The Simplest Approximations of Double Spherical Harmonics
Following from: Differential approximations, Two-flux approximation, P1 approximation of the spherical harmonics method
Leading to: Radiation of an isothermal plane-parallel layer, Radiative equilibrium in a plane-parallel layer, Radiation of a nonisothermal layer of scattering medium, Emissivity of combustion products in a solid-propellant rocket engine, Thermal microwave radiation of disperse systems on the sea surface
In the double spherical harmonics method, which is employed for one-dimensional problems, the spectral radiation intensity is presented as two separate series on the Legendre functions in forward and backward hemispheres; i.e., at = μ > 0 and μ < 0 (Davison, 1957). This expansion is sometimes preferable due to discontinuity of Iλ angular dependence on the free surface at μ = 0, whereas the angular dependences of Iλ are continuous at μ > 0 and μ < 0. The double spherical harmonics method (DPn approximation) was originally suggested by Yvon (1957) and employed first by Yvon (1957), Ziering and Schiff (1958), and Stewart and Zweifel (1958) to the problems of neutron physics. Solutions for cylinders and spheres were first reported by Drawbaugh and Noderer (1959) (see also additional comments by Schmidt and Gelbard, 1966). Long before Yvon’s sudy (1957), Sykes (1951) discussed the discontinuity of radiation intensity at μ = 0 and suggested using the so-called double Gaussian quadrature, which is equivalent to the double spherical harmonics (Case and Zweifel, 1967).
Note that an approximation similar to the DP1 can be derived as a combination of the moment method and the linear angular expansion used in P1. One is reminded of the half-range moment method formulated by Sherman (1967) for a plane-parallel layer and by Özişik et al. (1975) for a spherically symmetric enclosure. The double spherical harmonics method was also used by Wan et al. (1977) for radiative transfer in a slab with Rayleigh scattering.
It is clear that the zero approximation of this method, DP0, coincides with the above-discussed two-flux approximation. In other words, both the P1 and DP0 approximations are different versions of the diffusion approximation. One can expect that DP0 gives better results for optically thin layers of absorbing and scattering media and near the layer boundaries, whereas P1 is better for the radiation field at large optical distances from the boundaries or in the case of specific boundary conditions which do not lead to a sharp angular variation of the spectral radiation intensity. The latter statement will be confirmed by analysis of the accuracy of these approximations in the articles Radiation of an Isothermal Plane-Parallel Layer and Radiative Equilibrium in a Plane-Parallel Layer.
Following earlier studies conducted by Dombrovsky (1972, 1974a,b), we consider also the first approximation of the double spherical harmonics method, DP1, which presents the spectral radiation intensity as two linear functions of angles in two hemispheres. It is clear that this presentation of the angular dependence combines the advantages of P1 and DP0 approximations. Here, we will not give the derivation of the DP1 equations, but only the result for one-dimensional radiative transfer problems at linear anisotropic scattering:
where Nsymm = 1 corresponds to the cylindrical symmetry and Nsymm = 2 corresponds to the spherical symmetry. In the plane symmetry case, we have Nsymm = 0. The boundary conditions for the diffusely radiating wall at r = r0 are
where εw is the hemispherical emissivity of the wall and Tw is the wall temperature. Matrixes A and B depend on the wall reflection type:
The spectral radiation flux and the spectral radiation energy density are
As in the general case, the radiation energy balance takes place:
At the same time, diffusion equation (5) from the article Differential Approximations is not true, and the spectral radiation energy density Iλ0 satisfies the fourth-order differential equation. It is important that the equations of the DP1 approximation for linear anisotropic scattering are not the same as those for the transport scattering function.
One can show that only high-order approximations Pn and DPn are sensitive to the details of the scattering function. However, these approximations are not usually employed in engineering practice since the equivalent discrete ordinates method is more convenient for numerical calculations. The DP1 approximation is not as popular as the DP0 (two-flux) and P1 approximations. Nevertheless, the relatively accurate analytical solutions obtained in the DP1 approximation have been used by Dombrovsky and Ivenskikh (1973), Dombrovsky (1974a,b, 1976, 1979), and by Dombrovsky and Raizer (1992) to analyze radiative transfer in one-dimensional problems concerning the thermal radiation of disperse systems. The algorithm of the finite-difference solution for boundary-value problems formulated on the basis of the DP1 approximation is presented in the article Radiation of a Nonisothermal Layer of Scattering Medium. Note that the numerical solution of the DP1 equations was employed by Tsai (1991) in his study of combined heat transfer by conduction and radiation in a layer of absorbing, emitting, and anisotropically scattering material.
One should review some of the alternative attempts that employed the original idea by Krook (1955) on the moment method applied to separate solid angle subregions. Mengüç and Iyer (1988) used this procedure to formulate the DP1 approximation for a medium with linear-anisotropic scattering. They also gave the formulation for the octuple spherical harmonics (OP1) approximation for two-dimensional, axisymmetric cylindrical enclosures with a solid angle divided into eight subdomains. Iyer and Mengüç (1989) developed a similar hybrid model for two-dimensional rectangular enclosures by combining a four-flux method (which should not be confused with the generalization of the Kubelka-Munk two-flux model discussed in the article Two-flux Approximation) with the P1 approximation. It was shown in studies conducted by Mengüç and Iyer (1988) and Iyer and Mengüç (1989) that the computational models suggested work fairly well for particular problems considered by these authors. At the same time, the formal dividing of the solid angle cannot take into account the real angular structure of the radiation field. Therefore, this approach is not expected to work well with more complex problems. Remember that the classic spherical harmonics method is based on the angular expansion around the local radiation flux direction. In other words, it takes into account the local angular structure of the radiation field.
The preliminary discussion of the simplest approximations P1 and DP0 should be completed by a reference to a recent study by Brantley (2007), who suggested angularly adaptive P1-DP0 flux-limited diffusion solutions for time-dependent nonequilibrium radiative transfer problems. The well-known properties of P1 and DP0 have been used in developing the new computational model: the P1 approximation is predominant near thermodynamic equilibrium, whereas DP0 can more accurately capture the complicated angular dependence near a nonequilibrium radiation wave front. In addition, the DP0 approximation is more accurate in nonequilibrium optically thin regions where the positive and negative angular domains are largely decoupled.
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