ME 6127 Mechanics of Inviscid Compressible Fluid 6127-Lecture-01 (April 2016).pdf · The most...

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ME 6127

Mechanics of Inviscid Compressible Fluid

Lecture-01

10 May 2016

Dr. A.B.M. Toufique HasanAssociate Professor

Department of Mechanical Engineering, BUET

ME 6127: Mechanics of Inviscid Compressible Fluid 11

10 May 2016

M.Sc. Engg. Semester: April 2016

Compressibility

Compressibility of a fluid is basically a measure of the change of density that

will be produced in the fluid by a specific change in pressure.

In a fluid flow there are usually changes in pressure associated, for example, with

change in velocity in the flow. These pressure changes will in general induce

density changes, which will have an influence on the flow.density changes, which will have an influence on the flow.

If these density changes are important, the temperature changes are

becoming also important.

The study of flows in which the changes in density and temperature areimportant is basically what is known as compressible fluid flow or gas

dynamics.


dynamics.

It usually only being in gas flows that compressibility effects are important.

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Compressibility contd…

A change in the pressure applied to a certain amount of a substance (solid, liquid

or gas) always produces some change in its volume.

The proportionate change in volume of a particular material during the

compression is directly related to the change in the pressure.

The compressibility of fluid is defined by

dp

dv

v

1−=τ

Here dp represents a small increase in pressure applied to the material and dv the corresponding smallincrease in the original volume v. Since a rise in pressure always causes a decrease in volume, dv is alwaysnegative, and the minus sign is included in the equation to give a positive value of τ

Incase of isothermal process i.e. if the temperature of the fluid element is held

constant, the isothermal compressibility is defined by


T

Tp

v

v

∂

∂−=

1τ

For water, τT = 5x10-10 m2/N at 1 atm

air, τT = 10-5 m2/N at 1 atm (more than four orders of magnitude higher than water)

Mach Number

a

VM =

In fluid mechanics, the effect of compressibility in the flow field can beassessed by a number called the “Mach number”. This dimensionlessnumber is defined as

where

V is the flow velocity and a is the local speed of sound in the fluid.

Why speed of sound ? ? ?

This is the speed at which “signal” (disturbance) can travel through themedium. In case when an object moves through a fluid, it generatesdisturbances (infinitesimal pressure waves, which are sound waves) that


disturbances (infinitesimal pressure waves, which are sound waves) thatemanate from the object in all director.

When the speed of object becomes comparable or higher than the speedof sound, then the propagation and interaction of disturbance (signal)become complicated and different compared to low speed cases.

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Flow Classification

In aerodynamics, the following flow classes are classified roughlydepending on the Mach number-

M < 0.3 : Incompressible flow (density effects are negligible)M < 0.3 : Incompressible flow (density effects are negligible)

0.3< M < 0.8 : Subsonic flow, where density effects are important but

no appearance of shock waves

0.8< M < 1.2 : Transonic flow, where density shock waved first appear,

dividing the subsonic and supersonic flows.

1.2 < M < 3.0 : Supersonic flow, where shock waves are present but there

are no subsonic regions.


M > 3.0 : Hypersonic flow, where shock waves and other flow

changes are especially strong.

(Surface Chemistry, Plasma dynamics)

Applications of Compressible flow

The most obvious applications of compressible flow theory are in thedesign of high speed aircraft. These includes:

• Commercial civil aircraft• Military fighters• Ramjet vehicle• Scramjet vehicle• Scramjet vehicle• Rockets etc.

However, the knowledge of compressible fluid flow theory is required in thedesign and operation of many devices commonly encountered inengineering practice. Among these applications are:

• Gas turbines: The flow in the blading and nozzles is compressible.• Steam turbines: here the flow in the nozzles and blades must be


• : here the flow in the nozzles and blades must betreated as compressible.• Reciprocating engines: the flow of the gases through the valvesand in the intake and the exhaust systems must be treated ascompressible.• Combustion chambers: the study of combustion, in many casesrequires a knowledge of compressible fluid flow.

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Flight Envelope

Engines can operate only over a certain range of altitudes and velocities (Mach

numbers) which correspond to differing atmospheric pressure, temperature and

densities (all changes with altitude). This range is known as the engine’s flightenvelope.


Source: T. A. Ward, Aerospace Propulsion Systems, John Wiley & Sons (Asia), Pte Ltd., Singapore (2010)

Atmosphere


Reynolds number and Mach number significantly vary with altitude at the same flow velocity.

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Shock Waves

Shock waves are obvious in internal or external compressible flow.

The extremely thin region in which the transition from the initialThe extremely thin region in which the transition from the initialsupersonic velocity (M>1), relatively low-pressure state to the statethat involves a relatively low velocity (M<1) and high pressure istermed as a shock wave. The thickness is usually only a few meanfree paths.

Supersonic initial flow (M>1) is mandatory for the generation of ashock wave.


Physical Examples

the essential ingredients of these flows

shocks

shadowgraph of supersonic flowaround space crew modules: Mach

the essential ingredients of these flows

including: presence of multiple shockwaves, separated zones, and wakes, and

large scale structures.Flow


around space crew modules: Mach2.2 flow around an Apollo-like capsuleat 25° angle-of-attack

Source: T. B. Gatski & J P Bonnet, Compressibility, Turbulence and High Speed Flow, Elsevier, The Netherlands (2013)

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Physical Examples contd…

Flow



M>1


Shock waves in Compression ramp

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Flow



Flow

Symmetric shock waves

Flow

Flow

λ-shock wave


Shock wave structure in propulsion C-D nozzle

Asymmetric shock waves

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Schlieren photograph of high speed (transonic)

flow over an airfoil.

The nearly vertical shock wave is followed by


A generic missile body with Mach 5 embeds

many flow structures occurring simultaneously;

The nearly vertical shock wave is followed by

boundary layer separation that adversely

affects lift, drag, and other flight parameters.


many flow structures occurring simultaneously;

oblique shock wave at the tip of the body,

expansion fan at the shoulder, a dead airregion at the compression corner due to

shockwave- boundary layer interaction

(SWBLI).

A small scale X-15 placed in a NASA supersonic

wind tunnel produces an oblique shock wave at the

nose of the model (along with other shocks).


nose of the model (along with other shocks).


Re-entry vehicle Gun aerodynamics

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Test of a rocket nozzle F-16 Fighter plane


Shock diamond



Interaction of shock waves in compressor cascade

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Some Examples contd…


High speed flow (Compressible flow) in fan (a) and compressor blades (b)

(a) (b)

Course OutlineIntroduction to compressible flow

Review of thermodynamics of compressible flow

Analysis of subsonic and supersonic flow fieldsAnalysis of subsonic and supersonic flow fields

Normal shock waves

Oblique shock waves

Prandtle-Meyer Expansion waves

Shock Reflections and Interactions

Characteristic methods and perturbation technique

Reference Texts


Modern compressible flow- J D Anderson

Introduction to compressible fluid flow- P H Oosthuizen and W E Carscallen

Fundamentals of gas dynamics- R D Zucker and O Biblarz

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