Energy transfer and dissipation in forced isotropic turbulence

A model for the Reynolds-number dependence of the dimensionless dissipation rate $C_{\varepsilon }$ was derived from the dimensionless Kármán-Howarth equation, resulting in $C_{\varepsilon }=C_{\varepsilon ,\infty }+C/R_L+O(1/R_L^2)$, where $R_L$ is the integral scale Reynolds number. The coefficients $C$ and $C_{\varepsilon ,\infty }$ arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to $R_L=5875$ ($R_{\lambda }=435$), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law $R_L^n$ with exponent value $n=-1.000\pm 0.009$ and that this decay of $C_{\varepsilon }$ was actually due to the increase in the Taylor surrogate $U^3/L$. The model equation was fitted to data from the DNS, which resulted in the value $C=18.9\pm 1.3$ and in an asymptotic value for $C_{\varepsilon }$ in the infinite Reynolds-number limit of $C_{\varepsilon ,\infty }=0.468\pm 0.006$.

Figure 1

Variation of the dimensionless dissipation coefficient $C_{\varepsilon }$ with Taylor-Reynolds number $R_{\lambda }$ from our DNSs. Other investigations of forced turbulence are presented for comparison. The black line is a fit of the expression (44) to our data only.

Figure 2

Variation with Taylor-Reynolds number of the dissipation rate $\varepsilon$, maximum inertial transfer rate ${\varepsilon }_T$, and Taylor surrogate $U^3/L$, all scaled on the injection rate ${\varepsilon }_W$. The line corresponds to the fitted line in Fig. 1.

Figure 3

Dimensionless energy balance in the KHE, as expressed by Eq. (27). $R_{\lambda }=276$. The Taylor microscale is labeled for comparison. Note that the energy input is constant for scales $r<\lambda$.

Figure 4

Graph of the present DNS results for $C_{\varepsilon }$ against Reynolds number, once the estimate of the asymptote is subtracted. The effect of varying our estimate of the asymptote $C_{\varepsilon ,\infty }$ is shown by the three different symbols, where $C_{\varepsilon ,\infty }^{\prime }$ and $C_{\varepsilon ,\infty }^{\prime \prime }$ denote variations in the asymptote within one standard error. The dashed lines represent fits of the expression $C{R}_L^n$ to the data after subtracting the respective values of the asymptote.

Figure 5

The same data as in Fig. 4 plotted on logarithmic scales. The solid line represents a slope of $n=-1.000\pm 0.009$, obtained from a one-parameter fit of the expression $C{R}_L^n$ to the data points, after subtracting the asymptote $C_{\varepsilon ,\infty }=0.468$.

Figure 6

The expression given in Eq. (42) fitted to present DNS data resulting in $C_{\varepsilon ,\infty }=0.468$ and $C=18.9$.