Boundary layer Equations

Contents:
– Boundary Layer Equations;
– Boundary Layer Separation;
– Effect of londitudinal pressure gradient on boundary layer
evolution
– Blasius Solution
– Integral parameters: Displacement thickness and momentum
thickness
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Laminar Thin Boundary Layer Equations
(d<<x) over flat plate
flow, constant r and m.
p y  0 Streamlines slightly divergent
p x  dpe dx
 Steady

 2D
Navier-Stokes Equations along x direction:
u
u
1 p   2u  2u 
u v

   2  2 
x
y
r x  x y 
dpe dx
2004
Mecânica dos Fluidos II
 2u
Compared with
y 2
Prof. António Sarmento - DEM/IST
Laminar Thin Boundary Layer Equations
(d<<x) over flat plate
 Laminar
thin boundary layer equations (d<<x) for
flat plates
u
u
1 dpe
 2u
u v

 2
x
y
r dx
y
pe external pressure, can be calculated with Bernoulli’s Equation
as there are no viscous effects outside the Boundary Layer
Note 1. The plate is considered flat if d is lower then the local
curvature radius
Note 2. At the separation point, the BD grows a lot and is no
longer thin
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Turbulent Thin Boundary Layer
Equations (d<<x) over flat plate
 2D
Thin Turbulent Boundary Layer Equation
(d<<x) to flat plates:
u
u
1 dpe
 2u  uu uv uw 

u
v

 2  


x
y
r dx
y  x
y
z 
0
0
Resulting from
Reynolds Tensions
(note the w term)
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Boundary
Layer Separation: reversal of the flow by
the action of an adverse pressure gradient (pressure
increases in flow’s direction) + viscous effects
mfm: BL / Separation / Flow
over edges and blunt bodies
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Boundary
layer separation: reversal of the flow by the
action of an adverse pressure gradient (pressure increases
in flow’s direction) + viscous effects
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Bidimensional
(2D) Thin Boundary Layer (d<<x)
Equations to flat plates:
u
u
1 dpe
 2u
u v

 2
x
y
r dx
y

Close to the wall (y=0)
u=v=0 :
  2u 
1 dpe
 2  
 y  y 0 m dx

2004
Similar results to turbulent boundary layer - close to
the wall there is laminar/linear sub-layer region.
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Outside
 Close
 2u
0
2
y
Boundary layer:
to the wall (y=0)
u=v=0 :
  2u 
1 dpe
 2  
 y  y 0 m dx
 The
external pressure gradient can be:
o
dpe/dx=0 <–> U0 constant (Paralell outer streamlines):
o
dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):
dpe/dx<0 <–> U0 increases (Convergent outer streamlines):
o
2004
Same sign
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Zero
pressure gradient:
dpe/dx=0 <–> U0 constant (Paralell outer streamlines):
  2u 
 2   0
 y  y d
y
u
Curvature of velocity
profile is constant
No separation of boundary
layer
  2u 
 2   0
 y  y 0
Inflection point at the wall
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Favourable
pressure gradient:
dpe/dx<0 <–> U0 increases (Convergent outer streamlines):
  2u 
 2   0
 y  y d
y
No boundary layer
separation
  2u 
 2   0
 y  y 0
2004
Curvature of velocity profile
remains constant
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Adverse
pressure gradient:
dpe/dx>0 <–> U0 decreases (Divergent outer streamlines):
  2u 
 2   0
 y  y d
y
Boundary layer
Separation can occur
  2u 
 2   0
 y  y 0
P.I.
Curvature of velocity
profile can change
Separated Boundary Layer
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation
 Sum
of viscous forces:
 2u
 2
y
Become zero with velocity
Can not cause by itself the fluid stagnation (and
the separation of Boundary Layer)
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Separation

Effect of longitudinal pressure gradient:
dpe
0
dx
(Convergent outer
streamlines)
dpe
 0 (Divergent outer
streamlines)
dx
Viscous effects retarded
Viscous effects reinforced
u
1 1 dpe

 ...
Fuller velocity
x
u r dx
Less full velocity
profiles
profiles
Decreases BL growth
2004
Mecânica dos Fluidos II
Increases BL growths
Prof. António Sarmento - DEM/IST
Boundary Layer Separation

Effect of longitudinal pressure gradient:
Fuller velocity
profiles
u
1 1 dpe

 ...
x
u r dx
Less full velocity
profiles
Decreases BL growth
Increases BL growths
Fuller velocity profiles – more resistant
to adverse pressure gradients
Turbulent flows (fuller profiles)- more resistant to
adverse pressure gradients
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Boundary Layer Sepaation
Longitudinal and intense adverse pressure gradient does
not cause separation
=> there’s not viscous forces
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0
 Bidimensional
(2D) Thin Boundary Layer (d<<x)
Equations to flat plates:
u
u
 2u
u v
 2
x
y
y
u v

0
x y
 Boundary
2004
Condition: y=0
y=∞
Mecânica dos Fluidos II
u=v=0
u=U
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0
 Blasius
hypothesis:
u
 f   with
U

Ay
xn
The introdution of η corresponds to recognize that the
nondimension velocity profile is stabilized.
A and n are unknowns
 A 
 Remark:
e
 n 
y x
y
2004
Mecânica dos Fluidos II

nA
n
  n 1 y   
x
x
x
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Procedure:
oUsing current function:

u
y

v
x
o Replace u/U=f(η) e vatthe
  x boundary layer equation,
choose n such that the resulting equation does not depend on
x and A in order to simplify the equation.
.
o Remark:
n
n
n
d  
x
x
x

u
 Uf  
 U
f  d

d y y
A
A
A
F  
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0
n
x
 From:   U
F  
A

results:

 UF  
o u
y
u
nU

F  
o
x
x
u UA
 n F  
o
y x
 2u UA2
o
 2 n F  
2
y
x
o
2004



U
v
  nx n 1 F    nx n 1 F  
x
A
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0
 We
will obtain:
u
u
 2u
u v
 2
x
y
y
Unx2 n1
F  
FF   0
2
A
o Making n=1/2 and A  U  the equation comes:
2F    F  F    0 with
 Boundary
ux,0  0
vx,0  0
ux,   U
2004
U
y
x
Conditions:
UF 0  0
F   F 0  0
UF   U
Mecânica dos Fluidos II
F 0  0
F 0  0
F   1
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Graphical Solution:
U u
u  y x U  F F  
 F  
1,2
U
0,8
0,4
0
0
2004
2
0
1
2
3
4
5
6
7
8
Mecânica dos Fluidos II
0
0,3298
0,6298
0,8461

F0,9555
0,9916
0,999
6
4
0,999
U
1
y
 
x
0,3321
0,323
0,2668
0,1614
0,0642
0,0059
0,0024
8
0,0002
0,0001
Prof. António Sarmento - DEM/IST
10
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Solution:
U u
 F F  
y
x U
0
1
2
3
4
5
6
7
8
2004
0
0,3298
0,6298
0,8461
0,9555
0,9916
0,999
0,999
1
0,3321
0,323
0,2668
0,1614
0,0642
0,0059
0,0024
0,0002
0,0001
Mecânica dos Fluidos II
oShear stress at the wall
 u 
 0  m    mU U F 0
 y  y 0
x
o Friction coefficent
 0  2 F 0  0,664
c f  1 2 Ux
Re x
rU
2

Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Solution:
U u
 F F  
y
x U
0
1
2
3
4
5
6
7
8
0
0,3298
0,6298
0,8461
0,9555
0,9916
0,999
0,999
1
0,3321
0,323
0,2668
0,1614
0,0642
0,0059
0,0024
0,0002
0,0001
o Drag
L
D    0 dx 
o
1
U
mU
F 0
2
L
o Drag Coefficent
D
1,328
CD 

1
Re L
rU 2 L
2
Re L 
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
UL

Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Solution :
o Boundary layer thickness
U u
 F F  
y
x U
u y  d   0,99U
0
1
2
3
4
5
6
7
8
2004
0
0,3298
0,6298
0,8461
0,9555
0,9916
0,999
0,999
1
0,3321
0,323
0,2668
0,1614
0,0642
0,0059
0,0024
0,0002
0,0001
Mecânica dos Fluidos II
η=5
d
5
5


x
Ux 
Re x
o Shear stress at y=d
 d F 5

 1,8%
 0 F 0
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Displacement thickness:
1
dd 
U
1
*
d  dd 
U
d
 U  u dy
0
d
 U  u dy
Ud d  Ud   udy
0
0
Ideal Fluid flow
rate
Déficit of flow rate due to
velocity reduction at BD
U
d

Mecânica dos Fluidos II
Real Flow
rate
 U  u dy
0
2004

Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Displacement thickness :
1
dd 
U
d
 U  u dy
0
1
dd 
U

 U  u dy
0
d
Ud d  Ud   udy
0
Déficit of flow rate due to
velocity reduction at BD
Ideal Fluid flow
rate
q  U d  d d 
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Real Flow
rate
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Displacement thickness :
1
dd 
U
d
1
dd 
U

 U  u dy
0
d
 U  u dy
1
d d  d   udy
U0
0
Deviation of outer streamlines
Initial deviation of BD
δ δd
q/U
LC
Section where the streamline become part of boundary layer
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Blasius Solution for displacement thickness:
dd
x
ou

com
Re x 
Ux

dd
 0,344
d
δ dd
q/U
2004
1,72
Re x
LC
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Momentum thickness:
1
dm  2
U
1
  dm  2
U
d
 U  u udy

 U  u udy
0
0
d
d
d
d
0
0
0
0
U 2d m  U  udy   u 2 dy
d
2
2
u
dy

U
udy

U
dm


2
2
d  d d  d m 
u
dy

U

 U d  d d 
0
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Momentum flow rate through a section of BD:
d
qqmx   ru 2 dy  rU 2d  rU 2d d  rU 2d m
0
U rUd 
U rUd d 
Momentum flow
rate of uniform
profile
Reduction due to
deficit of flow
rate
2004
Mecânica dos Fluidos II
U rUd m 
Reduction due
to deficit
momentum flow
rate at BD
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Longitudinal momentum balance between the leading
edge and a cross section at x:
 
D  qqmx
x 0
 
 qqmx
x x
rU 2 d  d d 
rU 2d m
rU 2 d  d d  dm
δ dd
d-dd
LC
x
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Blasius Solution to momentum thickness:
dm
0,664

x
Re x
or
2004
with
Re x 
dm
 0.133
d
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Ux

Laminar Boundary Layer Equations

Contents:
– Thin Boundary Layer Equations with Zero Pressure
Gradient;
– Boundary Layer Separation;
– Effect of longitudinal pressure gradient on the evolution of
Boundary Layer
– Blasius Solution
– Local Reynolds Number and Global Reynolds Number
– Integral Parameters: displacement thickness and momentum
thickness
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Blasius Solution to Laminar Boundary Layer
Equation over a flat plate with dpe/dx=0

Recommended study elements:
– Sabersky – Fluid Flow: 8.3, 8.4
– White – Fluid Mechanics: 7.4 (sem método de Thwaites)
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Exercise
Large plate with neglectable thickness, lenght L=2m. Parallel and
non-disturbed air flow. (r=1,2 kg/m3, m=1,810-5 Pa.s) with U=2
m/s. Zero pressure gradient over the flat plate. Transition to turbulent
at Rex=106.
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
U=2m/s
dpe dx  0
L=2m
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
dpe dx  0
U=2m/s
L=2m
a) Find boundary layer thickness d at sections S1 and S2, at distance
x1=0,75 m and x2=1,5 m of the leading edge
Find xc:

5
1
,
8

10
1,2
6
xc  Re x c  10
 7,5m
U
2
Laminar Boundary layer at x1 and x2 – We can apply Blasius
Solution
d
5
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
x

Re x
Exercise
a) Find boundary layer thickness d at sections S1 and S2, at distance
x1=0,75 m and x2=1,5 m of the leading edge
Laminar Boundary layer at x1 and x2 – We can apply Blasius
Solution
d
5
x
2  0,75
5


10
1,5  105
x1  0,75m
Re x1
x2  1,5m
Re x 2  2 105
2004
Mecânica dos Fluidos II
d1 
0,75 5
5
Re x
 0,0119m
10
d 2  0,0168m
Prof. António Sarmento - DEM/IST

Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
dpe dx  0
U=2m/s
y=d(x)
L=2m
b) Check that it is a thin boundary layer.
A: Thin Blayer if d/x<<1:
Whyd/x at 2 is lower than
d/x at 1?
2004
Mecânica dos Fluidos II
0,0119
d 
 0,0159
  
0,75
 x 1
0,0168
d 

 0,0112
 
1,5
 x 2
Prof. António Sarmento - DEM/IST
Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
dpe dx  0
U=2m/s
Streamline
d
y1=?
x1=0,75m
L=2m
x2=1,5m
d) Find the value of y1 at x1 of the streamline passing through the coordinates
x2=1,5 and y2=d.
A: We have the same flow rate between the streamline and the plate at both cross sections
q  U d  d d 
Flow rate through a cross section of BD:
Flow rate through section 2:
Flow rate through section 1:
2004
Mecânica dos Fluidos II
q2  U d  d d 2
y1  d 2  d d 2  d d1
y1
d1
y1
0
0
d1
q1   udy   udy   udy  U d  d d 1 U  y1  d1 
Prof. António Sarmento - DEM/IST
Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
dpe dx  0
U=2m/s
Linha de corrente
y1=?
x1=0,75m
L=2m
d
x2=1,5m
d) Find the value of y1 at x1 of the streamline passing through the coordinates
x2=1,5 and y2=d.
A: We have the same flow rate between the streamline and the plate at both cross
sections
Laminar BD: d d  0,344d
y  d d d
1
2
d2
d1
0,0168m 0,0058m 0,0041m
2004
y1=0,0151m
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
dpe dx  0
U=2m/s
L=2m
e) Find the force per unit leght between sections S1 and S2.
A: There are no other forces applied except that imposed by the resistance (Drag) of plate:
The applied force between the leading edge and the cross section at x is:
Laminar BD:
d m2  0,133d 2  0,00223m
d m1  0,133d1  0,00158m
D0, x  rU 2 d m x
d m  0,133d
Drag force to section 2: D0,2=0,0107N/m
Drag force to section 1: D0,1=0,0076N/m
Drag force between 1 and 2: D1,2=D0,2-D0,1=0,0031N/m
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST
Exercise
r=1,2 kg/m3,
m=1,810-5 Pa.s
(Rex)c =106.
U=2m/s
dpe dx  0
L=2m
f) True or False?: ”Under the conditions of the problem, if the plate
was sufficiently long (L ), the boundary layer would
eventually separate?
False: The BD will separate only with adverse pressure
gradient. The drag forces will decrease with the velocity
over the plate. The drga forces are not able to stop the fluid
flow.
2004
Mecânica dos Fluidos II
Prof. António Sarmento - DEM/IST