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Three-Dimensional Transient Radiative Transfer Modeling Using Discontinuous Spectral Element Method

J. M. Zhao, L. H. Liu

2009
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Journal of thermophysics and heat transfer
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Nomenclature a = Anisotropy parameter of linear anisotropic scattering phase function A = Area, m 2 c = Speed of light, m/s G = Integrated intensity defined by Eq. (8a), W/m 2 h = One-dimensional standard nodal basis function = Three-dimensional standard nodal basis function H = Matrix defined in Eq. (7) I = Radiative intensity, W/(m 2 sr) 0 I = Amplitude of transient intensity, W/(m 2 sr) b I = Black body radiative intensity, W/(m 2 sr) p I = Transient intensity on the boundary, W/(m 2 sr) k =
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... Unit direction vector of z-direction K = General hexahedral element st K = Standard hexahedral element R L = Reference length scale, m M = Number of discrete ordinate directions M = Matrix defined in Eq. (7) w n = Unit normal vector of the wall K ∂ n = Unit normal vector at boundary of element K t N = Number of discretized time steps sk N = Number of solution nodes on each element N ϕ = Number of subdivisions for azimuthal angle N θ = Number of subdivisions for zenith angle p = Order of polynomial expansion z q = Heat flux of z-direction defined by Eq. (8b), W/m 2 r = Vector of spatial coordinates, ( , , ) x y z = r t = Time, s * t = Dimensionless time * / R t ct L = , Dimensionless time step *Ph.D, School of Energy Science and Engineering, 92 West Dazhi Street; jmzhaocn@gmail.com. † Professor, School of Energy Science and Engineering, 92 West Dazhi Street; lhliu@hit.edu.cn, (corresponding author). -2 -S = Function defined in Eq. (7d), W/m 3 T = Temperature, K u = Unit step function V = Volume, m 3 w = Weight of discrete ordinates approximation, sr , , x y z = Global coordinate system variables st x = Local coordinate vector, ( , , ) st s t s t s t x y z = x , , st st st x y z = Reference coordinate system variables Greek symbols β = Extinction coefficient ( ) a s β κ κ = + , m -1 β = Function defined in Eq. (7c), m -1 * t Δ = Dimensionless time step θ = Zenith angle a κ = Absorption coefficient, scattering coefficient, m -1 s κ = Scattering coefficient, m -1 ρ = Bidirectional reflection function σ = Stefan-Boltzmann constant, W/(m 2 K 4 ) L τ = Optical thickness, L L τ β = p τ = Transmissivity φ = Global nodal basis function ϕ = Azimuthal angle Φ = Scattering phase function ψ = Map function defined by Eq. (5) ω = Single scattering albedo Ω = Unit vector of radiation direction Ω = Solid angle, sr Subscripts n = Time step index i = Mapped one-dimensional index , , i j k ′ ′ ′ = Elemental spatial node index l = Node index of standard hexahedral element w = Value at wall Superscripts , m m′ = Index of discrete ordinate direction

doi:10.2514/1.39361
fatcat:74mkrir5dvgw7o4cnpdi4jpcem