Pierre Ricco

Critical assessment of turbulent drag reduction through spanwise wall oscillations

Direct numerical simulations of the incompressible Navier–Stokes equations are employed to study the turbulent wall-shear stress in a turbulent channel flow forced by lateral sinusoidal oscillations of the walls. The objective is to produce a documented database of numerically computed friction reductions. To this aim, the particular numerical requirements for such simulations, owing for example to the time-varying direction of the skin-friction vector, are considered and appropriately accounted for. A detailed analysis of the dependence of drag reduction on the oscillatory parameters allows us to address conflicting results hitherto reported in the literature. At the Reynolds number of the present simulations, we compute a maximum drag reduction of 44.7%, and we assess the possibility for the power saved to be higher than the power spent for the movement of the walls (when mechanical losses are neglected). A maximum net energy saving of 7.3% is computed. Furthermore, the scaling of the amount of drag reduction is addressed. A parameter, which depends on both the maximum wall velocity and the period of the oscillation, is found to be linearly related to drag reduction, as long as the half-period of the oscillation is shorter than a typical lifetime of the turbulent near-wall structures. For longer periods of oscillation, the scaling parameter predicts that drag reduction will decrease to zero more slowly than the numerical data. The same parameter also describes well the optimum period of oscillation for fixed maximum wall displacement, which is smaller than the optimum period for fixed maximum wall velocity, and depends on the maximum displacement itself.

Initial response of a turbulent channel flow to spanwise oscillation of the walls

The transient behaviour of a turbulent channel flow suddenly subjected to spanwise harmonic oscillations of the walls is numerically studied by means of direct numerical simulations of the incompressible Navier-Stokes equations. It is well known that this movement of the walls produces a sustained and significant reduction in turbulent friction; in this paper we focus on the early stages of the motion after the start of the oscillations when the fully developed state has not yet established. It is found that at the very beginning of the oscillatory motion the streamwise wall shear-stress remains constant for a short time interval, the length of which depends on the parameters defining the oscillation. A spanwise velocity profile starts to develop, almost coincident with the analytical laminar solution for the sudden start-up of harmonic oscillations of the wall. The spanwise flow fully adapts to the new forcing after about one oscillation period, whilst the longitudinal flow is still evolving towards its ...

Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

The Laminar Generalized Stokes Layer and Turbulent Drag Reduction

This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Both laminar and turbulent regimes for the streamwise flow are studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwise-modulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When the streamwise flow is turbulent, the laminar generalized Stokes layer solution describes well the space-averaged turbulent spanwise flow, provided that the phase speed of the waves is sufficiently different from the turbulent convection velocity, and that the time scale of the forcing is smaller than the life time of the near-wall turbulent structures. Under these conditions, the drag reduction is found to scale with the Stokes layer thickness, which renders the laminar solution instrumental for the analysis of the turbulent flow. A classification of the turbulent flow regimes induced by the waves is presented by comparing parameters related to the forcing conditions with the space and time scales of the turbulent flow.

Changes in turbulent dissipation in a channel flow with oscillating walls

Harmonic oscillations of the walls of a turbulent plane channel flow are studied by direct numerical simulations to improve our understanding of the physical mechanism for skin-friction drag reduction. The simulations are carried out at constant pressure gradient in order to define an unambiguous inner scaling: in this case, drag reduction manifests itself as an increase of mass flow rate. Energy and enstrophy balances, carried out to emphasize the role of the oscillating spanwise shear layer, show that the viscous dissipations of the mean flow and of the turbulent fluctuations increase with the mass flow rate, and the relative importance of the latter decreases. We then focus on the turbulent enstrophy: through an analysis of the temporal evolution from the beginning of the wall motion, the dominant, oscillation-related term in the turbulent enstrophy is shown to cause the turbulent dissipation to be enhanced in absolute terms, before the slow drift towards the new quasi-equilibrium condition. This mechanism is found to be responsible for the increase in mass flow rate. We finally show that the time-average volume integral of the dominant term relates linearly to the drag reduction.

Modification of near-wall turbulence due to spanwise wall oscillations

An experimental investigation of a turbulent boundary layer modified by spanwise wall oscillations is conducted in a water channel by means of the hydrogen-bubble technique. The purpose is to study the dynamics of near-wall turbulent structures to shed new light on the physical mechanisms characterizing the boundary layer perturbed by the wall motion and to comprehend how these changes cause a wall-shear stress reduction. It is likely that flow visualizations conducted at the highest values of maximum wall velocity describe the spatial transient evolution of the flow to the new modified state because of the limited extension of the oscillating wall section. When the oscillatory motion is imposed, the low-speed streaks shift laterally, and cyclically incline to an angle with respect to the streamwise direction. Flow visualizations from the end of the water channel distinctly show that the interaction between these low-velocity pockets and the overriding longitudinal vortices is strongly altered, the latter...

The pre-transitional Klebanoff modes and other boundary-layer disturbances induced by small-wavelength free-stream vorticity

The response of the Blasius boundary layer to free-stream vortical disturbances of the convected gust type is studied. The vorticity signature of the boundary layer is computed through the boundary-region equations, which are the rigorous asymptotic limit of the Navier–Stokes equations for low-frequency disturbances. The method of matched asymptotic expansion is employed to obtain the initial and outer boundary conditions. For the case of forcing by a two-dimensional gust, the effect of a wall-normal wavelength comparable with the boundary-layer thickness is taken into account. The gust viscous dissipation and upward displacement due to the mean boundary layer produce significant changes on the fluctuations within the viscous region. The same analysis also proves useful for computing to second-order accuracy the boundary-layer response induced by a three-dimensional gust with spanwise wavelength comparable with the boundary-layer thickness. It also follows that the boundary-layer fluctuations of the streamwise velocity match the corresponding free-stream velocity component. The velocity profiles are compared with experimental data, and good agreement is attained. The generation of Tollmien–Schlichting waves by the nonlinear mixing between the two-dimensional unsteady vorticity fluctuations and the mean flow distortion induced by localized wall roughness and suction is also investigated. Gusts with small wall-normal wavelengths generate significantly different amplitudes of the instability waves for a selected range of forcing frequencies. This is primarily due to the disparity between the streamwise velocity fluctuations in the free stream and within the boundary layer.

Wall heat transfer effects on Klebanoff modes and Tollmien-Schlichting waves in a compressible boundary layer

The influence of wall heat transfer on fluctuations generated by free-stream vortical disturbances in a compressible laminar boundary layer is investigated. These disturbances are thermal Klebanoff modes, namely low-frequency, streamwise-elongated laminar streaks of velocity and temperature, and oblique Tollmien–Schlichting waves, induced by a leading-edge adjustment receptivity mechanism. The flow is governed by the linearized unsteady boundary-region equations, which properly account for the nonparallel and spanwise diffusion effects, and for the continuous forcing of the free-stream convected gusts. Wall cooling stabilizes the laminar streaks when their spanwise wavelength is much larger than the boundary-layer thickness. For these conditions, the disturbances confine themselves in the outer edge layer further downstream, where the compressibility effects are marginal. Klebanoff modes for which the spanwise diffusion is comparable with the wall-normal diffusion possess an asymptotic solution similar to...

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 fre...

The influence of wall suction and blowing on boundary-layer laminar streaks generated by free-stream vortical disturbances

The effect of mean flow wall transpiration on boundary-layer fluctuations generated by free-stream disturbances of the convected gust type is investigated numerically. The theoretical frameworks of Leib et al. [J. Fluid Mech. 380, 169 (1999)] and Ricco [J. Fluid Mech. 638, 267 (2009)], based on the linearized unsteady boundary-region equations, are adopted. It is found that wall suction has a more significant attenuating effect on the low-frequency laminar streaks, while high-frequency disturbances are brought closer to the wall, but unaffected in magnitude. A simple asymptotic result, confirmed by the numerical calculations, shows that the characteristic peak of the laminar streaks in the core of the boundary layer may be suppressed completely by suction if this is sufficiently intense. Thought experiments of the modification induced by suction on existing wind-tunnel root-mean-square data are carried out. The findings are compared with other laboratory data with wall suction, and the reasons for discrep...