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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 gradient (dpe/dx=0);
– Virtual origin of the boundary layer;
– Hydraulically smooth and fully rough flat plates.
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Boundary Layer Introdution Transition from laminar to turbulent regime:
Ux
v
forcesiscosity
forces inertiaRe x – Distance to the leading edge
• Beginning of the BL 0x 0Re Laminar flow
•Sufficiently long plate : Re increases
Critical Re(5105)
Transition to turbulent
00 yyu very large
00 yyu 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
yu
y
u
uu
Experimental results from the law of the wall
0u
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-layer5* y ** yu
o Central region30* y 5.5log75.5 *10
* yu
o Transition layer530 * y
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Other approxmations for u=u(y)
o Take for any y5.5log75.5 *10
* yu
o Take - less reliable approximation, but easier to apply;
does not allow to calculate the shear stress in the wall.
7/1** 7.8 yu
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Bases:
–Von Kárman equation: mdx
dU 2
0
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): 71
7,8
yu
u
u
(Flat plates and ReL107)
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Shear stress on the wall:
71
7,8
yu
u
u
412
0 0227,0
UU
Note: this expression relates 0 with (still unknown).
y71
7,8
u
u
U
0u
81
1506,0
UUu
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
=7/72
71
7,8
yu
u
u y71
7,8
u
u
U
As we saw:
71
y
U
u
yd
U
u
U
um
1
0
1
a
am Conclusion:
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
7/72=0,0972<0,133 (Laminar BL)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
On the other hand:71
y
U
u
yd
U
ud
1
0
1
Form Factor:
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
8
1
d
29,1*
m
d 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: xo is the point where =0. In general we choose xo to be in the beginning of the turbulent BL.
Von Kármàn Equation: mdx
dU 2
0 dx
daU
2
Equation to 0:41
20 0227,0
UU
5441
0450 0284,0
Ua
xx
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
BL evolution on the flat plate:
xc x0
Laminar BL Turbulent BL
Transtion zone
cc Ux Re
(Rec5,5105) c
c
c xRe
5
0
c
Turbulent Boundary Layer on a flat plate (dpe/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>>xc): the BL is assumed to be turbulent from the beginning of the plate (x0=0=0).
5154
058,0
Ua
x
412
0 0227,0
UU
51
2 Re116,0
21
LD
a
LU
DC
2
51
0 Re0463,0 U
a
x
Valid if L>>xc (or ReL>>Rec). L is the plate lenght
Turbulent Boundary Layer on a flat plate (dpe/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
=> m0mc and x0=xc.
dx
dc m
f
2
From the Von Kármán equation
002 mcmtrans UD
mcm 0
cT
L
a
a 0
am
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
aL=0,133 (Blasius)aT=7/72
dxxcUDx
x
ftrans
c
0
2
2
1
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Virtual origin of the turbulent BL: xv
xv xc=xo
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
5154
058,0
Ua
x 41
4501,35
Uaxx cv
Would be as if the BL started turbulent from xv to reache 0 in x0.
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Case 2: calculation of the drag on the plate.
TL DDD
dxxdxxc
vv
x
x
L
x 00
BlasiusDc CxU 2
2
1
51
2
Re116,0
2
1
vxLv
axLU
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
51
2
Re116,0
2
1
vc xxvc
axxU
xv xc=xo
2004 Fluid Mechanics II Teacher António Sarmento - DEM/IST
Correlations for higher Re:
58,210 Relog
455,0
L
DC for Re109
LL
DCRe
60
88,1Relog
0776,02
10
for 3106 Re109
Turbulent Boundary Layer on a flat plate (dpe/dx=0)
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
Hidraulically smooth plates if 5
us
167,0
024,0
LCD
All the contents studied before are for smooth plates
Turbulent Boundary Layer
Hidraulicaly fully rough plates if 80
us
2004 Fluid Mechanics II Teacher António Sarmento - DEM/IST
Contents:– 1/7 Law of velocities;
– Turbulent boundary layer expressions with dpe/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 (dpe/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 (dpe/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, =1,1310-6 m2/s) with na undisturbed velocity of 6 m/s parallel to the plate. Rec=106. 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; =1,1310-6 m2/s; U= 6 m/s; Rec=106.
a) Thickness of the BL at x1=0,25 m?
Exercise: solution
610Re Uxc
c m 188,00 xxc m 00094,0Re5 ccc x
m 00129,0727
133,00 c
T
Lc a
a
If we had addmited that the BL grew turbulent from the beginning:
5441
014501 0284,0
Ua
xx
m 0056,0058,05154
11
Ua
x
m 0027,0
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; =1,1310-6 m2/s; U= 6 m/s; Rec=106.
b) Thickness of the BL at x2=1,9 m?
Exercise: solution
610Re Uxc
c m 188,00 xxc m 00094,0Re5 ccc x
m 00129,0727
133,00 c
T
Lc a
a
If we had addmited that the BL grew turbulent from the beginning:
5441
024502 0284,0
Ua
xx
m 0283,0058,05154
21
Ua
x
m 0264,0
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; =1,1310-6 m2/s; U= 6 m/s; Rec=106.
c) Total drag on the plate?
Exercise: solution
ba
xxa
xLxUDvcv xx
vcxL
v
c
c
5
1
5
1
2
ReRe116,0
Re
328,1
2
1
4145
01,35
Uaxx cv m 161,0For a 1/7 velocity law => a=7/72 =>
N 5,1452
If we had addmited that the BL grew turbulent from the beginning:
N 2,1489Re2
1116,0
5
1
2
b
aLUD
L
Difference of 2,5%
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106.
c) Total drag on the plate?
Exercise: solution
L % dif.2 8,23 5,14 3,85 36 2,5
10 1,5100 0,16
Difference between computing D taking into account the laminar BL or assuming turbulent
from the leading edge.
2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST
L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106.
d) Maximum roughness on the plate to be hidraulically smooth?
Exercise: solution
5
us
0u
41
0
20 0227,0
UU
It is necessary that: with
Where is 0 bigger? In the beginning of the turbulent BL
m/s 3,0
Pa 86,89
mm 0188,0
u
5