Fluid kinematics

Ministry of Higher Education & Scientific Research
Foundation of Technical Education
Technical College of Basrah
CH4: Fluid Kinematics
Training Package
in
Fluid Mechanics
Modular unit 4
Fluid Kinematics
By
Risala A. Mohammed
M.Sc. Civil Engineering
Asst. Lect.
Environmental & Pollution Engineering Department
2011

1- Over view
1-1 Target population
For the students of second class in
Environmental engineering Department in
Technical College

1-2 Rationale
A study of fluids in motion very important to
calculate the acceleration and speed of the fluid at
any point in the flow. And to the importance of flow
in many applied field such as flow of water in the
rivers, wind movement and flow of fluids in
industrial equipment

1-3 Central Idea
The main goal of this chapter are define the
following:
1- Fluid kinematic
2- Fluid velocity
3- Types of flow
4- Continuity equation

1-4 Instructions
1- Study over view thoroughly
2- Identify the goal of this modular unit
3- Do the Pretest and if you :-
*Get 9 or more you do not need to proceed
*Get less than 9 you have to study this modular
4- After studying the text of this modular unit , do the post test
and if you :-
*Get 9 or more , so go on studying modular unit five
*Get less than 9 , go back and study the modular unit four

1-5 Performance Objectives
At the end of this modular unit the student will be able to :
1- Define the fluid kinematic
2- Describe the fluid velocity and acceleration
3- Limit the type of fluid flow
4- Drive the continuity equation

2- Pre test
-
Q1)) Choose the Correct Answer (7 marks):
1- The motion of fluid particles may be described by which of the following methods?
a) Langrangian method (b) Eulerain method (c) both (a) and (b) (d) none of the above.
2- In which of the following methods, the observer concentrates on a point in the fluid system?
(a) Langrangian method b) Eulerian method (c) Any of the above (d) None of the above.
3- Ina steady flow the velocity
(a) does not change from place to place (b) at a given point does not change with time
(c) may change its direction but the magnitude remains unchanged (d) none of the above.
4- The flow in a pipe whose valve is being opened or closed gradually is an example of
(a) steady flow (b) unsteady flow c) rotational flow (d) compressible flow.
5- The type of flow in which the velocity at any given time does not change with respect to space is
called
(a) steady flow (b) compressible c) uniform flow d) rotational flow

Pre test
-
6- The flow in a river during the period of heavy rainfall is
(a) steady, non-uniform and three-dimensional (b) steady, uniform, two dimensional
(c) unsteady, uniform, three-dimensional (d) unsteady, non-uniform and three-dimensional.
7- If the flow is irrotational as well as steady it is known as
(a) non-uniform flow (b) one-dimensional flow (c) potential flow (d) none of the above.
Q2 (3 marks)
Not
Check your answers in key answer page

Introduction
-
Fluid Kinematics deals with the motion of fluids without considering
the forces and moments which create the motion.
 Lagrangian description:
 properties of individual fluid particles are defined as a function of time
as they move through the fluid; the overall fluid motion is found by
solving the EOMs for all fluid particles.
 Eulerian description:
 properties are defined at fixed points in space as the fluid flows past
these points; this is the most common description and yields the field
representation of fluid flow.

The Velocity Field
 Consider an array of sensors that can simultaneously
measure the magnitude and direction of fluid velocity at
many fixed points within the flow as a function of time; in
the limit of measuring velocity at all points within the flow,
we would have sufficient information to define the velocity
vector field:
k
j
i ˆ
)
,
,
,
(
ˆ
)
,
,
,
(
ˆ
)
,
,
,
( t
z
y
x
w
t
z
y
x
v
t
z
y
x
u 


V


The Velocity Field
 u, v, and w are the x, y, and z components of the velocity
vector
 The magnitude of the velocity, or speed, is denoted by V as
 Velocity field may be one- (u), two- (u,v)or three- (u,v,w)
dimensional
2
2
2
w
v
u
V 


 V


Visualization of Fluid Flow
Three basic types of lines used to illustrate fluid flow patterns:
Streamline: a line that is everywhere tangent to the local
velocity vector at a given instant.
Path line: a line that represents the actual path traversed
by a single fluid particle.
Streak line: a line that represents the locus of fluid
particles at a given instant that have earlier passed
through a prescribed point.

Acceleration Field
 Collecting derivative terms from all velocity components,
– The operator is termed the material, or substantial,
derivative; it represents the rate at which a variable (V
in this case) changes with time for a given fluid particle
moving through the flow field
 V
V
V
V
V
V
V
V


























t
z
w
y
v
x
u
t
Dt
D
Dt
D )
(

Acceleration Field
 The term is called the local acceleration; it represents
the unsteadiness of the fluid velocity and is zero for steady
flows.
 The terms are called convective
accelerations; they represent the fact that the velocity of the
fluid particle may vary due to the motion of the particle from
one point in space to another; it can occur for both steady
and unsteady flows.
t

V

z
y
x w
v
u 




 V
V
V



,
,

The Control Volume
 A control volume is a volume in space through which fluid
may flow; in some cases, the volume may move or deform
 The control volume has a boundary which separates it from
the surroundings and defines a control surface
 In the study of fluid dynamics, the control volume approach is
used to analyze fluid flow and fluid machinery
 The control volume approach is consistent with the Eulerian
description

Ideal Fluid in Motion
Four assumptions related to ideal flow:
Steady flow: the velocity of the moving fluid at any fixed
point does not change with time
Incompressible Flow: Fluid has constant density
Nonviscous flow: an object can move through the fluid at
constant speed- no resistive force within the fluid to moving
objects through it
Irrotational flow: Objects moving through the fluid do not
rotate about an axis through its center of mass

Types of Fluid Flow
Fluids may be classified as follows:
1. Steady and unsteady flows
2. Uniform and non-uniform flows
3. One, two and three dimensional flows
4. Rotational and irrotational flows
5.Laminar and turbulent flows
6.Compressible and incompressible flows.

Steady and unsteady flow
Steady flow.
The type of flow in which the fluid characteristics like velocity,
pressure, density,
Example. Flow through a prismatic. or non-prismatic conduit at a constant
flow rate Q m3/s is steady (A prismatic conduit has a constant size shape and
has a velocity equation in the form = ax2 + bx + c which is independent of
time t).

Steady and unsteady flow
Unsteady flow.
It is that type of flow in which the velocity, pressure or density at a
point inge w.r.t. time. Mathematically, we have
Example. The flow in a pipe whose valve is being opened or closed
gradually (velocity equation the form u = ax2 + bxt).

Uniform and non uniform flow
Uniform flow.
The type of flow, in which the velocity at any given time does not change
with respect to space is called uniform flow . Mathematically, we have
Change in velocity, and
Displacement in any direction
Example.
Flow through a straight prismatic conduit (i, e. flow through a straight pipe of
constant meter)

Uniform and non uniform flow
uniform flow.
-
Non
It is that type of flow in which the velocity at any given time changes
with respect to space . Mathematically,
Example.
(i) Flow through a non-prismatic conduit.
(ii) Flow around a uniform diameter pipe-bend or a canal bend.

One, two and three dimensional flow
One dimensional flow
It is that type of flow in which the flow parameter such as velocity is a
function of time and one space co-ordinate only. Mathematically.
u = f(x),v = 0 and w = 0
where u, v and ware velocity components in x, y and z directions respectively.
.
Example
Flow in a pipe where average flow parameters are considered for analysis

Two dimensional flow.
The flow in which the velocity is a function of time and two
rectangular space coordinates is called two dimensional flow.
Mathematically,
Examples.
(i) Flow between parallel plates of infinite extent.
(ii) Flow in the main stream of a wide river.

Three dimensional flow.
It is that type of flow in which the velocity is a function of time and three
mutually perpendicular directions. Mathematically,
Examples.
(i) Flow in a converging or diverging pipe or channel.
(ii) Flow in a prismatic open channel in which the width and the water depth are
of the same order of magnitude.

Rotational and irrotational flow
Rotational flow.
A flow is said to be rotational if the fluid particles while moving in the
direction of flow rotate about their mass centers. Flow near the solid
boundaries is rotational.
Example. Motion of liquid in a rotating tank.
Irrotational flow,
A flow is said to be irrotational if the fluid particles while moving in the
direction of flow do not rotate about their mass centers. Flow outside the
boundary layer is generally considered irrotationa1.
Example. Flow above a drain hole of a stationary tank or a wash basin.

Laminar and turbulent flow
Laminar flow.
A laminar flow is one in which paths taken by the individual particles do not
cross one another and move along well defined paths , This type of flow is also
called stream-line flow or viscous flow,
Examples.(i) Flow through a capillary tube .
. (ii) - Flow of blood in veins and arteries.
(iii} Ground water flow.
Turbulent flow.
A turbulent flow is that flow in which fluid particles move
in a zig zag way ,Example. High velocity flow in a conduit
of large size. Nearly all fluid flow problems encountered in
engineering practice have a turbulent character
For Reynolds number (Re) < 2000
For Reynolds number (Re) > 4000
For Re between 2000 and 4000
flow in pipes is laminar,
flow in pipes is turbulent
flow in pipes may be laminar or turbulent.

Compressible flow.
It is that type of flow in which the density (p) of the fluid changes from
point to point (or in other words density is not constant for this flow
Mathematically
Example. Flow of gases through orifices, nozzles, gas turbines, etc .
Incompressible flow
It is that type of flow in which density is constant for the fluid flow.
Liquids are generally considered flowing incompressible.
Mathematically,
Example. Subsonic aerodynamics
Compressible an incompressible flow

Example (1)
In a fluid, the velocity field is given by

Example(1)

Example(2)

Example (3)

Equation of Continuity
 Equation of Continuity: A relation
between the speed v of an ideal fluid
flowing through a tube of cross
sectional area A in steady flow state
 Since fluid is incompressible, equal
volume of fluid enters and leaves the
tube in equal time
 Volume V flowing through a tube
in time t is
V = A x =A vt
Then V = A1v1t =A2v2t
 A1v1=A2v2
 RV=A1v1=constant (Volume flow
rate)
 Rm=A1v1=constant (Mass flow
rate)

Example (4)
A pipe (J) 450 mm in diameter _
branches into two pipes (2 and 3) of diameters 300 mm and 200 mm respectively .If the average
velocity in 450 mm diameter pipe is 3 m/s find:
(i) Discharge through 450 mm diameter pipe;
(ii) Velocity in 200 mm diameter pipe if the average velocity in 300 mm pipe is 2.5 m/s.

Example(4)

Post test
A
Q1)) (5 mark)
Conical pipe diverges uniformly From 100 mm to 200 m diameter over a length „1
m. Determine the local and convective acceleration at the mid-section assuming
(i) Rate of flow is 0.12 m3/s and it remains constant;
(ii) Rate of flow varies uniformly from,0.12 m3/s to 0.24 m3ls in 5 sec., at t = 2 sec.
Q2)) (5 mark)
The diameter of a pipe at the section 1-1 and 2-2 are 200mm and 300mm respectively.
If the velocity of water flowing through the pipe section at 1-1 4msec, find
1) discharge through the pipe
2)velocity of water at section 2-2

Key answer
pre test
Q1))
1- ( c ) 2- ( d ) 3- ( b ) 4- ( b) 5- ( c ) 6- ( d ) 7- ( c )
Q2))

Key answer
post test
Q1))

Key answer

Key answer
Q2))

References
CH1: Fluid Properties
1. Evett, J., B. and Liu, C. 1989 “2500 solved problems in fluid mechanics and
hydraulics” Library of Congress Cataloging- in-Publication Data, (Schaum's
solved problems series) ISBN 0-07-019783-0
2. Rajput, R.,K. 2000 “ A Text Book of Fluid Mechanics and Hydraulic
Machines”. S.Chand & Company LTD.
3. White, F., M. 2000 “ Fluid Mechanics”. McGraw-Hill Series in Mechanical
Engineering.
4. Wily, S., 1983 “ Fluid Mechanics”. McGraw-Hill Series in Mechanical
Engineering.

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