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A Conditional Moment Closure Study of Chemical Reaction Source Terms in SCCI Combustion

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Abstract

The objective of this study is to evaluate conditional moment closure (CMC) approaches to model chemical reaction rates in compositionally stratified, autoigniting mixtures, in thermochemical conditions relevant to stratified charge compression ignition (SCCI) engines. First-order closure, second-order closure and double conditioning are evaluated and contrasted as options in comparison to a series of direct numerical simulations (DNSs). The two-dimensional (2D) DNS cases simulate ignitions in SCCI-like thermochemical conditions with compositionally stratified n-heptane/air mixtures in a constant volume. The cases feature two different levels of stratification with three mean temperatures in the negative-temperature coefficient (NTC) regime of ignition delay times. The first-order closure approach for reaction rates is first assessed using hybrid DNS-CMC a posteriori tests when implemented in an open source computational fluid dynamics (CFD) package known as OpenFOAM\(^{{\circledR }}\). The hybrid DNS-CMC a posteriori tests are not a full CMC but a DNS-CMC hybrid in that they compute the scalar and velocity fields at the DNS resolution, thus isolating the first-order reaction rate closure model as the main source of modelling error (as opposed to turbulence model, scalar probability density function model, and scalar dissipation rate model). The hybrid DNS-CMC a posteriori test reveals an excellent agreement between the model and DNS for the cases with low levels of stratification, whereas deviations from the DNS are observed in cases which exhibit high level of stratifications. The a priori analysis reveals that the reason for disagreement is failure of the first-order closure hypothesis in the model due to the high level of conditional fluctuations. Second-order and double conditioning approaches are then evaluated in a priori tests to determine the most promising path forwards in addressing higher levels of stratification. The a priori tests use the DNS data to compute the model terms, thus directly evaluating the model assumptions. It is shown that in the cases with a high level of stratification, even the second-order estimation of the reaction rate source term cannot provide a reasonably accurate closure. Double conditioning using mixture-fraction and sensible enthalpy, however, provides an accurate first-order closure to the reaction rate source term.

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Notes

  1. For clarity the hybrid DNS-CMC a posteriori and a priori concepts are defined here:

    • a posteriori: is referred to when the CFD-CMC model is integrated in time according to the model equations starting from the DNS initial condition. The results are then compared to DNS data.

    • a priori: is referred to when the DNS data are post-processed on their own without actually running the CMC model.

  2. Note that instead of feeding back the mean mass fractions, the density and other required properties such as the specific heats, viscosity, etc could be determined in the CMC domain and fed back. This would make the calculation cheaper but as a matter of convenience it is not implemented this way.

  3. The CMC solver may be invoked before the CFD solver, alternatively. For small enough time steps both approaches converge to the same solution.

  4. The number of bins was determined by a series of sensitivity tests not shown here.

  5. In effect, the ghost nodes hold the value of boundary nodes from reaction step. While in pinciple, the value of ghost nodes can be transported to the internal nodes via diffusion and convection, in the present cases here they will not affect the solution in the internal nodes. On the boundary nodes at ξ = 0 and ξ = 1 the value of N is fixed at zero by definition, therefore, no diffusion will take place from ghost nodes to internal nodes. As for convection, the \(\tilde {v}_{\xi }\) in Eq. 5 is an outgoing velocity at both boundaries. At the left boundary, \(\tilde {v}_{\xi }\) is negative (an outgoing wave from right to left), because at ξ = 0 the velocity becomes \(\tilde {v}_{\xi } = - \left (\partial Z_{\min } / \partial t \right ) / {\Delta } {\Theta }\) with \(\partial Z_{\min } / \partial t \ge 0\) in the present cases. At the right boundary, \(\tilde {v}_{\xi }\) is positive (an outgoing wave from left to right), because at ξ = 1 the velocity becomes \(\tilde {v}_{\xi } = - \left (\partial Z_{\max } / \partial t \right ) / {\Delta } {\Theta }\) with \(\partial Z_{\max } / \partial t \le 0\) in the present cases. For evolution of \(Z_{\min }\) and \(Z_{\max }\) in the present cases, see Fig. 5.

  6. As per Ref. [28, 29], if negative mixture-fractions are encountered upon initialisation, they are truncated to zero. In the HS cases, this operation shifts the mean mixture-fraction from Z 0 = 0.0195 to Z 0 = 0.0210.

  7. For example the relative error in τ 10% is found by the difference in τ 10% obtained from OF-CMC and DNS divided by τ 10% from DNS. The colours in Fig. 4 distinguish the positive values from negatives.

  8. The oscillations in the second-order results arise from statistical noise due to the finite sample. A larger sample would reduce this noise but it is unlikely to change the key findings.

  9. If \(\dot {\omega }\) is only a function of T, then Eq. 9 is still mathematically valid and the 〈⋅|〉 operator reduces to a normal averaging.

  10. In practice 5 bins are not enough and more is needed to capture the diffusive transport in the progress variable space which has to be accurately modelled.

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Acknowledgements

This work was supported by the Australian Research Council. The research benefited from computational resources provided through the National Computational Merit Allocation Scheme, supported by the Australian Government. The computational facilities supporting this project included the Australian NCI National Facility, the partner share of the NCI facility provided by Intersect Australia Pty Ltd., Pawsey Supercomputing Centre (with funding from the Australian Government and the Government of Western Australia), and the UNSW Faculty of Engineering.

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Authors and Affiliations

  1. School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    James Jalal Behzadi, Evatt R. Hawkes & Sanghoon Kook

  2. Department of Mechanical Engineering, University of Melbourne, VIC, 3010, Australia

    Mohsen Talei

  3. Department of Mechanical and Process Engineering, ETH Zürich, 8092, Zürich, Switzerland

    Michele Bolla

  4. School of Photovoltaic and Renewable Energy Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    Evatt R. Hawkes

  5. Dipartimento di Energia, Politecnico di Milano, 20156, Milan, Italy

    Tommaso Lucchini & Gianluca D’Errico

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  1. James Jalal Behzadi
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  2. Mohsen Talei
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  3. Michele Bolla
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  4. Evatt R. Hawkes
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  5. Tommaso Lucchini
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  6. Gianluca D’Errico
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  7. Sanghoon Kook
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Corresponding author

Correspondence to Evatt R. Hawkes.

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Funding

This research was supported under Australian Research Council under the Discovery Projects funding scheme (project numbers DP110104763 and DP150104393) and the Linkage Infrastructure, Equipment and Facilities scheme (project numbers LE140100002, LE160100002 and LE160100051). Evatt Hawkes is the recipient of an Australian Research Council Future Fellowship (project number FT100100536).

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The authors declare that they have no conflict of interest.

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Behzadi, J.J., Talei, M., Bolla, M. et al. A Conditional Moment Closure Study of Chemical Reaction Source Terms in SCCI Combustion. Flow Turbulence Combust 100, 93–118 (2018). https://doi.org/10.1007/s10494-017-9825-y

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