galerkin method for solving combined radiative and
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Galerkin method for solving combined radiative andconductive heat transfer
Mohamed Ghattassi, Jean Rodolphe Roche, Fatmir Asllanaj, MohamedBoutayeb
To cite this version:Mohamed Ghattassi, Jean Rodolphe Roche, Fatmir Asllanaj, Mohamed Boutayeb. Galerkin methodfor solving combined radiative and conductive heat transfer. International Journal of Thermal Sci-ences, Elsevier, 2016, 102, pp.122-136. 10.1016/j.ijthermalsci.2015.10.011. hal-01273062
Galerkin method for solving combined radiative and
conductive heat transfer
M. Ghattassia,b,∗, J. R. Rochea,b, F. Asllanajc,d, M. Boutayebe
aUniversite de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502,Vandoeuvre lesNancy, France
bCNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre les Nancy, France.cUniversite de Lorraine, Laboratoire d’Energetique et de Mecanique Theorique et
Appliquee, UMR 7563, Vandoeuvre les Nancy, France.dCNRS, Laboratoire d’Energetique et de Mecanique Theorique et Appliquee, UMR 7563,
Vandoeuvre les Nancy, France.eUniversite de Lorraine, Centre des Recherches en Automatique, UMR 7039, Vandoeuvre
les Nancy, France.
Abstract
This article deals with a numerical solution for combined radiation and con-duction heat transfer in a grey absorbing and emitting medium applied toa two-dimensional domain using triangular meshes. The radiative transferequation was solved using the high order Discontinuous Galerkin methodwith an upwind numerical flux. The energy equation was discretized usinga high order finite element method. Stability and error analysis were per-formed for the Discontinuous Galerkin method to solve radiative transferequation. A new algorithm to solve the nonlinear radiative-conductive heattransfer systems was introduced and different types of boundary conditionswere considered in numerical simulations. The proposed technique’s highperformance levels in terms of accuracy and stability are discussed in thispaper with numerical examples given.
Keywords: Radiative transfer equation, Energy equation, Galerkinmethod, Newton Method, Error estimation.
Nomenclature
∗Corresponding authorEmail address: [email protected] (M. Ghattassi)
Preprint submitted to International Journal of Thermal Sciences September 8, 2015
cp specific heat capacity, J kg−1K−1
C0(Ω) continuous functions which vanish at infinityD unit disk of R2: D = β ∈ R2 : |β| 6 1G? incident radiation intensity, W m−2
g dimensionless radiative boundary conditionI? radiation intensity, W m−2sr−1
Ib blackbody intensity, Wm2sr−1
hc convective exchange coefficient at the boundary, W m−2 K−1
kc thermal conductivity of the medium, W m−1 K−1
L thickness of the medium, mn refractive indexn outward unit normal to the boundaryNs conduction radiation parameterBi Biot numberQrad radiative source term, Wm−2
s? path length, ms dimensionless path lengthS?rad radiative source term, Wm−3
t time, sTh triangulation of ΩT ?, T ?0 temperature, initial temperature in the medium, KT, T0 dimensionless temperature, initial dimensionless temperature in the mediumTref reference temperature, K
Greek symbolsΩ dimensionless bounded domain in R2
Ω? bounded domain in R2, mε wall emission factorκ absorption coefficient, m−1
σB Stefan-Boltzmann constantρ diffuse reflectivity or density, kg m−3
β direction of propagation of radiation∆t,∆ξ time step s, dimensionless step timeΓ boundary of control volumeξ dimensionless time
θ =κ2L2
Ns
dimensionless constant
Subscripts
2
l direction angular directionh step of the mesh
1. Introduction
The heat transfer models in participating media for scientific purposeor engineering applications such as high temperature processing or thermalinsulation involve coupled radiation and conduction heat equations [1, 2, 3,4, 5, 6]. The most fundamental difficulties for the simulation of radiativeand conductive heat transfer problems are the nonlinear coupling terms andevolutive boundary conditions. Historically, many simplified models havebeen used to carry out numerical simulations of radiative transfer phenomena.Generally, two classes of numerical methods have been applied to attempt tosolve radiative heat transfer problems namely the Monte Carlo method [7, 8]and the deterministic methods [9, 10, 11].
In this study, we chose the Discontinuous Galerkin (DG) method, a de-terministic method, to solve the radiative transfer equation(RTE). The DGmethod is known to be locally conservative, stable and high-order accurate.It can also easily handle complex geometries, irregular meshes with hang-ing nodes and allows for different degrees of approximation for differentelements[12, 13]. The DG method is known to be a particularly powerfulnumerical scheme for the simulation of hyperbolic transport problems. Thenodal DG method can be easily combined with the well known finite ele-ment methods (FEM) or finite volume methods (FVM) to couple radiativeand conductive heat transfer. Spatial DG techniques applied to discrete-ordinates radiation transport have been pioneered by Reed and Hill [14] andLesaint and Raviart [15].
In [16], the DG method was extended to the solution of radiative heattransfer problems in absorbing, emitting and scattering media. In [16] the au-thors studies a centered numerical flux and implemented a parallel computingalgorithm based around the localized DG formulation. In [17] a 3-d modelwas reduced to 2-d using symmetry and the DG formulation. In [18, 19], theauthors added a stabilization term to penalize the jump of the solution acrossinterior faces of the triangulation and to obtain physical numerical solutionsin practical applications. For the transient state, many authors have consid-ered radiation and conduction coupled problem, see [20, 21, 22]. Mishra et al.[22], applied the lattice Boltzmann method to solve the energy equation of
3
a transient conduction-radiation problem in a 2-d rectangular enclosure andthe collapsed-dimension method was implemented to solve the RTE. Asllanajet al. [20], simulated transient heat transfer by radiation and conduction intwo-dimensional complex shaped domains with structured and unstructuredtriangular meshes working with an absorbing, emitting and non-scatteringgrey medium. To solve the RTE, these authors applied a modified FVMbased on a cell vertex scheme combined with a modified exponential schemewhere temperature was approximated by linear interpolation using nodalvalues. The PHAML (Parallel Hierarchical Adaptive Multi Level) code wasused to solve the heat conduction equation in space, with low or high orderfinite elements.
The objective of the work covered in this paper was to solve the nonlinearsystem which describes the heat transfer in a grey absorbing and emittingmedium. We introduced a high order DG method based on an upwind stablenumerical flux to solve the radiative heat transfer problem and a high orderfinite element method to solve the energy equation. Indeed, the high orderGalerkin method ensures a more accurate approach with a computationaltime gain. We can also generalize this method to solve combined radia-tion conduction heat transfer in irregular geometries. An adequate normwas introduced to prove the stability of DG method for solving RTE andwe shall also give an adapted error estimate which we found to be accu-rate in our simulations. A new numerical algorithm to solve the considerednonlinear-coupled radiative-conductive problem is given in this paper alongwith numerical results for high order scheme. We investigated a new type ofboundary conditions and obtained new numerical results for cases not cov-ered in previous publications. Finally we shall show that numerical stabilitywas verified in our simulations.
The outline of this paper is as follows. In the next section, the radiative-conductive heat transfer in the absorbing emitting medium will be brieflydescribed and the dimensionless form is introduced. The radiative boundarycondition and its corresponding mixed thermal boundary conditions will bepresented. Discretization of the RTE and energy equation by the discreteDG method and the finite element method respectively, are discussed indetail in Section 3 and also established the stability of the DG method withthe adequate norm. Finally, in Section 4, we shall investigate the coupledheat transfer problem and give results showing the effects of the conduction-radiation parameter and wall emissivity on radiative and conductive heatfluxes.
4
2. Problem statement
In this section, the radiative-conductive heat transfer system of partial dif-ferential equations in a two dimensional gray absorbing and emitting mediumis introduced. We denote
∂Ω?− = (s?, β) ∈ ∂Ω? ×D such that β.n < 0 . (1)
The unknown of the RTE is the radiation intensity denoted I?(t, s?, β) givenat time t, position s? and in the direction β. The unknown of the energyequation is the temperature T ?(t, s?) at time t and position s?. The RTE isgiven by (see,[20])
β.∇I?(t, s?, β) + κI?(t, s?, β) = κn2Ib(T?(t, s?)), for (t, s?, β) ∈ [0, τ?]× Ω? ×D, (2a)
I?(t, s?, β) = g?(t, s?, β), for (t, s?, β) ∈ [0, τ?]× ∂Ω?−, (2b)
where κ is the absorption coefficient of the medium and n is the refractiveindex. In this work, the refractive index and the absorption coefficient areassumed to be equal to one, n = 1, κ = 1m−1 . Ib(T
?) is the radiationintensity of the blackbody with, T ?, the temperature of the medium:
Ib(T?) =
σBπT ?4, (3)
where σB = 5.6698× 10−8 Wm−2K−4 is the Stefan-Boltzmann constant.The radiative boundary condition g?(t, s?, β) takes into account of two quan-tities, the emitted radiation intensity and the incoming radiation. For anopaque wall with specular reflection, we have
g?(t, s?, β) = εIb(T?(t, s?)) + (1− ε)I?(t, s?, βs) for (t, s?, β) ∈ [0, τ?]× ∂Ω?−, (4)
where ε is the wall emissivity (assumed to be constant), T ?(t, s?) is the tem-perature at the boundary of the medium and βs is the specular direction ofradiation defined by βs = β − 2(β.ni)ni where ni is the local outward surfacenormal. For an opaque wall with diffuse reflection, the boundary conditiontakes the following from
g?(t, s?, β) = εIb(T?(t, s?))+
(1− ε)π
∫ni.β
′<0
I?(t, s, β′)|ni.β
′|dβ
′(t, s?, β) ∈ [0, τ?]×∂Ω?−.
(5)
The function g? is split in two parts, the emission part g?T and the reflectedpart g?I .
g?(t, s?, β) = g?T (t, s?, β) + g?I (t, s?, β) (t, s?, β) ∈ [0, τ ?]× ∂Ω?
− (6)
5
whereg?T (t, s?, β) = εIb(T
?(t, s?)) (t, s?, β) ∈ [0, τ ?]× ∂Ω?−
and
g?I (t, s?, β) =
(1− ε)π
∫ni.β
′<0
I?(t, s, β′)|ni.β
′|dβ ′(t, s?, β) ∈ [0, τ ?]× ∂Ω?
−.
It should be underlined when the wall emissivity ε = 1 we have the blackwalls. The radiative heat flux is defined by
Qrad(t, s?) =
∫DI?(t, s?, β)β dβ for (t, s?) ∈ [0, τ?]× Ω?. (7)
Emission and absorption of radiation by the medium lead to a radiativesource term in the energy equation of the medium. It is defined by thefollowing relations:
S?rad(t, s?) = κ G?(t, s)− 4πIb(T
?(t, s?)) for (t, s?) ∈ [0, τ?]× Ω?, (8)
where G? is the incident radiation intensity,
G?(t, s?) =
∫DI?(t, s?, β) dβ for (t, s?) ∈ [0, τ?]× Ω?. (9)
The over all energy conservation links the three different modes of heat trans-fer, known as conduction, radiation and convection. In this study, convectionhas not been considered
ρcp∂T ?
∂t(t, s?)− kc4T ?(t, s?) = S?rad(t, s
?) for (t, s?) ∈ [0, τ?]× Ω? (10a)
T ?(0, s) = T ?0 (s) for all s? ∈ Ω?. (10b)
The data ρ, cp, and kc are the density, the specific heat capacity, and thethermal conductivity of the medium, respectively. In this work, they areassumed to be constant. The energy equation (10a)- (10b) and the RTE(2a)- (2b) are strongly coupled by the incident radiation intensity G? andthe temperature T ?.
The energy equation (10a) can be written as the dimensionless form, seeAppendix A:
∂T
∂ξ−4T + θT 4 = θG for (ξ, s) ∈ [0, τ ]× Ω (11a)
T (0, s) = T0(s) for all s ∈ Ω (11b)
6
where T is the dimensionless temperature. We consider the Dirichlet bound-ary conditions, where the temperature at the boundary is known
T (ξ, s) = f(ξ, s) (ξ, s) ∈ [0, τ ]× ∂Ω, (12)
and the Robin thermal conditions, the flux through the boundary is defined;
−∂T∂n
(ξ, s) = T (ξ, s)− f(ξ, s) (ξ, s) ∈ [0, τ ]× ∂Ω. (13)
Similarly, the dimensionless form of radiative transfer equation is given by,see Appendix A:
β.∇I(ξ, s, β) + I(ξ, s, β) = T 4(ξ, s), for (ξ, s, β) ∈ [0, τ ]× Ω×D, (14a)
I(ξ, s, β) = g(ξ, s, β), for (ξ, s, β) ∈ [0, τ ]× ∂Ω−. (14b)
3. Analysis of radiative conductive heat transfer
3.1. Radiative Transfer Equation.
To compute a numerical solution of the RTE, the classical SN−discreteordinate method is introduced [8]. The SN method consists of replacing theradiation intensity I(ξ, s, β) by a discrete radiation intensity (I1(ξ, s), I2(ξ, s),..., IN
β(ξ, s)), where I l(ξ, s) = I(ξ, s, βl) for all l ∈ 1, ..., Nβ. A quadrature
rule was chosen
(βl, wl) l = 1, ..., Nβ
, such that:
∫Df(β, s)dβ '
Nβ∑l=1
f(βl, s)wl and
Nβ∑l=1
wl = 4π. (15)
This quadrature formula preserves a symmetry for the opposite βl′ = −βl ofdiscrete ordinate βl in the quadrature set, the weights are equal wl′ = wl,see [8]. The discrete radiation intensity I l is the solution of a Nβ system ofpartial differential equations over Ω:
βl.∇I l(ξ, s) + I l(ξ, s) = T 4(ξ, s) (ξ, s) ∈ [0, τ ]× Ω
I l(ξ, s) = g(ξ, s, βl) (ξ, s) ∈ [0, τ ]× ∂Ωl−,(16)
where∂Ωl− = s ∈ ∂Ω such that βl.n < 0 . (17)
Let Hr(Ω) the classical Sobolev subspace of L2(Ω) functions, with normdenoted by ‖.‖r for r > 0, see [26].We denote by z(ξ, .) = T 4(ξ, .) and weassume that z(ξ, .) belongs to L2(Ω) for all ξ ∈ [0, τ ].We extend the boundarydatum g to ∂Ω by setting it to zero outside ∂Ωl
− and we assume that
g(ξ, ., βl) ∈L2(∂Ωl−) for all (ξ, l) ∈ [0, τ ]× 1, ..., Nβ .
7
Theorem 3.1 (Well-posedness, see [27]). The problem (16) is well-posed.Moreover, its unique solution I l(ξ, .) ∈ H1(Ω) satisfies (16) and I l(ξ, .) =g(ξ, ., βl) a.e. in ∂Ωl
− for all (ξ, l) ∈ [0, τ ]× 1, ..., Nβ.
LetI l(ξ, .)
=I1(ξ, .), I2(ξ, .), ..., INβ(ξ, .)
and
I l(ξ, .)
∈ Hr(Ω)Nβl=1,
where the space Hr(Ω)Nβl=1 is defined such that I l(ξ, .) ∈ Hr(Ω) for all(ξ, l) ∈ [0, τ ]× 1, ..., Nβ.
Now, we proceed with the spatial discretization of the SN transport equa-tion using the DG method . We consider a triangulation Th of Ω such that
Ω =⋃K∈Th
K,
• Each K is a triangle with nonempty internal;
•K1
⋂ K2 = ∅ for each distinct K1, K2 ∈ Th;
• F = K1
⋂K2 6= ∅ then F is a common face of K1 and K2;
• Interfaces are collected in the set F inh , and boundary faces are collectedin the set F bh. Henceforth, we set
Fh = F inh⋃F bh.
Moreover, for any mesh element K ∈ Th, the set
FK = F ∈ Fh|F ∈ ∂K ,
collects the mesh faces composing the boundary of K.
• diam(K) = h for each K ∈ Th (structured mesh).
Now, an approximation space Vh ⊂ L2(Ω) is introduced such that
Vh =Ih ∈ L2(Ω)/ ∀K ∈ Th, Ih|K
∈ Pk(K), (18)
where Pk(K) denotes the set of polynomial defined in K of degree less thanor equal to k. Define:
Vh = VhNβl=1 , (19)
and denote a generic element in Vh as Ih(ξ, .) =I lh(ξ, .)
Nβl=1
. Due to the
discontinuous nature of the spatial approximation, functions I lh ∈ Vh are
8
double-valued on interior faces. Consider an interior face F ∈ F inh separatingtwo mesh cells, K1 and K2. The mean value and jump of a function I lh(ξ, .) ∈Vh are defined as follows:
I lh(ξ, .)
=1
2(I l1(ξ, .) + I l2(ξ, .)), [[I lh(ξ, .)]] = (I l1(ξ, .)− I l2(ξ, .)),
for all ξ ∈ [0, τ ], where I l1(ξ, .) = I lh(ξ, .)|K1and I l2(ξ, .) = I lh(ξ, .)|K2
are the
restrictions of I lh(ξ, .) on the mesh cells K1 and K2, respectively. The DGformulation is obtained by multiplying the SN equation for the direction βlwith the test function wh ∈ Vh and applying the upwind numerical fluxesF(s) to approximate the quantity (βl.n)I l(ξ, .) on the elements boundary K:∫
K
I lh(ξ, .)wh − (βl.∇hwh)I lh(ξ, .) +
∫∂K
F(s)wh =
∫Ω
z(ξ, .)wh, ∀ξ ∈ [0, τ ]. (20)
The upwind numerical flux F(s) at the mesh interface F from K1 to K2 isgiven by:
F(s) = βl.nF
I l(ξ, s)
+
1
2|βl.nF|[[I l(ξ, s)]], ∀ξ ∈ [0, τ ], (21)
where nF is the outward unit normal to F at s. Summing over all cells,integrating by parts a second time, and separating volume and interfaceterms, we obtain a global formulation. Upon introducing the bilinear form
auph (Ih(ξ, .), wh) =
Nβ∑l=1
wl(∑K∈Th
[∫K
I lh(ξ, s)wlhds−∫K
I lh(ξ, s)(βl.∇wlh)ds
]+
∫∂Ω
(βl.n)⊕I lh(ξ, s)wlhdΓ(s) +∑
F∈Finh
∫F
(βl.nF)[[I lh(ξ, s)]]wlh(s)
dΓ(s)
+∑
F∈Finh
∫F
1
2|βl.nF|[[I lh(ξ, s)]][[wlh]]dΓ(s)−
∫∂Ω
(βl.n)gI(ξ, s, βl)wlhdΓ(s)
(22)
for all (ξ, wh) ∈ [0, τ ]×Vh, where for a real number x, we define its positiveand negative parts respectively as x⊕ := 1
2(|x| + x), x := 1
2(|x| − x). By
definition, the both quantities are nonnegative. Let consider the followingmap lh : Vh −→ R defined by
l(wh) =
Nβ∑l=1
wl
[∫Ω
z(ξ, s)wlhds+
∫∂Ω
(βl.n)gT (ξ, s, βl)wlhdΓ(s)
](23)
for all (ξ, wh) ∈ [0, τ ]× Vh.
9
The discrete ordinate DG problem is written as
Find Ih(ξ, .) ∈ Vh such as
auph (Ih(ξ, .), wh) = lh(wh) for all (ξ, wh) ∈ [0, τ ]× Vh.(24)
At the discrete level, we consider an approximation space given by thebroken polynomial space Vh = Pk(K), for all K ∈ Th. The global enumera-tion of the degrees of freedom is such that the local degrees of freedom arenumbered contiguously for each mesh element. This leads to a basis for Vhof the form
φKii∈DK
K∈Th
,
where the set DK =
1, ..., NKd
collects the local indices of the NK
d degreesof freedom for the mesh element K ( we assumed that all element has thesame degree NK
d ).The global matrix can be block-portioned in the form
ADGh =
AK1K1h AK1K2
h · · · AK1KNh
AK1K1h AK2K2
h · · · AK1K2h
.... . . . . .
...
AKNK1
h AKNK2
h · · · AKNKNh
where for all Kl, Km ∈ Th,
AKmKlh = [auph (φKmi , φKlj )] ∈ RNKd ×RNK
d . (25)
We split the discrete bilinear form auph into volume, interface, and boundaryface contributions as follows: For all Ih(ξ, .), wh ∈ Vh,
auph (Ih(ξ, .), wh) = avh(Ih(ξ, .), wh) + aifh (Ih(ξ, .), wh) + abh(Ih(ξ, .), wh) (26)
where
avh(Ih(ξ, .), wh) =
Nβ∑l=1
wl(∑K∈Th
[∫K
I lh(ξ, s)wlhds−
∫K
I lh(ξ, s)(βl.∇wlh)ds],
(27)
aifh (Ih(ξ, .), wh) =∑F∈Finh
∫F
(βl.nF)[[I lh(ξ, s)]]wlh(s)
dΓ(s)
+∑F∈Finh
∫F
1
2|βl.nF|[[I lh(ξ, s)]][[wlh]]dΓ(s)
(28)
10
and
abh(Ih(ξ, .), wh) =
∫∂Ω
(βl.n)⊕I lh(ξ, s)wlhdΓ(s)−
∫∂Ω
(βl.n)gI(ξ, s, βl)wlhdΓ(s).
(29)Each summation yields a loop over the corresponding mesh entities to as-
semble local contributions into the global matrix. The local matrix stemmingfrom the volume contribution of a generic mesh element K ∈ Th is
AK = [avh(φKi , φ
Kj )] ∈ RNK
d ×RNKd (30)
and it contributes to the diagonal block AKK of the global matrix A. Aninterface F ∈ F inh contributes to four blocks of the matrix A, and the localmatrix stemming from the interface contribution can be block-partitioned inthe form
AF =
[AK1K1h AK1K2
h
AK1K1h AK2K2
h
]where F = ∂K1
⋂∂K2,
AKmKnh = [aifh (φKn′
i , φKm
′
j )] ∈ RNKd ×RNK
d (31)
∀ m,n ∈ 1, 2 and n′,m
′ ∈ 1, ..., N are the indices of K1, K2 in the globalenumeration of mesh elements. Finally, a boundary face F ∈ F bh contributesthrough the local matrix
AF = [abh(φKi , φ
Kj )] ∈ RNK
d ×RNKd (32)
where F = ∂K1
⋂∂K2.
We can write (24) in the matrix form as
ADGh Ilh = FDGh , (33)
where the unknown Ilh of the system (33) is defined by
Ilh = (I1h,1, .., I
1h,NK
d, ..., INth,1, .., I
Nth,NK
d)T
for l ∈ 1, ..., Nβ. The second member FDGh is given by the first part ofthe map (23) and the emission part gT where the reflected part gI is takeninto account in the matrix ADG
h . A direct method for sparse matrices wasimplemented to solve linear system (33). Finally, we note that the matrix
11
assembly implementation and the basis functions selection in broken polyno-mial spaces are discussed in [27, p. 343, Appendix A: Implementation] and[13].
Now, to prove that problem (24) without the reflected boundary part hasa unique solution, we consider in Vh the following norm:
‖I‖2h =
Nβ∑l=1
wl
‖I l‖2L2(Ω) +
∫∂Ω
1
2|βl.n|I l
2dΓ(s) +
∑F∈Finh
∫F
1
2|βl.nF|[[I l]]2dΓ(s)
, (34)
for all I ∈ Vh. Then, we deduce the following stability result.
Lemma 3.2. Let ξ ∈ [0, τ ], then
auph (Ih(ξ, .), Ih(ξ, .)) = ‖Ih(ξ, .)‖2h for all Ih(ξ, .) ∈ Vh. (35)
Proof. Let ξ ∈ [0, τ ], by definition of auph (., .) we have for any Ih(ξ, .) ∈ Vh,
auph (Ih (ξ), Ih (ξ)) =
Nβ∑l=1
wl∑K∈Th
[∫K
(I lh(ξ, s))2ds−∫K
I lh(ξ, s)(βl.∇I lh(ξ, s))ds
]+
∫∂Ω
(βl.n)⊕(I lh(ξ, s))2dΓ(s) +∑
F∈Finh
∫F
(βl.nF)[[I lh(ξ, s)]]I lh(ξ, s)
dΓ(s)
+∑
F∈Finh
∫F
1
2|βl.nF|[[I lh(ξ, s)]][[I lh(ξ, s)]]dΓ(s) ∀(ξ, wh) ∈ [0, τ ]× Vh,
(36)
Using the integration by parts formula, it follows
auph (Ih (ξ), Ih (ξ)) =
Nβ∑l=1
wl∑K∈Th
[∫K
(I lh(ξ, s))2ds− 1
2
∫∂K
(βl.nK)(I lh(ξ, s))2ds
]+
∫∂Ω
(βl.n)⊕(I lh(ξ, s))2dΓ(s) +∑
F∈Finh
∫F
(βl.nF)[[I lh(ξ, s)]]I lh(ξ, s)
dΓ(s)
+∑
F∈Finh
∫F
1
2|βl.nF|[[I lh(ξ, s)]][[I lh(ξ, s)]]dΓ(s) ∀(ξ, wh) ∈ [0, τ ]× Vh,
(37)
The second term on the right hand side can be reformulated. Indeed, thecontinuity of βl across the interfaces leads to∑K∈Kh
∫∂K
1
2(βl.nK)(I l
2
h (ξ, s)) =∑
F∈Finh
∫F
1
2(βl.nF )[[I lh(ξ, s)
2]] +
∑F∈Fbh
∫F
1
2(β.nF )I l
2
h (ξ, s).
12
For all F ∈ F inh , where nK1 is the unit normal to F at s pointing from K1
to K2. I li(ξ, s) = I lh(ξ, s)|Ki, i ∈ 1, 2 it holds
1
2[[I lh
2(ξ, .)]] =
1
2(I l
2
1(ξ, .)− I l22(ξ, .)) =1
2(I l1(ξ, .)− I l2(ξ, .))(I l1(ξ, .) + I l2(ξ, .))
= [[I lh(ξ, .)]]I lh(ξ, .)
.
We obtain
auph (Ih(ξ, .), Ih(ξ, .)) =
Nβ∑l=1
wl‖I lh(ξ, .)‖2L2(Ω) +1
2
∫∂Ω
|βl.n|(I lh(ξ, s))2dΓ(s)
+∑
F∈Finh
∫F
1
2|βl.nF|[[I lh(ξ, s)]]2dΓ(s).
(38)
Then,auph (Ih(ξ, .), Ih(ξ, .)) = ‖Ih(ξ, .)‖h for all ξ ∈ [0, τ ]. (39)
Theorem 3.3. The problem (24) has a unique solution in Vh.
Proof. lh denotes the continuous map from Vh to R. Using the lemma 3.2we deduce that (24) has a unique solution.
To give an error estimate for the discrete ordinate and the DG methodconsidered in this work we applied the methodology used in [28].
LetI l(ξ, .)
∈ Hr(ΩNβl=1 for r > 0. If
I l(ξ, .)
satisfies:
auph (I l(ξ, .)
, wh) = lh(wh) for all (ξ, wh) ∈ [0, τ ]× Vh. (40)
The discrete-ordinate DG method has the following error estimate
‖I l(ξ, .)
− Ih(ξ, .) ‖h 6 Chmin r,k+1− 1
2
Nβ∑l=1
wl‖I l(ξ, .)
‖2r
12
for all ξ ∈ [0, τ ].,
(41)
with C a positive constant and k is the degree of the polynomial of approxi-mation. Now, we can give a L2(Ω) error estimate. To this end, we introducethe norm:
‖| I(ξ, .) − Ih(ξ, .) ‖| =
Nβ∑l=1
wl‖I l(ξ, .)
−I lh(ξ, .)
‖2L2(Ω)
12
for all ξ ∈ [0, τ ].
(42)
Thanks to the estimate (41) we deduce the following result
13
Lemma 3.4. Assume thatI l(ξ, .)
∈ Hr(ΩNβ for r > 0 is the solution of
(40), Ih ∈ Vh is solution of (24) and that the numerical quadrature satisfies(15). Then, the discrete-ordinate DG method, leads to
‖| I(ξ, .) − Ih(ξ, .) ‖| 6 C1hminr,k+1− 1
2
Nβ∑l=1
wl‖I l(ξ, .)
‖2r
12
for all ξ ∈ [0, τ ],
(43)
where C1 is a positive constant.
3.2. Energy equation
Let (., .)Ω and (., .)∂Ω be the inner product in L2(Ω) and L2(∂Ω) respec-tively. To consider a weak solution of the equation (11a), the following spaceis introduced :
V =T ∈ L2(0, τ,H1(Ω)) such that T (ξ, s) = f(ξ, s) for (ξ, s) ∈ [0, τ ]× ∂Ω
. (44)
The weak formulation of equation (11a) for Dirichlet boundary conditions(12) reads: Find T ∈ V such that
(d T
d ξ, ϕ)Ω + (∇T,∇ϕ)Ω + θ(T 4, ϕ)Ω = θ(G,ϕ)Ω, (45)
for all ϕ ∈ H10 (Ω) and satisfying the initial condition:
(T (0, s), ϕ)Ω = (T0, ϕ)Ω ∀ϕ ∈ L2(Ω). (46)
In the case of Robin boundary conditions (13), we have:Find T ∈ L2(0, τ,H1(Ω)) such that
(d T
d ξ, ϕ)Ω + (∇T,∇ϕ)Ω + θ(T 4, ϕ)Ω + (T, ϕ)∂Ω = θ(G,ϕ)Ω + (f, ϕ)∂Ω, (47)
for all ϕ ∈ H1(Ω) and satisfying the initial condition:
(T (0, s), ϕ)Ω = (T0, ϕ)Ω ∀ϕ ∈ L2(Ω). (48)
A proof of the existence and uniqueness of the solution of system (45)-(47)is given in [23].
3.3. Numerical approach of the energy equation
LetWh =
vh ∈ C0(Ω)/ ∀K ∈ Th, vh|K
∈ Pk(K), (49)
14
be a family of finite element subspaces of H1(Ω). The finite dimensional spaceWNbs = vect φ1, φ2, ..., φNbs. We denote byMi a node of the mesh Th. In theDirichlet boundary conditions case, we denote by E the set of points on theedge (E = i such that Mi ∈ ∂Ω), N be = Card(E) and I =
1, ..., N bs
\
E . To construct a finite element approximation of the temperature T , weintroduce:
Th(ξ, s) =
Nbs∑i=1
Thi(ξ)φi(s). (50)
Then, a semi-discrete finite element approximation of the weak formulation(45) is: Find Th : [0, τ ]→ Th(ξ, .) ∈ WNbs such that Th(ξ,Mj) = f(ξ,Mj) forall j ∈ E
(d Thd ξ
, φi)Ω + (∇Th,∇φi)Ω + θ(T 4h , φi)Ω = θ(Gh, φi)Ω ∀φi ∈WNbs , (51)
for all i ∈ I and satisfying the initial condition:
(Th, φi)Ω = (T0, φi)Ω ∀φi ∈W bsN . (52)
In the Robin boundary conditions case, a semi-discrete finite element approx-imation of the weak formulation (47) is: Find Th : [0, τ ] → Th(ξ, .) ∈ WNbs
such that
(d T
dξ, φi)Ω + (∇Th,∇φi)Ω+θ(Th
4, φi)Ω + (Th, φi)∂Ω
= θ(G,φi)Ω + (f, φi)∂Ω ∀φi ∈WNbs ,
(53)
and satisfying the initial condition:
(Th, φi)Ω = (T0, φi)Ω ∀φi ∈WNbs . (54)
We used the Crank-Nicolson scheme for time discretization, Let 4ξ =τ
N,
the equation (51) can be written in the following form
(Tn+1h − Tnh4ξ
, φi)Ω+(∇(Tn+1h + Tnh )
2,∇φi)Ω + θ(
Tn+1h
4+ Tnh
4
2, φi)Ω
= θ(Gn, φi)Ω ∀φi ∈WNbs .
(55)
The Newton method is employed for analyzing the nonlinearity,
T ?n+1h
4' Tnh
4 + 4Tnh3(Tn+1
h − Tnh ), (56)
15
then, the equation (55) can be written as
(Tn+1h − Tnh4ξ
, φi)Ω+(∇(Tn+1h + Tnh )
2),∇φi)Ω + θ((−Tnh
4 + 2Tnh3Tn+1h ), φi)Ω
= θ(Gn, φi)Ω ∀φi ∈WNbs .
(57)
In the same way, the equation (53) can be written as
(Tn+1h − Tnh4ξ
, φi)Ω + (∇(Tn+1h + Tnh )
2),∇φi)Ω + θ((−Tnh
4 + 2Tn3
h Tn+1h ), φi)Ω
+ ((Tn+1h + Tnh )
2, φi)∂Ω = θ(Gn, φi)Ω + (fn, φi)∂Ω ∀φi ∈WNbs ,
where T nh is the approximate value of Th at time ξn = n.4ξ , for all n ∈0, 1, ..., N.
Let us denote by T nh = [T nhi]Nbs
i=1 , Gn = [Gn
hi]Nbs
i=1 and F n = [fnhi]Nbs
i=1 whereGnhi and fnhi are given by
Gnhi = (Gn, φi)∂Ω fnhi = (fn, φi)∂Ω.
Then, we obtain the following system
Mh(Tn+1h − Tnh4ξ
) +Ah(Tn+1h + Tnh
2) + θMh(−Tn
4
h + 2Tn3
h Tn+1h )
+Mh(Tn+1h + Tnh
2) = θGn + Fn
(58)
for all n ∈ 0, 1, ..., N, where Mh and Ah denote respectively the massand stiffness matrix of finite elements method. A proof of the existence anduniqueness of the solution of discrete system (58) is given in [24]. A prioriestimates of numerical error are given in [25].
Given an initial condition T0, at each time step: first we solve the RTE,second we compute the the incident radiation intensityG and solve the energyequation. The described iteration are performed until a time τ1 such that thedifference in L2 norm between the solutions of the energy equation at timeτ1 and time τ1 +4ξ is less than 10−5.
4. Results and discussion
4.1. Radiative heat transfer in a square enclosure.
To show the performance of the DG method considered in this work wewill first present some test cases. The first case consists of calculating the L2
16
norm of the error for a particular form of the RTE solved for a prescribeddirection β in a square enclosure. This equation has the following form:
β.∇I + I = z in Ω.
I = 0 on ∂Ω−,(59)
where∂Ω− = s ∈ ∂Ω such that β.n < 0 , (60)
and β = (cos(
π
4), sin(
π
4))
z = cos(π
4)π
2sin(πs2)sin(
π
2(1− s1)) + sin(
π
4)πcos(πs2)cos(
π
2(1− s1))
+ sin(πs2)cos(π
2(1− s1)),
(61)
which has a smooth exact solution given by sin(πs2)cos(π
2(1−s1)) ∈ C∞(Ω),
see Figure 1.
Figure 1: The exact solution in mesh of unit square enclosure.
Figure 2 shows the L2 norm of the error for different orders of approxima-tion of the DG method. Our results were found to comply with the theoreticalresult (41) for the smooth solution I. When we took r sufficiently higher, the
17
10−2
10−1
100
10−12
10−10
10−8
10−6
10−4
10−2
100
h
Err
or
DG−P4
h9/2
DG−P3
h7/2
DG−P2
h5/2
DG−P1
h3/2
Figure 2: L2 norm of the error for the upwind DG method.
curve of the error with the polynomial approximation Pk for k ∈ 1, ..., 4was found to be proportional to hk+ 1
2 (k+ 12
= min (k + 12, r) for a sufficiently
higher value of r). In the second test case, we assumed that the medium tobe grey in a square enclosure. The enclosure was assumed to be filled withan absorbing medium at constant temperature Tm = 100K and the wallsassumed to be black and cold TN = TE = TS = TW = 0K where Tm wasthe temperature of the medium and TN , TE, TS, TW are the temperaturesat the boundary of the medium (North, East, South and West respectively).Numerical simulations were performed using the S2, S4 and S6 quadraturesand with a structured triangular grid composed of 441 nodes. Figure 3 shows
the dimensionless radiative heat fluxQrad
σBT 4ref
for three values of the absorp-
tion coefficient (κ = 10, κ = 1 and κ = 0.1 m−1). The numerical solutionwas found to be very close to the exact solution in passing S2 to S4 andfrom S4 to S6 respectively . Our results with S6 comply well with the exactsolution given in [29]. When the medium was more absorbing (κ = 10 m−1),the heat flux was found to be greater and tended towards the heat flux of theblackbody at temperature of the medium. When κ decreased, the heat fluxtended towards zero since the walls were cold. Our results show that the S6
quadrature already gives a better approximation than the results presented
18
in [30, 31]. In [31], the author shows that the error estimate was propor-tional to the inverse of the number of discrete directions Nβ. In figure 4 thenumerical results were obtained using the S4 angular discretization and fortwo values of the absorption coefficient κ = 2 m−1 and κ = 10 m−1. Figure 4shows the numerically computed dimensionless incident radiative heat flux.Accuracy was found to increase when the polynomial approximation degreewas higher which again complies with the theoretical results.
4.2. Combined radiative and conductive heat transfer in a square enclosure.
In this section, we shall present the numerical results for the coupledradiative-conductive heat transfer problem. In this case, the S4 angular dis-cretization was used to reduce the computational time while maintaining areasonable level of accuracy. The problems presented were solved with a con-stant dimensionless time step set to4ξ = 5.10−4. The reference temperatureTref was assumed to be the hot temperature Tref = 1000K. In addition, allcalculations were performed with the absorption coefficient κ = 1m−1 and inall cases, iterations were terminated when the steady state conditions wereachieved with a tolerance set to 10−5. For each test case, CPU time is indi-cated. Initially, the medium was assumed to be at a constant temperatureequal to T ?0 = 500K. The south surface (hot surface) and the other faceswere kept constant over time. The thermal boundary conditions consideredin the first test case were non-homogeneous Dirichlet conditions (12) andour aim in using these was to check our results against those in literatureon subject. The Robin boundary conditions (13), were also considered. Allcalculations were performed on a Mac OS X version 10.6.6 2.22GHz IntelCore 2 Duo using the MATLAB software.
4.2.1. Non-homogeneous thermal Dirichlet boundary conditions.
In this section, we used the finite element method with an order of ap-proximation equal to 2 (EFP2) to solve the energy equation and the DGmethod with an order of approximation equal to 2 (DGP2) was used to solvethe RTE. The simulations were performed for two types of mesh, a struc-tured triangular grid composed of 1600 nodes (mesh1, h=0.025), see figure5-(a) and a refined mesh in high-temperature region composed of 1534 nodes(mesh2), figure 5-(b). Under these conditions, the computational times re-quired by the program to converge to the steady state are approximately 2hwhen Ns = 1 and Ns = 0.1.
19
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
y(m)
Qrad/σ
BT
ref
4
S2
S4
S6
Exact solution
(a) κ = 10 m−1
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
y(m)
Qra
d/σ
BT
ref
4
S2
S4
S6
Exact solution
(b) κ = 1 m−1
0 0.2 0.4 0.6 0.8 10.05
0.06
0.07
0.08
0.09
0.1
0.11
y(m)
Qra
d/σ
BT
ref
4
S2
S4
S6
Exact solution
(c) κ = 0.1 m−1
Figure 3: Dimensionless radiative heat flux along the centerline of the south boundary,DG− P 1
20
0 0.2 0.4 0.6 0.8 10.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
y(m)
Qra
d/σ
B T
ref
4
DG−P1
DG−P2
DG−P3
DG−P4
(a) κ = 2 m−1
0 0.2 0.4 0.6 0.8 10.4
0.5
0.6
0.7
0.8
0.9
1
1.1
y(m)
Qra
d/
σBT
ref
4
DG−P1
DG−P2
DG−P3
DG−P4
Exact solution
(b) κ = 10 m−1
Figure 4: Dimensionless radiative heat flux with the upwind DG method and, S4
(a) (b)
Figure 5: The temperature distribution for Ns = 1 for tow types of mesh grid. (a) astructured triangular grid composed of 1600 nodes (h=0.025), (b) a refined mesh in high-temperature region composed of 1534 nodes.
21
4.2.1.a. Black surfaces.
In this first test case, the medium boundaries were assumed to be black sur-faces with imposed temperatures. This problem was investigated by Mishraet al. [22]. These authors used a Lattice Boltzmann method (LBM) to solvethe energy equation and the collapsed-dimension method (CDM) was usedto compute the radiative source term in the energy equation. This methodwas developed for 2-d simple geometries based on structured grids. Let usnote that the same transient problem was also studied by Asllanaj et al.[20]. In this latter work, the RTE was solved by using a FVM based on acell vertex scheme and associated to a modified exponential scheme. ThePHAML code was used to solve the energy equation using finite elementsmethod. The numerical solutions by Mishra et al. [22] were used as refer-ence to check those obtained with our numerical method. Figure 6 and figure7 show the evolution of the dimensionless temperature along the centerlineposition s1 = 0.5 at different times and for different number Ns = 0.1 andNs = 1. It should be noted that the computational time needed to achieveconvergence of our algorithm increases with Ns. Obviously, more importantthe radiative part is, more quickly steady state was reached. Our resultswere in agreement with those of Asllanaj et al. [20] and Mishra et al. [32],confirming the validity of our numerical method for black surfaces.
Table 1: computational time in seconds to obtain a converged solution for Ns = 1
EF-DGh P1 P2 P3 P4
0.1 167.77 282.92 789.65 1.492× 103
0.05 662.349 1.108× 103 3.219× 103 6.879× 103
0.025 2.5135× 103 5.183× 103 8.783× 103 12.43× 103
Table 1 shows the computational time taken by our program to convergeto the steady state solution using structured mesh 1, h = 0.025 correspondto the 1600 nodes and the other instead correspond to coarser mesh 1 grid.We tested the numerical stability for different temporal and spatial step dis-cretization. This guarantees the stability of our combined numerical scheme.
22
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y (m)
D
imen
sio
nle
ss T
em
pera
ture
Mishra and al.
Present study
ξ = 0.288
ξ = 0.015
ξ = 0.04ξ = 0.005
ξ = 0.001
Steady state
(a) Ns = 1
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y (m)
Dim
en
sio
nle
ss T
em
pera
ture
Mishra al.
Present study
Steady stateξ = 0.2395ξ = 0.015
ξ = 0.04
ξ = 0.005
ξ = 0.001
(b) Ns = 0.1
Figure 6: Dimensionless temperature along the centerline position using mesh 1 at differentdimensionless times and for two different values of the conduction-radiation parameter.
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
ξ=0.04
ξ=0.285Steady state
(a) Ns = 1
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
Steady state
ξ=0.04
ξ=0.238
(b) Ns = 0.1
Figure 7: Dimensionless temperature along the centerline position using mesh 2 at differentdimensionless times and for two different values of the conduction-radiation parameter.
23
4.2.1.b. Combined black and diffuse reflection opaque surfaces.
In this second test case, the hot (south) surface was assumed to be opaquewith diffuse reflection and the other three surfaces were black. The numberNs was set at 0.01. Figure 8 and figure 9 show the dimensionless temperaturedistributions along the centerline position for two values of the wall emissivityε equal to 0.5 and 0.1 respectively. These also show that the physical timerequired to obtain a converged solution decreases with ε. A steady statewas reached as dimensionless time ξ = 0.096 for ε = 0.5 and ξ = 0.1070 forε = 0.1 when we use mesh 1 and as dimensionless time ξ = 0.096 for ε = 0.5and ξ = 0.1070 for ε = 0.1 using mesh 2. Our results were found to complywith those of [32].
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
Present study
Mishra et AL.
Steady state
ξ = 0.005
ξ = 0.001
ξ=0.015 ξ=0.04
0.0960
(a) ε = 0.5
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
Present steady
Mishra et AL.
ξ=0.1070ξ=0.1070
ξ=0.001
ξ=0.005
ξ=0.04
ξ=0.015
Steady state
(b) ε = 0.1
Figure 8: Dimensionless temperature along the centerline position using mesh 1 at differentdimensionless times and for two different values of the emissivity coefficient.
24
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
ξ=0.04
ξ=0.096
Steady state
(a) ε = 0.5
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
ξ=0.04
ξ=0.1065
Steady state
(b) ε = 0.1
Figure 9: Dimensionless temperature along the centerline position using mesh 2 at differentdimensionless times and for two different values of the emissivity coefficient.
4.2.1.c. Opaque surfaces with combined specular and diffuse reflec-tion
The hot (south) surface was assumed to be opaque with specular reflectionand the other three surfaces to be opaque with diffuse reflection. To ourknowledge, this test case does not exist in literature in this area of research.For this instance we use a mesh 2. In figure 10 the dimensionless temperatureis presented along the centerline position for three values of the wall emis-sivity ε (0.1, 0.5 and 0.9) for all surfaces and with the conduction radiationnumber Ns equal to 1. The decrease in the emissivity coefficient was foundto delay the steady state. The convergence was reached as ξ = 0.086 forε = 0.9, ξ = 0.1065 for ε = 0.5 and ξ = 0.0998 for ε = 0.1.
4.2.2. Thermal Robin boundary conditions.
In this section we considered the Robin boundary conditions (13). Themedium boundaries were assumed to be black surfaces. To our knowledge,this is the first time this test case has been simulated. The simulation was
25
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
ξ=0.04
ξ=0.0998
Steady state
(a) ε = 0.3
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.005
ξ=0.015
ξ=0.04
ξ=0.1065
Steady state
(b) ε = 0.5
0 0.2 0.4 0.6 0.8 10.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
y(m)
Dim
en
sio
nle
ss T
em
pera
ture
ξ=0.001
ξ=0.0015
ξ=0.015
ξ=0.04
ξ=0.086
Steady state
(c) ε = 0.9
Figure 10: Dimensionless temperature along the centerline position at different dimen-sionless times and for three different values of the emissivity coefficient.
26
performed using a
D1 =
(x, y) ∈ R2; 1 6 x2 + y2 6 4, 1 6 |x| 6 2 and 1 6 |y| 6 2
and a refined mesh in high-temperature region composed of 1462 nodes. Weassume the medium wall to be transparent (the gain by radiation and theradiative losses are neglected). Then
−kc∂T ?
∂n= hc(T
? − T ?∞) (62)
where T ?∞ is the surrounding temperature and hc is the convective exchangecoefficient of the wall. We introduce the Biot number
Bi =hcL
kc.
Now, we give to the dimensionless Robin boundary condition
−∂T∂n
= Bi(T − T∞). (63)
In our numerical simulations, we take Bi equal to 1, this means that the heatconduction inside the body is lower than for the surface, see for instance [33].The dimensionless surrounding temperature T∞ is equal to 1 in
Γ1 =
(x, y) ∈ R2; x2 + y2 = 1, y 6 0
and is equal to 0.5 elsewhere in the boundary. The figure 11 shows the field ofthe temperature for three values of the conduction radiation number Ns equalto 1, 0.1 and 0.01. Decreasing the conduction radiation number was foundto also reduce the time of convergence to the steady state. The convergencewas reached as dimensionless time ξ = 1.9090 for Ns = 1, ξ = 1.7225 forNs = 0.1 and ξ = 1.6955 for Ns = 0.01.
5. Conclusion.
The aim of he work presented in this paper was to solve the combinedradiation and conduction heat transfer in a grey absorbing and emittingmedium. We used a high order DG method based on an upwind numeri-cal flux to solve the radiative heat transfer problem. The DG method was
27
(a) Ns = 1 (b) Ns = 0.1
(c) Ns = 0.01
Figure 11: The temperature distribution for different values of the conduction radiationnumber.
28
coupled with the high order finite element method to solve the energy equa-tion. Error estimates for the numerical solution were derived using suitablesolution regularity assumptions in the RTE and we showed the stability ofthe DG method. Using the Newton method, a new treatment for a couplednonlinear radiative-conductive system was implemented. We also investi-gated a new type of boundary conditions and showed that the numericalresult was consistent with the theoretical findings. Three types of radiativeboundary conditions were used -black walls, opaque walls with specular anddiffuse reflection- and we explored the Dirichlet and Robin thermal boundaryconditions.
The investigations of some basic aspects of the Continuous and Discontin-uous Galerkin methods indicate that these methods are promising for solv-ing the radiative-conductive heat transfer but much research still needs to bedone in the future such that extending the idea to complex three dimensionalgeometries and an effective parallel implementation of the algorithms.
Acknowledgements. This work was partially supported by the FrenchANR-EMERGENCE and the pole MATERALIA.
Appendix A. Dimensionless energy equation
Let ξ =αt
L2the dimensionless time where α =
kcρcp
is the thermal diffusiv-
ity. We also introduce the conduction radiation number Ns which satisfiesthe following expression:
Ns =kcκ
4σBT 3ref
.
Lets =
s?
L, τ =
ατ?
L2, T =
T ?
Tref,
T0(s) =T ?0 (s)
Tref, G =
G?
4σBT 4ref
and Srad =S?rad
4σBT 4ref
.
Hence, the energy equation (10a)-(10b) can be written as the dimensionlessform:
∂T
∂ξ−4T + θT 4 = θG, for (ξ, s) ∈ [0, τ ]× Ω,
T (0, s) = T0(s), for all s ∈ Ω.
where θ =κ2L2
Ns
is the dimensionless constant.
29
Dimensionless radiative transfer equationNow, we give the dimensional form of radiative transfer equation (2a)-
(2b):
β.∇I(ξ, s, β) + I(ξ, s, β) = T 4(ξ, s), for (ξ, s, β) ∈ [0, τ ]× Ω×D, (A.2a)
I(ξ, s, β) = g(ξ, s, β), for (ξ, s, β) ∈ [0, τ ]× ∂Ω− ×D, (A.2b)
whereIref =
σBπT 4ref , I =
I?
Irefand g =
g?
Iref
are the dimensionless constant.
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