Investigation of the inclination angles of wall-attached eddies for streamwise velocity and temperature fields in compressible turbulent channel flows

Townsend's attached eddy hypothesis (AEH), one of the most elegant models in incompressible wall turbulence, has been recently applied to compressible wall turbulence to explain the numerical observations and predict the scaling laws. Before a more profound extension can be established, a comprehensive investigation of the features of wall-attached eddies for streamwise velocity and temperature fields in compressible wall-bounded turbulence is required. In this work the AEH and the inner-outer interaction model [Marusic et al., Science 329, 193 (2010)] are combined to isolate the signature of attached eddies at a targeted length scale and then assess their inclination angles statistically based on the direct numerical simulation database. The inclination angle obtained in the streamwise velocity fluctuating fields, which approaches ${45}^{\circ }$ as the Reynolds number increases, shows a minor Mach-number influence within the Mach-number range included in this work. As for those in temperature fluctuations, a high statistical similarity can be seen to streamwise velocity fluctuations. This slight Mach-number effect indicates that a uniform model can be potentially developed for compressible wall-bounded turbulence with different Mach numbers.

Figure 1

Schematic diagram of a hierarchy of self-similar attached eddies in the $x\text{−}y$ plane. Four hierarchy levels are included and represented by trapezoids with different colors: the smallest eddies are in blue, and the highest ones are in purple. $y_i^{*}$ is the wall-normal distance of the grids adjacent to the wall. $y_o^{*}$ represents the wall-normal distance in the logarithmic region, and $y_o^{*}+▵y^{*}$ is the neighbor location of $y_o^{*}$ with a distance $▵y^{*}$, which is also in the logarithmic region.

Figure 2

Distance $▵y^{*}$ between two neighbor grids $y_o^{*}$ and $y_o^{*}+▵y^{*}$ in the wall-normal direction as a function of estimated eddy height $y_m^{*}=(y_o^{*}+y_o^{*}+▵y^{*})/2$.

Figure 3

Contours of LCF, ${\gamma }^2\left({\lambda }_x^{+}\right)$ as a function of the wall-normal distance $y^{*}$ and wavelength ${\lambda }_x^{+}$ computed from the streamwise velocity fluctuating $u^{″\prime }$ field in Table 1. ${\gamma }^2\left({\lambda }_x^{+}\right)$ represents the coherent degree of streamwise velocity fluctuations near the wall and at $y^{*}$ (the $y$ coordinate in the figure) in the logarithmic region. Black dotted lines indicate the region bounded by $y^{*}>100, {\lambda }_x/h=10$, and ${\lambda }_x/y=14$, which is supposed to show the self-similar property of attached eddies [32].

Figure 4

Contours of LCF, ${\gamma }^2\left({\lambda }_x^{+}\right)$ as a function of the wall-normal distance $y^{*}$ and wavelength ${\lambda }_x^{+}$ computed from the temperature fluctuating $T^{\prime }$ field in Table 1. ${\gamma }^2\left({\lambda }_x^{+}\right)$ represents the coherent degree of streamwise velocity fluctuations near the wall and at $y^{*}$ (the $y$ coordinate in the figure) in the logarithmic region. Black dotted lines indicate the region bounded by $y^{*}=100, {\lambda }_x/h=10$, and ${\lambda }_x/y=14$, which is supposed to show the self-similar property of attached eddies [32].

Figure 5

Gain factor with respect to the wavelength ${\lambda }_x^{+}$ and the corresponding impulse response function $h\left(▵x^{*}\right)=F_x^{-1}\left(H\left({\lambda }_x^{+}\right)\right)$ normalized by its maximum $h\left(▵x^{*}\right)/h{\left(▵x^{*}\right)}_{\max }$ at three selected wall-normal positions $y^{*}\approx 100$, 150, and 200 computed from the streamwise velocity fluctuating $u^{″\prime }$ field for the two highest Reynolds-number cases in Table 1. (a) $|H_L\left({\lambda }_x^{+}\right)|$ and (c) the normalized $h_L\left(▵x^{*}\right)$, which are imposed on velocity fluctuations in the logarithmic region to estimate the near-wall footprint; (b) $|H_W\left({\lambda }_x^{+}\right)|$ and (d) the normalized $h_W\left(▵x^{*}\right)$, which are imposed on near-wall velocity fluctuations to predict those in the logarithmic region.

Figure 6

Gain factor with respect to the wavelength ${\lambda }_x^{+}$ and the corresponding impulse response function $h\left(▵x^{*}\right)=F_x^{-1}\left(H\left({\lambda }_x^{+}\right)\right)$ normalized by its maximum $h\left(▵x^{*}\right)/h{\left(▵x^{*}\right)}_{\max }$ at three selected wall-normal positions $y^{*}\approx 100$, 150, and 200 computed from the temperature fluctuating $T^{\prime }$ field for the two highest Reynolds-number cases in Table 1. (a) $|H_L\left({\lambda }_x^{+}\right)|$ and (c) the normalized $h_L\left(▵x^{*}\right)$, which is imposed on velocity fluctuations in the logarithmic region to estimate the near-wall footprint; (b) $|H_W\left({\lambda }_x^{+}\right)|$ and (d) the normalized $h_W\left(▵x^{*}\right)$, which are imposed on near-wall velocity fluctuations to predict those in the logarithmic region.

Figure 7

Streamwise inclination angles ${\alpha }_{h_L}$ defined by Eq. (17) computed based on the peak of the impulse response function $h_L$ at three selected wall-normal positions $y^{*}\approx 100$, 150, and 200 using the two highest Reynolds-number cases in Table 1. (a) ${\alpha }_{h_L}$ in streamwise velocity fluctuations $u^{″\prime }$; (b) the same in temperature fluctuations $T^{\prime }$.

Figure 8

Comparison of premultiplied energy spectra of the original field and the attached eddies at a given scale in both the streamwise velocity $u^{″\prime }$ and temperature $T^{\prime }$ fluctuations of case M15R20K in the near-wall region and logarithmic region ($y^{*}=200$), where ${\tau }_w^{″\prime }$ and $q_w^{\prime }$ are the wall shear stress and wall heat flux, respectively. (a) Spectra from the streamwise velocity fluctuation and (b) results from the temperature fluctuations. Note that some energy spectra in (a) and (b) are amplified by a factor given in the legend since they are too small to present. To have a closer look at the relative energy distribution, the energy spectra are normalized by their maximum: (c) those from the streamwise velocity fluctuation; (d) those from the temperature fluctuations.

Figure 9

(a) Normal Reynolds stress ${\widetilde{u^{″}{}^2}}^{*}$ and ${\overline{u_{o,W}^{″\prime }{}^2}}^{+}$ vs $y_o/h$; (b) indicator $\mathrm{\Xi }$ defined in Eq. (19) evaluating the logarithmic dependence of ${\widetilde{u^{″}{}^2}}^{*}$ and ${\overline{u_{o,W}^{″\prime }{}^2}}^{+}$ on $y_o/h$. The green dotted lines present a linear fit of ${\overline{u_{o,W}^{″\prime }{}^2}}^{+}$ vs $y_o/h$ as a function of $y_o/h$, which has a slope of approximately 0.34. The red scatters present the results of case M08R17K, and the blue ones present those of case M15R20K.

Figure 10

(a) RMS of temperature fluctuations $T_{rms}^{\prime +}$ and ${T_{o,W,rms}^{\prime }}^{+}$ vs $y_o/h$; (b) the indicator ${\mathrm{\Xi }}_T$ following Eq. (19) with ${\widetilde{u^{″}{}^2}}^{*}$ replaced by $T_{rms}^{\prime +}$ or ${T_{o,W,rms}^{\prime }}^{+}$ evaluating their logarithmic dependence on $y_o/h$. The red scatters present the results of case M08R17K, and the blue ones present those of case M15R20K.

Figure 11

(a) Autocorrelation coefficient $R_{▵u_{o,W}^{″\prime }▵u_{o,W}^{″\prime }}\left(▵x^{*}\right)$ as a function of $▵x^{*}$ at three selected wall-normal positions $y_m^{*}\approx 100,150$, and 200 from cases M08R17K and M15R20K, which evaluates the scale of signatures in the logarithmic region induced by attached eddies of a given scale; the black dashed line in (a) denotes $R_{▵u_{o,W}^{″\prime }▵u_{o,W}^{″\prime }}=0.05$. (b) $▵x_{0.05}^{*}$ where $R_{▵u_{o,W}^{″\prime }▵u_{o,W}^{″\prime }}\left(▵x_{0.05}^{*}\right)\approx 0.05$ with respect to $y_m^{*}$. The black dashed line in (b) is $▵x_{0.05}^{*}=7.8·y_m^{*}$.

Figure 12

(a) Autocorrelation coefficient $R_{▵T_{o,W}^{\prime }▵T_{o,W}^{\prime }}\left(▵x^{*}\right)$ as a function of $▵x^{*}$ at three selected wall-normal positions $y_m^{*}\approx 100,150$, and 200 from cases M08R17K and M15R20K, which evaluates the scale of signatures in the logarithmic region induced by attached eddies of a given scale; the black dashed line in (a) denotes $R_{▵T_{o,W}^{\prime }▵T_{o,W}^{\prime }}=0.05$. (b) $▵x_{0.05}^{*}$ where $R_{▵T_{o,W}^{\prime }▵T_{o,W}^{\prime }}\left(▵x_{0.05}^{*}\right)\approx 0.05$ with respect to $y_m^{*}$. The black dashed line in (b) has a slope of 3.9, and the black dotted line has a slope of 5.6.

Figure 13

(a) Cross-correlation coefficient $R_{▵{\tau }_{w,L}^{″\prime }▵u_{o,W}^{″\prime }}$ as a function of $▵x/h$ at three selected wall-normal positions $y_m^{*}\approx 100,150$, and $y_m/h=0.3$ from case M15R20K, which evaluates the correlation between wall shear stress and signatures in the logarithmic region induced by attached eddies of a given scale. (b) Normalized $R_{▵{\tau }_{w,L}^{″\prime }▵u_{o,W}^{″\prime }}$ by its maximum $R_{\max }=\text{max}\left(R_{▵{\tau }_{w,L}^{″\prime }▵u_{o,W}^{″\prime }}\right)$ and wall-normal distance $y_m$ at the same positions from the same case.

Figure 14

(a) Cross-correlation coefficient $R_{▵q_{w,L}^{\prime }▵T_{o,W}^{\prime }}$ as a function of $▵x/h$ at three selected wall-normal positions $y_m^{*}\approx 100,150$, and $y_m/h=0.3$ from case M15R20K. (b) Normalized $R_{▵q_{w,L}^{\prime }▵T_{o,W}^{\prime }}$ by its maximum $R_{\max }=\max \left(R_{▵q_{w,L}^{\prime }▵T_{o,W}^{\prime }}\right)$ and wall-normal distance $y_m$ at the same positions from the same case.

Figure 15

Streamwise inclination angle ${\alpha }_m$ of wall-attached eddies under the assemble effect of multiscale eddies versus the wall-normal location $y^{*}$ where the inclination angle is evaluated in both the streamwise velocity $u^{″\prime }$ and temperature $T^{\prime }$ using the DNS database in Table 1. (a) Results from the streamwise velocity fluctuation; (b) those from the temperature fluctuations.

Figure 16

Streamwise inclination angle ${\alpha }_s$ of wall-attached eddies at a given height vs their height $y_m^{*}$ in both the streamwise velocity $u^{″\prime }$ and temperature $T^{\prime }$ using the DNS database in Table 1. (a) Results from the streamwise velocity fluctuation; (b) those from the temperature fluctuations. The black dashed lines indicate the average value of each case in the logarithmic region.

Figure 17

Variation of average values ${\alpha }_{s,m}$ of the streamwise inclination angle of wall-attached eddies at a given scale with Reynolds numbers ${\text{Re}}_{\tau }^{*}\left({\text{Re}}_{\tau }\right)$ in both the streamwise velocity $u^{″\prime }$ and temperature $T^{\prime }$ using the DNS database in Table 1. The corresponding results from incompressible flows computed in Cheng et al. [20] are also included for reference, which are denoted by black circles. From lower to higher average inclination angles, these four black circles are from incompressible channel flows at ${\text{Re}}_{\tau }\approx$ 547 [46], 934 [47], 2003 [48], and 4179 [49]. The black solid line is the theoretical prediction angle ${45}^{\circ }$, and the black dashed line denotes the asymptotic performance of ${\alpha }_{s,m}$ computed from incompressible flows.

Figure 18

Mean streamwise velocity profiles transformed by the Trettel-Larsson transformation [50] and the total-stress-based transformation [51]. The black solid line denotes the reference, which is taken from the incompressible channel flow with ${\text{Re}}_{\tau }\approx 5200$ by [52].