Decomposition of the mean friction drag in adverse-pressure-gradient turbulent boundary layers

In this study, we exploit the Renard-Deck identity [J. Fluid Mech. 790, 339 (2016)] to decompose the mean friction drag in adverse-pressure-gradient turbulent boundary layers (APG-TBLs) into three components, associated with viscous dissipation, turbulence kinetic energy production, and spatial growth of the flow, respectively. We consider adverse-pressure-gradient turbulent boundary layers developing on flat plates and airfoils, with friction Reynolds numbers in the range $200<{\text{Re}}_{\tau }<2000$, and with Rotta-Clauser pressure-gradient parameters ($\beta$) ranging from 0 to 50. The effects of Reynolds number, adverse pressure gradient, and the pressure-gradient history on the contributing components are individually investigated, and special attention is paid to the comparisons with zero-pressure-gradient turbulent boundary layers (ZPG-TBLs). Our results indicate that the inner peaks of the dissipation and production terms are located at $y^{+}\approx 6$ and $y^{+}\approx 16.5$, respectively, and their outer peaks scale with the $99\%$ boundary-layer thickness (${\delta }_{99}$), i.e., $y/{\delta }_{99}\approx 0.7$ and $0.53$, respectively. These results are independent of the friction Reynolds number, the magnitude of $\beta$, and its development history. Moreover, the spatial-growth component is negative in the investigated APG-TBLs, and its magnitude increases with $\beta$.

Figure 1

Contributions of the three decomposed components for flat-plate TBLs with constant $\beta =0$, $\approx 1$ and $\approx 2$, and ${\text{Re}}_{\tau }=670$.

Figure 2

Distributions of the premultiplied integrand in $C_{f1}/C_f$ (a), $C_{f2}/C_f$ (b), and $C_{f3}/C_f$ (c) as a function of $y^{+}$ for the flat-plate TBLs with constant $\beta =0$, $\approx 1$, $\approx 2$ and ${\text{Re}}_{\tau }=670$.

Figure 3

Profiles of mean streamwise velocity (a), wall-normal gradient of velocity (b), Reynolds shear stress (c), and production (d) as a function of $y^{+}$ for the flat-plate TBLs with constant $\beta =0$, $\approx 1$, $\approx 2$ and ${\text{Re}}_{\tau }=670$.

Figure 4

Distributions of the premultiplied integrand in $C_{f31}/C_f$ (a), $C_{f32}/C_f$ (b), and $C_{f33}/C_f$ (c) as a function of $y^{+}$ for the flat-plate TBLs with constant $\beta =0$, $\approx 1$, $\approx 2$, and ${\text{Re}}_{\tau }=670$.

Figure 5

Distributions of the premultiplied integrand in $C_{f1}/C_f$, $C_{f2}/C_f$, and $C_{f3}/C_f$ as a function of $y^{+}$ (a)–(c) and as a function of $y/{\delta }_{99}$ (d)–(f) for the APG-TBL with $\beta \approx 1.4$. The vertical dashed lines in (d)–(f) mark the position $y/{\delta }_{99}=0.7$, 0.53, and 0.65, respectively.

Figure 6

Profiles of mean convection $(〈u〉\partial 〈u〉/\partial x+〈v〉\partial 〈u〉/\partial y)/(U_e^2/{\delta }_{99})$ for the APG-TBL with $\beta \approx 1.4$. For legends, see caption in Fig. 5.

Figure 7

Distributions of the ($y/{\delta }_{99}$) premultiplied power of viscous and turbulent stress (a), viscous dissipation (b), turbulence kinetic energy production (c), and spatial growth (d) for the APG-TBL with $\beta \approx 1.4$.

Figure 8

(a) Distribution of $\beta$ as a function of $x/c$ on the suction side of the NACA4412 wing section for the three Reynolds numbers. The shaded areas are excluded from this study. (b) The ratio of each contribution to the total friction coefficient for the airfoil APG-TBLs.

Figure 9

Distributions of $\beta$ as a function of ${\text{Re}}_{\tau }$ on the suction side of the NACA4412 airfoil. The three values of $\beta$, i.e., $\beta =2$, 4 and 8, are represented by a circle, a square, and a triangle, respectively.

Figure 10

Distributions of the premultiplied integrand in $C_{f1}/C_f$ [(a), (d), (g)], $C_{f2}/C_f$ [(b), (e), (h)], and $C_{f3}/C_f$ [(c), (f), (i)] as a function of $y^{+}$ for the airfoil APG-TBLs at three different $\beta$. Top, middle, and bottom rows correspond to the conditions denoted by circle, square, and triangle in Fig. 9, respectively.

Figure 11

Distribution of the pressure-gradient parameter $\beta$ as a function of the friction Reynolds number ${\text{Re}}_{\tau }$ for various APG-TBL cases. The conditions $P_1$, $P_2$, and $P_3$ are denoted by a triangle, a circle, and a star, respectively.

Figure 12

Distribution of the premultiplied integrand in $C_{f1}/C_f$ [(a), (d), (g)], $C_{f2}/C_f$ [(b), (e), (h)], and $C_{f3}/C_f$ [(c), (f), (i)] as a function of $y^{+}$ for cases $P_1$ [(a)–(c)], $P_2$ [(d)–(f)], and $P_3$ [(g)–(i)].

Figure 13

Distributions of the premultiplied integrand in $C_{f1}/C_f$ [(a), (d)], $C_{f2}/C_f$ [(b), (e)], and $C_{f3}/C_f$ [(c), (f)] as a function of $y^{+}$ [(a)–(c)] and as a function of $y/{\delta }_{99}$ [(d)–(f)] for the APG-TBL with $\beta \approx 1$. The vertical dashed lines in (d)–(f) mark the position $y/{\delta }_{99}=0.7$, 0.53, and 0.65, respectively.

Figure 14

Distributions of the premultiplied integrand in $C_{f1}/C_f$ [(a), (d)], $C_{f2}/C_f$ [(b), (e)], and $C_{f3}/C_f$ [(c), (f)] as a function of $y^{+}$ [(a)–(c)] and as a function of $y/{\delta }_{99}$ [(d)–(f)] for the APG-TBL with $\beta \approx 2$. The vertical dashed lines in (d)–(f) mark the position $y/{\delta }_{99}=0.7$, 0.53, and 0.65, respectively.