uniform flow solution of potential flows 4 superposition s

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


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

= = − = −


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).


ln2

r

= or

2

=


2rV

r r r

= = =

and 1

0Vr r

= = − =


Source / Sink

(+) (–)


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

= + − =


Superposition (2)


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


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

= = − = −


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

=


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


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

= − =


Flow over a Cylinder (1)



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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.




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).


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)


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).



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).



Subcritical Flow

uniform flow solution of potential flows 4 superposition s

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