Aerodynamics 2017 fall - 1 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

Consider a closed curve C in a velocity field as shown in

the figure on the left. The instantaneous circulation around

curve C is defined by

C

sdV

Aerodynamics 2017 fall - 2 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

In 2-D, a line integral is counterclockwise by convention.

But aerodynamicists like to define circulation as positive

clockwise, hence the minus sign.

Aerodynamics 2017 fall - 3 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

Circulation is closely linked to the vorticity in the

flowfield. By Stokes’s Theorem

A AC

dAndAnVsdV ˆˆ

Aerodynamics 2017 fall - 4 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

In the 2-D xy plane, we have ξ=ξk and n=k, in which case

we have a simpler scalar form of the area integral.

)2( DindAA

Aerodynamics 2017 fall - 5 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

From this integral one can interpret the vorticity as

- circulation per area, or

dA

d

Aerodynamics 2017 fall - 6 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

Irrotational flows, for which ξ=0 by definition, therefore

have Γ=0 about any contour inside the flowfield.

Aerodynamic flows are typically of this type.

Aerodynamics 2017 fall - 7 -

Fundamental Principles & Equations

< 2.13. Circulation >

Circulation

The only restriction on this general principle is that the

contour must be reducible to a point while staying inside

the flowfield. A contours which contains a lifting airfoil,

for example, is not reducible, and will in general have a

nonzero circulation.

Aerodynamics 2017 fall - 8 -

Fundamental Principles & Equations

< 2.14. Stream function >

Definition

Consider defining the components of the 2-D mass flux

vector ρV as the partial derivatives of a scalar stream

function, denoted by : yx,

xv

yu

,

Aerodynamics 2017 fall - 9 -

Fundamental Principles & Equations

< 2.14. Stream function >

Definition

For low speed flows, ρ is just a known constant, and it is

more convenient to work with a scaled stream function

which then gives the components of the velocity vector V

yx,

xv

yu

,

Aerodynamics 2017 fall - 10 -

Fundamental Principles & Equations

< 2.14. Stream function >

Example

Suppose we specify the constant-density stream function to

be

which corresponds to a vortex flow around the origin.

2222 ln2

1ln, yxyxyx

2222,

yx

x

xv

yx

y

yu

which has a circular “funnel” shape

as shown in the figure. The implied

velocity components are then

Aerodynamics 2017 fall - 11 -

< 2.14. Stream function >

Streamline interpretation

The stream function can be interpreted in a number of

ways. First we determine the differential of ψ as follows.

Now consider a line along

which ψ is some constant ψ1.

Fundamental Principles & Equations

dxvdyud

dyy

dxx

d

1

1, yx

Aerodynamics 2017 fall - 12 -

Fundamental Principles & Equations

< 2.14. Stream function >

Streamline interpretation

Along this line, we can state that dψ=dψ1=d(constant)=0,

or

which is recognized as the equation for a streamline.

1

u

v

dx

dydxvdyu 0

Aerodynamics 2017 fall - 13 -

Fundamental Principles & Equations

< 2.14. Stream function >

Streamline interpretation

Hence, lines of constant ψ(x, y) are streamlines of the flow.

Similarly, for the constant-density case, lines of constant

ψ(x, y) are streamlines of the flow. In the example above,

the streamline defined by

can be seen to be a circle of

radius exp(ψ1).

1

22ln yx

Aerodynamics 2017 fall - 14 -

Fundamental Principles & Equations

< 2.14. Stream function >

Mass flow interpretation

Consider two streamlines along which ψ has constant

values of ψ1 and ψ2. The constant mass flow between these

streamlines can be computed by integrating the mass flux

along any curve AB spanning them.

12

Aerodynamics 2017 fall - 15 -

Fundamental Principles & Equations

< 2.14. Stream function >

Mass flow interpretation

The mass flow integration proceeds as follows

Hence, the mass flow between any two streamlines is given

simply by the difference of their stream function values.

B

A

B

A

B

AddxvdyudAnVm 12

ˆ

Aerodynamics 2017 fall - 16 -

Fundamental Principles & Equations

< 2.15. Velocity potential >

Definition

Consider defining the components of the velocity vector V

as the gradient of a scalar velocity potential function,

denoted by φ(x, y, z).

If we set the corresponding x, y, z components equal, we

have the equivalent definitions

zk

yj

xiV

ˆˆˆ

zw

yv

xu

,,

Aerodynamics 2017 fall - 17 -

Fundamental Principles & Equations

< 2.15. Velocity potential >

Example

Suppose we specify the potential function to be

y

xyx arctan,

which has a corkscrew shape

as shown in the figure.

Aerodynamics 2017 fall - 18 -

Fundamental Principles & Equations

< 2.15. Velocity potential >

Example

The implied velocity components are then

2222,

yx

x

yv

yx

y

xu

which corresponds to a vortex

flow around the origin.

Aerodynamics 2017 fall - 19 -

Fundamental Principles & Equations

< 2.15. Velocity potential >

Irrotationality

Any velocity field defined in terms of a velocity potential is

automatically an irrotational flow. Often the synonymous

term potential flow is also used.

0

0,0

022

V

yzzyyzzyz

v

y

w

zy

x

Aerodynamics 2017 fall - 20 -

Fundamental Principles & Equations

< 2.16. Stream function & Velocity potential >

Stream function

For a streamline, ψ(x, y) = constant.

Solve above equation for dy/dx,

0

dyudxv

dyy

dxx

d

u

v

dx

dy

const

Aerodynamics 2017 fall - 21 -

Fundamental Principles & Equations

< 2.16. Stream function & Velocity potential >

Velocity potential

Similarly, for an equipotential line φ(x, y) = constant.

Solve above equation for dy/dx,

0

dyvdxu

dyy

dxx

d

v

u

dx

dy

const

Aerodynamics 2017 fall - 22 -

Fundamental Principles & Equations

< 2.16. Stream function & Velocity potential >

Relationship

Hence, we have

constconst dxdydx

dy

/

1

This equation show that the

slope of a ψ=constant line is

the negative reciprocal of the

slope of a φ=constant line, i.e.,

streamlines and equipotential

lines are mutually

perpendicular.