uniform flow solution of potential flows 4 superposition s
TRANSCRIPT
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-3: List of Subjects
Problem Solutions? How?
Solution of Potential Flows
Uniform Flow
Source / Sink
Superposition
Doublet
Vortex
Potential Flow Summary
Flow over a Cylinder
Ideal v.s. Real Flow
Real Flow over a Cylinder
POTENTIAL FLOW ANALYSIS: A PURE THEORY
In this unit, we will attempt to solve the governing equation, called Laplace’s equation. This is called,
the potential flow analysis.
In order to analytically solve the governing equation of a flow field, we need to apply assumptions to
simplify the equation.
(1) What are the assumptions made to simplify the equation?
• Steady-state?
• Inviscid?
• No body forces?
• Incompressible?
• Irrotational?
(2) Because of the assumptions made, how your theoretical solution different (or apart) from the actual
(or real) flow field phenomena?
Unit B-3Page 1 of 17
Problem Solutions? How?
ELEMENTARY SOLUTIONS OF LAPLACE’S EQUATION
4 Elementary Flow Solutions:
(1) Uniform Flow
(2) Source/Sink Flow
(3) Doublet Flow
(4) Vortex Flow
Combinations of Elementary Flows:
• Uniform Flow + Source => Flow Around a Half-Rankine Body
• Uniform Flow + Source + Sink => Flow Around a Rankine Oval
• Uniform Flow + Doublet => Nonlifting Flow Around a Circular Cylinder
• Uniform Flow + Doublet + Vortex => Lifting Flow Around a Circular Cylinder
Unit B-3Page 2 of 17
Solution of Potential Flows
1. Solve Laplace’s equation to obtain or . This can be done by superimposing
the elementary solutions of Laplace’s equation.
2. Determine velocity field:
3. Determine pressure distribution:
uy
=
v
x
= −
1rV
r
=
V
r
= −
UNIFORM FLOW
Uniform flow with magnitude V and direction in positive x is the first elementary flow solution of
Laplace’s equation.
In 2-D Cartesian coordinate system:
V x = or V y =
Velocity field (2-D Cartesian) can be found as:
u Vx y
= = =
and 0v
y x
= = − =
In 2-D polar coordinate system:
cosV r = or sinV r =
Velocity field (2-D polar) can be found as: 1
cosrV Vr r
= = =
and
1sinV V
r r
= = − = −
Unit B-3Page 3 of 17
Uniform Flow
SOURCE/SINK FLOW
The source (+) / sink (−) flow is the second elementary flow solution of Laplace’s equation.
Sink / source flows can be characterized by a series of straight streamlines emanating from a single
point. The strength of source / sink ( ) is the volume flow rate (per unit depth).
In 2-D polar coordinate system:
ln2
r
= or
2
=
Velocity field (2-D polar) can be found as: 1
2rV
r r r
= = =
and 1
0Vr r
= = − =
Unit B-3Page 4 of 17
Source / Sink
(+) (–)
Unit B-3Page 5 of 17
Superposition (1)
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FLOW AROUND A RANKINE OVAL
Combining uniform flow + Source + Sink = flow around a Rankine oval
In 2-D Polar coordinate system:
1 2 1 2sin sin ( )2 2 2
V r V r
= + − = + −
There are two stagnation points (A and B) in the flow field. These stagnation points can be found by
setting =V 0 , such that:
2 bOA OB b
V
= = +
The equation of stagnation streamline (by skipping the details of derivation) is 0 = , hence:
( )1 2sin 02
V r
= + − =
Unit B-3Page 6 of 17
Superposition (2)
Unit B-3Page 7 of 17
Class Example Problem B-3-1
Related Subjects . . . “Superposition”
Consider the superposition of a uniform flow of strength and a source of strength .
The stagnation point of this flow (flow around a half-Rankine body) is:
Let and calculate the body surface (r / R) and the pressure coefficient
(Cp) over a given range of as follows:
),2(),( = Vr
(degrees) r / R Cp
30
45
90
135
150
180
/ 2R V =
V
5.236
3.332
1.571
1.111
1.047
1.0
−0.367
−0.514
−0.405
0.463
0.742
1.0
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DOUBLET FLOW
The doublet flow is the third elementary flow solution of Laplace’s equation.
The source ( ) and sink ( − ) pair with same strength at a single point, and the strength of doublet is
defined by: l
In 2-D polar coordinate system:
cos
2 r
= or
sin
2 r
= −
Velocity field (2-D polar) can be found as:
2
1 cos
2rV
r r r
= = = −
and
2
1 sin
2V
r r r
= = − = −
Unit B-3Page 8 of 17
Doublet
VORTEX FLOW
The vortex flow is the fourth elementary flow solution of Laplace’s equation. In 2-D polar coordinate
system:
2
= − or ln
2r
=
Velocity field (2-D polar) can be found as: 1
0rVr r
= = =
and
1
2V
r r r
= = − = −
Streamlines of concentric circles centered around a single point.
CIRCULATION OF THE VORTEX FLOW
The strength of vortex ( ) is called, the “circulation” defined as:
( )S
d = − V s = ??? (NO!)
( ) ( ) ( )2
0
ˆ ˆ 2C
d V e rd e V r
= − = − = − V s
Unit B-3Page 9 of 17
Vortex
+
NONLIFTING FLOW OVER A CYLINDER
Combining uniform flow + Doublet = nonlifting flow over a cylinder
In 2-D polar coordinate: 2
sinsin sin 1
2 2V r V r
r V r
= − = −
Let 2 2R V (R is the radius of the cylinder):
2
2sin 1
RV r
r
= −
The velocity field (in 2-D polar) can be obtained by:
( )2 2
2 2
1 1cos 1 1 cosr
R RV V r V
r r r r
= = − = −
( ) ( )2 2 2
3 2 2
2sin 1 sin 1 sin
R R RV V r V V
r r r r
= − = − + − = − +
STAGNATION POINTS AND STAGNATION STREAMLINE
The stagnation points can be obtained by setting 0=V : 2
21 cos 0
RV
r
− =
and
2
21 sin 0
RV
r
+ =
This will result in the two stagnation points: ( , ) ( ,0) and ( , )r R R =
The stagnation streamline is, therefore: 2
2sin 1 0
RV r
r
= − =
Unit B-3Page 10 of 17
Flow over a Cylinder (1)
Unit B-3Page 11 of 17
Flow over a Cylinder (2)
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PRESSURE COEFFICIENT OF THE FLOW OVER A CYLINDER
• The pressure coefficient on the surface of the cylinder is: 21 4sinpC = −
• The pressure coefficient on the surface of the cylinder indicates:
(1) pC distribution over the upper surface ( :0 → ) and lower surface ( : 2 → ) of the
cylinder are symmetrical.
(2) pC distribution over the front surface ( : 2 3 2 → ) and rear surface ( : 2 2 − → ) of
the cylinder are symmetrical.
D’ALEMBERT’S PARADOX
Lift coefficient of the flow over a circular cylinder is: , ,
1( )
2
R
l p lower p upper
R
c C C dxR
−
= −
Similarly, the drag coefficient can be given by: , ,
1( )
2
R
d p front p rear
R
c C C dyR
−
= −
• Since , , p lower p upperC C= and , , p front p rearC C= , these will lead to a conclusion of “no lift, no drag”
from the flow field analysis. This is called d’Alembert’s paradox.
Unit B-3Page 12 of 17
Flow over a Cylinder (3)
Unit B-3Page 13 of 17
Class Example Problem B-3-2
Related Subjects . . . “Flow over a Cylinder”
Consider the flow over a cylinder. Over the surface of the cylinder, calculate the
locations where the surface pressure (static) becomes equal to the freestream (static)
pressure.
?
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IDEAL V.S. REAL FLOW: CIRCULAR CYLINDER
The pressure distribution around circular cylinder (ideal flow) is symmetric, thus no lift and no drag will
be produced. This is known as d’ Alembert’s paradox.
Flow separation, due to the adverse pressure gradient, and resulting complex wake region induces the
non-symmetric pressure distribution, thus “pressure drag due to separation” is the main drag
contributor of real flows around circular cylinder
BASIC CONCEPT OF AERODYNAMIC PROFILE DRAG (DUE TO VISCOSITY)
As you can see, the presence of the viscosity will cause an aerodynamic drag. This is often called,
“aerodynamic profile drag.” The name implies that the drag is caused by the presence and the
behavior of “boundary layer” (due to viscosity). For detailed analysis on an aircraft wing’s cross section
(airfoil), the profile drag can further be categorized into the following two different types of drag:
• Df: Skin Friction (or “parasite”) Drag – the aerodynamic profile drag, caused by “attached”
boundary layer (friction). The actual amount of drag depends on flow type (either “laminar” or
“turbulent” flow). Obviously the turbulent flow is associated higher friction than laminar flow
(remember “pipe flow analysis” in Fluid Mechanics).
• Dp: Pressure Drag (or “drag due to separation”) – the aerodynamic profile drag, caused by
“separated” boundary layer (due to pressure difference, caused by separated flow).
Unit B-3Page 14 of 17
Ideal v.s. Real Flow
Flow Separation (wake)
Flow Separation (wake)
REAL FLOW OVER A CIRCULAR CYLINDER
(a) 0 Re 4 : pressure forces and friction forces balance each other: similar to the ideal flow (Stokes
Flow)
(b) 4 Re 40 : flow is separated on the back of the cylinder forming two stable vortices (separation
bubbles)
(c) 40 Re : the alternate shedding of vortices (von Karman vortex street) is observed (unsteady
flow) – as the Reynolds number is increased, the vortex street becomes turbulent and turns into a
wake: the DC is nearly constant ( 3 510 Re 3 10 )
(d) 5 63 10 Re 3 10 : the separation of the laminar boundary layer takes place on the forward face
of the cylinder (subcritical flow)
(e) 63 10 Re : the boundary layer transition from laminar to turbulent occurs – as the Reynolds
number is increased, the friction drag increases (supercritical flow)
Unit B-3Page 15 of 17
Real Flow over a Cylinder (1)
(a)
(b)
(c)(d)
(e)
Re = 1.54: Stokes flow
The flow field is very close to symmetrical. 10DC . The majority of drag in this flow regime is due to
viscosity (i.e., skin friction drag).
Re = 26: separation bubbles
The separation of the flow in the rearward face of the cylinder starts to occur, due to the adverse
pressure gradient. The separation is close to steady state in this flow regime, maintaining two
symmetrical separation “bubbles.” 1.0DC .
Re = 140: von Karman vortex street
The separation of the flow in the rearward face of the cylinder starts to become unsteady. 1.0DC .
The drag coefficient stays fairly constant over a wide range of Reynolds number, until the boundary
layer transitions to turbulent (Re 3105).
Unit B-3Page 16 of 17
Real Flow over a Cylinder (2)
Re = 1.54
Re = 26
Re = 140
SUBCRITICAL AND SUPERCRITICAL FLOWS
5 63 10 Re 3 10 : Subcritical flow
The early flow separation (laminar flow) creates large separation drag
63 10 Re : Supercritical flow
The turbulent flow delays flow separation, and reduces separation drag
BOUNDARY LAYER TRANSITION
Laminar flow: although the skin friction drag is smaller in laminar flow (than turbulent flow), the flow
separation occurs at early stage of the adverse pressure gradient region (thus, induces a large pressure
drag).
Turbulent flow: although the skin friction drag is larger than laminar flow, the flow separation moves
aft. Thus, the pressure drag is relatively small (smaller than laminar flow).
Unit B-3Page 17 of 17
Real Flow over a Cylinder (3)
Subcritical Flow