Conditional Source-term Estimation using dynamic ensemble selection and parallel iterative solution

Abstract

A modified version of the Least-Square QR-factorisation (LSQR) algorithm has been implemented in conjunction with Conditional Source-term Estimation (CSE) for lean, turbulent premixed methane–air combustion via Large Eddy Simulation (LES). The iterative solver can reduce computational times by an order of magnitude during the inversion phase of CSE in comparison with the conventional LU-decomposition method. The advantages of iterative and parallel iterative solvers become more prominent as the size of the system increases. The ensemble selection procedure for computing averages within localised regions of the simulation domain has also been updated to a dynamic routine. This allows for more flexible and efficient allocation of computational resources along with reduced input from the user, especially for complex geometries. Preliminary LES calculations have shown that the implementation of an iterative solver and a dynamic ensemble selection algorithm will reduce computational times significantly with negligible error contribution for one-condition CSE, which is applicable to purely premixed or non-premixed turbulent combustion problems. In addition, these algorithms provide the foundation for exceptional computational cost savings for the inversion in two-condition CSE, or Doubly Conditional Source-term Estimation (DCSE), which has shown promise for predicting partially-premixed combustion. Parallel computation of the inverse solution is particularly beneficial to DCSE as the computational cost of the inversion process is considerably larger than in one-condition CSE.

Keywords:

• Conditional Source-term Estimation
• turbulent premixed combustion
• LES
• LSQR
• CFD

Acknowledgements

The authors would like to thank Drs M.M. Salehi and N. Shahbazian for providing the LES parameters used in the setup of the Gülder burner.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Both and represent the conditional mass fraction of species k. The specification, , expresses the notion that the preceding quantity is a function of space and time. This simplified notation is applicable to all the scalar fields and conditional scalar fields in this section.

2. The PDF matrix is not sparse; however, the LSQR solver has proven to be robust and efficient for all the tests performed. These include standalone matrix inversions and CSE inversions in the context of LES.

3. The L₂-norm is defined as L₂ = [∑^l_{i = 1}(x_{i, LSQR} − x_{i, exact})²]^1/2, where l is the number of discrete bins in the tabulated PDF model, which is typically 50 for one-condition CSE problems.

4. Two-condition CSE is also referred to as Doubly Conditional Source-term Estimation (DCSE) in the literature [15].

5. We are primarily concerned with one-condition CSE for this study. However, the LSQR solver is independent of the inversion problem and it is meaningful to illustrate its potential to solve two-condition CSE matrices for future work.

6. The execution times for the 880,000 cell cases are averaged over two runs to eliminate possible network instability errors.

Additional information

Funding

The authors wish to express their gratitude to the Natural Sciences and Engineering Research Council of Canada (NSERC) for partial funding of the numerical work described herein.