Rapidly decorrelating velocity-field model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars

Light is shed upon Eulerian Monte Carlo methods and their application to the simulation of turbulent reactive flows. A rapid decorrelating velocity-field model is used to derive stochastic partial differential equations (SPDE’s) stochastically equivalent to the modeled one-point joint probability density function of turbulent reactive scalars. Those SPDE’s are shown to be hyperbolic, advection-reaction equations. They are dealt with in a generalized sense, so that discontinuities in the scalar fields can be treated. A numerical analysis is proposed and numerical tests are carried out. In particular, a comparison with the Lagrangian Monte Carlo method is performed.

Figure 1

Formation of discontinuities due to boundary conditions.

Figure 2

Test 1: no micromixing $\omega =0$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. Stochastic fields at time $t=10$.

Figure 3

Test 1: no micromixing $\omega =0$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. PDF at $t=10$ and $x=0.51$.

Figure 4

Test 1: no micromixing $\omega =0$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. Statistical convergence of the moments: error $e_p$ against the number of stochastic fields, $N$.

Figure 5

Test 1: no micromixing $\omega =0$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. Comparison of the moments computed with the EMC method against the analytic solution [Eq. (76)].

Figure 6

Test 1: no micromixing $\omega =0$, no source term $S=0$, $\Gamma =0.1$, $U_0=1$. Spatial convergence: error $e_p$ against the number of cells, $N_c$.

Figure 7

Test 2: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. (a) Stochastic fields at time $t=10$; (b) zoom-in the stochastic fields in the vicinity of $x=0$.

Figure 8

Test 2: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. Statistical convergence: error $e_p$ against the number of stochastic fields, $N$.

Figure 9

Test 2: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. Comparison of the moments computed with the EMC method against the reference solution (FD-Mom.). The reference solution is computed from the moment equation (77) with a finite difference (FD) method.

Figure 10

Test 2: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=1$. Spatial convergence: error $e_p$ against the number of cells, $N_c$.

Figure 11

Test 3: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1[1-14x^2{(1-x)}^2]$, $U_0=0$. Statistical convergence: error $e_p$ against the number of stochastic fields, $N$.

Figure 12

Test 3: IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1[1-14x^2{(1-x)}^2]$, $U_0=0$. Comparison of the moments computed with the EMC method against the reference solution (FD-Mom.). The reference solution is computed from the moment equation (77) with a finite-difference (FD) method.

Figure 13

Test 4: IEM model $\omega =20$, Arrhenius nonlinear source term, $\Gamma =0.1$, $U_0=1$. Stochastic fields at time $t=10$.

Figure 14

Test 4: IEM model $\omega =20$, Arrhenius nonlinear source term, $\Gamma =0.1$, $U_0=1$. Comparison of the moments computed with the EMC method against the reference solution (FD-PDF). The reference solution is computed from Eq. (79) with a finite-difference (FD) method.

Figure 15

Test 4: Langevin model $\omega =20$, $d_0=1$, Arrhenius nonlinear source term, $\Gamma =0.1$, $U_0=1$. Comparison of the moments computed with the EMC method against the reference solution (FD-PDF). The reference solution is computed from Eq. (79) with a finite-difference (FD) method.

Figure 16

Test 4: Langevin model $\omega =20$, $d_0=1$, Arrhenius nonlinear source term, $\Gamma =0.1$, $U_0=1$. Statistical convergence: error $e_p$ against the number of stochastic fields, $N$.

Figure 17

Comparison EMC/LMC-IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=0$. (a) Error $e_1$; (b) error $e_4$.

Figure 18

Comparison EMC/LMC-IEM model $\omega =20$, no source term $S=0$, $\Gamma =0.1$, $U_0=1$. (a) Error $e_1$; (b) error $e_2$.

Figure 19

Comparison EMC/LMC-Langevin model $\omega =20$, $d_0=1$, Arrhenius nonlinear source term, $\Gamma =0.1$, $U_0=1$. (a) Error $e_1$; (b) error $e_2$.