Editing Compressible flow (section)

==One-dimensional flow==
One-dimensional (1-D) flow refers to flow of gas through a duct or channel in which the flow parameters are assumed to change significantly along only one spatial dimension, namely, the duct length. In analysing the 1-D channel flow, a number of assumptions are made: 
*Ratio of duct length to width (L/D) is ≤ about 5 (in order to neglect [[friction]] and [[heat transfer]]), 
*[[Fluid dynamics|Steady vs. Unsteady Flow]],
*Flow is [[isentropic]] (i.e. a reversible adiabatic process), 
*[[Ideal gas law]] (i.e. P = ρRT)

===Converging-diverging Laval nozzles===
As the speed of a flow accelerates from the subsonic to the supersonic regime, the physics of [[nozzle]] and diffuser flows is altered. Using the conservation laws of fluid dynamics and thermodynamics, the following relationship for channel flow is developed (combined mass and momentum conservation):

:<math> dP\left(1 - M^2\right) = \rho V^2\left(\frac {dA} {A}\right) </math>, 
where dP is the differential change in pressure, M is the Mach number, ρ is the density of the gas, V is the velocity of the flow, A is the area of the duct, and dA is the change in area of the duct. This equation states that, for subsonic flow, a converging duct (dA < 0) increases the velocity of the flow and a diverging duct (dA > 0) decreases velocity of the flow. For supersonic flow, the opposite occurs due to the change of sign of (1 − M<sup>2</sup>). A converging duct (dA < 0) now decreases the velocity of the flow and a diverging duct (dA > 0) increases the velocity of the flow. At Mach = 1, a special case occurs in which the duct area must be either a maximum or minimum. For practical purposes, only a minimum area can accelerate flows to Mach 1 and beyond. See table of sub-supersonic diffusers and nozzles.

[[File:Sub-Supersonic Diffusers and Nozzles.png|thumb|center|Table showing the reversal in the physics of nozzles and diffusers with changing Mach numbers]]

Therefore, to accelerate a flow to Mach 1, a nozzle must be designed to converge to a minimum cross-sectional area and then expand. This type of nozzle – the converging-diverging nozzle – is called a [[de Laval nozzle]] after [[Gustaf de Laval]], who invented it. As subsonic flow enters the converging duct and the area decreases, the flow accelerates. Upon reaching the minimum area of the duct, also known as the throat of the nozzle, the flow can reach Mach 1. If the speed of the flow is to continue to increase, its density must decrease in order to obey conservation of mass. To achieve this decrease in density, the flow must expand, and to do so, the flow must pass through a diverging duct. See image of de Laval Nozzle.

[[File:Nozzle de Laval diagram.png|thumb|center|Nozzle de Laval diagram]]

===Maximum achievable velocity of a gas===
Ultimately, because of the energy conservation law, a gas is limited to a certain maximum velocity based on its energy content. The maximum velocity, ''V''<sub>max</sub>, that a gas can attain is:

:<math>V_\text{max} = \sqrt{2c_p T_t}</math>

where c<sub>p</sub> is the specific heat of the gas and T<sub>t</sub> is the [[stagnation temperature]] of the flow.

===Isentropic flow Mach number relationships===
Using conservations laws and thermodynamics, a number of relationships of the form

:<math> \frac{\text{property}_1}{\text{property}_2} = f(M,\gamma) </math>
 
can be obtained, where M is the Mach number and γ is the ratio of specific heats (1.4 for air). See table of isentropic flow Mach number relationships. 

[[File:Isentropic Flow Relations Table.PNG|thumb|center|Isentropic flow relationship table. Equations to relate the field properties in isentropic flow.]]

===Achieving supersonic flow===
As previously mentioned, in order for a flow to become supersonic, it must pass through a duct with a minimum area, or sonic throat. Additionally, an overall pressure ratio, P<sub>b</sub>/P<sub>t</sub>, of approximately 2 is needed to attain Mach 1. Once it has reached Mach 1, the flow at the throat is said to be ''[[choked flow|choked]]''. Because changes downstream can only move upstream at sonic speed, the mass flow through the nozzle cannot be affected by changes in downstream conditions after the flow is choked.

===Non-isentropic 1D channel flow of a gas - normal shock waves===
Normal shock waves are shock waves that are perpendicular to the local flow direction. These shock waves occur when pressure waves build up and coalesce into an extremely thin shockwave that converts kinetic energy into [[thermal energy]]. The waves thus overtake and reinforce one another, forming a finite shock wave from an infinite series of infinitesimal sound waves. Because the change of state across the shock is highly irreversible, [[Entropy in thermodynamics and information theory|entropy]] increases across the shock. When analysing a normal shock wave, one-dimensional, steady, and adiabatic flow of a perfect gas is assumed. Stagnation temperature and stagnation enthalpy are the same upstream and downstream of the shock.

[[File:Rankine-Hugoniot Relationships.PNG|thumb|center|The Rankine-Hugoniot equations relate conditions before and after a normal shock wave.]]

Normal shock waves can be easily analysed in either of two reference frames: the standing normal shock and the moving shock. The flow before a normal shock wave must be supersonic, and the flow after a normal shock must be subsonic. The Rankine-Hugoniot equations are used to solve for the flow conditions.