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ABSTRACT
The influences of differential diffusion of heat and mass on the Favre-filtered scalar dissipation rate (SDR) transport have been analyzed and modeled using a priori analysis of Direct Numerical Simulations (DNS) data of freely propagating statistically planar turbulent premixed flames with different values of global Lewis number, Le. The DNS data has been explicitly filtered using a Gaussian filter to obtain the unclosed terms of the Favre-filtered SDR transport equation, arising from turbulent transport (T1), density variation due to heat release (T2), strain rate contribution due to the alignment of scalar and velocity gradients (T3), correlation between the gradients of reaction rate and reaction progress variable (T4), molecular dissipation of SDR (−D2), and diffusivity gradients f(D). The statistical behaviors of these terms and their scaling estimates reported in a recent analysis have been utilized here to propose models for these unclosed terms in the context of Large Eddy Simulations (LES) and the performances of these models have been assessed using the values obtained from explicitly filtered DNS data. These newly proposed models are found to satisfactorily predict both the qualitative and quantitative behaviors of these unclosed terms for a range of filter widths Δ for all Le cases considered here.
Nomenclature
| c | = | reaction progress variable |
| cm | = | thermo-chemical parameter |
| CP | = | specific heat at constant pressure |
| CV | = | specific heat at constant volume |
| CF | = | model parameter |
| C3, C4 | = | model parameters |
| D | = | progress variable diffusivity |
| Dt | = | eddy diffusivity |
| Da | = | Damköhler number |
| D1 | = | molecular diffusion term |
| D2 | = | molecular dissipation term |
| fb | = | burning mode probability density function |
| = | model parameters | |
| f(D) | = | term due to diffusivity gradient |
| ksgs | = | sub-grid scale kinetic energy |
| = | thermo-chemical parameter | |
| Ka | = | Karlovitz number |
| Le | = | Lewis number |
| l | = | integral length scale |
| Ma | = | Mach number |
| Mi | = | ith component of resolved flame normal |
| Nc | = | scalar dissipation rate |
| Ni | = | ith component of flame normal |
| p | = | model parameter |
| Pr | = | Prandtl number |
| Q | = | general quantity |
| Ret | = | turbulent Reynolds number |
| ReΔ | = | sub-grid Reynolds number |
| = | unstrained laminar burning velocity | |
| t | = | time |
| tc | = | chemical time scale |
| tf | = | initial turbulent eddy turnover time |
| tsim | = | simulation time |
| T | = | instantaneous dimensional temperature |
| T+ | = | non-dimensional temperature |
| Tad | = | adiabatic flame temperature |
| T0 | = | reactant temperature |
| T1, T2, T3, T4 | = | terms in the transport equation of Favre-filtered scalar dissipation rate |
| ui | = | ith component of nondimensional fluid velocity |
| u′ | = | root mean square fluctuation of velocity |
| u′Δ | = | sub-grid velocity fluctuation |
| = | chemical reaction rate | |
| xi | = | ith Cartesian coordinate |
| YR | = | reactant mass fraction |
| YR0 | = | reactant mass fraction in unburned gas |
| YR∝ | = | reactant mass fraction in burned gas |
| αT | = | thermal diffusivity |
| αT0 | = | thermal diffusivity of the unburned gas |
| β | = | Zel’dovich number |
| β3, β′3 | = | model parameters |
| γ | = | ratio of specific heats (=CP/CV) |
| γ1, γ2 | = | model parameter |
| δth | = | thermal flame thickness |
| δz | = | Zel'dovich flame thickness |
| Δ | = | filter width |
| Γ | = | model parameter |
| μ | = | viscosity |
| μ0 | = | viscosity of unburned gas |
| ρ | = | density |
| ρ0 | = | unburned gas density |
| τ | = | heat release parameter |
| τij | = | viscous stress tensor |
| Φ′ | = | model parameter |
| = | LES-filtered value of a general quantity q | |
| = | Favre-filtered value of a general quantity q | |
| Subscripts | = | |
| 0 | = | unburned gas value |
| ∞ | = | burned gas value |
| res | = | resolved scale value |
| sg | = | sub-grid scale value |
| Acronyms | = | |
| DNS | = | direct numerical simulation |
| LES | = | large eddy simulation |
| = | probability density function | |
| SDR | = | scalar dissipation rate |
Nomenclature
| c | = | reaction progress variable |
| cm | = | thermo-chemical parameter |
| CP | = | specific heat at constant pressure |
| CV | = | specific heat at constant volume |
| CF | = | model parameter |
| C3, C4 | = | model parameters |
| D | = | progress variable diffusivity |
| Dt | = | eddy diffusivity |
| Da | = | Damköhler number |
| D1 | = | molecular diffusion term |
| D2 | = | molecular dissipation term |
| fb | = | burning mode probability density function |
| = | model parameters | |
| f(D) | = | term due to diffusivity gradient |
| ksgs | = | sub-grid scale kinetic energy |
| = | thermo-chemical parameter | |
| Ka | = | Karlovitz number |
| Le | = | Lewis number |
| l | = | integral length scale |
| Ma | = | Mach number |
| Mi | = | ith component of resolved flame normal |
| Nc | = | scalar dissipation rate |
| Ni | = | ith component of flame normal |
| p | = | model parameter |
| Pr | = | Prandtl number |
| Q | = | general quantity |
| Ret | = | turbulent Reynolds number |
| ReΔ | = | sub-grid Reynolds number |
| = | unstrained laminar burning velocity | |
| t | = | time |
| tc | = | chemical time scale |
| tf | = | initial turbulent eddy turnover time |
| tsim | = | simulation time |
| T | = | instantaneous dimensional temperature |
| T+ | = | non-dimensional temperature |
| Tad | = | adiabatic flame temperature |
| T0 | = | reactant temperature |
| T1, T2, T3, T4 | = | terms in the transport equation of Favre-filtered scalar dissipation rate |
| ui | = | ith component of nondimensional fluid velocity |
| u′ | = | root mean square fluctuation of velocity |
| u′Δ | = | sub-grid velocity fluctuation |
| = | chemical reaction rate | |
| xi | = | ith Cartesian coordinate |
| YR | = | reactant mass fraction |
| YR0 | = | reactant mass fraction in unburned gas |
| YR∝ | = | reactant mass fraction in burned gas |
| αT | = | thermal diffusivity |
| αT0 | = | thermal diffusivity of the unburned gas |
| β | = | Zel’dovich number |
| β3, β′3 | = | model parameters |
| γ | = | ratio of specific heats (=CP/CV) |
| γ1, γ2 | = | model parameter |
| δth | = | thermal flame thickness |
| δz | = | Zel'dovich flame thickness |
| Δ | = | filter width |
| Γ | = | model parameter |
| μ | = | viscosity |
| μ0 | = | viscosity of unburned gas |
| ρ | = | density |
| ρ0 | = | unburned gas density |
| τ | = | heat release parameter |
| τij | = | viscous stress tensor |
| Φ′ | = | model parameter |
| = | LES-filtered value of a general quantity q | |
| = | Favre-filtered value of a general quantity q | |
| Subscripts | = | |
| 0 | = | unburned gas value |
| ∞ | = | burned gas value |
| res | = | resolved scale value |
| sg | = | sub-grid scale value |
| Acronyms | = | |
| DNS | = | direct numerical simulation |
| LES | = | large eddy simulation |
| = | probability density function | |
| SDR | = | scalar dissipation rate |