Download BE105_20_fluid_dynamics_II

Lecture #20: Fluid Dynamics II
8. vorticity
translation
translation
rotation
velocity, u
(need way
to describe
rotation)
cylinder
spinning
in static
tank
w
2a
r
ut
Conservation of
angular momentum:
k = m r2 w
w = ut / r
k = m ut r (but m is constant)
k = ut r
ut = k / r (hyperbola)
w
ut = w r
r
ut
ut
ut = k / r
2a
a
r
rotation
motion
irrotational motion
is like pedals
of bicycle
ut
rotational motion
is like wheels
of bicycle
irrotation
motion
-a
a
r
vorticity is a measure of the rotational flow within a fluid,
i.e. when the fluid is acting like wheels, not pedals:
velocity, u(x,y)
x
y
ux
uy
u(x,y)
circulation, G
vorticity, w(x,y)
Duy
wz =
Dx
Dux
Dy
G=Sw
Drag
Force sensor
u
Force ~ d/dt (momentum)
~ (mass flux) x (velocity)
~rSuxu
~ r S u2
S
Force = ½ CD r S u2
CD = drag coefficient
Alternatively, from Bernoulli Equation,
C = H (total head) = ½ r u2 + P
dynamic
pressure
static
pressure
Force
Force coefficient = Dynamic
pressure
CD =
x area
Force
½ r u2 S
Drag coefficient is a dimensionless
measure of to what degree an object
removes energy from a flow.
Depends only on Reynolds number.
Why do some things create high drag and some things create low drag?
high velocity
low pressure
low velocity
high pressure
Pressure sums to zero
1
pressure
½ r u2
q
+
_
-1
q
‘Ideal’ flow around an object would generate
no net force (i.e. zero drag).
= D’Alembert’s paradox
+
wake
Can’t recover the
pressure on
downstream side of
the sphere
high velocity
low pressure
high velocity
low pressure
1
pressure
½ r u2
q
ideal
+
_
real
-1
q
low velocity
high pressure
wake
high velocity
low pressure
high velocity
low pressure
partial recovery on
downstream side
1
gentle
deceleration
q
P
ideal
+
_
-1
q
low velocity
high pressure
Ergo
‘streamlined’
shape
+
real
u
Drag
Drag has two physically different sources:
1) Pressure drag: asymmetry of upstream and downstream flows.
Caused by the inertial properties of fluid
Drag ~ r u2 S
2) Skin Friction: boundary layer effects
Caused by the viscous properties of fluid
Drag ~ m u S / L
Ratio of these two components
is exactly the Reynolds number:
Re =
r u2 S
muS/L
=uL/n
u
Drag
S
Drag = ½ CD r S u2
CD = 2 Drag / r S u2
Re =
L
100
intermediate
high (laminar)
high (turbulent)
10
low
CD
1
0.1
0.1
1
106
10
Reynolds number
r u2 S
muS/L
=uL/n
u
Drag
Drag
u
vs
Force coefficient, CD
Re
106
103
101
100
perpendicular
1
1
1.9
9.2
parallel
0.055
0.042
1.1
6.2
ratio
740
24
1.7
1.5
Conclusion: Shape (e.g. streamlining) is less important at low Re.