Misplaced Pages

Moving shock: Difference between revisions

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editContent deleted Content addedVisualWikitext
Revision as of 16:16, 31 January 2007 editGenick (talk | contribs)134 edits →External links← Previous edit Latest revision as of 15:27, 5 November 2021 edit undoSerols (talk | contribs)Extended confirmed users, Pending changes reviewers477,504 editsm Reverted edits by 154.237.140.178 (talk): unexplained content removal (HG) (3.4.10)Tag: Rollback 
(33 intermediate revisions by 21 users not shown)
Line 1: Line 1:
In ], a '''moving shock''' is a ] that is travelling through a ] (often ]eous) medium with a ] relative to the velocity of the fluid already making up the medium.<ref>Shapiro, Ascher H., ''Dynamics and Thermodynamics of Compressible Fluid Flow,'' Krieger Pub. Co; Reprint ed., with corrections (June 1983), {{ISBN|0-89874-566-7}}.</ref> As such, the ] relations require modification to calculate the properties before and after the moving shock. A knowledge of moving shocks is important for studying the phenomena surrounding ], among other applications.
''Moving 'Shock''' is situation where the ] is moving and do not stay in one location sometime people refers to it as ]. In many physical situations the shock is moving variable speed. However, in many situations this variable speed can be considered as a constant velocity to which many analytical tools can be applied. In this case, the moving shock can be transformed to a ``stationary normal shock" by attaching the coordinates to the shock. There are two broad categories of this shock one reffered to as "open valve" and "close valve". The open valve referred to situations when high speed gas run into slower moving gas. In the extreme case this open valve case is of high speed gas running into a still medium. The ``close valve'' referred to the case where shock is generated as a results of the shock reflecting from a solid (relatively hard material) and the shock is propagating upstream. Normally, when the two sides of the shock are moving the word ``partially'' is also applied.


==Theory==
The governing equations of the shock wave were well established well before World War Two. Yet, many cases no analytical solution was given. Recently, ] has derived solutions to many of the cases. For example, the shock speed as results of piston movement became simple analytical solution. These solutions show that for example when one open the valve in his garden and the pipe is empty flow initially contains a moving shock that creates extremely large temperature increase to a few milliseconds. The phenomenon of the moving shock has many industrial applications.
]
To derive the theoretical equations for a moving shock, one may start by denoting the region in front of the shock as subscript 1, with the subscript 2 defining the region behind the shock. This is shown in the figure, with the shock wave propagating to the right.
The velocity of the gas is denoted by ''u'', ] by ''p'', and the local ] by ''a''.
The speed of the shock wave relative to the gas is ''W'', making the total velocity equal to ''u<sub>1</sub>'' + ''W''.


Next, suppose a ] is then fixed to the shock so it appears stationary as the gas in regions 1 and 2 move with a velocity relative to it. Redefining region 1 as ''x'' and region 2 as ''y'' leads to the following shock-relative velocities:
⚫ == External links ==
*


:<math>\ u_y = W + u_1 - u_2,</math>
⚫ ]

:<math>\ u_x = W.</math>

With these shock-relative velocities, the properties of the regions before and after the shock can be defined below introducing the ] as ''T'', the ] as ''ρ'', and the ] as ''M'':

:<math>\ p_1 = p_x \quad ; \quad p_2 = p_y \quad ; \quad T_1 = T_x \quad ; \quad T_2 = T_y,</math>

:<math>\ \rho_1 = \rho_x \quad ; \quad \rho_2 = \rho_y \quad ; \quad a_1 = a_x \quad ; \quad a_2 = a_y,</math>

:<math>\ M_x = \frac{u_x}{a_x} = \frac{W}{a_1},</math>

:<math>\ M_y = \frac{u_y}{a_y} = \frac{W + u_1 - u_2}{a_2}.</math>

Introducing the ] as ''γ'', the ], density, and pressure ratios can be derived:

:<math>\ \frac{a_2}{a_1} = \sqrt{1 + \frac{2(\gamma - 1)}{(\gamma + 1)^2}\left},</math>

:<math>\ \frac{\rho_2}{\rho_1} = \frac{1}{1-\frac{2}{\gamma + 1}\left},</math>

:<math>\ \frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1}\left.</math>

One must keep in mind that the above equations are for a shock wave moving towards the right. For a shock moving towards the left, the ''x'' and ''y'' subscripts must be switched and:

:<math>\ u_y = W - u_1 + u_2,</math>

:<math>\ M_y = \frac{W - u_1 + u_2}{a_2}.</math>

==See also==
*]
*]
*]
*]
*]
*]
*]

==References==
{{Reflist}}

⚫ ==External links==
*

⚫ ]
]
]

Latest revision as of 15:27, 5 November 2021

In fluid dynamics, a moving shock is a shock wave that is travelling through a fluid (often gaseous) medium with a velocity relative to the velocity of the fluid already making up the medium. As such, the normal shock relations require modification to calculate the properties before and after the moving shock. A knowledge of moving shocks is important for studying the phenomena surrounding detonation, among other applications.

Theory

This diagram shows the gas-relative and shock-relative velocities used for the theoretical moving shock equations.

To derive the theoretical equations for a moving shock, one may start by denoting the region in front of the shock as subscript 1, with the subscript 2 defining the region behind the shock. This is shown in the figure, with the shock wave propagating to the right. The velocity of the gas is denoted by u, pressure by p, and the local speed of sound by a. The speed of the shock wave relative to the gas is W, making the total velocity equal to u1 + W.

Next, suppose a reference frame is then fixed to the shock so it appears stationary as the gas in regions 1 and 2 move with a velocity relative to it. Redefining region 1 as x and region 2 as y leads to the following shock-relative velocities:

  u y = W + u 1 − u 2 , {\displaystyle \ u_{y}=W+u_{1}-u_{2},}
  u x = W . {\displaystyle \ u_{x}=W.}

With these shock-relative velocities, the properties of the regions before and after the shock can be defined below introducing the temperature as T, the density as ρ, and the Mach number as M:

  p 1 = p x ; p 2 = p y ; T 1 = T x ; T 2 = T y , {\displaystyle \ p_{1}=p_{x}\quad ;\quad p_{2}=p_{y}\quad ;\quad T_{1}=T_{x}\quad ;\quad T_{2}=T_{y},}
  ρ 1 = ρ x ; ρ 2 = ρ y ; a 1 = a x ; a 2 = a y , {\displaystyle \ \rho _{1}=\rho _{x}\quad ;\quad \rho _{2}=\rho _{y}\quad ;\quad a_{1}=a_{x}\quad ;\quad a_{2}=a_{y},}
  M x = u x a x = W a 1 , {\displaystyle \ M_{x}={\frac {u_{x}}{a_{x}}}={\frac {W}{a_{1}}},}
  M y = u y a y = W + u 1 − u 2 a 2 . {\displaystyle \ M_{y}={\frac {u_{y}}{a_{y}}}={\frac {W+u_{1}-u_{2}}{a_{2}}}.}

Introducing the heat capacity ratio as γ, the speed of sound, density, and pressure ratios can be derived:

  a 2 a 1 = 1 + 2 ( γ − 1 ) ( γ + 1 ) 2 [ γ M x 2 − 1 M x 2 − ( γ − 1 ) ] , {\displaystyle \ {\frac {a_{2}}{a_{1}}}={\sqrt {1+{\frac {2(\gamma -1)}{(\gamma +1)^{2}}}\left}},}
  ρ 2 ρ 1 = 1 1 − 2 γ + 1 [ 1 − 1 M x 2 ] , {\displaystyle \ {\frac {\rho _{2}}{\rho _{1}}}={\frac {1}{1-{\frac {2}{\gamma +1}}\left}},}
  p 2 p 1 = 1 + 2 γ γ + 1 [ M x 2 − 1 ] . {\displaystyle \ {\frac {p_{2}}{p_{1}}}=1+{\frac {2\gamma }{\gamma +1}}\left.}

One must keep in mind that the above equations are for a shock wave moving towards the right. For a shock moving towards the left, the x and y subscripts must be switched and:

  u y = W − u 1 + u 2 , {\displaystyle \ u_{y}=W-u_{1}+u_{2},}
  M y = W − u 1 + u 2 a 2 . {\displaystyle \ M_{y}={\frac {W-u_{1}+u_{2}}{a_{2}}}.}

See also

References

  1. Shapiro, Ascher H., Dynamics and Thermodynamics of Compressible Fluid Flow, Krieger Pub. Co; Reprint ed., with corrections (June 1983), ISBN 0-89874-566-7.

External links

Categories:
Moving shock: Difference between revisions Add topic