Large Eddy Simulation of aTurbulent Diffusion Flame: Some Aspects of Subgrid Modelling Consistency

Abstract

In the context of Large Eddy Simulation (LES) solely for the momentum transport equation there may be found several models for the turbulent subgrid fluxes. Furthermore, among those relying on the eddy diffusivity approach, each model may be based on different invariants of the strain rate. Besides, when heat and mass transfer are also considered, closures for the subgrid turbulent scalar fluxes are also required. Hence, different model combinations may be considered. Additionally, when other physical phenomena are included, such as combustion, further subgrid modelling is involved. Therefore, in the present study a LES simulation of a turbulent diffusion flame is performed and different combination of subgrid models are used in order to analyse the numerical effects in the simulations. Several models for the turbulent momentum subgrid fluxes are considered, both constant and dynamically evaluated Schmidt numbers. Regarding combustion, in the context of the Flamelet/Progress-Variable (FPV) model, with an assumed probability density function for the turbulent-chemistry interactions and four different closures for the subgrid mixture fraction variance are considered. Hence, a large number of model combinations are possible. The present study highlights the need for a consistent closure of subgrid effects. It is shown that, in the context of an FPV modelling, incorrect capture of subgrid mixing results in a flame lift-off for the studied flame (DLR A diffusion flame), even though experimentally an attached flame was reported. It is found that a consistent formulation is required, that is, all subgrid closures should become active in the same regions of the domain to avoid modelling inconsistencies. In contrast, when the classical flamelet approach is used, no lift-off is observed. The reason is that the classical flamelet includes only a limited subset of the possible flame states, i.e. only includes burning flamelets and extinguished flamelets for scalar dissipation rates past the extinction one.

Appendix A

1.1 Subgrid Kinetic Energy Modelling

The VTE and STE subgrid mixing models are closed using a turbulent mixing time-scale. It has been argued that when an Smagorinsky-like turbulence model is used, closure for the subgrid dissipation and subgrid kinetic energy is consistent. However, when another model based on different invariants is used an inconsistency appears, such as in the WALE model. In the following an analysis of subgrid dissipation and subgrid kinetic energy modelling is developed for the WALE model, but an analogous process can be performed for any other subgrid viscosity model.

Modelling the relation between subgrid dissipation and subgrid kinetic energy [27] as

$$ \varepsilon_{sgs} = C_{\varepsilon} \frac{k_{sgs}^{3/2}}{l_{m}} = C_{\varepsilon} \frac{k_{sgs}^{3/2}}{\Delta} $$

(21)

where l _m represents the mixing length, which is taken to be the grid size Δ. Yoshizawa and Horiuti [49] suggest a value for the constant of C _ε = 1.8. The subgrid dissipation is

$$ \varepsilon_{sgs} = \widetilde{\tau}_{ij} \widetilde{S}_{ij} = 2 \nu_{t} \widetilde{S}_{ij} \widetilde{S}_{ij} $$

(22)

with $\widetilde {S}_{ij} = \frac {1}{2} \left (\frac {\partial \widetilde {u}_{i}}{\partial x_{j}}+\frac {\partial \widetilde {u}_{j}}{\partial x_{i}} \right )$. Combining the last two equations and introducing $|\widetilde {S}| = \sqrt {2 \widetilde {S}_{ij}\widetilde {S}_{ij}}$

$$ \nu_{t} |\widetilde{S}|^{2} = C_{\varepsilon} \frac{k_{sgs}^{3/2}}{\Delta} $$

(23)

In the following, subgrid kinetic energy is modelled as

$$ k= 2 C_{k} {\Delta}^{2} \widetilde{S}_{ij} \widetilde{S}_{ij} = C_{k} {\Delta}^{2} |\widetilde{S}|^{2} $$

(24)

where C _k is a model constant. For Smagorinsky-like subgrid viscosity models this is a constant value. However, for models based on invariants different than the strain, it is shown that this is not a true constant. Introducing (24) into (23) results in

$$\begin{array}{@{}rcl@{}} \nu_{t} |\widetilde{S}|^{2} &=& \frac{C_{\varepsilon}}{\Delta} \Bigl(C_{k} {\Delta}^{2} |\widetilde{S}|^{2} \Bigr)^{3/2} \\ \nu_{t} &=& C_{\varepsilon} (C_{k})^{3/2} {\Delta}^{2} |\widetilde{S}| \end{array} $$

(25)

Thus, next it is analysed the relation between the turbulent viscosity constant and the subgrid kinetic energy constant.

1.1.1 Smagorinsky-like subgrid viscosity

Beginning with the Smagorinsky model [4], the turbulent viscosity is

$$ \nu_{t} = {C_{s}^{2}} {\Delta}^{2} |\widetilde{S}| $$

(26)

Combining the latter with (25) results in

$$\begin{array}{@{}rcl@{}} {C_{s}^{2}} {\Delta}^{2} |S| &=& C_{\varepsilon} C_{k}^{3/2} {\Delta}^{2} |S| \\ {C_{s}^{2}} &=& C_{\varepsilon} C_{k}^{3/2} \end{array} $$

(27)

Hence, when the Smagorinsky model is used, the turbulent kinetic energy model constant (24) is a true constant. Consequently, subgrid dissipation and subgrid kinetic energy are modelled consistently.

1.1.2 WALE subgrid viscosity

Considering the WALE model [28], the turbulent viscosity is evaluated as

$$ \nu_{t} = {C_{w}^{2}} {\Delta}^{2} S_{d} $$

(28)

where S _d represents the operator of the WALE model

$$ S_{d} = \frac{\left( \mathcal{V}_{ij}\mathcal{V}_{ij}\right)^{3/2}}{\left( \widetilde{S}_{ij}\widetilde{S}_{ij} \right)^{5/2}+\left( \mathcal{V}_{ij}\mathcal{V}_{ij}\right)^{5/4}} $$

(29)

with

$$\mathcal{V}_{ij} = \frac{1}{2}\left( \left( \frac{\partial \widetilde{u}_{i}}{\partial x_{j}}\right)^{2}+ \left( \frac{\partial \widetilde{u}_{j}}{\partial x_{i}}\right)^{2} \right) - \frac{1}{3}\delta_{ij}\left( \frac{\partial \widetilde{u}_{k}}{\partial x_{k}}^{2} \right)^{2} $$

Introducing (28) into (25) yields

$$\begin{array}{@{}rcl@{}} {C_{w}^{2}} {\Delta}^{2} S_{d} &=& C_{\varepsilon} C_{k}^{3/2} {\Delta}^{2} |\widetilde{S}| \\ {C_{w}^{2}} S_{d} &=& C_{\varepsilon} C_{k}^{3/2} |\widetilde{S}| \end{array} $$

(30)

showing that the model constant for the subgrid kinetic energy should take into account the strain $|\widetilde {S}|$ and the operator used by the WALE model. Thus, the model constant in this case is

$$ C_{k}^{3/2} = \frac{{C_{w}^{2}}}{C_{\varepsilon}} \frac{S_{d}}{|\widetilde{S}|} $$

(31)

Consequently, if C _k is set to a specific value, a true constant, when the subgrid turbulent fluxes are modelled using the WALE model, there is a mismatch in (21). As a results, the modelling inconsistency previously described appears.