Embed Size (px)
Citation preview
AE301 Aerodynamics I
UNIT B: Theory of Aerodynamics
ROAD MAP . . .
B-1: Mathematics for Aerodynamics
B-2: Flow Field Representations
B-3: Potential Flow Analysis
B-4: Applications of Potential Flow Analysis
AE301 Aerodynamics I
Unit B-4: List of Subjects
Lifting Flow over a Cylinder
Real Flow over a Spinning Cylinder
Flow around an Airfoil
Real Flow: Starting Vortex
Kutta Condition
LIFTING FLOW OVER A CYLINDER
Combining nonlifting flow over a cylinder + vortex
=> lifting flow over a cylinder
(this is a similar flow filed around “spinning” circular cylinder)
In 2-D polar coordinate system:
2
2( sin ) 1 ln
2
R rV r
r R
= − +
, where R is the radius of the cylinder
The velocity field can be found by: 2
2
11 cosr
RV V
r r
= = −
2
21 sin
2
RV V
r r r
= − = − + −
Unit B-4Page 1 of 9
Lifting Flow over a Cylinder (1)
Nonlifting flow
over a cylinder Vortex of strength located
at the center of the cylinder
STAGNATION POINTS OF THE FLOW FIELD (1)
The stagnation points can be located by setting =V 0 , such that: 2
20 1 cosr
RV V
r
= = −
eqn. 1
2
20 1 sin
2
RV V
r r
= = − + −
eqn. 2
STAGNATION POINTS OF THE FLOW FIELD (2)
The first condition (eqn. 1) provides the following two solutions:
r R= (stagnation points are located on the surface of the cylinder)
or
2
= (stagnation points are located along the vertical axis)
If we combine these two solutions with the second condition (eqn. 2):
For r R= => 2 sin 02
VR
− − = => 1sin
4 V R
−
= −
For r R => ?
Unit B-4Page 2 of 9
Lifting Flow over a Cylinder (2)
STAGNATION POINTS OF THE FLOW FIELD (3)
For the case of r R= (stagnation points are on the surface of the cylinder)
=> 1sin4 V R
−
= −
There are two possible stagnation point solutions for a lifting flow over a circular cylinder. These are
based solely on the strength of the vortex (circulation):
(a) If 4 V R : two stagnation points (both at r R= )
(b) If 4 V R = : two stagnation points become one stagnation point ( r R= , 2 = − )
For the case of r R (stagnation points are NOT on the surface of the cylinder)
(c) If 4 V R : two stagnation points
• One inside: (this is trivial solution) and
• One outside of the cylinder, 2 = − )
From Kutta-Joukowski theorem, the lift force (per unit depth) of a spinning circular cylinder is related
to the circulation, such that:
'L V =
Unit B-4Page 3 of 9
Lifting Flow over a Cylinder (3)
Unit B-4Page 4 of 9
Class Example Problem B-4-1
Related Subjects . . . “Lifting Flow over a Cylinder”
Consider the lifting flow over a circular cylinder. Based on the potential flow
analysis, derive the relationship between lift coefficient (cl) and circulation ().
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Unit B-4Page 5 of 9
Real Flow over a Spinning Cylinder
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
Unit B-4Page 6 of 9
Flow around an Airfoil
'L V = V
C
L’
Circulation result by potential
flow analysis solution
1
Circulation should be “adjusted” based
on an “additional condition”
1 => 2
POTENTIAL FLOW FIELD AROUND AN AIRFOIL
Based on the potential flow analysis of a lifting flow around the circular cylinder. Let us now replace
the circular cylinder with an airfoil. This is the starting point of the potential flow analysis of a flow
over an airfoil.
• As we know from the analysis of a lifting flow around the circular cylinder, there are infinite
numbers of “valid” theoretical solutions, corresponding to infinite choices of circulation
Basically, the location(s) of stagnation points depend solely on the amount of circulation
• Therefore, a potential flow theory based flow field analysis over an airfoil, as we simplify many
“real” flow field characteristics (especially viscosity), will result in totally non-realistic solution . . .
• However, a given airfoil at a given angle of attack should produce a single value of lift that should
be more realistic against “real” flow field. We need an additional condition that fixes the amount
of circulation for a given airfoil at a given angle of attack . . .
Unit B-4Page 6 of 9
Flow around an Airfoil
'L V = V
C
L’
Unit B-4Page 6 of 9
Flow around an Airfoil
'L V = V
C
L’
!
?
STARTING VORTEX
Let us learn from “the nature” of the real flow field. Experimental observation for the development of
the flow field around an airfoil, which is set into motion from an initial state of rest, provides interesting
insight into the nature of real flow around an airfoil.
What happens, when the flow field is established around an airfoil . . . this is a classical “starting
vortex” formulation due to an “impulse start” of the real flow field.
(a) The flow tries to curl around the sharp trailing edge from the bottom surface to the top surface.
(b) The stagnation point on the upper surface starts moving toward the trailing edge.
(c) The flow is smoothly leaving the top and bottom surfaces of the airfoil at the trailing edge. Then,
the flow field achieves the steady-state condition.
Conclusion: an excess amount of circulation is “removed” from the circulation of the flow field so that
the flow at the trailing edge leaves smoothly. Can we somehow apply this as an additional condition
to the potential flow analysis over an airfoil?
Unit B-4Page 7 of 9
Real Flow: Starting Vortex
(a)
(b)
(c)
Unit B-4Page 6 of 9
Flow around an Airfoil
'L V = V
C
L’
Unit B-4Page 6 of 9
Flow around an Airfoil
'L V = V
C
L’
Circulation result by potential
flow analysis solution
1
Circulation “adjusted” based on
the “Kutta Condition”
1 => 2 (2 = 1 – shed “starting vortex”)
At the trailing edge, the flow
MUST leave smoothly
STARTING VORTEX AND THE KUTTA CONDITION
Starting vortex is the “nature’s own way” to adjust the flow field around an airfoil.
By shedding appropriate amount of starting vortex, the circulation of the flow field is “adjusted” so that
the flow at the trailing edge can leave smoothly. This is called, the Kutta condition.
For a trailing edge with finite angle:
1 2 0V V= = (thus, a stagnation point is formed at the trailing edge)
For a “cusped” trailing edge:
1 2V V= (but not 1 2 0V V= = )
• Potential flow analysis WITHOUT the Kutta Condition
• Potential flow analysis WITH the Kutta Condition
Unit B-4Page 8 of 9
Kutta Condition
Unit B-4Page 9 of 9
Class Example Problem B-4-2
Related Subjects . . . “Flow around an Airfoil”
Consider an airplane with a NACA airfoil (with
chord length of 1 m). The airplane’s airspeed is
50 m/s at the altitude of 4 km. The corresponding
lift coefficient of the airfoil is cl = 0.8.
(a) Determine the lift per unit span.
(b) Calculate the circulation around the airfoil.
+
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
_________________________________________________________
