Uniform blowing and suction applied to nonuniform adverse-pressure-gradient wing boundary layers

A detailed analysis of the effects of uniform blowing, uniform suction, and body-force damping on the turbulent boundary layer developing around a NACA4412 airfoil at moderate Reynolds number is presented. The flow over the suction and the pressure sides of the airfoil is subjected to a nonuniform adverse pressure gradient and a moderate favorable pressure gradient, respectively. We find that the changes in total skin friction due to blowing and suction are not very sensitive to different pressure-gradient conditions or the Reynolds number. However, when blowing and suction are applied to an adverse-pressure-gradient (APG) boundary layer, their impact on properties such as the boundary-layer thickness, the intensity of the wall-normal convection, and turbulent fluctuations are more pronounced. We employ the Fukagata-Iwamoto-Kasagi decomposition [K. Fukagata et al., Phys. Fluids 14, 73 (2002)] and spectral analysis to study the interaction between intense adverse pressure gradient and these control strategies. We find that the control modifies skin-friction contributions differently in adverse-pressure-gradient and zero-pressure-gradient boundary layers. In particular, the control strategies modify considerably both the streamwise-development and the pressure-gradient contributions, which have high magnitude when a strong adverse pressure gradient is present. Blowing and suction also impact the convection of structures in the wall-normal direction. Overall, our results suggest that it is not possible to simply separate pressure-gradient and control effects, a fact to take into account in future studies on control design in practical applications.

Figure 1

(Left) Section of the mesh for all the cases at ${\text{Re}}_c=200000$ and horizontal component of the instantaneous velocity in (top right) case C (intense blowing) and (bottom right) case E (intense suction) at an arbitrary time step. It is possible to appreciate the different thicknesses of the boundary layer. Velocity values from (blue) $\approx -0.2U_{\infty }$ to (red) $\approx 1.7U_{\infty }$. Black arrows and lines highlight the control region.

Figure 2

Summary of the cases considered in this study and relative control effects on the integrated lift ($C_l$), drag ($C_{d,f}$), and on the aerodynamic efficiency ($L/D$). Arrows, which point downwards or upwards, denote the increase or decrease of physical quantities, respectively. Green arrows denote control effects that are beneficial for the aerodynamic efficiency (e.g., lift increase or drag reduction), and red arrows denote control effects that are detrimental for the aerodynamic efficiency (e.g., lift reduction or drag increase). A black equal sign indicates only marginal changes.

Figure 3

Clauser pressure-gradient parameter $\beta$ for (left) cases BL2, SC2, BD2 and the suction side at ${\text{Re}}_c=200000$, (center) cases BL4 and SC4 and the suction side at ${\text{Re}}_c=400000$, and (right) case PB2 and the pressure side at ${\text{Re}}_c=200000$. The vertical dotted lines indicate the control region. Color code as in Table 1.

Figure 4

Summary of the qualitative effects of uniform blowing (cases BL2 and BL4), body-force damping (case BD2), and uniform suction (cases SC2 and SC4) on the development of the APG TBL on the suction side of the airfoil.

Figure 5

Local skin-friction coefficient $c_f$ for (left) cases BL2, SC2, BD2 and the suction side at ${\text{Re}}_c=200000$, (center) cases BL4 and SC4 and the suction side at ${\text{Re}}_c=400000$, and (right) case PB2 and the pressure side at ${\text{Re}}_c=200000$. The vertical dotted lines indicate the control region. Color code as in Table 1.

Figure 6

(Left) Absolute and (right) relative difference in local skin friction for blowing and suction with intensity $0.1\%U_{\infty }$ at (solid lines) ${\text{Re}}_c=200000$ and (symbols) ${\text{Re}}_c=400000$, and between the uncontrolled cases. Note that the horizontal axis is limited to the control region. The color code is the same as Table 1. For each control case, the difference is computed with respect to the uncontrolled case at the same Reynolds number (e.g., the solid blue line is $\mathrm{\Delta }{c}_f^{\mathrm{SC}2}=c_f^{\mathrm{SC}2}-c_f^{\mathrm{Ref}2}$), and the relative difference is normalized with $c_f$ of the uncontrolled case. The black dashed lines indicate the difference between the two uncontrolled cases at ${\text{Re}}_c=400000$ (Ref4) and ${\text{Re}}_c=200000$ (Ref2), and the relative difference is normalized with the $c_f$ of Ref2.

Figure 7

(Left) Inner-scaled mean wall-tangential velocity component $U_t^{+}$ as a function of the inner-scaled wall-normal distance $y_n^{+}$, and at streamwise location $x/c=0.75$ for cases BL2, SC2, BD2 and the uncontrolled suction side at ${\text{Re}}_c=200000$. Color code as in Table 1. (Right) Profiles of the mean velocity scaled following the Stevenson law [39] for the cases with blowing (BL2 and BL4), suction (SC2 and SC4), and the uncontrolled suction sides at both Reynolds numbers. Solid lines for ${\text{Re}}_c=200000$, dots for ${\text{Re}}_c=400000$, and black lines for reference data of ZPG TBL at similar Reynolds number [38]. The red lines in the background are $U_t^{+}=1/K(ln{y}_n^{+}+C)$, where $C=2.05$ and $K=0.41$.

Figure 8

(Left) Inner-scaled mean wall-normal velocity component $V_n^{+}$ as a function of the inner-scaled wall distance $y_n^{+}$ and at $x/c=0.75$, and (right) outer-scaled wall-normal velocity component $V_n/U_e$ as a function of the outer-scale wall distance $y_n/{\delta }_{99}$ at the same location. Profiles for cases BL2, SC2, BD2, PB2, the uncontrolled cases at ${\text{Re}}_c=200000$, and reference data of ZPG TBL at similar Reynolds number [38]. Color code as in Table 1, dashed line for the uncontrolled case on the pressure side (corresponding to case PB2) and black line for ZPG TBL.

Figure 9

(Left) Inner- and (right) outer-scaled fluctuations of wall-tangential velocity components ${\overline{u_t^2}}^{+}$ and $\overline{u_t{v}_n}$ as functions of the inner-scaled wall-normal distance $y_n^{+}$, for cases BL2, SC2, BD2 and the uncontrolled suction side at ${\text{Re}}_c=200000$ at location $x/c=0.75$. Color code as in Table 1, black lines for reference data of ZPG TBL at similar friction Reynolds number [38].

Figure 10

Summary of the qualitative APG and control effects on the turbulent statistics relative to ZPG TBL. Light arrows that point downwards and dark arrows that point upwards denote the increase or decrease of the physical quantity, respectively. Repeated arrows indicate stronger control effects in relative terms.

Figure 11

Relative contributions of the terms of the FIK decomposition to the total $c_f$ in the uncontrolled cases. By definition, $c_f^D=c_f^{D1}+c_f^{D2}+c_f^{D3}+c_f^{D4}$ and $c_f^{\delta }/c_f+c_f^T/c_f+c_f^D/c_f+c_f^P/c_f=100\%$. Positive and negative percentages denote positive and negative contributions, respectively. Colored cells denote the most relevant contributions (darker colors for higher values).

Figure 12

Streamwise development of the largest relative contributions to friction for the suction side of the reference case at ${\text{Re}}_c=400000$, (left) using the standard FIK formulation [Eq. (14)], (center) introducing the acceleration contribution [Eq. (15)], and (right) introducing the rotational-acceleration and Bernoulli contributions [Eq. (17)]. The relative contributions are normalized with the total $c_f$ and their sum is $\approx 0.9$ because $c_f^{\delta }$, $c_f^{D2}$, and $c_f^{D4}$ are neglected. The turbulent-fluctuation contribution, denoted by $c_f^T$ ($\blacksquare$), and the wall-normal-convection contribution, denoted by $c_f^{D1}$ ($\blacksquare$), are the same in all panels. The streamwise-development and the pressure-gradient contributions in the left panel are denoted by $c_f^{D3}$ ($\blacksquare$) and $c_f^P$ ($\blacksquare$), respectively. The acceleration contribution in the center panel is denoted by $c_f^A$ ($\blacksquare$). The rotational acceleration and the Bernoulli contributions in the right panel are denoted by $c_f^{AR}$ ($\blacksquare$) and $c_f^{AB}$ ($\blacksquare$), respectively. Note that $c_f^{AB}=-c_f^P$. The symbols in the left panel indicate the values at $x/c=0.75$, which are reported in Fig. 13.

Figure 13

Control effects on the terms of the FIK decomposition, normalized with the total $c_f$ of the reference case, e.g., $\mathrm{\Delta }{c}_f^{\delta }=(c_f^{\delta }-c_f^{\delta ,\mathrm{ref}.})/c_f^{\mathrm{ref}.}$. By definition, $\mathrm{\Delta }{c}_f^D=\mathrm{\Delta }{c}_f^{D1}+\mathrm{\Delta }{c}_f^{D2}+\mathrm{\Delta }{c}_f^{D3}+\mathrm{\Delta }{c}_f^{D4}$, and $\mathrm{\Delta }{c}_f^{\delta }+\mathrm{\Delta }{c}_f^T+\mathrm{\Delta }{c}_f^D+\mathrm{\Delta }{c}_f^P=\mathrm{\Delta }{c}_f$. Note that the differences are normalized with the reference $c_f$, so that the sum of all contributions gives the relative change of $c_f$. Colored cells denote the relevant changes in respect to the total $c_f$, and the darker color denotes the relevant changes which exhibit the dominant trend.

Figure 14

Inner-scaled premultiplied spanwise power-spectral density of the wall-tangential Reynolds-shear stress for cases BL4 and SC4 and the uncontrolled suction side at ${\text{Re}}_c=400000$, at location $x/c=0.75$, as functions of (left) the inner-scaled spanwise wavelength (${\lambda }_z^{+}$) and wall-normal distance ($y_n^{+}$), and (right) of the outer-scaled spanwise wavelength (${\lambda }_z/{\delta }_{99}$) and wall-normal distance ($y_n/{\delta }_{99}$). The contours illustrate the levels corresponding to the $25\%$ and the $75\%$ of the maximum power density in the inner region for each case, and the locations of these maxima are marked with stars. Color code as in Table 1, black lines for reference data of ZPG TBL at similar Reynolds number.

Figure 15

Inner-scaled premultiplied spanwise and temporal power-spectral density of the wall-tangential velocity fluctuations at (left) $y_n^{+}=15$ and (right) $y_n^{+}=150$ for cases BL4 and SC4 and the uncontrolled suction side at ${\text{Re}}_c=400000$, at location $x/c=0.75$. The contours illustrate the levels corresponding to the $25\%$ and the $75\%$ of the maximum power density for each case, and the locations of the maxima are marked with stars. Color code as in Table 1, for reference data of ZPG TBL at similar Reynolds number.

Figure 16

Profiles from Nickels formula [40] for (left) the inner-scaled wall-tangential velocity of the reference case at $R{e}_c=400,000$ and streamwise location $x/c=0.75$ and (right) the wall-tangential velocity scaled with the Stevenson law for blowing and suction at same $R{e}_c$ and $x/c$. The color code for the velocity profiles is the same as in in Table 1. The black lines denote the profiles from Nickels fit (solid and dashed line on the left plot denote blowing and suction, respectively). The red, orange, and purple lines denote the linear term (A1), the logarithmic term (A2), and the wake term (A3) for the reference case, respectively.

Figure 17

Comparison between relative contributions to the local skin-friction coefficient $c_f$ over the airfoil suction side, for the reference cases at (lines) ${\text{Re}}_c=200000$ and (symbols) ${\text{Re}}_c=400000$, computed using the (left) FIK and (right) RD decomposition. Contributions derived from similar terms of the governing equations are grouped together: ($\blacksquare$) Reynolds-shear stress ($c_f^T$ and $c_{f2}^{\mathrm{RD}}$), ($\blacksquare$) mean convection ($c_f^{D1}+c_f^{D3}$ and $c_{f31}^{\mathrm{RD}}$), ($\blacksquare$) streamwise derivatives ($c_f^{D2}+c_f^{D4}$ and $c_{f32}^{\mathrm{RD}}$), and ($\blacksquare$) pressure gradients ($c_f^P$ and $c_{f33}^{\mathrm{RD}}$).