Reaction analogy based forcing for incompressible scalar turbulence

We present a novel reaction analogy (RA) based forcing method for generating statistically stationary scalar fields in incompressible turbulence. The new method can produce more general scalar probability density functions (PDFs), for example, quasi-double-$\delta$ PDF, than current methods, while ensuring that scalar fields remain bounded, unlike existent forcing methodologies that can potentially violate naturally existing bounds. Such features are useful for generating initial fields in nonpremixed combustion, inlet conditions for spatially developing flows, or for studying non-Gaussian scalar turbulence. The RA method mathematically models hypothetical chemical reactions that convert reactants in a mixed state back into its pure unmixed components. Various types of chemical reactions are formulated and the corresponding mathematical expressions derived such that the reaction term is smooth in scalar space and is consistent with mass conservation. For large values of the scalar forcing rate, the method produces statistically stationary quasi-double-$\delta$ scalar PDFs. Quasiuniform, Gaussian, and stretched exponential scalar statistics are recovered for smaller values of the scalar forcing rate. The shape of the scalar PDF can be further controlled by changing the stoichiometric coefficients of the reaction. The ability of the new method to produce fully developed passive scalar fields with quasi-Gaussian PDFs is also investigated, by exploring the convergence of the scalar variance spectrum to the Obukhov-Corrsin scaling and of the third-order mixed structure function to the “four-thirds” Yaglom's law.

Figure 1

Snapshots of the scalar field, $\varphi$, produced by the proposed RA (with different target scalar dissipation rates), LS, and IMG forcing methods. The underlying turbulent field is the same for all four cases.

Figure 2

RA forcing terms for different stoichiometric coefficients versus $\varphi$.

Figure 3

The time variance over the stationary region of the volume averaged scalar dissipation normalized by ${\chi }^2$ for different values of ${\chi }_{\text{target}}$.

Figure 4

(a) Skewness S (circles) and kurtosis (flatness) K (crosses) scalar PDF moments obtained for various target dissipation rates, ${\chi }_{\text{target}}$, ranging from 0.00046 to 0.932. The Gaussian kurtosis of 3.0 is indicated with dashed lines. (b) Stationary scalar PDFs for different ${\chi }_{\text{target}}$ values.

Figure 5

(a) Effects of stochiometric coefficients on the scalar PDF with ${\chi }_{\text{target}}=0.01$. (b) Scalar PDFs obtained for ${\chi }_{\text{target}}$ close to its largest ${\chi }_{\text{target}}$ value supported on a ${256}^3$ mesh with $\eta k_{\text{max}}=3.0$ (${\chi }_{\text{target}}=0.932$) and $\eta k_{\text{max}}=1.5$ (${\chi }_{\text{target}}=0.17$). For comparison, scalar PDFs from ${512}^3$ and ${1024}^3$ simulations with ${\chi }_{\text{target}}=0.932$ are also shown.

Figure 6

(a) $L_{\infty }$ norm and (b) $L_2$ norm of the scalar field using the RA, IMG, and LS methods. The x axis time is scaled with T, the eddy turnover time ($\sim$ 3.0) (c) Scalar PDFs produced by the RA, IMG and LS methods. (d) Mechanical to scalar dissipation timescale ratio comparison for the three forcing methods. The leftmost IMG and LS datapoints were obtained with mean scalar gradient of 0.1 and scalar variance matching the RA forcing with ${\chi }_{\text{target}}=0.01$, respectively. The IMG and LS results are added to the plot based on their $\chi$ values.

Figure 7

Absolute values of the spectra of the terms contributing to spectral scalar variance equation. (a) ${\chi }_{\text{target}}=0.01$, $\eta k_{\text{max}}=1.5$, ${\text{Re}}_{\lambda }=410$, quasi-Gaussian scalar PDF, (b) ${\chi }_{\text{target}}=0.932$, $\eta k_{\text{max}}=3$, ${\text{Re}}_{\lambda }=255$, quasi-double-$\delta$ scalar PDF. (c) Forcing spectra normalized by the dissipation spectra in the dissipation range.

Figure 8

Compensated scalar variance spectra, ${\chi }^{-1}{\epsilon }^{1/3}{k}^{5/3}{E}_{\varphi }\left(k\right)$, versus the scaled wave number. Here, “filter” denotes a low wave number restriction of the RA forcing. The horizontal line corresponds to the Obukhov-Corrsin constant of 0.68.

Figure 9

MSF values for (a) ${\text{Re}}_{\lambda }=92$, with different forcing mechanisms; RA forcing corresponds to quasi-double-$\delta$ (${\chi }_{\text{target}}=0.932$), quasi-Gaussian (${\chi }_{\text{target}}=0.01$), and stretched (${\chi }_{\text{target}}=0.00046$) scalar PDFs, (c) RA forcing for quasi-double-$\delta$ (${\chi }_{\text{target}}=0.932$), ${\text{Re}}_{\lambda }=255$, and quasi-Gaussian (${\chi }_{\text{target}}=0.01$), ${\text{Re}}_{\lambda }=410$, scalar PDFs. Here, “filter” denotes a low wave number restriction of the RA forcing.