Numerical Investigation of Turbulent Kinetic Energy Dynamics in Chemically-Reacting Homogeneous Turbulence

Abstract

In this work, we investigate numerically the temporal evolution of Turbulent Kinetic Energy (TKE) of a chemically-reacting n-heptane and air mixture in statistically Homogeneous Isotropic Turbulence (HIT). Our specific focus is on the concurrent view of TKE evolution in both physical and scale (Fourier) spaces to identify the impact of reaction-induced heat release on turbulence. The simulation parameters are selected to represent the combustion characteristics of heavy hydrocarbon fuels under engine conditions. Results indicate that pressure dilatation work dominates the TKE evolution during the period of strong heat release and its dominance is attributed to the strong volumetric dilatation associated with the presence of reaction fronts in physical space. Viscous dissipation and viscous dilatation terms become much stronger with increasing heat release, primarily due to the increase in strain-rate and dilatation at the vicinity of the reaction fronts, but their magnitudes are still small compared to that of pressure dilatation work. In addition, the analysis in Fourier space shows that pressure dilatation work dominates the evolution of TKE not only in the mean, but also over a wide range of scales. The spectrum of pressure dilatation shows a power-law behavior, which is a direct consequence of the localized sheet-like reaction fronts in physical space. It is also shown that viscous dissipation spectrum initially removes kinetic energy at small scales when heat release is weak, but starts to remove kinetic energy at intermediate and later at large scales due to the presence of localized reaction fronts during the strong heat release period. More interestingly, it is observed that the inter-scale kinetic energy transfer spectrum moves energy from less dissipative scales (small scales) to scales where kinetic energy is more effectively removed by viscous dissipation work (large scales) during the period of strong heat release, which indicates possible up-scale kinetic energy transfer in Fourier space.

Appendix: Prognostic Equation for the Two-point Cross-correlation Tensor

In order to obtain the prognostic equation for the two-point cross-correlation tensor defined in Eq. 1, the Navier-Stokes equations are decomposed into Favre-mean and Favre-fluctuation. For the current flow configuration, three-dimensional homogeneity makes ensemble-averages equivalent to averages over the entire computational domain. Therefore, ensemble-averaged quantities are space-invariant and only time-dependent. The Favre-mean continuity equation and momentum equations are then simplified to, $\partial \overline {\rho }/ \partial t = 0$, and, $\partial \widetilde {u_{i}}/ \partial t = 0$, respectively, so that $\overline {\rho }$ and $\widetilde {u_{i}}$ are independent of space and time. Under these conditions, the governing equations for $\rho u_{i}^{"}$ and $u_{i}^{"}$ can be derived as,

$$ \frac{\partial u_{i}^{"}}{ \partial t} = -u_{j} \frac{\partial u_{i}^{"}}{ \partial x_{j}} -\frac{1}{\rho} \frac{\partial p} { \partial x_{i}} +\frac{1}{\rho} \frac{\partial \tau_{ij}} { \partial x_{j}}, $$

(22)

$$ \frac{\partial \rho u_{i}^{"}}{ \partial t} = -\frac{\partial \rho u_{i}^{"} u_{j}}{ \partial x_{j}} -\frac{\partial p} { \partial x_{i}} +\frac{\partial \tau_{ij}} { \partial x_{j}}. $$

(23)

The prognostic equation for the two-point cross-correlation tensor is obtained by combining (22) and (23). Denoting an arbitrary variable ϕ computed at locations x and y = x + r by (ϕ)_x and (ϕ)_y, respectively, we multiply the equation for $\left (u_{i}^{"}\right )_{\textbf {x}}$ by $\left (\rho u_{j}^{"}\right )_{\textbf {y}}$, the equation for $\left (u_{i}^{"}\right )_{\textbf {y}}$ by $\left (\rho u_{j}^{"}\right )_{\textbf {x}}$, the equation for $\left (\rho u_{i}^{"}\right )_{\textbf {x}}$ by $\left (u_{j}^{"}\right )_{\textbf {y}}$, and the equation for $\left (\rho u_{i}^{"}\right )_{\textbf {y}}$ by $\left (u_{j}^{"}\right )_{\textbf {x}}$. The prognostic equation for R_ij(r, t) is then obtained by adding the resulting four equations and taking the ensemble-average as,

$$ \frac{\partial R_{ij}(\textbf{r},t)}{ \partial t} = \tfrac{1}{4} \left[ R_{ij}^{A}(\textbf{r},t) + R_{ij}^{P}(\textbf{r},t) + R_{ij}^{V}(\textbf{r},t)\right]. $$

(24)

where,

$$ R_{ij}^{A}(\textbf{r},t) = -\overline{\left( u_{i}^{"}\right)_{\textbf{x}} \left( \frac{\partial \rho u_{i}^{"} u_{j}}{ \partial x_{j}}\right)_{\textbf{y}}}-\overline{\left( u_{i}^{"}\right)_{\textbf{y}} \left( \frac{\partial \rho u_{i}^{"} u_{j}}{ \partial x_{j}}\right)_{\textbf{x}}}-\overline{\left( \rho u_{i}^{"}\right)_{\textbf{x}} \!\left( u_{j} \frac{\partial u_{i}^{"}}{ \partial x_{j}}\right)_{\textbf{y}}} -\overline{\left( \rho u_{i}^{"}\right)_{\textbf{y}} \left( u_{j} \frac{\partial u_{i}^{"}}{ \partial x_{j}}\right)_{\textbf{x}}} , $$

(25)

$$ R_{ij}^{P}(\textbf{r},t) = -\overline{\left( u_{i}^{"}\right)_{\textbf{x}} \left( \frac{\partial p}{ \partial x_{i}}\right)_{\textbf{y}}}-\overline{\left( u_{i}^{"}\right)_{\textbf{y}} \left( \frac{\partial p}{ \partial x_{i}}\right)_{\textbf{x}}}-\overline{\left( \rho u_{i}^{"}\right)_{\textbf{x}} \left( \frac{1}{\rho} \frac{\partial p}{ \partial x_{i}}\right)_{\textbf{y}}} -\!\overline{\left( \rho u_{i}^{"}\right)_{\textbf{y}} \left( \frac{1}{\rho} \frac{\partial p}{ \partial x_{i}}\right)_{\textbf{x}}} , $$

(26)

$$ R_{ij}^{V}(\textbf{r},t) = \overline{\left( u_{i}^{"}\right)_{\textbf{x}} \left( \frac{\partial \tau_{ij}}{ \partial x_{x}}\right)_{\textbf{y}}}+\overline{\left( u_{i}^{"}\right)_{\textbf{y}} \left( \frac{\partial \tau_{ij}}{ \partial x_{j}}\right)_{\textbf{x}}}+\overline{\left( \rho u_{i}^{"}\right)_{\textbf{x}} \left( \frac{1}{\rho} \frac{\partial \tau_{ij}}{ \partial x_{j}}\right)_{\textbf{y}}} +\overline{\left( \rho u_{i}^{"}\right)_{\textbf{y}} \left( \frac{1}{\rho} \frac{\partial \tau_{ij}}{ \partial x_{j}}\right)_{\textbf{x}}} , $$

(27)

represent the two-point cross-correlation tensors between Favre-fluctuating momentum and velocity components, with the advection, pressure and viscous terms of the Favre-fluctuating momentum equations in conservative and non-conservative forms.