Higher-order spherical harmonics to model radiation in direct numerical simulation of turbulent reacting flows

Kshitij V. Deshmukh, Michael F. Modest, Daniel C. Haworth

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The exact treatment of the radiative transfer equation (RTE) is difficult even for idealized situations and simple boundary conditions. A number of higher-order approximations, such as the moment method, discrete ordinates method and spherical harmonics method, provide efficient solution methods. A statistical method, such as the photon Monte Carlo method, solves the RTE by simulating radiative processes such as emission, absorption, and scattering. Although accurate, it requires large computational resources and the solution suffers from statistical noise. The third-order spherical harmonics method (P3 approximation) used here decomposes the RTE into a set of 16 first-order partial differential equations. Successive elimination of spherical harmonic tensors reduces this set to six coupled second-order partial differential equations with general boundary conditions, allowing for variable properties and arbitrary three-dimensional geometries. The tedious algebra required to assemble the final form is offset by greater accuracy because it is a spectral method as opposed to the finite difference/finite volume approach of the discrete ordinates method. The radiative solution is coupled with a direct numerical solution (DNS) of turbulent reacting flows to isolate and quantify turbulence-radiation interactions. These interactions arise due to nonlinear coupling between the fluctuations of temperature, species concentrations, and radiative intensity. Radiation properties employed here correspond to a nonscattering fictitious gray gas with a Planck-mean absorption coefficient, which mimics that of typical hydrocarbon-air combustion products. Individual contributions of emission and absorption TRI have been isolated and quantified. The temperature self-correlation, the absorption coefficient-Planck function correlation, and the absorption coefficient-intensity correlation have been examined for small to large values of the optical thickness. Contributions from temperature self-correlation and absorption coefficient-Planck function correlation have been found to be significant for all the three optical thicknesses while absorption coefficient-intensity correlation is significant for optically thick cases, weak for optically intermediate cases, and negligible for optically thin cases.

Original languageEnglish (US)
Pages (from-to)207-230
Number of pages24
JournalComputational Thermal Sciences
Issue number2
StatePublished - 2009

All Science Journal Classification (ASJC) codes

  • Energy Engineering and Power Technology
  • Surfaces and Interfaces
  • Fluid Flow and Transfer Processes
  • Computational Mathematics


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