Potential Flow
Chapter Three
PART ONE
Dr. Hamdy A. Kandil
15 March 2016
The Stream Function, (x,y)
Consider the continuity equation for an incompressible 2D flow
u v
+
=0
x y
Substituting the clever transformation, (x,y)
Defined as:
Gives
u v
+
= (
) + (
)=0
This is true for any smooth
x y x y
y
x
function (x,y)
so that it always satisfies the continuity eq.
using stream
u
function
two unknows
v
} one
unknow
c
conservation of mass will always be satisfied
The Stream Function,
Why do this?
Single variable replaces (u,v).
Once is known, (u,v) can be determined.
Physical significance
1.
Curves of constant are streamlines of the
flow
2.
Difference in between streamlines is equal
to volume flow rate between streamlines
The Stream Function: Physical Significance
1. Curves of constant are streamlines of the flow
Recall that the streamline equation is
given by:
Change in along
streamline is zero
2. Difference in between streamlines is equal to volume
flow rate between streamlines
Let dq represent the volume rate of flow per unit width perpendicular
to the xy plane passing between the two streamlines.
From conservation of mass we know that the inflow, dq, crossing the
arbitrary surface AC must equal the net outflow through surfaces
AB and BC. Thus,
or in terms of the stream function
Thus, the volume rate of flow, q,
between two streamlines can be
determined by integration to yield
Example Stream Function
The velocity components in a steady, incompressible, twodimensional flow field are
u = 2y
v = 4x
Determine the corresponding stream function and show on a
sketch several streamlines.
Indicate the direction of flow along the streamlines.
From the definition of the stream function
u=
= 2y
y
= y 2 + f1 (x)
v=
= 4x
x
= 2x 2 + f 2 (y)
= 2x2 + y 2 + C
For simplicity, we set C = 0
Example Solution
=0
= 2x 2 + y 2
Streamlines can now be
determined by setting
=const. and plotting the
resulting curve.
With the above
expression for , the
value of at the origin is
zero so that the equation
of the streamline passing
through the origin is
0 = -2x2 + y2
For
y2 x 2
=1
/2
which we recognize as the
equation of a hyperbola
Rate of rotation (angular velocity)
Consider the rotation about z-axis of the rectangular element x-
y
The rotation of the side x
tan
Rate of rotation (angular velocity)
Angular Velocity of OA
lim
The rotation of the side y
tan
Angular Velocity of OB
lim
Rate of rotation (angular velocity)
The rotation of the element about the z axis is defined as the
average of the angular velocities of the two mutually
perpendicular lines OA and OB. If counterclockwise rotation is
considered to be positive, then:
Average rotation about z-axis
1 v
z =
2 x y
Average rotation about x-axis,
1 w
x =
2 y z
Average rotation about y-axis,
Rotation Vector
1 u w
2 z x
y =
r
1 w v u w v u
= x i + y j + z k =
i +
j + k
2 y z z x x y
Rotational and Irrotational Flows
The vorticity is defined as:
2 =
w v u w v u
i +
j + k
z
x
y
z
x y
For irrotational flow = = 0
Examples: Rotational flow:
Solid-Body Rotation
(Forced Vortex): u = r
Rotational and Irrotational Flows
Examples: Irrotational flow:
Free Vortex: u = K/r
Mathematical Representation
Vorticity is the curl of the velocity vector
For 3-D vorticity in Cartesian coordinates:
v
i
v
V
x
u
v
j
y
v
v
k
v w v
= i
+
z
y z
w
v u w v v u
j
+ k
z x x y
The horizontal relative vorticity (about z axis) is found by
eliminating terms with vertical ( ) components:
v
i
v
v
k ( V )
x
u
v
j
y
v
v
k
v u
=
z x y
w
Circulation ( )
The circulation ( ) is a measure of rotation and is defined as
the line integral of the tangential component of the velocity
taken around a closed curve in the flow field.
r
r r
= V dl = V cos dl
NOTE:
The flow is irrotational if
=0,
=0,
=0
For 2-D Cartesian Coordinates
d = udx + ( v +
v
u
) dxdy
y
x
= z . area
v
u
dx ) dy ( u +
dy ) dx vdy
x
y
= (
= . area
Potential Function,
Irrotational approximation:
vorticity is negligibly small
In general, inviscid regions are
also irrotational, but there are
situations where inviscid flow
are rotational, e.g., solid body
rotation.
What are the implications of irrotational approximation. Look at
continuity and momentum equations.
Use the vector identity where is a scalar function
Since the flow is irrotational where
is a scalar potential function
Irrotational Flow Approximation
Therefore, regions of irrotational flow are also called regions of
potential flow.
From the definition of the gradient operator
Cartesian
Cylindrical
,
x
,
ur =
r
u=
, w=
z
y
1
, uz =
u =
r
z
v=
Substituting into the continuity equation for incompressible flow
gives:
Irrotational Flow Approximation
This means we only need to solve 1 linear scalar equation to
determine all 3 components of velocity!
Laplace Equation
Luckily, the Laplace equation appears in numerous fields of
science, engineering, and mathematics. This means there are
well developed tools for solving this equation.
Momentum equation
If we can compute from the Laplace equation (which came from
continuity) and velocity from the definition
, why do we
need the NSE? the answer: To compute Pressure.
To begin analysis, apply irrotational approximation to viscous term
of the incompressible NSE
=0
Irrotational Flow Approximation
Therefore, the NSE reduces to the Euler equation for
irrotational flow
Instead of integrating to find P, use vector identity to derive
Bernoulli equation
Irrotational Flow Approximation
This allows the steady Euler equation to be written as
This form of Bernoulli equation is valid for inviscid and irrotational
flow since weve shown that NSE reduces to the Euler equation.
However,
Inviscid
Irrotational ( = 0)
Irrotational Flow Approximation
Therefore, the process for irrotational flow
1.
Calculate from Laplace equation (from continuity)
2.
Calculate velocity from definition
3.
Calculate pressure from Bernoulli equation (derived from
momentum equation)
2
p V2
p V
+
+ gz =
+
+ gz
2
2
Valid for 3D or 2D
Irrotational Flow Approximation
2D Flows
For 2D flows, we can also use the stream function
Recall the definition of stream function for planar (x-y) flows
u=
v=
Since vorticity is zero for irrotational flow,
z =
v u
=0
x y
= 2
This proves that the Laplace equation holds for the stream
function and the velocity potential
Irrotational Flow Approximation
2D Flows
Constant values of :
streamlines
Constant values of :
equipotential lines
and are mutually
orthogonal
is defined by continuity;
2 results from
irrotationality
is defined by irrotationality;
2 results from continuity
Flow solution can be achieved by solving either 2 or 2,
however, BCs are easier to formulate for .
Relation between and lines
If a flow is incompressible, irrotational, and two dimensional, the
velocity field may be calculated using either a potential function or a
stream function.
Using the potential function, the velocity components in Cartesian
coordinates are
u=
v=
d =
dx +
dy = udx + vdy
x
y
And
For lines of constant potential (d = 0), which are called
equipotential lines:
u
dy
=
v
dx =c
Since a streamline is everywhere tangent to the local velocity, the
slope of a streamline, which is a line of constant , is
Relation between and lines
v
dy
=
dx =c u
Comparing equations of slopes yields:
1
dy
=
(dy dx) =c
dx =c
The slope of an equipotential line is the negative reciprocal of
the slope of a streamline.
Therefore, streamlines ( = constant) are everywhere
orthogonal (perpendicular) to equipotential lines ( = constant).
This observation is not true, however, at stagnation points,
where the components vanish simultaneously.
Irrotational Flow Approximation
2D Flows
Similar derivation can be performed for cylindrical coordinates
(except for 2 for axisymmetric flow)
Planar, cylindrical coordinates: flow is in (r,) plane
Axisymmetric, cylindrical coordinates : flow is in (r,z) plane
Planar
Axisymmetric
Irrotational Flow Approximation
2D Flows
Potential flows Visualization
Flow fields for which an incompressible fluid is assumed to be frictionless and
the motion to be irrotational are commonly referred to as potential flows.
Paradoxically, potential flows can be simulated by a slowly moving, viscous
flow between closely spaced parallel plates.
For such a system, dye
injected upstream reveals an
approximate potential flow
pattern around a streamlined
airfoil shape.
Similarly, the potential flow
pattern around a bluff body is
shown. Even at the rear of the
bluff body the streamlines
closely follow the body shape.
Generally, however, the flow
would separate at the rear of
the body, an important
phenomenon not accounted for
with potential theory.
Irrotational Flow Approximation
2D Flows
Method of Superposition
1.
Since 2 = 0 is linear, a linear combination of two or more
solutions is also a solution, e.g., if 1 and 2 are solutions,
then (A1), (A+1), (1+2), (A1+B2) are also solutions
2.
3.
Also true for in 2D flows (2 =0)
Velocity components are also additive
Irrotational Flow Approximation
2D Flows
Given the principal of superposition, there are several
elementary planar irrotational flows which can be
combined to create more complex flows.
Elementary Planar Irrotational Flows
Uniform stream
Line source/sink
Line vortex
Doublet
Elementary Planar Irrotational Flows
Uniform Stream
u = U=constant , v = 0, w = 0
In Cartesian coordinates
= U x , = U y
Conversion to cylindrical
coordinates can be achieved
using the transformation
= U r cos , = U r sin
Stagnation Flow
The flow is an incoming far field flow which is
perpendicular to the wall, and then turns its direction near
the wall
The origin is the stagnation point of the flow. The velocity
is zero there.
Application: Stagnation Flow
For a stagnation flow,
Hence,
= u = Bx,
x
r
V = Bx i By j
= v = By
y
Therefore,
B 2
B 2
2
(
x
y
)
=
r cos 2
And
2
2
= u = Bx,
= v = By
y
x
Therefore
B 2
= Bxy = r sin 2
2
Elementary Planar Irrotational Flows
Line Source/Sink
vr
Lets consider fluid flowing
radially outward from a line
through the origin
perpendicular to x-y plane
from mass conservation:
The volume flow rate per unit
thickness is K
This gives velocity
components
vr =
K
and v = 0
2r
K
=
=
2r r r
=
and v = 0 =
r
r
vr =
Stream function and potential function
= 0 and
r
K
=
r 2r
By integration:
K
ln r
=
2
=0
r
K
K
and
=
=
r 2r
2
By integration:
K
=
Equations are for a source/sink
at the origin
Elementary Planar Irrotational Flows
Line Source/Sink
If source/sink is moved
to (x,y) = (a,b)
K
K
K
Elementary Planar Irrotational Flows
Line (potential) Vortex
A potential vortex is defined as a
singularity about which fluid flows
with concentric streamlines
Vortex at the origin. First look at
velocity components
1 $
=
=0
"
$
1
2%
"
& =
Equations are for a line vortex
at the origin where the
arbitrary integration constants
are taken to be zero at
(r,)=(1,0)
These can be integrated to give
and
&
Free Vortex
The potential represents a flow swirling around origin with a
constant circulation .
The magnitude of the flow decreases as 1/r.
Line Vortex
now we consider situation when the
stream lines are concentric circles i.e. we
interchange potential and stream
functions:
circulation
= V ds =
C
= K
= K ln r
ds = d = 0
in case of vortex the circulation is zero
along any contour except ones enclosing
origin
2
K
(rd ) = 2 K
r
ln r
2
Stokess Theorem
The transformation from a line integral to a surface integral in
three-dimensional space is governed by Stokess theorem:
where n dA is a vector normal to the surface, positive when pointing
outward from the enclosed volume, and equal in magnitude to the
incremental surface area.
In words, the integral of the normal component of the curl of the
velocity vector over any surface A is equal to the line integral of
the tangential component of the velocity around the curve C which
bounds A.
Stokess theorem is valid when A
represents a simply connected region in
which VS is continuously differentiable.
Thus, it is not valid if the area A contains
regions where the velocity is infinite.
Potential Vortex
The curl of the velocity vector for the potential vortex can be
found using the definition for the curl of V S in cylindrical
coordinates
But & = 0 and
=
2%
which simplifies in two dimensions to:
Although the flow is irrotational ( =0), we must remember that
the velocity is infinite at the origin (i.e., when r = 0 ).
In fact, the flow field at the origin is rotational and vorticity
exists there.
We will now calculate the circulation around a closed curve C1
which encloses the origin. We can choose a circle of radius r1
The circulation is
Recall that Stokess theorem, is not valid if the region contains
points where the velocity is infinite, which is true for vortex
flow at the origin.
However, if we calculate the circulation around a closed curve C2,
which does not enclose the origin, such as that shown in Fig. 3.13
b, we find that
Therefore, the circulation around a closed curve not containing
the origin is zero.
Paths for the calculation of the circulation for a potential vortex: (a)
closed curve C1, which encloses origin; (b) closed curve C2, which does
not enclose the origin.
Shape of the free surface
=
=
1
=
" 2%
Bernollis equation
p V2
+
+ gz = const
2
at the free surface p=0:
V12 V2 2
=
+z
2g 2g
2
z= 2 2
8 r g
Elementary Planar Irrotational Flows
Line Vortex
If vortex is moved to (x,y) =
(a,b)
