Scaling of turbulence intensities up to ${\text{Re}}_{\tau }={10}^6$ with a resolvent-based quasilinear approximation

A minimal form of quasilinear approximation (QLA), recently proposed with a stochastic forcing and proper orthogonal decomposition modes [Hwang and Ekchardt, J. Fluid Mech. 894, A23 (2020)], has been extended by employing a resolvent framework. A particular effort is made to reach an extremely high Reynolds number by carefully controlling the approximation without loss of the general scaling properties in the spectra, while setting out the main limitations and accuracy of the proposed QLA with possibility of further improvement. The QLA is subsequently applied to turbulent channel flow up to ${\text{Re}}_{\tau }={10}^6$ (${\text{Re}}_{\tau }$ is the friction Reynolds number). While confirming that the logarithmic wall-normal dependence in streamwise and spanwise turbulence intensities robustly appears, it reveals some nontrivial difference from the scaling of the classical attached eddy model based on inviscid flow assumption. First, the spanwise wave number spectra do not show any clearly visible inverse-law behavior due to the viscous wall effect prevailing in a significant portion of the lower part of the logarithmic layer. Second, the near-wall peak streamwise and spanwise turbulence intensities are found to deviate from $ln{\text{Re}}_{\tau }$ scaling for ${\text{Re}}_{\tau }\gtrsim {10}^4$. Importantly, the near-wall streamwise turbulence intensity is inversely proportional to $1/U_{cl}^{+}$ ($U_{cl}^{+}$ is the inner-scaled channel centreline velocity), consistent with the scaling obtained from an asymptotic analysis of the Navier-Stokes equations [Monkewitz and Nagib, J. Fluid Mech. 783, 474 (2015)]. The same behavior was also observed for the streamwise turbulence intensity in the logarithmic region, as was predicted with the asymptotic analysis. Finally, the streamwise turbulence intensity in the logarithmic region is found to become greater than the near-wall one at ${\text{Re}}_{\tau }\simeq O\left({10}^5\right)$. It is shown that this behavior originates from the near-wall spectra associated with large-scale inactive motions, the intensity of which gradually decays as ${\text{Re}}_{\tau }\to \infty$.

Figure 1

The weight function $W\left(k_z\right)$ and the preconditioned one ${\tilde{W}}_z\left(k_z\right)$ in Eq. (16): (a), (b) optimal and (c), (d) broadband forcing.

Figure 2

Second-order turbulence statistics for optimal/broadband QLAs at ${\text{Re}}_{\tau }=5000$ and for DNS [10] at ${\text{Re}}_{\tau }=5186$: (a) $-\overline{u^{\prime }{v}^{\prime }}/u_{\tau }^2$; (b) $u_{\mathrm{rms}}^{\prime }/u_{\tau }$; (c) $v_{\mathrm{rms}}^{\prime }/u_{\tau }$; (d) $w_{\mathrm{rms}}^{\prime }/u_{\tau }$. Here DNS (—), optimal forcing QLA with $N=2$ ($---$), $N=8$ ($-·-$), and $N=16$ ($···$), and broadband forcing QLA with $N=2$ ($---$), $N=8$ ($-·-$), and $N=16$ ($···$).

Figure 3

Premultiplied spanwise wave number spectra of wall-normal velocity from (a) DNS [10] at ${\text{Re}}_{\tau }=5186$ and from (b)–(d) the resolvent QLA at ${\text{Re}}_{\tau }=5\times {10}^3$. In (b) $N=16$; (c) $N=8$; (d) $N=2$. The contours are normalized by their maximum value.

Figure 4

Premultiplied spanwise wave number spectra at ${\text{Re}}_{\tau }=5\times {10}^3$ from (a), (d), (g) resolvent QLA with optimal forcing and two leading resolvent modes ($N=2$), from (b), (e), (h) stochastic QLA with two leading POD modes [33] and from (c), (f), (i) DNS [10] (at ${\text{Re}}_{\tau }=5186$): (a)–(c) streamwise velocity, (d)–(f) wall-normal velocity, (g)–(i) spanwise velocity.

Figure 5

Second-order turbulence statistics for optimal QLA, time-domain QLA [33] both at ${\text{Re}}_{\tau }=5\times {10}^3$ and DNS [10] at ${\text{Re}}_{\tau }=5186$: (a) $-\overline{u^{\prime }{v}^{\prime }}/u_{\tau }^2$, (b) $u_{\mathrm{rms}}^{\prime }/u_{\tau }$, (c) $v_{\mathrm{rms}}^{\prime }/u_{\tau }$, (d) $w_{\mathrm{rms}}^{\prime }/u_{\tau }$. Here DNS (—), optimal $N=2$ ($---$), stochatsic $N_{\mathrm{POD}}=2$ ($---$).

Figure 6

Premultiplied spanwise wave number spectra from (a), (d), (g), (j) DNS [10] and (b), (c), (e), (f), (h), (i), (k), (l) resolvent QLA in the inner-scaled coordinates: (a)–(c) streamwise velocity, (d)–(f) Reynolds shear stress, (g)–(i) wall-normal velocity, (j)–(l) spanwise velocity. Here ${\text{Re}}_{\tau }=1001,1995,5186$ for DNS, ${\text{Re}}_{\tau }={10}^3,2\times {10}^3,5\times {10}^3$ for (b), (e), (h), (k) and ${\text{Re}}_{\tau }={10}^4,{10}^5,{10}^6$ for (c), (f), (i), (l). Here the spectra are normalized by their maximum value, and the contour level is spaced with 20% of the maximum.

Figure 7

Premultiplied spanwise wave number spectra from (a), (d), (g), (j) DNS [10] and (b), (c), (e), (f), (h), (i), (k), (l) resolvent QLA in the outer-scaled coordinates: (a)–(c) streamwise velocity, (d)–(f) Reynolds shear stress, (g)–(i) wall-normal velocity, (j)–(l) spanwise velocity. Here ${\text{Re}}_{\tau }=1001,1995,5186$ for DNS ${\text{Re}}_{\tau }={10}^3,2\times {10}^3,5\times {10}^3$ for (b), (e), (h), (k), and ${\text{Re}}_{\tau }={10}^4,{10}^5,{10}^6$ for (c), (f), (i), (l). Here the spectra are normalized by their maximum value, and the contour level is spaced with 20% of the maximum.

Figure 8

Premulitplied spanwise wave number spectra of streamwise velocity in the (a), (c), (e) $h$- and (b), (d), (f) $y$-scaled wave number coordinates: (a), (b) the resolvent QLA at ${\text{Re}}_{\tau }={10}^6$ and (c), (d) at ${\text{Re}}_{\tau }=5\times {10}^3$; (e), (f) DNS at ${\text{Re}}_{\tau }=5186$ [10]. In panels (a) and (b) $y/h=0.0008,0.001,0.002,0.005,0.01$ ($y^{+}\simeq 800,1000,2000,5000,10000$), whereas in panels (c)–(f) $y/h=0.04,0.06,0.08,0.1$ ($y^{+}\simeq 200,300,400,500$).

Figure 9

Reynolds stress profiles from (a), (c), (e), (g) DNS and (b), (d), (f), (h) resolvent QLA in the inner-scaled wall-normal coordinate. Here ${\text{Re}}_{\tau }=543,1001,1995,5186$ for DNS, and ${\text{Re}}_{\tau }={10}^3,2\times {10}^3,5\times {10}^3,{10}^4,2\times {10}^4,{10}^5,2\times {10}^5,5\times {10}^5,{10}^6$ for QLA.

Figure 10

Reynolds normal stress profiles from (a), (c), (e), (g) DNS and (b), (d), (f), (g) resolvent QLA in the outer-scaled wall-normal coordinate. Here ${\text{Re}}_{\tau }=543,1001,1995,5186$ for DNS and ${\text{Re}}_{\tau }={10}^3,2\times {10}^3,5\times {10}^3,{10}^4,2\times {10}^4,{10}^5,2\times {10}^5,5\times {10}^5,{10}^6$ for QLA.

Figure 11

Growth of the near-wall streamwise turbulence intensity ($y^{+}=30$) with (a) $ln{\text{Re}}_{\tau }$ (b) $1/U_{cl}^{+}$ where $U_{cl}^{+}$ denotes the mean centreline velocity. In panels (a) and (c), the red straight lines indicate a fit in the form of $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2=a_1+b_{1}ln\left({\text{Re}}_{\tau }\right)$ where constants $a_1$ and $b_1$ are obtained from the data at ${\text{Re}}_{\tau }=1\times {10}^3,2\times {10}^3$. In panels (b) and (d) the red straight lines indicate a fit in the form of $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2=a_2+b_2/U_{cl}^{+}$ where constants $a_2$ and $b_2$ are obtained from the data at ${\text{Re}}_{\tau }=5\times {10}^5,2\times {10}^5$. In panels (e) and (f) the near-wall streamwise turbulence intensity is investigated at $y^{+}=y_{\max }^{+}$, where $y_{\max }^{+}$ represents the wall-normal location of the peak streamwise turbulent intensity, for $1\times {10}^3\le {\text{Re}}_{\tau }\le 2\times {10}^4$ and $y^{+}=30$ for $1\times {10}^5\le {\text{Re}}_{\tau }\le 1\times {10}^6$ in order to account for the emergence of the outer peak at large ${\text{Re}}_{\tau }$.

Figure 12

Behavior of streamwise turbulence intensity in the log-layer for $y/h=0.05$ (—), $y/h=0.1$ ($---$), and $y/h=0.2$ ($·$ - $·$). The colored lines represent linear fits given in the form of $\overline{u^{\prime }{u}^{\prime }}/u_{\tau }^2=a_2+b_2/U_{cl}^{+}$ where constants $a_2$ and $b_2$ are obtained from the data at ${\text{Re}}_{\tau }=5\times {10}^5,2\times {10}^5$.

Figure 13

Premultiplied spanwise wave number spectra of streamwise velocity at (a) $y/h=0.05$ and (b) $y/h=0.1$ for ${\text{Re}}_{\tau }={10}^4,{10}^5,{10}^6$.

Figure 14

Premultiplied spanwise wave number spectra of streamwise velocity at $y^{+}=30$ for ${\text{Re}}_{\tau }={10}^4,{10}^5,{10}^6$.

Figure 15

Sensitivity to the optimization parameter, $p$, used in order to mitigate the sharp decay in $k_z$ of the singular values of the resolvent operator. Here ${\epsilon }_1$ ($\square$) refers to the $Q$-norm error, ${‖\overline{u^{\prime }{v}^{\prime }}-\langle u^{\prime }{v}^{\prime }\rangle ‖}_Q$ defined in Sec. 2b and ${\epsilon }_2$ ($\circ$) denotes the standard ${\mathcal{L}}_2$-norm error.

Figure 16

Sensitivity of the modified and original weight functions, defined in Eq. (16), as $p$ is varied. Here $p=0.1$ (—), $p=0.2$ ($---$), $p=0.3$ ($·$ - $·$), $p=0.4$ ($···$), $p=0.5$ ($---$), $p=0.6$ ($·$ - $·$).

Figure 17

Energy contents of the resolvent modes for $k_x=0$ and $\omega =0$: (a) the contribution of the first 10 modes (${\sigma }_j^2/V$) to the energy of all modes ($V={\sum }_{\forall j}{\sigma }_j^2$) for ${\lambda }_z=3.5h$ ($•$) and ${\lambda }_z^{+}=80$ ($\circ$) at ${\text{Re}}_{\tau }=5\times {10}^3$. In panels (b) and (c) the contribution of the two most energetic modes is investigated over (b) the outer- and (c) inner-scaled spanwise wavelengths for ${\text{Re}}_{\tau }=5\times {10}^3$ (—), ${\text{Re}}_{\tau }=2\times {10}^4$ ($---$), and ${\text{Re}}_{\tau }=5\times {10}^5$ ($·$ - $·$).