Drag Reduction and Energy Saving by Spanwise Traveling Transversal Surface Waves for Flat Plate Flow

Wall-resolved large-eddy simulations are performed to study the impact of spanwise traveling transversal surface waves in zero-pressure gradient turbulent boundary layer flow. Eighty variations of wavelength, period, and amplitude of the space- and time-dependent sinusoidal wall motion are considered for a boundary layer at a momentum thickness based Reynolds number of $Re_\theta = 1000$. The results show a strong decrease of friction drag of up to $26\,\%$ and considerable net power saving of up to $10\,\%$. However, the highest net power saving does not occur at the maximum drag reduction. The drag reduction is modeled as a function of the actuation parameters by support vector regression using the LES data. A substantial attenuation of the near-wall turbulence intensity and especially a weakening of the near-wall velocity streaks are observed. Similarities between the current actuation technique and the method of a spanwise oscillating wall without any normal surface deflection are reported. In particular, the generation of a directional spanwise oscillating Stokes layer is found to be related to skin-friction reduction.


Introduction
Surface friction in turbulent wall-bounded flows is one of the major contributors to the overall drag of flow over slender bodies in general and passenger planes at cruise flight in particular. Lowering turbulent friction drag is therefore essential to meet future CO 2 reduction goals. Besides preventing fully turbulent flow and benefiting from the considerably lower laminar drag [45], there is substantial past and ongoing research in the field of turbulent drag reduction. Unlike active techniques, which require energy introduction to the system, passive techniques such as riblets [50,4,51,15,14] and compliant surfaces [7,23,31,53] yield reduced skin friction without any added energy. However, compared to passive approaches, which are optimized for single operating conditions, active techniques are adaptive and, at least for some techniques, can achieve higher net power saving. These results, however, hold mostly in canonical flow setups like turbulent channel flows under laboratory conditions, i.e., at extremely low technology readiness levels.
In the following, we briefly review active flow control techniques that are most relevant to the current study. Particularly, we focus on methods that use either in-plane wall motion, such as forcing parallel to the wall or out of plane wall motion. In this study, we employ actuation that belongs to the latter category. We discuss its similarities and peculiarities over existing techniques, and present its drag reduction and net power saving potentials, which reach 26% and 10% compared to the unactuated flow.
Inspired by turbulence suppression by temporary pressure gradient variations [35], Jung et al. [22] performed the first simulations of spanwise wall oscillations which resulted in significantly lowered friction drag. The method was subsequently investigated in detail in the following years for Poiseuille flow [6,38] and turbulent boundary layer flow [41,52,29]. Detailed analyses indicated an interaction of the oscillating spanwise shear with the near-wall velocity streaks [49,1]. Furthermore, it was found that the maximum drag reduction in turbulent boundary layer flow is moderately lower than in turbulent channel flow and is reached at a significantly lower oscillation period [29]. Motivated by this simple but effective approach, other forms of spatio-temporal forcing have been developed, which is excellently discussed by Quadrio [37]. A modified variant of the purely temporal oscillations of spanwise velocity [49] are spanwise traveling waves of spanwise forcing [10,11] or spanwise traveling waves of a flexible surface [54]. Although the techniques are different in their actuation principle, the effect of introducing oscillating spanwise shear close to the wall is alike.
Another actuation variant is spanwise traveling transversal surface waves [20]. Instead of directly introducing spanwise velocity, the surface is wavily deflected in the wall-normal direction to generate a secondary flow field of periodic wallnormal and spanwise fluctuations. Positive drag reduction using this technique was achieved experimentally [20,47,30] and numerically for channel flow [48], boundary layer flow [26,28,27,19], and airfoil flow [2]. Tomiyama and Fukagata [48] observed a possible shielding effect of quasi-streamwise vortices from the wall by the wave-like deformations and showed that a combination of the thickness of the Stokes layer, i.e., the actuation period, and the actuation velocity amplitudes scales reasonably well with drag reduction.
However, the question remains what happens at higher amplitudes and wavelengths and lower periods, especially considering the vast gap between the mostly relatively short wavelength setups in numerical simulations and the large wavelengths in all experimental setups limited by mechanical actuator constraints. We will investigate if the trend of higher drag reduction for longer wavelengths [11] can be confirmed. Furthermore, an optimum forcing period T + in inner scaling was not determined for this technique and it remains an open question if one exists and if so if it is in the range of other techniques, e.g., T + ≈ 70 for spanwise oscillating wall in turbulent boundary layer flow [29]. In this study, we address these questions. We investigate the higher reduction trends with longer wavelengths and examine the flow sensitivities over the space spanned by the three actuation parameters, i.e., wavelength, wave period, and wave amplitude, using high-resolution large-eddy simulations (LES) of turbulent boundary layer flow. In total, 80 configurations are computed. The objective is to achieve drag reduction and net energy saving in the range of other actuation techniques and to compare the flow response to that from pure spanwise oscillations.
The paper has the following structure. First, the numerical method is concisely described in section 2. Then the flow setup and all flow and actuation parameters are specified in section 3. The results are discussed in section 4. Finally, the essential results are summarized in section 5.

Numerical method
The actuated turbulent boundary layer flow is computed by solving the unsteady compressible Navier-Stokes equations by a large-eddy simulation (LES) formulation. To capture the temporal variation of the geometry, the equations are written in the Arbitrary Lagrangian-Eulerian (ALE) formulation [17] such that the actuated wall can be represented by an appropriate mesh deformation. Additional volume fluxes are determined to satisfy the Geometry Conservation Law (GCL).
The discrete solution is based on a finite-volume approximation on a structured body-fitted mesh. A second-order accurate formulation of the inviscid fluxes using the advection upstream splitting method (AUSM) is applied. The cell-surface values of the flow quantities are reconstructed from the surrounding cell-center values using a Monotone Upstream Scheme for Conservation Laws (MUSCL) type strategy. The viscous fluxes are discretized by a modified cell-vertex scheme at second-order accuracy. The time integration is performed by a second-order accurate five-stage Runge-Kutta scheme, rendering the overall discretization secondorder accurate.
The subgrid scales in the LES are implicitly modeled following the monotonically integrated large-eddy simulation approach [5], i.e., the numerical dissipation of the AUSM scheme models for the viscous dissipation of the high wavenumber turbulence spectrum [33]. Thus, the small-scale structures are not explicitly resolved in the whole flow domain and the grid is used as a spatial filter resolving the large energy-containing structures in the inertial subrange.
The numerical method has thoroughly been validated by computing a wide variety of internal and external flow problems [43,3,39,46]. Analyses of drag reduction have been performed for riblet structured surfaces [24] and for traveling transversal surface waves in canonical turbulent boundary layer flow [26,27,28,34] and in turbulent airfoil flow [2]. The quality of the results confirms the validity of the approach for the current flow problem.

Computational Setup
The zero-pressure gradient (ZPG) turbulent boundary layer flow over a wall actuated by a sinusoidal wave motion is defined in a Cartesian domain with the x-axis in the main flow direction, the y-axis in the wall-normal direction, and the z-axis in the spanwise direction. The velocity vector in the Cartesian frame of reference x = (x, y, z) is denoted by u = (u, v, w), the pressure is given by p, and the density by ρ. The flow variables are non-dimensionalized using the flow quantities at rest, the speed of sound a 0 , and the momentum thickness of the boundary layer at x 0 = 0 such that θ(x 0 = 0) = 1. The momentum thickness based Reynolds number is Re θ = u ∞ θ/ν = 1, 000 at x 0 where u ∞ is the freestream velocity and ν is the kinematic viscosity. The Mach number is M = 0.1, i.e., the flow is nearly incompressible. Note that unlike standard ZPG turbulent boundary layer flow, the actuated flow is statistically three-dimensional due to the wave propagating in the z-direction.
An overview of the setup is given in Fig. 1. The dimensions of the physical domain are L x = 190 θ, L y = 105 θ in the streamwise and wall-normal direction. For the spanwise direction, five domain widths, L z ∈ [21.65 θ, 25.98 θ, 34.64 θ, 38.97 θ, 64.95 θ] are used. The mesh resolution is ∆x + = 12.0 in the streamwise direction, ∆y + wall = 1.0 in the wall-normal direction with gradual coarsening off the wall up to ∆y + = 16.0 at the boundary layer edge, and ∆z + = 4.0 in the spanwise direction. This yields a DNS-like resolution near the wall. Away from the wall, the resolution requirements are lower such that overall an LES resolution is achieved.
At the inflow of the domain, the reformulated synthetic turbulence generation (RSTG) method by Roidl et al. [42] is used to prescribe a fully turbulent inflow distribution with an adaptation length of less than five boundary-layer thicknesses δ 99 . A fully turbulent boundary layer is achieved at x 0 , which marks the onset of the actuation. Characteristic outflow conditions are applied at the downstream and upper boundaries, whereas periodic conditions are used in the spanwise direction. On the wall, no-slip conditions are imposed and the wall motion is described by where A + = Au τ /ν is the amplitude, λ + = λu τ /ν is the wavelength, and T + = T u 2 τ /ν is the period. If not otherwise stated an inner scaling is used for all wave parameters, i.e., the quantities are scaled by the kinematic viscosity ν and the friction velocity u τ of the non-actuated reference case N 1 .
In total, 80 variations of A + ∈ [0, 78], T + ∈ [20,120], and λ + ∈ [200, 3000] are simulated. A detailed list of all parameter combinations can be found in Tab. A1 in the appendix. Note that the narrowest domain has a spanwise extent of L + z = 1000 such that for all wavelengths λ + < 1000 multiple wavelengths are considered. A sketch of all wavelengths and the respective maximum amplitude at each wavelength is illustrated in Fig. 2. To enable a smooth spatial transition from the stationary flat plate to the deflected wall and vice versa, the piecewise defined function is used in Eq. 1. The drag is integrated over the wall surface within the streamwise interval x ∈ [50.0, 100.0] and over the entire spanwise extent. This area is colored in Fig. 1. Hence, the drag is only computed in the region where the flow is fully influenced by the traveling wave actuation. The actuated boundary layer is not impacted by the flow upstream and downstream of the actuated surface.
The computing strategy is such that first, the non-actuated reference case is simulated for tu ∞ /θ ≈ 650 convective times until a quasi-steady state is observed in the integrated drag. All actuated cases are then initialized by the flow field of the reference case and the transition from a flat plate to an actuated wall flow is performed via a temporal decay controlled by 1 − cos(t). Once a new quasi-steady state is observed all simulations are averaged over tu ∞ /θ ≈ 1250 times.

Results
In the following, the results of the parameter study will be investigated in detail. First, a grid convergence study is performed in Sec. 4.1. Then, the wall-shear stress reductions as a function of the wave parameters are thoroughly discussed in Sec. 4.2. The findings are compared with data from the literature for the same and similar drag reduction techniques. This analysis is followed in Sec. 4.3 by a discussion of the variation of the total drag, i.e., the wall-shear stress multiplied by the wetted surface, since the wetted surface changes for different parameter setups. Support vector regression is used to predict drag reductions and to examine the drag reduction sensitivities for varying actuation settings is concisely presented in Sec. 4.4. The statistics of the non-actuated and actuated flow field are compared, and links to drag reduction mechanisms are drawn in Sec. 4.5. The spanwise shear distribution in the near-wall region with special focus on the periodic Stokes shear and its relevance for drag reduction is analyzed in Sec. 4.6. Finally, the net energy balance results are presented in Sec. 4.7.

Grid convergence
To ensure a sufficient grid resolution for the large-eddy simulation of the actuated turbulent boundary layer flow a grid convergence study is conducted. The data of the three meshes that are compared are listed in Tab. 1. Besides the mesh data, the drag reduction ∆c d , which is defined in Sec. 4.3, is given. The actuation parameters in inner coordinates are λ + = 1000, T + = 40, and A + = 40. They define case N 24 in Tab. A1 in the appendix for which a moderate drag reduction is   achieved. It is evident from the results in Tab. 1 that the drag reduction values on the standard and the fine grid are nearly identical. On the coarse grid, however, a clear deviation is determined. A comparison of the symmetric stresses and the shear-stress component of the Reynolds stress tensor at the streamwise location of x/θ = 50, i.e., on the non-actuated surface at Re θ = 1033, is shown in Fig. 3. The distributions of the standard and the fine mesh nearly collapse for the non-actuated reference case and for the actuated case N 24 for all four components. Note that similar results were determined for other actuation parameter configurations. Only on the coarse mesh larger deviations are obtained. Furthermore, the data of the non-actuated  Table 1: Summary of the grid spacings ∆x + , ∆y + wall , and ∆z + , the number of cells inside the boundary layer N BL , the number of cells in the coordinate directions, the total number of cells, and the computed drag reduction ∆c d for the coarse, standard, and fine grid.
reference case for the standard and the fine mesh shows good agreement with DNS data [44] of a turbulent boundary layer at a similar Reynolds number, i.e., Re θ = 1006.
In conclusion, the analysis of the data shows that the resolution of the standard grid can be considered sufficient to accurately predict actuated turbulent boundary layer flow.

Wall-shear stress reductions
The skin-friction reduction ∆c f is defined in percent by where the wall-shear stress τ w is averaged over the shaded surface A surf in Fig. 1.
The values for ∆c f of the 80 cases are listed in Tab. A1 in the appendix. The dependence of ∆c f on the various parameters, i.e., the wavelength, period, amplitude, and amplitude velocity, is summarized in Fig. 4. The highlighted and numbered distributions are mainly from cases of the upper envelope of the wall-shear stress reduction, i.e., the maximum ∆c f values for the wavelength, the period, and the amplitude are emphasized. The discussion and illustration in Fig. 4 summarize the pronounced varying dependence of the wall-shear stress on the different actuation parameters. In Fig. 4(a) a quasi-linear increase of the skin-friction reduction ∆c f as a function of the wavelength λ + is observed. Note, however, that this quasi-linear distribution is achieved by changing simultaneously the amplitude A + and the period T + . Especially the latter has to undergo quite a non-linear variation to obtain such an approximately linear ∆c f growth.
The dependence of ∆c f on the wave period T + at various A + and λ + is presented in Fig. 4(b). Due to the coupling between the forcing strength and the actuation period, which is in contrast to other actuation methods like spanwise traveling waves of spanwise forcing [11] and traveling waves of flexible wall [54], the optimum period T + is determined by an internal, i.e., fluid mechanical, and an external, i.e., actuator, related condition. The ideal T + is defined by the streak formation time scale [49], i.e., for oscillatory spanwise forcing the period must be small enough to disrupt the reorganization of the streaks, and a sufficient strength of the forcing, which increases with decreasing period, is required. The dependence of the skin-friction reduction on the period in Fig. 4(b) shows that the optimum   Fig. 3: Comparison of the wall-normal distributions of the symmetric stresses and the shear-stress component of the Reynolds stress tensor on the coarse, standard, and fine grids for the non-actuated reference case and the actuated case N 24 with DNS data [44].
period among all 80 cases is on the order of T + = O (50), which is slightly lower than the optimum period T + ≈ 70 of a spanwise oscillating wall in turbulent boundary layer flow [29].
Note that likewise tendencies can be found for spanwise traveling oscillatory forcing [11] such as increased drag reduction with higher wavelengths (cf. Fig.4(a)). The longest wavelength considered in this study (λ + = 3000) is comparable to that used in the experimental setups by Tamano and Itoh [47] and Li et al. [30]. Although their lowest investigated period is T + ≈ 110 and thus considerably higher than the optimum found in this study, their results corroborate the tendency of higher wall-shear stress reduction at lower periods in the regime 110 ≤ T + ≤ 302.5.
The distribution of the skin-friction decrease as a function of the amplitude in Fig. 4(c) shows that the maximum skin-friction reduction is directly coupled to the amplitude. This is to some extent expected since the velocity and thus, the strength  of the actuation V + = 2πA + /T + is directly related to the amplitude. That is, at a given period the strength of the actuation is determined by the amplitude. This is confirmed by the experimental findings of Li et al. [30]. They obtain in a lower amplitude range a monotonic increase of the skin-friction reduction for increasing amplitude. Figure 4(d) presents the skin-friction reductions as a function of the velocity amplitude of the actuation V + , where the scaling shows a quasi-linear behavior for larger wavelengths λ + > 1000. A similar scaling for ∆c f was proposed by Tomiyama and Fukagata [48] by combining the amplitude of the actuation velocity V + (cf. Fig. 4(d)) and the thickness of the Stokes layer T + /(2π) such that ∆c f = f (A + 2π/T + ) which is plotted in Fig. 5. For shorter wavelengths (cf. Fig. 5(a)), i.e., λ + ≤ 1000, a  linear scaling is only observed for small values A + 2π/T + < 10. Note that the current overall reductions are lower than those in [48] which is likely due to the higher Reynolds number in this study (Re τ = 360 vs. Re τ = 180 in [48]) and due to a generally lower skin-friction reduction efficiency in turbulent boundary layers compared to turbulent channel flow [40]. For higher scaling factor values, the distribution is more scattered. For larger wavelengths λ + > 1000 (cf. Fig. 5(b)), the skin-friction reduction scales almost linearly over the entire range. Above a certain value A + 2π/T + 20, however, the linear behavior of the skin-friction reduction deteriorates. We believe the reason for this degradation is the large momentum injection into the boundary layer via too high a velocity amplitude. This increases the spanwise velocity component which leads to an amplified turbulent exchange.

Drag reduction
Introducing a wave motion of the surface means that the area of the moving wall increases with the amplitude and the wavelength of the wave. The data in Tab. A1 in the appendix shows that this change of the wetted surface ∆A surf can be quite substantial, especially at small wavelengths and high amplitudes. At high wavelengths, this variation becomes rather small. Since the friction drag is defined by the product of the wall-shear stress and the surface interacting with the fluid, the variation of the wetted surface has to be taken into account leading to a non-linear relation between wall-shear stress and friction drag reduction. The averaged drag reduction is defined as  where c d is the drag coefficient computed by an integration over the shaded surface A surf in Fig. 1, The quantity n denotes the unit normal vector of the surface, e y is the unit vector in the y-direction, and A ref = 1 is the reference surface. The data in Tab. A1 in the appendix evidences the differences between ∆c f and ∆c d , especially at small wavelengths. The highest drag reduction is ∆c d = 26 % for a wavelength of λ + = 3000, a period of T + = 50, and an amplitude of A + = 78. Due to the large wavelength, the increase of the wetted surface is only ∆A surf = 0.7 %. The highest drag increase ∆c d = −27 % with the highest corresponding skin-friction coefficient increase ∆c f = −7 % is observed for λ + = 200, T + = 20, and A + = 30. As stated before, this configuration with a small wavelength suffers considerably from a drastic increase of the wetted surface ∆A surf = 19.4 %.
The temporal distributions of the instantaneous drag of the "best" and "worst", i.e., highest and lowest drag reduction N 80 ∆c d = 26 % and N 2 ∆c d = −27 %, which is a massive drag increase, are compared exemplarily in Fig. 6, with the instantaneous drag of the reference case. The temporal fluctuations of the drag appear stronger for the "worst" case. Note that due to the larger wavelength the drag of the "best" case is numerically integrated over a three times larger spanwise extent. Due to the spanwise averaging, this leads to the temporally smoother distribution of the N 80 (∆c d = 26 %) case compared to the N 2 (∆c d = −27 %) case. Its solid portion is interpolated using all LES data. The dotted ridgeline between points A and B extrapolates a better actuation response at given λ + for amplitudes A + > 78, i.e. beyond the investigated parameter range. Point C corresponds to the actuation parameters of the LES simulation N 80 with the largest drag reduction.

Drag reduction modeling and sensitivity analysis
In the following, the drag reduction ∆c d is modeled as a function of the actuation parameters λ + , T + , and A + . For this task, LES simulations provide only a sparse data set comprising 80 points. This amount would roughly correspond to a Cartesian discretization of a three-dimensional data space with only 3 × 3 × 3 points. However, the modeling is further complicated by the fact that these points are far from regularly distributed. A dense coverage of the actuation space using expensive LES simulations is hardly feasible.
The modeling is performed using a powerful regression solver from machine learning: support vector regression (SVR) [9]. The algorithm is chosen for its prediction accuracy and its smooth response distribution. SVR is a supervised learning algorithm that constructs a mapping between features or inputs and a known response. Here, SVR maps the actuation parameters λ + , T + , and A + on the averaged drag reduction ∆c d . Note that due to the highly non-linear response behavior and the scarcity of data points at very low wavelengths, cases with wavelengths λ + < 500 are ignored during the modeling process. This range is also of little interest as the best drag reduction is found at larger wavelengths. This data exclusion effectively yields 71 data points instead of 80. Overfitting is prevented with a 5-fold cross-validation. The SVR model has a coefficient of determination of R 2 = 0.93 indicating an excellent prediction accuracy.
The SVR model from the LES data is employed to visualize a continuous actuation response in the investigated parameter range of A + ∈ [0, 78], T + ∈ [20,120], and λ + ∈ [500, 3000]. Fig. 7 shows the isosurfaces of three drag reduction levels: ∆c d = 15, 20, and 25%. Within this parameter range, the best performance of 26.5 % is achieved at λ + = 3000, T + = 38 and A + = 78, which is slightly higher than the best simulated LES configuration N 80 with ∆c d = 26.3 %. This location indicates that better drag reduction could be achieved by increasing amplitude and wavelength.
An extrapolation of better performance outside the investigated parameter range is obtained with a ridgeline. In every λ + = const plane, the drag reduction ∆c d features a single maximum (A + r , T + r ) with respect to the actuation amplitude A + and period T + . The curve of (A + , T + , λ + ) connecting all these λ + -dependent ∆c d maxima is the ridgeline, which is illustrated as a thick black curve in Fig. 7. Variables on this ridgeline are denoted by the subscript 'r'.
In the range λ + ∈ [560, 1865], this maximum is inside the modeled T + and A + data range. It is illustrated by the solid black curve. However, the ridgeline leaves this modeled data range through the exit point A at the top surface A + = 78 near λ + ≈ 1865. The rigdeline is extrapolated outside the data range and depicted as dotted curve between points A and B. The extrapolation method is detailed in [12]. Along the ridgeline, ∆c d monotonously increases from 7% at λ + = 560 to the maximum of 27.9% at λ + = 3000 (point B). Note that this point is outside the current LES parameter range.
The ridgeline defines a 'skeleton' of the parametric behavior as illustrated in Fig. 8. Fig. 8a shows its projection in the λ + − T + and λ + − A + planes. Like in Fig. 7, the dotted sections correspond to the extrapolated ridgeline between points A to B. The amplitude A + r and period T + r along the ridgeline monotonously increase with the wavelength λ + . The period asymptotes rapidly towards 44. The amplitude continually increases with the wavelength although at a decreasing rate.
An intriguing physical insight about the drag-reduction mechanism is revealed in Fig. 8b complementing Fig. 5b. The relative drag reduction ∆c d along the ridgeline is shown as a function of the scaling parameter proposed by Tomiyama and Fukagata [48] based on a Stokes layer of a transverse wall oscillation. ∆c d clearly exhibits a linear behavior along the ridgeline in the scaling parameter range between 15 and 30. Away from the ridgeline, the scaling shows scatter on the order of that observed in Fig. 5(b).

Turbulent flow statistics
In the following, the mean statistics of a drag reduced flow will be investigated in detail. For this analysis, the case with the highest drag reduction, i.e., the case N 80 , will be considered. If data from other cases is used, it is explicitly indicated. All presented wall-normal distributions are considered at the streamwise position x = 90 θ, which is located in the actuated region. The actuated fully developed turbulent flow possesses a momentum based Reynolds number Re θ = 1077. The flow field of the actuated cases is phase averaged in the spanwise direction. Therefore,  [48]. The solid circles marked A, B, and C correspond to the likewise marked points in Fig. 7.
a triple decomposition [18] of the flow variables is used where φ is the temporal and spanwise average,φ are periodic fluctuations, φ = φ +φ are phase averaged quantities, and φ are stochastic fluctuations. Using this decomposition, φ = f (x, y) represents phase independent quantities,φ = f (x, y, z) are the periodic fluctuations generated through the actuation, i.e., the secondary flow field, and φ = f (x, y, z, t) are turbulent fluctuations. Spanwise averages are obtained along lines of constant distance from the wall, i.e., along the curved mesh lines. This calculation of the spanwise average suffers from some uncertainty for short wavelengths with high amplitudes, where the traveling wave massively intrudes into the boundary layer. For spanwise averages of long wavelengths as in the N 80 case, where the local perturbation of the viscous sublayer and the buffer layer is less drastic, this problem does not occur. A first overall impression of the impact of the wave actuation on the turbulent coherent structures is given in Fig. 9 by comparing contours of the λ 2 -criterion [21] for the random velocity fluctuations u i . It is evident that the total number of vortical structures in the near-wall region is significantly reduced for the actuated flow. Extended regions of little to hardly any structures occur in Fig. 9(b). A closer look evidences that unlike the structures of the non-actuated reference flow in Fig. 9(a), the structures of the actuated flow are inclined to the left and right depending on the phase angle of the traveling wave. It will be discussed in Sec. 4.6 that this wave determined orientation of the flow structures is an important feature related to drag reduction [49].
To highlight the influence of the actuation on the instantaneous flow field, Fig. 10 shows contours of the instantaneous random fluctuations of the velocity That is, the distinctive structure of thin meandering streaks is considerably alleviated compared to the non-actuated reference case. The wall-normal distributions of the phase averaged streamwise velocity u above the wave crest and in the wave trough and the mean velocity u are shown in Fig. 11. The scaling with the friction velocity of the non-actuated reference case u τ,ref in Fig. 11(a) illustrates the decrease of the velocity in the near-wall region. The wall-normal gradient at the wall is lowered, which results in drag reduction. Scaling the velocities with the friction velocity u τ of the actuated wall in Fig. 11(b) leads to an offset of the velocity profiles in the logarithmic region with respect to the non-actuated reference case by ∆B + ≈ 3.8. Based on the idea of the analysis of the impact of roughness on fully turbulent flow [36,8], Gatti and Quadrio [16] suggested the offset ∆B + to predict drag reduction at higher Reynolds numbers Fig. 10: Contours of the random streamwise velocity fluctuations for u + = −3 (blue) and u + = 3 (red) in the near-wall region 0 < y + < 20 of (a) the nonactuated reference case and (b) the actuated case with the highest drag reduction N 80 , i.e., λ + = 3000, T + = 50, and A + = 78.
Note, however, that this equation cannot be further simplified since for actuated turbulent boundary layer flow the term Re τ Re τ,0 is neither constant, as for constant pressure gradient turbulent channel flow, nor can it be substituted by the drag reduction rate, as for constant flow rate turbulent channel flow. Thus, ∆c f cannot be directly determined by equation (4). Nevertheless, using the local values of c f,0 , ∆c f , Re τ , and Re τ,0 at x = 90θ the calculated offset from equation (4) is ∆B + = 4.07, which reasonably agrees with the result ∆B + = 3.8 shown in Fig. 11(b). The velocity profiles in Fig. 11 show that for the current actuation neither the non-actuated nor the actuated friction velocity scaling -regardless from crest, trough, or spanwise averaged wall shear scaling-result in a collapsed distribution over the entire boundary layer. In other words, an inner scaling does not hold over the entire boundary layer.
Next, the components of the Reynolds stress tensor are depicted in Fig. 12. Through the actuation, the symmetric Reynolds stresses u i u i and the Reynolds shear stress u v shown in Fig. 12(a) are significantly lowered with only minor phase variations. Considering all cases for λ + > 1000, a good correlation of the decrease of the skin-friction with the decrease of the peak of the streamwise velocity fluctuations is computed (R = 0.90). For the case N 80 , the reductions at y + = 14.2, which defines the location of the peak of the streamwise fluctuations and the location of the maximum streamwise velocity streak intensity, are approx. 39 % for the streamwise component and 62 % for the shear-stress component. This suggests that the turbulent motion close to the wall is massively damped. Touber and Leschziner [49] have reported similarly large reductions in this region for spanwise wall oscillations without normal deflection. They emphasize the importance of the reduced near-wall Reynolds shear stress and drag, as characterized by the Fukagata, Iwamoto, Kasagi (FIK) identity [13], i.e., for the shear-stress contribution c f,RSS ∼ δ 99 0 (1 − y)(−u v )dy. The structural property of the turbulent motion is evidenced by the anisotropy invariant map [32] in Fig. 12(b). The stronger suppression of the streamwise fluctuations compared to the other components is illustrated by the shift of the actuated distribution away from onedimensional turbulence in the upper right vertex to isotropic turbulence in the lower vertex.
The distributions of the joint probability density function (PDF) of the streamwise and the wall-normal stochastic velocity fluctuations u and v are presented in Fig. 13. High values in the upper left quadrant (negative u and positive v ) denote ejections of fluid from the near-wall region towards the outer flow, whereas high values in the lower right quadrant (positive u and negative v ) indicate sweeps of high-speed fluid from the outer flow towards the near-wall region. As can be seen in Fig. 13, an overall attenuation of the fluctuations is observed with a strong damping of the sweeps and ejections in the second and the fourth quadrant. Again, this is in agreement with the results from spanwise oscillating walls without normal deflection [1,49].  Spanwise premultiplied energy spectra of the velocity fluctuations κE u i u i , where κ = 2π/l z is the wavenumber, are presented in Fig. 14. Each spectrum is normalized by the total resolved energy of the corresponding velocity component and the related case, i.e., the non-actuated reference case and N 80 . No general decrease in the energy peak of the actuated case is observed, only a shift in the energy distribution as a function of the structural wavelength and wall-normal coordinate can be seen. A comparison between the two differently normalized spectra for the streamwise component (cf. Fig. 14(a) and Fig. 14(b)) shows an energy decrease especially for the small scales and in the near-wall region. In other words, for the actuated case N 80 the energy is accumulated further off the wall in the larger scale turbulent structures. The peak of the non-actuated reference case at λ + ≈ 100, which is associated with the typical spacing of the near-wall streaks of l + z = O(100), becomes less pronounced for the actuated case N 80 and is shifted off the wall. This observation corroborates the visual impression from Fig. 10 of a strong reduction of the near-wall streaks for the actuated case. Similar tendencies are observed for the wall-normal (cf. Fig. 14(c) and Fig. 14(d)) and the spanwise velocity component (cf. Fig. 14(e) and Fig. 14(f)). Additionally, a stronger concentration of the energy in the length scale range of the near-wall streaks is observed for the spanwise velocity component. There is a sharper peak for the actuated case in comparison to a broader energy distribution in the nonactuated case.
In Fig. 15 the phase averaged and spanwise averaged distributions of the vorticity fluctuations ω i ω i are presented as a function of the wall-normal distance. The comparison of the profiles of each component shows that the major attenuation is observed in the wall-normal and the spanwise components, whereas the streamwise component shows only minor changes. Generally, for all cases with λ + > 1000 a good correlation between the decrease of the skin-friction and the decrease of the peak of the wall-normal (R = 0.96) and spanwise (R = 0.98) vorticity fluctuations was found. Again, similar vorticity trends were reported for spanwise oscillating walls [49]. The drag reduction was discussed to be caused by the weakening of velocity streaks near y + ≈ 10 [1]. That is, at least for actuation with large wavelengths λ + > 1000, a direct interference with quasi-streamwise vortices [48] has a minor effect on drag reductions.
The comparison of the vorticity fluctuation contours for four cases, i.e., the non-actuated reference case, the case with the highest drag increase N 2 , a case with moderate drag reduction N 24 , and the case with the highest drag reduction N 80 , in Fig. 16 shows details about the phase dependence of the overall structure of the vorticity field. It is obvious that the drag increase is associated with strongly enlarged and highly phase dependent vorticity contours. Due to the high amplitude and short wavelength sinusoidal wall motion, the boundary layer flow is massively perturbed. This is completely different for the two drag reduction cases, where the overall boundary layer structure is maintained but with reduced values of the wall-normal and spanwise vorticity component. Phase variations occur especially for the wall-normal component with the highest decrease above the wave crest and the lowest decrease in the wave trough. Note, however, that despite the clear phase variations, the overall lowered vorticity levels are maintained throughout the entire actuation period, as shown in Fig. 15.

Spanwise shear
To investigate the secondary flow strength and its effect on the near-wall turbulent structures, the Stokes strain ∂w + /∂y + , i.e., the rate of change in the wall-normal direction of the periodic fluctuations of the spanwise velocity component, is considered in Fig. 17 for cases with wavelengths λ + = 200, λ + = 1000, λ + = 1800, and λ + = 3000. Based on the data summarized in Tab A1 in the appendix, for each λ + = const set the cases with the highest, medium, and lowest skin-friction reduction are shown. The Stokes strain is used to characterize the Stokes layer that develops above an oscillating wall without any wall-normal deflection. However, similar to a configuration with pure spanwise oscillating walls [22,49] a Stokeslike layer is also generated by a transversal wave motion of the surface. Through the introduction of a periodic wall-normal velocityṽ, a periodic spanwise velocity componentw is induced via mass conservation resulting in a wall-normal shear distribution defining a Stokes layer. Cases with a high drag reduction generally show a high level of symmetric, i.e., positive and negative, spanwise shear very close to the wall, whereas less symmetric shear distributions yield lower drag reduction. When the Stokes drag significantly increases near the wall, i.e., a singular-like distribution occurs, the drag reduction reduces drastically. This observation supports the assumption that the drag reduction mechanism is strongly linked to spanwise oscillations which are generated either by wave oscillations [49], traveling waves of spanwise forcing [11], spanwise velocity at the wall [54], or spanwise transversal surface waves [25,28].
The assumption of the importance of the oscillating spanwise shear for skinfriction reduction also yields an explanation for the increasing skin-friction reduction with growing wavelength, which agrees with an observation of Du et al. [11]. For short wavelengths, e.g., λ + = 200, the integration of the spanwise shear distribution over the spanwise width of a near-wall streak, i.e., l + z = O(100), results in only a minor force in the spanwise direction acting on the streaks. For wavelengths λ + > 1000, however, the spanwise shear varies only slightly over the width of a streak, therefore a considerable spanwise force is determined by the integration over the spanwise streak width. Note that this behavior does not occur for spanwise wall oscillations, since the periodic spanwise shear does not depend on the spanwise coordinate.
A comparison of spanwise and streamwise shear is shown in Fig. 18. Touber and Leschziner [49] discuss a certain optimal scenario for oscillatory forcing in turbulent channel flow, where the ratio of spanwise to streamwise shear reaches values of up to ∂w + /∂y + ∂u + /∂y + ≈ 3. Fig. 18(b) shows that a similar value of this ratio is obtained for the case with the highest skin-friction reduction N 80 , whereas a lower ratio is obtained for the cases with medium (N 82 ) and low (N 83 ) skin-friction reduction. For all cases, the change of the skin-friction is well correlated, i.e., R = 0.84 at the wave crest and R = 0.85 in the wave trough, with a spanwise-to-streamwise shear ratio of ∂w + /∂y + ∂u + /∂y + = 3.1. Overall, the results for the spanwise traveling transversal waves in Fig. 17 underline the similarities to other drag reduction techniques based on periodic spanwise forcing. The results of the cases with lower wavelength in combination with high amplitude and high frequency deviate from this observation due to the increased wetted surface.

Energy saving analysis
The previous discussion has shown that considerable drag reduction rates have been obtained. However, drag reduction is not the only metric of interest. From a prospective application point of view, the question of net energy saving and its relation to drag reduction must be addressed. The ideal net energy saving is defined as where P d,ref/act = u ∞ (F f + F p ) is the power necessary to overcome the friction F f and pressure forces F p of the non-actuated P d,ref and actuated surface P d,act in the streamwise direction. The power spent on control P control , i.e., on deflecting the surface for the traveling wave, is computed by where e xz = (1, 0, 1) T is a combination of the unit vectors in the streamwise and wall-normal direction. Hence, P control is a combination of the viscous and pressure forces effective in the y-direction multiplied by the speed of the wall motion in the y-direction. The values of P control and ∆P net are depicted in Fig. 19 for all cases.  N80, ∂w + /∂y + crest N80, ∂u + /∂y + crest N80, ∂w + /∂y + trough N80, ∂u + /∂y + trough (a) The data for ∆P net are also listed in Tab. A1 in the appendix. Fig. 19(a) shows the expected approximately linear dependence between the power spent P control and the actuation velocity cubed (V + ) 3 = 2πA + /T + 3 .
It is evident from Fig. 19(b) that net power saving is only obtained for a few cases with a maximum of ∆P net = 10 % for case N 84 . Most cases clearly show no net power saving but net power loss. For instance, for N 20 ∆P net is ∆P net = −289 %, i.e., almost the fourfold P d,ref has to be invested.
A closer look at the net-power-saving cases in Fig. 19(c) shows that high drag reduction rates are no indicator for high net power saving. That is, there is no linear relation between drag reduction and net power saving. Instead, a high value of the scaling parameter A + 2π/T + obtained by a low amplitude speed 2πA + /T + leads to positive net power saving. Thus, as expected, there is a trade-off between the minimum power input to effectively influence the turbulent boundary layer and a maximum power input above which the energy costs grow tremendously. In the current parameter range, the optimum energy saving solution, i.e., ∆P net = 10 %, is achieved for the N 84 case with λ + = 3000, T + = 90, and A + = 66, which possesses just a medium drag reduction of ∆c d = 16 %. Note that the parameters that result in high net energy saving are in the upper range of the interval. Furthermore, the data in Tab. A1 in the appendix indicates that the sensitivity of ∆P net is less pronounced for larger wavelength and above a wave period of 60. (c) Fig. 19: Dependence of (a) the power spent P control and (b) the net power saving ∆P net on the cube of the actuation velocity amplitude (V + ) 3 = (2πA + /T + ) 3 ; a zoom of the red rectangle in (b) is shown in (c). To indicate which cases possess net power saving the notation for three selected cases is given in (c).

Conclusions
To analyze drag reducing effects and the net energy saving potential of spanwise traveling transversal surface waves high-resolution large-eddy simulations were conducted. The parameter space defined by the wave amplitude, wave period, and wavelength was investigated based on 80 wave parameter setups for purely spanwise traveling waves. The variation of skin-friction reduction, i.e., mean wallshear stress alteration, drag reduction, i.e., surface integrated wall-shear stress, and net energy saving was analyzed. In brief, a maximum drag reduction and net energy saving of 26 % and 10 % was found.
The highest skin-friction reduction was achieved for a period of T + ≈ 50, which is lower than the one reported for spanwise oscillating wall and within the range of the streak formation time scale. Larger wavelengths and amplitudes yielded higher skin-friction reduction. For wavelengths larger than 1000 plus units, a scaling with the Stokes layer height and the velocity amplitude was found to predict skin-friction reduction reasonably well. Additionally, the difference between skinfriction reduction and drag reduction, i.e., the increase of the wetted surface was taken into account, was found to be substantial for short wavelengths in combination with high amplitudes. A drag-reduction model was derived from the sparse dataset using optimized support vector regression. From the model, a tendency to an asymptotic behavior of amplitude and period could be identified, supporting the assumption of an optimum period in the range 40 ≤ T + ≤ 50 for large wavelengths. Moreover, a ridgeline behavior of optimum drag reduction in the high wavelength regime was extracted from the model.
The statistical results of the turbulent flow field confirmed this result for high wavelength configurations, where similar effects of the actuation on the near-wall region compared to spanwise oscillating walls were observed. That is, considerable reductions of the near-wall velocity streak strength were found for the cases with high drag reduction. For the highest drag reduction case, the smaller wall-shear stress was coupled to a substantial decrease of the Reynolds shear stress in the near-wall region. Generally, for large wavelength cases λ + > 1000 the decrease of the wall-normal and spanwise vorticity fluctuations strongly correlated with skinfriction reduction and drag reduction. A comparison among several configurations revealed that for unfavorable combinations of short wavelength and high amplitude, a considerable increase of the turbulent exchange resulting in skin-friction and drag increase was observed, whereas large wavelengths circumvented this effect and led to drag reduction. The periodic secondary flow field generated by the wavy surface motion approximated that of Stokes flow. Similar oscillating spanwise shear distributions were observed for many drag reducing cases, although no perfectly symmetrical oscillatory excitation of the near-wall structures is achieved.
No linear relationship between drag reduction and net energy saving was determined. That is, due to the non-linear response of the near-wall flow to the actuation the highest drag reduction does not result in the highest net energy saving. The maximum net energy saving ∆P net = 10 % was achieved for a drag reduction of ∆c d = 16 %, which is clearly lower than the maximum drag reduction of ∆c d = 26 %. A high value of the product of the actuation amplitude speed and the thickness of the Stokes layer at low amplitude speed results in positive net energy saving. The susceptibility of ∆P net is less pronounced for larger wavelength, which is a promising observation for prospective applications.

Conflict of Interests
The authors declare that they have no conflicts of interest.  Table A1: Actuation parameters of the turbulent boundary layer simulations, where each setup is denoted by a case number N . The quantity λ + is the spanwise wavelength of the traveling wave, T + is the period, and A + is the amplitude, all given in inner units, i.e., non-dimensionalized with the kinematic viscosity ν and the friction velocity u τ . Each block includes setups with varying period and amplitude for a constant wavelength. The list includes the values of the averaged relative drag reduction ∆c d , the averaged relative skin friction reduction ∆c f , the relative increase of the wetted surface ∆A surf , and the net power saving ∆P .