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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
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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.
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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
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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
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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.
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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
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• : 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.
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Source: T. A. Ward, Aerospace Propulsion Systems, John Wiley & Sons (Asia), Pte Ltd., Singapore (2010)
Atmosphere
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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.
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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
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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
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Physical Examples contd…
M>1
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Shock waves in Compression ramp
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Physical Examples contd…
Flow
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Physical Examples contd…
Flow
Symmetric shock waves
Flow
Flow
λ-shock wave
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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
Physical Examples contd…
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.
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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).
Physical Examples contd…
nose of the model (along with other shocks).
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Re-entry vehicle Gun aerodynamics
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Physical Examples contd…
Test of a rocket nozzle F-16 Fighter plane
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Shock diamond
Physical Examples contd…
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Interaction of shock waves in compressor cascade
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Some Examples contd…
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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
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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