Compressible Fluid Mechanics

Compressible Fluid Mechanics - Glossary (Under Construction)

Compressibility starts to become significant for Mach Numbers greater than 0.3 - all supersonic flows are compressible

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Adiabatic Processes

Calorifically Perfect Gas
A perfect gas where c_v and c_p are constants.
Compressibility
The compressibility, tau, is given by

link effect on speed of sound
Energy Equation

Enthalpy
where e is internal energy, p is pressure and v is the volume per unit mass, i.e. the reciprocal of density.
For a perfect gas, enthalpy is a function of temperature only

h = h(T) For a calorically perfect gas, this becomes
h = c_pT where c_p is the specific heat at constant pressure
The total enthalpy for a steady, adiabatic, inviscid flow is given at any point by in other words, the total enthalpy is the sum of the static enthalpy and the kinetic energy at that particular point.
The total enthalpy is constant along a streamline.
If all the streamlines of a flow originate from a common uniform freestream, then the total enthalpy is the same for each streamline. Consequently, for a steady, adiabatic flow, the total enthalpy is constant throughout the entire flow.
From

we can see that under these same conditions, the total temperature will also be constant throughout the flow.
Entropy
Isentropic Mach Number The Mach number is a measure of the directed motion of the gas compared with the random thermal motion of the molecules. Mach Wave Shock Wave A shock wave will only occur in supersonic flows - it is an extremely thin region, in practise, about 10 - 5 cm across which flow proplerties can change enormously. In practise, it usual mathematically to model shocks as a line discontinuity. In general these properties will increase across a shockwa wave pressure density temperature entropy whereas as the following properties will go down velocity The flow in front of the shock waave will always be supersonic. Behind a normal shock the flow will always be subsonic, although with an oblique shock wave, the flow can be iether subsonic or supersonic. Shock Wave (Normal) Shock Wave (Oblique) Second Law of Thermodynamics Speed of Sound related to speed of molecular movement. two a pressure and density are related via the perfect gas equation of state. i.e. funtion of temperature only lower compressibility higher speed of sound
First Law of Thermodynamics
is equivalent to the Principle of Conservation of Energy.
The heat added and work done on a system will raise the temperature

For a reversible process
which produces
Internal Energy
For a perfect gas , the internal energy is a function of temperature only.

e = e(T) For a calorically perfect gas, this becomes
e = c_vT where c_v is the specific heat at constant volume
Isentropic
An isentropic process is both adiabatic and reversible.
Isentropic relations can be summarized as

Mach Number
The Mach number is a measure of the directed motion of the gas compared with the random thermal motion of the molecules.
It is given by
where u is the flow velocity and a is the speed of sound
Mach Wave

Shock Wave
A shock wave will only occur in supersonic flows - it is an extremely thin region, in practise, about 10^{- 5} cm. across which flow proplerties can change enormously. In practise, it usual mathematically to model shocks as a line discontinuity. In general these properties will increase across a shock wave

pressure
density
temperature
entropy
whereas as the following properties will go down

velocity
The flow in front of the shock wave will always be supersonic. Behind a normal shock the flow will always be subsonic, although with an oblique shock wave, the flow can be iether subsonic or supersonic.
Shock Wave (Normal)

Shock Wave (Oblique)

Reversible Processes
Inviscid, frictionless
Second Law of Thermodynamics

Specific Heat at constant pressure
Staring off from

c_p - c_v = R
we can show that

Specific Heat at constant volume
Staring off from

c_p - c_v = R
we can show that

Speed of Sound
The speed of sound increases proportionally to the absolute temperature.

where c is speed of sound, k is a constant and T is absolute temperature
Since temperature varies within the troposphere, the speed of sound will vary with altitude also. At sea level it is 1224 km/hr and at 9100 meters altitude it is 1090,8 km/hr. The Earth's troposphere ends near 11000 meters, and since the temperature in the stratosphere is fairly constant, the speed of sound is also approximately a constant - 1060 km/hr.
related to speed of molecular movement. two a pressure and density are related via the perfect gas equation of state. i.e. funtion of temperature only lower compressibility higher speed of sound