Stochastic dynamical model for space-time energy spectra in turbulent shear flows

Space-time energy spectra describe the distribution of energy density over space and timescales, which are fundamental to studying dynamic coupling at spatial and temporal scales and turbulence-generated noise. The present paper develops a dynamic autoregressive (DAR) random forcing model for space-time energy spectra in turbulent shear flows. This model includes the two essential mechanisms of statistical decorrelation: the convection proposed by Taylor's model and the random sweeping proposed by the Kraichnan-Tennekes model. The new development is that DAR random forcing is introduced to represent the random sweeping effect. The resulting model can correctly reproduce the convection velocity and spectral bandwidths, while a white-in-time random forcing model makes erroneous predictions on spectral bandwidths. The DAR model is further combined with linear stochastic estimation (LSE) to reconstruct the near-wall velocity fluctuations of the desired space-time energy spectra. Direct numerical simulation of turbulent channel flows is used to validate the DAR model and evaluate the Werner-Wengle wall model and the LSE approach. Both the wall model and LSE incorrectly estimate the spectral bandwidths.

Figure 1

Comparison of the space-time energy spectra obtained from the DNS and the WW wall model for the streamwise velocity fluctuations in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) Contours obtained from the DNS at $y_W^{+}=5$ (colored shades with solid lines) and the WW wall model at $y_W^{+}=5$ (dashed lines). (b) Contours obtained from the DNS at $y_O^{+}=92$ (colored shades with solid lines) and the WW wall model at $y_W^{+}=5$ (dashed lines).

Figure 2

(a) Wave-number-dependent convection velocities and (b) spectral bandwidths for the streamwise velocity fluctuations in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. The results from the DNS at $y_W^{+}=5$ are denoted by red solid lines, the results from the DNS at $y_O^{+}=92$ are denoted by blue dashed lines and the results from the WW model at $y_W^{+}=5$ are denoted by green dash-dotted lines.

Figure 3

Comparisons of the (a) space-time energy spectra, (b) convection velocities, and (c) spectral bandwidths obtained from the DNS and LSE for the streamwise velocity fluctuations in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) The contours from the DNS at $y_W^{+}=5$ are denoted as colored shades with solid lines and the results from LSE at $y_W^{+}=5$ are denoted as dashed lines. (b,c) The results from the DNS at $y_W^{+}=5$ are denoted by red solid lines, the results from the DNS at $y_O^{+}=92$ are denoted by blue dashed lines and the results from the LSE at $y_W^{+}=5$ are denoted by green dash-dotted lines.

Figure 4

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 5

(a) Wave-number-dependent convection velocities and (b) spectral bandwidths for the streamwise velocity fluctuations at $y^{+}=270$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. The results from the DNS are denoted by red solid lines, and the results from the DAR model are denoted by green dash-dotted lines.

Figure 6

Comparison of the space-time correlations obtained from the DNS and the DAR model for 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 correlations at three different temporal separations obtained from the DNS (colored solid lines) and the DAR model (dashed lines with dots).

Figure 7

Temporal evolutions of the streamwise velocity fluctuations at $y_W^{+}=5$ in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. (a) The DNS results. (b) The LSE results. (c) The results from the DAR model. (d) The results from the DAR model with LSE.

Figure 8

Scatter plot of the normalized temporal derivatives $-(h/U_b^2){\partial }_{t}u$ vs. the normalized spatial derivatives $(h/U_b){\partial }_{x}u$ for $y_W^{+}=5$ at ${\mathrm{Re}}_{\tau }=550$. (a) The DNS result. (b) The result from the DAR model with LSE.

Figure 9

Comparison of the space-time energy spectra obtained from the DNS and the DAR model with LSE for the streamwise velocity fluctuations at $y_W^{+}=5$ in turbulent channel flows. (a) ${\mathrm{Re}}_{\tau }=550$. Contours obtained from the DNS (colored shades with solid lines) and the combination model (dashed lines with dots). (b) ${\mathrm{Re}}_{\tau }=180$. Contours obtained from the DNS (colored shades with solid lines) and the combination model (dashed lines with dots). (c) ${\mathrm{Re}}_{\tau }=550$. Cuts through space-time energy spectra at three different wave numbers obtained from the DNS (colored solid lines) and the combination model (dashed lines). (d) ${\mathrm{Re}}_{\tau }=180$. Cuts through space-time energy spectra at three different wave numbers obtained from the DNS (colored solid lines) and the combination model (dashed lines).

Figure 10

The Hellinger distance between the model and the DNS results at ${\mathrm{Re}}_{\tau }=550$ and ${\mathrm{Re}}_{\tau }=180$.

Figure 11

(a) Wave-number-dependent convection velocities and (b) spectral bandwidths for the streamwise velocity fluctuations in turbulent channel flows at ${\mathrm{Re}}_{\tau }=550$. The results from the DNS at $y_W^{+}=5$ are denoted by red solid lines, the results from the DNS at $y_O^{+}=92$ are denoted by blue dashed lines, and the results from the DAR model with LSE at $y_W^{+}=5$ are denoted by green dash-dotted lines.

Figure 12

(a) Wave-number-dependent convection velocities and (b) spectral bandwidths for the streamwise velocity fluctuations in turbulent channel flows at ${\mathrm{Re}}_{\tau }=180$. The results from the DNS at $y_W^{+}=5$ are denoted by red solid lines, the results from the DNS at $y_O^{+}=80$ are denoted by blue dashed lines and the results from the DAR model with LSE at $y_W^{+}=5$ are denoted by green dash-dotted lines.

Figure 13

Comparison of the space-time correlations obtained from the DNS and the DAR model with LSE for streamwise velocity fluctuations at $y_W^{+}=5$ in turbulent channel flows. (a) ${\mathrm{Re}}_{\tau }=550$. Contours obtained from the DNS (colored shades with solid lines) and the DAR model with LSE (dashed lines with dots). (b) ${\mathrm{Re}}_{\tau }=180$. Contours obtained from the DNS (colored shades with solid lines) and the DAR model with LSE (dashed lines with dots). (c) ${\mathrm{Re}}_{\tau }=550$. Cuts through space-time correlations at three different temporal separations obtained from the DNS (colored solid lines) and the DAR model with LSE (dashed lines with dots). (d) ${\mathrm{Re}}_{\tau }=180$. Cuts through space-time correlations at three different temporal separations obtained from the DNS (colored solid lines) and the DAR model with LSE (dashed lines with dots).