Large-Eddy Simulation of High-Speed Vaporizing Liquid-Fuel Spray Using Mixed Gradient-Type Structural Subgrid-Scale Model

ABSTRACT

The accuracy of the large-eddy simulation (LES) of high-speed vaporizing sprays is highly dependent on the choice of subgrid-scale (SGS) models. This study adopts a nonlinear SGS stress model to simulate a well-known high-speed non-reacting vaporizing liquid-fuel spray case, Spray A. An advantage of using the nonlinear structural SGS model is that it can provide reverse energy transfer from small to large scales, and thus, the model can be more accurate than traditional eddy-viscosity models at capturing cross-scale interactions between turbulence and spray. Also, the adopted gradient-type approach is very computationally efficient. To understand the performance of the nonlinear LES framework, this study presents comprehensive comparisons against experimental data and results obtained using the Smagorinsky model and the sub-grid kinetic energy model. In particular, we compare the general structure of the flows, liquid and vapor penetrations, and mass fraction profiles. Results using the nonlinear structural SGS stress model show minimal dependence on the grid size regarding the capture of flow structures. Using the same mesh resolution, the statistics of the simulated field exhibit both good agreement with measurements and a significant improvement over simulations based on the other two types of models. Results support the assertion that the proposed nonlinear LES framework can accurately achieve direct injection simulations with reasonable computational costs.

KEYWORDS:

• Large-eddy simulation
• spray
• sub-grid scale

Acknowledgments

This research was supported by the National Natural Science Foundation of China (51776082). Computing resources were provided by the National Supercomputer Center in Guangzhou.

A Derivation of the gradient formation of ${\mathit{u}}_{\mathit{s}\mathit{g}\mathit{s}}$

The scale-similarity form of the $u_{sgs}$ introduced by Bharadwaj et al. [26] is written as

(17) $u_{sgs,i}=2{\tilde{u}}_i-3{\tilde{\tilde{u}}}_i+{\widetilde{\tilde{\tilde{u}}}}_i.$(17)

Considering the Gaussian filtering on the resolved field [29], the filtered field can be expanded in the series

(18) ${\tilde{\tilde{u}}}_i={\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i+O({\tilde{\mathrm{\Delta }}}^4).$(18)

Follow a similar procedure, one can get

(19) ${\widetilde{\tilde{\tilde{u}}}}_i={\tilde{\tilde{u}}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{\tilde{u}}}_i+O({\tilde{\mathrm{\Delta }}}^4)={\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i+O({\tilde{\mathrm{\Delta }}}^4)={\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{12}{\mathrm{\nabla }}^2{\tilde{u}}_i+O({\tilde{\mathrm{\Delta }}}^4).$(19)

Hence, the expression of $u_{sgs}$ can be written as

(20) $u_{sgs,i}=2{\tilde{u}}_i-3\left({\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i\right)+\left({\tilde{u}}_i+\frac{{\tilde{\mathrm{\Delta }}}^2}{12}{\mathrm{\nabla }}^2{\tilde{u}}_i\right)+O({\tilde{\mathrm{\Delta }}}^4)=-\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i+O({\tilde{\mathrm{\Delta }}}^4).$(20)

As shown in equation 16(16) $u_{sgs,i}=-\frac{{\tilde{\mathrm{\Delta }}}^2}{24}{\mathrm{\nabla }}^2{\tilde{u}}_i.$(16) , we use the leading term of the series as an approximation of $u_{sgs}$.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [51776082].