Use of eddy viscosity in resolvent analysis of turbulent channel flow

The predictions obtained from resolvent analysis with and without an eddy viscosity model for turbulent channel flow at ${\text{Re}}_{\tau }=550$ are compared to direct numerical simulation data to identify the scales and wave speeds for which resolvent analysis provides good predictions. The low-rank behavior of the standard resolvent identifies energetic regions of the flow whereas the eddy resolvent is low rank when the resulting projection of the leading eddy resolvent mode onto the leading mode from spectral proper orthogonal decomposition is maximum. The highest projections are obtained for structures that are associated with the near-wall cycle and structures that are energetic at $z=\pm 0.5$. It is argued that these types of structures are likely to be correctly predicted for any friction Reynolds number due to the inner and outer scaling of the Cess eddy viscosity profile. The eddy resolvent also correctly identifies the most energetic wave speed for these two scales. For all other scales, neither analysis reliably predicts the most energetic wave speed or mode shapes. The standard resolvent tends to overestimate the most energetic wave speed while the eddy resolvent underestimates it. The resulting eddy resolvent modes are overly “attached” to the wall since the wall-normal gradient of the eddy viscosity overestimates the transport of energy towards the wall. These observations have direct implications for future work towards estimating turbulent channel flows using resolvent analysis and suggest that the Cess profile can be further optimized for individual scales to provide better low-order models of turbulent channel flows.

Figure 1

(a) Mean velocity profile and (b) Reynolds stress profiles from the present study (solid lines) and Ref. [39] (open squares).

Figure 2

Low-rank maps for (a) SPOD, (b) the standard resolvent, and (c) the eddy resolvent for a fixed wave speed of $c^{+}=18.9$. Contours of the turbulent kinetic energy spectrum at $z=-0.5$ are denoted in black.

Figure 3

Projection of the leading SPOD mode onto the leading pair of resolvent modes for various wave speeds. Contours of the turbulent kinetic energy spectrum at the wall-normal location where $c^{+}=U^{+}$ are denoted in black.

Figure 4

Projection of the leading SPOD mode onto the leading pair of eddy modes for various wave speeds. Contours of the turbulent kinetic energy spectrum at the wall-normal location where $c^{+}=U^{+}$ are denoted in black.

Figure 5

Most energetic wave speed for $(k_x,k_y)=(4,30)$ as a function of different strengths of the eddy viscosity as denoted by $\mathcal{S}$.

Figure 6

Wave speed corresponding to the maximum ${\lambda }_1$ computed from (a) DNS compared to the most amplified wave speed predicted by the (b) standard resolvent and (c) eddy resolvent.

Figure 7

Difference between the most amplified wave speed predicted by the (a) standard resolvent and (b) eddy resolvent compared to the wave speed corresponding to the maximum ${\lambda }_1$ computed from DNS.

Figure 8

The (a) streamwise, (b) spanwise, and (c) wall-normal components of the leading SPOD, resolvent, and eddy modes for mode 0 for which $\mathit{k}=(0,4,\omega =0)$.

Figure 9

The (a) streamwise, (b) spanwise, and (c) wall-normal components of the leading SPOD, resolvent, and eddy modes for mode 1 for which $\mathit{k}=(1,2,18.5)$.

Figure 10

The (a) streamwise, (b) spanwise, and (c) wall-normal components of the leading SPOD, resolvent, and eddy modes for mode 2 for which $\mathit{k}=(2,8,16.8)$.

Figure 11

The (a) streamwise, (b) spanwise, and (c) wall-normal components of the leading SPOD, resolvent, and eddy modes for Mode 3 for which $\mathit{k}=(4,30,10.1)$.

Figure 12

Wall-normal profiles of (a) ${\text{Re}}_{\tau }{\nu }_T$ and (b) ${\nu }_T$ for various ${\text{Re}}_{\tau }$.

Figure 13

Mode shapes for $\mathit{k}=(1,2,U_{\mathrm{CL}}^{+}-2)$ from (a) standard resolvent analysis, (b) eddy resolvent analysis, and (c) eddy resolvent analysis setting ${\text{Re}}_T=12.5$. The Reynolds numbers range from ${\text{Re}}_{\tau }=180$ up to ${\text{Re}}_{\tau }=20000$.

Figure 14

Mode shapes for $(k_x^{+},k_y^{+},c^{+})=(2\pi /1000,2\pi /100,10)$ from (a) standard resolvent analysis, (b) eddy resolvent analysis, and (c) eddy resolvent analysis setting ${\text{Re}}_T=12.5$. The Reynolds numbers range from ${\text{Re}}_{\tau }=180$ up to ${\text{Re}}_{\tau }=20000$.

Figure 15

Wall-normal profiles of the eddy viscosity gradient ${\nu }_t^{\prime }$ for various ${\text{Re}}_{\tau }$.

Figure 16

(a) Time-averaged nonlinear transfer from DNS for $(k_x,k_y)=(0,4)$ compared to (b) $\widehat{Edd}\left(z\right)$ and (c) its components for $\omega =0$.

Figure 17

(a) Time-averaged nonlinear transfer from DNS for $(k_x,k_y)=(4,30)$ compared to (b) $\widehat{Edd}\left(z\right)$ and (c) its components for $c^{+}=10$.

Figure 18

Leading eddy modes for $\mathit{k}=(0,4,0)$ for various strengths of the eddy viscosity gradient $\mathcal{G}$.

Figure 19

Leading eddy modes for $\mathit{k}=(1,2,18.5)$ for various strengths of the eddy viscosity gradient $\mathcal{G}$.

Figure 20

Comparison of the mean velocity, eddy viscosity, and wall-normal of the eddy viscosity profiles from DNS to the Cess profiles.

Figure 21

Comparison of the first eddy resolvent modes using the DNS and Cess eddy viscosity profiles for mode 1.