2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
von Kárman Equation for flat plates (dpe/dx≠0)
For laminar or turbulent flows: in the turbulent case we take time-average velocity and pressure.
Procedure: Mass and momentum balance of the following control volume:
dx
dxdx
d
x x+dx
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
dx
xmdxxm
m
dxudydx
dudym
x
dxx
00
Mass balance: dxxxVC mmmMdt
d
Steady Flow
o Flow rate: xmx
x udym
0
o Flow rate : dxxm
dxudydx
d
0
von Kárman Equation forflat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
dx
xmdxxm
m
dxudydx
dUmUq
xqm
0
o x Momentum flow rate through y=δ:
dxudydx
dm
0
Mass balance :
von Kárman Equation for flat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
dxxqmxqxqmxq
qmxq
VCxx
VC
xdxqmxqmxqmxx Fqqq
dt
dK
o Difference : xqmxdxxqmx qq
dxdyudx
d
0
2
x momentum balance:
Steady flow
von Kárman Equation for flat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
xqmxqdxxqmxq
qmxq
qmxqmxdxqmxx qqqF
xxVC
x momentum balance:
dxdyudx
d
0
2 dxudydx
dU
0
von Kárman Equation for flat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
dxddppddpppFVCx 02
1
Forces along x:
p+dpp
p+1/2dp
τ0
dxdx
dpF
VCx
0
dx
dUU
dx
dpe
von Kárman Equation for flat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
00
0 dyuUdx
dUudyuU
x
Final result:
Using the definition of d and δm:
dx
dUUU
dx
ddm 2
0
von Kárman Equation for flat plates (dpe/dx≠0)
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
When dpe/dx=0 (dU/dx=0):
dx
dU m 2
0 dx
d
Uc m
f
2
21 2
0
von Kárman Equation for flat plates (dpe/dx≠0)
When dpe/dx=0 (dU/dx=0) we have m=a
(a takes different values in laminar and turbulent flow):
dx
dac f
2 Boundary layer grows faster
when Cf is higher
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Approximate solutions for laminar boundary layer for dpe/dx=0
Blasius solution shows that fU
u
xx
yRewith
andxx Re
5
yx
x
y5
5
y
fU
u
a – constant along the BD
1
0
1
yd
U
u
U
um
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Approximate solutions for laminar boundary layer for dpe/dx=0
von Kárman Equation:dx
dU m 2
0
but
0
0
y
y
u
β - constant
dx
daU
2
0
yyd
UudU
U
dx
daU
U
2xax Re
12 00
Integrating
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Approximate solutions for laminar boundary layer for dpe/dx=0
Remark: a and β depend on the velocity profile, however δ/x, cf and CD do not vary much with profile shape
xf
ac
Re
2
xax Re
12
U0
We have
and
LD
aC
Re
22
UUc f
2
220
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Approximate profiles for Laminar BD
a β c f
Linear 0,167 1
Parabolic 0,133 2
Sinusoidal 0,137
Blasius
0dxdp e
y
U
u
2
2
yy
U
u
y
U
u
2sin
xRe
789,4
xRe
484,5
xRe
461,3
xRe
5
2
xRe
578,0
xRe
729,0
xRe
656,0
xRe
664,0
x
Approximate solutions for laminar boundary layer for dpe/dx=0
ma 0 yydUud
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Contents:– von Kármàn Equation;
– Simplification for ;
– Approximate solutions for laminar Boundary Layers with
zero pressure gradient .
Blasius Solution for Laminar Boundary Layer Equation over a flat plate with dpe/dx=0
0dx
dpe
0dx
dpe
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Recommended Study Elements:– Sabersky – Fluid Flow: 8.6, 8.7
– White – Fluid Mechanics: 7.3, 7.4
Von Kárman Equation for a flat plate
2004 Mecânica dos Fluidos II Prof. António Sarmento - DEM/IST
Problem on the Von Kármàn Equation