Numerical study of turbulent channel flow perturbed by spanwise topographic heterogeneity: Amplitude and frequency modulation within low- and high-momentum pathways

We have studied the effects of topographically driven secondary flows on inner-outer interaction in turbulent channel flow. Recent studies have revealed that large-scale motions in the logarithmic region impose an amplitude and frequency modulation on the dynamics of small-scale structures near the wall. This led to development of a predictive model for near-wall dynamics, which has practical relevance for large-eddy simulations. Existing work on amplitude modulation has focused on smooth-wall flows; however, Anderson [J. Fluid Mech. 789, 567 (2016)] addressed the problem of rough-wall turbulent channel flow in which the correlation profiles for amplitude modulation showed trends similar to those reported by Mathis et al. [Phys. Fluids 21, 111703 (2009)]. For the present study, we considered flow over surfaces with a prominent spanwise heterogeneity, such that domain-scale turbulent secondary flows in the form of counter-rotating vortices are sustained within the flow. (We also show results for flow over a homogeneous roughness, which serves as a benchmark against the spanwise-perturbed cases.) The vortices are anchored to the topography such that prominent upwelling and downwelling occur above the low and high roughness, respectively. We have quantified the extent to which such secondary flows disrupt the distribution of spectral density across constituent wavelengths throughout the depth of the flow, which has direct implications for the existence of amplitude and frequency modulation. We find that the distinct outer peak associated with large-scale motions—the “modulators”—is preserved within the upwelling zone but vanishes in the downwelling zone. Within the downwelling zones, structures are steeper and shorter. Single- and two-point correlations for inner-outer amplitude and frequency modulation demonstrate insensitivity to resolution across cases. We also show a pronounced crossover between the single- and two-point correlations, a product of modulation quantification based upon Parseval's theorem (i.e., spectral density, but not the wavelength at which energy resides, defines the strength of modulation).

Figure 1

Illustration of the topographies considered in present study: (a) case 1, (b) case 2, and (c) case 3 (cases 4 and 7 are higher-resolution versions of case 1, and so on, which precludes the need to provide images of the additional cases). The panels include annotation of the reference heights, $z_{\mathrm{Ref}.}$, needed to compute correlations for amplitude and frequency modulation. The panels also include annotation of virtual towers over which time-series measurements of fluctuating velocity was recorded (in (a), we say simply “Tower 1” and “Tower 2,” since this case is a homogeneous roughness without any “Trough” or “Crest”). Table 1 contains additional simulation attributes.

Figure 2

Color flood contour of instantaneous streamwise velocity fluctuations, ${\tilde{u}}^{\prime }(x,y_l,z,t_l)$, in the $x\text{−}z$ plane at discrete spanwise location, $y_l$, at arbitrarily selected local time, $t_l$: (a)–(c) crest and (d)–(f) trough (see also Fig. 1). Vectors of $\{{\tilde{u}}^{\prime }(x,y_l,z,t_l),{\tilde{w}}^{\prime }(x,y_l,z,t_l)\}$ have been superimposed, for reference. (a), (d) Case 1, (b), (e) case 2, and (c), (f) case 3 (see also Table 1).

Figure 3

Color flood contour visualization of time- and $x$-averaged swirl strength, ${\langle {\lambda }_{ci}\rangle }_{x,T}(y,z)\left({\langle {\tilde{\omega }}_x\rangle }_{x,T}(y,z)/{\langle \tilde{\mathbf{\omega }}\rangle }_{x,T}(y,z)\right)$, in the $y\text{−}z$ plane, with $\{{\langle \tilde{v}\rangle }_{x,T}(y,z),{\langle \tilde{w}\rangle }_{x,T}(y,z)\}$ vectors superimposed. (a) Case 1, (b) case 2, and (c) case 3 (see also Table 1). In (b) and (c), we have added annotation for the location of topographically driven LMPs and HMPs.

Figure 4

Time- and streamwise-averaged vertical profiles of (a)–(f) first-order and (g), (h) second-order flow statistics. Profiles are derived from above the (a), (c), (e), (g) crest and (b), (d), (f), (h) trough, respectively. Profiles are normalized by (a), (b) the maximum advective velocity and (c)–(h) friction velocity, $u_{\tau }$. Black, case 1; dark gray, case 2; and light gray, Case 3. Dashed black lines in (c), (d) denote the logarithmic profile with roughness length, $z_0$, selected to optimally fit the LES data set. The horizontal dashed black lines in (g), (h) denote $h_{\max }/H$ (see also Table 1).

Figure 5

Spatial correlation map of fluctuating streamwise velocity, ${\rho }_{xx}(\delta x,y,z;z_{\mathrm{Ref}.})$ [Eq. (14)], in the streamwise lag–wall-normal plane for (a), (b), (d), (f) $z_{\mathrm{Ref}.}/H=0.01$, (c) $z_{\mathrm{Ref}.}/H=0.056$, and (e) $z_{\mathrm{Ref}.}/H=0.12$, above the (a), (c), (e) crest and (b), (d), (f) trough, respectively. Superimposed on the ${\rho }_{xx}(\delta x,y,z;z_{\mathrm{Ref}.})$ color flood contours is the streamwise lag corresponding with maximum correlation, $\delta x_m(z;z_{\mathrm{Ref}.})$ [Eq. (15)]. (a), (b) Case 1, (c), (d) case 2, and (e), (f) case 3. $\delta x_m(z;z_{\mathrm{Ref}.})$ for cases 1–3 above the (g) crest and (h) trough, respectively, where solid horizontal lines denote the reference heights used in Eqs. (14) and (15), while the inclined solid lines are used to interpret the LES data points. Symbol colors correspond with case 1 (solid black), case 2 (solid dark gray), and case 3 (solid light gray).

Figure 6

Color flood contours of Fourier-mode spectrograms of $\tilde{u}/u_{\tau }$. (a, b) Case 1, (c, d) case 2, and (e, f) case 3 above the (a, c, e) crest and (b, d, f) trough. Annotations have been added for the LES grid-filter width $\mathrm{\Delta }/H$, domain length $L_x/H$, and separation wavelength 2; we have also added annotations for two reference locations, $z_{\mathrm{Ref}{.}_1}$ and $z_{\mathrm{Ref}{.}_2}$, which are used in Sec. 3e to determine two-point modulation of small-scale amplitude and frequency. For cases 2 and 3 above the crest, vertical dashed lines denote $h_{\max }/H$.

Figure 7

Color flood contours of wavelet-based spectrograms of $\tilde{u}/u_{\tau }$. (a), (b) Case 1, (c), (d) case 2, and (e), (f) case 3 above the (a), (c), (e) crest and (b), (d), (f) trough. Annotations have been added for the shear-normalized LES grid-filter frequency $f_{\min }^{*}H/U_0$, domain length $f_{\max }^{*}H/U_0$, and shear-normalized separation frequency $f_{c}H/U_0=0.5$; we have also added annotations for two reference locations, $z_{\mathrm{Ref}{.}_1}$ and $z_{\mathrm{Ref}{.}_2}$, which are used in Sec. 3e to determine two-point modulation of small-scale amplitude and frequency. For cases 2 and 3 above the crest, vertical dashed lines denote $h_{\max }/H$.

Figure 8

Vertical profiles for small-scale amplitude modulation by large scale, ${\tilde{u}}_L^{\prime }(x_l,y_l,z,t)$, at discrete streamwise-spanwise locations, $\{x_l,y_l\}$, corresponding with the (a), (c), (e), (g) crest and (b), (d), (f), (h) trough. Dashed and solid profiles correspond with single-point [Eq. (7)] and two-point [Eq. (9)] correlation, respectively, and corresponding values for both are shown by the left and right figure ordinates. (a), (b), (e), (f) Black, dark gray, and light gray correspond with cases 1, 4, and 7, respectively; (c), (d), (g), (h) black, dark gray, and light gray correspond with cases 3, 6, and 9, respectively. The reference heights used for (a)–(d) and (e)–(h) are $z_{\mathrm{Ref}{.}_1}/H=0.5$ (red) and $z_{\mathrm{Ref}{.}_2}/H=0.25$ (blue), respectively, and both were included on the spectrograms (Figs. 6 and 7).

Figure 9

Vertical profiles for small-scale frequency modulation by large scale, ${\tilde{u}}_L^{\prime }(x_l,y_l,z,t)$, at discrete streamwise-spanwise locations, $\{x_l,y_l\}$, corresponding with the (a), (c), (e), (g) crest and (b), (d), (f), (h) trough. Dashed and solid profiles correspond with single-point [Eq. (8)] and two-point [Eq. (10)] correlation, respectively, and corresponding values for both are shown by the left and right figure ordinates. (a), (b), (e), (f) Black, dark gray, and light gray correspond with cases 1, 4, and 7, respectively; (c), (d), (g), (h) black, dark gray, and light gray correspond with cases 3, 6, and 9, respectively. The reference heights used for (a)–(d) and (e)–(h) are $z_{\mathrm{Ref}{.}_1}/H=0.5$ (red) and $z_{\mathrm{Ref}{.}_2}/H=0.25$ (blue), respectively, and both were included on the spectrograms (Figs. 6 and 7).