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A PARABOLIC FORMULATION OF THE DISCRETE ORDINATES METHOD FOR THE TREATMENT OF COMPLEX GEOMETRIES

DOI: 10.1615/ICHMT.1995.RadTransfProc.60

Rainer Koch
Lehrstuhl und Institut für Thermische Strömungsmaschinen Unversität Karlsruhe, Karlsruhe, Germany

W. Krebs
Lehrstuhl und Institut für Thermische Strömungsmaschinen Unversität Karlsruhe, Karlsruhe, Germany

Sigmar Wittig
Lehrstuhl und Institut für Thermische Strömungsmaschinen Unversität Karlsruhe, Karlsruhe, Germany

Raymond Viskanta
Heat Transfer Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, USA

Résumé

Among the various methods proposed for numerically solving the radiative transfer equation, the discrete ordinates method is presently judged as one of the most promising. The usual procedure in the method is to solve one first order differential equation for each of the discrete directions. The numerical treatment of these first order differential equations is well known for its convenient programming and its small computer memory requirements. However, this approach possesses some major shortcomings. The most serious one is due to the nature of the differential equations (first order, hyperbolic type). The method is difficult to implement in advanced finite volume or finite element codes for combusting flows which are designed to handle complex geometries.
In order to overcome this disadvantage, an alternate methodology based on the even parity formulation of the discrete ordinates equations is proposed. This approach leads to a set of second order differential equations of the parabolic type. Their structure is formally similar to the differential equation describing a diffusion process. Hence, this formulation of the discrete ordinates equations is compatible with the numerical schemes employed by computer codes for combusting flows.
The parabolic formulation of the discrete ordinates method has been implemented in a computer program for three-dimensional combusting flows developed at the University of Karlsruhe. The code is based on a finite volume formulation and can handle complex geometries by using body-fitted, non-orthogonal grids.
The major objective of the paper is to demonstrate the capabilities of the method. Its accuracy is evaluated by sample calculations of benchmark solutions. The test cases are defined on a Cartesian coordinates system. The results reveal that the achievable accuracy of the method is comparable to the conventional discrete ordinates method.
Special emphasis is placed on the various effects encountered when of curvilinear, body-fitted grids are used. As a typical studied example, the rotation of the grid with respect to the principal coordinates system has been, and it was found to have an unexpected effect on the radiative flux distribution. A close examination reveals that this effect is related to the directional biasing inherent to the angular quadrature scheme.
It is well known that the conventional discrete ordinates method tends to exhibit ray effects. Our study reveals that ray effects are also present within the parabolic formulation of the discrete ordinates method, and that they may be suppressed by applying higher order quadrature schemes.

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