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24 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Contents
:

1/7 velocity law;

Equations for the turbulent boundary layer with zero pressure
dp
e
/dx=0)
;

Virtual origin of the boundary layer;

Hydraulically smooth and fully rough flat plates.

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Boundary Layer Introdution

Transition from laminar to turbulent regime:

x

Beginning of the BL

Laminar
flow

Sufficiently long plate :

Re increases

Critical Re

(

5

10
5
)

Transition to
turbulent

very large

decreases

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Boundary Layer Introdution

Turbulent regions of the BL:

Linear sub
-
layer
(no turbulence)
;

Transition layer;

Central region

logaritmic profile zone
(turbulence
not affected by the wall)
;

External zone

(turbulent vortices mixed with non
-
turbulent outside flow)
.

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Law of the wall

Experimental results from the law of the wall

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Law of the wall

Characteristics of the velocity profile
u
*
=f(y
*
)
:

o

Linear, laminar or viscous sub
-
layer

o

Central region

o

Transition layer

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Other approxmations for
u=u(y)

o
Take

for any
y

o
Take

-

less reliable approximation,

but easier to apply;

does not allow to calculate the shear stress in the wall.

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Bases:

V
on Kárman equation:

Note 1: the velocity profile in the BL follows the law of the
wall , but this law has a less convenient form.

Note 2: as we saw in the laminar case, the integral parameters
of the BL are little affected by the shape of the velocity profile

Velocity profile (empirical):

(Flat plates and Re
L

10
7
)

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Shear stress on the wall:

Note: this expression relates

0

with

(still unknown)
.

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

=
7/72

As we saw:

a

Conclusion:

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

7/72=0,0972<0,133 (Laminar BL)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

On the other hand:

Form Factor:

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

Laminar BL => 2,59

The fuller the velocity profile is, closer to 1
the Form Factor is.

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Note:
x
o

is the point where

=

0
. In general we choose
x
o

to
be in the beginning of the turbulent BL.

Von Kármàn Equation:

Equation to

0
:

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

BL evolution on the flat plate:

x
c

x
0

Laminar BL

Turbulent BL

Transtion
zone

(Re
c

5,5

10
5
)

0

c

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Case 1

the section of interest is very far away from the
critical section
(x>>x
c
)
: the BL is assumed to be turbulent
from the beginning of the plate
(x
0
=

0
=
0
)
.

Valid if
L>>x
c
(or
Re
L
>>Re
c
)
. L

is the
plate lenght

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Case 2

the section of interest is not very faraway from the
critical section: the transition zone is not considered

=>

m0

mc

and

x
0
=x
c
.

From the Von Kármán equation

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

a
L
=0,133 (Blasius)

a
T
=7/72

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Virtual origin of the turbulent BL:
x
v

x
v

x
c
=x
o

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

Would be as if the BL started turbulent from
x
v

to reache

0

in
x
0
.

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Case 2: calculation of the drag on the plate
.

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

x
v

x
c
=x
o

2004

Fluid Mechanics II Teacher António Sarmento
-

DEM/IST

Correlations for higher Re:

for Re

10
9

for 3

10
6

Re

10
9

Turbulent Boundary Layer on a flat plate
(
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

Hidraulically smooth plates if

All the contents studied before are for smooth plates

Turbulent Boundary Layer

Hidraulicaly fully rough plates if

2004

Fluid Mechanics II Teacher António Sarmento
-

DEM/IST

Contents
:

1/7 Law of velocities;

Turbulent boundary layer expressions with
dp
e
/dx

null above
a flat plate;

Virtual origin of the boundary layer;

Hydraulically smooth and fully rough plates.

Turbulent Boundary Layer on a flat
plate (
dp
e
/
dx
=0)

2004

Fluid Mechanics II Teacher António Sarmento
-

DEM/IST

Sources
:

Sabersky

Fluid Flow: 8.9

White

Fluid Mechanics: 7.4

Turbulent Boundary Layer on a flat
plate (
dp
e
/
dx
=0)

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

A plate is 6 m long and 3 m wide and is immersed in a water
flow (

=1000 kg/m3,

=1,13

10
-
6

m
2
/s) with na undisturbed
velocity of 6 m/s parallel to the plate.
Re
c
=10
6
. Compute:

a) The thickness of the BL at
x
=0,25 m;

b) The thickness of the BL at
x
=1,9 m;

c) The total drag on the plate;

d) The maximum roughness on the plate for it to be hydraulically
smooth.

Exercise

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

L
= 6 m;
b
=3 m;

=1000 kg/m3;

=1,13

10
-
6

m
2
/s;
U
= 6 m/s;
Re
c
=10
6
.

a) Thickness of the BL at
x
1
=0,25 m?

Exercise: solution

If we had addmited that the BL grew turbulent from the beginning:

In this case, the result would
be significantly different

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

L
= 6 m;
b
=3 m;

=1000 kg/m3;

=1,13

10
-
6

m
2
/s;
U
= 6 m/s;
Re
c
=10
6
.

b) Thickness of the BL at
x
2
=1,9 m?

Exercise: solution

If we had addmited that the BL grew turbulent from the beginning:

In this case, the result would
have a much smaller
difference

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

L
= 6 m;
b
=3 m;

=1000 kg/m3;

=1,13

10
-
6

m
2
/s;
U
= 6 m/s;
Re
c
=10
6
.

c) Total drag on the plate?

Exercise: solution

For a 1/7 velocity law =>
a
=7/72

=>

If we had addmited that the BL grew turbulent from the beginning:

Difference of 2,5%

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

L
= 6 m;
b
=3 m;

=1000 kg/m3;

=1,13

10
-
6

m
2
/s;
U
= 6 m/s;
Re
c
=10
6
.

c) Total drag on the plate?

Exercise: solution

Difference between computing
D

taking into
account the laminar BL or assuming turbulent

2004

Fluid Mechanics II Prof. António Sarmento
-

DEM/IST

L
= 6 m;
b
=3 m;

=1000 kg/m3;

=1,13

10
-
6

m
2
/s;
U
= 6 m/s;
Re
c
=10
6
.

d) Maximum roughness on the plate to be hidraulically smooth?

Exercise: solution

It is necessary that:

with

Where
is

0
bigger?

In the beginning of the turbulent BL