Transcript Fluid Kinematics

ENT 257/4
FLUID MECHANICS
Prof. Madya Dr. Ghulam Abdul Quadir
School of Mechatronic Engineering
Email: [email protected]
H/P: 0174789452
Fluid Kinematics
• Fluid kinematics refers to features of a
fluid in motion.
• Consideration of velocity, acceleration,
flow rate, nature of flow and flow
visualisation are taken up in fluid
kinematics.
There are two ways to analyse fluid motion:
1)
Lagrangian method, and 2) Eulerian method
In the Lagrangian method, a particle or an element of fluid is identified
and followed during the course of motion with time, as shown in Fig.
1.
•
The identified lump of fluid may change in shape, size and state as it
moves.
•
The laws of Mechanics must be applied to it at all times.
u  u ( x, y, z, t ); v  v( x, y, z, t )
z  z ( x, y , z , t )
•
•
Difficulty in tracing the lump of fluid rules out the possibility of applying
the Lagrangian approach
In the Eulerian method, the fluid is observed by setting up fixed
stations or observatories in the flow field. Motion of the fluid is
specified by velocity components expressed as functions of space
and time,
• In the Eulerian approach, the fluid motion at all points in the flow
field is determined by applying the laws of mechanics at all fixed
stations.
• This is considerably easier than the Lagrangian approach and is
followed in the study of Fluid Mechanics.
Dimensions of Flow
A fluid flow is said to be one, two, or three dimensional depending
upon the number of independent space coordinates, i.e. one, two, or
three respectively, required to describe the flow.
One dimensional Flow: When the dependent variables are functions
of only one space coordinate, say x, it is one dimensional flow.
Examples: Flow through pipes, channels and variable area ducts if
the velocity distribution is considered constant at each cross section.
Dependent variables such as velocity, pressure, density,
temperature vary only with x, the only independent variable in one
dimensional flow as shown in Fig. 2.
The axis of the passage does not have to be a straight line for a flow
to be one dimensional. For example, the flow shown in Fig. 2(b) is
one dimensional if s, the streamwise direction is chosen as the
independent coordinate.
The axis of the passage does not have to be a straight line for a flow to
be one dimensional. For example, the flow shown in Fig. 2(b) is one
dimensional if s, the streamwise direction is chosen as the independent
coordinate.
Two dimensional Flow:
When the dependent variables in a fluid flow vary with only two
space coordinates, the flow is said to be two dimensional. The flow
does not vary along the third coordinate direction
Example: The flow around a circular cylinder of infinite length (as
shown in Fig. 2c) is two dimensional in the x-y plane.
Axisymmetric Flow: A flow is said to be axisymmetric if the velocity
profile is symmetrical about the axis of symmetry. In other words, the
velocity profile is the same at different diametral planes drawn
through the passage.
Example: Velocity profiles at two locations for an axisymmetric flow
through a conical passage are shown in Fig. 2(d).
Steady Flow: If the dependent fluid variables at any point in the flow
do not change with time, the flow is said to be steady flow. Thus

(dependent fluid var iables )  0
t
u v w
p 
 
0
 ,etc
t t t
t t
It follows that

(dependent fluid var iables )  0
t
Unsteady Flow: If the dependent fluid variables change with the
passage of time at a position in the flow, the flow is called unsteady
flow. Thus
• Steadiness of flow means that the flow-pattern does not change
with time whereas unsteadiness refers to changing flow-pattern with
the passage of time at the same point in space.
Uniform Flow: If the velocity, in magnitude, direction and sense is
identical throughout the flow field, the flow is said to be uniform
flow. This requires the velocity components to be the same at
different positions in the flow.
In other words, the space rate of change of velocity components at
that time must vanish.
It follows that

(dependent fluid var iables )  0
s
u
u u v v w
 0      , etc
x
y z x y y
for uniform flow.
It is conventional to define the term “uniform flow” only
in terms of the velocity components rather than in
terms of other dependent fluid variables.
Further, a flow may be considered uniform over the
cross sections although it may not be uniform
longitudinally.
Non-uniform Flow: If the velocity components at
different locations are different at the same instant of
time, the flow is said to be non-uniform.
Note: Steadiness refers to ‘no change with time’ and
uniformity refers to ‘no change in space’. Therefore, a
flow can be steady or unsteady quite independent of
its being uniform or non-uniform. All the four
combinations are possible.
Acceleration in Fluid Flow: The velocity components in a fluid
flow are, in general, functions of space and time
u  u ( x, y, z , t ), v  v( x, y, z , t ),
w  w( x, y, z , t )
From differential calculus, an infinitesimal change in u is given by
u
u
u
u
du  dx  dy  dz  dt
x
y
z
t
And the acceleration components are given by
du Du
u u
u u
ax  
u v w 
dt Dt
x y
z t
dv Dv
v v
v v
ay  
u v w 
dt Dt
x y
z t
dw Dw
w w
w w
az  
u v w 
dt Dt
x
y
z t
Total acceleration = Convective acceleration + Local
acceleration
Total acceleration is also called as substantial, material or
particle derivative.
• For steady flow: Only Local acceleration components are zero.
• For uniform flow: Only Convective acceleration components
vanish.
• For steady uniform flow: Total acceleration becomes zero and
the flow is non-accelerating in space and time.
Expressions for the components of acceleration in cylindrical
coordinates for an axisymmetric flow are given below:
ur
ur u
ur
ar  u r
 uz


r
z
r
t
u
u ur u u
a  ur
 uz


r
z
r
t
u z
u z u z
a z  ur
 uz

r
z
t
2
Streamlines, Pathlines and Streaklines
A streamline in a fluid flow is a line tangent to which at any point is
in the direction of velocity at that point at that instant.
• Streamlines are, therefore, equivalent to an instantaneous snap-shot
indicating the directions of velocity in the entire flow field as shown
in Fig. 4(a) and (b).
• There can be no flow across a streamline since the component of
velocity normal to a streamline is zero.
• A streamline cannot intersect itself nor can any streamline intersect
another streamline.
• In steady flow, the same streamline-pattern should hold at all times,
since the velocity vector cannot change with time.
Consider, first a streamline in a plane flow in the x-y plane as shown
in Fig. 3. By definition, the velocity vector U at a point P must be
tangential to the streamline at that point. It follows that
dy/dx = tan θ = v/u or u dy – v dx = 0
where u and v are the velocity components along x and y directions
respectively.
It may also be noticed that the velocity vector is expressed as
U = U (s, t)
which shows that the velocity may vary along a streamline direction
s as well as with the passage of time.
Consider next an elementary displaced element δs along a general
streamline where the velocity is U such that
U = u i+ v j + w k
and
δs = δx i+ δy j + δz k
By the definition of a streamline, U must be directed along δs. For
collinear vectors, the cross product must vanish.
Hence,
U x δs = 0
i
u
j
v
k
w 0
x y z
Or,
(vδz – wδy) i– (uδz – wδx) j + (uδy – vδx) k = 0
δx/u = δy/v = δz/w
whence
which is the equation of a streamline in a three dimensional flow,
steady or unsteady, uniform or non-uniform, viscous 0r inviscid,
compressible or incompressible.
A strteam surface is generated by a large number of closely
spaced streamlines which pass through an arbitrary curve AB as
in Fig. 4(c).
•
A stream surface can be plane, axisymmetric or spatial.
•
If the arbitrary line constitutes a closed curve, e.g., ABCD
as in Fig. 4(d), the closely spaced
streamlines passing
through the curve constitute a tubular stream surface called a
stream tube.
•
No flow can penetrate a stream surface or a stream tube.
•
•
•
•
A pathline in a fluid flow is the trajectory of a fluid particle, say P1 as
it advances with the passage of time, say from ti to final time tf as in
Fig. 4(e).
Pathlines are, therefore, history lines of individual fluid particles
over a period of time.
Tangent at a point on a pathline must be in the direction of
velocity at that point at the time
when the particle passes that
point.
A pathline, therefore, intersect itself at different times and in
general pathlines may
constitute a pattern of intersecting zigzag lines.
In steady flow, the streamlines and pathlines coincide.
A streakline is the locus of locations, Fig. 4(f), at an instant of time,
of all the fluid particles that have passed through a fixed point in a
flow field.
A little consideration will show that the streamlines, pathlines and
streaklines are coincident in a steady flow.
Example:
Given the velocity field
U = 5 x3 i– 15 x2 y j
Obtain the equation of the streamlines.
Law of Conservation of mass: Already explained in ENT 255/4.
However, for completeness, I am giving you the derivation of the
continuity equation again for ready reference.
Conservation of mass or Continuity Equation
• Consider a control volume within the boundary layer as shown
in Fig. 7(b) and assume that steady state conditions prevail.
• There are no gradients in the z direction and the fluid is
incompressible.
Rate of mass flow out of the
C.V. in the y-direc.ion
Rate of mass flow into the
control volume in the xdirection
7
Rate of mass flow out of the
control volume in the xdirection
Rate of mass flow into the C.V. in the
y-direction
7
From Fig. 7, the net mass flow into the element in the x direction is,
u
  dxdy
x
Similarly, the net mass flow into the control volume in the y direction is,
v

dydx
y
Since the net mass flow rate out of the control volume must be zero, we
obtain
 (
u v
 )dxdy  0
x y
From which it follows that in two-dimensional steady flow, conservation of
mass requires that
u v

0
x y
(13)
Example:
Does a velocity field given by
U = 5 x3 i – 15 x2 y j +t k
Represent a possible fluid motion of an incompressible fluid ?
Example:
Two components of velocity in an incompressible fluid flow are given by
u = x3 – y3
and
v = z3 – y3
Determine the third component, assuming that the origin is a stagnation
point.
Example:
Do the following velocity components represent a possible flow ?
Discuss the possibilities.
u = 2 x + 3 y and v = 3 y – 4 x.
Stream Function for Two-Dimensional Incompressible Flow
• Since no fluid can cross a streamline, the flow occurring
between two streamlines must remain unchanged.
• For example, in Fig. 6, the mass flow between streamlines 1
and 2 taken across A1 A2 must be the same as across B1B2 or
A1B2.
• The volumetric flow rate between them must also be constant
for ρ = Constant.
• A value of stream function ψ is assigned to each streamline
such that the volumetric flow rate is given by ψ2 – ψ1.
Mathematically, the streamfunction for a flow in the x-y plane is
defined as a function of x and y such that, the velocity
components are given by:
u

y
and
v

x
Let us examine the continuity equation for two dimensional incompressible
flow in the light of the definition of ψ, the stream function, defined in terms
of the velocity components. We get
 2
 2
 2
 2



0
xy yx xy xy
• Which shows that the continuity equation is identically satisfied.
• Hence, existence of ψ for a flow implies a possible flow.
• Conversely, for any possible flow, a function ψ must exist.
In the cylindrical polar coordinates, the stream function ψ is a function
of r and θ such that
1 
ur 
r 
and

u  
r
Here again, it may be seen that the velocity components identically
satisfy the continuity equation.
Example;
a) Does the stream function ψ = x y represent a physically possible flow ?
If so, determine the velocity at a point (2,3).
b) Does a stream function exist for a flow field described by u = 2 y and v
= - 2 x ? If so, determine it and plot a set of streamlines.
c) The velocity in a flow field is given by u = 3 m/s and v = 6 m/s.
Determine the equation of the streamline passing through the origin
and the one passing through a point (2m, 3m).
Kinematics of a Fluid element
A fluid element may move in a flow and undergo
(a) Pure or irrotational translation
(b) Pure rotation or rotational translation
(c) Pure distortion or deformation; angular or linear.
These are illustrated in Fig. 9 (a), (b) and (c) respectively by
taking an upright cubical fluid element to start with in each case.
Irrotational Flow
• For irrotational flow, each component of rotation or of vorticity
(rotation = ½ vorticity) must vanish, i.e., ωx = 0 = ωy = ωz.
• In particular, for plane flow in x-y plane,
•
δu/δy - δv/δx = 0
is the condition of irrotationality.

 u,
x
and

v
y
• A function Φ = Φ(x,y) can be defined with an attempt to satisfy the
condition of irrotationality identically.
• For a two-dimensional flow in the x-y plane,
u v

 0;
y x
 2
 2

0
yx xy
satisfy the condition of irrotationality identically
• In general, the space derivatives of Φ give the velocity components
along the corresponding space coordinates.
• The function Φ is called the velocity potential function.
• It may be stressed that if a flow is irrotational, a velocity potential
function Φ must exist.
• Conversely, if a function Φ is specified then the flow represented by
the velocity components obtained from Φ must be irrotational.
Example
d) In a two
dimensional incompressible flow the fluid velocity
components are given by u = x - 4y and v = - y – 4x.
Show that the flow satisfies the continuity equation.
Obtain the expression for the stream function.
If the flow is potential, obtain also the expression for the velocity
potential.
Soln.
•
Verify the continuity equation.
•
To obtain the stream function, use its definition in terms of the
velocity components; integrate u component definition of it w.r.t. y
and find the constant of integration from the boundary condition
(say, ψ = 0 at x = 0 and y = 0). To find the functional constant,
differentiate the stream function w.r.t. x and equate it to v
component of velocity. Simplify and you will get ψ = 2 x2 + xy –
2y2.
•
To check whether the flow is potential, there are two possible
approaches:
(a)
From the vorticity component in x direction
(b)
Laplace’s equation must be satisfied
.
•
•
•
To obtain the velocity potential, use the its definition in terms of
the velocity component u; integrate it w.r.t. x. You will get an
expression for velocity potential with functional constant f(y).
Differentiate the expression w.r.t. y and equate to v.
The expression for velocity potential will be Φ = x2/2 – 4yx - y2/2.
e) The velocity components in a two dimensional flow-field for an
incompressible fluid are expressed as:
U = y3/3 + 2x – x2y
and
v = xy2 – 2y – x3/3
i)
Show that these functions represent a possible case of an
irrotational flow
ii)
Obtain an expression for the stream function and
iii)
Obtain an expression for the velocity potential
• EULER’S EQUATION OF MOTION ALONG A STREAMLINE
To derive a relation between velocity, pressure, elevation and
density along a streamline.
• Fig. shows a short section of a streamtube surrounding the
streamline and having a small cross sectional area so that
velocity be considered constant over the cross section.
• AB and CD are two cross sections separated by a short
distance δs.
• At AB, the area is A, velocity v, pressure p and elevation z, while
at CD, the corresponding values are, A+ δA, v+ δv, p+ δp and z+
δz.
• The surrounding fluid will exert a pressure pside on the sides of
the element.
• The fluid is assumed to be inviscid. Thus, there will be no shear
stresses on the sides of the element and pside will act normally.
• The weight of the element mg will act vertically downward at an
angle θ to the streamline.
Mass per unit time flowing
= ρAv
Rate of increase of momentum from AB to CD = ρAv[(v+ δv) – v)] =
ρAv δv
The forces acting to produce this increase of momentum in the
direction of motion are:
Force due to p in direction of motion
= pA
Force due to p+ δp opposing motion
= (p+ δp)(A+ δA)
Force due to pside producing a component
In the direction of motion
= pside δA
Force due to mg producing a component
Opposing motion
= mgcosθ
Resultant force in the direction of motion pA – (P+ δp)(A+ δA)+pside
δA-mgcosθ
The value of pside will vary from p at AB to p+ δp at CD and can be
taken up as p+k δp, where k is a fraction.
Weight of element, mg = ρg*Volume = ρg(A+½ δA) δs,
cosθ = δz/ δs
Resultant force in the direction of motion = -p δA – A δ p - δp δA + p
δA + k δp. δA – ρg(A+½ δA) δs.(δz/ δs)
Neglecting products of small quantities,
Resultant force in the direction of motion = -A δp - ρgA δz
Applying Newton’s second law of motion, we get,
ρAv δv = -A δp - ρgA δz
Dividing by ρAδs, we get
1 p
v
z
v  g 0
 s
s
s
Or in the limit as δs
0,
1 dp
dv
dz
v  g
0
 ds
ds
ds
(A)
• This is known as Euler’s Equation, giving, in differential form,
the relationship between pressure p, velocity v, density ρ
and elevation z along a streamline for steady flow.
v2
  gz  const
 2
p
• It cannot be integrated until the relationship between density ρ and
pressure p is known.
• For an incompressible flow, for which ρ is constant, integration of
the above equation (A) along the streamline, w.r.t. to s, gives
p
v2

 z  const
g 2 g
• In which the terms represent the energy per unit weight. The
above equation is known as Bernoulli’s Equation and states
the relationship between pressure, velocity and elevation for
steady flow of a frictionless fluid of constant density
• An alternative form is:
v2

 gz  const
 2
p
In which the terms represent the energy per unit volume.
• These equations apply to a single streamline. The sum of the three
terms is constant along any streamline, but the value of the constant
may be different for different streamlines in a given stream.
• If the earlier integration of Equation (A) is carried along the
streamline between any two points indicated by suffixes 1 and 2,
2
2
then
p1 v1
p2 v2

 z1 

 z2
g 2 g
g 2 g
• For a compressible fluid, the integration of Equation (A) can only be
partially completed, to give

dp
v2

z  H
g
2g