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| [[Image:X-15 Model in Supersonic Tunnel - GPN-2000-001272.jpg|thumb|200px|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).]]An '''oblique shock''' wave, unlike a [[normal shock]], is inclined with respect to the incident upstream flow direction. It will occur when a supersonic flow encounters a corner that effectively turns the flow into itself and compresses. The upstream streamlines are uniformly deflected after the shock wave. The most common way to produce an oblique shock wave is to place a wedge into [[supersonic]], [[compressible flow]]. Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly [[discontinuous]] changes in the thermodynamic properties of a gas occur. While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave.
| | == the actual physics Nike Shox Online == |
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| It is always possible to convert an oblique shock into a normal shock by a [[Galilean transformation]].
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| == Oblique Shock Wave Theory ==
| | <li>[http://enseignement-lsf.com/spip.php?article64#forum24232301 http://enseignement-lsf.com/spip.php?article64#forum24232301]</li> |
| [[Image:Obliqueshock.PNG|thumb|400px|Supersonic flow encounters a wedge and is uniformly deflected forming an oblique shock.]][[Image:ObliqueShockAngleRelation.png|thumb|400px|This chart shows the oblique shock angle, β, as a function of the corner angle, θ, for a few constant M<sub>1</sub> lines. The blue line separates the strong and weak solutions. The chart assumes <math>\gamma</math>=1.4, which is valid for an ideal diatomic gas.]] | | |
| For a given [[Mach number]], M<sub>1</sub>, and corner angle, θ, the oblique shock angle, β, and the downstream Mach number, M<sub>2</sub>, can be calculated. M<sub>2</sub> is always less than M<sub>1</sub>. Unlike after a normal shock, M<sub>2</sub> can still be supersonic (weak shock wave) or subsonic (strong shock wave). Weak solutions are often observed in flow geometries open to atmosphere (such as on the outside of a flight vehicle). Strong solution may be observed in confined geometries (such as inside a nozzle intake). Strong solution is required when the flow need to match the downstream high pressure condition. Discontinuous changes also occur in the pressure, density and temperature, which all rise downstream of the oblique shock wave.
| | <li>[http://www.atzjk.com/bbs/forum.php?mod=viewthread&tid=193407&extra= http://www.atzjk.com/bbs/forum.php?mod=viewthread&tid=193407&extra=]</li> |
| | | |
| === the θ-β-M equation ===
| | <li>[http://www.metransparent.com/spip.php?article10827&lang=ar&id_forum=14263/ http://www.metransparent.com/spip.php?article10827&lang=ar&id_forum=14263/]</li> |
| Using the [[continuity equation]] and the fact that the [[Speed#Tangential_speed|tangential velocity component]] does not change across the shock, trigonometric relations eventually lead to the θ-β-M equation which shows θ as a function of M<sub>1</sub> β, and ɣ, where ɣ is the [[Heat capacity ratio]]. <ref>http://www.aerostudents.com/files/aerodynamicsC/obliqueShockWaves.pdf</ref>
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| | | <li>[http://enseignement-lsf.com/spip.php?article64#forum23778602 http://enseignement-lsf.com/spip.php?article64#forum23778602]</li> |
| <math>\tan \theta =
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| 2\cot\beta\frac{M_1^2\sin^2\beta-1}{M_1^2(\gamma+\cos2\beta)+2}</math>
| | </ul> |
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| It is more intuitive to want to solve for β as a function of M<sub>1</sub> and θ, but this approach is more complicated, the results of which are often contained in tables or calculated through an applet.
| |
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| <!--Commenting out the material below. I could not find this stuff in any standard textbooks (J. D. Anderson, John & Keith, Zucker & Biblarz etc.) Please remove the comments only after adding proper sources.
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| Here is a way to calculate β knowing θ and M<sub>1</sub>.
| |
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| First define λ:
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| <math>\lambda =
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| \sqrt{(M_1^2-1)^2-3(1+\frac{\gamma-1}{2}M_1^2)(1+\frac{\gamma+1}{2}M_1^2)\tan^2\theta}
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| </math> | |
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| Then define χ:
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| <math> | |
| \chi =
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| \frac{(M_1^2-1)^3-9(1+\frac{\gamma-1}{2}M_1^2)(1+\frac{\gamma-1}{2}M_1^2+\frac{\gamma+1}{4}M_1^4)\tan^2\theta}{\lambda^3}
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| </math>
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| Finally β is given by the relation:
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| <math>\tan \beta =
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| \frac{M_1^2-1+2\lambda\cos(\frac{4\pi\delta+\arccos(\chi)}{3})}{3(1+\frac{\gamma-1}{2}M_1^2)\tan \theta}
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| </math>
| |
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| where δ=0 corresponds to the strong shock solution and δ=1 to the weak one.
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| -->
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| === Maximum deflection angle ===
| |
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| Within the θ-β-M equation, a maximum corner angle, θ<sub>MAX</sub>, exists for any upstream Mach number. When θ > θ<sub>MAX</sub>, the oblique shock wave is no longer attached to the corner and is replaced by a detached [[bow shock (aerodynamics)|bow shock]]. A θ-β-M diagram, common in most compressible flow textbooks, shows a series of curves that will indicate θ<sub>MAX</sub> for each Mach number. The θ-β-M relationship will produce two β angles for a given θ and M<sub>1</sub>, with the larger angle called a strong shock and the smaller called a weak shock. The weak shock is almost always seen experimentally.
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| The rise in pressure, density, and temperature after an oblique shock can be calculated as follows:
| |
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| <math>\frac{p_2}{p_1} =
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| 1+\frac{2\gamma}{\gamma+1}(M_1^2\sin^2\beta-1)</math>
| |
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| <math>\frac{\rho_2}{\rho_1} =
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| \frac{(\gamma+1)M_1^2\sin^2\beta}{(\gamma-1)M_1^2\sin^2\beta+2}</math>
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| <math>\frac{T_2}{T_1} =
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| \frac{p_2}{p_1}\frac{\rho_1}{\rho_2}.</math>
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| M<sub>2</sub> is solved for as follows:
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| <math>M_2 =
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| \frac{1}{\sin(\beta-\theta)}\sqrt{\frac{1+\frac{\gamma-1}{2}M_1^2 \sin^2 \beta}{\gamma M_1^2 \sin^2 \beta- \frac{\gamma-1}{2}}}.</math>
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| == Oblique Shock Wave Applications ==
| |
| [[File:Grumman F-14 Tomcat 2.JPG|thumb|F-14D Tomcat showing wedge-shaped inlets]]
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| Oblique shock waves are used predominantly in engineering applications when compared with normal shock waves. This can be attributed to the fact that using one or a combination of oblique shock waves results in more favorable post-shock conditions (lower post-shock temperature, etc.) when compared to utilizing a single normal shock. An example of this technique can be seen in the design of supersonic aircraft engine inlets, which are wedge-shaped to compress air flow into the combustion chamber while minimizing thermodynamic losses. Early supersonic aircraft jet engine inlets were designed using compression from a single normal shock, but this approach caps the maximum achievable Mach number to roughly 1.6. The wedge-shaped inlets are clearly visible on the sides of the [[F-14 Tomcat]], which has a maximum speed of Mach 2.34.
| |
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| Many supersonic aircraft wings are designed around a thin diamond shape. Placing a diamond-shaped object at an angle of attack relative to the supersonic flow streamlines will result in two oblique shocks propagating from the front tip over the top and bottom of the wing, with [[Prandtl-Meyer expansion fan]]s created at the two corners of the diamond closest to the front tip. When correctly designed, this generates lift.
| |
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| == Oblique Shock Waves and the Hypersonic Limit ==
| |
| As the Mach number of the upstream flow becomes increasingly hypersonic, the equations for the pressure, density, and temperature after the oblique shock wave reach a mathematical [[limit (mathematics)|limit]]. The pressure and density ratios can then be expressed as:
| |
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| <math>\frac{p_2}{p_1} \approx | |
| \frac{2\gamma}{\gamma+1}M_1^2\sin^2\beta</math>
| |
| | |
| | |
| <math>\frac{\rho_2}{\rho_1} \approx | |
| \frac{\gamma+1}{\gamma-1}.</math>
| |
| | |
| For a perfect atmospheric gas approximation using γ = 1.4, the hypersonic limit for the density ratio is 6. However, hypersonic post-shock dissociation of O<sub>2</sub> and N<sub>2</sub> into O and N lowers γ, allowing for higher density ratios in nature. The hypersonic temperature ratio is:
| |
| | |
| <math>\frac{T_2}{T_1} \approx
| |
| \frac{2\gamma(\gamma-1)}{(\gamma+1)^2}M_1^2\sin^2\beta.</math>
| |
| | |
| == See also ==
| |
| * [[Bow shock (aerodynamics)]]
| |
| * [[Gas dynamics]]
| |
| * [[Mach reflection]]
| |
| * [[Moving shock]]
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| * [[Shock wave]]
| |
| * [[Shock polar]]
| |
| | |
| == References ==
| |
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| <references /> | |
| * {{cite book
| |
| | last = Liepmann | first = Hans W. | coauthors = Roshko, A. | |
| | title = Elements of Gasdynamics | origyear = 1957
| |
| | publisher = [[Dover Publications]] | year = 2001
| |
| | isbn = 0-486-41963-0 }}
| |
| * {{cite book
| |
| | last = Anderson | first = John D. Jr.
| |
| | title = Fundamentals of Aerodynamics | origyear = 1984 | edition = 3rd Edition
| |
| | publisher = [[McGraw-Hill|McGraw-Hill Science/Engineering/Math]] |date=January 2001
| |
| | isbn = 0-07-237335-0 }}
| |
| * {{cite book
| |
| | last = Shapiro | first = Ascher H.
| |
| | title = The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1 | origyear = 1953
| |
| | publisher = [[Ronald Press]]
| |
| | isbn = 978-0-471-06691-0 }} | |
| | |
| ==External links==
| |
| * [http://www.grc.nasa.gov/WWW/K-12/airplane/oblique.html NASA oblique shock wave calculator] ([[Java applet]]).
| |
| * [http://www.youtube.com/watch?v=iNBZBChS2YI Supersonic wind tunnel test demonstration (Mach 2.5) with flat plate and wedge creating an oblique shock(Video)]
| |
| | |
| [[Category:Fluid dynamics]]
| |
| [[Category:Aerodynamics]]
| |
| [[Category:Shock waves]]
| |
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