# Redistribution of Energy during Interaction of a Shock Wave with a Temperature Layered Plasma Region at Hypersonic Speeds

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method and Statement of the Problem

_{1}= 0.41 (120K) was used. It was assumed that the coefficient of thermal conductivity k depends on temperature as

^{0.5}.

_{n}= 0.01205 kg/m

^{3}and p

_{n}= 1013.25 Pa, and the Reynolds number Re = 9500 corresponds to the conditions of the experiments in [24].

_{s}= αρ

_{∞}. Here, the parameter α characterizes the medium rarefaction, α < 1. The pressure in the source layers was equal to its undisturbed value, so, the temperature was increased in these layers. The gaps between the layers had a width equal to a half of a layer width. Below, for brevity, such a heated area is referred to as an energy source. This heated area is set at the initial time instant at some distance from the shock wave (SW), x

_{sw}is the initial shock wave coordinate, and x

_{s}is the stratified energy source boundary coordinate (Figure 1a).

^{6}nodes in the calculation area, counting the central node). Location of the stratified energy source on difference grid in enlarged form is presented in Figure 1b,c.

## 3. Analysis of the Grid Convergence

_{max}, calculated on these four difference grids, are also quite close (Figure 2b). These facts indicate the presence of grid convergence for the considered grids. Note that we used the finest grid (Grid1) for the simulations.

## 4. An Example of the Thermally Layered Gas Media Formation

## 5. Results of the Simulation

_{sw}= 1.5 and the stratified energy source boundary coordinate x

_{s}= 1.4.

_{max}in the obtained shock-wave structures is presented for different rarefaction parameters α for all considered Mach numbers of the shock wave. Here, the maximum value of the internal energy is calculated over the coordinates in the region of the obtained shock-wave structure with instabilities. This value has a sense of the greatest local increase in the internal energy due to the formation of such a structure as a result of the action of a stratified energy source. It can be seen that these dependences have maxima on t at the beginning stage of the interaction. These maximum values are larger for smaller α and for larger M. Figure 11 shows these dependencies for different M for all considered α. It can be seen that at the first stage, the dynamics of ε

_{max}is changing stronger for smaller α and larger M. Note that fluctuations in internal energy are directly connected with fluctuations in the temperature of the gas in the flow, therefore, from the results shown in Figure 10 and Figure 11, similar conclusions follow for the gas temperature [25].

_{max}in the obtained unstable shock-wave structures. This value has a sense of the greatest local increase in the kinetic energy due to the formation of a shock-wave structure as a result of the use of a stratified energy source. In Figure 12, the dependences E

_{max}(t) are presented for different rarefaction parameter α for the considered shock wave’s Mach numbers. One can see that the dependences have maxima on t which are larger for larger α and larger M. Figure 13 shows these dependencies for different M for all considered α It can be seen from these dependences that the growth of the kinetic energy occurs quicker for larger α and larger M.

_{sw}= 1.5 and the source boundary coordinate x

_{s}= 1.4. For the study of kinetic energy, this time interval turned out to be not enough (because its maximum value is reached at the middle stage in time). Here, we used a different geometry of the calculation domain with the position of the shock wave x

_{sw}= 2.25 and the boundary of the energy source x

_{s}= 2.15 with the same vertical dimensions. This made it possible to study the time history of kinetic energy in the time interval up to t = 0.25 for the considered shock wave Mach numbers.

_{i}and N

_{j}—the amounts of grid nodes in i- and j-directions, ε

_{averaged}and ε

_{h averaged}are the averaged values of internal energy in the stratified and homogeneous sources, accordingly.

_{h}and E

_{h}are close to constant.

_{ε}and in the kinetic energy η

_{E}are evaluated as follows:

_{h}and E

_{h}are the values inside the region between the shock wave and contact discontinuity. The values η

_{ε}and η

_{E}have the meaning of the maximum relative difference of the considered types of energy over time between their values for stratified and homogeneous energy sources.

_{ε}and η

_{E}on the rarefaction parameter α are presented for the considered Mach numbers. It is seen that the relative deviation in the local peaks of ε can achieve up to 29% (for M = 6, α = 0.1) and increases with decreasing α and M (Figure 15a). The relative deviation in the local peaks of E can achieve up to 8.3 times (for M = 6, α = 0.1) and also increases with decreasing α and M (Figure 15b). It should be pointed out that this value depends on the temperature of the layers in the energy source, more so than on the Mach number of the shock wave. Note that the difference in the behavior of η

_{ε}for M = 6 and for M = 8–10 is connected with a different location of the maximum value ε

_{max}in the considered shock-wave structures.

## 6. Manifestation of the Richtmyer-Meshkov Instability

## 7. Conclusions

- The use of stratified energy sources gives the possibility to redistribute the energy behind the shock wave front in such a way that results in the distortion or complete disappearance of the shock wave fronts (in density fields).
- The mechanism of shock wave blurring is associated with multiple manifestations of the Richtmyer-Meshkov instabilities.
- The use of stratified energy sources gives the possibility to obtain the local zones of specific internal energy and volume kinetic energy behind the shock wave with the relative difference exceeding up to 29% and 8.3 times, accordingly, these values for the homogeneous energy source with the same total energy; these relative differences increase with decreasing α and M.
- The dependences of specific internal energy have maxima on t at the beginning stage of the interaction which are larger for smaller α and for larger M, and the internal energy is changing more significantly for these α and M. The dependences of kinetic energy have maxima on t which are larger for larger α and larger M, and the growth of the kinetic energy occurs more quickly for these α and M.
- The values of the considered types of energy in the obtained shock-wave structures can be controlled at hypersonic speeds via the rarefaction parameter (or temperature) in the layers of the stratified energy source.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of the problem statement (

**a**); location of the stratified energy source on difference grid (enlarged) (

**b**); location of one layer of on difference grid (enlarged) (

**c**).

**Figure 2.**Comparison of the calculations on four different grids, M = 8, α = 0.4: (

**a**) isolines of ε, t = 0.12 (

**b**) dynamics of ε

_{max}.

**Figure 3.**Examples of large-scale (

**a**) and small-scale (

**b**) strata of an ionization-unstable discharge.

**Figure 4.**Dynamics of the density field in isochores: M = 8, α = 0.1, t = 0.017, 0.07 and 0.12 (superposed).

**Figure 16.**Manifestation of the Richtmyer-Meshkov instabilities in a case of wider layers, density, M = 3, α = 0.4.

Difference Grid | A Number of Working Nodes in the Calculation Area | The Space Steps Values |
---|---|---|

Grid 1 | 1.6 × 10^{6} | h_{x} = h_{y} = 0.0005 |

Grid 2 | 1.2 × 10^{6} | h_{x} = h_{y} = 0.000571 |

Grid 3 | 0.9 × 10^{6} | h_{x} = h_{y} = 0.000667 |

Grid 4 | 0.63 × 10^{6} | h_{x} = h_{y} = 0.0008 |

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**MDPI and ACS Style**

Azarova, O.A.; Lapushkina, T.A.; Krasnobaev, K.V.; Kravchenko, O.V.
Redistribution of Energy during Interaction of a Shock Wave with a Temperature Layered Plasma Region at Hypersonic Speeds. *Aerospace* **2021**, *8*, 326.
https://doi.org/10.3390/aerospace8110326

**AMA Style**

Azarova OA, Lapushkina TA, Krasnobaev KV, Kravchenko OV.
Redistribution of Energy during Interaction of a Shock Wave with a Temperature Layered Plasma Region at Hypersonic Speeds. *Aerospace*. 2021; 8(11):326.
https://doi.org/10.3390/aerospace8110326

**Chicago/Turabian Style**

Azarova, O. A., T. A. Lapushkina, K. V. Krasnobaev, and O. V. Kravchenko.
2021. "Redistribution of Energy during Interaction of a Shock Wave with a Temperature Layered Plasma Region at Hypersonic Speeds" *Aerospace* 8, no. 11: 326.
https://doi.org/10.3390/aerospace8110326