Wave-number–frequency spectrum for turbulence from a random sweeping hypothesis with mean flow

We derive the energy spectrum in wave-number–frequency space for turbulent flows based on Kraichnan’s idealized random sweeping hypothesis with additional mean flow, which yields the instantaneous energy spectrum multiplied by a Gaussian frequency distribution. The model spectrum has two adjustable parameters, the mean flow velocity and the sweeping velocity, and has the property that the power-law index of the wave-number spectrum translates to the frequency spectrum, invariant for arbitrary choices of the mean velocity and sweeping velocity. The model spectrum incorporates both Taylor’s frozen-in flow approximation and the random sweeping approximation in a natural way and can be used to distinguish between these two effects when applied to real time-resolved multipoint turbulence data. Evaluated in real space, its properties with respect to space-time velocity correlations are discussed, and a comparison to the recently introduced elliptic model is drawn.

Figure 1

Streamwise wave-number–frequency spectrum for varying sweeping velocity for various ratios $U/V$ of the mean to sweeping velocity where the mean velocity is set at $U=0.01\mathrm{m}/\mathrm{s}$. Left: $U/V=1.0$, middle: $U/V=5.0$, and right: $U/V=20.0$. The units of the spectrum (contour scale bar) are $\left[E(k_z,\omega )\right]={\mathrm{m}}^3/\mathrm{s}$.

Figure 2

Streamwise wave-number–frequency spectrum for varying sweeping velocity for various ratios $U/V$ of the mean to sweeping velocity, where the sweeping velocity is set at $V=0.01\mathrm{m}/\mathrm{s}$. Left: $U/V=2.0$, middle: $U/V=0.5$, and right: $U/V=0.0$.

Figure 3

Upper panel: streamwise energy spectrum and lower panel: frequency spectrum for the choice of parameters of Fig. 1, left panel.

Figure 4

Streamwise space-time correlation evaluated from Eq. (25) in the wave-number range from 0.1 to 100.0 rad/m for three cases: left panel: $U=0.0$ and $V=0.01\mathrm{m}/\mathrm{s}$; middle panel: $U=V=0.01\mathrm{m}/\mathrm{s}$; and right panel: $U=0.01$ and $V=0.002\mathrm{m}/\mathrm{s}$. The solid black lines define the isocorrelation lines (27) of the elliptic model.