Separating adverse-pressure-gradient and Reynolds-number effects in turbulent boundary layers

Zero pressure-gradient (ZPG) turbulent boundary layers (TBLs) at high Reynolds numbers (Re) and TBLs with adverse pressure gradients (APGs) share some similarities such as the emergence of an outer peak in the streamwise variance profile related to the enhancement of large-scale energy. Reynolds-number and pressure-gradient effects tend to cause such an energizing of the outer-layer structures, which makes it difficult to distinguish them, mainly because both effects are usually coupled. Using two experimental data sets of ZPG and APG TBLs in which Re and pressure gradient are analyzed independently, the present paper shows that the two effects have different imprints on the flow. In particular, the analysis shows that (1) the small-scale energy in APG TBLs is, contrary to canonical ZPG TBLs, not universal, but is dependent on the APG strength, (2) the APG enhances both the small- and large-scale energy in the outer region, and (3) the outer-peak location and the amplitude of the streamwise variance scale differently for increasing Re and APG magnitude. These findings may have ramifications on the development of turbulence models since the pressure gradient has a direct influence on the near-wall dynamics.

Figure 1

Mean streamwise variance profiles at matched ${\text{Re}}_{\tau }\approx 4400$, nondimensionalized by (a) the friction velocity $u_{\tau }$ and viscous length ${\ell }^{★}$ and (b) the Zagarola–Smits (ZS) length scale $U_{\mathrm{ZS}}$ defined below and the boundary-layer thickness ${\delta }_{99}$. Colors represent ($––$) $\beta \approx 0$ (experimental data from Refs. [32, 33]), ($\square$) $\beta \approx 0.75$, ($○$) $\beta \approx 1.1$, and ($◇$) $\beta \approx 2.2$. Gray dashed line indicates raw hot-wire data without applying the correction from Ref. [34].

Figure 2

Streamwise evolution of the Rotta-Clauser pressure-gradient parameter $\beta$ as function of the Reynolds number based on the displacement thickness ${\text{Re}}_{{\delta }^{*}}$. The colors and symbols are listed in Table 1. Reference black dash lines at $\beta =1.1\pm 25\%$ are included. Filled symbols indicated the region bounded by 25% deviation from $\beta \approx 1.1$, which henceforth will be denoted approximately constant-$\beta$ region.

Figure 3

Inner-scaled premultiplied energy spectra of the streamwise velocity at ${\text{Re}}_{\tau }\approx 4400$ with contour levels at ${\kappa }_x^{+}{\mathrm{\Phi }}_{uu}^{+}=$ 0.25, 0.4, 0.575, 0.775, 0.95, 1.2, 1.6, 2.0, for pressure-gradient strengths (a) $\beta \approx 0$ (experimental data from Ref. [32]), (b) $\beta \approx 0.75$, (c) $\beta \approx 1.1$, (d) $\beta \approx 2.2$. Auxiliary red symbols are located at the following coordinates: ($+$) $(y^{+}=15,{\lambda }_x^{+}=1000)$, ($\circ$) $(y^{+}=3.9{\text{Re}}_{\tau }^{0.5},{\lambda }_x/{\delta }_{99}=6)$ for all the cases. Additionally red ($\square$) symbols are located at $(y^{+}=y_{\max ,\mathrm{outer}}^{+},{\lambda }_x/{\delta }_{99}=3)$ for all the APG cases and $(y/{\delta }_{99}=0.3,{\lambda }_x/{\delta }_{99}=3)$ for the ZPG case. An auxiliary horizontal dashed line is placed at $\lambda /{\delta }_{99}=1$.

Figure 4

(a) Decomposition of the velocity fluctuations (solid lines) into a small-scale (dotted lines) and a large-scale (dashed lines) component for ${\text{Re}}_{\tau }\approx 4400$. (b) Small-scale variance conditioned on a large-scale fluctuation value of $u_L^{+}=0$, ${\overline{u_S^2}}^{+}(u_L=0)$, for ${\text{Re}}_{\tau }\approx 4400$ together with the small-scale (dotted lines) component from (a). The inset represents the ratio between the APG cases and the ZPG case, i.e., ${\overline{u_S^2}}^{+}{(u_L=0)}_{\mathrm{APG}}/{\overline{u_S^2}}^{+}{(u_L=0)}_{\mathrm{ZPG}}$. Gray ($––$) represents $\beta \approx 0$, blue ($––$) $\beta \approx 0.75$, red ($––$) $\beta \approx 1.1$, and green ($––$) $\beta \approx 2.2$.

Figure 5

Dominant scale-decomposed skewness term, $3\overline{\overline{u_L^{+}{u^2}_S^{+}}}$ for ${\text{Re}}_{\tau }\approx 4400$ (right). Gray ($––$) represents $\beta \approx 0$, blue ($––$) $\beta \approx 0.75$, red ($––$) $\beta \approx 1.1$, and green ($––$) $\beta \approx 2.2$.

Figure 6

(a) Instantaneous example of fluctuating signal (${u^{\prime }}^{+}$) and (b) large-scale fluctuations component ($u_L^{+}$) computed based on a filter timescale of $t{U}_c/{\delta }_{99}=1$ at $y^{+}\approx 15$. Example case at ${\text{Re}}_{\tau }\approx 4400$ and pressure-gradient strength $\beta \approx 1.1$. Red filled symbols represent signal segments corresponding to $u_L^{+}>0$ and blue empty symbols to $u_L^{+}<0$.

Figure 7

(a) Streamwise variance calculated by signal segments with positive large-scale fluctuations $\overline{u^{2^{+}}}{|}_{u_L^{+}>0}$ (red), negative large-scale fluctuations $\overline{u^{2^{+}}}{|}_{u_L^{+}<0}$ (blue), and with the full signal length $\overline{u^{2^{+}}}$ (black). (b) Difference between the streamwise variance calculated by signal segments with positive and negative large-scale fluctuations $\overline{u^{2^{+}}}{{|}_{u_L^{+}>0}-\overline{u^{2^{+}}}|}_{u_L^{+}<0}$ (black dashed line) and dominant scale-decomposed skewness term, $3\overline{\overline{u_L^{+}{u^2}_S^{+}}}$ (blue line). Example case at ${\text{Re}}_{\tau }\approx 4400$ and pressure-gradient strength $\beta \approx 1.1$. The dashed vertical lines indicate approximately the cross-over point between the $\overline{u^{2^{+}}}{|}_{u_L^{+}>0}$ and $\overline{u^{2^{+}}}{|}_{u_L^{+}<0}$ curves, i.e., $\overline{u^{2^{+}}}{{|}_{u_L^{+}>0}-\overline{u^{2^{+}}}|}_{u_L^{+}<0}=0$.

Figure 8

Maximum value of the streamwise variance normalized with (a) $U_{\mathrm{ZS}}$ and (b) $U_{\mathrm{edge}}$ as a function of ${\text{Re}}_{{\delta }^{*}}$. Black filled symbols represent the near-wall peak and open symbols the outer peak. Magenta symbols indicate the outer peak values for the ZPG cases from Ref. [32]. Wall-normal position of the outer-peak location normalized with (c) ${\delta }_{99}$ and (d) ${\delta }^{*}$ as a function of ${\text{Re}}_{\tau }$. The data correspond to the approximately constant-$\beta$ region for $\beta \approx 1.1$ (Fig. 2). The symbols for the APG cases are reported in Table 1. Magenta symbols indicate the outer peak location for the ZPG cases from Ref. [32].

Figure 9

Maximum value of the streamwise variance at the outer peak location normalized with (a) $U_{\mathrm{ZS}}$ and (b) $U_{\mathrm{edge}}$ against $\beta$. Wall-normal position of the outer peak location normalized with (c) ${\delta }_{99}$ and (d) ${\delta }^{*}$ against $\beta$. The data shown correspond to the region $3.1\mathrm{m}\le x\le 4.4\mathrm{m}$. The colors and symbols for the APG cases are reported in Table 1. Magenta symbols indicate the ZPG outer peak location and value for the ZPG cases from Ref. [32]. Color varies from light magenta to dark magenta with increasing Re.