# The Post-Shock Nonequilibrium Relaxation in a Hypersonic Plasma Flow Involving Reflection off a Thermal Discontinuity

## Abstract

**:**

_{2}gas heated to 5000 K/10,000 K across the interface and M = 3.5, the relaxation time determined for the transmitted wave is up to 50% shorter and the relaxation depth for both waves is significantly reduced, thus resulting in a weakened wave structure. The results of the extension into larger values of heating strength and the shock Mach numbers are discussed. The findings can be useful in the areas of research involving strong shocks interacting with optical discharges or other heated media on the scale where the shock structure becomes important.

## 1. Introduction

_{2}to tens of cm in O

_{2}, and a minuscule amount of water vapor present in the gas may result in the shock structure of a completely different scale [8].

## 2. Reflection of a Shock Wave Having an Extended Structure

_{i}is incident normally, from left to right, on a discontinuous plane interface (dashed line) separating two gases of different properties. A portion of the incident shock energy is reflected back (“r”) into the medium

**1**(left of the interface) and the remainder of it is transmitted (“t”) into the medium.

_{0}in both media is constant and uniformly distributed, and p

_{0}= p

_{atm}. The gas is heated to T

_{01}= 5000 K from the incident shock side of the interface, and to T

_{02}= 10,000 K on the other side. Considering diatomic nitrogen as the medium on both sides, there will be no significant dissociation at this temperature. For the benefit of clarity in the following calculations, the discontinuous interface is chosen as reflecting more efficiently. However, a smooth interface can be readily included in considerations using, for ex., the model [9].

_{01}≠ T

_{02}, the shock refraction equation can be set using the Rankine–Hugoniot relation together with the gas pressure and flow speed continuity conditions across the interface, p

_{1}= p

_{2}, u

_{1}= u

_{2}, as was done in [10], where u

_{1}and u

_{2}denote the speed of flow in the media 1 and 2, accordingly. The continuous distribution of gas parameters in the post-shock nonequilibrium zone will be described with a set of functions f (x, t), each being specific to the temperature, pressure, density, etc. distributions.

_{i}(x, t) corresponds to the temperature distribution established on the high-pressure side of the shock during the relaxation period (to be determined). The distribution function f

_{T}(x, t) is defined as the current value of the temperature in the zone normalized to its value immediately behind the shock front. Similarly, the pressure distribution is described with a function f

_{P}(x, t), and consequently:

_{i}

^{(0)}is the pressure jump in the flow immediately behind the front (at x = 0), and the time t is counted starting from the moment the incident shock strikes the interface. The pressure continuity across the interface:

_{1}= λ

_{2,}can be used to simplify Equation (2). The reason for this is the fact that the specific heat ratio γ in the gas is affected strongly up to approximately 2500 K and then, as the temperature increases to 10,000 K, it has a tendency to saturate at a constant value of approximately γ =1.2860 [16]. Thus, when the gas heating is intense on both sides of the interface, the difference in the parameters becomes insignificant. The graph of Figure 2 illustrating this property is plotted assuming excitation of vibrations only and no dissociation in the gas.

_{1}= γ

_{2}, and considering non-dissociating gas, μ

_{1}= μ

_{2}, the parameter θ is then reduced to the temperature ratio ${T}_{21}^{\left(0\right)}={T}_{02}/{T}_{01}$(the heating strength across the interface).

## 3. Flow Parameter Profiles in the Relaxation Zone of the Incident Shock

_{r}(x, t) in Equation (2) requires known parameter distribution functions in the relaxation zone of the incident shock, f

_{T}(x, t) and f

_{P}(x, t). To determine f

_{T}(x, t), the approach [13] developed for a plain shock of moderate to strong intensities propagating free in a uniformly heated medium can be utilized. The model is based on experimental data used in the enthalpy expressions via the active and inert components of the energy content factors ${\beta}_{a}$ and ${\beta}_{i}$ taken on both sides of the front. Assuming that the gas in front of the incident shock is in equilibrium, the enthalpy ${H}_{1}={\beta}_{a}\frac{{P}_{1}}{{\rho}_{1}}$, and a similar expression holds for the inert component, ${H}_{2}={\beta}_{i}\frac{{P}_{2}}{{\rho}_{2}}$ in the gas experiencing nonequilibrium relaxation. Then, the temperature distribution T

_{x}

_{1}in the nonequilibrium zone in the flow behind the shock is obtained by integrating the following expression:

_{1}, and, in front of the shock, 0 < x < λ

_{T}

_{,}λ

_{T}is the relaxation zone length for the temperature profile. The coefficients δ

_{1}through δ

_{3}and ε

_{1}through ε

_{10}are the functions of the Mach number, incident medium temperature, molecular weight μ, the specific heat ratio γ, the energy content factors ${\beta}_{a}$ and ${\beta}_{i}$, the correction σ, and the coefficients in the fit function $\lambda ={c}_{3}{T}_{x1}+{c}_{4}$ for the relaxation length data. The factor $z=h\nu /{k}_{B}T$, where h is Plank’s constant, k

_{B}is the Boltzmann constant, and ν is the vibrational frequency (cm

^{−1}) in the harmonic oscillator mode approximation. The expressions for the coefficients δ and ${\epsilon}_{1}$ through ${\epsilon}_{10}$ are given in Appendix A.

_{T}can be read from the temperature profile directly, or determined analytically as the distance ${\lambda}_{T}=I\left({T}_{31}\right)-I\left({T}_{21}\right)$, where the system reaches its new equilibrium temperature T

_{31}asymptotically (Figure 3).

_{1}= 5000 K, M

_{i}= 3.5, and with estimated σ = 0.9204, the solution in Equation (7) takes the form illustrated in Figure 3. The horizontal coordinates of the most left (at x = 0) and right (x = λ

_{T}) points of the curve give the asymptotic normalized temperatures T

_{21}and T

_{31}, accordingly, T

_{21}> T

_{31}, and the difference between the point’s vertical coordinates is the length of the relaxation zone λ

_{T}of 0.0890 cm. In further calculations, for simplicity, the numerical data obtained for the solution in Figure 3 can be closely fitted with a bell-type function, $x\ge 0$:

_{i}= 3.5 in N

_{2}at 5000 K is approximately 0.20 μs, which corresponds to the relaxation length of 0.89 mm.

## 4. The Relaxation Dynamics in the Zone during Shock Reflection

_{0}) is given in terms of a distance x from the front, and is defined up to the moment when the shock front hits the interface. As the front advances into the hotter medium, the continued relaxation in the gas in front of the interface results in the parameter distribution being dependent on the interaction time t. The temporal change of the initial distribution function f

_{0}at the point of reflection is equivalent to its shift to the right with the shock wave speed V

_{sw}at the distance ∆ = V

_{sw}t (Figure 4). As seen in the diagram, at a current time t and a location x within the relaxation layer, the shift results in a transition from the initial function f

_{0}(x, t) to that at the distance x

_{i}= x + ∆, i.e., f

_{0}(x, t) → f(x

_{i}, t) = f

_{0}(x + ∆, t), thus yielding a new distribution:

_{r}in the reflected wave can be found. The temporal change in the distribution π

_{r}occurs during the transitional period $0>t>{t}_{\lambda}$, where t

_{λ}is the relaxation time. After that, a stationary state corresponding to a new equilibrium established at the end of the relaxation zone in the gas takes place again.

_{t}using the pressure continuity condition (4), or the solution can be obtained from already determined π

_{r}utilizing the same continuity condition.

_{i}= 3.5 in N

_{2}at 5000 K, for which the incident shock temperature profile (5) can be utilized. Numerical results for the flow parameter distribution in the relaxation zones across the head of the rarefaction wave fan (in the reflected wave) and across the front of the transmitted shock are presented in Figure 5. The pressure distribution across the transmitted wave π

_{t}is obtained using condition (4) and the temperature distribution T

_{21}

^{(t)}using relationship (9).

_{21}–T

_{31}, and pressures, π

_{2}–π

_{3}, attributable to the relaxation depth.

_{r}. Then for the two waves of interest

_{b}[13]. The presence of such a boundary will not affect the flow until the time when the head of the rarefaction wave reaches the left-hand boundary at the time t

_{b}= x

_{b}/c

_{0}, where c

_{0}is the speed of sound in the media in front of the reflected wave. Then, the model relations are applicable if the relaxation time for the incident shock is shorter than the time t

_{b}

_{.}In this case, the expression for the flow speed at the head of the expansion fan in the relaxation zone of the reflected rarefaction wave takes the following form:

_{i}and u

_{t}, and that for the rarefaction wave u

_{r}at the head of the expansion fan. Similar to the structure typical for the temperature and pressure profiles in Figure 5, the flow speed for both waves emerging on the interface shows reduced relaxation depths compared to that in the incident wave, and most notably, significant shortening of around 20% of the relaxation time in the transmitted wave.

_{r}and normalized temperature profile in the nonequilibrium zone behind the incident shock f

_{T}(x),

^{+}characteristics [13] are applied to a gas in any thermodynamic state, but later the correction to the speed of sound value can be done in the expression for the flow speed u. The factor σ in the adjustment includes the specific heat correction due to the temperature, the virial correction takes into account inter-molecular interactions that are the function of temperature and pressure, and the correction accounting for relaxation processes leading to acoustical dispersion is a function of temperature, pressure, and frequency [10].

_{λ}, and the relaxation depth D

_{T}. The latter is defined through the gas parameters at the two ends of the relaxation zone, immediately behind the shock front (index “2”) and at the location where a new state of equilibrium is established (index ”3”). Then, for the extended temperature jump determined by Equation (6)

_{T}= 0.432, Table 1 and Figure 3a) and a moderate gas compression (D

_{p}= 0.103, Figure 3b,c), the feature commonly observed in real gases [17]. The effect mostly shows up in the gas density, for which the relaxation depth is almost five times stronger than for the pressure (0.103 vs. 0.490 accordingly), and similar to that for the temperature. The additional compression built up across the zone drives an increase in the flow speed u

_{i}with time (0.061), in a degree comparable to that for the pressure. The relaxation time of around 0.150 μs is common for all three gas parameters.

_{r}, also increases with time (D

_{π}

^{(i)}= 0.048), thus fortifying the wave, along with a very slight decrease in the temperature (D

_{T}

^{(r)}= 0.004). A stronger effect on the flow speed u

_{r}(D

_{u}

^{(r)}= 0.141) is the result of an increase in time compression in the zone acting in the same direction as the motion of the gas flow within the rarefaction wave structure.

_{t}= 0.186 vs. 0.432) resulting in additional weaker compression in the zone (D

_{p}= 0.059 vs. 0.103) drives the flow speed acceleration (D

_{u}= 0.038 vs. 0.061). Another remarkable effect on relaxation in the transmitted wave is significantly reduced, by 20 to 50%, relaxation times in the profiles (120–100 μs vs 150 μs). However, accounting for acceleration of the transmitted shock in the hotter gas, the relaxation length in this medium makes up its original value in the zone of the incident wave.

_{kinetic}= ∆x

_{visc}+ ∆x

_{vib}, ∆x

_{visc}<< ∆x

_{vib}. The shock width ∆x is then determined by dominating process depending on the temperature and thermodynamic properties of a specific gas or mixture, ∆x = ∆x

_{visc}+ ∆x

_{vib}, + ∆x

_{rad}, usually with ∆x

_{kin}<< ∆x

_{rad}. The resonant radiation transfer that is typically present in partially ionized gases at elevated pressure, especially in the presence of strong density gradients, can contribute via an increase in the density of electronically excited atoms [18].

## 5. Extension of the Model into a Wider Range of Parameters

_{02}, the temporal profiles of pressure ratio π

_{r}can be calculated. In the absence of significant dissociation and the heat capacity ratios being close to equal, γ

_{1}≈ γ

_{2}, θ is reduced to the initial gas temperature ratio, θ = T

_{02}/T

_{01.}If keeping the incident shock Mach number fixed, the extended structure of the incident wave (5) will not change. Then, the pressure jump in the reflected portion of the wave will be described with the same relation (2), but now in the interval of θ values. The key variable in the calculations, π

_{r}, then can be used to determine all the remaining variables for the reflected and transmitted waves using relations (9), (12), (13) and (15).

_{r}are plotted vs. the parameter θ. Each curve corresponds to a different coordinate x within the relaxation zone, from x = 0 to x = 0.025 cm, through equal intervals of 0.005 cm. The coordinate x increases from the upper to the lower curve. In another plot, Figure 7b, the temporal profiles for π

_{r}are plotted for a number of θ values, between 1.0 and 2.6 through the equal interval of 0.2. The parameter θ increases from the upper curve to the lower curve.

_{r}(θ) converge at the same point at θ = 1 thus confirming that, in the absence of sizable difference in the heat capacity ratios and insignificant dissociation in the gas during the transitional period, no any shift in the transition point determining the character of the reflected wave is produced. The shifts are not large naturally and become noticeable in real gases when there is a significant difference in the gas temperatures across the interface resulting in a diversion in the gas properties [9,16].

_{i}, while keeping the parameter θ fixed. The procedure can be split into two parts.

_{01}, and M

_{i,}and thus all the changes during the transitional period are considered relative to that in the incident wave. Thus the processes of formation of the shock structure and reflection, being independent of each other, can be considered separately.

_{i}to determine the reflected and transmitted wave intensities at x = 0, and then the extension zone profile can be reconstructed using the method described above. For this, first, the transcendental Equation (2) can be analyzed to find a solution for π

_{r}(M

_{i}, θ, x = 0) as a function of the incident Mach number (with θ as a fixed parameter), and then for the transmitted shock strength π

_{t}(M

_{i}, θ, x = 0), with the quantities defined as the reflection and transmission coefficients, accordingly:

_{i}, θ)/π

_{i}(M

_{i}, θ), t = π

_{t}(M

_{i}, θ)/π

_{i}(M

_{i}, θ), x = 0}

_{r}as a function of the Mach number.

_{0}, υ

_{1}, υ

_{3}, υ

_{4}, υ

_{5}, and υ

_{6}are the functions of M

_{i}, θ, and λ, and their expressions are given in Appendix A. A similar equation can be obtained for π

_{t}(M

_{i}, θ, x = 0) using the differential Equation (19), the transformation relation between the variables, and condition (4), where π

_{i}(x = 0) is a known function of M

_{i}. Alternatively, if a solution for π

_{r}(M

_{i}, θ, x = 0) is obtained first, then the function π

_{t}(M

_{i}, θ, x = 0) is determined directly from the condition (4):

_{r}(x = 0)]

^{−1}first quickly increases with the Mach number and then saturates at the value of 1/π

_{r}≈ 1.54 after the Mach number reaches M

_{i}= 10. Accounting for a simultaneous and quick increase in the pressure jump across the incident shock π

_{i}(M

_{i}, x = 0) with M

_{i}, the shock reflectivity r = π

_{r}(M

_{i}, θ)/π

_{i}(M

_{i}, θ) diminishes to negligible values. Thus, at the interface with a fixed value of θ, the increase in the shock Mach number results in diminishing relative losses due to reflection. The results are not surprising since in this case, the internal energy in the gas due to heating across the interface becomes less and less comparable to the work done by compression in the incident shock of increasing strength. Similarly, for the transmission coefficient $t={\left|{\pi}_{t}/{\pi}_{i}\right|}_{x=0}$, the conclusions reciprocal to the above can be made by applying relationship (4), from which it follows that stronger shocks transmit interfaces more effectively.

## 6. Summary and Conclusions

_{2}) and elevated temperatures on both sides of the interface (5000 K and 10,000 K) were the conditions enabling noticeable excitation of molecular vibrations, while the dissociation term levels were still insignificant. A much slower relaxation rate for vibrations relative to that for the active degrees of freedom results in the delay in establishing the equilibrium in the gas past the shock front, leading to a relaxation zone of significant width with a continuous distribution of the parameters inside of it.

_{02}, the relaxation length makes up its original value for the incident shock. Thus, if the transmitted shock structure is registered in experiments in terms of time, the shortening of the relaxation zone, along with a lower contrast level (reduced relaxation depths) the structure will be displayed as being significantly less pronounced. However measuring the structure in terms of distance may not show a visible difference, except for the intensity. After the transitional period limited with the incident shock relaxation time, the gas state past the transmitted wave will be adjusted accordingly to the new temperature of the medium and can be determined with the relations (5)–(9), the experimental data for the energy content factors ${\beta}_{a}$, ${\beta}_{i}$, and the corresponding factor σ.

_{i}increases. Thus, attenuation of stronger shocks by means of reflection becomes less efficient.

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1

#### Appendix A.2

#### Appendix A.3

_{0}, υ

_{1}, υ

_{3}, υ

_{4}, υ

_{5}, and υ

_{6}appearing in Equation (19).

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**Figure 1.**On the space–time diagram, a plane shock of an extended structure (“i”) incident normally on a plane interface (dashed line), from left to right, is partially reflected (“r”) and partially transmitted (“t”) into the media of higher temperature T

_{02}.

**Figure 3.**The temperature (

**a**), pressure (

**b**), and density (

**c**) temporal profiles in the relaxation zone of the incident shock. Diatomic nitrogen at 5000 K, M

_{i}= 3.5.

**Figure 4.**Temporal change of the distribution function at the point of reflection (x = 0). The shock is moving from left to right.

**Figure 5.**Flow parameter distribution in the relaxation zone across reflected (

**a**,

**b**) and transmitted (

**c**,

**d**) waves. M

_{i}= 3.5. Incident medium is non-dissociating N

_{2}at 5000 K.

**Figure 6.**The flow speed distribution in the transitional layer across the incident shock, (

**a**) u

_{i}, the transmitted shock, (

**b**) u

_{t}, and at the head of the expansion fan of the rarefaction waves (

**c**) u

_{r}. M

_{i}= 3.5, incident medium is non-dissociating N

_{2}at 5000 K.

**Figure 7.**(

**a**) The pressure jump π

_{r}vs. parameter θ. Each curve corresponds to a different coordinate x within the relaxation zone, from x = 0 to x = 0.025 cm, through equal intervals of 0.005 cm. The coordinate x increases from the upper to the lower curve. (

**b**) π

_{r}vs. time t. Each line corresponds to various θ between 1.0 and 2.6 through the equal interval of 0.2. The parameter θ increases from the upper curve to the lower curve.

**Figure 8.**The pressure jump in the reflected wave π

_{r}(x = 0) vs. incident shock Mach number M

_{i}in non-dissociating nitrogen at 5000 K. The heating strength is fixed at the value $\theta =\sqrt{2}$.

**Table 1.**The relaxation times ${t}_{\lambda}$ and relaxation depths ${D}_{\lambda}$ in the relaxation zone for the temperature, pressure, density, and flow speed distributions in the incident, reflected, and transmitted waves.

Relaxation Time (μs) and Depth | T_{21} | π | u | |
---|---|---|---|---|

Incident: | ${t}_{\lambda}^{\left(i\right)}$, μs | 0.150 | 0.150 | 0.150 |

${D}^{\left(i\right)}$ | −0.432 | +0.103 | + 0.061 | |

Reflected: | ${t}_{\lambda}^{\left(r\right)}$, μs | 0.150 | 0.150 | 0.150 |

${D}^{\left(r\right)}$ | +0.004 | −0.048 | −0.141 | |

Transmitted: | ${t}_{\lambda}^{\left(t\right)}$, μs | 0.100 | 0.100 | 0.120 |

${D}^{\left(t\right)}$ | −0.186 | +0.059 | + 0.036 |

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**MDPI and ACS Style**

Markhotok, A.
The Post-Shock Nonequilibrium Relaxation in a Hypersonic Plasma Flow Involving Reflection off a Thermal Discontinuity. *Plasma* **2023**, *6*, 181-197.
https://doi.org/10.3390/plasma6010014

**AMA Style**

Markhotok A.
The Post-Shock Nonequilibrium Relaxation in a Hypersonic Plasma Flow Involving Reflection off a Thermal Discontinuity. *Plasma*. 2023; 6(1):181-197.
https://doi.org/10.3390/plasma6010014

**Chicago/Turabian Style**

Markhotok, Anna.
2023. "The Post-Shock Nonequilibrium Relaxation in a Hypersonic Plasma Flow Involving Reflection off a Thermal Discontinuity" *Plasma* 6, no. 1: 181-197.
https://doi.org/10.3390/plasma6010014