North East Aircraft Museum

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

Condition for adiabatic conditions

Calorifically Perfect Gas

A
perfect gas where cv and cp are constants.

Compressibility

The compressibility, tau, is given by

Formula for Compressibility
link effect on speed of sound

Energy Equation

Enthalpy

Enthalpy Formula
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 = cpT
where cp 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

First Law of Thermodynamics

For a reversible process

Formula for work in reversible processes
which produces
First Law of Thermodynamics for Reversible Processes

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 = cvT
where cv is the specific heat at constant volume

Isentropic

An isentropic process is both
adiabatic and reversible.

Isentropic relations can be summarized as

Isentropic Relations

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

Formula for Mach Number
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

cp - cv = R

we can show that

Formula for Specific Heat at Constant Pressure

Specific Heat at constant volume

Staring off from

cp - cv = R

we can show that

Formula for Specific Heat at Constant Volume

Speed of Sound

The speed of sound increases proportionally to the absolute temperature.

Relation between Speed of Sound and 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


Brian Daugherty