LES-TPDF Methods And Their Applications For High-speed Turbulent Combustion

The Lagrangian PDF methods are investigated to extend to high-speed turbulent combustion,especially to supersonic combustion.Theoretical analysis,mathematical modeling,numerical tests and case verification are used for this purpose.The Lagrangian PDF is coupled with the finite-difference LES for high-speed reactive flows,among which basic issues such as physical assumptions and numerical calculations are studied.A comprehensive method along with robust numerical calculation has been established.The PDF transport equations for high-speed flows are derived and the issues during the coupling process with LES are analyzed.Based on the concept of particle system,it is proved that the particle interpolation scheme should maintain velocity divergence to obtain good scalar consistency between LES and PDF.Furthermore,three basic principles have been put forward to improve the consistency of LES-PDF.It is also observed that the modeling and numerical calculation of the conditional subgrid high-speed source terms have a profound impact on the energy consistency of LES-PDF.The conditional filtered high-speed source terms have been theoretically analyzed and a new model has been proposed.In this new model,the PDF conditional density is modeled as the Monte Carlo particle density in PDF,rather than the filtered density in LES.Numerical tests have been carried to evaluate the new mode.Results show that the energy consistency of LES-PDF has been improved by this new model and the particle velocity correction can promote the energy consistency with this new model.Numerical accuracy of the above LES-PDF method deteriorate for supersonic flows,especially in the vicinity of shock waves.Numerical errors mainly come from the calculation of the resolved part of the high-speed source terms.This problem is further considered.Based on the conservation laws of hyperbolic systems,conservative terms have been defined.Numerical difficulties stem from the non-conservative property of the high-speed source terms.Based on these analyses,some basic laws have been given to choose the PDF energy variable.The total non-chemical enthalpy is proposed as the energy variable so that the high-speed source terms are stringently conservative.Besides,the conditional subgrid high-speed source terms for the total non-chemical enthalpy is supposed to be much smaller than those for the sensible enthalpy.Then,the transport equations and models of the total non-chemical enthalpy-species mass fractions PDF have been derived and established.Numerical results show that the energy consistency of LES-PDF has been further improved by this new method and the energy consistency at subgrid level is also elevated.The scalar-pressure PDF is also considered.Different from the previous pressure PDF with calorically perfect gas assumption,this pressure PDF is based on the perfect gas assumption.The transport equations and the corresponding models are also discussed.Numerical tests show that considerable numerical oscillation exists around the contact surface for the PDF pressure field with such scalar-pressure PDF.This numerical issue is further analyzed.The main reason is that the basic transport equation is a non-conservative form,which is not consistent with the conservative hyperbolic systems used in LES.For such a question,developing general numerical methods for non-conservative hyperbolic system may be of some help.An alternative way is the scalar-fluctuation pressure PDF method.The filtered pressure is obtained from LES,while the fluctuation pressure is computed by PDF.The particle models are developed and numerical tests are carried to verify this method.The particle velocity interpolation method and the particle velocity correction(PVC)scheme are the keys of maintaining the particle mass/density consistency of LES-PDF.For supersonic flows,the particle velocity interpolation methods are expected to have desirable velocity divergence and upwind properties.To avoid free parameters in the existing particle velocity correction schemes,we devise a general PVC formulation from which the existing schemes can be recovered.Based on this general formulation,we propose an improved PVC scheme with spatial smoothing operators applying to the PDF fields.Different definitions of smoothing operators are proposed and compared to obtain the most distinguished one.The improved PVC scheme has been verified in numerical tests and shown a good generality.The proposed LES-PDF methods and the corresponding numerical schemes are used to simulate the hydrogen/air supersonic reactive temporally mixing layer.Based on the statistics of the flow fields,the proposed LES-PDF methods show considerable improvements than those of the LES with Arrhenius limited reaction rate model,which suggest that the proposed LES-PDF methods are suitable for supersonic reactive flows.