yaa
DESCRIPTION
argaTRANSCRIPT
FLUID MECHANCIS II
BDA 30203
CHAPTER 2
POTENTIAL FLOW
DR. AZMAHANI SADIKIN
C16-101-09
Ideal flow
Fluid element kinematics
Linear motion and deformation
Angular motion and deformation
Vorticity
Rotational and Irrotational
Stream function
Velocity potential
Basic simple plane – potential flow
Uniform flow
Source and sink
Vortex
Types of Motion or Deformation of
Fluid Elements
In fluid mechanics, an element may undergo four
fundamental types of motion or deformation:
(a) translation, (b) rotation,
(c) linear strain (also called extensional strain), and
(d) shear strain.
All four types of motion or deformation usually occur
simultaneously.
It is preferable in fluid dynamics to describe the motion
and deformation of fluid elements in terms of rates
such as
velocity (rate of translation),
angular velocity (rate of rotation),
linear strain rate (rate of linear strain), and
shear strain rate (rate of shear strain).
In order for these deformation rates to be useful in the
calculation of fluid flows, we must express them in
terms of velocity and derivatives of velocity.
Fundamental types of fluid
element motion or
deformation: (a) translation,
(b) rotation, (c) linear strain,
and (d) shear strain.
Types of Motion or Deformation of
Fluid Elements
Types of Motion or Deformation of
Fluid Elements
A vector is required in order to fully describe the rate of translation in three
dimensions. The rate of translation vector is described mathematically as
the velocity vector.
Linear Motion and
deformation
Linear Motion and
deformation
Fluid Deformation: Linear Deformation
Rate of rotation (angular velocity) at a point: The average rotation rate of two
initially perpendicular lines that intersect at that point.
Rate of rotation of fluid
element about point O
For a fluid element that
translates and deforms as
sketched, the rate of
rotation at point O is
defined as the average
rotation rate of two
initially perpendicular
lines (lines a and b).
Angular Motion and
deformation
So, for x and y axes
Angular Motion and
deformation
The rate of rotation vector is equal to the angular velocity vector.
Angular Motion and
deformation
Vorticity and Rotationality
Another kinematic property of great importance to the analysis of fluid flows is
the vorticity vector, defined mathematically as the curl of the velocity vector
Vorticity is equal to twice the
angular velocity (rotation vector)
of a fluid particle
Vorticity and Rotationality
The direction
of a vector
cross product
is determined
by the right-
hand rule.
The vorticity vector is equal to
twice the angular velocity vector
of a rotating fluid particle.
Vorticity and Rotationality
• If the vorticity at a point in a flow field is nonzero, the fluid particle
that happens to occupy that point in space is rotating; the flow in
that region is called rotational.
• Likewise, if the vorticity in a region of the flow is zero (or
negligibly small), fluid particles there are not rotating; the flow in
that region is called irrotational.
• Physically, fluid particles in a rotational region of flow rotate end
over end as they move along in the flow.
The difference between rotational and
irrotational flow: fluid elements in a
rotational region of the flow rotate, but
those in an irrotational region of the
flow do not.
Vorticity and Rotationality
Vorticity and Rotationality
For a two-dimensional flow in the
xy-plane, the vorticity vector
always points in the z- or −z-
direction. In this illustration, the
flag-shaped fluid particle rotates
in the counterclockwise direction
as it moves in the xy-plane;
its vorticity points in the positive
z-direction as shown.
A simple analogy can be made
between flow A and a merry-go-
round or roundabout, and flow B
and a Ferris wheel.
As children revolve around a
roundabout, they also rotate at
the same angular velocity as that
of the ride itself. This is analogous
to a rotational flow.
In contrast, children on a Ferris
wheel always remain oriented in
an upright position as they trace
out their circular path. This is
analogous to an irrotational flow.
A simple analogy: (a) rotational
circular flow is analogous to a
roundabout, while (b) irrotational
circular flow is analogous to a
Ferris wheel. (Çengel et al. 2011)
Stream Function
The Stream Function in Cartesian Coordinates
Incompressible, two-dimensional stream
function in Cartesian coordinates:
stream
function
There are several definitions of
the stream function, depending
on the type of flow under
consideration as well as the
coordinate system being used.
Stream Function and Streamlines
Stream function automatically satisfies
the continuity equation, and only in 2
dimensional flow
Streamlines are lines of
constant stream function
If streamlines get closer,
their gradient becomes larger,
and the velocity increases in turn
If we make one additional assumption—that the flow is
irrotational—the analysis of inviscid flow problems is
further simplified.
Irrotational Flow
The vorticity, is zero for the irrotational flow field.
For the irrotational flow field, x V = 0 ; therefore the
rotation about the z axis becomes zero;
Therefore,
02
1
y
u
x
vz
y
u
x
v
Irrotational Flow
A general flow field would not
satisfy these three equations.
However, a uniform flow does.
Since u = U (a constant), v = 0
and w = 0; so all equations in
light green boxes are satisfed.
Therefore, a uniform flow field
(in which there are no velocity
gradients) is certainly an
example of an irrotational flow.
And at y and z axis, z
v
y
w
x
w
z
u
Uniform flow in a x direction (Munson
and Young, 2010)
Irrotational Flow
Various regions of flow: (a) around bodies, (b) through channel
(Munson and Young, 2010)
For an irrotational flow the velocity gradients are
related through Eqs. in the light green boxes for
irrotational flows.
The velocity components can be expressed in
terms of a scalar function as (x,y,z,t). So that the
velocity components can be expressed as
is called the velocity potential
The Velocity Potential
zw
yv
xu
In vector form so for irrotational flow the velocity
is expressible as the gradient of a scalar function .
V
The Velocity Potential
The velocity potential satisfies the irrotationality of the flow
field, whereas the stream function satisfy the conservation of
mass.
The velocity potential can be defined for a general three-
dimensional flow, but the stream function is restricted to two-
dimensional flows.
For an incompressible fluid, we know from conservation of
mass that
Therefore for incompressible, irrotational flow
(with ), it follows that
0 . V
V 02
where is the Laplacian
operator. In Cartesian coordinates
This is called Laplace’s Equation and
inviscid, incompressible, irrotational flow
are governed by this equation. This type
of flow is called potential flow.
The Velocity Potential
) ( . ) (2
02
2
2
2
2
2
zyx
Basic/Simple Plane Potential Flow
Laplace’s equation is linear partial
differential equation.
For potential flow, basic solutions can be
simply added to obtain more complicated
solutions.
Consider a two-dimensional (plane)
flows using cartesian coordinates as
below, for potential flow;
yv
xu
Basic/Simple Plane Potential Flow
We can define a stream function for plane flow too, so,
where we know that stream function satisfy the continuity
equation. Now, for irrotational flow,
and in terms of the stream function
or
Thus, for a plane irrotational flow, we can use either the velocity
potential or the stream function – both must satisfy the
Laplace’s equation in two dimensions.
xv
yu
x
v
y
u
xxyy
0
2
2
2
2
yx
Uniform Flow
The simplest plane flow is one for which the streamlines are
all straight and parallel, and the magnitude of velocity is
constant. We call this flow – Uniform flow
Consider a uniform flow in the positive x direction as shown
in Figure a below. So, u = U and v = 0, and in terms of
velocity potential,
These two equations can be integrated to yield
0
yU
x
CUx
Uniform flow: a) In the x direction b) in an arbitary direction,
Uniform Flow
C can be set to zero, so for a uniform flow in the x positive
direction,
Ux
Source and Sink
A source or sink represents a purely radial flow.
For velocity potential,
If m is positive,
the flow is radially outward
– source flow
If m is negative,
the flow is toward the origin
– sink flow
The m, the flowrate, is the strength of the source or sink
rm
ln2
The streamline pattern for a source
Source and Sink
But, when r = 0, the velocity becomes infinite, and
impossible. So, does not exist in real fluid.
In real flows, we can approximate away from the
origin as source and sink.
For stream function
So, the streamlines (lines of = constant) are
radial lines, the equipotential lines (lines of =
constant) are concentric circles at the origin.
2
m
Vortex
Consider a flow field in which the streamlines are
concentric circles. Interchange the velocity potential
and stream function for the source.
Then, let = K and = -K ln r , where K is
constant. In this case, the streamlines are concentric
circles, with vr = 0
Tangenial velocity varies inversley with the distance
from the origin, occur at r = 0
r
K
rrv
1
Vortex
Vortex motion is irrotational because the flow field is
described by the a velocity potential.
Rotation refers to orientation of fluid element, and not path
followed by the element.
Irrotational vortex is called free vortex
Rotational vortex cannot be described as a velocity potential
– forced vortex
A combined vortex is one with a forces vortex as a central
core and a velocity distribution corresponding to that of a
free vortex outside the core. ( drain in a bathtub)
Streamline pattern for a vortex
Vortex
Motion of fluid element from A to B: (a) for irrotational (free vortex) b)
for rotational (forced) vortex
Vortex
Satellite image of a hurricane near the Florida coast; water droplets
move with the air, enabling us to visualize the counterclockwise
swirling motion. However, the major portion of the hurricane is actually
irrotational, while only the core (the eye of the storm) is rotational.
Vortex
Circulation
Circulation, is defined as the line integral of the
tangential component of the velocity taken around
a closed curve in the flow field.
Circulation around various paths in a free vortex
rln2
2
cds .V