Real-Space Manifestations of Bottlenecks in Turbulence Spectra

An energy-spectrum bottleneck, a bump in the turbulence spectrum between the inertial and dissipation ranges, is shown to occur in the nonturbulent, one-dimensional, hyperviscous Burgers equation and found to be the Fourier-space signature of oscillations in the real-space velocity, which are explained by boundary-layer-expansion techniques. Pseudospectral simulations are used to show that such oscillations occur in velocity correlation functions in one- and three-dimensional hyperviscous hydrodynamical equations that display genuine turbulence.

Figure 1

Plots for the DHB Eq. (1). (a) Log-log plots of the compensated energy spectrum $k^{2}E(k)$ versus $k$ for $\alpha =2$ (black squares), $\alpha =4$ (blue filled-circles), $\alpha =8$ (green diamonds), and $\alpha =16$ (red hexagons). (b) Plots of the steady-state solution $u(x)$ for the same values of $\alpha$ as in (a); these are indistinguishable from $u^o(x)$ (thick magenta line) away from $x=\pi$. Inset: Plots of $u^d(x)$ versus $x$ around $x=\pi$. (c) Semilog plot of $A_{16}$ (red filled circles) versus $\pi -x$; the black line is the fit.

Figure 2

(a) Log-log plot of the compensated energy spectrum $k^{5/3}E(k)$ versus $k$ for the SHB equation with $\alpha =8$ and (inset) a plot of its correlation function $D_{\mathrm{SHB}}^o$ showing oscillations of wavelength ${\lambda }_8^{\mathrm{SHB}}$, which is inversely related to the wave number of the bottleneck. (b) The compensated energy spectrum $k^{5/3}E(k)$ for the 3D HNS equation ($\alpha =4$) with a bottleneck peak at wave number $K_{b,\alpha }^{\mathrm{HNS}}=40$. (c) A plot of the correlation function $D_{\mathrm{HNS}}(r)$ versus $r/\eta$, where $\eta$ is the Kolmogorov scale [16], for the 3D HNS equation. Inset: Oscillations in a plot of the function $D_{\mathrm{HNS}}^o(r)$.