Fundamentals of Fluid Flow

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

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

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

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.

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.

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.

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)

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)

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

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.,

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,


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.


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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.


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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.


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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.


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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

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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Statement of conservation

Control

Volume

General

Quantity In

General

Quantity Out Quantity

Generation

Change in

Money In - Out GenerationMoney Transfer

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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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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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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


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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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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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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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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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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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

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

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

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

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

Dr. Mohan Jagadeesh Kumar Mandapati

Associate Professor

SMBS, VIT University

[email protected]

Fundamentals of Fluid Flow

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