Part A | 7.3
160 Part A Fundamentals
In two-dimensional Cartesian components, the velocity
can be written as the curl of the stream function O
k
u D u O i C v O j D
@
@y
O i
@
@x
O j :
In two-dimensional polar coordinates, this is
u D u r O
r C u  O
 D
1
r
@
@Â
O
r
@
@r
O
 :
Likewise, the velocity potential ensures the irrotationality of the flow
r u D r .r / D 0 :
In two-dimensional Cartesian components, the velocity
can be written as the gradient of a scalar potential
u D u O i C v O j D
@@
@x
O i C
@@
@y
O j :
In two-dimensional polar coordinates, this is
u D u r O
r C u  O
 D
@@
@r
O
r C
1
r
@@
@Â
O
 :
For irrotational, incompressible flow, both the velocity
potential and stream function satisfy Laplace’s equation
r
2
D 0
and
r
2
D 0 ;
which is linear, such that solutions can be superposed
r
2
. 1 C 2 / D r
2
1 C r
2
2
and
r
2
.. 1 C 2 / D r
2
1 C r
2
2 :
Solutions for and can be superposed in the complex
plane to find a complex velocity potential w , such that
w D C i , where i D
p 1 is the imaginary number
and
dw
dz
D u iv ;
gives the velocity component in two-dimensional Cartesian coordinates. Here, the complex coordinate z D x C
iy D re
i .
As Laplace’s equation is second order, solutions
generally require two boundary conditions in each of
the independent variables (either x and y in Cartesian
coordinates or r and  in polar coordinates):
1. The velocity of the fluid at solid surfaces must
match the velocity of the solid surface. Without loss
of generality the problem can be formulated such
that the coordinate system is in the reference frame
of the surface, such that the velocity is zero. With O
n
as a unit vector perpendicular to the surface and O
s as
a unit vector tangent to the surface, we have
@@
@n
D 0 and
@
@s
D 0 :
2. Far from the surface of the body, the velocity must
approach its freestream value U, such that
@@
@x
D U and
@
@y
D U :
The flow velocity can be found from solutions to
Laplace’s equation for the velocity potential and stream
function. However, these solutions are based purely on
the geometry of the flow (kinematic) and cannot be directly used to determine the forces acting on objects
embedded in the flow. In order to solve for these fluidic
forces, Bernoulli’s equation must be used to determine
the pressure around an object in the flow – by integrating the pressure around an object, one can then
determine the fluidic forces acting on it.
The steady, irrotational, and incompressible form
of Bernoulli’s equation is often used to solve potential
flow problems
u
2
2
C gz C
p
D
.r /
2
2
C gz C
p
D B ;
(7.63)
where B is constant everywhere in the flow field.
Catalog of Flow Fields
Complex flows around objects can be built from the
superposition of very simple flows. Here, a reference
list of some of the most basic flows is provided as
a guide:
1. Simple flows:
a) Uniform flow (Fig. 7.55)
The velocity potential and stream function can
be determined by quadrature @@=@x D u D U.
We can integrate this with respect to x to find
D Ux C f 1 .y/. Then, v D @@=@y D df 1 =dy D
0. Integrating with respect to y, we find that
f 1 .y/ D C, where C is a constant. Without loss
of generality we can let C D 0. Therefore, D
Ux is the velocity potential for a uniform flow.
Similarly, we can solve for the stream function
by integrating the relation @ =@y D u D U to
160 Part A Fundamentals
In two-dimensional Cartesian components, the velocity
can be written as the curl of the stream function O
k
u D u O i C v O j D
@
@y
O i
@
@x
O j :
In two-dimensional polar coordinates, this is
u D u r O
r C u  O
 D
1
r
@
@Â
O
r
@
@r
O
 :
Likewise, the velocity potential ensures the irrotationality of the flow
r u D r .r / D 0 :
In two-dimensional Cartesian components, the velocity
can be written as the gradient of a scalar potential
u D u O i C v O j D
@@
@x
O i C
@@
@y
O j :
In two-dimensional polar coordinates, this is
u D u r O
r C u  O
 D
@@
@r
O
r C
1
r
@@
@Â
O
 :
For irrotational, incompressible flow, both the velocity
potential and stream function satisfy Laplace’s equation
r
2
D 0
and
r
2
D 0 ;
which is linear, such that solutions can be superposed
r
2
. 1 C 2 / D r
2
1 C r
2
2
and
r
2
.. 1 C 2 / D r
2
1 C r
2
2 :
Solutions for and can be superposed in the complex
plane to find a complex velocity potential w , such that
w D C i , where i D
p 1 is the imaginary number
and
dw
dz
D u iv ;
gives the velocity component in two-dimensional Cartesian coordinates. Here, the complex coordinate z D x C
iy D re
i .
As Laplace’s equation is second order, solutions
generally require two boundary conditions in each of
the independent variables (either x and y in Cartesian
coordinates or r and  in polar coordinates):
1. The velocity of the fluid at solid surfaces must
match the velocity of the solid surface. Without loss
of generality the problem can be formulated such
that the coordinate system is in the reference frame
of the surface, such that the velocity is zero. With O
n
as a unit vector perpendicular to the surface and O
s as
a unit vector tangent to the surface, we have
@@
@n
D 0 and
@
@s
D 0 :
2. Far from the surface of the body, the velocity must
approach its freestream value U, such that
@@
@x
D U and
@
@y
D U :
The flow velocity can be found from solutions to
Laplace’s equation for the velocity potential and stream
function. However, these solutions are based purely on
the geometry of the flow (kinematic) and cannot be directly used to determine the forces acting on objects
embedded in the flow. In order to solve for these fluidic
forces, Bernoulli’s equation must be used to determine
the pressure around an object in the flow – by integrating the pressure around an object, one can then
determine the fluidic forces acting on it.
The steady, irrotational, and incompressible form
of Bernoulli’s equation is often used to solve potential
flow problems
u
2
2
C gz C
p
D
.r /
2
2
C gz C
p
D B ;
(7.63)
where B is constant everywhere in the flow field.
Catalog of Flow Fields
Complex flows around objects can be built from the
superposition of very simple flows. Here, a reference
list of some of the most basic flows is provided as
a guide:
1. Simple flows:
a) Uniform flow (Fig. 7.55)
The velocity potential and stream function can
be determined by quadrature @@=@x D u D U.
We can integrate this with respect to x to find
D Ux C f 1 .y/. Then, v D @@=@y D df 1 =dy D
0. Integrating with respect to y, we find that
f 1 .y/ D C, where C is a constant. Without loss
of generality we can let C D 0. Therefore, D
Ux is the velocity potential for a uniform flow.
Similarly, we can solve for the stream function
by integrating the relation @ =@y D u D U to