Uneven surface heating of anisotropic spherical layer


Аuthors

Zarubin V. S.*, Zarubin V. S., Leonov V. V.**

Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia

*e-mail: zarubin@bmstu.ru
**e-mail: lv-05@mail.ru

Abstract

When the surface of the layer of thermal protection coating with a low value of thermal conductivity factor is unevenly heated, an area with elevated temperature, which may exceed the allowable value for the coating material, may occur on this surface. One of the possible ways to reduce the effect of intense local heating on the performance of the layer of thermal protection coating is to use an anisotropic material, the thermal conductivity of which, for example, is higher in the tangential direction than in the direction normal to the surface.

Numerical analysis of the effect of anisotropy on the temperature state of a flat layer of thermal protection coating showed that with a high degree of anisotropy, when the thermal conductivity of the material of layers in the longitudinal direction exceeds the thermal conductivity in the transverse direction by more than an order of magnitude, a cooling effect occurs on some part of the streamlined surface, i.e. a change in the direction of the heat flow occurs on this part of the surface due to the increase in its temperature.

It should be expected that the manifestation of a similar effect is possible in case of a flow around anisotropic thermal protection coating on an axisymmetric body with a dulling in the form of a spherical surface fragment. In order to quantify this effect, a model problem of stationary thermal conductivity in anisotropic spherical layer of a thermal protection coating is considered with varying degrees of non-uniformity of its surface heating.

When analyzing the temperature state of a spherical layer of a thermal protection coating, the steady-state temperature distribution can be described by the function T(r,θ), which satisfies Laplace’s differential equation. In the model problem under consideration, the T0 temperature is set for the inner surface of the layer, and the heat flux of q(θ) density is supplied to its outer surface. The adopted variant of setting the thermal effect on the anisotropic spherical layer allows, based on the problem solution, to compare the effect of uniform heating of the layer on the outer surface with the localized thermal effect occurring, for example, at the front critical point behind the detached shock wave in case of hypersonic flow around a spherical dull. In order to solve the problem, the method of separation of variables (Fourier method) with the use of Legendre polynomials is applied.

As a result of solving the problem, it has been established that an increase in the degree of anisotropy of the spherical layer material leads to a decrease in the temperature of the most heated surface point. Also, quantitative estimates have been obtained for the degree of influence of changes in the heterogeneity of the heat flux density distribution and the relative thickness of the layer of thermal protection material on the temperature of the most heated point.

The obtained solution makes it possible to choose the characteristics of anisotropic material of a thermal protection coating that make it possible to reduce the intensity of thermal effect on the surface depending on the parameters of the supplied heat flux.

Keywords:

spherical layer, thermal protection coating, thermal conductivity anisotropy

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