(5)
As described in Dr. Sepri’s Note, the conditions leading to a similar solution
are:
(i)
B.C need to be similar
→
(
ρ
v
)
0
restricted in form
(ii)
I.C is similar, that is, can’t accept an arbitrary
f
0
(
η
)
Advanced Fluid Mechanics
Chapter 521
(iii)
External pressure gradient must comply with
β
= const.
(iv)
Density profile is similar.
As
m
= 0.091,
0
=
∂
∂
y
y
u
=
U
∞
0
=
η
ηη
f
= 0, therefore, the separation occurs. We
conclude that
If
m
> 0
0
>
dX
dU
e
,
⇒
dx
dp
e
=
－
ρ
U
e
dX
dU
e
<0
⇒
accelerating flow
If
m
< 0 (but
－
1/2 <
m
)
dX
dU
e
< 0,
⇒
dx
dp
e
> 0
⇒
decelerating flow
In this course, the flow is taken as incompressible
; therefore, the flow is a accelerated
as it past a wedge and decelerated as it past a corner.
0
>
dx
du
e
nozzle)
(subsonic
0
<
dx
du
e
diffuser)
(subsonic
However, as the flow is compressible
, it will be different, e.g.
1
M
2
M
shock
)
M
(M
2
1
>
expansion
1
M
2
M
)
M
(M
2
1
<
2
1
T
T
>
2
1
T
T
but
<
diffser)
c
(supersoni
nozzle)
c
(supersoni
Since M=
RT
V
γ
→
V=M
RT
γ
It hard to tell whether V
1
>V
2
or V
1
<V
2
But normally V
1
<V
2
Advanced Fluid Mechanics
Chapter 522
(In xy plane, no gravity force acting)
0
>
m
0
<
m
.
0
→
u
.
∞
→
u
)
(
X
)
(
V
∞
U
0
>
dx
dU
e
0
<
dx
dU
e
seperation
,
0
0
=
∂
∂
=
y
y
u
e
U
1
2
3
4
5
.
.
For curve 14, 24 & 34, the velocity profile at 5 will not be the same.
(Cannot determine the
dx
dU
e
>0 or
dx
dU
e
<0 from the
slope of the local surface w.r.t the freestream direction.
It normally further upstream as shown)
Advanced Fluid Mechanics
Chapter 523
Problem:
Show that (
δ
*
/
τ
w
) d
p
/d
x
represents the ratio of pressure force to wall friction force
in the fluid in a boundary layer. Show that it is constant for any of the FalknerSkan
wedge flows. (J. schetz. P. 92, prob. 4.6)
Advanced Fluid Mechanics
Chapter 524
5.4 Flow in the wake of Flat Plate at zero incidence
Preface: the B.L. equation can be applied not only in the region near a solid wall, but
also in a region where the influence of friction is dominating exists in the
interior of a fluid. Such a case occurs when two layers of fluid with different
velocities meet, such as:
wake and jet
.
Consider the flow in the wake of a flat plate at zero incidences
l
y
∞
U
∞
U
∞
U
δ
1
A
A
1
B
B
x
h
surface
control
≠
=
∞
0
v
U
u
)
,
(
y
x
u
Want to find out: (1) the velocity profile in the wake
.
Assume: d
p
/d
x
= 0
For the mass flow rate: (
Σ
= 0)
At
AA
1
section =
ρ
∫
h
0
U
∞
d
y
(entering)
At
BB
1
section =

ρ
∫
h
0
u
d
y
(leaving)
At
AB
section = 0
At
A
1
B
1
section =

ρ
∫
h
0
(
U
∞

u
)d
y
←
(To keep
Σ
mass
= 0)
Actually along
A
1

B
1
, the
u
=
U
∞
, the mass must be more
out to satisfy continuity
m
&
=

ρ
∫
B
A
v
(
x
,
h
)d
x
Advanced Fluid Mechanics
Chapter 525
For the
x
momentum floe rate:
At
AA
1
section =
ρ
∫
h
0
U
∞
2
d
y
(entering)
At
BB
1
section =

ρ
∫
h
0
u
2
d
y
(leaving)
At
AB
section = 0
At
A
1
B
1
section =
AB
m
&
U
∞
=
U
∞
[

ρ
∫
h
0
(
U
∞

u
)d
y
] =

ρ
∫
h
0
U
∞
(
U
∞

u
)d
y
Drag on the upper surface =
Σ
Rate of change of
x
momentum in A
1
B
1
BA
=
ρ
∫
h
0
u
(
U
∞

u
)d
y
(5.17)
In order to calculate the velocity profile, let us first assume a velocity defect
u
1
(
x, y
)
as
u
1
(
x, y
) =
U
∞

u
(x, y)
(5.18)
and
u
1
<<
U
∞
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 Spring '14
 AndreaVacca(P)
 Fluid Dynamics, Fluid Mechanics, The Land