Reynolds-number effects on the outer region of adverse-pressure-gradient turbulent boundary layers

We study the Reynolds-number effects on the outer region of moderate adverse-pressure-gradient (APG) turbulent boundary layers (TBLs) and find that their small-scale (viscous) energy reduces with increasing friction Reynolds number (${\mathrm{Re}}_{\tau }$). The trend is based on analyzing APG TBL data across $600\lesssim {\mathrm{Re}}_{\tau }\lesssim 7000$ and contrasts with the negligible variation in small viscous-scaled energy noted for canonical wall flows. The data sets considered include those from a well-resolved numerical simulation [Pozuelo et al., J. Fluid Mech. 939, A34 (2022)], which provides access to an APG TBL maintained at near-equilibrium conditions across $1000\lesssim {\mathrm{Re}}_{\tau }\lesssim$ 2000, with a well-defined flow history, and a new high-${\mathrm{Re}}_{\tau }$ ($\sim 7000$) experimental study from the large Melbourne wind tunnel, with its long test section modified to permit development of an APG TBL from a “canonical” upstream condition. The decrease in small-scale energy with ${\mathrm{Re}}_{\tau }$ is revealed via decomposing the streamwise normal stresses into small- and large-scale contributions, based on a sharp spectral cutoff. The origin for this trend is traced back to the production of turbulent kinetic energy in an APG TBL, the small-scale contribution to which is also found to decrease with ${\mathrm{Re}}_{\tau }$ in the outer region. The conclusion is reaffirmed by investigating attenuation of streamwise normal stresses due to changing spatial resolutions of the numerical grid or hotwire sensors, which reduces with increasing ${\mathrm{Re}}_{\tau }$ and is found to be negligible at ${\mathrm{Re}}_{\tau }\sim 7000$ in this study. The results emphasize that new scaling arguments and spatial-resolution corrections should be tested rigorously across a broad ${\mathrm{Re}}_{\tau }$ range, particularly for pressure gradient TBLs.

Figure 1

(a) Schematic overview of the large Melbourne wind-tunnel test section modified to set up an APG TBL (not to scale). Markings in red indicate the modifications made for the APG setup, which include (i) controlling the air bleeding from the ceiling in the upstream section ($x<9$ m) to enforce a nominal ZPG, and (ii) introducing outflow screens at the test-section exit to increase tunnel static pressure. (b) Mean pressure coefficient, $C_P$, measured at various streamwise locations, $x$, for different screen no. configuration 0, 1, 2, or 3, with 0 corresponding to the scenario when ZPG is maintained across the entire tunnel test section. (c) Photograph of an aluminium outflow screen installed at the test-section outlet. The screen had circular holes of 9.5 mm diameter at a pitch of 12.7 mm, resulting in a porosity of 51%. Dashed gray line in (b) refers to the $x$ location up to which a nominal ZPG was maintained, while dashed gold lines indicate locations for hotwire measurements.

Figure 2

Parameter space $\beta$ vs ${\mathrm{Re}}_{\tau }$ using a combination of previously published data [8, 13, 15, 52] and new experimental data. See Tables 2 and 3 for information on respective data sets. $\beta$ vs ${\mathrm{Re}}_{\tau }$ is plotted across the full domain of the LES-APG data set [8] to indicate the upstream history for the selected locations (given by symbols).

Figure 3

Viscous-scaled (a), (c), (d), (e) mean velocity profiles and (b) streamwise normal stresses corresponding to the Exp-ZPG and Exp-APG data sets with $l^{+}\approx$ 10 (Table 2). In (a), the high-${\mathrm{Re}}_{\tau }$ experimental data are compared with low-${\mathrm{Re}}_{\tau }$ ZPG TBL data set of Sillero et al. [53] and the well-resolved LES-APG data set (Table 3). Panels (c)–(e) depict the enlarged views of data in specific $z^{+}$ ranges in (a), indicated by yellow boxes. Solid lines correspond to complete velocity profiles from simulation data sets, and open symbols correspond to experimental data acquired across the TBL. Filled symbols in (a) and (c) are from the dedicated experiments conducted very close to the wall to estimate skin-friction velocity, $U_{\tau }$. Dash-dotted black line in (a) and (d) corresponds to the well-known log law of the mean velocity following Marusic et al. [6], with $\kappa =$ 0.39 and $A=4.3$.

Figure 4

Viscous-scaled premultiplied energy spectra of the streamwise velocity fluctuations ($k_x{\varphi }_{uu}^{+}$), as a function of ${\lambda }_x^{+}$ and $z^{+}$, computed from the Exp-ZPG and Exp-APG data sets with $l^{+}\approx$ 10 (Table 2). Dashed yellow and dash-dotted magenta lines indicate ${\lambda }_x^{+}$ = $3{\delta }_{99}^{+}$ and ${\lambda }_x^{+}$ = $6{\delta }_{99}^{+}$, respectively.

Figure 5

(a), (b) Streamwise normal stress, ${\overline{u^2}}^{+}$ and (c), (d) constant energy contours of premultiplied $u$ spectra for (a), (c) ZPG TBL and (b), (d) APG TBL data sets at various ${\mathrm{Re}}_{\tau }$. Estimates for LES-ZPG and LES-APG data sets correspond to their original (i.e., well-resolved) grid resolution ($\mathrm{\Delta }{y}^{+}$; Table 3), while those for Exp-ZPG and Exp-APG are from hotwires with $l^{+}\approx 10$ (Table 2). Symbols and color scheme for each data set are as documented in Sec. 2. White and green background in (c) and (d) indicate small-scale (${\overline{u_S^2}}^{+}$; ${\lambda }_y^{+}\le 300$) and large-scale contributions (${\overline{u_L^2}}^{+}$; ${\lambda }_y^{+}>300$), respectively.

Figure 6

Streamwise normal stress decomposed into viscous-scaled (a), (c) large-scale (${\overline{u_L^2}}^{+}$) and (b), (d) small-scale contributions (${\overline{u_S^2}}^{+}$) for (a), (b) LES-ZPG and (c), (d) LES-APG data sets at various ${\mathrm{Re}}_{\tau }$. Estimates for the simulation data sets correspond to their original (i.e., well-resolved) grid resolution ($\mathrm{\Delta }{y}^{+}$; Table 3).

Figure 7

(a), (d) Dominant contribution to production of turbulent-kinetic energy, $-{\overline{uw}}^{+}\partial U^{+}/\partial z^{+}$, decomposed into viscous-scaled (b), (e) large-scale ($-{\overline{u_L{w}_L}}^{+}\partial U^{+}/\partial z^{+}$) and (c), (f) small-scale contributions ($-{\overline{u_S{w}_S}}^{+}\partial U^{+}/\partial z^{+}$) for (a)–(c) LES-ZPG and (d)–(f) LES-APG data sets at various ${\mathrm{Re}}_{\tau }$. Estimates for the simulation data sets correspond to their original grid resolution ($\mathrm{\Delta }{y}^{+}$; Table 3). Note that the profiles are premultiplied with $z^{+}$. Also, the range of the vertical axis for (a), (b), (c), and (f) is different than for (d) and (e).

Figure 8

(a), (c), (e), (f) Streamwise normal stress, ${\overline{u^2}}^{+}$ and (b), (d), (f), (h) constant-energy contours of premultiplied $u$ spectra estimated for LES-APG data sets with various spatial resolutions ($\mathrm{\Delta }{y}^{+}$) at ${\mathrm{Re}}_{\tau }\approx$ (a), (b) 585, (c), (d) 1020, (e), (f) 1508, and (g), (h) 1959. Symbols and color scheme for each data set are as documented in Table 3. Dashed orange lines indicate $z^{+}$ associated with the outer peak (i.e., $z_{OP}^{+}$). $\mathrm{\Delta }{\overline{u^2}}^{+}\left(z_{OP}^{+}\right)$ in (a), (c), (e), (g) corresponds to the attenuation in ${\overline{u^2}}^{+}$ at $z_{OP}^{+}$ as $\mathrm{\Delta }{y}^{+}$ increases from the well-resolved to worst-resolved scenario. White and green background in (b), (d), (f), and (h) indicate small-scale (${\overline{u_S^2}}^{+}$; ${\lambda }_y^{+}\le 300$) and large-scale contributions (${\overline{u_L^2}}^{+}$; ${\lambda }_y^{+}>300$), respectively.

Figure 9

Variation in attenuation of ${\overline{u^2}}^{+}$ at $z_{OP}^{+}$ [i.e., $\mathrm{\Delta }{\overline{u^2}}^{+}\left(z_{OP}^{+}\right)$] with ${\mathrm{Re}}_{\tau }$. Here $\mathrm{\Delta }{\overline{u^2}}^{+}\left(z_{OP}^{+}\right)$ is the difference between ${\overline{u^2}}^{+}(z_{OP}^{+}$) as $\mathrm{\Delta }{y}^{+}$ increases from the well-resolved to worst-resolved scenario (Tables 2 and 3). Note that $z_{OP}^{+}$ is assumed at $0.3{\delta }_{99}^{+}$ for the Exp-APG data following Sanmiguel Vila et al. [30]. Dashed green line indicates linear fit based purely on the LES-APG data.

Figure 10

(a)–(c) Streamwise turbulent-kinetic energy, ${\overline{u^2}}^{+}$ and (d)–(f) constant-energy contours of premultiplied $u$ spectra estimated for Exp-APG data sets with various spatial resolutions ($l^{+}$) at $\beta \approx$ (a), (d) 0.9, (b), (e) 1.3, and (c), (f) 1.7. Symbols and color scheme for each data set are as documented in Table 2.

Figure 11

(a) $U^{+}$ and ${\overline{u^2}}^{+}$ profiles measured in the Melbourne wind tunnel at matched $x$ locations and free-stream velocity conditions for two different outflow screen configurations, Screen no. 0 and no. 3, as described in Sec. 2. (b) Mean streamwise velocity profiles measured for high ${\mathrm{Re}}_{\tau }$ Exp-APG data sets with hotwires of various spatial resolutions ($l^{+}$) at various $\beta , 0.9\lesssim \beta \lesssim 1.7$. Symbols and color scheme for each data set are as documented in Table 2. Data are separated by a shift in the $U^{+}$ axis for clarity.

Figure 12

(a) ${\overline{u^2}}^{+}$ decomposed into (b) large-scale (${\overline{u_L^2}}^{+}$) and (c) small-scale contributions (${\overline{u_S^2}}^{+}$) for DNS data set of a turbulence channel flow [56] at various ${\mathrm{Re}}_{\tau }$. Energy decomposition is based on using ${\lambda }_{y,c}^{+}=300$, the same as in Fig. 6.

Figure 13

Small-scale contribution to (a) streamwise normal stress (${\overline{u_S^2}}^{+}$) and (b) production of turbulent kinetic energy ($-{\overline{u_S{w}_S}}^{+}\frac{\partial U^{+}}{\partial z^{+}}$) compared for LES data set of Bobke et al. [15] and Pozuelo et al. [8]. The comparison is made at similar ${\mathrm{Re}}_{\tau }$ and $\beta$ but different $\overline{\beta }$, to understand the influence of upstream history. Spectral decomposition is based on ${\lambda }_{y,c}^{+}=300$. Estimates for the simulation data sets correspond to their original (i.e., well-resolved) grid resolution ($\mathrm{\Delta }{y}^{+}$, Table 3).