Mathematical Simulation of Thermal Shock Wave Dynamics in Nonlinear Local Non-Equilibrium Media

Abstract

We performed a mathematical simulation of heat transfer in a local non-equilibrium medium whose transfer characteristics are functions of the temperature distribution. A homogeneous polynomial of arbitrary degree represents nonlinearities in thermal conductivity and thermal diffusivity. The mathematical model consists of a hyperbolic nonlinear heat transfer wave equation, initial conditions and nonlinear boundary conditions of the second and first kind. To solve this problem, we used a conservative homogeneous finite-difference scheme along the upper time grid line (implicitly). We then used the tridiagonal matrix algorithm of the second order in the spatial variable and of the first order in time to solve the resulting system of linearised algebraic equations. A periodic series of rectangular temperature or heat flux pulses form the boundary conditions of the first and second kind. Computation results reveal ultimate propagation rates of temperature and heat fronts featuring pronounced first-kind discontinuities with attenuating magnitudes. As the process unfolds, the initial pulses heat the region between the boundary and the heat wave front, while the subsequent pulses traverse this region at a higher velocity due to thermal diffusivity being a function of temperature, their fronts "catching up" with the previous fronts, increasing the discontinuity magnitude at the initial pulse front, that is, forming a thermal shock wave front similar to that of a shock wave in gas dynamics. We obtained such thermal shock waves for boundary conditions of both the first and the second kind. We also analysed kinematic and dynamic characteristics of thermal waves

The study was supported by the Russian Science Foundation (grant no. 22-21-00776)

Please cite this article in English as:

Formalev V.F., Garibyan B.A., Kolesnik S.A. Mathematical simulation of thermal shock wave dynamics in nonlinear local non-equilibrium media. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 4 (103), pp. 80--94 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2022-4-80-94

References

[1] Lykov A.V. Using methods of irreversible processes thermodynamics for study on heat and mass transfer. Inzhenerno-fizicheskiy zhurnal, 1965, vol. 9, no. 3, pp. 287--304 (in Russ.).

[2] Sobolev S.L. Local non-equilibrium transport models. Phys. Usp., 1997, vol. 40, no. 10, pp. 1043--1053. DOI: https://doi.org/10.1070/PU1997v040n10ABEH000292

[3] Zarubin V.S., Kuvyrkin G.N. Ignition of a propane-air mixture for a reflected shock wave at high pressures. High Temp., 2003, vol. 41, no. 2, pp. 257--265. DOI: https://doi.org/10.1023/A:1023390021091

[4] Zarubin V.S., Leonov V.V., Zarubin V.S. Jr. Temperature state of the anisotropic spherical layer during convective heat exchange with the environment. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, no. 4 (95), pp. 40--55 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2019-4-40-55

[5] Lykov A.V. Teplomassoobmen [Heat and Mass Transfer]. Moscow, Energiya, 1978.

[6] Shashkov A.G., Bubnov A.V., Yanovskiy S.Yu. Volnovye yavleniya teploprovodnosti [Wave phenomena of heat conduction]. Moscow, Editorial URSS Publ., 2004.

[7] Formalev V.F., Kartashov É.M., Kolesnik S.A. On the dynamics of motion and reflection of temperature solitons in wave heat transfer in limited regions. J. Eng. Phys. Thermophy., 2020, vol. 93, no. 1, pp. 10--15. DOI: https://doi.org/10.1007/s10891-020-02085-9

[8] Formalev V.F., Kolesnik S.A., Kuznetsova E.L. Wave heat transfer in the orthotropic half-space under the action of a nonstationary point source of thermal energy. High Temp., 2018, vol. 56, no. 5, pp. 727--731. DOI: https://doi.org/10.1134/S0018151X18050073

[9] Formalev V.F., Kartashov É.M., Kolesnik S.A. Simulation of nonequilibrium heat transfer in an anisotropic semispace under the action of a point heat source. J. Eng. Phys. Thermophy., 2019, vol. 92, no. 6, pp. 1537--1547. DOI: https://doi.org/10.1007/s10891-019-02074-7

[10] Kartashov E.M. Mathematical models of heat conduction with a two-phase lag. J. Eng. Phys. Thermophy., 2016, vol. 89, no. 2, pp. 346--356. DOI: https://doi.org/10.1007/s10891-016-1385-9

[11] Sobolev S.L. Discrete space-time model for heat conduction: application to size dependet thermal conductivity in nano-films. Int. J. Heat Mass Transf., 2017, vol. 108, part A, pp. 933--939. DOI: https://doi.org/10.1016/j.ijheatmasstransfer.2016.12.051

[12] Kudinov V.A., Eremin A.V., Kudinov I.V., et al. Strongly nonequilibrium model of thermal ignition with account for space-time nonlocality. Combust. Explos. Shock Waves, 2018, vol. 54, no. 6, pp. 649--653. DOI: https://doi.org/10.1134/S0010508218060035

[13] Kirsanov Yu.A. The influence of thermal relaxation and thermal damping on transient processes with cyclic boundary conditions. High Temp., 2017, vol. 55, no. 4, pp. 535--540. DOI: https://doi.org/10.1134/S0018151X17030130

[14] Kudinov I.V., Eremin A.V., Kudinov V.A., et al. Mathematical model of damped elastic rod oscillations with dual-phase-lag. Int. J. Solids Struct., 2020, vol. 200-201, pp. 231--241. DOI: https://doi.org/10.1016/j.ijsolstr.2020.05.018

[15] Formalev V.F., Kolesnik S.A., Garibyan B.A. Heat transfer with absorption in anisotropic thermal protection of high-temperature products. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2019, no. 5 (86), pp. 35--49 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2019-5-35-49

[16] Formalev V.F., Kolesnik S.A., Garibyan B.A. Analytical solution of the problem of conjugate heat transfer between a gasdynamic boundary layer and anisotropic strip. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2020, no. 5 (92), pp. 44--59 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2020-5-44-59

[17] Formalev V.F., Kolesnik S.A., Garibyan B.A. Mathematical modeling of heat and mass transfer during aerodynamic heating of the nose parts of hypersonic aircraft. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 1 (100), pp. 107--121 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2022-1-107-121