“STUDY OF CONVERGENT-DIVERGENT NOZZLE”

A MINOR PROJECT REPORT

Submitted by

A. RAHUL RAO 0206ME061001ANIKET SINGH 0206ME061005KAPIL PATEL 0206ME061019MADAN DHAKAL 0206ME061021MANISH R. BHATIA 0206ME073D03NAVEEN KUMAR NAMDEO 0206ME061025

DEPARTMENT OF MECHANICAL ENGINEERING

GYAN GANGA INSTITUTE OF TECHNOLOGY & SCIENCES

JABALPUR (M.P.)

RAJIV GANDHI PRODYOGIKI VISHWAVIDYALAYA,

BHOPAL (M.P.)

DECEMBER- 2009

TABLE OF CONTENT

CHAPTER NO. TITLE PAGE NO.

CERTIFICATE I DECLARATION II ACKNOWLEDGEMENT III ABSTRACT IV LIST OF TABLES V LIST OF FIGURES V

1. INTRODUCTION 1

2. LITERATURE REVIEW 2

2.1 FUNCTION OF PATTERNS 2

2.2 TYPES OF PATTERNS 2

2.3 PATTERN DESIGN AND

CONSIDERATIONS 8

2.4 SELECTION OF PATTERN MATERIAL 9

2.4.1 MATERIALS USED FOR PATTERNS

2.4.1.1 WOOD 9

2.4.1.2 METAL 10

2.4.1.3 POLYSTERENE 10

2.4.1.4 PLASTER 15

2.4.1.5 WAX 16

2.5 WOOD USED FOR PATTERN 16

2.6 PATTERN CONSTRUCTION AND STEPS 19

INVOLVED

2.6.1 PATTERN LAYOUT 19

2.6.2 PATTERN COLORS 19

2.7 PATTERN ALLOWANCES 20

2.8 CALCULATIONS 24

CERTIFICATE

This is to certify that the Minor Project report entitled “STUDY OF

CONVERGENT-DIVERGENT NOZZLE” is submitted by A. RAHUL

RAO, ANIKET SINGH, KAPILPATEL, MADAN DHAKAL, NAVEEN

KUMAR NAMDEO & MANISH R. BHATIA in Mechanical Engineering

from “RAJIV GANDHI PROUDYOGIKI VISHWAVIDYALAYA,

BHOPAL (M.P.)”

Prof. P. K. Sahu Prof. K.L. KanojiaGuide HOD Dept. of Mechanical Engg. Dept. of Mechanical Engg.

CERTIFICATE

This is to certify that the Minor Project report entitled “STUDY OF

CONVERGENT-DIVERGENT NOZZLE” is submitted by A. RAHUL

RAO,ANIKET SINGH, KAPIL PATEL, MADAN DHAKAL, NAVEEN

KUMAR NAMDEO & MANISH R. BHATIA in Mechanical Engineering

from “RAJIV GANDHI PROUDYOGIKI VISHWAVIDYALAYA,

BHOPAL (M.P.)”

Internal Examiner External Examiner

Date: Date:

DECLARATION

We hereby declare that the project entitled “STUDY OF CONVERGENT-

DIVERGENT NOZZLE” which is being submitted to “RAJIV GANDHI

PROUDYOGIKI VISHWAVIDYALAYA, BHOPAL (M.P.)” is an

authentic record of our own work done under the guidance of Prof. P. K.

SAHU, Department of Mechanical Engineering, GYAN GANGA

INSTITUTE OF TECHNOLOGY & SCIENCES, JABALPUR.

The matter reported in this Project has not been submitted earlier

for the award of any other degree.

Dated : / / A. RAHUL RAO ANIKET SINGH Place : Jabalpur KAPILPATEL MADAN DHAKAL NAVEEN KUMAR NAMDEO MANISH R. BHATIA

ACKNOWLEDGEMENT

We sincerely express indebtedness to esteemed and revered guide “Prof. P. K. Sahu”, Professor in Mechanical Department for his invaluable guidance, supervision and encouragement throughout the work. Without his kind patronage and guidance the project would not have taken shape.

We take this opportunity to express deep sense of gratitude to “Prof. K.L. Kanojia”, Head of “Mechanical Department” for his encouragement and kind approval. We would like to express our sincere regards to him for advice and counseling from time to time.

We owe sincere thanks to all the lecturers in “Mechanical Department” for their advice and counseling time to time.

Dated: / / A. RAHUL RAOPlace: Jabalpur ANIKET SINGH KAPILPATEL MADAN DHAKAL NAVEEN KUMAR NAMDEO MANISH R. BHATIA

ABSTRACT

This project is a effort by our team towards the Study of convergent-divergent nozzle.

The purpose of this is to simulate the operation of a converging-diverging nozzle, perhaps

the most important and basic piece of engineering hardware associated with propulsion

and the high speed flow of gases. This device was invented by Carl de Laval toward the

end of the l9th century and is thus often referred to as the 'de Laval' nozzle. This applet is

intended to help students of compressible aerodynamics visualize the flow through this

type of nozzle at a range of conditions.

A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is

basically a tube that is pinched in the middle, making an hourglass-shape. It is used as a

means of accelerating the flow of a gas passing through it. It is widely used in some types

of steam turbine and is an essential part of the modern rocket engine and supersonic jet

engines. It is used in rocketry, where its use can achieve speeds of 4 km/s or more; jet

engines, and in wind tunnels where it is capable of producing smoother airflow than other

techniques.

We have also included the basics of Computational fluid Dynamics. Computational Fluid

Dynamics (CFD) is a computer-based tool for simulating the behavior of systems

involving fluid flow, heat transfer, and other related physical processes.

LIST OF FIGURES:-

Figure1:-Diagram of Solid Pattern

Figure2:-Diagram of Split Pattern

Figure3:-Diagram of Match Plate Pattern

Figure4:-Diagram of Gated Pattern

Figure5:-Diagram of Sweep Pattern

Figure6:-Wooden Pattern

LIST OF TABLES:

Table1:- Strength of different woods

Table2:- Shrinkage Allowance

Nozzle

A nozzle is a mechanical device designed to control the characteristics of a fluid flow as

it exits (or enters) an enclosed chamber or pipe via an orifice. A nozzle is often a pipe or

tube of varying cross sectional area and it can be used to direct or modify the flow of a

fluid (liquid or gas). Nozzles are frequently used to control the rate of flow, speed,

direction, mass, shape, and/or the pressure of the stream that emerges from them.

Types of nozzles

Jets

A gas jet, fluid jet, or hydro jet is a nozzle intended to eject gas or fluid in a coherent

stream into a surrounding medium. Gas jets are commonly found in gas stoves, ovens, or

barbecues. Gas jets were commonly used for light before the development of electric

light. Other types of fluid jets are found in carburetors, where smooth calibrated orifices

are used to regulate the flow of fuel into an engine, and in jacuzzis or spas. Another

specialized jet is the laminar jet. This is a water jet that contains devices to smooth out

the flow, and gives laminar flow, as its name suggests. This gives better results for

fountains. Nozzles used for feeding hot blast into a blast furnace or forge are called

tuyeres.

High velocity nozzles

Frequently the goal is to increase the kinetic energy of the flowing medium at the

expense of its pressure and internal energy.

Nozzles can be described as convergent (narrowing down from a wide diameter to a

smaller diameter in the direction of the flow) or divergent (expanding from a smaller

diameter to a larger one). A de Laval nozzle has a convergent section followed by a

divergent section and is often called a convergent-divergent nozzle ("con-di nozzle").

Convergent nozzles accelerate subsonic fluids. If the nozzle pressure ratio is high enough

the flow will reach sonic velocity at the narrowest point (i.e. the nozzle throat). In this

situation, the nozzle is said to be choked.

Increasing the nozzle pressure ratio further will not increase the throat Mach number

beyond unity. Downstream (i.e. external to the nozzle) the flow is free to expand to

supersonic velocities. Note that the Mach 1 can be a very high speed for a hot gas; since

the speed of sound varies as the square root of absolute temperature. Thus the speed

reached at a nozzle throat can be far higher than the speed of sound at sea level. This fact

is used extensively in rocketry where hypersonic flows are required, and where propellant

mixtures are deliberately chosen to further increases the sonic speed.

Divergent nozzles slow fluids, if the flow is subsonic, but accelerate sonic or supersonic

fluids. Convergent-divergent nozzles can therefore accelerate fluids that have choked in

the convergent section to supersonic speeds. This CD process is more efficient than

allowing a convergent nozzle to expand supersonically externally. The shape of the

divergent section also ensures that the direction of the escaping gases is directly

backwards, as any sideways component would not contribute to thrust.

Propelling nozzles

A jet exhaust produces a net thrust from the energy obtained from combusting fuel which

is added to the inducted air. This hot air is passed through a high speed nozzle, a

propelling nozzle which enormously increases its kinetic energy.

For a given mass flow, greater thrust is obtained with a higher exhaust velocity, but the

best energy efficiency is obtained when the exhaust speed is well matched with the

airspeed. However, no jet aircraft can sustain flight while exceeding its exhaust jet speed,

due to momentum considerations. Supersonic jet engines, like those employed in fighters

and SST aircraft (e.g. Concorde), need high exhaust speeds. Therefore supersonic aircraft

very typically use a CD nozzle despite weight and cost penalties. Subsonic jet engines

employ relatively low, subsonic, exhaust velocities. They thus employ simple convergent

nozzles. In addition, bypass nozzles are employed giving even lower speeds.

Rocket motors use convergent-divergent nozzles with very large area ratios so as to

maximize thrust and exhaust velocity and thus extremely high nozzle pressure ratios are

employed. Mass flow is at a premium since all the propulsive mass is carried with

vehicle, and very high exhaust speeds are desirable.

Magnetic nozzles

Magnetic nozzles have also been proposed for some types of propulsion, such as

VASIMR, in which the flow of plasma is directed by magnetic fields instead of walls

made of solid matter.

Spray nozzles

Many nozzles produce a very fine spray of liquids.

Atomizer nozzles are used for spray painting, perfumes, carburettors for internal

combustion engines, spray on deodorants, antiperspirants and many other uses.

Air-Aspirating Nozzle-uses an opening in the cone shaped nozzle to inject air

into a stream of water based foam (CAFS/AFFF/FFFP) to make the concentrate

"foam up". Most commonly found on foam extinguishers and foam handlines.

Swirl nozzles inject the liquid in tangentially, and it spirals into the center and

then exits through the central hole. Due to the vortexing this causes the spray to

come out in a cone shape.

Vacuum nozzles

Vacuum cleaner nozzles come in several different shapes.

Shaping nozzles

Some nozzles are shaped to produce a stream that is of a particular shape. For example

Extrusion molding is a way of producing lengths of metals or plastics or other materials

with a particular cross-section. This nozzle is typically referred to as a die.

CONVERGENT-DIVERGENT NOZZELS

Laval nozzle is a Convergent-divergent nozzle in which subsonic flow prevails in the

converging section, critical od transonic conditions in the throat and supersonic flow in

the diverging section. The opening narrows then opens back up, as you can see in the

picture below. These are also known as De Leval nozzles (named after de Laval, the

Swedish scientist who invented it). This type of nozzle is an hourglass shape, which

accelerates the exhaust to very high velocities. The convergent portion of the nozzle

accelerates the exhaust. This acceleration produces much more thrust than the engine

would without a nozzle. This device was invented by Carl de Laval toward the end of the

l9th century and is thus often referred to as the 'de Laval' nozzle

It consists of three main parts which are:

a.) Short Convergent Part (Velocity increase towards the end of this part)

b.) Short throat part. (Here Maximum Mass Flow rate is achieved.)

c.) Divergent Part i.e. diffuser (It is designed in the manner that it doesn’t have effect on

the mass flow rate achieved at throat.

Mach number

Mach number (Ma or M) (is the speed of an object moving through air, or any fluid

substance, divided by the speed of sound as it is in that substance. It is commonly used to

represent an object's (such as an aircraft or missile) speed, when it is traveling at (or at

multiples of) the speed of sound.

An F/A-18 Hornet at transonic speed and displaying the Prandtl–Glauert singularity just

before reaching the speed of sound

Where,

is the Mach number

is the speed of the source (the object relative to the medium) and

is the speed of sound in the medium

The Mach number is named after Czech/Austrian physicist and philosopher Ernst Mach.

Because the Mach number is often viewed as a dimensionless quantity rather than a unit

of measure, with Mach, the number comes after the unit; the second Mach number is

"Mach 2" instead of "2 Mach" (or Machs). This is somewhat reminiscent of the early

modern ocean sounding unit "mark" (a synonym for fathom), which was also unit-first,

and may have influenced the use of the term Mach. In the decade preceding man's flying

faster than sound, aeronautical engineers referred to the speed of sound as Mach's

number, never "Mach 1".

The Mach number is commonly used both with objects traveling at high speed in a fluid,

and with high-speed fluid flows inside channels such as nozzles, diffusers or wind

tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At a

temperature of 15 degrees Celsius, the speed of sound is 340.3 m/s (1225 km/h, or 761.2

mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed represented by

Mach 1 is not a constant; for example, it is dependent on temperature and atmospheric

composition. In the stratosphere it remains constant irrespective of altitude even though

the air pressure varies with altitude.

Since the speed of sound increases as the temperature increases, the actual speed of an

object traveling at Mach 1 will depend on the fluid temperature around it. Mach number

is useful because the fluid behaves in a similar way at the same Mach number. So, an

aircraft traveling at Mach 1 at sea level (340.3 m/s, 761.2 mph, 1,225 km/h) will

experience shock waves in much the same manner as when it is traveling at Mach 1 at

11,000 m (36,000 ft), even though it is traveling at 295 m/s (654.6 mph, 1,062 km/h, 86%

of its speed at sea level).

High-speed flow around objects

Flight can be roughly classified in five categories:

Subsonic: M < 1

Sonic: M=1

Transonic: 0.8 < M < 1.2

Supersonic: 1.2 < M < 5

Hypersonic: M > 5

For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = M

25.4 in air at high altitudes. The speed of light in vacuum corresponds to a Mach number

of approximately 880,000 (relative to air at sea level).

At transonic speeds, the flow field around the object includes both sub- and supersonic

parts. The transonic period begins when first zones of M>1 flow appear around the

object. In case of an airfoil (such as an aircraft's wing), this typically happens above the

wing. Supersonic flow can decelerate back to subsonic only in a normal shock; this

typically happens before the trailing edge. (Fig. a)

As the speed increases, the zone of M>1 flow increases towards both leading and trailing

edges. As M=1 is reached and passed, the normal shock reaches the trailing edge and

becomes a weak oblique shock: the flow decelerates over the shock, but remains

supersonic. A normal shock is created ahead of the object, and the only subsonic zone in

the flow field is a small area around the object's leading edge. (Fig. b)

(a) (b)

Mach number in transonic airflow around an airfoil; M<1 (a) and M>1 (b).

When an aircraft exceeds Mach 1 (i.e. the sound barrier) a large pressure difference is

created just in front of the aircraft. This abrupt pressure difference, called a shock wave,

spreads backward and outward from the aircraft in a cone shape (a so-called Mach cone).

It is this shock wave that causes the sonic boom heard as a fast moving aircraft travels

overhead. A person inside the aircraft will not hear this. The higher the speed, the more

narrow the cone; at just over M=1 it is hardly a cone at all, but closer to a slightly

concave plane.

At fully supersonic speed, the shock wave starts to take its cone shape and flow is either

completely supersonic, or (in case of a blunt object), only a very small subsonic flow area

remains between the object's nose and the shock wave it creates ahead of itself. (In the

case of a sharp object, there is no air between the nose and the shock wave: the shock

wave starts from the nose.)

As the Mach number increases, so does the strength of the shock wave and the Mach

cone become increasingly narrow? As the fluid flow crosses the shock wave, its speed is

reduced and temperature, pressure, and density increase. The stronger the shock, the

greater the changes. At high enough Mach numbers the temperature increases so much

over the shock that ionization and dissociation of gas molecules behind the shock wave

begin. Such flows are called hypersonic.

It is clear that any object traveling at hypersonic speeds will likewise be exposed to the

same extreme temperatures as the gas behind the nose shock wave, and hence choice of

heat-resistant materials becomes important.

High-speed flow in a channel

As a flow in a channel crosses M=1 becomes supersonic, one significant change takes

place. Common sense would lead one to expect that contracting the flow channel would

increase the flow speed (i.e. making the channel narrower results in faster air flow) and at

subsonic speeds this holds true. However, once the flow becomes supersonic, the

relationship of flow area and speed is reversed: expanding the channel actually increases

the speed.

The obvious result is that in order to accelerate a flow to supersonic, one needs a

convergent-divergent nozzle, where the converging section accelerates the flow to M=1,

sonic speeds, and the diverging section continues the acceleration. Such nozzles are

called de Laval nozzles and in extreme cases they are able to reach incredible, hypersonic

speeds (Mach 13 at sea level).

An aircraft Mach meter or electronic flight information system (EFIS) can display Mach

number derived from stagnation pressure (pitot tube) and static pressure.

Calculating Mach number

Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic

compressible flow is derived from Bernoulli's equation for M<1:

where:

is Mach number

is impact pressure and

is static pressure

is the ratio of specific heat of a gas at a constant pressure to heat at a constant

volume (1.4 for air).

The formula to compute Mach number in a supersonic compressible flow is derived from

the Rayleigh Supersonic Pitot equation:

Where:

is now impact pressure measured behind a normal shock.

Shock wave

Whenever a supersonic flow (compressible) abruptly changes to subsonic flow, a shock

wave (analogous to hydraulic jump in an open channel) is produced, resulting in a sudden

rise in pressure, density, temperature and entropy. This occurs due to pressure

differentials and when the Mach number of the approaching flow M1>1. A shock wave is

a pressure wave of finite thickness, of the order of 10-2 to 104mm in the atmospheric

pressure. A shock wave takes place in the diverging section of a nozzle, in a diffuser,

throat of a supersonic wind tunnel, in front of sharp-nosed bodies.

Shock waves are of two types:

1. Normal shocks which are almost perpendicular to the flow.

2. Oblique shocks which are inclined to the flow direction.

Back pressure

Back pressure usually refers to the pressure exerted on a moving fluid by obstructions or

tight bends in the confinement vessel along which it is moving, such as piping or air

vents, against its direction of flow. For example, an automotive exhaust muffler with a

particularly high number of twists, bends, turns and right angles could be described as

having particularly high back pressure. Back pressure, in the exhaust sense of the term, of

a four-stroke engine is usually termed as being a "bad thing" for performance; however,

in the interest of reducing exhaust sound to levels allowable by public noise ordinances,

back pressure can be regulated using systems from simple butterfly valves to fully

computer controlled units sensing pressure in the exhaust pipe itself.

In a two-stroke engine however, a certain amount of exhaust backpressure is needed to

prevent unburned fuel/air mixture to pass right through the cylinders into the exhaust.

Fig 1.1 illustrates the effects of varying back pressure on a converging–diverging nozzle.

The series of cases labeled a through j is considered next

1. Let us first discuss the cases designated a, b, c, and d. Case a corresponds to pB = pE =

po for which there is no flow. When the back pressure is slightly less than po (case b),

there is some flow, and the flow is subsonic throughout the nozzle. In accordance with

the discussion of Fig. 1.2, the greatest velocity and lowest pressure occur at the throat,

and the diverging portion acts as a diffuser in which pressure increases and velocity

decreases in the direction of flow. If the back pressure is reduced further, corresponding

to case c, the mass flow rate and velocity at the throat are greater than before. Still, the

flow remains subsonic throughout and qua –litatively the same as case b. As the back

pressure is reduced, the Mach number at the throat increases, and eventually a Mach

number of unity is attained there (case d). As before, the greatest velocity and lowest

pressure occur at the throat, and the diverging portion remains a subsonic diffuser.

However, because the throat velocity is sonic, the nozzle is now choked: The maximum

mass flow rate has been attained for the given stagnation conditions. Further reductions in

back pressure cannot result in an increase in the mass flow rate.

2. When the back pressure is reduced below that corresponding to case d, the flow

through the converging portion and at the throat remains unchanged. Conditions within

the diverging portion can be altered, however, as illustrated by cases e, f, and g. In case e,

the fluid passing the throat continues to expand and becomes supersonic in the diverging

portion just downstream of the throat; but at a certain location an abrupt change in

properties occurs. This is called a normal shock. Across the shock, there is a rapid and

irreversible increase in pressure, accompanied by a rapid decrease from supersonic to

subsonic flow. Downstream of the shock, the diverging duct acts as a subsonic diffuser in

which the fluid continues to decelerate and the pressure increases to match the back

pressure imposed at the exit. If the back pressure is reduced further (case f), the location

of the shock moves downstream, but the flow remains qualitatively the same as in case e.

With further reductions in back pressure, the shock location moves farther downstream of

the throat until it stands at the exit (case g). In this case, the flow throughout the nozzle is

isentropic, with subsonic flow in the converging portion, M =1 at the throat, and

supersonic flow in the diverging portion. Since the fluid leaving the nozzle passes

through a shock, it is subsonic just downstream of the exit plane.

3. Finally, let us consider cases h, i, and j where the back pressure is less than that

corresponding to case g. In each of these cases, the flow through the nozzle is not

affected. The adjustment to changing back pressure occurs outside the nozzle. In case h,

the pressure decreases continuously as the fluid expands isentropically through the nozzle

and then increases to the back pressure outside the nozzle. The compression that occurs

outside the nozzle involves oblique shock waves. In case i, the fluid expands

isentropically to the back pressure and no shocks occur within or outside the nozzle. In

case j, the fluid expands isentropically through the nozzle and then expands outside the

nozzle to the back pressure through oblique expansion waves. Once M = 1 is achieved at

the throat, the mass flow rate is fixed at the maximum value for the given stagnation

conditions, so the mass flow rate is the same for back pressures corresponding to cases d

through j. The pressure variations outside the nozzle involving oblique waves cannot be

predicted using the one-dimensional flow model. In case of ideal nozzle isentropic

expansion is considered, with nozzle efficiency at unity. And flow at divergent (diffuser)

section as frictionless, but in actual it’s not feasible.

Choked flow

At the "throat", where the cross sectional area is a minimum, the gas velocity locally

becomes sonic (Mach number = 1.0), a condition called choked flow. Choked flow of a

fluid is a Fluid dynamics condition caused by the Venturi effect. When a flowing fluid at

a certain pressure and temperature flows through a restriction into a lower pressure

environment, under the conservation of mass the fluid velocity must increase for initially

subsonic upstream conditions as it flows through the smaller...

. As the nozzle cross sectional area increases the gas begins to expand and the gas flow

increases to supersonic velocities where a sound wave will not propagate backwards

through the gas as viewed in the frame of reference of the nozzle.

The Fluid Flow in Duct

During the fluid flow in a duct, the properties of fluid changes along the stream line. In

most situations this can be treated as a one-dimensional flow

*The Pressure and Velocity

From conservation of energy

dh= - cdc

du +d(pv)= - cdc

du+pdv+vdp= -cdc

From the first law of thermodynamics:

δq=du+pdv=0

Then: du+pdv+vdp= -cdc

vdp= - cdc

- vdp= cdc

To increase the velocity of fluid (dc>0), the pressure must be decreased. This kind of

duct is called Nozzle Or to decrease the velocity (dc<0) of fluid to obtain a high pressure

in a duct flow. This kind of duct is called diffuser.

Velocity and the Cross Section Area of Duct

From conservation of mass

From the process equation

v

dvk

p

dp

As to a nozzle dc>0

c

dcMa

A

dA)1( 2

(1)If the fluid velocity is subsonic, then (Ma2-1) <0 Therefore: dA<0 The nozzle’s shape should be as following:

(2)If the fluid velocity is ultrasonic, then (Ma2-1)>0 Therefore: dA>0 The nozzle’s shape should be as following:

(3)If the nozzle’s inlet velocity is subsonic, but outlet velocity ultrasonic, then:

dA<0 → dA=0 → dA>0

The nozzle’s shape should be as following:

The Critical Compression Ratio

Ultrasonic flow

Ultrasonic flow

divergent nozzle

Subsonic flow Ultrasonic flow

Convergent-divergent nozzle

Throat

For convergent-divergent nozzle, the velocity at throat keeps as sound-velocity. This state

is called critical state. The flux of this kind of nozzle is depending on that of throat.

The Compression Ratio which is low enough to make the air flow at sound-velocity at the

exit of nozzle is called the critical compression Ratio.

It is denoted by εc

Effect of Area Variation on Flow Properties in Isentropic Flow

In considering the effect of area variation on flow properties in isentropic flow, we shall

concern ourselves primarily with the velocity and pressure. We shall determine the effect

of change in area, A, on the velocity V, and the pressure p.

From Bernoulli's equation, we can write

1

1

2

k

k

c k

εc=

0.528 For air

0.546 For saturated steam

0.577 For superheated steam

Dividing by , we obtain

(19.1)

A convenient differential form of the continuity equation

Substituting from Eq. (19.1),

Invoking the relation ( ) for isentropic process in Eq. (19.2), we get

From Eq. (19.3), we see that for Ma<1 an area change causes a pressure change of the

same sign, i.e. positive dA means positive dp for Ma<1. For Ma>1, an area change causes

a pressure change of opposite sign.

Again, substituting from Eq.(19.1) into Eq. (19.3), we obtain

From Eq. (19.4), we see that Ma<1 an area change causes a velocity change of opposite

sign, i.e. positive dA means negative dV for Ma<1. For Ma>1, an area change causes a

velocity change of same sign.

These results are summarized in Fig.19.1, and the relations (19.3) and (19.4) lead to the

following important conclusions about compressible flows:

1.   At subsonic speeds (Ma<1) a decrease in area increases the speed of flow. A

subsonic nozzle should have a convergent profile and a subsonic diffuser should

possess a divergent profile. The flow behavior in the regime of Ma<1 is therefore

qualitatively the same as in incompressible flows.

In supersonic flows (Ma>1), the effect of area changes are different. According to Eq.

(19.4), a supersonic nozzle must be built with an increasing area in the flow direction. A

supersonic diffuser must be a converging channel. Divergent nozzles are used to produce

supersonic flow in missiles and launch vehicles

Shapes of nozzles and diffusers in subsonic and supersonic regimes

Suppose a nozzle is used to obtain a supersonic stream staring from low speeds at the

inlet (Fig.19.2). Then the Mach number should increase from Ma=0 near the inlet to

Ma>1 at the exit. It is clear that the nozzle must converge in the subsonic portion and

diverge in the supersonic portion. Such a nozzle is called a convergent-divergent nozzle.

A convergent-divergent nozzle is also called a de Laval nozzle, after Carl G.P. de Laval

who first used such a configuration in his steam turbines in late nineteenth century (this

has already been mentioned in the introductory note). From Fig.19.2 it is clear that the

Mach number must be unity at the throat, where the area is neither increasing nor

decreasing. This is consistent with Eq. (19.4) which shows that dV can be non-zero at the

throat only if Ma=1. It also follows that the sonic velocity can be achieved only at the

throat of a nozzle or a diffuser.

A convergent-divergent nozzle

The condition, however, does not restrict that Ma must necessarily be unity at the throat,

According to Eq. (19.4), a situation is possible where at the throat if dV=0 there.

For an example, the flow in a convergent-divergent duct may be subsonic everywhere

with Ma increasing in the convergent portion and decreasing in the divergent portion with

at the throat (see Fig.19.3). The first part of the duct is acting as a nozzle, whereas

the second part is acting as a diffuser. Alternatively, we may have a convergent-divergent

duct in which the flow is supersonic everywhere with Ma decreasing in the convergent

part and increasing in the divergent part and again at the throat (see Fig. 19.4).

Convergent-divergent duct with at throat

Convergent-divergent duct with at throat

Question- At some section in the convergent-divergent nozzle in which air is flowing,

pressure, velocity, temperature and cross-sectional area are 22 kN/m2 , 170 m/s, 200º C

and 1000 mm2 respectively. If the flow conditions are iscentropic, determine:

(i) Stagnation temperature and stagnation pressure.

(ii) Sonic velocity and Mach number at this section.

(iii) Velocity, Mach number and flow area at outlet section where pressure is 110

kN/m2.

(iv) Pressure, temperature, velocity and flow at throat of the nozzle.

Take for air R=287 J/kg K, Cp=1.0 kJ/kg K and γ =1.4

Solution: Let subscripts 1, 2 and t refers to the condition at given section, outlet section

and throat section of the nozzle respectively.

Pressure in the nozzle, p1= 200 kN/m2

Velocity of air, V1=170 m/s

Temperature, T1 = 200+273=473 K

Cross sectional area, A1 = 1000 mm2 = 1000×10-6 = 0.001 m2

For air: R = 287 J/kg K, Cp=1.0 kJ/kg K , γ =1.4

(i) Stagnation temperature (Ts) and stagnation pressure (ps):

Stagnation temperature, Ts = T1+ V12

2×Cp

=473+ 170 2

2×(1.0×1000)

= 487.45 K (or 214.45 ºC)

Also, PS/P1 = (TS/T1)γ/γ-1 = (487.45 / 473)1.4/1.4-1 = 1.111

Stagnation pressure, PS = 200*1.111 = 222.2 kN/m2

(ii) Sonic velocity and mach number at this section:

Sonic velocity, C1 = √γRT1 = √1.4*287*473 = 435.9 m/s

Mach number, M1 = V1/C1 = 170/435.9 = 0.39

(iii) Velocity, Mach number and flow area at outlet section where pressure is

110kN/m2

Pressure at outlet section, p2 = 110 kN/m2

PS/P2 = [1+ (1.4-1/2)M22]γ/γ-1

222.2/110 = [1+ {(1.4-1)/2} M20] 1.4/1.4-1

M2 = 1.05

(T2/TS) = (p2/pS)γ-1/γ

T2= 398.7 K

Sonic velocity at outlet section, C2 = √γRT2 = 400.25 m/s

Velocity at outlet section, V2 = M2 * C2 = 420.26 m/s

Now, mass flow at the given section = mass flow at outlet section (exit)

…..continuity equation

i.e. ρ1A1V1 = ρ2A2V2 or

p1 * A1V1 = p2 *A2V2

RT1 RT2

Flow area at the outlet section,

A2 = p1A1V1T2 = 200*0.001*170*398.7 = 6.199*10-4 m2

T1p2V2 473*110*420.26

Hence, A2= 6.199*10-4 m2 or 619.9 mm2

(iv) Pressure(pt), temperature (Tt), Velocity(Vt), and flow area (At) at throat

of the nozzle:

At throat, critical conditions prevail, i.e. the flow velocity becomes equal to the sonic

velocity and Mach number attains a unit value.

From equation, TS = [1+ (γ-1/2) M2t]

Tt

Tt = 406.2 K (or 133.2 ºC)

Also, pt = (Tt/Ts) γ/γ-1

pt = 117.32 kN / m2

Sonic velocity (corresponding to throat conditions),

Ct = √γRT1 = √1.4*287*406.2 = 404 m/s

Flow Velocity, Vt = Mt* Ct = 1*404 = 404 m/s

By continuity equation, we have : ρ1A1V1 = ρtAtVt

p1 * A1V1 = pt *AtVt

RT1 RTt

Flow area of throat, At = p1A1V1Tt = 200*0.001*170*406.2 = 6.16*10-4 m2

T1ptVt 473*117.32*404

Hence, A2= 6.16*10-4 m2 or 6.16 mm2

APPLICATION OF CONVERGENT-DIVERGENT NOZZLES:

Rocket

Ram-jet engine

Space Shuttle

Diffuser (compressor)

Combustion chamber

Nozzle

COMPUTATIONAL FLUID DYNAMICS

Computational fluid dynamics (CFD) is one of the branches of fluid mechanics that uses

numerical methods and algorithms to solve and analyze problems that involve fluid

flows. Computers are used to perform the millions of calculations required to simulate the

interaction of liquids and gases with surfaces defined by boundary conditions. Even with

high-speed supercomputers only approximate solutions can be achieved in many cases.

Ongoing research, however, may yield software that improves the accuracy and speed of

complex simulation scenarios such as transonic or turbulent flows.

Computational Fluid Dynamics (CFD) is a computer-based tool for simulating the

behavior of systems involving fluid flow, heat transfer, and other related physical

processes. It works by solving the equations of fluid flow (in a special form) over a

region of interest, with specified (known) conditions on the boundary of that region.

THE HISTORY OF COMPUTATIONAL FLUID DYNAMICS [CFD]

Computers have been used to solve fluid flow problems for many years.

Numerous programs have been written to solve either specific problems, or specific

classes of problems. From the mid-1970’s, the complex mathematics required to

generalize the algorithms began to be understood, and general purpose CFD solvers were

developed. These began to appear in the early 1980’s and required what were then very

powerful computers, as well as an in-depth knowledge of fluid dynamics, and large

amounts of time to set up simulations. Consequently, CFD was a tool used almost

exclusively in research.

Recent advances in computing power, together with powerful graphics and

interactive 3D manipulation of models have made the process of creating a CFD model

and analyzing results much less labor intensive, reducing time and, hence, cost.

Advanced solvers contain algorithms which enable robust solutions of the flow field in a

reasonable time As a result of these factors, Computational Fluid Dynamics is now an

established industrial design tool, helping to reduce design time scales and improve

processes throughout the engineering world. CFD provides a cost-effective and accurate

alternative to scale model testing, with variations on the simulation being performed

quickly, offering obvious advantages.

Uses of CFD

CFD is used by engineers and scientists in a wide range of fields. Typical applications include:

• Process industry: Mixing vessels, chemical reactors

• Building services: Ventilation of buildings, such as atriums

• Health and safety: Investigating the effects of fire and smoke

• Motor industry: Combustion modeling, car aerodynamics

• Electronics: Heat transfer within and around circuit boards

Computational Fluid Dynamics: CFD Methodology

• Environmental: Dispersion of pollutants in air or water

• Power and energy: Optimization of combustion processes

• Medical: Blood flow through grafted blood vessels

The Mathematics of CFD

The set of equations which describe the processes of momentum, heat and mass transfer

are known as the Navier-Stokes equations. These partial differential equations were

derived in the early nineteenth century and have no known general analytical solution but

can be discretized and solved numerically .Equations describing other processes, such as

combustion, can also be solved in conjunction with the Navier-Stokes equations. Often,

an approximating model is used to derive these additional equations, turbulence models

being a particularly important example.

There are a number of different solution methods which are used in CFD codes. The most

common, and the one on which ANSYS CFX is based, is known as the finite volume

technique. In this technique, the region of interest is divided into small sub-regions,

called control volumes. The equations are discretized and solved iteratively for each

control volume. As a result, an approximation of the value of each variable at specific

points throughout the domain can be obtained. In this way, one derives a full picture of

the behavior of the flow. Additional information on the Navier-Stokes equations and

other mathematical aspects of the ANSYS CFX software suite is available

Technicalities

The most fundamental consideration in CFD is how one treats a continuous fluid in a

discretized fashion on a computer. One method is to discretize the spatial domain into

small cells to form a volume mesh or grid, and then apply a suitable algorithm to solve

the equations of motion (Euler equations for inviscid, and Navier-Stokes equations for

viscous flow). In addition, such a mesh can be either irregular (for instance consisting of

triangles in 2D, or pyramidal solids in 3D) or regular; the distinguishing characteristic of

the former is that each cell must be stored separately in memory. Where shocks or

discontinuities are present, high resolution schemes such as Total Variation Diminishing

(TVD), Flux Corrected Transport (FCT), Essentially NonOscillatory (ENO), or MUSCL

schemes are needed to avoid spurious oscillations (Gibbs phenomenon) in the solution.

If one chooses not to proceed with a mesh-based method, a number of alternatives exist,

notably :

Smoothed particle hydrodynamics (SPH), a Lagrangian method of solving fluid

problems,

Spectral methods, a technique where the equations are projected onto basis

functions like the spherical harmonics and Chebyshev polynomials,

Lattice Boltzmann methods (LBM), which simulate an equivalent mesoscopic

system on a Cartesian grid, instead of solving the macroscopic system (or the real

microscopic physics).

It is possible to directly solve the Navier-Stokes equations for laminar flows and for

turbulent flows when all of the relevant length scales can be resolved by the grid (a Direct

numerical simulation). In general however, the range of length scales appropriate to the

problem is larger than even today's massively parallel computers can model. In these

cases, turbulent flow simulations require the introduction of a turbulence model. Large

eddy simulations (LES) and the Reynolds-averaged Navier-Stokes equations (RANS)

formulation, with the k-ε model or the Reynolds stress model, are two techniques for

dealing with these scales.

In many instances, other equations are solved simultaneously with the Navier-Stokes

equations. These other equations can include those describing species concentration

(mass transfer), chemical reactions, heat transfer, etc. More advanced codes allow the

simulation of more complex cases involving multi-phase flows (e.g. liquid/gas, solid/gas,

liquid/solid), non-Newtonian fluids (such as blood), or chemically reacting flows (such as

combustion).

Navier-Stokes Equations

Newton's Second Law

 

The integral form of the linear momentum equation was discussed in the Linear

Momentum Integral Equation section. Recall, Newton's second law on a differential fluid

element is

     δF = δm a

where δF is the resultant force acting on the fluid element (mass = δm). a is the

acceleration of the fluid element, and it is given by

     

Expanding into its Cartesian components yields

     

There are two types of forces acting on the fluid element: body force (δFB) and surface

force (δFS).

     δF = δFB + δFS

The only body force considered here is the weight of the fluid element. That is,

     δFB = δm g = δm (gx i + gy j + gz k)

Generally, gravity only acts in one direction, but since the coordinate system is not set, all

three terms are included for the general case. Other body forces, such as those due to the

magnetic and electric fields, can be fit into this framework, but they are not covered in

this eBook.

     

Notations for the Stresses

Surface Forces in the x-direction

 

The surface forces are due to the stresses exerted on the sides of the fluid element. There

are two types of stresses applied on the surface: normal stress (σij) and shear stress (τij).

Normal stress acts perpendicular to the surface while shear stress is tangential to the

surface. The subscript i refer to the axis normal to the surface, and the subscript j

represents the direction of the stress. The notation of the stresses is illustrated further in

the animation shown on the left.

All surface forces acting in the x-direction on the fluid element are shown in the figure. A

summation of the surface forces in the x-direction yields

     

Note that the stresses are multiplied by the area to obtain the surface forces. Similarly, the

total surface forces in the y- and z-directions (not shown in figure) are obtained as

     

The resultant surface force is then given as

     δFS = δFSx i + δFSy j + δFSz k

The mass of the fluid element can be expressed in terms of its volume and fluid density

(δm = ρ δxδyδz), so that the linear momentum equation in Cartesian coordinates reduces

to

                                                         

For Newtonian fluids, such as water, oil and air, the shear stress field is symmetric, and it

is related to the rate of shear strain in a linear fashion. The results are (derivation details

not given here),

     

     

     

      

     

     

      

where μ is the viscosity of the fluid. Notice, the pressure term, p, only acts normal to the

surface for each element face. Also, the pressure is assumed to be the same on all three

faces, i.e. hydrostatic pressure.

For incompressible flow, the term is zero based on the continuity equation. The

linear momentum equations thus become

     

                             

         

The above equations are generally referred to as the Navier-Stokes equations, and

commonly written as a single vector form,

Although the vector form looks simple, this equation is the core fluid mechanics

equations and is an unsteady, nonlinear, 2nd order, partial differential equation. It is

extremely hard to solve, and only simple 2D problems have been solved. Computational

Fluid Dynamics (CFD) is most often used to solve the Navier-Stokes equations.

Navier–Stokes equations

The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel

Stokes, describe the motion of fluid substances that is substances which can flow. These

equations arise from applying Newton's second law to fluid motion, together with the

assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the

gradient of velocity), plus a pressure term.

They are exceptionally useful because they describe the physics of many things of

academic and economic interest. They may be used to model the weather, ocean currents,

water flow in a pipe, the air's flow around a wing, and motion of stars inside a galaxy.

The Navier–Stokes equations in their full and simplified forms help with the design of

aircraft and cars, the study of blood flow, the design of power stations, the analysis of

pollution, and many other things. Coupled with Maxwell's equations they can be used to

model and study magneto hydrodynamics.

The Navier–Stokes equations are also of great interest in a purely mathematical sense.

Somewhat surprisingly, given their wide range of practical uses, mathematicians have not

yet proven that in three dimensions solutions always exist (existence), or that if they do

exist they do not contain any infinities, singularities or discontinuities (smoothness).

These are called the Navier–Stokes existence and smoothness problems. The Navier–

Stokes equations dictate not position but rather velocity. A solution of the Navier–Stokes

equations is called a velocity field or flow field, which is a description of the velocity of

the fluid at a given point in space and time. Once the velocity field is solved for, other

quantities of interest (such as flow rate or drag force) may be found. This is different

from what one normally sees in classical mechanics, where solutions are typically

trajectories of position of a particle or deflection of a continuum. Studying velocity

instead of position makes more sense for a fluid; however for visualization purposes one

can compute various trajectories.

Properties

Nonlinearity

The Navier–Stokes equations are nonlinear partial differential equations in almost every

real situation. In some cases, such as one-dimensional flow and Stokes flow (or creeping

flow), the equations can be simplified to linear equations. The nonlinearity makes most

problems difficult or impossible to solve and is the main contributor to the turbulence that

the equations model.

The nonlinearity is due to convective acceleration, which is an acceleration associated

with the change in velocity over position. Hence, any convective flow, whether turbulent

or not, will involve nonlinearity, an example of convective but laminar (nonturbulent)

flow would be the passage of a viscous fluid (for example, oil) through a small

converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly

studied and understood.

Turbulence

Turbulence is the time dependent chaotic behavior seen in many fluid flows. It is

generally believed that it is due to the inertia of the fluid as a whole: the culmination of

time dependent and convective acceleration; hence flows where inertial effects are small

tend to be laminar (the Reynolds number quantifies how much the flow is affected by

inertia). It is believed, though not known with certainty, that the Navier–Stokes equations

describe turbulence properly.

The numerical solution of the Navier–Stokes equations for turbulent flow is extremely

difficult, and due to the significantly different mixing-length scales that are involved in

turbulent flow, the stable solution of this requires such a fine mesh resolution that the

computational time becomes significantly infeasible for calculation (see Direct numerical

simulation). Attempts to solve turbulent flow using a laminar solver typically result in a

time-unsteady solution, which fails to converge appropriately. To counter this, time-

averaged equations such as the Reynolds-averaged Navier-Stokes equations (RANS),

supplemented with turbulence models (such as the k-ε model), are used in practical

computational fluid dynamics (CFD) applications when modeling turbulent flows.

Another technique for solving numerically the Navier–Stokes equation is the Large-eddy

simulation (LES). This approach is computationally more expensive than the RANS

method (in time and computer memory), but produces better results since the larger

turbulent scales are explicitly resolved.

Applicability

Together with supplemental equations (for example, conservation of mass) and well

formulated boundary conditions, the Navier–Stokes equations seem to model fluid

motion accurately; even turbulent flows seem (on average) to agree with real world

observations.

The Navier–Stokes equations assume that the fluid being studied is a continuum not

moving at relativistic velocities. At very small scales or under extreme conditions, real

fluids made out of discrete molecules will produce results different from the continuous

fluids modeled by the Navier–Stokes equations. Depending on the Knudsen number of

the problem, statistical mechanics or possibly even molecular dynamics may be a more

appropriate approach.

Another limitation is very simply the complicated nature of the equations. Time tested

formulations exist for common fluid families, but the application of the Navier–Stokes

equations to less common families tends to result in very complicated formulations which

are an area of current research. For this reason, the Navier–Stokes equations are usually

written for Newtonian fluids.

Almost universally the equations are written for a simple class of fluids (which most

liquids and all known gases belong to) known as Newtonian fluids. Studying such fluids

is "simple" because the viscosity model ends up being linear; truly general models for the

flow of other kinds of fluids (such as blood) do not, as of 2009, exist.

Derivation and description

The derivation of the Navier–Stokes equations begins with an application of Newton's

second law: conservation of momentum (often alongside mass and energy conservation)

being written for an arbitrary control volume. In an inertial frame of reference, the

general form of the equations of fluid motion is:[2]

where is the flow velocity, ρ is the fluid density, p is the pressure, is the (deviatoric)

stress tensor, and represents body forces (per unit volume) acting on the fluid and is

the del operator. This is a statement of the conservation of momentum in a fluid and it is

an application of Newton's second law to a continuum; in fact this equation is applicable

to any non-relativistic continuum and is known as the Cauchy momentum equation.

This equation is often written using the substantive derivative, making it more apparent

that this is a statement of Newton's second law:

The left side of the equation describes acceleration, and may be composed of time

dependent or convective effects (also the effects of non-inertial coordinates if present).

The right side of the equation is in effect a summation of body forces (such as gravity)

and divergence of stress (pressure and stress).

Convective acceleration

An example of convection. Though the flow is steady (time independent), the fluid

decelerates as it moves down the diverging duct (when the flow is subsonic), hence there

is acceleration.

A very significant feature of the Navier–Stokes equations is the presence of convective

acceleration: the effect of time independent acceleration of a fluid with respect to space,

represented by the nonlinear quantity:

which may be interpreted either as or as with the tensor

derivative of the velocity vector Both interpretations give the same result, independent

of the coordinate system — provided is interpreted as the covariant derivative.

Interpretation as (v· )v∇

The convection term is often written as

Where the advection operator is used. Usually this representation is preferred

because it is simpler than the one in terms of the tensor derivative

Interpretation as v·( v)∇

Here is the tensor derivative of the velocity vector, equal in Cartesian coordinates to

the component by component gradient. The convection term may, by a vector calculus

identity, be expressed without a tensor derivative:[4][5]

The form has use in irrotational flow, where the curl of the velocity (called vorticity)

is equal to zero.

Regardless of what kind of fluid is being dealt with, convective acceleration is a

nonlinear effect. Convective acceleration is present in most flows (exceptions include

one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping

flow (also called Stokes flow) .

Stresses

The effect of stress in the fluid is represented by the and terms, these are gradients

of surface forces, analogous to stresses in a solid. is called the pressure gradient and

arises from the isotropic part of the stress tensor. This part is given by normal stresses

that turn up in almost all situations, dynamic or not. The anisotropic part of the stress

tensor gives rise to , which conventionally describes viscous forces; for

incompressible flow, this is only a shear effect. Thus, is the deviatoric stress tensor, and

the stress tensor is equal to

where is the 3×3 identity matrix. Interestingly, only the gradient of pressure matters, not

the pressure itself. The effect of the pressure gradient is that fluid flows from high

pressure to low pressure.

The stress terms p and are yet unknown, so the general form of the equations of motion

is not usable to solve problems. Besides the equations of motion—Newton's second law

—a force model is needed relating the stresses to the fluid motion. For this reason,

assumptions on the specific behavior of a fluid are made (based on natural observations)

and applied in order to specify the stresses in terms of the other flow variables, such as

velocity and density.

The Navier–Stokes equations result from the following assumptions on the deviatoric

stress tensor :

the deviatoric stress vanishes for a fluid at rest, and – by Galilean invariance –

also does not depend directly on the flow velocity itself, but only on spatial

derivatives of the flow velocity

in the Navier–Stokes equations, the deviatoric stress is expressed as the product of

the tensor gradient of the flow velocity with a viscosity tensor , i.e. :

the fluid is assumed to be isotropic, as valid for gases and simple liquids, and

consequently is an isotropic tensor; furthermore, since the deviatoric stress

tensor is symmetric, it turns out that it can be expressed in terms of two scalar

dynamic viscosities μ and μ”: where

is the rate-of-strain tensor and is the

rate of expansion of the flow

the deviatoric stress tensor has zero trace, so for a three-dimensional flow

2μ + 3μ” = 0

As a result, in the Navier–Stokes equations the deviatoric stress tensor has the following

form:

with the quantity between brackets the non-isotropic part of the rate-of-strain tensor

The dynamic viscosity μ does not need to be constant – in general it depends on

conditions like temperature and pressure, and in turbulence modelling the concept of

eddy viscosity is used to approximate the average deviatoric stress.

The pressure p is modelled by use of an equation of state. For the special case of an

incompressible flow, the pressure constrains the flow in such a way that the volume of

fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with

Incompressible flow of Newtonian fluids

A simplification of the resulting flow equations is obtained when considering an

incompressible flow of a Newtonian fluid. The assumption of incompressibility rules out

the possibility of sound or shock waves to occur; so this simplification is invalid if these

phenomena are important. The incompressible flow assumption typically holds well even

when dealing with a "compressible" fluid — such as air at room temperature — at low

Mach numbers (even when flowing up to about Mach 0.3). Taking the incompressible

flow assumption into account and assuming constant viscosity, the Navier–Stokes

equations will read, in vector form:

Here f represents "other" body forces (forces per unit volume), such as gravity or

centrifugal force. The shear stress term becomes the useful quantity when the

fluid is assumed incompressible and Newtonian, where is the dynamic viscosity.

It's well worth observing the meaning of each term (compare to the Cauchy momentum

equation):

Note that only the convective terms are nonlinear for incompressible Newtonian flow.

The convective acceleration is an acceleration caused by a (possibly steady) change in

velocity over position, for example the speeding up of fluid entering a converging nozzle.

Though individual fluid particles are being accelerated and thus are under unsteady

motion, the flow field (a velocity distribution) will not necessarily be time dependent.

Another important observation is that the viscosity is represented by the vector Laplacian

of the velocity field. This implies that Newtonian viscosity is diffusion of momentum,

this works in much the same way as the diffusion of heat seen in the heat equation (which

also involves the Laplacian).

If temperature effects are also neglected, the only "other" equation (apart from

initial/boundary conditions) needed is the mass continuity equation. Under the

incompressible assumption, density is a constant and it follows that the equation will

simplify to:

This is more specifically a statement of the conservation of volume (see divergence).

These equations are commonly used in 3 coordinates systems: Cartesian, cylindrical, and

spherical. While the Cartesian equations seem to follow directly from the vector equation

above, the vector form of the Navier–Stokes equation involves some tensor calculus

which means that writing it in other coordinate systems is not as simple as doing so for

scalar equations (such as the heat equation).

Cartesian coordinates

Writing the vector equation explicitly,

Note that gravity has been accounted for as a body force, and the values of gx,gy,gz will

depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads:

The velocity components (the dependent variables to be solved for) are typically named

u, v, w. This system of four equations comprises the most commonly used and studied

form. Though comparatively more compact than other representations, this is a nonlinear

system of partial differential equations for which solutions are difficult to obtain.

Cylindrical coordinates

A change of variables on the Cartesian equations will yield the following momentum

equations for r, φ, and z:\

The gravity components will generally not be constants, however for most applications

either the coordinates are chosen so that the gravity components are constant or else it is

assumed that gravity is counteracted by a pressure field (for example, flow in horizontal

pipe is treated normally without gravity and without a vertical pressure gradient). The

continuity equation is:

This cylindrical representation of the incompressible Navier–Stokes equations is the

second most commonly seen (the first being Cartesian above). Cylindrical coordinates are

chosen to take advantage of symmetry, so that a velocity component can disappear. A

very common case is axisymmetric flow with the assumption of no tangential velocity (uφ

= 0), and the remaining quantities are independent of φ:

Spherical coordinates

In spherical coordinates, the r, θ, and φ momentum equations are (note the convention

used: θ is colatitude):

Mass continuity will read:

These equations could be (slightly) compacted by, for example, factoring 1 / r2 from the

viscous terms. This isn't done to preserve the structure of the Laplacian and other

quantities.

Stream function formulation

Taking the curl of the Navier–Stokes equation results in the elimination of pressure. This

is especially easy to see if 2D Cartesian flow is assumed (w = 0 and no dependence of

anything on z), where the equations reduce to:

Differentiating the first with respect to y, the second with respect to x and subtracting the

resulting equations will eliminate pressure and any potential force. Defining the stream

function ψ through

results in mass continuity being unconditionally satisfied (given the stream function is

continuous), and then incompressible Newtonian 2D momentum and mass conservation

degrade into one equation:

where is the (2D) biharmonic operator and ν is the kinematic viscosity, . This

single equation together with appropriate boundary conditions describes 2D fluid flow,

taking only kinematic viscosity as a parameter. Note that the equation for creeping flow

results when the left side is assumed zero.

In axisymmetric flow another stream function formulation, called the Stokes stream

function, can be used to describe the velocity components of an incompressible flow with

one scalar function.

Compressible flow of Newtonian fluids

There are some phenomena that are closely linked with fluid compressibility. One of the

obvious examples is sound. Description of such phenomena requires more general

presentation of the Navier–Stokes equation that takes into account fluid compressibility.

If viscosity is assumed a constant, one additional term appears, as shown here:

where μv is second viscosity coefficient. It is related to volume viscosity or bulk

viscosity. This additional term disappears for incompressible fluid, when the divergence

of the flow equals zero.

Euler's Equations

For inviscid flow (μ = 0), the Navier-Stokes equations reduce to

     

The above equations are known as Euler's equations. Note that the equations governing

inviscid flow have been simplified tremendously compared to the Navier-Stokes

equations; however, they still cannot be solved analytically due to the complexity of the

nonlinear terms (i.e., u ∂u/∂x, v ∂u/∂y, w ∂u/∂z, etc.). Hence, in the study of fluid

mechanics, numerical methods such as the finite element and finite difference methods

(along with the use of computers) are often used to approximate the fluid flow problems.

Euler's equations can be written in vector form as

Methodology

In all of these approaches the same basic procedure is followed.

During preprocessing

o The geometry (physical bounds) of the problem is defined.

o The volume occupied by the fluid is divided into discrete cells (the mesh).

The mesh may be uniform or non uniform.

o The physical modeling is defined - for example, the equations of motions

+ enthalpy + radiation + species conservation

o Boundary conditions are defined. This involves specifying the fluid

behaviour and properties at the boundaries of the problem. For transient

problems, the initial conditions are also defined.

The simulation is started and the equations are solved iteratively as a steady-state

or transient.

Finally a postprocessor is used for the analysis and visualization of the resulting

solution.

Discretization methods

The stability of the chosen discretization is generally established numerically rather than

analytically as with simple linear problems. Special care must also be taken to ensure that

the discretization handles discontinuous solutions gracefully. The Euler equations and

Navier-Stokes equations both admit shocks, and contact surfaces.

Some of the discretization methods being used are:

FINITE ELEMENT METHOD

The finite element method (FEM) (sometimes referred to as finite element analysis) is

a numerical technique for finding approximate solutions of partial differential equations

(PDE) as well as of integral equations. The solution approach is based either on

eliminating the differential equation completely (steady state problems), or rendering the

PDE into an approximating system of ordinary differential equations, which are then

numerically integrated using standard techniques such as Euler's method, Runge-Kutta,

etc.

In solving partial differential equations, the primary challenge is to create an equation

that approximates the equation to be studied, but is numerically stable, meaning that

errors in the input data and intermediate calculations do not accumulate and cause the

resulting output to be meaningless. There are many ways of doing this, all with

advantages and disadvantages. The Finite Element Method is a good choice for solving

partial differential equations over complex domains (like cars and oil pipelines), when the

domain changes (as during a solid state reaction with a moving boundary), when the

desired precision varies over the entire domain, or when the solution lacks smoothness.

For instance, in a frontal crash simulation it is possible to increase prediction accuracy in

"important" areas like the front of the car and reduce it in its rear (thus reducing cost of

the simulation); Another example would be the simulation of the weather pattern on

Earth, where it is more important to have accurate predictions over land than over the

wide-open sea.

History

The finite-element method originated from the need for solving complex elasticity and

structural analysis problems in civil and aeronautical engineering. Its development can be

traced back to the work by Alexander Hrennikoff (1941) and Richard Courant (1942).

While the approaches used by these pioneers are dramatically different, they share one

essential characteristic: mesh discretization of a continuous domain into a set of discrete

sub-domains, usually called elements.

Hrennikoff's work discretizes the domain by using a lattice analogy while Courant's

approach divides the domain into finite triangular subregions for solution of second order

elliptic partial differential equations (PDEs) that arise from the problem of torsion of a

cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier

results for PDEs developed by Rayleigh, Ritz, and Galerkin.

Development of the finite element method began in earnest in the middle to late 1950s

for airframe and structural analysis and gathered momentum at the University of Stuttgart

through the work of John Argyris and at Berkeley through the work of Ray W. Clough in

the 1960s for use in civil engineering. By late 1950s, the key concepts of stiffness matrix

and element assembly existed essentially in the form used today. NASA issued request

for proposals for the development of the finite element software NASTRAN in 1965. The

method was provided with a rigorous mathematical foundation in 1973 with the

publication of Strang and Fix's An Analysis of The Finite Element Method has since been

generalized into a branch of applied mathematics for numerical modeling of physical

systems in a wide variety of engineering disciplines, e.g., electromagnetism and fluid

dynamics.

Application

A variety of specializations under the umbrella of the mechanical engineering discipline

(such as aeronautical, biomechanical, and automotive industries) commonly use

integrated FEM in design and development of their products. Several modern FEM

packages include specific components such as thermal, electromagnetic, fluid, and

structural working environments. In a structural simulation, FEM helps tremendously in

producing stiffness and strength visualizations and also in minimizing weight, materials,

and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the

distribution of stresses and displacements. FEM software provides a wide range of

simulation options for controlling the complexity of both modeling and analysis of a

system. Similarly, the desired level of accuracy required and associated computational

time requirements can be managed simultaneously to address most engineering

applications. FEM allows entire designs to be constructed, refined, and optimized before

the design is manufactured.

This powerful design tool has significantly improved both the standard of engineering

designs and the methodology of the design process in many industrial applications. The

introduction of FEM has substantially decreased the time to take products from concept

to the production line. It is primarily through improved initial prototype designs using

FEM that testing and development have been accelerated. In summary, benefits of FEM

include increased accuracy, enhanced design and better insight into critical design

parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive

design cycle, increased productivity, and increased revenue.

Finite volume method

The finite volume method is a method for representing and evaluating partial differential

equations in the form of algebraic equations. Similar to the finite difference method,

values are calculated at discrete places on a meshed geometry. "Finite volume" refers to

the small volume surrounding each node point on a mesh. In the finite volume method,

volume integrals in a partial differential equation that contain a divergence term are

converted to surface integrals, using the divergence theorem. These terms are then

evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a

given volume is identical to that leaving the adjacent volume, these methods are

conservative. Another advantage of the finite volume method is that it is easily

formulated to allow for unstructured meshes. The method is used in many computational

fluid dynamics packages.

1D example

Consider a simple 1D advection problem defined by the following partial differential equation

Here, represents the state variable and represents the flux or flow of . Conventionally, positive represents flow to the right whilst negative

represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, , into finite volumes or cells with cell centres indexed as . For a particular cell, , we can define the volume average value

of at time and , as

and at time as,

where and represent locations of the upstream and downstream faces or edges

respectively of the cell.

Integrating equation (1) in time, we have:

where .

To obtain the volume average of at time , we integrate over the

cell volume, and divide the result by , i.e.

We assume that is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension

, we can apply the divergence theorem, i.e. , and

substitute for the volume integral of the divergence with the values of evaluated at

the cell surface (edges and ) of the finite volume as follows:

where .

We can therefore derive a semi-discrete numerical scheme for the above problem with

cell centres indexed as , and with cell edge fluxes indexed as , by differentiating (6) with respect to time to obtain:

where values for the edge fluxes, , can be reconstructed by interpolation or extrapolation of the cell averages. It should be noted that equation (7) is exact for the volume averages; i.e., no approximations have been made during its derivation.

General hyperbolic problem

We can also consider a general hyperbolic problem, represented by the following PDE,

Here, represents a vector of states and represents the corresponding flux tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, , we take the volume integral over the total volume of the cell, , which gives,

On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields

where represents the total surface area of the cell and is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (7), i.e.

Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. MUSCL reconstruction is often used in high resolution schemes where shocks or discontinuities are present in the solution.

Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, one cell's loss is another cell's gain!

Finite difference method.

This method has historical importance and is simple to program. It is currently only used in few specialized codes. Modern finite difference codes make use of an embedded boundary for handling complex geometries making these codes highly efficient and accurate. Other ways to handle geometries are using overlapping-grids, where the solution is interpolated across each grid.

Where Q is the vector of conserved variables, and F, G, and H are the fluxes in the x, y, and z directions respectively.

Boundary element method. The boundary occupied by the fluid is divided into surface mesh.

High-resolution schemes are used where shocks or discontinuities are present. To capture sharp changes in the solution requires the use of second or higher order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of flux limiters to ensure that the solution is total variation diminishing.

Introduction to ANSYS CFXANSYS CFX is general purpose Computational Fluid Dynamics (CFD) software suite

that combines an advanced solver with powerful pre- and post-processing capabilities.

ANSYS CFX is capable of modeling:

• Steady-state and transient flows

• Laminar and turbulent flows

• Subsonic, transonic and supersonic flows

• Heat transfer and thermal radiation

• Buoyancy

• Non-Newtonian flows

• Transport of non-reacting scalar components

• Multiphase flows

• Combustion

• Flows in multiple frames of reference

• Particle tracking.

ANSYS CFX includes the following features:

• An advanced coupled solver that is both reliable and robust.

• Full integration of problem definition, analysis, and results presentation.

• An intuitive and interactive setup process, using menus and advanced graphics.

The Benefits of Fluid Simulation from ANSYS

ANSYS fluid flow analysis technology allows for an in-depth analysis of the fluid

mechanics in many types of products and processes. Not only does it reduce the need for

expensive prototypes, it provides comprehensive data that is not easily obtainable from

experimental tests. Fluid simulation can be used to complement physical testing. Some

designers use it to analyze new systems before deciding which validation tests, and how

many, need to be performed. When troubleshooting, problems are solved faster and more

reliably because fluid dynamics analysis highlights the root cause, not just the effect.

When optimizing new equipment designs, many what-if scenarios can be analyzed in a

short time. This can result in improved performance, reliability and product consistency.

ANSYS will continue to innovate and integrate so that customers can replace more of

their traditional capital-intensive design processes with a Simulation Driven Product

Development method.

The ANSYS AdvantageANSYS is a major product for computer based Prototyping. While material Prototyping

is as old as mankind, the computerized approach is relatively new and has on offer certain

major advantages over the classical approach. In the following some info in regard.

Before we start using or selling a actual product we want to make sure it works properly

w/o premature failures.

The earliest method to achieve a working design was (and is) 'make and brake'. If it

doesn't break it is a valid design, otherwise we need to think it over, to redesign wrt

dimensions, material, and manufacturing method - remove evident design flaws. Make

and break does have some draw backs. The make of a new prototype may be costly and

time consuming, while still not indicating with certainty were a flaw really hides..

In the last decades a new method came to maturity, which adds more flexibility, cost

efficiency and, especially, more insight into prototyping. This more recent method is

known under names like FEA (finite element analysis), simulation or virtual (digital)

prototyping, and moves the material prototype, the experimental verification, into the

computer.

Simulation is a great step ahead, allows precise prognosis and optimization of the

performance of parts and products, with all the flexibility of a computerized model -

changing dimensions, materials, loads w/o the necessity to create a new material

prototype. The range of physics problems that can be analyzed is basically unlimited - be

it mechanical or thermal loads, be it a fluid dynamic question, a acoustics setting or

electromagnetic device, simulation can handle it (keyword multiphysics).

One other great advantage, besides the flexibility of computer based prototyping is the

ability to evaluate the designs response in any location of the part. While material

prototyping needs sensors to measure quantities like strain, temperature, flow speed or

another physical quantity (if you are able to arrive with the sensor in a certain position,

and the sensor doesn't disturb the measurement by it's very presence) simulation models

provide these quantities w/o effort at every time and point on the model (section the

model at will, plot contours of quantities, graph, list, i.e. view results any way suits).

How does one build a computerized simulation model? Basically the part(s) of interest

are modeled in a 3D CAD system, usually. This CAD is then further processed in a

'preprocessor', i.e. loading and any other material data, physical data and the like are

specified, further 'solution' settings for solving the numerical model are defined. After

solving the model it gets 'post-processed' in that results are accessed for verification and

reporting.

The great advantage of computer based prototyping is evident: modifications (e.g.

updating material data, changing some dimension) and extensions (adding a new load

scenario or a result quantity to evaluate) to the actual model are possible at any stage of

simulation.

CONCLUSION

REFERENCES

1. Rajput R. K., Fluid Mechanics and Hydraulic Machines, 2006, Dhanpat Rai Publications, Page no. 725 to 753

2. Jain R. K. , Production Technology, 2003, Khanna Publishers, Page no. 138 to 148

3. Guidelines from “Metal & Casting Industry, Madan Mehel , Jabalpur”

4. http://www.engineeringtoolbox.com/