Numerical forcing scheme to generate passive scalar mixing on the centerline of turbulent round jets in a triply periodic box

Numerical studies of passive scalars in three-dimensional (3D) periodic box turbulence have often used arbitrary scalar forcing schemes to sustain the variance. These existing methods represent certain flow configurations, but they have not been derived using specific velocity and scalar profiles. In this work, a forcing technique is devised to generate centerline scalar mixing of round jets in a triply periodic box. It is derived from the scalar transport equation using a Reynolds-like decomposition of the scalar field. The equation is closed by applying the known mean velocity and scalar profiles of axisymmetric jets. The result is a combination of a mean gradient term and a linear scalar term. Direct numerical simulations at different ${\text{Re}}_{\lambda }$ have been performed with these source terms for unity Schmidt numbers. Scalar flux values and scaling exponents of energy spectra from simulations are comparable to experimental values. In addition, a dimensional analysis shows that the normalized scalar statistics, such as variance, flux, and dissipation rate, should only be a function of Reynolds number; indeed, such quantities computed from our simulations approach constant values as the Reynolds number increases. The effects of velocity forcing on scalar fields are also investigated; changing velocity forcing terms may result in unstable scalar fields even under the same scalar forcing. It may indicate that an appropriate relation between the velocity and scalar forcing schemes can help producing a proper scalar mixing environment.

Figure 1

Scaling coefficients for the variance [(a), Eq. (31)], the scalar flux [(b), Eq. (32)], and the dissipation rate [(c), Eq. (33)] from DNS1-5. The dashed lines are weighted least-squares fits of a functional form $h\left({\text{Re}}_{\lambda }\right)=a_1-a_{2}exp(-a_3{\text{Re}}_{\lambda })$.

Figure 2

Numerical dissipation defined as ${\alpha }_{\mathrm{num}}\equiv 2({\alpha }_1+{\alpha }_2)-{\alpha }_3$ for DNS1-5.

Figure 3

Contours of scalar fields from DNS2 in $xy$ (top) and $yz$ (bottom) planes with three different forcing methods: the linear scalar (LS) forcing (left), the jet centerline (JC) forcing (middle), and the mean gradient (MG) forcing (right).

Figure 4

Normalized scalar flux $\frac{\langle \langle u_x^{*}{C}^{*}\rangle \rangle }{\sqrt{\langle \langle {u_x^{*}}^2\rangle \rangle \langle \langle {C^{*}}^2\rangle \rangle }}$ and its corresponding experimental quantity. Round jet experiments with $\text{Sc}\approx 1$ are from Anderson and Bremhorst [23], Darisse et al. [24], and Chevray and Tutu [32], from low to high Reynolds numbers. Round jet experiments with $\text{Sc}\gg 1$ are from Webster et al. [21] and Antoine et al. [33], from low to high Reynolds numbers. The figure displays DNS1-5 results with the original forcing terms, $\frac{U_c}{x_o}{C}^{*}+\frac{C_c}{x_o}{u}_x^{*}$ (red triangles); with only the linear scalar (LS) term, $\frac{U_c}{x_o}{C}^{*}$ (black triangles); and with only the mean gradient (MG) term, $\frac{C_c}{x_o}{u}_x^{*}$ (blue triangles). The mean of DNS1-5 values with the original forcing terms is 0.52, shown as the dashed red line.

Figure 5

Scalar energy spectrum: $\varphi \left({\kappa }_1\right)$ computed from DNS5 at ${\mathrm{Re}}_{\lambda }=129$, and its least-squares fit result with model spectrum, Eq. (40).

Figure 6

Scaling exponent $n$ for the scalar energy spectra, $\varphi \sim {\kappa }^{-n}$, calculated from DNS1-6. Experimental values are taken from two round jets [34, 35] and other shear flows [36]. The dashed line is a fit provided by Ref. [37]. The dash-dotted line shows $n=5/3$.

Figure 7

Timescale ratio $r_{\chi }$ [Eq. (43)] from DNS1-5, Iso1-2, and Mag1-2. The mean of DNS1-5 values is shown as the dashed red line.

Figure 8

Normalized scalar flux from DNS1-5, Iso1-2, and Mag1-2. The mean of DNS1-5 values is shown as the dashed red line.

Figure 9

Volume-averaged ratio, $2\frac{U_c}{x_o}\frac{\langle k\rangle }{\langle \varepsilon \rangle }$, from DNS1, Iso1-2, and Mag1-2, plotted as a function of the time normalized by the eddy timescale, $\tau =k_o/{\varepsilon }_o$. The horizontal dashed line indicates the time-averaged value of each simulation.

Figure 10

Volume-averaged scalar flux, $\langle u_x^{*}{C}^{*}\rangle$, from DNS2 plotted as a function of the time normalized by the eddy timescale, $\tau$. The horizontal dashed line indicates its time-averaged value.

Figure 11

Averaging error ${\widehat{\delta }}^2$ as a function of $n$, for $\langle u_x^{*}{C}^{*}\rangle$ data presented in Fig. 10. The solid line represents the least-squares fit result with ${\widehat{\delta }}^2$ for $n\ge 920$: ${\widehat{\delta }}^2=0.00309/n-0.911/n^2$.

Figure 12

$f\left({N;\mathit{B}|}_{n_k}\right)$ from Eq. (A10) is computed as a function of $n_k$.

Figure 13

The plot for timescale ratio [Eq. (43)] reproduced using summations of uncertainties for four parameters, $\langle \chi \rangle , \langle {C^{*}}^2\rangle , \langle \varepsilon \rangle$, and $\langle k\rangle$.