Actuation response model from sparse data for wall turbulence drag reduction

We compute, model, and predict drag reduction of an actuated turbulent boundary layer at a momentum-thickness-based Reynolds number of $R{e}_{\theta }=1000$. The actuation is performed using spanwise traveling transversal surface waves parametrized by wavelength, amplitude, and period. The drag reduction for the set of actuation parameters is modeled using 71 large-eddy simulations (LESs). This drag model allows us to extrapolate outside the actuation domain for larger wavelengths and amplitudes. The modeling novelty is based on combining support vector regression for interpolation, a parametrized ridgeline leading out of the data domain, a scaling for the drag reduction, and a discovered self-similar structure of the actuation effect. The model yields high prediction accuracy outside the training data range.

Figure 1

Overview of the physical domain of the actuated turbulent boundary layer flow, where $L_x$, $L_y,$, and $L_z$ are the domain dimensions in the Cartesian directions, $\lambda$ is the wavelength of the spanwise traveling wave, and $x_0$ marks the actuation onset. The shaded red surface $A_{\mathrm{surf}}$ marks the integration area of the wall-shear stress ${\tau }_w$.

Figure 2

Analytical response surface of Eq. (2). Also shown is the ridgeline (black) in the interpolated (solid line) and the extrapolated (dotted line) regimes. The red lines denote lines of steepest ascent seeded at various domain locations.

Figure 3

(d) The drag reduction model ${\widehat{\mathrm{\Delta }c}}_D({\lambda }^{+},T^{+},A^{+})$. The gray surfaces represent three drag reduction levels: 20%, 23%, and 25%. The ridgeline (black) is displayed in the interpolated (solid line) and the extrapolated (dotted line) regimes. The ridgeline leaves the investigated domain at point $A$ and predicts the optimal drag reduction for ${\lambda }^{+}\le 5000$ at point $B$ (${\lambda }^{+}=5000$, $T^{+}=44$, and $A^{+}=99$). The red curves denote lines of steepest ascent seeded at various domain locations. The contour distributions at the top [panels (a)–(c)] represent scaled-drag reductions. These reductions are the shape functions $F$ defined in Eq. (13) and modeled in Eq. (14). The three slices correspond to the blue-framed $T^{+}$–$A^{+}$ rectangles in the three-dimensional plot.

Figure 4

Projection of the ridgeline onto the ${\lambda }^{+}\text{−}{T}^{+}$ and ${\lambda }^{+}\text{−}{A}^{+}$ planes, as well as the drag reduction along the ridge as function of ${\lambda }^{+}$. The solid lines are interpolated with SVR, whereas the dotted lines are obtained by Eqs. (10) (for $T_r^{+}$ and $A_r^{+}$), and (11) (for $\mathrm{\Delta }{c}_{d,r}$). Points $A$ and $B$ are the same as those in Fig. 3. The vertical gray dashed line separates the interpolation and extrapolation regions.

Figure 5

Drag reduction along the ridgeline as function of the Tomiyama and Fukagata scaling. The figure shows a linear behavior starting at ${\lambda }^{+}\approx 1000$. The solid line is obtained from data interpolated with SVR. The dotted line is obtained with Eq. (11). Points $A$ and $B$ are the same as those in Fig. 3.

Figure 6

(a) The reference $F$ distribution, and (b) the nonlinear model $\widehat{F}$. The model faithfully represents the reference response.

Figure 7

The cross-validated mean square error for a range of models with increasing complexity $N_{\xi }$.