fundamentals of fluid flow
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Fundamentals of Fluid Flow
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To study the behaviour of fluid in motion we need to
consider several factors.
Whereas, behaviour of fluid in rest is influenced by its
specific weight only.
The science of study which deals with fluid motion and
forces causing its motion is known as “Fluid Kinetics”
Introduction
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In fluid kinetics it is necessary to observe the fluid behaviour
with respect to space as well as with time.
Lagrangian Approach: An individual fluid particle is selected
and attention is focussed on the behaviour of particle during its
course of motion through out the space.
Eulerian Approach: A fixed volume in space is selected and
attention is focussed on the fluid behaviour with respect to this
volume.
Introduction
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Velocity of a fluid particle is the change in displacement with
respect to change in time.
Velocity of a fluid particle will have three components and in
vector notation it is represented as,
Velocity components are in general function of x, y, z and t.
Velocity of Fluid Particles
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0t
sV Lt
t
ˆˆ ˆV ui vj wk
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Velocity components are in general function of x, y, z and t.
When x, y and z are constants the velocity represent
velocity of fluid particle at a point for different time
interval.
When t is constant then it represents velocity at different
location in the flow field for a given instant of time.
Velocity of Fluid Particles
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Steady and unsteady flow,
Uniform and non-uniform flow,
One dimensional and multi dimensional flow,
Rotational and irrotational flow, and
Laminar and Turbulent flow.
Types of fluid flow
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Steady flow:
When the fluid properties like velocity, pressure, density
etc., which describe the fluid behaviour at a point, doesn’t
change with respect to time.
Types of fluid flow
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0, 0, in gener l 0a u v
tt t
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Unsteady flow:
When the fluid properties like velocity, pressure, density
etc., which describe the fluid behaviour at a point, change
with respect to time.
Types of fluid flow
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0, 0, in gener l 0a u v
tt t
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Flow of liquids under pressure through long pipe line of
constant diameter is an example of such type of flow.
Types of fluid flow
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0, 0, in gener l 0a u v V
Sx y
Uniform flow:
When the fluid velocity either in magnitude or in direction,
doesn’t change with respect to position at a particular instant
time.
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Flow of liquids under pressure through long pipe line of
varying diameter is an example of such type of flow.
Types of fluid flow
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0, 0, in gener l 0a u v V
Sx y
Non-Uniform flow:
When the fluid velocity changes with respect to position at a
particular instant time.
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Types of fluid flow
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Note:
All the flows can exist independent of each other, so that
any of the four types of combinations of flows are possible.
Flow through constant diameter pipes leads to uniform flow.
Flow with constant rate leads to steady flow.
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Types of fluid flow
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Multi dimensional flow:
When the fluid velocity is function of more than one
principal coordinates and independent of time.
V = f(x, y, z) or V = f(x, y) or V = f(y, z) or V = f(x, z)
Steady Multi dimensional flow:
When the fluid velocity is function of more than one
principal coordinates and dependent of time.
V = f(x, y, z, t) or V = f(x, y, t) or V = f(y, z, t) or V = f(x, z, t)
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Types of fluid flow
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One dimensional flow:
When the fluid velocity is function of any one of the
principal coordinates and independent of time.
V = f(x) or V = f(y) or V = f(z)
Steady One dimensional flow:
When the fluid velocity is function of any one of the
principal coordinates and dependent of time.
V = f(x, t) or V = f(y, t) or V = f(z, t)
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Types of fluid flow
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Rotational flow:
Fluid particles rotate about their mass centres while they are
moving.
Hence they will have some angular velocity, ω
Irrotational flow:
Fluid particles do not rotate about their mass centres while
they are moving.
Hence they will not have any angular velocity, ω
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Types of fluid flow
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Laminar flow:
Fluid particles move in layers with one layer of fluid sliding
smoothly over the adjacent layer.
Flow of viscous fluid is an example of laminar flow.
Turbulent flow:
Fluid particles move in an entirely disorderly manner, that results
in rapid and continuous mixing of fluid leading to momentum
transfer across the flow.
Flow in natural steams, artificial channels, water supply pipes etc.,
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Fluid flow can be described by:
Steam lines,
Stream tubes,
Path lines, and
Steak lines.
Fluid flow description
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Steam lines:
It is an imaginary line drawn in the flow field such that its tangent
at a point will gives us velocity of fluid at that point.
The pattern of fluid flow is represented by a series of steam lines.
The differential equation for a steam line in multi dimensional
coordinates is,
Fluid flow description
x y z
u v w
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Steam lines:
Across a steam line there can be no fluid flow.
For steady flow, steam line pattern doesn’t change with respect to
time.
For unsteady flow, steam line pattern changes with respect to time.
Fluid flow description
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Steam tubes:
It is an imaginary tube formed by a group of steam lines passing
through a small closed curve, which may or may not be circular.
There will not be any flow across the boundary of stream tube.
Fluid will enter or leave the steam tube through its ends only.
For flow pattern analysis stream tubes are used.
Fluid flow description
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Path Lines:
It is a line traced by a single fluid particle as it moves over a
period of time.
It indicates the direction of velocity of fluid particle at successive
instant of time.
In steady flow the path line and steam lines are identical.
In unsteady flow both are different.
Fluid flow description
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Streak Lines:
It is a line traced by a fluid particles having same velocity between
different path lines at the instant of time.
In steady flow streak line, path line and steam lines are identical.
In unsteady flow they are different.
Fluid flow description
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Principle of conservation of mass:
Mass is conserved.
Principle of conservation of energy:
Energy conserved.
Principle of conservation of momentum:
Momentum is conserved.
Basic principles of fluid flow
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Fundamental Principles of Conservation
Principles which govern basic equations used in CFD as
well as in analytical fluid dynamics
What basically conservation talks about
Bank Account
Money In Money Out Balance
+
Interest
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Fundamental Principles of Conservation
Statement of conservation
Control
Volume
General
Quantity In
General
Quantity Out Quantity
Generation
Change in
Money In - Out GenerationMoney Transfer
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Fundamental Principles of Conservation
Control Volume, CV:
Identified region in space across which changes in a
particular quantity are found
When we’ve a CV we can write a balance for the CV
Difficulty lies in utilization of principles of basic
mechanics
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Fundamental Principles of Conservation
Newton’s law of motion are developed based on
“Lagrangian description”
Sets of particles n their trajectory which r governed by
Newton’s law of motion
Where as in “Eulerian description” we’ve CV n we r
interested to study transport phenomena across CV
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Reynold’s Transport Theorm
Convert a system from “Lagrangian description” to
“Eulerian description”
System is essentially an identified collection of particles
of fixed mass and identity
System at „t‟ System at „t + ∆t‟
III I II
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Reynold’s Transport Theorm
System at „t‟ System at „t + ∆t‟
III I II
Consider a property of the system “N”
t I IIt t
t t II IIIt t t t
N N N
N N N
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Reynold’s Transport Theorm
Rate of change of „N’ of the system
II II III It t t t t t t t t
sys
N N N NN NdN
dt t t t t
• Rate of outflow of ‘N’ across CV III t tN
t
• Rate of inflow of ‘N’ across CV I tN
t
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Reynold’s Transport Theorm
Rate of change of „N’ of the system
0
Rate of
Out flow-In flow
t t t
tsys CV
N NdN NLt
dt t t
CS
n V dA V
• Rate of {out flow – in flow}
dA
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Reynold’s Transport Theorm
Conservation of the property
sys CV CS
dN Nn V dA
dt t
• RTT relates total rate of change of a property related to a
system of fixed mass & identity with the rate of change of
the property w.r.t a CV and a balance of outflow n inflow
across the system boundary/surface.
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Conservation of Mass
Here we consider, N = m, → n = 1
sys CV CS
dm mV dA
dt t
• Assumptions:
– A non deformable control volume {volume ≠ f (t)}
– Control volume to be stationary {rel. vel. = abs vel.}
– When we consider system of fixed mass LHS will
automatically equal to zero
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Conservation of Mass
The earlier expression changes to
0
. 0
CV CS
CV CS
dV V dAt
dV V dAt
• Convert surface integral to volume integral by using
divergence theorem
. .CS CV
V dA V dV
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Conservation of Mass
The earlier expression changes to
. 0
. 0
CV CV
CV
dV V dVt
V dVt
• Since choice of elemental volume is arbitrary, hence the
value of integrand itself is zero
. 0Vt
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Conservation of Mass
Finally we tried to convert integral form of conservation
of mass into differential form
• This is well known “Continuity Equation”
. 0Vt
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In a two dimensional fluid flow the component of
the velocity along the x axis is given as follows
Determine the component of the velocity along the y-
axis for the condition of continuity of flow.
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Exercise 01
2 33 2u x x y y
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In a three dimensional fluid flow the velocity
components given as follows
State the whether flow is continuous or not.
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Exercise 02
2 2 2 2, , 2u x y v y z w x y z
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In a three dimensional fluid flow the velocity
components given as follows
Find the velocity component in z direction if it
satisfies the continuity of fluid flow.
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Exercise 03
2 2 2 22 8, 2 6u x z v y z
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Show that two dimensional fluid flow is continuous
with the following velocity components
Also find whether the flow is irrotational or not?
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Exercise 04
2 26 , 3 3u xy v x y
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In a two dimensional fluid flow x velocity component is
given by
Find the y velocity component of fluid flow with a
condition u = 0 at y = 0. Assume that the continuity is
satisfied.
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Exercise 05
2 2u ax bxy cy