# Mach no. should be constant for existence of similarity solution

I am reading some research papers on Taylor-Sedov type self-similar solution to the blast wave problems...

It is written that for the existence of similarity solutions Mach number should be constant...please explain this?

and Please explain when it is possible to find self-similar flows with variable Mach numbers?

Also Please tell me range of Mach and Knudsen numbers which are applicable in the interstellar medium ?

• Just a note: The Mach number is entirely dependent upon the speed of the "piston" relative to the medium and type of "sound" speed relevant to that medium (i.e., in space plasmas, there are many "sound" speeds). – honeste_vivere Mar 4 '16 at 14:39
• @ honeste_vivere...fluid velocity is variable in the blast wave problems and sound velocity is also variable...then why Mach No. is constant... – user45799 Mar 5 '16 at 4:37
• The Mach number would only be constant over a specific "short" period of time, where "short" is defined specific to a given problem. – honeste_vivere Mar 5 '16 at 12:33
• @ honeste_vivere...please explain it more....and please suggest some references.... – user45799 Mar 6 '16 at 4:27
• Without some form of energy input (i.e., something constantly pushing the piston), the shock wave will have to slow down by definition since it dissipates energy. – honeste_vivere Mar 7 '16 at 15:32

# Quick Note

According to Wikipedia, a similarity solution is a form of scale invariance such that, in the case of fluid flow, the flow "looks" the same independent of time or length scale.

# Question 1

It is written that for the existence of similarity solutions Mach number should be constant...please explain this?

In the case of Taylor-Sedov-von Neumann like solutions [e.g., see pages 192-196 in Whitham, 1999], the relevant parameter is the position of the shock wave at $r = R\left( t \right)$, given by: $$R\left( t \right) = k \left( \frac{ E }{ \rho_{up} } \right)^{1/5} \ t^{2/5} \tag{1}$$ where $t$ is time from the initial release of energy $E$ from a point source, $\rho_{up}$ is the ambient gas mass density, and $k$ is a dimensionless parameter used for scaling. These solutions are founded upon two assumptions given as follows:

1. the explosion resulted from a sudden release of energy $E$ from a point source and $E$ is the only dimensional parameter introduced by the explosion;
2. the disturbance is so strong that the ambient air/gas pressure and speed of sound can be neglected compared to those in the blast wave.

The second assumption implies the strong shock limit, namely that the downstream parameters are given by (in shock frame): \begin{align} U_{dn} & = \left( \frac{ 2 }{ \gamma + 1 } \right) U_{up} \tag{2a} \\ \rho_{dn} & = \left( \frac{ \gamma + 1 }{ \gamma - 1 } \right) \rho_{up} \tag{2b} \\ P_{dn} & = \left( \frac{ 2 }{ \gamma + 1 } \right) \rho_{up} \ U_{up}^{2} \tag{2c} \end{align} where subscript $up$($dn$) corresponds to upstream(downstream) averages, $\gamma$ is the ratio of specific heats, $U_{j}$ is bulk flow speed in region $j$ (i.e., $up$ or $dn$), and $P_{j}$ is the pressure (here just using dynamic or ram pressure) in region $j$.

It can be seen from Equation 1 the only parameter of the ambient air/gas relevant is the density, $\rho_{up}$. Note that the shock speed, $U_{up}$, will then be given by $dR/dt$ (i.e., the time derivative of Equation 1), or: \begin{align} U_{up}\left( t \right) & = \frac{ 2 \ k }{ 5 } \left( \frac{ E }{ \rho_{up} } \right)^{1/5} \ t^{-3/5} \tag{3a} \\ & = \frac{ 2 \ k^{5/2} }{ 5 } \sqrt{ \frac{ E }{ \rho_{up} } } \ R^{-3/2} \tag{3b} \end{align} Since the upstream density and pressure are assumed constant, then the upstream sound speed, $C_{s,up}$, must be constant as well.

Answer 1: The Mach number would be proportionally related to time as $M \propto t^{-3/5} \propto R^{-3/2}$, which is not constant but can be expressed independently of temporal or spatial scales. Over very small radial distances one can, of course, approximate the Mach number as being roughly constant but this is not a good approximation in many cases unless the shock is "old."

# Question 2

and Please explain when it is possible to find self-similar flows with variable Mach numbers?

The self-similarity of the above relations arise because there are no independent length or time scales in the solutions. This is because we can rewrite Equations 2a and 2c solely in terms of $E$ and $R$ such that they do not explicitly depend upon length or time. We can solve Equation 1 for $t$ and use it to invert our pressure and flow speed expressions. These expressions are given as: \begin{align} U_{dn} & = \left( \frac{ 4 \ k }{ 5 \left( \gamma + 1 \right) } \right) \left( \frac{ E }{ \rho_{up} } \right)^{1/5} \ t^{-3/5} \tag{4a} \\ & = \left( \frac{ 4 \ k^{5/2} }{ 5 \left( \gamma + 1 \right) } \right) \sqrt{ \frac{ E }{ \rho_{up} } } \ R^{-3/2} \tag{4b} \\ P_{dn} & = \left( \frac{ 8 \ \rho_{up} \ k^{2} }{ 25 \left( \gamma + 1 \right) } \right) \left( \frac{ E }{ \rho_{up} } \right)^{2/5} \ t^{-6/5} \tag{4c} \\ & = \left( \frac{ 8 \ E \ k^{2} }{ 25 \left( \gamma + 1 \right) } \right) R^{-3} \tag{4d} \end{align}

Thus you can see that $U_{dn}$ and $P_{dn}$ are only functions (explicitly) of $E$ and $R$. Since $E/\rho_{up}$ has units of $L^{5}/T^{2}$, we can define a dimensionless parameter $\zeta = E t^{2}/\rho_{up} r^{5}$ that all relevant parameters (e.g., $U_{dn}$) must depend upon in some way, where $r$ is just a radial position. We can further define the dimensionless parameter $\xi = r/R\left( t \right) \propto \zeta^{-1/5}$, upon which all of our characteristics must depend.

Answer 2: The Mach number can be variable (i.e., it depends upon radial position), but one can construct a set of equations whereby it does not explicitly depend upon a length or time scale. I think you may have been confusing constant with independence from time and length scales.

# Question 3

Also Please tell me range of Mach and Knudsen numbers which are applicable in the interstellar medium ?

Wikipedia has a list of relevant parameters for the interstellar medium (ISM). You can use the expressions found in this answer to determine relevant sound speeds. To determine the Knudsen number, $K_{T}$, you can estimate the Coulomb collision mean free path and use the $L_{T} = \lvert d \ln{T_{e}}/dx \rvert^{-1}$ as the characteristic scale length, where $T_{e}$ is the electron temperature (you could use a total temperature too) and $d/dx$ is just a one-dimensional spatial gradient.

Answer 3: In the solar wind $K_{T}$ ~ 0.01-10 [e.g., Bale et al., 2013; Horaites et al., 2015; Landi et al., 2014] and $K_{T}$ ~ 1 for the local interstellar medium or LISM [e.g., Baranov, 2009]. For other ISM regions, the uncertainty in $K_{T}$ will be large since it must be inferred from electromagnetic radiation measurements alone, which require numerous assumptions in order to invert the spectrum into a particle velocity distribution.

As for the Mach number in the ISM, that depends upon with respect to what object or reference frame. If you are looking for the Mach number of, say, the heliosphere with respect to the LISM, there are numerous papers on the heliospheric bow shock (it's actually under debate at the moment whether the flow of the LISM is large enough to produce a bow shock here).

# References

• Bale, S.D., et al. (2013) "Electron Heat Conduction in the Solar Wind: Transition from Spitzer-Härm to the Collisionless Limit," Astrophys. J. Lett. 769(2), L22, doi:10.1088/2041-8205/769/2/L22.
• Baranov, V.B. (2009) "Kinetic-Fluid Perspective on Modeling the Heliospheric/Interstellar Medium Interface," Space Sci. Rev. 143, pp. 449-464, doi:10.1007/s11214-008-9392-6.
• Horaites, K., et al. (2015) "Self-Similar Theory of Thermal Conduction and Application to the Solar Wind," Phys. Rev. Lett. 114(24), 5003, doi:10.1103/PhysRevLett.114.245003.
• Landi, S., et al. (2014) "Electron Heat Flux in the Solar Wind: Are We Observing the Collisional Limit in the 1 AU Data?," Astrophys. J. Lett. 790(1), L12, doi:10.1088/2041-8205/790/1/L12.
• Whitham, G.B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.