Wall Forcing Research Articles

Turbulent drag reduction by spanwise wall forcing. Part 1. Large-eddy simulations

Turbulent drag reduction (DR) through streamwise travelling waves of the spanwise wall oscillation is investigated over a wide range of Reynolds numbers. Here, in Part 1, wall-resolved large-eddy simulations in a channel flow are conducted to examine how the frequency and wavenumber of the travelling wave influence the DR at friction Reynolds numbers $Re_\tau = 951$ and $4000$ . The actuation parameter space is restricted to the inner-scaled actuation (ISA) pathway, where DR is achieved through direct attenuation of the near-wall scales. The level of turbulence attenuation, hence DR, is found to change with the near-wall Stokes layer protrusion height $\ell _{0.01}$ . A range of frequencies is identified where the Stokes layer attenuates turbulence, lifting up the cycle of turbulence generation and thickening the viscous sublayer; in this range, the DR increases as $\ell _{0.01}$ increases up to $30$ viscous units. Outside this range, the strong Stokes shear strain enhances near-wall turbulence generation leading to a drop in DR with increasing $\ell _{0.01}$ . We further find that, within our parameter and Reynolds number space, the ISA pathway has a power cost that always exceeds any DR savings. This motivates the study of the outer-scaled actuation pathway in Part 2, where DR is achieved through actuating the outer-scaled motions.

Turbulent drag reduction by spanwise wall forcing. Part 2. High-Reynolds-number experiments

We present measurements of turbulent drag reduction (DR) in boundary layers at high friction Reynolds numbers in the range of$4500 \le Re_\tau \le 15\ 000$. The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusicet al.(Nat. Commun., vol. 12, 2021) for high-$Re_\tau$flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic particle image velocimetry to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, DR of up to 25 % was achieved, but mostly with net power saving (NPS) losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive NPS of 5–10 % for moderate DRs of 5–15 %. The results suggest that OSA is an attractive pathway for energy-efficient DR in high-Reynolds-number applications.

Linear impulse response in three-dimensional oscillatory boundary layers formed by periodic modulation of a rotating disk

A numerical investigation is undertaken on the development of linear disturbances in the rotating disk boundary layer, in which a time-periodic modulation is applied to the disk rotation rate. The model gives a prototypical example of a three-dimensional oscillatory boundary layer, by adding a Stokes layer to the von Kármán flow that develops on a steady disk. The study extends the Floquet analysis of Morgan et al. (J. Fluid Mech., vol. 925, 2021, A20), who showed that disk modulation stabilises the stationary convective instabilities found on the steady rotating disk. Using a radial homogeneous flow approximation, whereby the radial dependence of the basic state is ignored, disturbance development is simulated for several modulation settings, with flow conditions matched to both convective and absolute forms of linear instability. Disturbances excited via a stationary periodic wall forcing display behaviour consistent with that found using Floquet theory; time-periodic modulation stabilises the cross-flow instability by reducing the radial growth rate. In addition, convective and absolute instabilities, generated by an impulsive wall forcing, are both stabilised by the introduction of modulation to the disk rotation rate. Modulation establishes a significant reduction in both the temporal growth rate and the disturbance amplitude as it propagates away from the impulse origin. Moreover, greater stabilising control benefits are realised as the modulation amplitude increases.

Stochastic wall model for turbulent pipe flow using Immersed Boundary Method and Large Eddy Simulation

A hybrid IBM-LES method is presented with the objective to simulate high-Reynolds number pipe flows on coarse Cartesian meshes. The IBM method is first used to simulate a laminar pipe flow and results have shown to converge with second order accuracy to the exact solution. A new forcing scheme inside the IBM wall thickness improves significantly numerical accuracy and provides an interesting way to control the fluid–solid interaction. Based on this new modeling of the IBM wall boundary condition, turbulent pipe flows for Reynolds numbers in the range 50,000 to 500,000 are then considered. The IBM wall forcing under these conditions is developed based on the classical turbulent wall laws, namely the log-law and the power-law, able to reproduce the mean velocity profile. We show that adjusting the control parameters of these two models makes possible to recover the correct bulk velocity and mean velocity profile. In order to improve the fluctuations level and spatial distribution of turbulent structures inside the pipe, we propose to extend the log-law modeling using local and unsteady value of the wall shear stress obtained from a stochastic model. The latter preserves spatiotemporal correlations of the wall friction and enhances the reliability of the simulations in terms of both mean bulk flow and fluctuations. The effects of both the Reynolds number and the grid resolution are also discussed and empiric correlations for the model parameters are proposed.

Drag Reduction of Turbulent Boundary Layers by Travelling and Non-Travelling Waves of Spanwise Wall Oscillations

Turbulence control in the form of a streamwise travelling wave of transverse wall motion was studied numerically by employing direct numerical simulations (DNS). Both total and phase averaging were utilised to examine the statistical behaviour of the turbulence affected by the wall forcing, with a focus on the skin friction. Comparison with results from pure temporal and spatial wall forcing are conducted, and a compilation of data is used to explore analogies with drag-reduced channel flow.

A vortex–wave interaction theory describing the effect of boundary forcing on shear flows

A strongly nonlinear theory describing the effect of small amplitude boundary forcing in the form of waves on high Reynolds number shear flows is given. The interaction leads to an $O(1)$ change in the unperturbed flow and is relevant to a number of forcing mechanisms. The cases of the shear flow being bounded or unbounded are both considered and the results for the unbounded case apply to quite arbitrary flows. The instability criterion for unbounded flows is expressed in terms of the wall forcing and the friction Reynolds number. As particular examples we investigate wall transpiration or surface undulations as sources of the forcing and both propagating and stationary waves are considered. Results are given for propagating waves with crests perpendicular to the flow direction and for stationary waves with crests no longer perpendicular to the flow direction. In the first of those situations we find the instability induced by transpiration waves is independent of the propagation speed. For wavy walls downstream propagation completely stabilises the flow at a critical speed whereas upstream propagation greatly destabilises the flow. For stationary oblique waves we find that the instability is enhanced and a much wider range of unstable wavenumbers exists. For the bounded case with a wall of fixed wavelength we identify a critical wavelength where the most dangerous mode switches from the aligned to the oblique configuration. For the transpiration problem in the oblique configuration a strong resonance occurs when the vortex wavelength coincides with the spanwise wavelength of the forcing.

Global linear instability of rotating-cone boundary layers in a quiescent medium

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles (ψ∈[20∘:80∘]) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

Wall Oscillation Induced Drag Reduction Zone in a Turbulent Boundary Layer

Spanwise oscillation applied on the wall under a turbulent boundary layer flow is investigated using direct numerical simulation. The temporal wall-forcing produces a considerable drag reduction (DR) over the region where oscillation occurs. Three simulations with identical oscillation parameters have been performed at different Reynolds numbers with one of them replicating the experiment by Ricco and Wu (Exp. Therm. Fluid Sci. 29, 41–52, 2004). The downstream development of DR in the numerical simulation and experiment is nearly identical. The velocity profiles and the indicator function are investigated with respect to the variation in DR and Reynolds number. The DR affects the slope of the logarithmic part of the velocity profile in accordance with previous theoretical findings. Low speed streaks are visualized and the bending of longitudinal vortices related to the drag reduction phenomenon is discussed. In addition, the visualization is compared with the corresponding results from the experiments. The spatial transient of the DR before reaching its maximum value is analyzed and is found to vary linearly with the oscillation period. An analysis of the energy budget is presented and the fundamental differences compared to the streamwise homogeneous channel flow are elucidated. While the power budget improves with increasing Reynolds number, it is shown that the net power remains negative for the wall forcing parameters considered here, even under ideal conditions. On the other hand, the analysis together with channel and boundary layer flow data in the literature provides an estimation of net energy saving for boundary layer flows which depends on the streamwise extent of the oscillating zone.

Streamwise-travelling viscous waves in channel flows

The unsteady viscous flow induced by streamwise-travelling waves of spanwise wall velocity in an incompressible laminar channel flow is investigated. Wall waves belonging to this category have found important practical applications, such as microfluidic flow manipulation via electro-osmosis and surface acoustic forcing and reduction of wall friction in turbulent wall-bounded flows. An analytical solution composed of the classical streamwise Poiseuille flow and a spanwise velocity profile described by the parabolic cylinder function is found. The solution depends on the bulk Reynolds number R, the scaled streamwise wavelength lambda , and the scaled wave phase speed U. Numerical solutions are discussed for various combinations of these parameters. The flow is studied by the boundary-layer theory, thereby revealing the dominant physical balances and quantifying the thickness of the near-wall spanwise flow. The Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theory is also employed to obtain an analytical solution, which is valid across the whole channel. For positive wave speeds which are smaller than or equal to the maximum streamwise velocity, a turning-point behaviour emerges through the WKBJ analysis. Between the wall and the turning point, the wall-normal viscous effects are balanced solely by the convection driven by the wall forcing, while between the turning point and the centreline, the Poiseuille convection balances the wall-normal diffusion. At the turning point, the Poiseuille convection and the convection from the wall forcing cancel each other out, which leads to a constant viscous stress and to the break down of the WKBJ solution. This flow regime is analysed through a WKBJ composite expansion and the Langer method. The Langer solution is simpler and more accurate than the WKBJ composite solution, while the latter quantifies the thickness of the turning-point region. We also discuss how these waves can be generated via surface acoustic forcing and electro-osmosis and propose their use as microfluidic flow mixing devices. For the electro-osmosis case, the Helmholtz–Smoluchowski velocity at the edge of the Debye–Hückel layer, which drives the bulk electrically neutral flow, is obtained by matched asymptotic expansion.

Modal and non-modal stability of boundary layers forced by spanwise wall oscillations

Modal and non-modal perturbation growth in boundary layers subjected to time-harmonic spanwise wall motion are examined. The superposition of the streamwise Blasius flow and the spanwise Stokes layer can lead to strong modal amplification during intervals of the base-flow period. Linear stability analysis of frozen phases of the base state demonstrates that this growth is due to an inviscid instability, which is related to the inflection points of the spanwise Stokes layer. The generation of new inflection points at the wall and their propagation towards the free stream leads to mode crossing when tracing the most unstable mode as a function of phase. The fundamental mode computed in Floquet analysis has a considerably lower growth rate than the instantaneous eigenfunctions. Furthermore, the algebraic lift-up mechanism that causes the formation of Klebanoff streaks is examined in transient growth analyses. The wall forcing significantly weakens the wall-normal velocity perturbations associated with lift-up. This effect is attributed to the formation of a pressure field which redistributes energy from the wall-normal to the spanwise velocity perturbations. The results from linear theory explain observations from direct numerical simulations of breakdown to turbulence in the same flow configuration by Hack & Zaki (J. Fluid Mech., vol. 760, 2014a, pp. 63–94). When bypass mechanisms are dominant, the flow is stabilized due to the weaker non-modal growth. However, at high amplitudes of wall oscillation, transition is promoted due to fast growth of the modal instability.

Drag Reduction of a Turbulent Boundary Layer over an Oscillating Wall and Its Variation with Reynolds Number

Spanwise oscillation applied on the wall under a spatially developing turbulent boundary layer flow is investigated using direct numerical simulation. The temporal wall forcing produces a considerable drag reduction over the region where oscillation occurs. Downstream development of drag reduction is investigated from Reynolds number dependency perspective. An alternative to the previously suggested power-law relation between Reynolds number and peak drag reduction values, which is valid for channel flow as well, is proposed. Considerable deviation in the variation of drag reduction with Reynolds number between different previous investigations of channel flow is found. The shift in velocity profile, which has been used in the past for explaining the diminishing drag reduction at higher Reynolds number for riblets, is investigated. A new predictive formula is derived, replacing the ones found in the literature. Furthermore, unlike for the case of riblets, the shift is varying downstream in the case of wall oscillations, which is a manifestation of the fact that the boundary layer has not reached a new equilibrium over the limited downstream distance in the simulations. Taking this into account, the predictive model agrees well with DNS data. On the other hand, the growth of the boundary layer does not influence the drag reduction prediction.

Laminar Streaks in Oscillating Boundary Layers

The growth of laminar streaks in a flat-plate boundary layer is investigated in the presence of both spanwise wall oscillations and steady streamwise-dependent spanwise wall forcing. The laminar streaks are forced by free-stream vortical disturbances, which interact with the boundary layer to produce disturbances, which first grow and then decay. The oscillating plate produces a base flow that matches Blasius boundary layer in the streamwise and wall-normal directions, but produces a generalized Stokes layer (GSL) in the spanwise direction. This depends on the Blasius flow and tends to the classical Stokes layer (CSL) in the asymptotic limit of high oscillation frequency or large downstream distance. Spanwise oscillations of the plate and spanwise wall forcing can both reduce the total energy and maximum amplitude of the disturbances. Their relative effects are discussed.

Scaling of the velocity profile in strongly drag reduced turbulent flows over an oscillating wall

Scaling analysis of the velocity profiles in strongly drag reduced flows reveals that the slope of the logarithmic part depends on the amount of drag reduction (DR). Unlike DR due to polymeric fluids, the slope changes gradually and can be predicted by the analysis. Furthermore, the intercept of the profiles is found to vary linearly with the DR. Two velocity scales are utilized: the reference (undisturbed) and the actual friction velocity. The theory is based on the assumption that the near-wall linear region is only governed by the actual friction velocity, while the outer part is governed by the reference friction velocity. As a result, logarithmic part is influenced by both velocity scales and the slope of the velocity profile is directly linked to the DR. The theoretically obtained results are verified by data from six previously performed direct numerical simulations (DNSs) of boundary layers over spatial and temporal wall oscillations, with a wide range of resulting DR. The theory is further supported by data from numerous investigations (DNSs as well as experiments) of wall-bounded flows forced by various forms of oscillating wall-motion. The assumption that the outer part is unaffected by the actual friction velocity limits the validity of the proposed log-law to flows not fully adapted to the imposed wall forcing, hence the theory provides a measure of the level of adjustment. In addition, a fundamental difference in the applicability of the theory to spatially developing boundary flow and infinite channel flow is discussed.

Turbulent drag reduction through oscillating discs

AbstractThe changes in a turbulent channel flow subjected to sinusoidal oscillations of wall flush-mounted rigid discs are studied by means of direct numerical simulations (DNS). The Reynolds number is ${Re}_{\tau }=180$, based on the friction velocity of the stationary-wall case and the half-channel height. The primary effect of the wall forcing is the sustained reduction of wall-shear stress, which reaches a maximum of 20 %. A parametric study on the disc diameter, maximum tip velocity, and oscillation period is presented, with the aim of identifying the optimal parameters which guarantee maximum drag reduction and maximum net energy saving, the latter computed by taking into account the power spent to actuate the discs. This may be positive and reaches 6 %. The Rosenblat viscous pump flow, namely the laminar flow induced by sinusoidal in-plane oscillations of an infinite disc beneath a quiescent fluid, is used to predict accurately the power spent for disc motion in the fully developed turbulent channel flow case and to estimate localized and transient regions over the disc surface subjected to the turbulent regenerative braking effect, for which the wall turbulence exerts work on the discs. The Fukagata–Iwamoto–Kasagi identity is employed effectively to show that the wall-friction reduction is due to two distinguished effects. One effect is linked to the direct shearing action of the near-wall oscillating-disc boundary layer on the wall turbulence, which causes the attenuation of the turbulent Reynolds stresses. The other effect is due to the additional disc-flow Reynolds stresses produced by the streamwise-elongated structures which form between discs and modulate slowly in time. The contribution to drag reduction due to turbulent Reynolds stress attenuation depends on the penetration thickness of the disc-flow boundary layer, while the contribution due to the elongated structures scales linearly with a simple function of the maximum tip velocity and oscillation period for the largest disc diameter tested, a result suggested by the Rosenblat flow solution. A brief discussion on the future applicability of the oscillating-disc technique is also presented.

Performance losses of drag-reducing spanwise forcing at moderate values of the Reynolds number

A fundamental problem in the field of turbulent skin-friction drag reduction is to determine the performance of the available control techniques at high values of the Reynolds number Re. We consider active, predetermined strategies based on spanwise forcing (oscillating wall and streamwise-traveling waves applied to a plane channel flow), and explore via Direct Numerical Simulations (DNS) up to Reτ = 2100 the rate at which their performance deteriorates as Re is increased. To be able to carry out a comprehensive parameter study, we limit the computational cost of the simulations by adjusting the size of the computational domain in the homogeneous directions, compromising between faster computations and the increased need of time-averaging the fluctuating space-mean wall shear-stress. Our results, corroborated by a few full-scale DNS, suggest a scenario where drag reduction degrades with Re at a rate that varies according to the parameters of the wall forcing. In agreement with already available information, keeping them at their low-Re optimal value produces a relatively quick decrease of drag reduction. However, at higher Re the optimal parameters shift towards other regions of the parameter space, and these regions turn out to be much less sensitive to Re. Once this shift is accounted for, drag reduction decreases with Re at a markedly slower rate. If the slightly favorable trend of the energy required to create the forcing is considered, a chance emerges for positive net energy savings also at large values of the Reynolds number.

Comparison between spatial and temporal wall oscillations in turbulent boundary layer flows

AbstractDirect numerical simulations have been performed to study the drag reduction resulting from spatial oscillations of a segment of the wall under a turbulent boundary layer. The oscillating motion is imposed by utilizing a streamwise modulated spanwise wall forcing. The results are compared with earlier simulations using temporal oscillations with an identical segment and forcing amplitudes, and with a frequency related to the wavelength through a convective velocity. Two different oscillation amplitudes with equal oscillation wavelength have been used, which allows for a direct comparison between a relatively weak and strong forcing of the flow. The weaker forcing results in 25 % drag reduction while the stronger forcing, with twice the amplitude, yields 41 % drag reduction. Comparison with the temporal cases reveals drastically improved energy savings for the spatial oscillation technique, in accordance with earlier channel flow investigations. The streamwise variation of spanwise shear is shown to follow the analytical solution to the laminar Navier–Stokes equations derived under the assumption of constant friction velocity. Furthermore, the spanwise velocity profiles at various phases are compared with the analytical solution, and show very good agreement. The downstream development of the spatial Stokes layer thickness is theoretically estimated to be ${\sim }{x}^{1/ 15} $, in general agreement with the simulation data. The spatial variation of the spanwise Reynolds stress is investigated and compared with the variation in time for the temporal wall forcing cases. The controversy regarding a zero or non-zero production of spanwise Reynolds stress in the temporal case is elucidated. In addition, comparison with the spatial case reveals that a second production term originating from the downstream variation of the spanwise wall velocity has a negative contribution to the production, and hence relates to the larger drag reduction in the case of spatial forcing.

Linear stability of plane Poiseuille flow over a generalized Stokes layer

Linear stability of plane Poiseuille flow subject to spanwise velocity forcing applied at the wall is studied. The forcing is stationary and sinusoidally distributed along the streamwise direction. The long-term aim of the study is to explore a possible relationship between the modification induced by the wall forcing to the stability characteristic of the unforced Poiseuille flow and the signifcant capabilities demonstrated by the same forcing in reducing turbulent friction drag. We present in this paper the statement of the mathematical problem, which is considerably more complex that the classic Orr-Sommerfeld-Squire approach, owing to the streamwise-varying boundary condition. We also report some preliminary results which, although not yet conclusive, describe the effects of the wall forcing on modal and non-modal characteristics of the flow stability.

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

AbstractThe generalizations of the Onsager model for the radial boundary layer and the Carrier–Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier–Maslen models, because they are not restricted to the limit$\mathscr{A}\gg 1$, though they are restricted to the limit$\mathit{Re}\gg 1$and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter$\mathscr{A}= \sqrt{m{\Omega }^{2} {R}^{2} / (2{k}_{B} T)} $. This parameter$\mathscr{A}$is the ratio of the peripheral speed,$\Omega R$, to the most probable molecular speed,$ \sqrt{2{k}_{B} T/ m} $, the Reynolds number$\mathit{Re}= {\rho }_{w} \Omega {R}^{2} / \ensuremath{\mu} $, where$m$is the molecular mass,$\Omega $and$R$are the rotational speed and radius of the cylinder,${k}_{B} $is the Boltzmann constant,$T$is the gas temperature,${\rho }_{w} $is the gas density at wall, and$\ensuremath{\mu} $is the gas viscosity. In the case of wall forcing, analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier–Maslen equation for the velocity potential. For the case of mass/momentum/energy insertion into the flow, the separation-of-variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier–Maslen equation) are determined. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used ‘diffuse reflection’ boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a ‘temperature slip’ (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 10 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio (length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier–Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.

Laminar streaks with spanwise wall forcing

The influence of steady sinusoidal oscillations of spanwise wall velocity on the Klebanoff modes, i.e. unsteady streaky fluctuations induced by free-stream turbulence in the pre-transitional Blasius boundary layer, is investigated numerically. The wall motion induces a spanwise boundary layer which grows downstream as x1/6 and has an asymptotic analytical solution at large downstream distances. While the forcing has no effect on the initial growth of the streaks, their intensity eventually increases or decreases substantially depending on the relative magnitude between the forcing wavelength and the characteristic length scales of the streaks. The wall actuation enhances the streak intensity if the streak spanwise length scale is much larger than the Blasius boundary layer thickness. The streak energy is instead attenuated when the spanwise viscous diffusion effects play a key role. Wall pressure fluctuations may also be significantly damped in this case. The Klebanoff modes generated by full-spectrum free-stream turbulence are predicted to be attenuated by the wall motion. The asymptotic scaling analysis reveals that there exists an optimal forcing wavelength for full-spectrum streak attenuation as long as the spanwise length scales of the dominant streaks are as large as or smaller than the Blasius boundary layer thickness, a common scenario encountered in experiments. The optimal forcing wavelength is found to be comparable with the streak streamwise length scale. As the amplitude of the wall forcing increases, the reduction of streak intensity grows monotonically. The streaks are completely suppressed in the limit of large amplitude.

Streamwise oscillation of spanwise velocity at the wall of a channel for turbulent drag reduction

Steady forcing at the wall of a channel flow is studied via direct numerical simulation to assess its ability of yielding reductions in turbulent friction drag. The wall forcing consists of a stationary distribution of spanwise velocity that alternates in the streamwise direction. The idea behind the forcing builds on the existing technique of the spanwise wall oscillation and exploits the convective nature of the flow to achieve an unsteady interaction with turbulence. The analysis takes advantage of the equivalent laminar flow, which is solved analytically to show that the energetic cost of the forcing is unaffected by turbulence. In a turbulent flow, the alternate forcing is found to behave similarly to the oscillating wall; in particular an optimal wavelength is found which yields a maximal reduction in turbulent drag. The energetic performance is significantly improved, with more than 50% of maximum friction saving at large intensities of the forcing, and a net energetic saving of 23% for smaller intensities. Such a steady, wall-based forcing may pave the way to passively interacting with the turbulent flow to achieve drag reduction through a suitable distribution of roughness, designed to excite a selected streamwise wavelength.