Space-time energy spectra in turbulent shear flows

This article presents a review and perspectives on the models for space-time energy spectra in turbulent shear flows. The Taylor, Kraichnan-Tennekes, and elliptic approximation (EA) models are re-examined in terms of the picture of turbulent passage, which is proposed by Taylor's frozen-flow hypothesis and the Kraichnan-Tennekes random sweeping hypothesis; the stochastic dynamic models for reproduction of space-time energy spectra, such as dynamic autoregression model, are discussed; and the statistical models for reconstruction of space-time energy spectra from incomplete data sets in experimental measurements are revisited. We present three distinct approaches of successive approximation for developing the models of space-time energy spectra and use the conditional moments of energy distribution to characterize space-time energy spectra, such as propagation velocities and spectral bandwidths.

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Figure 1

Space-time energy spectrum $\mathrm{\Phi }(k_x,\omega )$ of the streamwise velocity at $y^{+}=92$ obtained from DNS of turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) The colored surface shows the energy density in the frequency wave-number domain. The green lines denote the cuts of the surface with a fixed energy density. The red solid lines denote the isospectral contours that are approximately ellipses. The blue dashed line indicates the preferential direction of the contours, and its slope is related to the convection velocity. The blue solid line shows the spatial energy spectrum at a fixed frequency. For comparison, the space-time energy spectrum predicted by Taylor's model is represented by the gray curve, which exhibits vanishing spectral bandwidths. (b) The spatial energy spectrum at a fixed frequency (blue solid line) shown in panel (a) is plotted against wave numbers. The red dashed line denotes its mean wave number. The black solid line with arrows denotes its spectral bandwidth.

Figure 2

Schematic representation of turbulent passage and the space-time correlation models. Turbulent passage is pictured as the downstream movement of flow patterns at a propagation speed with a certain distortion. In the EA model, space-time correlations are determined by the propagation and distortion of flow patterns. The former is represented by the propagation velocity $U$ and the latter is represented by the sweeping velocity $V$. The EA model becomes Taylor's model if $V$ is zero and becomes the Kraichnan-Tennekes random sweeping model if $U$ is zero.

Figure 3

Comparison of the space-time energy spectra obtained from the DNS and the DAR model for the streamwise velocity fluctuations at $y^{+}=270$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) Contours obtained from the DNS (colored shades with solid lines) and the DAR model (dashed lines with dots). (b) Cuts through space-time energy spectra at three different wave numbers obtained from the DNS (colored solid lines) and the DAR model (dashed lines). Figure adapted from Ref. [27].

Figure 4

Comparison of the space-time energy spectra obtained from the DNS and the DAR model with LSE for the streamwise velocity fluctuations at $y^{+}=5$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) Contours obtained from the DNS (colored shades with solid lines) and the DAR model (dashed lines with dots). (b) Cuts through space-time energy spectra at three different wave numbers obtained from the DNS (colored solid lines) and the DAR model (dashed lines). Figure adapted from Ref. [27].

Figure 5

Temporal evolutions and wave-number-dependent convection velocities of the streamwise velocity fluctuations at $y^{+}=5$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. Top: The LSE results. Middle left: Subtraction of LSE from DNS. Middle right: The DAR model. Bottom left: The DNS results. Bottom right: The DAR model with LSE.

Figure 6

Comparison of the space-time energy spectra obtained from the DNS and the LMW model for the streamwise velocity fluctuations at $y^{+}=92$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) Contours obtained from the DNS (colored shades with solid lines) and the LMW model (dashed lines with dots). (b) Cuts through space-time energy spectra at three different frequencies obtained from the DNS (colored solid lines) and the LMW model (dashed lines).