Editing Compressible flow (section)

==Two-dimensional flow==
Although one-dimensional flow can be directly analysed, it is merely a specialized case of two-dimensional flow. It follows that one of the defining phenomena of one-dimensional flow, a normal shock, is likewise only a special case of a larger class of [[oblique shock]]s. Further, the name "normal" is with respect to geometry rather than frequency of occurrence. Oblique shocks are much more common in applications such as: aircraft inlet design, objects in supersonic flight, and (at a more fundamental level) supersonic nozzles and diffusers. Depending on the flow conditions, an oblique shock can either be attached to the flow or detached from the flow in the form of a [[bow shock (aerodynamics)|bow shock]].

{|style="margin: 0 auto;"
| [[File:X-15 Model in Supersonic Tunnel - GPN-2000-001272.jpg|thumb|left|Attached shock wave shown on a X-15 Model in a supersonic wind tunnel]]
|
| [[File:Bowshock example - blunt body.jpg|thumb|right|Bowshock example for a blunt body]]
|}

===Oblique shock waves===
{{Main|Oblique shock}}
[[File:Oblique Shock Wave.JPG|thumb|Diagram of obstruction]]
Oblique shock waves are similar to normal shock waves, but they occur at angles less than 90° with the direction of flow.  When a disturbance is introduced to the flow at a nonzero angle (δ), the flow must respond to the changing boundary conditions. Thus an oblique shock is formed, resulting in a change in the direction of the flow.

====Shock polar diagram====
[[File:Shock polar diagram.jpg|thumb|Shock polar diagram]]
Based on the level of flow deflection (δ), oblique shocks are characterized as either strong or weak. Strong shocks are characterized by larger deflection and more entropy loss across the shock, with weak shocks as the opposite. In order to gain cursory insight into the differences in these shocks, a shock polar diagram can be used. With the static temperature after the shock, T*, known the speed of sound after the shock is defined as,
:<math> A^* = \sqrt{\gamma RT^*} </math>

with R as the gas constant and γ as the specific heat ratio. The Mach number can be broken into Cartesian coordinates
:<math>\begin{align}
  M^*_{2x} &= \frac{V_x}{a^*} \\
  M^*_{2y} &= \frac{V_y}{a^*}
\end{align}</math>

with V<sub>x</sub> and V<sub>y</sub> as the x and y-components of the fluid velocity V. With the Mach number before the shock given, a locus of conditions can be specified. At some {{not a typo|δ<sub>max</sub>}}, the flow transitions from a strong to weak oblique shock. With δ = 0°, a normal shock is produced at the limit of the strong oblique shock and the Mach wave is produced at the limit of the weak shock wave.

====Oblique shock reflection====
Due to the inclination of the shock, after an oblique shock is created, it can interact with a boundary in three different manners, two of which are explained below.

=====Solid boundary=====
Incoming flow is first turned by angle δ with respect to the flow. This shockwave is reflected off the solid boundary, and the flow is turned by – δ to again be parallel with the boundary. Each progressive shock wave is weaker and the wave angle is increased.

=====Irregular reflection=====
An irregular reflection is much like the case described above, with the caveat that δ is larger than the maximum allowable turning angle. Thus a detached shock is formed and a more complicated reflection known as Mach reflection occurs.

===Prandtl–Meyer fans===
{{Main|Prandtl-Meyer expansion fan}}
Prandtl–Meyer fans can be expressed as both compression and expansion fans. Prandtl–Meyer fans also cross a boundary layer (i.e. flowing and solid) which reacts in different changes as well. When a shock wave hits a solid surface the resulting fan returns as one from the opposite family while when one hits a free boundary the fan returns as a fan of opposite type.

====Prandtl–Meyer expansion fans====
[[File:Prandtl-Meyer Expansion.jpg|thumb|Prandtl–Meyer expansion fan diagram]]
To this point, the only flow phenomena that have been discussed are shock waves, which slow the flow and increase its entropy. It is possible to accelerate supersonic flow in what has been termed a [[Prandtl–Meyer expansion fan]], after Ludwig Prandtl and Theodore Meyer. The mechanism for the expansion is shown in the figure below.

As opposed to the flow encountering an inclined obstruction and forming an oblique shock, the flow expands around a convex corner and forms an expansion fan through a series of isentropic Mach waves. The expansion "fan" is composed of Mach waves that span from the initial Mach angle to the final Mach angle. Flow can expand around either a sharp or rounded corner equally, as the increase in Mach number is proportional to only the convex angle of the passage (δ). The expansion corner that produces the Prandtl–Meyer fan can be sharp (as illustrated in the figure) or rounded. If the total turning angle is the same, then the P-M flow solution is also the same.

The Prandtl–Meyer expansion can be seen as the physical explanation of the operation of the Laval nozzle. The contour of the nozzle creates a smooth and continuous series of Prandtl–Meyer expansion waves.

====Prandtl–Meyer compression fans====
[[File:Prandtl-Meyer Compression Fan.JPG|thumb|Basic PM compression diagram]]
A Prandtl–Meyer compression is the opposite phenomenon to a Prandtl–Meyer expansion. If the flow is gradually turned through an angle of δ, a compression fan can be formed. This fan is a series of Mach waves that eventually coalesce into an oblique shock. Because the flow is defined by an isentropic region (flow that travels through the fan) and an [[anisentropic]] region (flow that travels through the oblique shock), a slip line results between the two flow regions.