of laminar
boundary layer is only 0.5mm
♦ water at 3m/s ⇒ δmax = 1.3mm
the verification of the validity of the solution obtained is
typically done by the drag force measurement
Let us define the coefficient of friction
cf =
total drag force
F ⋅b
=
1
1
ρ U ∞2 ⋅ area
ρ U ∞2 x0 b
2
2
(2.15)
where b is the width of the plate
theoretical value is determined by substituting F and δ from
eqs. (2.10) and (2.14) giving
U x 
c f = 1.46 ∞ 0 
 ν 
−
1
2
= 1.46 Re −0.5
( 2.16)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
Experimental verification of eq. (2.15):
1. Measurement of drag force (the numerator) by means of an
aerodynamic scales
2. Measurement of kinetic pressure (the denominator) by Pitot
tube situated in undisturbed flow
measurement of drag force must be done with the use of the plate
of the length L meeting the condition
U∞L
< 5 ⋅105
(2.17)
ν
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
which ensures the laminar character of the boundary layer along
whole plate
experimental results of cf show good agreement with
theoretically predicted values
⇓
assumption that shear stress varies linearly with coordinate
is true
comment:
from practical (engineer’s) point of view the calculations of
laminar boundary layer are of minor importance because of the
small contribution of drag forces acting in this part of boundary
layer
example:
air at 320km/h – laminar/turbulent transition takes place at
6cm from the leading edge
water at 3m/s – 17cm
under favourable conditions the above lengths may be 4 times
greater
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
2.3. FRICTIONAL DRAG FORCES IN TURBULENT
BOUDARY LAYER
2.3.1. Transition from laminar to turbulent boundary layer
1. At some distance from the leading edge laminar flow breaks
down giving irregularly oscillating velocities
2. Intermittent character of the flow:
flow is steady for some time, then interval with unsteady
variations occurs
intermittency factor
∆t i
∑
I=
∈ ⟨0;1⟩
T
3. The share of unsteadiness increases downstream the leading
edge (I ↑)
4. At some distance random fluctuations are present all the time
⇒ turbulent boundary layer (I = 1)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
The properties of the transition boundary layer are partly those
of laminar and partly of turbulent flow
⇓
experimental and theoretical studies are very difficult but in
engineering work (when drag forces are of the greatest
importance) we may not take into account the transition
boundary layer due to small part of surface exposed to it
2.3.2. Turbulent boundary layer
from the viewpoint of engineering work turbulent boundary
layer is of the most importance; in many practical situations it is
responsible for the majority of drag forces
example:
boundary layer at the stern of a 300m long ship travelling at
30knots (15.4 m/s) is about 1.4m thick if the steel plate is
smooth (for rough surfaces δ is even thicker)
Frictional drag force
F = ρ ∫ (U ∞ − U ) U dy
δ
( 2.9)
0
for turbulent boundary layers (due to their thickness) there are a
lot of experimental results of sufficient accuracy
⇓
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
velocity distribution across the layer may be described by a power
law of the following form
U  y
= 
U∞ δ 
n
( 2.18)
the value of n varies from 1/5 (for conditions near the transition
from laminar flow) to 1/7 (for fully developed turbulent
boundary layer)
putting the above formula into the equation (2.9) and integrating
we have
7
F = ρU ∞2 δ
( 2.19)
72
As for laminar boundary layer the drag force F equals to the
summation of the varying stresses τ0 all along the surface from
the leading edge
F = ∫ τ 0 dx =
x0
7
ρ U ∞2 δ
(2.20)
72
0
The above relation ignores the laminar part of the boundary
layer. To solve the problem we have to eliminate stress on the
surface. Shear stress τ0 cannot be determined from Newtonian
law
dU
τ =µ
( 2.1)
dy
So τ0 is taken from experiment, e.g. for flow in circular pipe
which is wholly occupied by boundary layer
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
Flow within a pipe may be regarded as a boundary layer on a flat
plate which has been wrapped round the axis (the outer edge)
It is easy to measure τ0 since the friction drag force causes
measurable pressure drop along the pipe
∆p ~ F ~ τ 0
shear stress for the pipe flow
τ0 =
2 

0.023 ρ U max
(2.21)
4

U
⋅
r
 max 
( 2.22)
and U max = U ∞
(2.23)
ν
1
for boundary layer in flat plate
r =δ
putting the above into the integral formula (2.20) for drag force we
have
 ν
7
 dx
ρU ∞2 δ = ∫ 0.023 ρ U ∞2 
72
U
δ
 ∞ 
0
x0
1
4
( 2.24)
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Advanced Fluid
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integration gives
21
2. F ri c t i onal D rag F orc es i n
Boundary Layer
 U ∞ x0 

= 0.376 
x0
 ν 
δ
−
1
5
(2.25)
experimental verification of the resulting formula is not easy by
measurement of boundary layer thickness (as for laminar
conditions), so it is expressed in the form
 U ∞ x0 

c f = 0.073 

 ν 
−
1
5
(2.26)
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2. F ri c t i onal D rag F orc es i n
Boundary Layer
formula (2.26) may be easily verified
The results of experimental studies are in very good agreement
with theoretical predictions (the derivation is based on the
assumption of velocity distribution taken from experiment)
up to Reynolds number
Re =
U ∞ x0
ν
= 20 ⋅ 10 6
experimental results do not differ from theoretical ones more than
2%
for higher Re errors become unacceptably large; for the previously
mentioned example of the ship
Re =
15.4 ⋅ 300
100 ⋅ 10 − 6
≈ 4 ⋅ 109
error of calculated cf is more than 30%
conclusion:
the theoretical description of the turbulent boundary layers
should be developed