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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.