ON THE FEATURES OF HEAT TRANSFER IN ANISOTROPIC REGIONS WITH DISCONTINUOUS THERMAL-PHYSICAL CHARACTERISTICS

Vladimir F. Formalev1, Sergey A. Kolesnik2, Ekaterina L. Kuznetsova3§, Lev N. Rabinskiy4 1,2Department of Computing Mathematics and Programming Moscow Aviation Institute National Research University 125993, Volokolamsk Highway 4, Moscow, RUSSIAN FEDERATION 3Department of Space Technology Moscow Aviation Institute National Research University 125993, st. Autumn 22, Moscow, RUSSIAN FEDERATION 4Department of Applied Mechanics Moscow Aviation Institute National Research University 125993, Volokolamsk Highway 4, Moscow, RUSSIAN FEDERATION


Introduction
Mathematical simulation of heat transfer in anisotropic regions and at their boundaries (both internal and external) contacting with high-temperature media contains considerable difficulties.They are related with (i) a necessity to simulate the multidimensional problems only, (ii) the fact that heat conduction in such cases is described by tensors of rank two rather than by scalars, which involves presence of mixed derivatives in the equations of heat conduction, and (iii) absence of information about publications on formulation of edge conditions with derivatives at the boundaries of anisotropic media.The development of theory of heat conduction in anisotropic bodies was considered in two monographs [1] and [2] by one of the authors of the present article, as well as in a section of the monograph by Carslaw and Jaeger [3] and in a number of publications (see, e.g., [4], [5] and [6]) by the authors of the present article.However, we do not know the works on the theory of heat conduction in anisotropic regions with discontinuous thermal-physical characteristics (TPC).
Here we simulate heat conduction in anisotropic regions with ideal contacts at the internal TPC discontinuity boundaries (mating boundaries) under heat exchange at the external boundaries.In this case, not only the tensor components but also the angles orientating the principal axes of the heat conduction tensor are discontinuous at the mating boundaries of two anisotropic bodies.

Problem Statement
It is known that in anisotropic bodies (contrary to isotropic ones) the heat flux density vectors are not oriented along the normals to isotherms owing to the presence of off-diagonal components of the heat conduction tensor [1].This indicates that there are normal as well as tangential components of the heat flux density vectors on isotherms in anisotropic space.Especially that fact occurs at the interfaces between two contacting anisotropic bodies.
Let us consider a contact boundary for two anisotropic bodies with TPC ñI , ρ I , K I and c II , ρ II , K II (heat capacities, densities and heat conduction tensors, respectively).Then transfer of potential (temperature) at the TPC discontinuity boundaries occurs only along the normal to the boundary.According to the first law of thermodynamics, the normal components of the heat flux density vectors, (K I gradT I , n 0 ) and (K II gradT II , n 0 ), are continuous at these boundaries, i.e., the boundary conditions of the fourth kind proposed by Lykov are valid [7]: Here n 0 is the normal unit vector at a point of the TCP discontinuity boundary; scalar products of the heat flux density vectors and normal unit vectors are in parentheses.
Tangential components of the heat flux density vectors on opposite sides of the TCP discontinuity boundary do not participate in potential transfer through that boundary because they do not cross it.Their values are formed mainly by the prehistory of heat exchange along the mating boundary, so tangential components of the heat flux density vectors are discontinuous at each point, i.e.
(K I gradT I , τ 0 ) = (K II gradT II , τ 0 ), where τ 0 is the unit vector of tangential to the TCP discontinuity boundary.Therefore, the heat flux densities at the TCP discontinuity boundaries are of discontinuous character in anisotropic bodies.At first glance such an analysis leads to a paradoxical conclusion on violation of the first law of thermodynamics, since the input heat flux density at a point of the TCP discontinuity boundary is not equal to the output one.If, however, the total input and output heat fluxes to the whole TCP discontinuity boundary are considered, then continuity of heat fluxes is ensured.
In the 2D case, if the smooth TCP discontinuity boundary is described by the function y = f (x), then after expansion Eq. (1) takes the following form: where λ I 11 , λ I 12 = λ I 21 , λ I 22 and λ II 11 , λ II 12 = λ II 21 , λ II 22 are components of the heat conduction tensors K I and K II , respectively; f ′ (x) = tgα, where α is the angle between the vector n 0 and local axis Oy.
For curvilinear boundaries of anisotropic bodies, it is reasonable to introduce the concept of heat conduction λ n along the normal to the curvilinear boundary.Then it follows from Eqs. ( 1) and (2) that After dividing the above expression by and taking into account that we obtain Eq. ( 4) allows for components of the heat conduction tensors and behavior of the boundary y = f (x).For a rectangular plate y = const(y ′ (x) = tgα = 0) λ s n = λ s 22 , s = I, 2; then the normal component of the heat flux density is It involves all the components of temperature gradient and not the only component, as in the case of an isotropic plate.

Physico-Mathematical Model
We considered the following problem for a 2D region as a two-layer curvilinear body shown in Figure 1, with TPC discontinuity boundary w5: ∂T ∂y ∂T ∂y ∂T ∂y The components K I , K II of heat conduction tensors are determined from the following equations [1]: where λ s ξ s , λ s η s are the principal components of tensors K I , K II and φ s are the orientation angles of principal axes Oξ I , Oξ II about axis Ox.The components from Eq. ( 17) also characterize different orientations of the principal axes of heat conduction tensors at the TPC discontinuity boundaries.
One can see from the edge conditions Eqs. ( 8)-( 14) that conductive heat fluxes at the boundaries involve all temperature gradient components.It follows from Eqs. ( 14) and (17) that the tensors K I , K II (and consequently tangential components of the heat flux densities) are discontinuous at the boundary w5, even if the principal components of tensors K I , K II both equal each other (i.e., λ I ξ I = λ II ξ II and λ I η I = λ II η II ) and the orientation angles of principal axes, φ I and φ II , are not equal to each other.

Solution Method
Equations ( 6)-( 17) are solved numerically on the mesh by using the alternating directions method with time extrapolation [2].The integro-interpolation method is applied for boundary nodes to retain the second order [8].
For regular nodes on the mesh given by Eq. (18), the above method gives where Here Λ 11 i+1,m − T n i+1,m , m = j − 1, j, j + 1.The scheme of Eqs. ( 19) and (20) completely approximates the differential Eqs. ( 6) and ( 7) in the case of constant components of the heat conduction tensor, with , and is absolutely stable [2].In Eqs. ( 19) and ( 20) are mesh functions on the corresponding time layers t n , t n+1/2 , t n+1 .The scheme of Eqs. ( 19) and ( 20) is economical because it is realized by scalar sweeps along the coordinate axes Ox and Oy only.To retain the order of approximation for regular nodes, let us discuss approximation of boundary conditions with derivatives using the integro-interpolation method [8].
Our consideration will be performed for nodes at the layer mating boundary (an approach to other irregular nodes is similar).Let us write down Eqs. ( 6) and (7) as where q s x , q s y are components of the heat flux density vector.For a plane TPC discontinuity boundary and the edge mating condition Eq. ( 14) at the TPC discontinuity boundary is Let us choose an arbitrary node (x i , y j = l 2 ) at the boundary w5 and integrate Eqs. ( 21) on an interval x ∈ [x i−1/2 , x i+1/2 ] with respect to variable x and then with respect to variable y (at first on an interval y ∈ [y j−1/2 , y j ] and then on an interval y ∈ [y j , y j+1/2 ]).After this, by applying the quadrature formulae for rectangles with the second degree of accuracy on the steps h 1 , h 2 we obtain By adding Eqs. ( 24) for s = 1 and s = 2 and using Eqs.( 21), we obtain the expressions that approximate the initially stated problem at the nodes at the mating boundary w5 with components q s x , q s y of the heat flux density vector accurate to the second degree: The terms (q I y ) i,j h 1 and (q II y ) i,j h 1 in Eq. (25) cancel because of Eq. ( 23), so the mating conditions are automatically obeyed.
Let us apply the scheme Eq. ( 19) of the alternating directions method with extrapolation, by approximating the derivatives in Eq. (25) with using Eq. ( 22) towards axis Ox on the n + 1/2 -th temporal half layer at the nodes , s = I, 2; (28) 2 ) A procedure of the integro-interpolation method with usage of boundary conditions Eqs. (10)-( 13) is applied separately at the nodes at i = 0 and i = I to close the system.
After solving the set of linear algebraic equations by the scalar sweep method, we get distribution of temperature T n+1/2 i,j at the boundary y = l 2 .The systems of algebraic equations using the sub-scheme Eq. (20) relative to T n+1 i,j are obtained similarly.

Results of Numerical Experiments
Using the above numerical methods, we developed a program complex that realizes the mathematical model Eqs.( 6)-(17).This program complex makes it possible to calculate temperature fields and components of heat flux density vector, in particular, at the TPC discontinuity boundary.Shown in Figure 2 are temperature fields in a moment of t = 200 s in an anisotropic two-layer plate with sizes of l 1 = 0.05m, l 2 = 0.02m and l 3 = 0.02.One can see from Figure 2 that the temperature field in the body remains continuous at the mating boundary, while the temperature derivatives and tangential components of the heat flux densities have discontinuities of the first kind at this boundary.This can be seen more clearly in Figure 3 that presents (with the same input data) the distributions of normal (q y ) and tangential (q z ) components of the heat flux density vector along the variable y, at fixed x = 0.004.One can clearly see in Figure 3 that normal component q y of the heat flux density along axis Oy remains continuous, while tangential component q x has discontinuities of the first kind at the TPC discontinuity boundary.

Figure 1 :
Figure 1: Mating boundary w5 between two anisotropic media with different thermal-physical characteristics.

Figure 2 :
Figure 2: Temperature distribution in an anisotropic two-layer body with discontinuous characteristics of heat transfer.

Figure 3 :
Figure 3: Behavior of components q x and q y of the heat flux density vectors at the mating boundary of anisotropic bodies.