Constant-energetics control-based forcing methods in isotropic helical turbulence

To realize high Reynolds number turbulent flow in a numerical simulation, an artificial energy input is required. In this study, we propose a deterministic forcing method and a stochastic forcing method to maintain the mean turbulence kinetic energy (TKE) and helicity at ideal values. The proposed forcing methods control both the mean TKE and helicity by introducing two tuning parameters: $\alpha$ and $\beta$, where $\alpha$ is the helicity injection rate and $\beta$ is the convergence of target statistical quantities to ideal values. We carried out 24 direct numerical simulations to investigate the effects of the forcing methods, including the tuning parameters, on turbulence. When we controlled the forcing term based on the proposed methods, the temporal correlation between the TKE and energy dissipation rate was artificially suppressed in the deterministic forcing method, and for higher $\beta$ in the stochastic forcing method. No differences were found between the proposed deterministic and stochastic forcing methods in terms of one- and two-point statistics. There are also some common characteristics between the energy and helicity dissipation rates, including the normalized mean energy and helicity dissipation rates, scale-by-scale budgets, and stretched exponential form of probability density functions. The numerical results support the joint cascade scenario of energy and helicity.

Figure 1

The shape of ${\epsilon }_K^f\left(t\right)/{\epsilon }_{\infty }$.

Figure 2

The temporal evolutions of the statistical quantities for case 1. (a) TKE for nonhelical turbulence, (b) helicity for nonhelical turbulence, (c) TKE for helical turbulence, and (d) helicity for helical turbulence. The insets show the changes in mean energy and mean helicity dissipation rates. The insets in (a) and (c) show the temporal evolution of the TKE and mean energy dissipation rate. The interval between two successive circles indicates $T_{L,0}$.

Figure 3

Same as Fig. 2 but for case 2.

Figure 4

Same as Fig. 2 but for case 3.

Figure 5

Lumley's anisotropy-invariant map for velocity fluctuations in case 1. The scatter plots are shown every $T_{L,0}$. The black line represents the bounded curve given as $-I_2^f=3{\left(|I_3^f|/2\right)}^{\frac{2}{3}}$. The inset in the upper left shows Lumley's anisotropy map for vorticity fluctuations. The insets in the lower right show the changes in the anisotropy-invariant function.

Figure 6

Same as Fig. 5 but for case 2.

Figure 7

Same as Fig. 5 but for case 3.

Figure 8

(a) Normalized energy spectra and (b) normalized helicity spectra. Insets in (a) and (b) show the compensated spectra and relative helicity spectra, respectively. The dash-dotted lines in (a) and (b) show $C_K{k}_{*}^{-\frac{5}{3}}$ and $C_H{k}_{*}^{-\frac{5}{3}}$, where $C_K=1.68$ and $C_H=1.1$. The dash-dotted line in the relative helicity spectrum shows the $k^{-1}$ slope.

Figure 9

(a) Scale-by-scale energy budget, (b) scale-by-scale helicity budget, and (c) third-order structure function normalized by the Kolmogorov velocity $v_{\eta }={\left(\nu \epsilon \right)}^{\frac{1}{4}}$. The inset in (c) is the ratio between $S_3^H\left(r\right)$ and $S_3^E\left(r\right)$. The black dash-dotted lines in (a) and (b) show $4/3$; the black dash-dotted line in (c) shows $4/3r/\eta$.

Figure 10

Probability density functions of the energy dissipation rates for the (a) nonhelical case and (b) helical case, and those of the enstrophy for the (c) nonhelical case and (d) helical case. The black solid lines in the insets fit the curves (28).

Figure 11

Probability density functions of $2\nu R_{ij}{S}_{ji}$ for the (a) nonhelical case and (b) helical case; those of $\nu h_s$ for the (c) nonhelical case and (d) helical case; and those of $\nu A_{ij}{\mathrm{\Omega }}_{ij}$ for the (e) nonhelical case and (f) helical case.

Figure 12

Volume rendering of $Q$ in (a) A2 and (b) B2.

Figure 13

Isosurface of $Q$ colored by relative helicity and superhelicity, where the red (blue) colored structure represents the positive (negative) region. $Q$ colored by (a) relative helicity in A2, (b) relative superhelicity in A2, (c) relative helicity in B2, and (d) relative helicity in B2.